Single neuron dynamics models linking theory and experiment

Single neuron dynamics models linking theory and experiment

[15] neural ensembles can generate oscillatory activity endogenously through local interactions between excitatory and inhibitory neurons. alternatively, the integration of the non-stationary solutions of the fokker-planck equation (equation 20) also describes the dynamical behavior of the network, and this would allow the explicit simulation of neuronal and cortical activity (single cells, eeg, fmri) and behavior (e. in terms of the above expressions, the normalized correlation function and the coherence function, which are both used widely in the literature, are(77,78)respectively. as such, the frequency of large-scale oscillations does not need to match the firing pattern of individual neurons. bottom panel: spike-train autocorrelation: thick gray line depicts the experimentally determined autocorrelation and the thin black line is calculated from the isi distribution under the assumption of a renewal process. modes and massesthe fokker-planck equation, (equation 1), is a rather beautiful and simple expression that prescribes the evolution of ensemble dynamics, given any initial conditions and equations of motion that embed our neuronal model. the dynamics of these ion channels have been captured in the well-established hodgkin–huxley model that describes how action potentials are initiated and propagated by means of a set of differential equations. the coefficients ναβ represent the synaptic density between excitatory (e) and inhibitory (i) populations or from the stochastic/noise (n) or sensory (s) inputs.: deco g, jirsa vk, robinson pa, breakspear m, friston k (2008) the dynamic brain: from spiking neurons to neural masses and cortical fields. inversion of these models has also furnished estimates of underlying physiological parameters and their variations across the brain, in different states of arousal and pathophysiology [86],[93],[94]. these experiments exploit the fact that (i) cortical neurons have long apical dendrites and are easily polarized by an electric field, and (ii) that epileptiform bursts can be initiated by stimulation. neural mass models can be generalized to neural field models by making the expectations a function of space, thereby furnishing wave equations that describe the spatiotemporal evolution of expected neuronal states over the cortical surface. addition to fast direct synaptic interactions between neurons forming a network, oscillatory activity is modulated by neurotransmitters on a much slower time scale. recent developments (see the heterogeneous connectivity in neural fields section) now allow elucidation of the impact of biologically relevant connection heterogeneities on the stability and conduction of cortical activity. This makes dynamic models critical in integrating theory and experiments. the particular case that the drift is linear and the diffusion coefficient, σ2(t), is given by a constant, the fokker-planck equation describes a well-known stochastic process called the ornstein-uhlenbeck process [19]. in particular, some forms of bci allow users to control a device by measuring the amplitude of oscillatory activity in specific frequency bands, including mu and beta rhythms. frequencies and amplitudes were derived using a complex morlet wavelet decomposition of the real and simulated time series. previous work has shown that many properties of neuronal dynamics can be obtained by regarding activity changes as perturbations of a steady state [86]. although this permits small scale nonlinear activity to coincide with and influence stochastic macroscopic activity, it requires a somewhat elaborate framework. in fact, we can write the density dynamics in terms of a linear operator or jacobian q(3). g, rolls e (2006) decision-making and weber's law: a neurophysiological model. is commonly accepted that the information processing underlying brain functions, like sensory, motor, and cognitive functions, is carried out by large groups of interconnected neurons [1]–[4]. hierarchical nature of the system is embodied by the targeted nature of the sensory inputs and the separate parameterization of parameters that couple masses to or within the sheet, csens and csheet, respectively. lm, david o, friston kj (2005) stochastic models of neuronal dynamics. at the microscopic scale, connectivity is dense, concentrated equally in vertical and horizontal directions and, more or less isotropic when considered across different cortical regions. spike-train variability under constant stimulation with pure tones that differed in their sound frequencies and intensities. pa (2006) patchy propagators, cortical dynamics, and the generation of spatially structured gamma oscillations. are different cortical representations integrated to form a coherent stream of perception, cognition, and action?-yishai r, bar-or l, sompolinsky h (1995) theory of orientation tuning in visual cortex. r, hernandez a, zainos a (2004) neuronal correlates of a perceptual decision in ventral premotor cortex. the secondary effect of the appearance of limit cycle dynamics is to suppress the impact of the spatially uncorrelated stochastic inputs. one of the key outstanding problems is to reconcile the apparent discrepancy between proposals involving a key role of nonlinear dynamics (see also [118]) and the apparent success of mean-field models to predict measured evoked responses, without recourse to nonlinear dynamics. the recurrent arrows indicate recurrent connections between the different neurons in a population. an exemplary approach, owing to boltzmann and maxwell, is the approximation of the motion of molecules in a gas by mean-field terms such as temperature and pressure. indeed, eeg signals change dramatically during sleep and show a transition from faster frequencies to increasingly slower frequencies such as alpha waves. isolated cortical neurons fire regularly under certain conditions, but in the intact brain cortical cells are bombarded by highly fluctuating synaptic inputs and typically fire seemingly at random. this makes dynamic models critical in integrating theory and experiments. neurons are the cells responsible for encoding, transmitting, and integrating signals originating inside or outside the nervous system. this allows one to construct bifurcation diagrams to understand the nonlinear mechanisms underlying equilibrium dynamics. "synchronization between motor cortex and spinal motoneuronal pool during the performance of a maintained motor task in man". local recurrent circuitry has received the most attention, but other theoretical mechanisms for the maintenance of persistent activity, including local recurrent synaptic feedback and intrinsic cellular bistability [58],[59], have been put forward. g, pollatos o, zihl j (2002) the time course of selective visual attention: theory and experiments. alternatively, the integration of the non-stationary solutions of the fokker-planck equation (equation 20) also describes the dynamical behavior of the network, and this would allow the explicit simulation of neuronal and cortical activity (single cells, eeg, fmri) and behavior (e. model spike trains are compared with measured spike trains, it is notable that for sharp, high peaks in the am of the stimulus, the model tends to exhibit greater spike-timing precision than the investigated receptor neurons (see fig. in brain slices, these waves can take the form of synchronous discharges, as seen during epileptic seizures [80], and spreading excitation associated with sensory processing [81]. the main source of randomness is from fluctuations in recurrent currents (resulting from “quenched” randomness in the connectivity and transmission delays) and fluctuations in the external currents. perceiving different odors leads to different subsets of neurons firing on different sets of oscillatory cycles. the resulting dynamics can be seen to emerge from the interplay of stochastic dispersion and flow-determined ensemble contraction. because there are about 60–80 receptor neurons per ear, which can be subdivided into 4 classes with different frequency-tuning characteristics (römer 1976), population averages could be used to achieve reliable mean responses despite significant spike-time variability on the single-cell level. nature and strength of neuronal connectivity varies markedly when considered across the heirarchy of spatial scales. for example, one of the best known types is the spike and wave oscillation, which is typical of generalized or absence epileptic seizures, and which resembles normal sleep spindle oscillations. hence, these cells generate the normal sinus rhythm and are called pacemaker cells as they directly control the heart rate. the computational units of these circuits are spiking neurons, which transform a large set of inputs, received from different neurons, into an output spike train that constitutes the output signal of the neuron. fact matches our finding that the variability of both isis and spike count strongly depends on the stimulus. this furnishes realistic simulations of the population activity of hippocampal pyramidal neurons, based on something known as the refractory density equation and a single-neuron threshold model. simulations give insight into the rich neural ensemble dynamics at different spatial scales that arise spontaneously, are evoked by sensory inputs, or follow changes in state parameters. the mean-field approximation is used extensively in statistical physics and is essentially a technique that finesses an otherwise computationally or analytically intractable problem. g, lee ts (2002) a unified model of spatial and object attention based on inter-cortical biased competition. these include binning the phase space and using a discrete approximation to a continuous density. the transmission of information within and between neurons involves changes in the so-called resting membrane potential, the electrical potential of the neurons at rest, when compared to the extracellular space. the first discovered and best-known frequency band is alpha activity (7. the underlying dynamics of such networks can be described explicitly by the set of coupled differential equations (equation 4) above. Computational models at different space–time scales help us understand the fundamental mechanisms that underpin neural processes and relate these processes to neuroscience data. results of simulating an ensemble of 250 neurons with sensory evoked synaptic currents to all neurons between t = 1,000 ms and t = 3,000 ms. intrinsic or intracortical fibers are confined to cortical gray matter in which the cortical neurons reside; these intrinsic connections define the local connectivity within an area. before turning to neural mass models, we consider some applications of mean-field modeling that will be reprised in the last section. simulations: ensemble activity from neuronal to whole brain scalesthis section illustrates neuronal ensemble activity at microscopic, mesoscopic, and macroscopic spatial scales through numeric simulations. k, schiff s, gluckman b (2005) propagating activation during oscillations and evoked responses in neocortical slices.., from brain stem inputs) and are modeled as a constant flow with a superimposed poisson train of discrete pulses. "a dendritic mechanism for decoding traveling waves: principles and applications to motor cortex". we further assessed the quality of the fano factor prediction by computing the mean absolute deviation of the predicted from the measured fano factors, scaled by the error of the experimental fano factor (see methods). the resulting differential equation describing the temporal evolution of the population density is called the fokker-planck equation, and reads(20). this level of description is usually framed as a (stochastic) differential equation (langevin equation) that describes how the states evolve as functions of each other and some random fluctuations with(2)where, d = ½σ2 and ω is a standard wiener process; i. of characterising the density dynamics explicitly, one can summarize it in terms of coefficients parameterising the expression of modes:(34)where μ = η−p, η− being the generalized inverse of the matrix encoding the basis set of modes. a, knight b, sirovich l (2000) on the simulation of large populations of neurons. this approximation is also valid if α and β are interpreted as effective values, averaged over subpopulations. mean-field models are suited to data which reflect the behavior of a population of neurons, such as the electroencephalogram (eeg), magnetoencephalogram (meg), and fmri. model is based on the assumption that spikes are generated stochastically at time t with a probability that depends on the product of 2 terms only: the strength q(t) of an “effective stimulus” at that very moment and a term that depends on the length of the interval δ since the last spike (generated at time tlast = t − δ). finally, in the section entitled cognitive and clinical applications, we illustrate applications of neural ensemble modeling in health and disease; namely, decision-making, auditory scene analysis, and absence seizures. multiple initial conditions, the time series of y(t) are plotted for the three regimes, one stream only (top), bistable (middle), and two streams only (bottom). mass models can be regarded as a special case of ensemble density models, where we summarize our description of the ensemble density with a single number..,(26)and(27)in addition, the probability mass leaving the threshold at time t has to be re-injected at the reset potential at time t+tref (where tref is the refractory period of the neurons), which can be accommodated by rewriting equation 22 as follows:(28)where h(. if we neglect the local dynamics f(μ), , and use an exponential kernel as in equation 39, we recover equations 39 and 40. the smooth spatiotemporal dispersion of the evoked cortical response and its time delayed corticothalamic volley are evident.., not topographically specific) connections between populations to be handled straightforwardly, simply by increasing rab while reducing γab, thereby allowing influences to propagate long distances with little damping..,(10)in the derivation of the last equation, we have assumed that p(ν′,t) and ρ(ε| ν′) are infinitely many times differentiable in ν. this furnishes realistic simulations of the population activity of hippocampal pyramidal neurons, based on something known as the refractory density equation and a single-neuron threshold model. an alternative approach is to recursively enslave micro- and mesoscopic activity to predicted macroscopic field oscillations by driving them with the predicted mean-field synaptic currents. we first explore the salient properties of these neurons by analyzing their responses to constant stimuli (pure tones of constant intensity). this is determined by the proportion of active neurons by counting the number of spikes nspikes(t,t+dt) in a small time interval dt and dividing by n and by dt [18]; i. the coupling term, hc, incorporates both the nature of the (all-to-all) within-ensemble coupling and the epsp with parametric strength c. s, fellous jm, whitmer d, tiesinga p, and sejnowski tj. conclusion, we have seen that statistical descriptions of neuronal ensembles can be formulated in terms of a fokker-planck equation, a functional differential equation prescribing the evolution of a probability density on some phase space. in particular, inhibitory interneurons play an important role in producing neural ensemble synchrony by generating a narrow window for effective excitation and rhythmically modulating the firing rate of excitatory neurons. the interaction between neurons can give rise to oscillations at a different frequency than the firing frequency of individual neurons. have identified some intrinsic neuronal properties that play an important role in generating membrane potential oscillations. s (1977) dynamics of pattern formation in lateral-inhibition type neural fields. efficient discrimination of temporal patterns by motion-sensitive neurons in primate visual cortex. model developed so far provides a simple and compact description of the variability of isis in response to constant stimuli. representation of acoustic communication signals by insect auditory receptor neurons. stimulus sequences (top) and its resulting neural field dynamics (bottom). central question in understanding neural coding principles therefore concerns the nature, origin, and computational implications of neural variability. these large-scale oscillations can also be measured outside the scalp using electroencephalography (eeg) and magnetoencephalography (meg). in this case, one can ignore the k dependence in the other propagators, and it becomes possible to express the transfer function with elements of the form(81)where is typically a complicated expression depending on the various jab(ω). the ion channels open or close depending on the membrane potential and on substances released by the neurons, namely neurotransmitters, which bind to receptors on the cell's membrane and hyperpolarize or depolarize the cell. further analysis of the 3 hz (absence) bifurcation in a reduced model argues that interactions between the reticular and specific nuclei of the thalamus contribute importantly to the absence seizure waveform [135]. the underlying dynamics of such networks can be described explicitly by the set of coupled differential equations (equation 4) above. in the absence of extrinsic neural and hormonal control, cells in the sa node will rhythmically discharge. evoked afferent pulse densities are shown because they reflect more accurately the expected synaptic currents, through their action on postsynaptic neurons. there are two subtypes of excitatory population, namely: specific and nonselective. lm, david o, friston kj (2005) stochastic models of neuronal dynamics. experimental evidence, supporting the existence of neural fields, has been accumulated (see [53] for a summary). this figure shows the average firing rate as a function of f1 and f2, obtained with the spiking simulations (diamond points correspond to the average values over 200 trials, and the error bars to the standard deviation). if we fourier transform the resulting set of linear equations, we find for the fluctuating parts(56,57,58,59,60)where is given by equation 50 and we have assumed that all the parameters of the equations (but not the fields of activity) are constant on the timescales of interest. is commonly accepted that the information processing underlying brain functions, like sensory, motor, and cognitive functions, is carried out by large groups of interconnected neurons [1]–[4]. assuming a gaussian distribution of individual neuronal firing thresholds, one obtains a symmetric sigmoid-shaped function for ςa as per the mean-field model section. these include binning the phase space and using a discrete approximation to a continuous density. the resulting dynamics can be seen to emerge from the interplay of stochastic dispersion and flow-determined ensemble contraction. because the eigenvalues are complex (due to the fact that the jacobian is not symmetric), the decay is oscillatory in nature, with a frequency that is proportional to the imaginary part of the eigenvalue and a rate constant proportional to the real part. to underscore the observation that the mean synaptic currents evidence an emergent phenomenon, and not merely the super-position of a bursting neuron, the time series of a single neuron is provided in figure 6c. imagine a very large number of neurons that populate phase space with a density p(ν,t). cv values also vary with stimulus intensity and sound frequency (fig.-train variability of auditory neurons in vivo: dynamic responses follow predictions from constant stimuli. 2001), and grasshopper communication signals were excluded, because such stimuli cause strong fluctuations in the adaptation level of locust auditory receptor neurons (benda 2002). coupling between theta and gamma activity is thought to be vital for memory functions, including episodic memory. moreover, in systems with mixed (excitatory and inhibitory) connectivity or excitatory systems with adaptive currents, solitary traveling pulses are also possible. are different cortical representations integrated to form a coherent stream of perception, cognition, and action? in general, these can vary in space, due to differences among brain regions, and in time, due to effects like habituation, facilitation, and adaptation. in steady state, the two-point correlation function can be obtained from equation 73 via the wiener-khinchtine theorem, giving(74)in the case where the system is statistically uniform, equation 74 depends only on the separation r = r′−r, giving(75)where(76)has been used and the arguments of t and n have been shown for emphasis. the fundamental principles underlying higher brain functions requires the integration of different levels of experimental investigation in cognitive neuroscience (from single neurons, neuroanatomy, neurophysiology, and neuroimaging, to neuropsychology and behavior) via a unifying theoretical framework that captures the neural dynamics inherent in the elaboration of cognitive processes. This makes dynamic models critical in integrating theory and experiments. it would also be possible to increase the degree of forward and backward asymmetry by incorporating purely ampa-like kinetics for the former and nmda-like kinetics for the latter, as has been proposed as a mechanism for perceptual inference [116],[117]. several studies have focused on spatially continuous neural fields, which describe the temporal change of neural activity on local scales, typically within a brain area (see [4],[54],[55] for reviews), assuming homogeneous connectivity and time delays. response to input, a neuron or neuronal ensemble may change the frequency at which it oscillates, thus changing the rate at which it spikes. right panel: isi distributions of the recording (gray) and the poisson process (black).), nonlinear dynamics and chaos: where do we go from here? if the αab and βab are independent of b (which is not generally the case), then the subscript b on dab can be omitted and va itself satisfies equation 44 with the right side of equation 44 replaced by the sum of pab over b. effective input q of the model neuron depends on the intensity i of the acoustic stimulus and was determined such that the observed and the predicted mean firing rates matched. in fact, bistability and hysteresis turn out to be properties of the classification process rather than properties of the neural field dynamics. d, bojak i (2005) understanding the transition to seizure by modeling the epileptiform activity of general anesthetic agents. in this particular case, the classification system y(t) does traverse from the positive (two streams) to the negative (one stream) fixed point and back. here the neural fields maintain the firing of its neurons to represent any location along a continuous physical dimension such as head direction, spatial location, or spatial view. a commonly used approximation is(48)where θa is the firing threshold for channels of type a and is the standard deviation of the threshold over the population. modeling at the single neuron level is necessary because this is the level at which information is exchanged between the computing elements of the brain; the neurons. hierarchical nature of the system is embodied by the targeted nature of the sensory inputs and the separate parameterization of parameters that couple masses to or within the sheet, csens and csheet, respectively.., [39]–[41]) and has been used recently as the basis of a generative model for event-related potentials that can be inverted using real data [42]. also, and (but for simplicity, we drop the tilde in from now on). in doing this, we assume for simplicity that all the model parameters are constant in time and space, although it is possible to relax this assumption at some cost in complexity. p, sacerdote l, tomassetti f (1995) on the comparison of feller and ornstein-uhlenbeck models for neural activity. the generic form of neural field dynamics can be written as (see also [53]):(41)where μ = μ(x,t) is the neural field, capturing the neural mass activity at time t and position x. Each level of description relates uniquely to neuroscience data, from single-unit recordings, through local field potentials to functional magnetic resonance imaging (fMRI), electroencephalogram (EEG), and magnetoencephalogram (MEG). in general, neurons with the same state v(t) at a given time t have a different history because of random fluctuations in the input current i(t). however, following a small increase in the constant flow term, due to a sensory input, isensory, the quiescent state becomes unstable and the neuron evolves on a (noise-modulated) limit cycle. as noted above, the fokker-planck equation summarizes the flow and dispersion of states over phase space in a way that is a natural summary of population dynamics in genetics (e. an example is functional magnetic resonance imaging (fmri), measuring regional changes in metabolism and blood flow associated with changes in brain activity.

Neural oscillation - Wikipedia

consequently, the dynamics are described by the evolution of the probability density function:(8)which expresses the population density, which is the fraction of neurons at time t that have a membrane potential v(t) in the interval [ν,ν+dν]. in the simplest case, we can use a single mode that can be regarded as encoding the location of a probability mass, hence neural mass models.., [127],[128]) is: (i) when the frequency separation is relatively small and/or the rate is relatively slow, listeners perceive a single integrated melody (or stream) and can accurately report the ordering of the tones, and (ii) when the frequency separation is relatively large and/or the rate relatively fast, people clearly perceive two segregated auditory streams, one with a higher pitch than the other. however, if the probability of a large group of neurons is rhythmically modulated at a common frequency, they will generate oscillations in the mean field (see also figure at top of page). there are n′ neural populations and j′ stimulus sources, and we assume that there is no feedback of stimuli on themselves, or of the brain on stimuli, then we can write equation 61 as(63,64,65)where the sum on the left of equation 63 extends only over populations in the brain, while the sum on the right covers only stimulus sources, denoted by j. inhibitory postsynaptic currents (ipscs) to both excitatory and inhibitory neurons are mediated by gaba receptors. models attempt to model the dynamics of large (theoretically infinite) populations of neurons. nevertheless, the integrate-and-fire (if) model is not only defined by the subthreshold dynamics but includes a reset after each spike generation, which makes the whole dynamics highly nonlinear. the latter intrinsic fibers have a transmission speed of chom and a transmission delay . d, amitai y (1997) propagating neuronal discharges in neocortical slices: computational and experimental study. although individual neuron activities cannot be recovered through non-invasive bci because the skull damps and blurs the electromagnetic signals, oscillatory activity can still be reliably detected. inferences can also be made about the parameters of the nonlinear if response at the cell body, and about speeds, ranges, and time delays of subsequent axonal propagation, both within the cortex and on extracortical paths (e. new synthetic data sets, numbering 100, were constructed from the original data (isis or spike counts) by drawing randomly with replacement. comparison of the measured (fexp) and predicted (fmodel) fano factors for all cells and all counting window sizes. the thick gray line depicts the experimentally measured firing rate; the thin black line denotes the firing rate of the model. frequency of ongoing oscillatory activity is increased between t1 and t2. thus, neurons sharing the same state v(t) in a population are indistinguishable. field modelsthe density dynamics and neural mass models above covered state the attributes of point processes, such as eeg sources, neurons, or neuronal compartments. in terms of the above expressions, the normalized correlation function and the coherence function, which are both used widely in the literature, are(77,78)respectively. e (2007) dynamical systems in neuroscience: the geometry of excitability and bursting. for the investigated auditory receptors, the cell dynamics are captured by a renewal process under constant stimulation so that each neuron can be characterized by one unique recovery function w(t − tlast).., [39]–[41]) and has been used recently as the basis of a generative model for event-related potentials that can be inverted using real data [42]. synchronization can be modulated by task constraints, such as attention, and is thought to play a role in feature binding,[53] neuronal communication,[2] and motor coordination. however, if isis are independent under constant stimulation, the underlying neural dynamics correspond to a renewal process (cox 1962), which allows a compact mathematical description. Modeling at the single neuron level is necessary because this is the level at which information is exchanged between the computing elements of the brain; the neurons. mean-field approach has been applied to biased-competition and cooperation networks and has been used to model single neuronal responses, fmri activation patterns, psychophysical measurements, effects of pharmacological agents, and effects of local cortical lesions [6], [24]–[33]. neither test provided strong hints against assuming independent isis, we can proceed with this assumption and systematically explore simple renewal models to describe the measured spike-train variability. groff d, neelakanta p, sudhakar r, aalo v (1993) stochastical aspects of neuronal dynamics: fokker-planck approach. g, kleinfeld d (2001) traveling electrical waves in cortex: insights from phase dynamics and speculation on a computational role. the inputs one neuron receives at the synapses from other neurons cause transient changes in its resting membrane potential, called postsynaptic potentials. the highly irregular firing of cortical cells is inconsistent with temporal integration of random epsps. from intrinsic properties of neurons, network properties are also an important source of oscillatory activity. phase resetting in medicine and biology: stochastic modelling and data analysis. the present case, application of the model was facilitated by the fact that responses from primary sensory neurons were analyzed. sequences vary in presentation rate and the frequency difference between the tones. "relevance of nonlinear lumped-parameter models in the analysis of depth-eeg epileptic signals". in the section entitled cognitive and clinical applications, we present one of these examples, in the context of decision-making. hz)[5] that can be detected from the occipital lobe during relaxed wakefulness and which increases when the eyes are closed. expected mean states of the ensemble excitatory neurons μe = 〈ve〉 are, as above, based upon morris lecar planar dynamics, with slow potassium channel dynamics. model of a biological neuron is a mathematical description of the properties of nerve cells, or neurons, that is designed to accurately describe and predict its biological processes. equations 93, 94, and 95 define the dynamics of a stream classification model in one of its simplest forms. propagation of pulses within and between populations determines the values of φab. pfurtscheller and colleagues found a reduction in alpha (8–12 hz) and beta (13–30 hz) oscillations in eeg activity when subjects made a movement. (note that va is linearly related to the current reaching the soma, and to μ in earlier sections. schaettefind this author on google scholarfind this author on pubmedsearch for this author on this siteview author's works on this sitetim gollischfind this author on google scholarfind this author on pubmedsearch for this author on this siteview author's works on this siteandreas v. inhibitory postsynaptic currents (ipscs) to both excitatory and inhibitory neurons are mediated by gaba receptors. g, rolls et (2003) attention and working memory: a dynamical model of neuronal activity in the prefrontal cortex. we have tried to show the conceptual and mathematical links among the ensuing levels of description and how these models can be used to characterize key dynamical mechanisms in the brain. spike trains can form all kinds of patterns, such as rhythmic spiking and bursting, and often display oscillatory activity. this specifies how the coefficients of probability modes would evolve from any initial state or following a perturbation to the neuronal states. and theoretical arguments propose that the onset of a seizure reflects a bifurcation in cortical activity from damped stochastic activity—where peaks in the power spectrum reflect damped linear resonances—to high amplitude nonlinear oscillations arising from activity on a limit cycle or chaotic attractor [86], [130]–[134]. at macroscopic scales, connectivity is sparser, can be considered exclusively horizontal, and is predominantly excitatory in nature. mesoscopic models tell us how neural elements interact to yield emergent behavior at the level of microcolumns and cortical columns. the effect of the input is to effect a bifurcation in each neuron from stochastic to limit cycle dynamics (phase locking), suppressing the impact of the spatially uncorrelated stochastic inputs. one approach is to construct a multiscale hierarchy, with self-consistent evolution equations at each scale and to couple the emergent dynamics from fine scales into the activity at coarser scales [114]. this function takes values between 0 and 1 and describes the influence of the last spike at tlast on the generation of a spike at time t. (right) shows the results of a simulated seizure in the corticothalamic field model described in the recent developments in neural field models section. we then illustrate examples of spatiotemporal dynamics occurring in corticothalamic loops during absence seizures. carry out the first step, the experimental relation f(i) is needed. here, population dynamics are described by a set of one-dimensional partial differential equations in terms of the distributions of the refractory density (where the refractory state is defined by the time elapsed since the last action potential). this pattern will change as a function of its own spatial configuration, φk(x), the connections (whom and νij), and, last but not least, the transmission times of information exchange, and d/c. l, sacerdote l (1979) the ornstein-uhlenbeck process as a model for neuronal activity. based on this view [21], and neurophysiological evidence [22], it has been hypothesized that each cortical area represents a set of alternative hypotheses, encoded in the activities of cell assemblies. these are referred to as neural field models and are discussed in the following sections. short test stimuli are then inserted to evaluate the response of the adapted neuron to intensity changes. in order to simplify the analysis, we neglect the dynamics of the afferent neurons (see [10] for extensions considering detailed synaptic dynamics such as ampa, nmda, and gaba). in numerical simulations: ensemble activity from neuronal to whole brain scale, we provide numerical simulations of neuronal ensemble dynamics across a hierarchy of spatial and temporal scales. this approximation is also valid if α and β are interpreted as effective values, averaged over subpopulations. of a hindmarsh–rose neuron showing typical bursting behavior: a fast rhythm generated by individual spikes and a slower rhythm generated by the bursts. we compare this with an explicit model of neural mass dynamics at the mesoscopic scale in the subsequent section. s, terry j, breakspear m (2006) on the genesis of spike-wave oscillations in a mean-field model of human thalamic and corticothalamic dynamics. this approximation is valid when the axonal delays contribute mostly to the dynamics, for instance in large-scale networks, when the local dynamics are much faster than the network dynamics. frequencies and amplitudes were derived using a complex morlet wavelet decomposition of the real and simulated time series. the fundamental principles underlying higher brain functions requires the integration of different levels of experimental investigation in cognitive neuroscience (from single neurons, neuroanatomy, neurophysiology, and neuroimaging, to neuropsychology and behavior) via a unifying theoretical framework that captures the neural dynamics inherent in the elaboration of cognitive processes. g, rolls et (2004) a neurodynamical cortical model of visual attention and invariant object recognition. points correspond to the average values over 200 trials, and the error bars to the standard deviation. hence the flow terms in the neural mass model contribute to the expression of aperiodic dynamics in addition to the stochastic inputs. these models are covered in the neural field models section. seed neuron is chosen at random and the interneuron spike difference for all other neurons is plotted each time it spikes. all three anatomical attributes undergo characteristic changes during the development of the human brain and its function, as well changing in the aged and diseased brain (see [9] for an overview). expected mean states of the ensemble excitatory neurons μe = 〈ve〉 are, as above, based upon morris lecar planar dynamics, with slow potassium channel dynamics. g, rolls et (2004) a neurodynamical cortical model of visual attention and invariant object recognition. this can be done in experimental animals using implanted electrodes to record the rates and timing of action potentials. (2006) dynamic causal modeling of evoked responses in eeg and meg. key forms for neural field equations were proposed and analysed by [45]–[47]. negative interspike-interval correlations increase the neuronal capacity for encoding time-dependent stimuli. experimental evidence, supporting the existence of neural fields, has been accumulated (see [53] for a summary). ms), and the parameter τr describing the relative refractory period lay between 0. in the absence of mean-field effects, the biorthogonality of the eigenfunctions effectively uncouples the dynamics of the modes they represent(35). to account for neural refractoriness, recovery functions (berry and meister 1998; johnson 1996) are incorporated into the framework. we argue that elaborating principled and informed models is a prerequisite for grounding empirical neuroscience in a cogent theoretical framework, commensurate with the achievements in the physical sciences. anatomical connectivity, w = whom+whet, comprising homogeneous and heterogeneous connections. hence this system permits an exploration of the relative impact of the flow (deterministic) and diffusive (stochastic) effects as embodied at the ensemble level by the fokker-planck equation (equation 20) at the neuronal network level. a problem here concerns the resulting emergence of sustained oscillations within mesoscopic activity and the possible causal inconsistency that this may entail. (2003) epilepsies as dynamical diseases of brain systems: basic models of the transition between normal and epileptic activity. symposium on robotics and cybernetics: computational engineering in systems applications. g, kleinfeld d (2001) traveling electrical waves in cortex: insights from phase dynamics and speculation on a computational role. summary, for any model of neuronal dynamics, specified as a stochastic differential equation, there is a deterministic linear equation that can be integrated to generate ensemble dynamics. the observed renewal-process characteristics of the receptor neurons could be beneficial for this discrimination task because they enhance the coding possibilities on such short timescales. oscillations recorded from multiple cortical areas can become synchronized to form large scale brain networks, whose dynamics and functional connectivity can be studied by means of spectral analysis and granger causality measures. neurons communicate with one another via synapses and affect the timing of spike trains in the post-synaptic neurons. m, terry j, friston k (2003) modulation of excitatory synaptic coupling facilitates synchronization and complex dynamics in a biophysical model of neuronal dynamics. single bump solutions have been used for neural modeling of the head-direction system [61]–[64], place cells [65]–[68], movement initiation [69], and feature selectivity in visual cortex, where bump formation is related to the tuning of a particular neuron's response [70]. in fact, the ability to show multistable pattern formation is the only relevant property of the network and can be realized in multiple network architectures as discussed in previous sections. functional specialization of the brain emerges from the collective network dynamics of cortical circuits. b (2000) dynamics of encoding in neuron populations: some general mathematical features. in two dimensions, many other interesting patterns can occur, such as spiral waves [71], target waves, and doubly periodic patterns. each cell was stimulated with 5 to 24 intensities covering the entire dynamic range of the neurons. m, kirsch h, szeri a (2005) pathological pattern formation and cortical propagation of epileptic seizures. intuitively, auditory streaming or stream segregation is like listening to bass and soprano vocalists singing simultaneously. mesoscopic models tell us how neural elements interact to yield emergent behavior at the level of microcolumns and cortical columns. oscillations are sensitive to several drugs influencing brain activity; accordingly, biomarkers based on neural oscillations are emerging as secondary endpoints in clinical trials and in quantifying effects in pre-clinical studies. the constant γ controls the rise time of voltage, in response to inputs (see also the neural field models section). in brain slices, these waves can take the form of synchronous discharges, as seen during epileptic seizures [80], and spreading excitation associated with sensory processing [81]. in fact, bistability and hysteresis turn out to be properties of the classification process rather than properties of the neural field dynamics. it would also be possible to increase the degree of forward and backward asymmetry by incorporating purely ampa-like kinetics for the former and nmda-like kinetics for the latter, as has been proposed as a mechanism for perceptual inference [116],[117]. we can derive directly the predicted ensemble mean response by simply summing over all neurons. purely theoretical formulations of the binding-by-synchrony hypothesis were proposed first,[59] but subsequently extensive experimental evidence has been reported supporting the potential role of synchrony as a relational code. our objective is to highlight some of the key notions of ensemble dynamics and to illustrate relationships between dynamics at different spatial scales. the 4 different types of locust auditory receptor neurons differ in their attachment site to the tympanum (gray 1960), and their characteristic frequencies (römer 1976), but not in their cytoanatomy (gray 1960). in our case, the derived transfer function, φ, corresponds consistently to the assumptions of the simple lif model described in the from spiking-neurons to mean-field models section. if whet describes the connectivity of n areas, then it can always be written as a sum of two-point connections via(83)where νij ∈ ℜ again represents the coupling strength between areas at xi and xj. qj is written as nj to make the distinction between population firing rates and incoming stimulus rates absolutely clear. this allowed us to calculate the effective stimulus strength q(t) directly from the applied input and to determine the recovery functions from responses to constant stimuli. moreover, in systems with mixed (excitatory and inhibitory) connectivity or excitatory systems with adaptive currents, solitary traveling pulses are also possible. this multiregional brain-on-a-chip is electrically active and reacts to drug applications in a brain region dependent way. before turning to neural mass models, we consider some applications of mean-field modeling that will be reprised in the last section. matching the resulting firing-rate fluctuations as well as the isi variability therefore presents a demanding test for the model framework. oscillatory activity in the brain is widely observed at different levels of organization and is thought to play a key role in processing neural information. evaluate the importance of the recovery functions within the model framework, we compared it with 2 simpler, reduced model variants that are widely used in the analysis of spike-train data: rate-modulated poisson processes without and with (absolute) refractory period. a major area of research in neuroscience involves determining how oscillations are generated and what their roles are. naturally, it is useful to understand how neuronal activity unfolds on the spatially continuous cortical sheet. fokker-planck equation describing the ornstein-uhlenbeck process, with μ = 〈j〉j nq(t) and σ2 = 〈j2〉j nq(t), can be rewritten as a continuity equation:(22)where f is the flux of probability defined as follows:(23)the stationary solution should satisfy the following boundary condition:(24)and(25)which expresses the fact that the probability current at threshold gives, by a self-consistent arguments, the average firing rate, q, of the population. is a naturally recurring state characterized by reduced or absent consciousness and proceeds in cycles of rapid eye movement (rem) and non-rapid eye movement (nrem) sleep. a problem here concerns the resulting emergence of sustained oscillations within mesoscopic activity and the possible causal inconsistency that this may entail. the experimentally determined and theoretically predicted autocorrelation functions closely match (fig. neural network model describes a population of physically interconnected neurons or a group of disparate neurons whose inputs or signalling targets define a recognizable circuit. the coupling term, hc, incorporates both the nature of the (all-to-all) within-ensemble coupling and the epsp with parametric strength c. solve the equations defined by equation 32 for all i, we integrate the differential equation below, describing the approximate dynamics of the system, which has fixed-point solutions corresponding to equation 32:(33). the subthreshold membrane potential of each neuron evolves according to a simple rc circuit, with a time constant τ = rc given by the following equation:(4)where ii(t) is the total synaptic current flow into the cell i and vl is the leak or resting potential of the cell in the absence of external afferent inputs. by comparing the response to that of a mesoscopic neural mass model, we show what is gained and what is lost by abstracting to a more tractable set of evolution equations. 14 shows numerical simulations corresponding to the response of vpc neurons during the comparison period (to be contrasted with the experimental results shown in figure 2 of [121]). recovery function w(δ) is a unique function for each neuron and can be obtained from the cell's response to a constant stimulus, as explained in methods. in this section, we overview this modal approach to ensemble dynamics, initially in the general setting and then in the specific case, where the dynamics can be captured by the activity of a single node. the other hand, neural response characteristics depend sensitively on the temporal features of the stimulus pattern (mainen and sejnowski 1995), which is of special importance for sensory neurons receiving highly structured dynamic stimuli (de ruyter van steveninck et al. ruyter van steveninck rr, lewen gd, strong ss, koberle r, and bialek w.[76] it has been proposed that motor commands in the form of travelling waves can be spatially filtered by the descending fibres to selectively control muscle force. as the state of each neuron evolves, the points will flow through phase space, and the ensemble density p(ν,t) will evolve until it reaches some steady state or equilibrium. ms and conditional spike probability b · ρ(t|tlast) obtained from eq. for each of the 4 different counting window lengths, the mean fano factors of the model (m) are very close to the experimental (e) mean fano factors (10 ms: m: 0. if the steady state is stable for all k, spectra and other properties of the linear perturbations can be self-consistently defined; otherwise a fully nonlinear analysis is needed. three different levels have been widely recognized: the micro-scale (activity of a single neuron), the meso-scale (activity of a local group of neurons) and the macro-scale (activity of different brain regions). if the αab and βab are independent of b (which is not generally the case), then the subscript b on dab can be omitted and va itself satisfies equation 44 with the right side of equation 44 replaced by the sum of pab over b. if the perturbation is due to a stochastic train, then the neuron fires randomly at an average rate proportional to the stochastic inputs. the frequency of these oscillations was in the range of 40 hz and differed from the periodic activation induced by the grating, suggesting that the oscillations and their synchronization were due to internal neuronal interactions. in the full nonlinear fokker-planck formulation, different phase functions or probability density moments could couple to each other; both within and between populations or ensembles.

Single Neuron Dynamics — Models Linking Theory and Experiment

Single Neuron Dynamics — Models Linking Theory and Experiment

Single neuron dynamics and computation

hence the dynamics at this scale mirror those within the microscopic ensemble, which each node in this simulation is constructed to represent. functional specialization of the brain emerges from the collective network dynamics of cortical circuits. furthermore, the accuracy and reliability of responses in the investigated cells has been shown to be strongly influenced by the stimulus statistics (machens et al.-train variability of auditory neurons in vivo: dynamic responses follow predictions from constant stimuli. conditional probability analyses of the spike activity of single neurons. the solution of equation 28 satisfying the boundary conditions (equations 24–27) is:(29)taking into account the fraction of neurons, qtref, in the refractory period and the normalization of the mass probability,(30)finally, substituting equation 29 into equation 30, and solving for q, we obtain the population transfer function, φ, of ricciardi [13]:(31)where . is linearly added to ongoing oscillatory activity between t1 and t2. as a consequence, those signal components that are the same in each single measurement are conserved and all others, i. on the other hand, it has been demonstrated for some systems that intrinsic isi correlations (nonrenewal behavior) can enhance information transmission (chacron et al. models attempt to model the dynamics of large (theoretically infinite) populations of neurons. discrepancies between the measured and predicted fano factors often coincide with mismatches between the respective firing rates (see fig. the dynamics of the neural field μ(x,t) are given by the wave equation 39, which has been extended to accommodate auditory inputs s(x,t) as follows:(93)where, as a reminder, γ = c/r, c is the speed of spike propagation, and r parameterizes the spatial decay of lateral interactions. at the microscopic scale, we simulate an entire array of spiking neurons in response to a sensory-evoked synaptic current. because our experimental data are consistent with a renewal process, the recovery function can be derived from a single interspike-interval histogram obtained under constant stimulation. under instantaneous interactions, c→∞, single population models with locally excitatory and laterally inhibitory connectivity can support global periodic stationary patterns in one dimension as well as single or multiple localized solutions (bumps and multi-bumps) [47]. neuron is modeled as a planar reduction of the hodgkin-huxley model [109],[110], namely,(87)where fion introduces conductance-determined transmembrane currents through voltage-dependent channels, ion = {na+,k+} and i are synaptic currents. and clinical applicationsin this section, we present three distinct applications of neural ensemble modeling. indeed, eeg studies suggest that visual perception is dependent on both the phase and amplitude of cortical oscillations. r, salinas e (2003) flutter discrimination: neural codes, perception, memory and decision making. among the most important are harmonic (linear) oscillators, limit cycle oscillators, and delayed-feedback oscillators. the subject area "single neuron function" applicable to this article? spatially localized bump solutions are equivalent to persistent activity and have been linked to working memory in prefrontal cortex [56],[57]. by targeting either na+ or ca++ currents and including (postsynaptic) voltage-dependent effects, this function can incorporate, to a first-order approximation, a variable proportion of ampa or nmda-like kinetics [52]. brain's network dynamics depend on the connectivity within individual areas, as well as generic and specific patterns of connectivity among cortical and subcortical areas [4],[9],[98]. oscillations have been most widely studied in neural activity generated by large groups of neurons. broadly speaking, models are used to generate data, to study emergent behaviors, or they can be used as forward or observation models, which are inverted given empirical data. of characterising the density dynamics explicitly, one can summarize it in terms of coefficients parameterising the expression of modes:(34)where μ = η−p, η− being the generalized inverse of the matrix encoding the basis set of modes. fokker-planck equation describing the ornstein-uhlenbeck process, with μ = 〈j〉j nq(t) and σ2 = 〈j2〉j nq(t), can be rewritten as a continuity equation:(22)where f is the flux of probability defined as follows:(23)the stationary solution should satisfy the following boundary condition:(24)and(25)which expresses the fact that the probability current at threshold gives, by a self-consistent arguments, the average firing rate, q, of the population. a, mcnaughton b (1997) path integration and cognitive mapping in a continuous attractor neural network model. this is not possible in the (planar) single neural dynamics of the microscopic system because chaotic dynamics require at least three degrees of freedom. field models are another important tool in studying neural oscillations and are a mathematical framework describing evolution of variables such as mean firing rate in space and time. although this permits small scale nonlinear activity to coincide with and influence stochastic macroscopic activity, it requires a somewhat elaborate framework. for example, when a person looks at a tree, visual cortex neurons representing the tree trunk and those representing the branches of the same tree would oscillate in synchrony to form a single representation of the tree. these neurons reflect the implementation of the perceptual comparison process and may underlie the process of decision-making. the inputs one neuron receives at the synapses from other neurons cause transient changes in its resting membrane potential, called postsynaptic potentials. quantitative models can estimate the strength of neural oscillations in recorded data. linear oscillators and limit-cycle oscillators qualitatively differ in terms of how they respond to fluctuations in input. "coherent oscillations: a mechanism of feature linking in the visual cortex? the equations, their derivation, and relevant references are provided in the recent developments in neural field models section. m, sleigh j, steyn-ross d, steyn-ross m (2006) general anesthetic-induced seizures can be explained by a mean-field model of cortical dynamics. if the perturbation is due to a stochastic train, then the neuron fires randomly at an average rate proportional to the stochastic inputs. hr, cowan jd (1973) a mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. "driving fast-spiking cells induces gamma rhythm and controls sensory responses". such stimuli generally result in different firing rates because of the neuron's frequency tuning. when the postsynaptic potential reaches a threshold, the neuron produces an impulse. low response variability in simultaneously recorded retinal, thalamic and cortical neurons. 7e); the fano factor is overestimated when the firing rate is underestimated, and vice versa. to underscore the observation that the mean synaptic currents evidence an emergent phenomenon, and not merely the super-position of a bursting neuron, the time series of a single neuron is provided in figure 6c. local recurrent circuitry has received the most attention, but other theoretical mechanisms for the maintenance of persistent activity, including local recurrent synaptic feedback and intrinsic cellular bistability [58],[59], have been put forward. the transmission of information within and between neurons involves changes in the so-called resting membrane potential, the electrical potential of the neurons at rest, when compared to the extracellular space.., a cortical column has o(104)−o(108) neurons) which are massively interconnected (on average, a neuron makes contact with o(104) other neurons). this phenomenon is best seen in local field potentials which reflect the synchronous activity of local groups of neurons, but has also been shown in eeg and meg recordings providing increasing evidence for a close relation between synchronous oscillatory activity and a variety of cognitive functions such as perceptual grouping. let us assume that n neurons synapse onto cell i and that jij is the efficacy of synapse j, then the total synaptic afferent current is given by(5)where is the emission time of the kth spike from the jth presynaptic neuron. in particular, [119]–[124] have studied the neural mechanisms underlying perceptual comparison by measuring single-neuron responses in monkeys trained to compare two mechanical vibrations applied sequentially to the tip of a finger; the subjects have to report which of the two stimuli has the higher frequency. compare the results for the time course of the firing rate and the fano factor to the recovery-function model, we need to provide the model with a temporally modulated input q(t), which is derived at every instant from the sound intensity i(t) (see fig. key forms for neural field equations were proposed and analysed by [45]–[47]. (2006) dynamic causal modeling of evoked responses in eeg and meg. if we indicate the firing rate qa for the cell type a by a subscript, then φab can be expressed in terms of the firing rate at other locations and earlier times. at the microscopic scale, we simulate an entire array of spiking neurons in response to a sensory-evoked synaptic current. instead the model is based on an instantaneous relationship between the intensity of the acoustic stimulus i and the effective stimulus strength q so that q(t) = q[i(t)]. in a typical streaming experiment, two sequences are created using sets of high and low tones. the length of an isi generated by a renewal process depends only on the stimulus, and not on the preceding isi, thus maximizing the number of potential output signals in the neural code. as discussed above, this process underscores the dynamical growth in the mean-field oscillations and the interdependent contraction of the interneuron spike timing variance shown in figures 6 and 7. We argue that elaborating principled and informed models is a prerequisite for grounding empirical neuroscience in a cogent theoretical framework, commensurate with the achievements in the physical sciences. any given spatial wavenumber, k, and temporal frequency, ω, equations 56–60 can be rearranged to obtain(61,62)where the gains are defined by gab = ρaνab. 6 shows the results of simulating an ensemble of 250 neurons with a sensory input to all neurons between t = 1,000 ms to t = 3,000 ms. finally, we extend the framework to time-varying stimuli (am pure tones) and compare the quantitative model predictions with measured test data. nonlinear dynamics of such models have also been discussed in the literature, resulting in successful predictions of epileptic dynamics, for example [86],[92], but are not considered here (but see the cognitive and clinical applications section). to compare the model predictions to the isi distributions derived from the test stimuli, only the stimulus strength q was adjusted so that the mean isis of the model and the recording matched. the latter intrinsic fibers have a transmission speed of chom and a transmission delay . the sound intensities thus cover the steepest region of the neuron's rate–intensity function so that the neuron is most sensitive to amplitude modulations. h is a constant threshold value and γ is the spatial domain of the neural field, where x ∈ γ = [0,l]. these changes in potential are mediated by the flux of ions between the intracellular and extracellular space. however, the stationary solutions of the fokker-planck equation (equation 20) represent the stationary solutions of the original if neuronal system. as a first step, we can split the connectivity function, w, into two parts, the homogeneous connectivity, whom(|x−y|), which depends only on the distance, and the heterogeneous connectivity, whet(x,y), which captures the effects of the extrinsic fiber system (for an alternative approach with applications to visual gamma phenomena, see [100]–[102]). giudice p, fusi s, mattia m (2003) modeling the formation of working memory with networks of integrate-and-fire neurons connected by plastic synapses. thus the dimension reduction afforded by the neural mass approximation allows the introduction of more complex intrinsic dynamics, permitting dynamical chaos. the coefficients ναβ represent the synaptic density between excitatory (e) and inhibitory (i) populations or from the stochastic/noise (n) or sensory (s) inputs.(49)where, as per equation 40, rab is the characteristic range of axons, including dendritic arborization, cab is the characteristic velocity of signals in these axons, and γab = cab / rab is the resulting temporal damping coefficient in the absence of pulse regeneration. 7b, the time-dependent firing rates of the model (black lines) and experiment (gray lines) closely coincide most of the time. a, sejnowski tj (2001) thalamocortical assemblies: how ion channels, single neurons and large scale networks organize sleep oscillations. these latter patterns take the form of stripes and checkerboard-like patterns, and have been linked to drug-induced visual hallucinations [72]. in this paper, we review and integrate, in a unifying framework, a variety of computational approaches that have been used to characterize the dynamics of the cortex, as evidenced at different levels of measurement. a key aim in modeling is to strike a balance between having too few parameters to be realistic, and too many for the data to be able to constrain them effectively. "human memory strength is predicted by theta-frequency phase-locking of single neurons". Computational models at different space–time scales help us understand the fundamental mechanisms that underpin neural processes and relate these processes to neuroscience data. cortex is a complex system, characterized by its dynamics and architecture, which underlie many functions such as action, perception, learning, language, and cognition. we will consider the relationship between density dynamics and neural mass models and how these can be extended to cover spatiotemporal dynamics in the brain. the most striking achievement in this regard is the reduction of a large population of spiking neurons to a distribution function describing their probabilistic evolution—that is, a function that captures the likely distribution of neuronal states at a given time.[61] since then, numerous studies have replicated these findings and extended them to different modalities such as eeg, providing extensive evidence of the functional role of gamma oscillations in visual perception. Macroscopic models can inform us about whole brain dynamics and interactions between large-scale neural systems such as cortical regions, the thalamus, and brain stem. thus the dimension reduction afforded by the neural mass approximation allows the introduction of more complex intrinsic dynamics, permitting dynamical chaos. left panel: time course of the observed (thick gray line) and the predicted (thin black line) fano factors. propagation of pulses within and between populations determines the values of φab.(44,45,46)where νab is a coupling strength, nab is the mean number of synapses on neuron a from neurons b, sab is the mean time-integrated strength of the response of v per incoming spike, and θab is the average rate of incoming spikes (allowing for the possibility of a discrete time delay, τab, between populations b and a in addition to any delays due to spreading within populations). in other words, equation 85 identifies quantitatively how a particular neural activation is impacted by its local and global connectivity in a biologically realistic environment, including signal exchange with finite and varying (intracortical versus corticocortical) transmission speeds. these models aim to describe how the dynamics of neural circuitry arise from interactions between individual neurons. b, manin d, sirovich l (1996) dynamical models of interacting neuron populations. the distinction relates to that between localisationism and connectionism that dominated thinking about cortical function in the nineteenth century. its structural architecture has been studied for more than a hundred years; however, its dynamics have been addressed much less thoroughly. this implicitly encodes variability in the postsynaptic depolarisation, relative to the potential at which the neuron would fire. theses biomarkers are often named "eeg biomarkers" or "neurophysiological biomarkers" and are quantified using quantitative electroencephalography (qeeg). first term represents recurrent feedback from neurons within the ensemble due to their own firing. this simple mathematical model can be extended naturally to accommodate multiple populations and cortical sheets, spike frequency adaptation, neuromodulation, slow ionic currents, and more sophisticated forms of synaptic and dendritic processing as described in the review articles [4],[54],[55]. in what follows, we will explain in detail the arguments that take us from the spiking behavior of individual neurons to the mean-field dynamics described by the fokker-planck equation. once the stimulus ends, there is a brief quiescent phase because all of the neurons have just fired and require a short train of stochastic inputs before they commence firing again. this makes dynamic models critical in integrating theory and experiments. however, the mechanisms are quite distinct: individual neurons within the microscopic ensemble fired stochastically, but at uncorrelated times. as above, the cortex is approximated as a 2-d sheet and r is assumed to be the actual position in the case of the cortex; other structures, such as the thalamus, are linked to the cortex via a primary topographic map.[30] harmonic oscillations appear very frequently in nature—examples are sound waves, the motion of a pendulum, and vibrations of every sort. each attribute induces a dimension in the phase space of a neuron; in our example the phase space would be three dimensional and the state of each neuron would correspond to a point ν = {v,i,t} ∈ℜ3 or particle in phase space. class i neurons can generate action potentials with arbitrarily low frequency depending on the input strength, whereas class ii neurons generate action potentials in a certain frequency band, which is relatively insensitive to changes in input strength. the fano factor f(t) is derived from the spike-count distribution for a certain counting time t by dividing the variance of the spike count σn(t)2 by its average 〈n(t)〉 (2) the errors of the measured cv values and fano factors were determined using the bootstrap method. simulations: ensemble activity from neuronal to whole brain scalesthis section illustrates neuronal ensemble activity at microscopic, mesoscopic, and macroscopic spatial scales through numeric simulations.[28] a number of nuclei in the brainstem have diffuse projections throughout the brain influencing concentration levels of neurotransmitters such as norepinephrine, acetylcholine and serotonin. m (1999) attractor neural network models of spatial maps in hippocampus. low firing rates, on the other hand, can lead to more than 10-fold higher fano factors for the same counting time. cortex is a complex system, characterized by its dynamics and architecture, which underlie many functions such as action, perception, learning, language, and cognition. these rules are reflected in the abstractions and refinements of the models which address the different scales. this fiber system is myelinated, which increases the transmission speed by an order of magnitude, and is not invariant under spatial translations (heterogeneous); in fact it is patchy [99]. smooth spatiotemporal dispersion of the evoked cortical response and its time delayed corticothalamic volley are evident. r, hernandez a, zainos a (2004) neuronal correlates of a perceptual decision in ventral premotor cortex. the main source of randomness is from fluctuations in recurrent currents (resulting from “quenched” randomness in the connectivity and transmission delays) and fluctuations in the external currents. it was argued that the study of these bifurcations provides a parsimonious explanation of the unique time course, symmetry, onset, and offset of both absence and tonic clonic seizures, capturing their similarities and the differences. plos comput biol 4(8):Introductionthe brain appears to adhere to two fundamental principles of functional organization, functional integration and functional specialization, where the integration within and among specialized areas is mediated by connections among them. in fact, the ability to show multistable pattern formation is the only relevant property of the network and can be realized in multiple network architectures as discussed in previous sections. when averaged over a population of neurons, with normal response characteristics, a reasonable approximation for the firing rate, q, is(47)where qamax is the maximum firing rate and sa is a monotonic increasing sigmoidal function that approaches zero as va→−∞ and unity as va→∞. we can now write equation 63 in matrix form as(66)where a is an n′×n′ matrix, q is an n′-element column vector, b is an n′×j′ matrix, and n is a j′-element column vector. summary of the notation for all the main dynamical variables and physiological parameters is given in table 1. a key aim in modeling is to strike a balance between having too few parameters to be realistic, and too many for the data to be able to constrain them effectively. g, rolls et, horwitz b (2004) ‘what’ and ‘where’ in visual working memory: a computational neurodynamical perspective for integrating fmri and single-neuron data. equations 93, 94, and 95 define the dynamics of a stream classification model in one of its simplest forms. indeed, for appropriate inputs, individual spikes can be highly reliable and precisely timed (berry et al. a, knight b, sirovich l (2000) on the simulation of large populations of neurons. gollisch: department of molecular and cellular biology, harvard university, cambridge, ma 02138.., a cortical column has o(104)−o(108) neurons) which are massively interconnected (on average, a neuron makes contact with o(104) other neurons). here the neural fields maintain the firing of its neurons to represent any location along a continuous physical dimension such as head direction, spatial location, or spatial view. in numerical simulations: ensemble activity from neuronal to whole brain scale, we provide numerical simulations of neuronal ensemble dynamics across a hierarchy of spatial and temporal scales. "collective frequencies and metastability in networks of limit-cycle oscillators with time delay". and more physiological constraints have been incorporated into neural field models of the type discussed here (see equations 39 and 40). if the steady state is stable for all k, spectra and other properties of the linear perturbations can be self-consistently defined; otherwise a fully nonlinear analysis is needed. top row: first return map for the cloud interspike delay over five consecutive time steps, before (a) and following (b) synaptic input. in fourier space, this gives(50,51)where k = (kx,ky) is the wave vector and ω is the angular frequency. w, schöner g (2002) dynamic field theory of movement preparation. the first two moments in the kramers-moyal expansion are called drift and diffusion coefficients, respectively, and they are given by:(16)(17)in general, keeping only the leading term linear in dt, it is easy to prove that for k>1,(18)and hence,(19). consequently, the dynamics are described by the evolution of the probability density function:(8)which expresses the population density, which is the fraction of neurons at time t that have a membrane potential v(t) in the interval [ν,ν+dν]. the most successful and widely used model of neurons, the hodgkin–huxley model, is based on data from the squid giant axon.., a single neuron or neural ensemble) by its circular phase alone and hence ignores the amplitude of oscillations (amplitude is constant). s, deco g (2002) large-scale neural model for visual attention: integration of experimental single cell and fmri data. the smooth spatiotemporal dispersion of the evoked cortical response and its time delayed corticothalamic volley are evident. this implies that neurons are spontaneously at rest (quiescent) but depolarize with a small perturbation. the recovery parameters of the type i and the type ii receptor neurons were not significantly different (p = 0. the absolute recovery period was estimated as the shortest experimental isi encountered during an experiment, and the input relations q(i) were determined by matching mean firing rates as described in methods. characterize the receptor dynamics within a general theoretical framework, we first assess the spike-train variability in response to constant-intensity stimuli over a large range of sound frequencies and intensities. these experiments showed that groups of spatially segregated neurons engage in synchronous oscillatory activity when activated by visual stimuli. we have demonstrated for these cells that the mean response and its fluctuations can be predicted with a model that contains a stimulus encoder based on firing rates and a simple stochastic spike generator.

Neural oscillation - Wikipedia

The Dynamic Brain: From Spiking Neurons to Neural Masses and

the neurons in the two specific populations additionally receive external inputs encoding stimulus specific information. any computation and behavioral decision has to be based on this representation. instead of modelling individual neurons, this approach approximates a group of neurons by its average properties and interactions. neural field is illustrated by the rectangular box showing the neural activity μ(x,t) composed of inhibitory and excitatory neurons. weakly coupled oscillators can generate a range of dynamics including oscillatory activity. these changes in potential are mediated by the flux of ions between the intracellular and extracellular space.. the power spectral density of ψ at k and ω is(70)the frequency and wavenumber spectra are then(71,72)a position-dependent frequency cross-spectrum can be calculated from equation 70:(73)where the angle brackets denote an average over multiple trials and/or over the phase of the exogenous stimuli that drive the system. depending on the properties of the connection, such as the coupling strength, time delay and whether coupling is excitatory or inhibitory, the spike trains of the interacting neurons may become synchronized. in: methods in neuronal modeling, edited by koch c and segev i. neurons can generate multiple action potentials in sequence forming so-called spike trains. 13 shows a biophysically realistic computational model for a probabilistic decision-making network that compares two mechanical vibrations applied sequentially (f1 and f2). the electric potentials generated by single neurons are far too small to be picked up outside the scalp, and eeg or meg activity always reflects the summation of the synchronous activity of thousands or millions of neurons that have similar spatial orientation. simulation of a network of if neurons allows one to study the dynamical behavior of the neuronal spiking rates. m, sleigh j, steyn-ross d, steyn-ross m (2006) general anesthetic-induced seizures can be explained by a mean-field model of cortical dynamics. in this particular case, the classification system y(t) does traverse from the positive (two streams) to the negative (one stream) fixed point and back. a: cv as a function of stimulus intensity for a receptor neuron stimulated with 3-, 5-, 7-, 9-, and 11-khz tones (3 repetitions of each stimulus; error bars: error measure for cv values determined using bootstrap methods). in a future extension of the model, however, one could easily include a detailed adaptation model that captures cell-intrinsic currents (benda and herz 2003) and dynamics of the mechanical stimulus coupling (gollisch and herz 2004). pa, rennie ca, wright jj (1997) propagation and stability of waves of electrical activity in the cerebral cortex. when the postsynaptic potential reaches a threshold, the neuron produces an impulse. for the longer timescales, experimental and model spike-timing reliability are in close agreement. work in this area has resulted in numerous quantitatively verified predictions about brain electrical activity, including eeg time series [86],[87],[90], spectra [50],[86],[87],[90],[91], coherence and correlations, evoked response potentials (erps) [87], and seizure dynamics [86],[90],[92]. the network is partitioned into populations of neurons whose input currents share the same statistical properties and fire spikes independently at the same rate. the bold line separating stable and unstable regions indicates the course of the critical surface as the time delay changes. if c(δ) and ciid(δ) differ significantly, the observed isis cannot be assumed independent. the generic form of neural field dynamics can be written as (see also [53]):(41)where μ = μ(x,t) is the neural field, capturing the neural mass activity at time t and position x. "generative models of cortical oscillations: neurobiological implications of the kuramoto model". examples are walking, breathing, and swimming,[55] most evidence for central pattern generators comes from lower animals, such as the lamprey, but there is also evidence for spinal central pattern generators in humans. note that in this section we simulate dynamics at the scale of coupled individual neurons. nature and strength of neuronal connectivity varies markedly when considered across the heirarchy of spatial scales. these rhythmic outputs are produced by a group of interacting neurons that form a network, called a central pattern generator. a statistical description of each population is given by a probability density function that expresses the distribution of neuronal states (i. spontaneous activity is usually considered to be noise if one is interested in stimulus processing; however, spontaneous activity is considered to play a crucial role during brain development, such as in network formation and synaptogenesis. subpotentials, vab, respond in different ways to incoming spikes, depending on their synaptic dynamics (ion-channel kinetics, diffusion in the synaptic cleft, etc. note that in this section we simulate dynamics at the scale of coupled individual neurons. a positive (negative) electric field applied across the slice increased (decreased) the speed of wave propagation, consistent with the theoretical predictions of neural field theory, assuming that a positive (negative) electric field reduces (increases) the threshold, h, in equation 42. more than 50 years later, intrinsic oscillatory behavior was encountered in vertebrate neurons, but its functional role is still not fully understood. the introduction of propagation delays leads to dynamics that are very reminiscent of those observed empirically. a: cv values predicted for n = 14 cells are plotted against the experimentally measured values. on the other hand, the corticocortical (extrinsic) fiber system contains fibers which leave the gray matter and connect distant areas (up to 20 cm [84]). the measurement was repeated 20 times, and averages were taken (see fig.-train variability of auditory neurons in vivo: dynamic responses follow predictions from constant stimuli. the mean-field technique allows us to discard the index denoting the identity of any single neuron and express the infinitesimal change, dv(t), in the membrane potential of all neurons as:(14)where n is the number of neurons, and q(t) is the mean population firing rate. the mathematical analysis of the neural field models is typically performed with linear stability theory, weakly nonlinear perturbation analysis, and numerical simulations. such simulations are illustrative, they are computationally intensive; even when limited to just 250 neurons at <5 s of integration time. dorsal part of the thorax was opened to expose the metathoracic ganglion and the auditory nerve, which was fixed with a custom-made forceps mounted on a micromanipulator. there are two different types of population: excitatory and inhibitory. predicted fano factors (black line) agree well with the experimentally determined fano factors (thick gray line). Mean-field and related formulations of dynamics also play an essential and complementary role as forward models that can be inverted given empirical data. within the sheet, the coupling drops in proportion to spatial separation and is hence scale-free:(91)where f incorporates all intrasystem dynamics as per equation 89 and the indices numerate either the sensory node {sens} or the nodes within the sheet {sheet}. b, rit v (1995) electroencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns. models that accurately describe neural variability over a wide range of stimulation and response patterns are therefore highly desirable, especially if they can explain this variability in terms of basic neural observables and parameters such as firing rate and refractory period. in modeling the activity of large numbers of neurons, the central idea is to take the density of neurons to the continuum limit, resulting in spatially continuous neural networks. neurodynamical model for a probabilistic decision-making network that performs the comparison of two mechanical vibrations applied sequentially (f1 and f2)..,(26)and(27)in addition, the probability mass leaving the threshold at time t has to be re-injected at the reset potential at time t+tref (where tref is the refractory period of the neurons), which can be accommodated by rewriting equation 22 as follows:(28)where h(. we have introduced parameterisation in terms of probability modes because it provides a graceful link to neural mass models. although individual neurons exhibit nonlinear dynamics, the ensemble mean dynamics are (linearly) stable to the stochastic inputs until the background current is increased. work in this area has resulted in numerous quantitatively verified predictions about brain electrical activity, including eeg time series [86],[87],[90], spectra [50],[86],[87],[90],[91], coherence and correlations, evoked response potentials (erps) [87], and seizure dynamics [86],[90],[92]. for time-dependent stimuli, the latter property is no longer true and one rather speaks of a “modulated renewal process” (reich et al. computational models at different space–time scales help us understand the fundamental mechanisms that underpin neural processes and relate these processes to neuroscience data. modes and massesthe fokker-planck equation, (equation 1), is a rather beautiful and simple expression that prescribes the evolution of ensemble dynamics, given any initial conditions and equations of motion that embed our neuronal model. its structural architecture has been studied for more than a hundred years; however, its dynamics have been addressed much less thoroughly. figure 20 presents an example of a bifurcation arising from a 3 hz oscillatory instability in the corticothalamic neural field model of the recent developments in neural field models section. in this case, all moments higher than two become negligible, in relation to the drift (μ) and diffusion (σ2) coefficients.., eigen vectors) of the fokker-planck operator, q [17], where qη = ηλ⇒η−qη = λ and λ is a leading-diagonal matrix of eigenvalues. this reduction is justified by the fact that the integration time of the receptor neurons is on the timescale of about 1 ms (gollisch and herz 2005; prinz and ronacher 2002), a timescale shorter than the minimal observed isi. Models of the cortex can establish which types of large-scale neuronal networks can perform computations and characterize their emergent properties. s, terry j, breakspear m (2006) on the genesis of spike-wave oscillations in a mean-field model of human thalamic and corticothalamic dynamics. We argue that elaborating principled and informed models is a prerequisite for grounding empirical neuroscience in a cogent theoretical framework, commensurate with the achievements in the physical sciences. the average firing rate of the population f1

Spike-Train Variability of Auditory Neurons In Vivo: Dynamic

important phenomena have been studied by linearizing these models around their steady state solutions. simulation of a network of if neurons allows one to study the dynamical behavior of the neuronal spiking rates. then the spatially continuous soma potential, va, is the sum of contributions, vab, arriving as a result of activity at each type of (mainly) dendritic synapse b, where b indexes both the incoming neural population and the neurotransmitter type of the receptor. in this case, all moments higher than two become negligible, in relation to the drift (μ) and diffusion (σ2) coefficients. on in vivo recordings from locust auditory receptor neurons, a model system for the auditory periphery of insects (michelsen 1971; römer 1976; ronacher and krahe 2000; stumpner and von helversen 2001), we systematically explore a phenomenological description that captures the spike-train statistics at different average firing rates and that applies to constant as well as temporally modulated sound stimuli. jj, liley dtj (1996) dynamics of the brain at global and microscopic scales: neural networks and the eeg. we followed this approach and then used w(δ) to predict the shape of isi distributions for different sound intensities and mean firing rates through eq. this simple mathematical model can be extended naturally to accommodate multiple populations and cortical sheets, spike frequency adaptation, neuromodulation, slow ionic currents, and more sophisticated forms of synaptic and dendritic processing as described in the review articles [4],[54],[55]. benefits and drawbacks of such mechanistic but multicomponent frameworks—as opposed to phenomenological one-step models such as those proposed by berry and meister (1998) or kara et al. and 1 ms, spike timing of the model spike trains is more reliable than that of the experimental spike trains., we first discuss how to find the steady states of neural field models. linear predictions from neural field models have accounted successfully for a range of experimental phenomena, as mentioned above. a number of neurophysiological experiments on decision-making reveal the neural mechanisms underlying perceptual comparison, by characterising the neuronal correlates of behavior [119]–[121]. within the sheet, the coupling drops in proportion to spatial separation and is hence scale-free:(91)where f incorporates all intrasystem dynamics as per equation 89 and the indices numerate either the sensory node {sens} or the nodes within the sheet {sheet}. neural oscillations and synchronization have been linked to many cognitive functions such as information transfer, perception, motor control and memory. a key feature of recent models is that they use parameters that are of functional significance for eeg generation and other aspects of brain function; for example, synaptic time constants, amount of neurotransmitter release or reuptake, and the speed of signal propagation along dendrites., neural field models can be construed as a spatiotemporal convolution (c. velazquez jl, cortez ma, snead oc, wennberg r (2003) dynamical regimes underlying epileptiform events: role of instabilities and bifurcations in brain activity. by employing population-specific fields and parameters, it allows each population to generate a family of outgoing fields that propagate to different populations in different ways. mathematics of the hodgkin–huxley model are quite complicated and several simplifications have been proposed, such as the fitzhugh–nagumo model and the hindmarsh–rose model. as evident in figure 6b, the increased firing synchrony leads in turn to a marked increase in the simulated local field potentials as individual neurons begin to contribute concurrent ion currents. in two dimensions, many other interesting patterns can occur, such as spiral waves [71], target waves, and doubly periodic patterns. laurent and colleagues showed that oscillatory synchronization has an important functional role in odor perception.. the power spectral density of ψ at k and ω is(70)the frequency and wavenumber spectra are then(71,72)a position-dependent frequency cross-spectrum can be calculated from equation 70:(73)where the angle brackets denote an average over multiple trials and/or over the phase of the exogenous stimuli that drive the system. for both constant and dynamic stimuli the variability could be modeled using the same stochastic process, which indicates that there is no principal difference between the responses to constant or strongly time varying stimuli. the key assumption in the population density approach is that the afferent input currents impinging on neurons in one population are uncorrelated. in the full nonlinear fokker-planck formulation, different phase functions or probability density moments could couple to each other; both within and between populations or ensembles. the slightly negative correlation at lag 1 indicates a small effect of adaptation induced by the previous spike (for comparison also see brandman and nelson 2002; chacron et al. the most striking achievement in this regard is the reduction of a large population of spiking neurons to a distribution function describing their probabilistic evolution—that is, a function that captures the likely distribution of neuronal states at a given time. we tested different parameterizations (including single and double exponentials) and obtained best results with michaelis–menten type sigmoid functions (see e. intracortical connections are illustrated as densely connected fibers in the upper sheet and define the homogeneous connectivity whom. using bifurcation analysis, different oscillatory varieties of these neuronal models can be determined, allowing for the classification of types of neuronal responses. neurons derived from different brain regions are inherently different in vitro: a novel multiregional brain-on-a-chip. spatially localized bump solutions are equivalent to persistent activity and have been linked to working memory in prefrontal cortex [56],[57]. r, hernandez a, zainos a, salinas e (2003) correlated neuronal discharges that increase coding efficiency during perceptual discrimination.[36] in particular, it describes how the activity of a group of interacting neurons can become synchronized and generate large-scale oscillations. as noted above, the fokker-planck equation summarizes the flow and dispersion of states over phase space in a way that is a natural summary of population dynamics in genetics (e. giudice p, fusi s, mattia m (2003) modeling the formation of working memory with networks of integrate-and-fire neurons connected by plastic synapses. "a mechanism for cognitive dynamics: neuronal communication through neuronal coherence". an exemplary approach, owing to boltzmann and maxwell, is the approximation of the motion of molecules in a gas by mean-field terms such as temperature and pressure. jy, guan l, tsau y (1999) propagating activation during oscillations and evoked responses in neocortical slices.., not topographically specific) connections between populations to be handled straightforwardly, simply by increasing rab while reducing γab, thereby allowing influences to propagate long distances with little damping. p, rennie c, rowe d (2002) dynamics of large-scale brain activity in normal arousal states and epileptic seizures. (2006) bold responses to stimuli: dependence on frequency, stimulus form, amplitude, and repetition rate. that is, although individual neurons exhibit nonlinear dynamics, the ensemble mean dynamics are (linearly) stable to the stochastic inputs until the background current is increased. the simplest such formulation is [95](52)where x is the evolving parameter, y is the quantity that drives the evolution, x(0) and y(0) are steady state values, and x(1) is a constant that describes the strength of feedback. in equation 7, h(t) is the heaviside function (h(t) = 1 if t>0, and h(t) = 0 if t<0). pa (2006) patchy propagators, cortical dynamics, and the generation of spatially structured gamma oscillations.[44] it is very common in single neurons where spike timing is adjusted to neuronal input (a neuron may spike at a fixed delay in response to periodic input, which is referred to as phase locking[10]) and may also occur in neuronal ensembles when the phases of their neurons are adjusted simultaneously. in neural mass models, we ignore this possibility because we can only couple the expectations or first moments. frequency locking is not surprising in a model with spatial coupling, the amplitude modulation is a novel, emergent property of the nonlinear dynamics.[14] different neural ensembles are coupled through long-range connections and form a network of weakly coupled oscillators at the next spatial scale. direct simulations of these equations yield a complex spatiotemporal pattern, covering the individual trajectory of the internal state of each neuron in the network. then the fixed point state is rendered unstable by the stimulus current and large amplitude oscillations occur. this memory term, or “recovery function” (berry and meister 1998; for a comparison with the “hazard function,” see gerstner and kistler 2002; johnson 1996), captures the influence of refractoriness on the generation of the next action potential. the symbol ⊗ indicates a convolution of the driver with the temporal response function h(t), which incorporates the time constants of the dynamics. a: acoustic stimulus; the envelope is an instance of gaussian white noise with 400-hz cutoff frequency, 36-db spl mean intensity, and 3-db sd. in contrast to this, spike counts for the investigated receptor neurons display far more precision (mean fano factor for 10-ms counting windows: 0. this inversion allows one to select the best model (given some data) and make probabilistic comments about the parameters of that model. the final term, isensory, models sensory input, consisting of a constant synaptic current to a subset of neurons, whenever the sensory stimulus is present. the ion channels open or close depending on the membrane potential and on substances released by the neurons, namely neurotransmitters, which bind to receptors on the cell's membrane and hyperpolarize or depolarize the cell.[35] interactions amongst these oscillators are introduced by a simple algebraic form (such as a sine function) and collectively generate a dynamical pattern at the global scale. inversion of these models has also furnished estimates of underlying physiological parameters and their variations across the brain, in different states of arousal and pathophysiology [86],[93],[94]. for example, neuronal activity generated by two populations of interconnected inhibitory and excitatory cells can show spontaneous oscillations that are described by the wilson-cowan model. of kuramoto model showing neural synchronization and oscillations in the mean field. therefore we used a comparison of the autocorrelation function c(δ) of the observed spike train (mean not subtracted) to the autocorrelation function ciid(δ) of a spike train with the same isi distribution and independent, identically distributed (iid) isis as a second test for independence of isis. as the state of each neuron evolves, the points will flow through phase space, and the ensemble density p(ν,t) will evolve until it reaches some steady state or equilibrium. "distinct mechanisms for synchronization and temporal patterning of odor-encoding neural assemblies". a, sejnowski tj (2001) thalamocortical assemblies: how ion channels, single neurons and large scale networks organize sleep oscillations.[47] because induced responses may have different phases across measurements and therefore would cancel out during averaging, they can only be obtained using time-frequency analysis. this specifies how the coefficients of probability modes would evolve from any initial state or following a perturbation to the neuronal states. Models of the cortex can establish which types of large-scale neuronal networks can perform computations and characterize their emergent properties. cortex is a complex system, characterized by its dynamics and architecture, which underlie many functions such as action, perception, learning, language, and cognition. m (1999) attractor neural network models of spatial maps in hippocampus. each of these parameters is constrained by physiological and anatomical measurements, or, in a few cases, by other types of modeling. models adopt a variety of abstractions in order to describe complex oscillatory dynamics observed in brain activity. p, abbott l (2002) theoretical neuroscience: computational and mathematical modeling of neural systems. this pattern will change as a function of its own spatial configuration, φk(x), the connections (whom and νij), and, last but not least, the transmission times of information exchange, and d/c. recent developments (see the heterogeneous connectivity in neural fields section) now allow elucidation of the impact of biologically relevant connection heterogeneities on the stability and conduction of cortical activity.. resonance behavior that does not result in action potentials, may also contribute to oscillatory activity by facilitating synchronous activity of neighboring neurons. in the lif model, each neuron i can be fully described in terms of a single internal variable, namely the depolarization vi(t) of the neural membrane. a statistical description of each population is given by a probability density function that expresses the distribution of neuronal states (i. combining this stochastic spike generator with a deterministic stimulus encoder allows us to calibrate the model neurons with independent measurements of the receptors' input–output relation. we then illustrate healthy and pathological activity in neural field models. we compare this with an explicit model of neural mass dynamics at the mesoscopic scale in the subsequent section. as explained in detail in methods, this relation between q and i is based on the relation between i and the firing rate f (obtained from previous measurements with constant stimuli; see fig. tremor is an involuntary, somewhat rhythmic, muscle contraction and relaxation involving to-and-fro movements of one or more body parts. the mean synaptic currents evidence an emergent phenomenon, and not merely the superposition of a bursting neuron, as can be seen in (c): clearly no burst is evident at this scale. we will consider the relationship between density dynamics and neural mass models and how these can be extended to cover spatiotemporal dynamics in the brain. "in vitro neurons in mammalian cortical layer 4 exhibit intrinsic oscillatory activity in the 10- to 50-hz frequency range". the final term, isensory, models sensory input, consisting of a constant synaptic current to a subset of neurons, whenever the sensory stimulus is present. sequences vary in presentation rate and the frequency difference between the tones. oscillatory activity in groups of neurons generally arises from feedback connections between the neurons that result in the synchronization of their firing patterns. apart from slight asymmetries, no systematic differences between the predictions and the experimental data are observed. the summation is assumed to be linear, and all potentials are measured relative to the resting potential [95]. receptor cells were recorded intracellularly from the axon in the auditory nerve with standard glass microelectrodes (borosilicate, gc100f10; harvard apparatus, edenbridge, uk), filled with a 1 m kcl solution (30–60 mω resistance). w (1979) nonlinear dynamics of paleocortex manifested in the olfactory eeg. values for the seed neuron used in figure 7 are plotted in red. there are two subtypes of excitatory population, namely: specific and nonselective. the lines correspond to the mean-field calculations: the black line indicates f1f2 (f2 = f1−8 hz). d, seamans j, sejnowski t (2000) neurocomputational models of working memory. the mean-field approach ensures that these dynamics will converge to a stationary attractor that is consistent with the steady-state dynamics we require [10],[20]. under the diffusion approximation, equation 21 can also be interpreted (by means of the central limit theorem), as the case in which the sum of many poisson processes (equation 5) becomes a normal random variable with mean μ(t) and variance σ2. g, rolls et (2005) attention, short term memory, and action selection: a unifying theory. the excitability of neurons can be subdivided in class i and ii. these fibers are myelinated and hence to be distinguished from the typically unmyelinated (hence slower) intracortical fibers. for τa; 2-sample t-test), and the parameters for the single type iii (γ = 3. black indicates f1f2 (f2 = f1−8 hz). pe, blair ht, cho j (2001) the anatomical and computational basis of the rat head-direction cell signal. mathematically, spike generation is thus described by a probability per unit time (the “hazard”) ρ(t|tlast) that is conditional on the last spike occurring at time tlast (7) the values of the recovery function w range between zero and unity. these rules are reflected in the abstractions and refinements of the models which address the different scales. further extension for more complex and realistic models are possible. the mean-field approach ensures that these dynamics will converge to a stationary attractor that is consistent with the steady-state dynamics we require [10],[20]. bursting neurons have the potential to serve as pacemakers for synchronous network oscillations, and bursts of spikes may underlie or enhance neuronal resonance. f, bellanger j, bartolomei f, chauvel p (2000) relevance of nonlinear lumped parameter models in the analysis of depth- eeg epileptic signals. vk, haken h (1997) a derivation of a macroscopic field theory of the brain from the quasi-microscopic neural dynamics. then the fixed point state is rendered unstable by the stimulus current and large amplitude oscillations occur. in this paper, we review and integrate a variety of computational approaches that have been used to characterize the dynamics of the cortex, as evidenced at different levels of measurement. as evident in figure 6b, the increased firing synchrony leads in turn to a marked increase in the simulated local field potentials as individual neurons begin to contribute concurrent ion currents. jy, guan l, tsau y (1999) propagating activation during oscillations and evoked responses in neocortical slices. previous work has shown that many properties of neuronal dynamics can be obtained by regarding activity changes as perturbations of a steady state [86]. spatially uniform steady states can be obtained by solving the preceding equations with all time and space derivatives set to zero, assuming that the parameters are spatially constant. a, mcnaughton b (1997) path integration and cognitive mapping in a continuous attractor neural network model. single neurons and groups of neurons can generate oscillatory activity spontaneously. based on this view [21], and neurophysiological evidence [22], it has been hypothesized that each cortical area represents a set of alternative hypotheses, encoded in the activities of cell assemblies. thus, the incoming presynaptic δ-pulse from other neurons is basically low-pass filtered to produce an epsp or ipsp in the post-synaptic cell. compare firing rates and fano factors with model predictions, spike trains were generated by a renewal process with a bin size of b = 0. inserting this expansion in equation 9, and replacing the time derivative in ν′ by the equivalent time derivative in ν, we obtain(11,12)where 〈…〉ν denotes the average with respect to ρ(ε| ν) at a given ν. that the brain contains multiple populations of neurons, indexed by the subscript a, which labels simultaneously the structure in which a given population lies (e. r, pierce p, connors b (1988) periodicity and directionality in the propagation of epileptiform discharges across neortex. subpotentials, vab, respond in different ways to incoming spikes, depending on their synaptic dynamics (ion-channel kinetics, diffusion in the synaptic cleft, etc. when the voltage across the capacitor reaches a threshold θ, the circuit is shunted (reset) and a δ pulse (spike) is generated and transmitted to other neurons. different types of coding schemes have been proposed, such as rate coding and temporal coding. stimulus sequences (top) and its resulting neural field dynamics (bottom). this is particularly true for cortical neurons, which typically discharge with high variability under in vivo conditions (buracas et al. "the rhythms of steady posture: motor commands as spatially organized oscillation patterns". there are two subtypes of excitatory population: namely, specific and nonselective. input is commonly construed to be a firing rate (or pulse density) and is a sigmoid function, ς, of mean voltage of the same or another ensemble. the secondary effect of the appearance of limit cycle dynamics is to suppress the impact of the spatially uncorrelated stochastic inputs. in the first test, correlations between an interspike interval isik and any of its successors isik+j are analyzed by calculating serial correlation coefficients rj for lags j up to 20 (j ≥ 0) (5) where σisi2 denotes the variance of the isis, and 〈 · 〉 stands for the average over k. these seizures are transient signs and/or symptoms of abnormal, excessive or hypersynchronous neuronal activity in the brain.[10] class ii neurons are also more prone to display sub-threshold oscillations in membrane potential. as above, synaptic currents are induced by the pulse density of the presynaptic neurons, rather than directly via individual presynaptic depolarization. if we fourier transform the resulting set of linear equations, we find for the fluctuating parts(56,57,58,59,60)where is given by equation 50 and we have assumed that all the parameters of the equations (but not the fields of activity) are constant on the timescales of interest. we then show that the measured spike trains are compatible with a renewal process (cox 1962) that includes a recovery function to describe neural refractoriness (berry and meister 1998; gerstner and kistler 2002). pe, blair ht, cho j (2001) the anatomical and computational basis of the rat head-direction cell signal. the model implements a dynamical competition between neurons: the model enables a formal description of the transients (nonstationary) and probabilistic character of behavior (performance) by the explicit use, at the microscopic level, of spiking and synaptic dynamics of one-compartment if neuron models. "cyclic variations in eeg during sleep and their relation to eye movements, body motility and dreaming". acoustic stimuli were presented by loudspeakers [esotec d-260, dynaudio (skanderborg, denmark) on a dca 450 amplifier (denon electronic gmbh, ratingen, germany)]. the introduction of propagation delays leads to dynamics that are very reminiscent of those observed empirically. once the stimulus ends, there is a brief quiescent phase because all of the neurons have just fired and require a short train of stochastic inputs before they commence firing again. it is a set of nonlinear ordinary differential equations that approximates the electrical characteristics of a neuron, in particular the generation and propagation of action potentials. the average over all pairs of spike trains is then taken as the reliability rcorr (4) this measure takes values between 0 (minimum reliability) and 1 (maximum reliability). (note that va is linearly related to the current reaching the soma, and to μ in earlier sections..,(10)in the derivation of the last equation, we have assumed that p(ν′,t) and ρ(ε| ν′) are infinitely many times differentiable in ν. g, rolls et (2003) attention and working memory: a dynamical model of neuronal activity in the prefrontal cortex.

  • Categorization, prototype theory and neural dynamics.

    assuming a gaussian distribution of individual neuronal firing thresholds, one obtains a symmetric sigmoid-shaped function for ςa as per the mean-field model section. computational models at different space–time scales help us understand the fundamental mechanisms that underpin neural processes and relate these processes to neuroscience data. g, lee ts (2002) a unified model of spatial and object attention based on inter-cortical biased competition. d, seamans j, sejnowski t (2000) neurocomputational models of working memory. the model implements a dynamical competition between neurons: the model enables a formal description of the transients (nonstationary) and probabilistic character of behavior (performance) by the explicit use, at the microscopic level, of spiking and synaptic dynamics of one-compartment if neuron models. its function is the classification of the peripheral spatiotemporal neural field dynamics. as discussed in the previous section, early attempts include neural field theories which approximate the large-scale components of the connectivity matrix as translationally invariant and decaying over space [45],[48],[50]. these models are covered in the neural field models section. r, salinas e (2001) touch and go: decision-making mechanisms in somatosensation. by targeting either na+ or ca++ currents and including (postsynaptic) voltage-dependent effects, this function can incorporate, to a first-order approximation, a variable proportion of ampa or nmda-like kinetics [52]. a single fiber connects the two distant regimes (a) and (b) and contributes to the heterogeneous connectivity, whet, whereas regime (c) has only homogeneous connections. oscillatory activity may respond by increases or decreases in frequency and amplitude or show a temporary interruption, which is referred to as phase resetting. h is a constant threshold value and γ is the spatial domain of the neural field, where x ∈ γ = [0,l]. illustrate ensemble dynamics from first principles, we directly simulate a network of coupled neurons which obey deterministic evolution rules and receive both stochastic and deterministic inputs. thus not only absolute but also relative refractoriness is required to understand the variability and reliability of the measured spike trains. the present purpose, we simulate a single mass with both excitatory and inhibitory neurons [52],[112]. in this case, one can ignore the k dependence in the other propagators, and it becomes possible to express the transfer function with elements of the form(81)where is typically a complicated expression depending on the various jab(ω). local interactions between neurons can result in the synchronization of spiking activity and form the basis of oscillatory activity. the standard deviation of the noise signal around the mean intensity i0 was either 3 db (5 cells) or 5 db (one cell). mr, jirsa vk (2007) neural field dynamics with heterogeneous connection topology. however, the mechanisms are quite distinct: individual neurons within the microscopic ensemble fired stochastically, but at uncorrelated times. b, manin d, sirovich l (1996) dynamical models of interacting neuron populations. all three anatomical attributes undergo characteristic changes during the development of the human brain and its function, as well changing in the aged and diseased brain (see [9] for an overview). however, following a small increase in the constant flow term, due to a sensory input, isensory, the quiescent state becomes unstable and the neuron evolves on a (noise-modulated) limit cycle. by using this site, you agree to the terms of use and privacy policy. the nernst potentials, conductances, and background current are set so that, in the absence of noise and sensory inputs, each neuron rests just below a saddle-node bifurcation to a limit cycle [109]. every treatment of the interplay of anatomical connectivity (local and global connections) and functional connectivity (network dynamics) will have to be represented in the form of equation 85 or a variation thereof. each of these parameters is constrained by physiological and anatomical measurements, or, in a few cases, by other types of modeling. second, the effective model input q is calculated from f such that when applied to the model neuron as a constant stimulus, q causes the correct mean firing rate f. these include features such as separate excitatory and inhibitory neural populations (pyramidal cells and interneurons), nonlinear neural responses, synaptic, dendritic, cell-body, and axonal dynamics, and corticothalamic feedback [38], [43], [44], [48], [50], [83]–[87]. the average firing rate of the population f11,(18)and hence,(19). illustrate the effects of interplay between anatomical and functional connectivity, we discuss a simple example following [103],[104]. m, almeida r, deco g, stetter m (2004) cooperation and biased competition model can explain attentional filtering in the prefrontal cortex. for simplicity, we consider the stationary solution μ0(x) = 0 to be the rest state and consider its deviations μ(x,t) = ξk′(t)φk′ (x)+c. in what follows, we consider stationary solutions for ensemble dynamics. in addition to this competition-centered view, a cooperation-centered picture of brain dynamics, where global representations find their neural correlate in assemblies of coactivated neurons, has been formulated [21],[23].[1] the possible roles of neural oscillations include feature binding, information transfer mechanisms and the generation of rhythmic motor output. "coherent oscillations in monkey motor cortex and hand muscle emg show task-dependent modulation". "prestimulus oscillatory activity in the alpha band predicts visual discrimination ability". for the present purposes, the epsp consists of a brief steady current whenever the presynaptic neuron is depolarized. auditory receptor neurons encode vibrations of the tympanic membrane, the animal's eardrum, in their spike trains. this derivation is a little dense but illustrates the approximating assumptions and level of detail that can be captured by density dynamics. w, schöner g (2002) dynamic field theory of movement preparation. the intention is to demonstrate concrete examples of ensemble dynamics under varying influences of flow and dispersion. now provide brief illustrations of sensory evoked and nonlinear activity as modeled by macroscopic field equations. in this section, we try to clarify some key concepts and show how they relate to each other. these models are particularly attractive because the density dynamics conform to a simple equation: the fokker-planck equation(1)this equation comprises a flow and a dispersion term; these terms embed the assumptions about the dynamics (phase flow, f(ν,t)) and random fluctuations (dispersion, d(ν,t)) that constitute our model at the neuronal level.(49)where, as per equation 40, rab is the characteristic range of axons, including dendritic arborization, cab is the characteristic velocity of signals in these axons, and γab = cab / rab is the resulting temporal damping coefficient in the absence of pulse regeneration.
  • Plasticity in single neuron and circuit computations : Article : Nature

    the mean-field approximation is used extensively in statistical physics and is essentially a technique that finesses an otherwise computationally or analytically intractable problem. the subthreshold dynamical equation 4, given the input current (from equation 5), can be integrated, and yields(6,7)if the neuron i is initially (t = 0) at the resting potential (vi(0) = vl). in particular, it aims to relate dynamic patterns of brain activity to cognitive functions such as perception and memory. stability of equation 86 is obtained if re[λ]<0 and is lost, according to [105], at re[λ] = 0, that is λ = iω. in doing this, we assume for simplicity that all the model parameters are constant in time and space, although it is possible to relax this assumption at some cost in complexity. "oscillatory properties of guinea-pig inferior olivary neurones and their pharmacological modulation: an in vitro study". resetting also permits the study of evoked activity, a term used in electroencephalography and magnetoencephalography for responses in brain activity that are directly related to stimulus-related activity.[23][24] like pacemaker neurons in central pattern generators, subtypes of cortical cells fire bursts of spikes (brief clusters of spikes) rhythmically at preferred frequencies.[25] neurons are locally connected, forming small clusters that are called neural ensembles. equation 49 is also satisfied if φab is replaced by the free propagator and the right side is replaced by a source of the form δ(r−r′)δ(t−t′). Mesoscopic models tell us how neural elements interact to yield emergent behavior at the level of microcolumns and cortical columns. In this paper, we review and integrate, in a unifying framework, a variety of computational approaches that have been used to characterize the dynamics of the cortex, as evidenced at different levels of measurement. low spike-count variances facilitate signal detection, indicating that the refractoriness of the investigated receptor neurons might be helpful for discriminating signals, such as grasshopper calling songs from different males. figure 6a shows a raster plot of the neural spike timing whilst figure 6b shows the simulated local field potential from the ensemble ( = total current flow across all neurons). in steady state, the two-point correlation function can be obtained from equation 73 via the wiener-khinchtine theorem, giving(74)in the case where the system is statistically uniform, equation 74 depends only on the separation r = r′−r, giving(75)where(76)has been used and the arguments of t and n have been shown for emphasis. the term mass action model was coined by [38] as an alternative to density dynamics. the symbol ⊗ indicates a convolution of the driver with the temporal response function h(t), which incorporates the time constants of the dynamics. in the latter case, the limit cycle dynamics dominate, although the stochastic inputs modulate the depolarization amplitude. the present purpose, we simulate a single mass with both excitatory and inhibitory neurons [52],[112]. these approaches have been successful in capturing key phenomena of large-scale brain dynamics, including characteristic eeg power spectra [45],[50], epilepsy [92], and meg activity during sensorimotor coordination [49]. experimental results are from the same cell as in fig. an important extension of these models speaks to the fact that neuronal dynamics play out on a spatially extended cortical sheet. the lack of isis below some minimum value and the subsequent increase reflect the influence of absolute and relative refractoriness, respectively. as in b, the absolute refractory period was chosen as the shortest isi of the recorded neuron. macroscopic models can inform us about whole brain dynamics and interactions between large-scale neural systems such as cortical regions, the thalamus, and brain stem. we assume that there exists only a single corticocortical fiber with terminals at locations x1 and x2, that is n = 2. inserting this expansion in equation 9, and replacing the time derivative in ν′ by the equivalent time derivative in ν, we obtain(11,12)where 〈…〉ν denotes the average with respect to ρ(ε| ν) at a given ν. spike-timing reliability of experimental (gray) and model (black) spike trains, averaged over 6 cells (error bars denote ±1se). figure 6a shows a raster plot of the neural spike timing whilst figure 6b shows the simulated local field potential from the ensemble ( = total current flow across all neurons). the element taj is the response of qa to a change in nj at the same frequency and wave vector. by employing population-specific fields and parameters, it allows each population to generate a family of outgoing fields that propagate to different populations in different ways. broadly speaking, models are used to generate data, to study emergent behaviors, or they can be used as forward or observation models, which are inverted given empirical data. input is commonly construed to be a firing rate (or pulse density) and is a sigmoid function, ς, of mean voltage of the same or another ensemble. by recurrently biasing each others' competitive internal dynamics, the neocortical system arrives at a global representation in which each area's state is maximally consistent with those of the other areas. at the microscopic scale, connectivity is dense, concentrated equally in vertical and horizontal directions and, more or less isotropic when considered across different cortical regions. the mathematical analysis of the neural field models is typically performed with linear stability theory, weakly nonlinear perturbation analysis, and numerical simulations. neural mass models can be generalized to neural field models by making the expectations a function of space, thereby furnishing wave equations that describe the spatiotemporal evolution of expected neuronal states over the cortical surface. average firing rate of a neuron as a function of f1 and f2, obtained with the spiking simulations of the response of vpc neurons during the comparison period (to be contrasted with the experimental results shown in figure 2 of [121]). they are assumed to originate from the somatosensory area s2 and from the pfc, encoding the frequency of both stimuli f1 (stored) and f2 (present) to be compared during the comparison period, i. intuitively, auditory streaming or stream segregation is like listening to bass and soprano vocalists singing simultaneously. g, rolls et (2005) attention, short term memory, and action selection: a unifying theory. these neurons reflect the implementation of the perceptual comparison process and may underlie the process of decision-making. an example is functional magnetic resonance imaging (fmri), measuring regional changes in metabolism and blood flow associated with changes in brain activity. there are n′ neural populations and j′ stimulus sources, and we assume that there is no feedback of stimuli on themselves, or of the brain on stimuli, then we can write equation 61 as(63,64,65)where the sum on the left of equation 63 extends only over populations in the brain, while the sum on the right covers only stimulus sources, denoted by j. note that, in comparision to the use of the terms μν and ς(μν) in equation 40, the present wave-equation is formalized in relationship to population-specific pulse densities, φab, and firing rates, qa,b. the constant γ controls the rise time of voltage, in response to inputs (see also the neural field models section). "oscillatory gamma activity in humans and its role in object representation". critical surface, at which the equilibrium state undergoes an instability, is plotted as a function of the real and imaginary part of the eigenvalue of its connectivity, w. as above, synaptic currents are induced by the pulse density of the presynaptic neurons, rather than directly via individual presynaptic depolarization. thus, neurons sharing the same state v(t) in a population are indistinguishable. this is one perspective on why these simple mean-field models are called neural mass models.-field modelsthis section provides an overview of mean-field models of neuronal dynamics and their derivation from models of spiking neurons.-train variability of auditory neurons in vivo: dynamic responses follow predictions from constant stimuli. top row: first return map for the cloud interspike delay over five consecutive time steps, before (a) and following (b) synaptic input. and theoretical arguments propose that the onset of a seizure reflects a bifurcation in cortical activity from damped stochastic activity—where peaks in the power spectrum reflect damped linear resonances—to high amplitude nonlinear oscillations arising from activity on a limit cycle or chaotic attractor [86], [130]–[134]. g, cowan j (1979) a mathematical theory of visual hallucination patterns. this allows the simulation of a large number of interconnected neurons that form a neural network. the final sort of model we will consider is the generalisation of neural mass models that allow for states that are functionals of position on the cortical sheet. d, robinson p, rennie c (2004) estimation of neurophysiological parameters from the waking eeg using a biophysical model of brain dynamics. shown by our data, responses of receptor neurons may allow an even simpler framework with a clear separation between external stimulus and stimulus-independent cell dynamics (see also brenner et al. in particular, models of interacting pyramidal cells and inhibitory interneurons have been shown to generate brain rhythms such as gamma activity. figure 6a shows the combined data from all recorded cells (n = 14: 11 from the second set of experiments plus 3 with sufficient number of repetitions from the first set), each measured at 5–24 different intensities. most of these results are furnished by slice studies of pharmacologically treated tissue, taken from the cortex [75]–[77], hippocampus [78], and thalamus [79]. the recurrent arrows indicate recurrent connections between the different neurons in a population. in the section entitled neural modes and masses, we return to the full probability distribution function and show how it can be represented by a set of scalars that parameterize it parsimoniously. types i–iii have their greatest sensitivity in the lower-frequency range between 3 and 8 khz with differences in absolute sensitivity and the exact location of the sensitivity maximum; type iv is most sensitive at high frequencies around 15 khz. solve the equations defined by equation 32 for all i, we integrate the differential equation below, describing the approximate dynamics of the system, which has fixed-point solutions corresponding to equation 32:(33). direct simulations of these equations yield a complex spatiotemporal pattern, covering the individual trajectory of the internal state of each neuron in the network. this hence fails to capture some of the cardinal properties of the microscopic ensemble, namely the coupling between the first and second moments (mean and variance). numerous experimental studies support a functional role of neural oscillations; a unified interpretation, however, is still lacking. conclusion, we have seen that statistical descriptions of neuronal ensembles can be formulated in terms of a fokker-planck equation, a functional differential equation prescribing the evolution of a probability density on some phase space. r, hernandez a, zainos a, lemus l, brody c (2002) neural correlates of decision making in secondary somatosensory cortex. k, schiff s, gluckman b (2005) propagating activation during oscillations and evoked responses in neocortical slices. that is, although individual neurons exhibit nonlinear dynamics, the ensemble mean dynamics are (linearly) stable to the stochastic inputs until the background current is increased. imagine a very large number of neurons that populate phase space with a density p(ν,t)., for a spatially uniform sinusoidal drive, the resulting steady state evoked potential (ssep) is obtained by using(80)where ω0 is the drive frequency and φ is its phase. velazquez jl, cortez ma, snead oc, wennberg r (2003) dynamical regimes underlying epileptiform events: role of instabilities and bifurcations in brain activity. a key feature of recent models is that they use parameters that are of functional significance for eeg generation and other aspects of brain function; for example, synaptic time constants, amount of neurotransmitter release or reuptake, and the speed of signal propagation along dendrites. r, salinas e (2003) flutter discrimination: neural codes, perception, memory and decision making. us pause for a moment and reflect upon the significance of equation 85.-yishai r, bar-or l, sompolinsky h (1995) theory of orientation tuning in visual cortex. this affords a considerable simplification of the dynamics and allows one to focus on the behavior of a large number of ensembles, without having to worry about an explosion in the number of dimensions or differential equations one has to integrate. and more physiological constraints have been incorporated into neural field models of the type discussed here (see equations 39 and 40). they can result from postsynaptic potentials from synchronous inputs or from intrinsic properties of neurons. any given spatial wavenumber, k, and temporal frequency, ω, equations 56–60 can be rearranged to obtain(61,62)where the gains are defined by gab = ρaνab. neural responses were amplified (bramp-01, npi electronis, tamm, germany) and recorded by a data-acquisition board (national instruments, pci-mio-16e-1) with a sampling rate of 10 khz. legs, wings, head, and gut were removed to immobilize the animals and to facilitate access to the metathoracic ganglion and auditory nerve. mass models can be regarded as a special case of ensemble density models, where we summarize our description of the ensemble density with a single number. this type of direct simulation is computationally expensive, making it very difficult to analyze how the underlying connectivity relates to various dynamics. points correspond to the average values over 200 trials, and the error bars to the standard deviation. figure 20 presents an example of a bifurcation arising from a 3 hz oscillatory instability in the corticothalamic neural field model of the recent developments in neural field models section.., [127],[128]) is: (i) when the frequency separation is relatively small and/or the rate is relatively slow, listeners perceive a single integrated melody (or stream) and can accurately report the ordering of the tones, and (ii) when the frequency separation is relatively large and/or the rate relatively fast, people clearly perceive two segregated auditory streams, one with a higher pitch than the other. the inserted histogram depicts the distribution of differences between fmodel and fexp. "magnetoencephalography - theory, instrumentation, and applications to noninvasive studies of the working human brain". s, deco g (2002) large-scale neural model for visual attention: integration of experimental single cell and fmri data. for the present purposes, the epsp consists of a brief steady current whenever the presynaptic neuron is depolarized. mean-field and related formulations of dynamics also play an essential and complementary role as forward models that can be inverted given empirical data. here, population dynamics are described by a set of one-dimensional partial differential equations in terms of the distributions of the refractory density (where the refractory state is defined by the time elapsed since the last action potential). "the oscillation score: an efficient method for estimating oscillation strength in neuronal activity". m, stam c (2005) dynamics of a neural system with a multiscale architecture. f(μ) captures the local dynamics of the neural field, and tc = t−|x−x′|/c is the time delay due to signal propagation. this level of description is usually framed as a (stochastic) differential equation (langevin equation) that describes how the states evolve as functions of each other and some random fluctuations with(2)where, d = ½σ2 and ω is a standard wiener process; i. the bold line separating stable and unstable regions indicates the course of the critical surface as the time delay changes. the distinction relates to that between localisationism and connectionism that dominated thinking about cortical function in the nineteenth century. in this paper, we address how distributed and specialized neuronal responses are realized in terms of microscopic brain dynamics; we do this by showing how neuronal systems, with many degrees of freedom, can be reduced to lower dimensional systems that exhibit adaptive behaviors. as above, the cortex is approximated as a 2-d sheet and r is assumed to be the actual position in the case of the cortex; other structures, such as the thalamus, are linked to the cortex via a primary topographic map. it was argued that the study of these bifurcations provides a parsimonious explanation of the unique time course, symmetry, onset, and offset of both absence and tonic clonic seizures, capturing their similarities and the differences. recent examples include studies about the responses of retinal ganglion cells to random flicker (berry and meister 1998) and responses of retinal ganglion cells, lgn neurons, and v1 neurons to moving gratings (kara et al. the key assumption in the population density approach is that the afferent input currents impinging on neurons in one population are uncorrelated. in this work, we analyze the response variability recorded in vivo from locust auditory receptor neurons under acoustic stimulation. they generally arise when a physical system is perturbed by a small degree from a minimum-energy state, and are well-understood mathematically. m, almeida r, deco g, stetter m (2004) cooperation and biased competition model can explain attentional filtering in the prefrontal cortex. above equations contain a number of parameters encoding physiology and anatomy (e. stimuli were again 1 s long, separated by 1-s–long pauses, and now repeated 60 times. in the simplest case, we can use a single mode that can be regarded as encoding the location of a probability mass, hence neural mass models. kuramoto model of coupled phase oscillators[34] is one of the most abstract and fundamental model used to investigate neural oscillations and synchronization. in what follows, we consider stationary solutions for ensemble dynamics. the element taj is the response of qa to a change in nj at the same frequency and wave vector. we have tried to show the conceptual and mathematical links among the ensuing levels of description and how these models can be used to characterize key dynamical mechanisms in the brain. m, stam c (2005) dynamics of a neural system with a multiscale architecture. each level of description relates uniquely to neuroscience data, from single-unit recordings, through local field potentials to functional magnetic resonance imaging (fmri), electroencephalogram (eeg), and magnetoencephalogram (meg). "synchronization of cortical activity and its putative role in information processing and learning". our results are therefore direct predictions from the stimulus and do not rely on an observed poststimulus time histogram (psth). the dispersion relation of linear waves in the system is given by(69)and the system is stable at a particular real k if all the frequency roots of this equation have negative imaginary parts.., the inhibitory mean activity) to the dynamics enables the expression of chaotic dynamics [52],[112]., motivation, and emotion systems associated with early-stage intense romantic love. in summary, we can formulate the ensemble dynamics of any neuronal system, given its equations of motion, using the equation above. the corresponding second order equations of motion are a neural wave equation (see [48],[49] and below)(40)where γ = c/r and ▽2 is the laplacian.–6 khz), and 2 type iv receptors (cf ≅ 12–20 khz), identified by cf and best-response threshold (römer 1976). we assume that there exists only a single corticocortical fiber with terminals at locations x1 and x2, that is n = 2. the simplest such formulation is [95](52)where x is the evolving parameter, y is the quantity that drives the evolution, x(0) and y(0) are steady state values, and x(1) is a constant that describes the strength of feedback.[82][83] tight coordination of single-neuron spikes with local theta oscillations is linked to successful memory formation in humans, as more stereotyped spiking predicts better memory. one approach is to construct a multiscale hierarchy, with self-consistent evolution equations at each scale and to couple the emergent dynamics from fine scales into the activity at coarser scales [114]. the experimental protocol complied with german law governing animal care. stability of equation 86 is obtained if re[λ]<0 and is lost, according to [105], at re[λ] = 0, that is λ = iω. did not observe differences for the parameters of the recovery function for the different receptor cell types, although we cannot completely rule out the possibility that type iii and type iv cells make an exception because too few cells of these types were recorded. frequency changes are also commonly observed in central pattern generators and directly relate to the speed of motor activities, such as step frequency in walking. at the level of neural ensembles, synchronized activity of large numbers of neurons can give rise to macroscopic oscillations, which can be observed in an electroencephalogram. one group of 8 cells was stimulated with at least 4 different sound frequencies (between 3 and 11 khz for low-frequency receptors; between 12 and 25 khz for high-frequency receptors). for example, retinal waves are thought to have properties that define early connectivity of circuits and synapses between cells in the retina. excitatory connectivity is plotted in (a); purely inhibitory in (b); center-on, surround-off in (c); and center-off, surround-on in (d). this derivation is a little dense but illustrates the approximating assumptions and level of detail that can be captured by density dynamics.[7] consequently, neural oscillations have been linked to cognitive states, such as awareness and consciousness. in this section, we overview this modal approach to ensemble dynamics, initially in the general setting and then in the specific case, where the dynamics can be captured by the activity of a single node. g, rolls et, horwitz b (2004) ‘what’ and ‘where’ in visual working memory: a computational neurodynamical perspective for integrating fmri and single-neuron data. the general model accurately accounts for spike-train variability in response to a variety of both constant and dynamic stimuli.[49][50] this model implies that slow event-related responses, such as asymmetric alpha activity, could result from asymmetric brain oscillation amplitude modulations, such as an asymmetry of the intracellular currents that propagate forward and backward down the dendrites. central pattern generators are neuronal circuits that—when activated—can produce rhythmic motor patterns in the absence of sensory or descending inputs that carry specific timing information. the first set of experiments, stimuli were pure tones of constant intensity and 1-s duration, followed by a 1-s–long quiet pause. f(μ) captures the local dynamics of the neural field, and tc = t−|x−x′|/c is the time delay due to signal propagation. the central theme of this review is that the activity in populations of neurons can be understood by reducing the degrees of freedom from many to few, hence resolving an otherwise intractable computational problem. field modelsthe density dynamics and neural mass models above covered state the attributes of point processes, such as eeg sources, neurons, or neuronal compartments. these models have a long history spanning a half-century (e. e, lopes da silva fh (1999) electroencephalography: basic principles, clinical applications, and related fields. click the target next to the incorrect subject area and let us know. these results demonstrate that key ingredients of the stochastic response dynamics of a sensory neuron are faithfully captured by a simple stochastic model framework.
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    • Frontiers | Using a hybrid neuron in physiologically inspired models

      values for the seed neuron used in figure 7 are plotted in red. if we neglect the local dynamics f(μ), , and use an exponential kernel as in equation 39, we recover equations 39 and 40. to investigate these aspects, it is essential to advance stochastic descriptions of neural response dynamics that are valid over a broad spectrum of conditions including different neural activation strengths and different types of temporal stimulus modulation. one can then perform full nonstationary simulations using these parameters in the full if scheme to generate true dynamics. the dynamics are thus of the form(89)(90)where the function g represents the coupling between mean firing rates and induced synaptic currents. in other words, states like the depolarisation of an excitatory ensemble in the granular layer of cortex can be regarded as a continuum or field, which is a function of space, x, and time, μ(t)→μ(x,t). this means that the spatiotemporal spike patterns produced by neural circuits convey information among neurons; this is the microscopic level on which the brain's representations and computations rest [6].[15] oscillatory activity in single neurons can also be observed in sub-threshold fluctuations in membrane potential. in particular, [119]–[124] have studied the neural mechanisms underlying perceptual comparison by measuring single-neuron responses in monkeys trained to compare two mechanical vibrations applied sequentially to the tip of a finger; the subjects have to report which of the two stimuli has the higher frequency. 6 shows the results of simulating an ensemble of 250 neurons with a sensory input to all neurons between t = 1,000 ms to t = 3,000 ms. in general, these can vary in space, due to differences among brain regions, and in time, due to effects like habituation, facilitation, and adaptation., addiction, and emotion regulation systems associated with rejection in love.[39] the temporal evolution of resting state networks is correlated with fluctuations of oscillatory eeg activity in different frequency bands. whilst additional dimensions could be added to the microscopic neurons, this would add to an already significant computational burden. neural oscillations could create periodic time windows in which input spikes have larger effect on neurons, thereby providing a mechanism for decoding temporal codes. g, cowan j (1979) a mathematical theory of visual hallucination patterns. n, agam o, bialekw, and de ruyter van steveninck r. an important extension of these models speaks to the fact that neuronal dynamics play out on a spatially extended cortical sheet. mean-field and related formulations of dynamics also play an essential and complementary role as forward models that can be inverted given empirical data. hence the neurons show an evolution towards phase locking, which was not present prior to the stimulus. oscillations are observed throughout the central nervous system at all levels, and include spike trains, local field potentials and large-scale oscillations which can be measured by electroencephalography (eeg). this allows one to formulate the dynamics of the expected field in terms of partial differential equations in space and time. as discussed in the mean-field model section, it is possible to study a reduced model representing only the mean ensemble dynamics. inhibitory neurons, μi, respond passively to input from excitatory neurons and feedback to induce additional outward (rectifying) currents in excitatory cells.[51] under this assumption, asymmetries in the dendritic current would cause asymmetries in oscillatory activity measured by eeg and meg, since dendritic currents in pyramidal cells are generally thought to generate eeg and meg signals that can be measured at the scalp.[26] long-range connections between different brain structures, such as the thalamus and the cortex (see thalamocortical oscillation), involve time-delays due to the finite conduction velocity of axons. it is the most common of all involuntary movements and can affect the hands, arms, eyes, face, head, vocal cords, trunk, and legs. the average firing rate of the population f1f2 (f2 = f1−8 hz). are several interesting aspects to these modeling initiatives, which generalize the variants discussed in earlier sections: (i) synaptic and dendritic dynamics and summation of synaptic inputs to determine potentials at the cell body (soma), (ii) generation of pulses at the axonal hillock, and (iii) propagation of pulses within and between neural populations. in brief, time-dependent effects can be included in neural field models by adding dynamical equations for the evolution of the parameters. intracortical fibers are mostly unmyelinated and extend laterally up to 1 cm (in the human brain) with excitatory and inhibitory connections. in general, neurons with the same state v(t) at a given time t have a different history because of random fluctuations in the input current i(t). these are referred to as neural field models and are discussed in the following sections. figure 5 shows a stochastically driven neuron (a) compared to a noise-modulated periodic neuron (b). measuring adapted f–i curves: in this example, the receptor neuron is first adapted to a background intensity i0 of 57 db spl. resetting occurs when input to a neuron or neuronal ensemble resets the phase of ongoing oscillations. this can be addressed with neural field models; involving differential operators with both temporal and spatial terms. as a first step, we can split the connectivity function, w, into two parts, the homogeneous connectivity, whom(|x−y|), which depends only on the distance, and the heterogeneous connectivity, whet(x,y), which captures the effects of the extrinsic fiber system (for an alternative approach with applications to visual gamma phenomena, see [100]–[102]). amari also identified criteria to determine if only one bump, multiple bumps, or periodic solutions exist and if they are stable. models of spike generation based on recovery functions have been successfully applied to sensory neurons (e. linear predictions from neural field models have accounted successfully for a range of experimental phenomena, as mentioned above. first term represents recurrent feedback from neurons within the ensemble due to their own firing. although we consider neural field models last, they were among the first mean-field models of neuronal dynamics [43],[44]. a group of neurons engages in synchronized oscillatory activity, the neural ensemble can be mathematically represented as a single oscillator. in the microscopic system considered above, interneuron coupling was via a direct pulse during presynaptic depolarization. because the receptors are the first stage of the auditory system, the experimentally observed response variability is not inherited from other neurons, but must be caused by intrinsic processes. r, salinas e (2001) touch and go: decision-making mechanisms in somatosensation. in the present simulation, the 3 hz seizure has inherently aperiodic dynamics, as shown in the right panel of figure 20. what follows, we derive the fokker-planck equation for neuronal dynamics that are specified in terms of spiking neurons. one can then perform full nonstationary simulations using these parameters in the full if scheme to generate true dynamics. however, these simulations are computationally expensive and their results probabilistic, which makes them unsuitable for systematic explorations of parameter space. important phenomena have been studied by linearizing these models around their steady state solutions. f, bellanger j, bartolomei f, chauvel p (2000) relevance of nonlinear lumped parameter models in the analysis of depth- eeg epileptic signals. in short, we replace the full ensemble density with a mass at a particular point and then summarize the density dynamics by the location of that mass. "is gamma-band activity in the local field potential of v1 cortex a "clock" or filtered noise? by recurrently biasing each others' competitive internal dynamics, the neocortical system arrives at a global representation in which each area's state is maximally consistent with those of the other areas.[4] neuronal oscillations became a hot topic in neuroscience in the 1990s when the studies of the visual system of the brain by gray, singer and others appeared to support the neural binding hypothesis. Its structural architecture has been studied for more than a hundred years; however, its dynamics have been addressed much less thoroughly. because such long-time effects could lead to systematic errors in the analysis of short-time spike-count fluctuations, this nonstationarity was compensated as follows: the stimulus intensity used in the model was reduced by a time-dependent term such that average model firing rates in the time windows from 300 to 400 and 800 to 900 ms matched those of the recording. for instance, the amplitude and phase of alpha activity at the moment of visual stimulation predicts whether a weak stimulus will be perceived by the subject. altogether, this might yield a biophysically motivated and simple, yet highly accurate description of stimulus encoding for this insect auditory model system. symposium on robotics and cybernetics: computational engineering in systems applications. for example, an extended mean-field framework, which is consistent with the if and realistic synaptic equations that considers both the fast and slow glutamatergic excitatory synaptic dynamics (ampa and nmda) and the dynamics of gaba inhibitory synapses, can be found in [10]. summary of the notation for all the main dynamical variables and physiological parameters is given in table 1. to test which parameters actually govern the isi variability, as measured by the coefficient of variation (cv), we stimulated the receptors with pure tones of at least 4 different frequencies and various intensity levels. Mean-field and related formulations of dynamics also play an essential and complementary role as forward models that can be inverted given empirical data. the summation is assumed to be linear, and all potentials are measured relative to the resting potential [95]. to study brain function in humans, techniques allowing the indirect study of neuronal activity have been developed. functions of neural oscillations are wide ranging and vary for different types of oscillatory activity. neurodynamical model for a probabilistic decision-making network that performs the comparison of two mechanical vibrations applied sequentially (f1 and f2). in addition to this competition-centered view, a cooperation-centered picture of brain dynamics, where global representations find their neural correlate in assemblies of coactivated neurons, has been formulated [21],[23]. this class of models are also sometimes referred to as continuous attractor neural networks (cann). e, lopes da silva fh (1999) electroencephalography: basic principles, clinical applications, and related fields. these models were generalized by [48],[49] who, critically, considered delays in the propagation of spikes over space. and clinical applicationsin this section, we present three distinct applications of neural ensemble modeling. note that, in comparision to the use of the terms μν and ς(μν) in equation 40, the present wave-equation is formalized in relationship to population-specific pulse densities, φab, and firing rates, qa,b. that the brain contains multiple populations of neurons, indexed by the subscript a, which labels simultaneously the structure in which a given population lies (e. an alternative approach is to recursively enslave micro- and mesoscopic activity to predicted macroscopic field oscillations by driving them with the predicted mean-field synaptic currents. this implies that neurons are spontaneously at rest (quiescent) but depolarize with a small perturbation. 1997; reinagel and reid 2002), resulting in spike-count variances far below the mean spike count (de ruyter van steveninck et al. "resting gaba concentration predicts peak gamma frequency and fmri amplitude in response to visual stimulation in humans". us pause for a moment and reflect upon the significance of equation 85. by comparing the response to that of a mesoscopic neural mass model, we show what is gained and what is lost by abstracting to a more tractable set of evolution equations. cv values near unity occur for low firing rates (<50 hz) and approach values near 0. the envelope of the stimulus was gaussian white noise with a cutoff frequency of 400 hz and sd of 3 or 5 db. the power ratio and the interval map: spiking models and extracellular recordings. "oscillatory activity in sensorimotor cortex of awake monkeys: synchronization of local field potentials and relation to behavior". in general, eeg signals have a broad spectral content similar to pink noise, but also reveal oscillatory activity in specific frequency bands. note that both model and measurement variability are anticorrelated with the firing rate (fig. we argue that elaborating principled and informed models is a prerequisite for grounding empirical neuroscience in a cogent theoretical framework, commensurate with the achievements in the physical sciences. specifically, in the absence of a sensory input, neurons fire sporadically due to background stochastic inputs. in summary, we can formulate the ensemble dynamics of any neuronal system, given its equations of motion, using the equation above. one of the key outstanding problems is to reconcile the apparent discrepancy between proposals involving a key role of nonlinear dynamics (see also [118]) and the apparent success of mean-field models to predict measured evoked responses, without recourse to nonlinear dynamics. has recently been proposed that even if phases are not aligned across trials, induced activity may still cause event-related potentials because ongoing brain oscillations may not be symmetric and thus amplitude modulations may result in a baseline shift that does not average out. the equations, their derivation, and relevant references are provided in the recent developments in neural field models section. the oscillatory dynamics of neuronal spiking as identified in the hodgkin–huxley model closely agree with empirical findings. click the target next to the incorrect subject area and let us know. hence the flow terms in the neural mass model contribute to the expression of aperiodic dynamics in addition to the stochastic inputs. are several interesting aspects to these modeling initiatives, which generalize the variants discussed in earlier sections: (i) synaptic and dendritic dynamics and summation of synaptic inputs to determine potentials at the cell body (soma), (ii) generation of pulses at the axonal hillock, and (iii) propagation of pulses within and between neural populations. in other words, states like the depolarisation of an excitatory ensemble in the granular layer of cortex can be regarded as a continuum or field, which is a function of space, x, and time, μ(t)→μ(x,t). g, rolls et (2005) neurodynamics of biased competition and cooperation for attention: a model with spiking neurons. r, hernandez a, zainos a, lemus l, brody c (2002) neural correlates of decision making in secondary somatosensory cortex.., the inhibitory mean activity) to the dynamics enables the expression of chaotic dynamics [52],[112]. these models are particularly attractive because the density dynamics conform to a simple equation: the fokker-planck equation(1)this equation comprises a flow and a dispersion term; these terms embed the assumptions about the dynamics (phase flow, f(ν,t)) and random fluctuations (dispersion, d(ν,t)) that constitute our model at the neuronal level. also, and (but for simplicity, we drop the tilde in from now on). the impulses or spikes, called action potentials, are characterized by a certain amplitude and duration and are the units of information transmission at the interneuronal level. a, graham l (2007) population model of hippocampal pyramidal neurons, linking a refractory density approach to conductance-based neurons. we then illustrate examples of spatiotemporal dynamics occurring in corticothalamic loops during absence seizures. these include features such as separate excitatory and inhibitory neural populations (pyramidal cells and interneurons), nonlinear neural responses, synaptic, dendritic, cell-body, and axonal dynamics, and corticothalamic feedback [38], [43], [44], [48], [50], [83]–[87]. whilst additional dimensions could be added to the microscopic neurons, this would add to an already significant computational burden. in the section entitled cognitive and clinical applications, we present one of these examples, in the context of decision-making. in a typical streaming experiment, two sequences are created using sets of high and low tones. the most prevalent models of neuronal populations or ensembles are based upon something called the mean-field approximation. noise-driven harmonic oscillators realistically simulate alpha rhythm in the waking eeg as well as slow waves and spindles in the sleep eeg. induced activity generally reflects the activity of numerous neurons: amplitude changes in oscillatory activity are thought to arise from the synchronization of neural activity, for instance by synchronization of spike timing or membrane potential fluctuations of individual neurons. macroscopic models can inform us about whole brain dynamics and interactions between large-scale neural systems such as cortical regions, the thalamus, and brain stem. mean-field models are suited to data which reflect the behavior of a population of neurons, such as the electroencephalogram (eeg), magnetoencephalogram (meg), and fmri. each level of description relates uniquely to neuroscience data, from single-unit recordings, through local field potentials to functional magnetic resonance imaging (fmri), electroencephalogram (eeg), and magnetoencephalogram (meg). any single neuron could have a number of attributes; for example, post-synaptic membrane depolarization, v, capacitive current, i, or the time since the last action potential, t. the particular case that the drift is linear and the diffusion coefficient, σ2(t), is given by a constant, the fokker-planck equation describes a well-known stochastic process called the ornstein-uhlenbeck process [19]. these experiments exploit the fact that (i) cortical neurons have long apical dendrites and are easily polarized by an electric field, and (ii) that epileptiform bursts can be initiated by stimulation. these models were generalized by [48],[49] who, critically, considered delays in the propagation of spikes over space. cvs or fano factors were calculated for each synthetic data set, and the sds of these reevaluated quantities were taken as an estimate of the error of the original cv/fano factor. at the mesoscopic scale, neuronal coupling is via neural firing pulse densities, ςa, which capture the expected neuronal firing rate, given the mean neuronal transmembrane potential ς(μa) for a = e,i. w (1979) nonlinear dynamics of paleocortex manifested in the olfactory eeg. reliability of a fly motion-sensitive neuron depends on stimulus parameters. differences in the encoding quality may therefore simply arise from the specific usage of the neuron's dynamic range and encoding capacity by the particular stimulus. responses to time-varying stimuli: experimental and model results for a sample cell. on the other hand, it has been shown that even single auditory receptor neurons contain sufficient information to discriminate conspecific communication signals, with single spikes carrying significant amounts of information at high temporal resolution (machens et al. a single fiber connects the two distant regimes (a) and (b) and contributes to the heterogeneous connectivity, whet, whereas regime (c) has only homogeneous connections. to avoid artifacts arising from finite sampling of the isi distributions, recovery functions were parameterized using a class of standard sigmoid functions (see eq. the latter will be captured by specific choices of the local dynamics, f(μ), in equation 41; for instance, [60] choose a cubic-shaped function of the firing rate, which, under appropriate parameters, allows for intrinsic bistability. models of the cortex can establish which types of large-scale neuronal networks can perform computations and characterize their emergent properties. the solution of equation 28 satisfying the boundary conditions (equations 24–27) is:(29)taking into account the fraction of neurons, qtref, in the refractory period and the normalization of the mass probability,(30)finally, substituting equation 29 into equation 30, and solving for q, we obtain the population transfer function, φ, of ricciardi [13]:(31)where . evoked potentials and event-related potentials are obtained from an electroencephalogram by stimulus-locked averaging, i. d, amitai y (1997) propagating neuronal discharges in neocortical slices: computational and experimental study. black indicates f1f2 (f2 = f1−8 hz). the kuramoto model is widely used to study oscillatory brain activity and several extensions have been proposed that increase its neurobiological plausibility, for instance by incorporating topological properties of local cortical connectivity.., from brain stem inputs) and are modeled as a constant flow with a superimposed poisson train of discrete pulses. we then illustrate healthy and pathological activity in neural field models.
    • Temporal pairwise spike correlations fully capture single-neuron

      in this approach, individual if neurons are grouped together into populations of statistically similar neurons. other models relevant for systems neuroscience can be found in [4],[8],[9]. we can derive directly the predicted ensemble mean response by simply summing over all neurons. instead, the probability of firing is rhythmically modulated such that neurons are more likely to fire at the same time, which gives rise to oscillations in their mean activity (see figure at top of page). the excitatory recurrent postsynaptic currents (epscs) are mediated by ampa (fast) and nmda-glutamate (slow) receptors, whereas external epscs imposed on the network are driven by ampa receptors only. mr, jirsa vk (2007) neural field dynamics with heterogeneous connection topology. models of the cortex can establish which types of large-scale neuronal networks can perform computations and characterize their emergent properties. intrinsic or intracortical fibers are confined to cortical gray matter in which the cortical neurons reside; these intrinsic connections define the local connectivity within an area. and 1 ms, the spike timing of model spike trains is greater than that of the experimental spike trains. a positive (negative) electric field applied across the slice increased (decreased) the speed of wave propagation, consistent with the theoretical predictions of neural field theory, assuming that a positive (negative) electric field reduces (increases) the threshold, h, in equation 42. the intention is to demonstrate concrete examples of ensemble dynamics under varying influences of flow and dispersion. s (1977) dynamics of pattern formation in lateral-inhibition type neural fields. we can then rewrite the neural field equation as follows:(82)where tc = t−|x−y| /c and c is the propagation speed through the heterogeneous corticocortical or extrinsic connections. in individual neurons, oscillations can appear either as oscillations in membrane potential or as rhythmic patterns of action potentials, which then produce oscillatory activation of post-synaptic neurons. firing of neurons also forms the basis of periodic motor commands for rhythmic movements. the corresponding second order equations of motion are a neural wave equation (see [48],[49] and below)(40)where γ = c/r and ▽2 is the laplacian. p, rennie c, rowe d (2002) dynamics of large-scale brain activity in normal arousal states and epileptic seizures. Modeling at the single neuron level is necessary because this is the level at which information is exchanged between the computing elements of the brain; the neurons. neural tissue can generate oscillatory activity in many ways, driven either by mechanisms within individual neurons or by interactions between neurons. any single neuron could have a number of attributes; for example, post-synaptic membrane depolarization, v, capacitive current, i, or the time since the last action potential, t. nevertheless, the integrate-and-fire (if) model is not only defined by the subthreshold dynamics but includes a reset after each spike generation, which makes the whole dynamics highly nonlinear. let us assume that n neurons synapse onto cell i and that jij is the efficacy of synapse j, then the total synaptic afferent current is given by(5)where is the emission time of the kth spike from the jth presynaptic neuron. in fact, we can write the density dynamics in terms of a linear operator or jacobian q(3). the network is partitioned into populations of neurons whose input currents share the same statistical properties and fire spikes independently at the same rate. figure 5 shows a stochastically driven neuron (a) compared to a noise-modulated periodic neuron (b). some types of neurons have the tendency to fire at particular frequencies, so-called resonators. ck, stemmler mb, prinz p, krahe r, ronacher b, and herz avm. neurons can generate rhythmic patterns of action potentials or spikes. its function is the classification of the peripheral spatiotemporal neural field dynamics. the impulses or spikes, called action potentials, are characterized by a certain amplitude and duration and are the units of information transmission at the interneuronal level. many models are used in the field, each defined at a different level of abstraction and trying to model different aspects of neural systems. this inversion allows one to select the best model (given some data) and make probabilistic comments about the parameters of that model. "breaking the silence: brain-computer interfaces (bci) for communication and motor control". in our case, the derived transfer function, φ, corresponds consistently to the assumptions of the simple lif model described in the from spiking-neurons to mean-field models section. the high dimensionality and complexity of these fokker-planck formalisms can be finessed with a mean-field approximation to give nonlinear fokker-planck equations, describing the evolution of separable ensembles that are coupled by mean-field effects. the model captures the spike-count variability and salient features of the fine temporal structure in response to dynamic stimuli, although the recovery functions were always calculated from isi distributions obtained under constant stimulation. during behavioral tasks, this persistent elevated neuronal firing can last for tens of seconds after the stimulus is no longer present. stationary dynamics of each population can be described by the population transfer function, which provides the average population rate as a function of the average input current. Its structural architecture has been studied for more than a hundred years; however, its dynamics have been addressed much less thoroughly. a second set of experiments, stimuli were pure tones at the cell's characteristic frequency whose amplitudes were modulated by gaussian white noise with a cutoff frequency of 400 hz. way to investigate the biological basis of information processing in the brain is to study the response of neurons to stimulation. they are assumed to originate from the somatosensory area s2 and from the pfc, encoding the frequency of both stimuli f1 (stored) and f2 (present) to be compared during the comparison period, i. although individual neurons exhibit nonlinear dynamics, the ensemble mean dynamics are (linearly) stable to the stochastic inputs until the background current is increased. models based on these principles have been used to provide mathematical descriptions of neural oscillations and eeg rhythms. often, a neuron's firing rate depends on the summed activity it receives. to study brain function in humans, techniques allowing the indirect study of neuronal activity have been developed. the bifurcation structure of traveling waves in neural fields can be analysed using a so-called evans function and has recently been explored in great detail [74]. above equations contain a number of parameters encoding physiology and anatomy (e. if we indicate the firing rate qa for the cell type a by a subscript, then φab can be expressed in terms of the firing rate at other locations and earlier times. seed neuron is chosen at random and the interneuron spike difference for all other neurons is plotted each time it spikes. the cv values from the measurements and the model match closely over the whole range of values. every treatment of the interplay of anatomical connectivity (local and global connections) and functional connectivity (network dynamics) will have to be represented in the form of equation 85 or a variation thereof. the subthreshold dynamical equation 4, given the input current (from equation 5), can be integrated, and yields(6,7)if the neuron i is initially (t = 0) at the resting potential (vi(0) = vl). in this section, we try to clarify some key concepts and show how they relate to each other. oscillations are also thought be involved in the sense of time[66] and in somatosensory perception. neuron is modeled as a planar reduction of the hodgkin-huxley model [109],[110], namely,(87)where fion introduces conductance-determined transmembrane currents through voltage-dependent channels, ion = {na+,k+} and i are synaptic currents. neurons are the cells responsible for encoding, transmitting, and integrating signals originating inside or outside the nervous system. it then rises monotonically to account for the relative refractory period during which the neuron relaxes back to its normal level of excitability. comparison of the model performance with 2 widely used alternative models. this allows one to formulate the dynamics of the expected field in terms of partial differential equations in space and time. oscillations are commonly studied from a mathematical framework and belong to the field of "neurodynamics", an area of research in the cognitive sciences that places a strong focus upon the dynamic character of neural activity in describing brain function. there are two subtypes of excitatory population: namely, specific and nonselective. counting windows are 10 ms (a), 25 ms (b), 50 ms (c), and 100 ms (d). they range from models of the short-term behaviour of individual neurons, through models of how the dynamics of neural circuitry arise from interactions between individual neurons, to models of how behaviour can arise from abstract neural modules that represent complete subsystems. each attribute induces a dimension in the phase space of a neuron; in our example the phase space would be three dimensional and the state of each neuron would correspond to a point ν = {v,i,t} ∈ℜ3 or particle in phase space. this is determined by the proportion of active neurons by counting the number of spikes nspikes(t,t+dt) in a small time interval dt and dividing by n and by dt [18]; i. this means that questions about neuronal function are generally addressed by inference on models or their parameters, where the model links neuronal processes that are hidden from our direct observation. if whet describes the connectivity of n areas, then it can always be written as a sum of two-point connections via(83)where νij ∈ ℜ again represents the coupling strength between areas at xi and xj. a commonly used approximation is(48)where θa is the firing threshold for channels of type a and is the standard deviation of the threshold over the population. qj is written as nj to make the distinction between population firing rates and incoming stimulus rates absolutely clear.: deco g, jirsa vk, robinson pa, breakspear m, friston k (2008) the dynamic brain: from spiking neurons to neural masses and cortical fields. hr, cowan jd (1972) excitatory and inhibitory interactions in localized populations of model neurons. minimal stable regions for the equilibrium state of a neural field as a function of its connectivity and time delay τ = d/c. stationary dynamics of each population can be described by the population transfer function, which provides the average population rate as a function of the average input current. at the mesoscopic scale, neuronal coupling is via neural firing pulse densities, ςa, which capture the expected neuronal firing rate, given the mean neuronal transmembrane potential ς(μa) for a = e,i. the mean synaptic currents evidence an emergent phenomenon, and not merely the superposition of a bursting neuron, as can be seen in (c): clearly no burst is evident at this scale. pa, rennie ca, wright jj (1997) propagation and stability of waves of electrical activity in the cerebral cortex. during behavioral tasks, this persistent elevated neuronal firing can last for tens of seconds after the stimulus is no longer present. on the other hand, the corticocortical (extrinsic) fiber system contains fibers which leave the gray matter and connect distant areas (up to 20 cm [84]). curves are similar, but slopes, curvatures, and offsets may vary considerably between individual cells. that is, neuronal activity depends on its current state as well as spatial gradients, which allow its spread horizontally across the cortical surface. 14 shows numerical simulations corresponding to the response of vpc neurons during the comparison period (to be contrasted with the experimental results shown in figure 2 of [121]). for example, an extended mean-field framework, which is consistent with the if and realistic synaptic equations that considers both the fast and slow glutamatergic excitatory synaptic dynamics (ampa and nmda) and the dynamics of gaba inhibitory synapses, can be found in [10]. g, pollatos o, zihl j (2002) the time course of selective visual attention: theory and experiments.[13] it considers the brain a dynamical system and uses differential equations to describe how neural activity evolves over time. plos comput biol 4(8):Introductionthe brain appears to adhere to two fundamental principles of functional organization, functional integration and functional specialization, where the integration within and among specialized areas is mediated by connections among them. at macroscopic scales, connectivity is sparser, can be considered exclusively horizontal, and is predominantly excitatory in nature. Macroscopic models can inform us about whole brain dynamics and interactions between large-scale neural systems such as cortical regions, the thalamus, and brain stem.., stryer 2002) (10) here τa denotes the absolute refractory period, γ determines the maximal curvature of the recovery function, and τr is a measure of the duration of the relative refractory period in that w(δ) has reached 50% of its maximum value at time δ = τa + τr. "who reads temporal information contained across synchronized and oscillatory spike trains? a renewal process, it is possible to compute the recovery function directly from the isi distribution pisi(δ) obtained under constant stimulation [q(t) = q], as has been discussed in the literature (berry and meister 1998; gerstner and kistler 2002; johnson 1996) (8) equation 8 can be inverted to calculate the isi distribution from the recovery function (9) recovery functions determined from experimental data according to eq. the variable discharge of cortical neurons: implications for connectivity, computation, and information coding. further extension for more complex and realistic models are possible. c, robinson p, wright j (2002) unified neurophysical model of eeg spectra and evoked potentials. e (2007) dynamical systems in neuroscience: the geometry of excitability and bursting. m, terry j, friston k (2003) modulation of excitatory synaptic coupling facilitates synchronization and complex dynamics in a biophysical model of neuronal dynamics. naturally, it is useful to understand how neuronal activity unfolds on the spatially continuous cortical sheet. (2006) bold responses to stimuli: dependence on frequency, stimulus form, amplitude, and repetition rate. summary, for any model of neuronal dynamics, specified as a stochastic differential equation, there is a deterministic linear equation that can be integrated to generate ensemble dynamics. when the voltage across the capacitor reaches a threshold θ, the circuit is shunted (reset) and a δ pulse (spike) is generated and transmitted to other neurons. c: raster plot of experimental spikes in response to 30 stimulus presentations. the most prevalent models of neuronal populations or ensembles are based upon something called the mean-field approximation. phase resetting is fundamental for the synchronization of different neurons or different brain regions[9][26] because the timing of spikes can become phase locked to the activity of other neurons. illustrate the effects of interplay between anatomical and functional connectivity, we discuss a simple example following [103],[104]. this hence fails to capture some of the cardinal properties of the microscopic ensemble, namely the coupling between the first and second moments (mean and variance). of neuronal firing may serve as a means to group spatially segregated neurons that respond to the same stimulus in order to bind these responses for further joint processing, i. in fourier space, this gives(50,51)where k = (kx,ky) is the wave vector and ω is the angular frequency. during the experiments, preparations were kept at a fixed temperature of about 30°c. functional role of synchronized oscillatory activity in the brain was mainly established in experiments performed on awake kittens with multiple electrodes implanted in the visual cortex. the variability of interspike intervals (isis) was analyzed using the coefficient of variation (cv), which is defined as the ratio between the standard deviation σisi of the isi distribution and its average 〈isi〉 (1) to quantify the spike-count variability, fano factors (fano 1947) were calculated. illustrate ensemble dynamics from first principles, we directly simulate a network of coupled neurons which obey deterministic evolution rules and receive both stochastic and deterministic inputs. we can then rewrite the neural field equation as follows:(82)where tc = t−|x−y| /c and c is the propagation speed through the heterogeneous corticocortical or extrinsic connections. ck, schütze h, franz a, kolesnikova o, stemmler mb, ronacher b, and herz avm. activity is brain activity in the absence of an explicit task, such as sensory input or motor output, and hence also referred to as resting-state activity. the neurons are fully connected (with synaptic strengths as specified in the text).: we only request your email address so that the person you are recommending the page to knows that you wanted them to see it, and that it is not junk mail. however, these simulations are computationally expensive and their results probabilistic, which makes them unsuitable for systematic explorations of parameter space. under instantaneous interactions, c→∞, single population models with locally excitatory and laterally inhibitory connectivity can support global periodic stationary patterns in one dimension as well as single or multiple localized solutions (bumps and multi-bumps) [47]. under the diffusion approximation, equation 21 can also be interpreted (by means of the central limit theorem), as the case in which the sum of many poisson processes (equation 5) becomes a normal random variable with mean μ(t) and variance σ2. the alternative models were tuned to this cell using the same procedures as those for the full model. the scaled deviation was then averaged over time to yield the mean relative prediction error (3) to quantify spike timing reliability in experimental and model spike trains, we used a correlation-based measure introduced by schreiber et al. further analysis of the 3 hz (absence) bifurcation in a reduced model argues that interactions between the reticular and specific nuclei of the thalamus contribute importantly to the absence seizure waveform [135]. g, rolls et (2005) neurodynamics of biased competition and cooperation for attention: a model with spiking neurons. as discussed in the section entitled recent developments in neural field models, these incorporate synaptic filtering and axonal conduction delays, in addition to the population-wide conversion of membrane potentials into firing densities [50]. as a next step, we analyzed the variability in response to time-varying stimuli and tested whether the model framework also captures response variability for this larger class of more complex stimuli. these latter patterns take the form of stripes and checkerboard-like patterns, and have been linked to drug-induced visual hallucinations [72]. f, warland d, de ruyter van steveninck r, and bialek w. for simplicity, we consider the stationary solution μ0(x) = 0 to be the rest state and consider its deviations μ(x,t) = ξk′(t)φk′ (x)+c. stimulus generation, spike detection, and data analysis were performed using custom-made software. in addition to local synchronization, oscillatory activity of distant neural structures (single neurons or neural ensembles) can synchronize.[11] these oscillations are also observed in motor output of physiological tremor[79] and when performing slow finger movements. these sorts of models have been extremely useful in modeling spatiotemporally extended dynamics (e. frequency locking is not surprising in a model with spatial coupling, the amplitude modulation is a novel, emergent property of the nonlinear dynamics. this suggests that the different response classes of receptors are formed by neurons of a uniform electrophysiological type. we can now write equation 63 in matrix form as(66)where a is an n′×n′ matrix, q is an n′-element column vector, b is an n′×j′ matrix, and n is a j′-element column vector. parallel processing: coping with the variability of neuronal messages in directional hearing of insects. stimulus and recovery dependence of cat cochlear nerve fiber spike discharge probability. the neurons in the two specific populations additionally receive external inputs encoding stimulus specific information. models are essential for neuroscience, in the sense that the most interesting questions pertain to neuronal mechanisms and processes that are not directly observable. the neurons are fully connected (with synaptic strengths as specified in the text).[87] for example, a non-invasive bci interface can be created by placing electrodes on the scalp and then measuring the weak electric signals. most of these results are furnished by slice studies of pharmacologically treated tissue, taken from the cortex [75]–[77], hippocampus [78], and thalamus [79]. curves are displaced relative to each other and illustrate that there is no simple functional relation between the variability and stimulus frequency or intensity. the bifurcation structure of traveling waves in neural fields can be analysed using a so-called evans function and has recently been explored in great detail [74]. accounts of the variability observed in neural spike trains are a prerequisite for the proper interpretation of neural dynamics and coding principles. in this paper, we review and integrate, in a unifying framework, a variety of computational approaches that have been used to characterize the dynamics of the cortex, as evidenced at different levels of measurement. the shortest isi encountered during the entire experiment was taken as the absolute refractory period (here: 1. in neural mass models, we ignore this possibility because we can only couple the expectations or first moments. in the microscopic system considered above, interneuron coupling was via a direct pulse during presynaptic depolarization. "functional role of gamma and theta oscillations in episodic memory". this can be done in experimental animals using implanted electrodes to record the rates and timing of action potentials. in this paper, we address how distributed and specialized neuronal responses are realized in terms of microscopic brain dynamics; we do this by showing how neuronal systems, with many degrees of freedom, can be reduced to lower dimensional systems that exhibit adaptive behaviors. Mesoscopic models tell us how neural elements interact to yield emergent behavior at the level of microcolumns and cortical columns. c, robinson p, wright j (2002) unified neurophysical model of eeg spectra and evoked potentials. theta rhythms are very strong in rodent hippocampi and entorhinal cortex during learning and memory retrieval, and they are believed to be vital to the induction of long-term potentiation, a potential cellular mechanism for learning and memory..,(39)where |x−x′| is the distance between the spatial locations x and x′, c is the characteristic speed of spike propagation, and γ reflects the spatial decay of lateral interactions.