Propagator theory of brain dynamics

Analyzing the brain's dynamic response to targeted stimulation using generative modeling

Generative models of brain activity have been instrumental in testing hypothesized mechanisms underlying brain dynamics against experimental datasets. Beyond capturing the key mechanisms underlying spontaneous brain dynamics, these models hold an exciting potential for understanding the mechanisms underlying the dynamics evoked by targeted brain stimulation techniques. This paper delves into this emerging application, using concepts from dynamical systems theory to argue that the stimulus-evoked dynamics in such experiments may be shaped by new types of mechanisms distinct from those that dominate spontaneous dynamics. We review and discuss (a) the targeted experimental techniques across spatial scales that can both perturb the brain to novel states and resolve its relaxation trajectory back to spontaneous dynamics and (b) how we can understand these dynamics in terms of mechanisms using physiological, phenomenological, and data-driven models. A tight integration of targeted stimulation experiments with generative quantitative modeling provides an important opportunity to uncover novel mechanisms of brain dynamics that are difficult to detect in spontaneous settings.

Unified theory of alpha, mu, and tau rhythms via eigenmodes of brain activity

A compact description of the frequency structure and topography of human alpha-band rhythms is obtained by use of the first four brain activity eigenmodes previously derived from corticothalamic neural field theory. Just two eigenmodes that overlap in frequency are found to reproduce the observed topography of the classical alpha rhythm for subjects with a single, occipitally concentrated alpha peak in their electroencephalograms. Alpha frequency splitting and relative amplitudes of double alpha peaks are explored analytically and numerically within this four-mode framework using eigenfunction expansion and perturbation methods. These effects are found to result primarily from the different eigenvalues and corticothalamic gains corresponding to the eigenmodes. Three modes with two non-overlapping frequencies suffice to reproduce the observed topography for subjects with a double alpha peak, where the appearance of a distinct second alpha peak requires an increase of the corticothalamic gain of higher eigenmodes relative to the first. Conversely, alpha blocking is inferred to be linked to a relatively small attention-dependent reduction of the gain of the relevant eigenmodes, whose effect is enhanced by the near-critical state of the brain and whose sign is consistent with inferences from neural field theory. The topographies and blocking of the mu and tau rhythms within the alpha-band are explained analogously via eigenmodes. Moreover, the observation of three rhythms in the alpha band is due to there being exactly three members of the first family of spatially nonuniform modes. These results thus provide a simple, unified description of alpha band rhythms and enable experimental observations of spectral structure and topography to be linked directly to theory and underlying physiology.

Method for quantifying arousal and consciousness in healthy states and severe brain injury via EEG-based measures of corticothalamic physiology

BACKGROUND

Characterization of normal arousal states has been achieved by fitting predictions of corticothalamic neural field theory (NFT) to electroencephalographic (EEG) spectra to yield relevant physiological parameters.

NEW METHOD

A prior fitting method is extended to distinguish conscious and unconscious states in healthy and brain injured subjects by identifying additional parameters and clusters in parameter space.

RESULTS

Fits of NFT predictions to EEG spectra are used to estimate neurophysiological parameters in healthy and brain injured subjects. Spectra are used from healthy subjects in wake and sleep and from patients with unresponsive wakefulness syndrome, in a minimally conscious state (MCS), and emerged from MCS. Subjects cluster into three groups in parameter space: conscious healthy (wake and REM), sleep, and brain injured. These are distinguished by the difference X-Y between corticocortical (X) and corticothalamic (Y) feedbacks, and by mean neural response rates α and β to incoming spikes. X-Y tracks consciousness in healthy individuals, with smaller values in wake/REM than sleep, but cannot distinguish between brain injuries. Parameters α and β differentiate deep sleep from wake/REM and brain injury.

COMPARISON WITH EXISTING METHODS

Other methods typically rely on laborious clinical assessment, manual EEG scoring, or evaluation of measures like Φ from integrated information theory, for which no efficient method exists. In contrast, the present method can be automated on a personal computer.

CONCLUSION

The method provides a means to quantify consciousness and arousal in healthy and brain injured subjects, but does not distinguish subtypes of brain injury.

Analytic Model for Feature Maps in the Primary Visual Cortex

A compact analytic model is proposed to describe the combined orientation preference (OP) and ocular dominance (OD) features of simple cells and their mutual constraints on the spatial layout of the combined OP-OD map in the primary visual cortex (V1). This model consists of three parts: (i) an anisotropic Laplacian (AL) operator that represents the local neural sensitivity to the orientation of visual inputs; and (ii) obtain a receptive field (RF) operator that models the anisotropic spatial projection from nearby neurons to a given V1 cell over scales of a few tenths of a millimeter and combines with the AL operator to give an overall OP operator; and (iii) a map that describes how the parameters of these operators vary approximately periodically across V1. The parameters of the proposed model maximize the neural response at a given OP with an OP tuning curve fitted to experimental results. It is found that the anisotropy of the AL operator does not significantly affect OP selectivity, which is dominated by the RF anisotropy, consistent with Hubel and Wiesel's original conclusions that orientation tuning width of V1 simple cell is inversely related to the elongation of its RF. A simplified and idealized OP-OD map is then constructed to describe the approximately periodic local OP-OD structure of V1 in a compact form. It is shown explicitly that the OP map can be approximated by retaining its dominant spatial Fourier coefficients, which are shown to suffice to reconstruct its basic spatial structure. Moreover, this representation is a suitable form to analyze observed OP maps compactly and to be used in neural field theory (NFT) for analyzing activity modulated by the OP-OD structure of V1. Application to independently simulated V1 OP structure shows that observed irregularities in the map correspond to a spread of dominant coefficients in a circle in Fourier space. In addition, there is a strong bias toward two perpendicular directions when only a small patch of local map is included. The bias is decreased as the amount of V1 included in the Fourier transform is increased.

Generalized neural field theory of cortical plasticity illustrated by an application to the linear phase of ocular dominance column formation in primary visual cortex

Physiologically based neural field theory (NFT) is extended to encompass cortical plasticity dynamics. An illustrative application is provided which treats the evolution of the connectivity of left- and right-eye visual stimuli to neuronal populations in the primary visual cortex (V1), and the initial, linear phase of formation of approximately one-dimensional (1D) ocular dominance columns (ODCs) that sets their transverse spatial scale. This links V1 activity, structure, and physiology within a single theory that already accounts for a range of other brain activity and connectivity phenomena, thereby enabling ODC formation and many other phenomena to be interrelated and cortical parameters to be constrained across multiple domains. The results accord with experimental ODC widths for realistic cortical parameters and are based directly on a unified description of the neuronal populations involved, their connection strengths, and the neuronal activity they support. Other key results include simple analytic approximations for ODC widths and the parameters of maximum growth rate, constraints on cortical excitatory and inhibitory gains, elucidation of the roles of specific poles of the V1 response function, and the fact that ODCs are not formed when input stimuli are fully correlated between eyes. This work provides a basis for further generalization of NFT to model other plasticity phenomena, thereby linking them to the range multiscale phenomena accounted for by NFT.

Modal-Polar Representation of Evoked Response Potentials in Multiple Arousal States

An expansion of the corticothalamic transfer function into eigenmodes and resonant poles is used to derive a simple formula for evoked response potentials (ERPs) in various states of arousal. The transfer function corresponds to the cortical response to an external stimulus, which encodes all the information and properties of the linear system. This approach links experimental observations of resonances and characteristic timescales in brain activity with physically based neural field theory (NFT). The present work greatly simplifies the formula of the analytical ERP, and separates its spatial part (eigenmodes) from the temporal part (poles). Within this framework, calculations involve contour integrations that yield an explicit expression for ERPs. The dominant global mode is considered explicitly in more detail to study how the ERP varies with time in this mode and to illustrate the method. For each arousal state in sleep and wake, the resonances of the system are determined and it is found that five poles are sufficient to study the main dynamics of the system in waking eyes-open and eyes-closed states. Similarly, it is shown that six poles suffice to reproduce ERPs in rapid-eye movement sleep, sleep state 1, and sleep state 2 states, whereas just four poles suffice to reproduce the dynamics in slow wave sleep. Thus, six poles are sufficient to preserve the main global ERP dynamics of the system for all states of arousal. These six poles correspond to the dominant resonances of the system at slow-wave, alpha, and beta frequencies. These results provide the basis for simplified analytic treatment of brain dynamics and link observations more closely to theory.

Gamma-band correlations in the primary visual cortex

This paper generalizes and extends previous work on using neural field theory to quantitatively analyze the two-dimensional (2D) spatiotemporal correlation properties of gamma-band (30-70 Hz) oscillations evoked by stimuli arriving at the primary visual cortex, and modulated by patchy connectivities that depend on orientation preference (OP). Correlation functions are derived analytically for general stimulus and measurement conditions. The theoretical results reproduce a range of published experimental results. These include (i) the existence of two-point oscillatory temporal cross correlations with zero time lag between neurons with similar OP; (ii) the influence of spatial separation of neurons on the strength of the correlations; and (iii) the effects of differing stimulus orientations. They go beyond prior work by incorporating experimentally observed patchy projection patterns to predict the 2D correlation structure including both OP and ocular dominance effects, thereby relaxing assumptions of translational invariance implicit in prior one-dimensional analysis.

Neural field theory of evoked response potentials in a spherical brain geometry

Evoked response potentials (ERPs) are calculated in spherical and planar geometries using neural field theory of the corticothalamic system. The ERP is modeled as an impulse response and the resulting modal effects of spherical corticothalamic dynamics are explored, showing that results for spherical and planar geometries converge in the limit of large brain size. Cortical modal effects can lead to a double-peak structure in the ERP time series. It is found that the main difference between infinite planar geometry and spherical geometry is that the ERP peak is sharper and stronger in the spherical geometry. It is also found that the magnitude of the response decreases with increasing spatial width of the stimulus at the cortex. The peak is slightly delayed at large angles from the stimulus point, corresponding to group velocities of 6-10 m s^{-1}. Strong modal effects are found in the spherical geometry, with the lowest few modes sufficing to describe the main features of ERPs, except very near to spatially narrow stimuli.

Physical brain connectomics

Brain connectivity and structure-function relationships are analyzed from a physical perspective in place of common graph-theoretic and statistical approaches that overwhelmingly ignore the brain's physical structure and geometry. Field theory is used to define connectivity tensors in terms of bare and dressed propagators, and discretized representations are implemented that respect the physical nature and dimensionality of the quantities involved, retain the correct continuum limit, and enable diagrammatic analysis. Eigenfunction analysis is used to simultaneously characterize and probe patterns of brain connectivity and activity, in place of statistical or phenomenological patterns. Physically based measures that characterize the connectivity are then developed in coordinate and spectral domains; some of which generalize or rectify graph-theoretic measures to implement correct dimensionality and continuum limits, and some replace graph-theoretic quantities. Traditional graph-based measures are shown to be highly prone to artifacts introduced by discretization and threshold, often because essential physical constraints have not been imposed, dimensionality has not been included, and/or distinctions between scalar, vector, and tensor quantities have not been considered. The results can replace them in ways that converge correctly and measure properties of brain structure, rather than of its discretization, and thus potentially enable physical interpretation of the many phenomenological results in the literature. Geometric effects are shown to dominate in determining many brain properties and care must be taken not to interpret geometric differences as differences in intrinsic neural connectivity. The results demonstrate the need to use systematic physical methods to analyze the brain and the potential of such methods to obtain new insights from data, make new predictions for experimental test, and go beyond phenomenological classification to dynamics and mechanisms.

Spectrum of connectivity fluctuations including the effect of activity-dependent feedback

The spatiotemporal spectrum of feedback-driven fluctuations of brain connectivity is investigated using nonlinear neural field theory of the corticothalamic system. Weakly nonlinear dynamics of neural feedbacks are expanded in terms of first order perturbations of neural activity relative to a fixed point. Susceptibilities are used to quantify the change in connectivity per unit change in presynaptic or postsynaptic activity caused by nonlinear feedbacks such as facilitation, depression, sensitization, potentiation, and the effects of discrete eigenmode structure are included for a spherical brain geometry. Spectral signatures such as resonances are identified that allow the presence of particular presynaptic and postsynaptic feedback effects to be inferred. These include additional resonances at high frequencies and shifts of existing spectral peaks, mostly visible in the lowest spatial modes of the response.

NFTsim: Theory and Simulation of Multiscale Neural Field Dynamics

A user ready, portable, documented software package, NFTsim, is presented to facilitate numerical simulations of a wide range of brain systems using continuum neural field modeling. NFTsim enables users to simulate key aspects of brain activity at multiple scales. At the microscopic scale, it incorporates characteristics of local interactions between cells, neurotransmitter effects, synaptodendritic delays and feedbacks. At the mesoscopic scale, it incorporates information about medium to large scale axonal ranges of fibers, which are essential to model dissipative wave transmission and to produce synchronous oscillations and associated cross-correlation patterns as observed in local field potential recordings of active tissue. At the scale of the whole brain, NFTsim allows for the inclusion of long range pathways, such as thalamocortical projections, when generating macroscopic activity fields. The multiscale nature of the neural activity produced by NFTsim has the potential to enable the modeling of resulting quantities measurable via various neuroimaging techniques. In this work, we give a comprehensive description of the design and implementation of the software. Due to its modularity and flexibility, NFTsim enables the systematic study of an unlimited number of neural systems with multiple neural populations under a unified framework and allows for direct comparison with analytic and experimental predictions. The code is written in C++ and bundled with Matlab routines for a rapid quantitative analysis and visualization of the outputs. The output of NFTsim is stored in plain text file enabling users to select from a broad range of tools for offline analysis. This software enables a wide and convenient use of powerful physiologically-based neural field approaches to brain modeling. NFTsim is distributed under the Apache 2.0 license.

Neural field theory of perceptual echo and implications for estimating brain connectivity

Neural field theory is used to predict and analyze the phenomenon of perceptual echo in which random input stimuli at one location are correlated with electroencephalographic responses at other locations. It is shown that this echo correlation (EC) yields an estimate of the transfer function from the stimulated point to other locations. Modal analysis then explains the observed spatiotemporal structure of visually driven EC and the dominance of the alpha frequency; two eigenmodes of similar amplitude dominate the response, leading to temporal beating and a line of low correlation that runs from the crown of the head toward the ears. These effects result from mode splitting and symmetry breaking caused by interhemispheric coupling and cortical folding. It is shown how eigenmodes obtained from functional magnetic resonance imaging experiments can be combined with temporal dynamics from EC or other evoked responses to estimate the spatiotemporal transfer function between any two points and hence their effective connectivity.

Theory of corticothalamic brain activity in a spherical geometry: Spectra, coherence, and correlation

Corticothalamic neural field theory is applied to a spherical geometry to better model neural activity in the human brain and is also compared with planar approximations. The frequency power spectrum, correlation, and coherence functions are computed analytically and numerically. The effects of cortical boundary conditions and resulting modal aspects of spherical corticothalamic dynamics are explored, showing that the results of spherical and finite planar geometries converge to those for the infinite planar geometry in the limit of large brain size. Estimates are made of the point at which modal series can be truncated and it is found that for physiologically plausible parameters only the lowest few spatial eigenmodes are needed for an accurate representation of macroscopic brain activity. A difference between the geometries is that there is a low-frequency 1/f spectrum in the infinite planar geometry, whereas in the spherical geometry it is 1/f^{2}. Another difference is that the alpha peak in the spherical geometry is sharper and stronger than in the planar geometry. Cortical modal effects can lead to a double alpha peak structure in the power spectrum, although the main determinant of the alpha peak is corticothalamic feedback. In the spherical geometry, the cross spectrum between two points is found to only depend on their relative distance apart. At small spatial separations the low-frequency cross spectrum is stronger than for an infinite planar geometry and the alpha peak is sharper and stronger due to the partitioning of the energy into discrete modes. In the spherical geometry, the coherence function between points decays monotonically as their separation increases at a fixed frequency, but persists further at resonant frequencies. The correlation between two points is found to be positive, regardless of the time lag and spatial separation, but decays monotonically as the separation increases at fixed time lag. At fixed distance the correlation has peaks at multiples of the period of the dominant frequency of system activity.

Spectral signatures of activity-dependent neural feedback in the corticothalamic system

The modulation of neural quantities by presynaptic and postsynaptic activities via local feedback processes is investigated by incorporating nonlinear phenomena such as relative refractory period, synaptic enhancement, synaptic depression, and habituation. This is done by introducing susceptibilities, which quantify the response in either firing threshold or synaptic strength to unit change in either presynaptic or postsynaptic activity. Effects on the power spectra are then analyzed for a realistic corticothalamic model to determine the spectral signatures of various nonlinear processes and to what extent these are distinct. Depending on the feedback processes, there can be enhancements or reductions in low-frequency and/or alpha power, splitting of the alpha resonance, and/or appearance of new resonances at high frequencies. These features in the power spectra allow processes to be fully distinguished where they are unique, or partly distinguished if they are common to only a subset of feedbacks, and can potentially be used to constrain the types, strengths, and dynamics of feedbacks present.

Critical dynamics of Hopf bifurcations in the corticothalamic system: Transitions from normal arousal states to epileptic seizures

A physiologically based corticothalamic model of large-scale brain activity is used to analyze critical dynamics of transitions from normal arousal states to epileptic seizures, which correspond to Hopf bifurcations. This relates an abstract normal form quantitatively to underlying physiology that includes neural dynamics, axonal propagation, and time delays. Thus, a bridge is constructed that enables normal forms to be used to interpret quantitative data. The normal form of the Hopf bifurcations with delays is derived using Hale's theory, the center manifold theorem, and normal form analysis, and it is found to be explicitly expressed in terms of transfer functions and the sensitivity matrix of a reduced open-loop system. It can be applied to understand the effect of each physiological parameter on the critical dynamics and determine whether the Hopf bifurcation is supercritical or subcritical in instabilities that lead to absence and tonic-clonic seizures. Furthermore, the effects of thalamic and cortical nonlinearities on the bifurcation type are investigated, with implications for the roles of underlying physiology. The theoretical predictions about the bifurcation type and the onset dynamics are confirmed by numerical simulations and provide physiologically based criteria for determining bifurcation types from first principles. The results are consistent with experimental data from previous studies, imply that new regimes of seizure transitions may exist in clinical settings, and provide a simplified basis for control-systems interventions. Using the normal form, and the full equations from which it is derived, more complex dynamics, such as quasiperiodic cycles and saddle cycles, are discovered near the critical points of the subcritical Hopf bifurcations.

K-complexes, spindles, and ERPs as impulse responses: unification via neural field theory

To interrelate K-complexes, spindles, evoked response potentials (ERPs), and spontaneous electroencephalography (EEG) using neural field theory (NFT), physiology-based NFT of the corticothalamic system is used to model cortical excitatory and inhibitory populations and thalamic relay and reticular nuclei. The impulse response function of the model is used to predict the responses to impulses, which are compared with transient waveforms in sleep studies. Fits to empirical data then allow underlying brain physiology to be inferred and compared with other waves. Spontaneous K-complexes, spindles, and other transient waveforms can be reproduced using NFT by treating them as evoked responses to impulsive stimuli with brain parameters appropriate to spontaneous EEG in sleep stage 2. Using this approach, spontaneous K-complexes and sleep spindles can be analyzed using the same single theory as previously been used to account for waking ERPs and other EEG phenomena. As a result, NFT can explain a wide variety of transient waveforms that have only been phenomenologically classified to date. This enables noninvasive fitting to be used to infer underlying physiological parameters. This physiology-based model reproduces the time series of different transient EEG waveforms; it has previously reproduced experimental EEG spectra, and waking ERPs, and many other observations, thereby unifying transient sleep waveforms with these phenomena.

Inference of direct and multistep effective connectivities from functional connectivity of the brain and of relationships to cortical geometry

BACKGROUND

The problem of inferring effective brain connectivity from functional connectivity is under active investigation, and connectivity via multistep paths is poorly understood.

NEW METHOD

A method is presented to calculate the direct effective connection matrix (deCM), which embodies direct connection strengths between brain regions, from functional CMs (fCMs) by minimizing the difference between an experimental fCM and one calculated via neural field theory from an ansatz deCM based on an experimental anatomical CM.

RESULTS

The best match between fCMs occurs close to a critical point, consistent with independent published stability estimates. Residual mismatch between fCMs is identified to be largely due to interhemispheric connections that are poorly estimated in an initial ansatz deCM due to experimental limitations; improved ansatzes substantially reduce the mismatch and enable interhemispheric connections to be estimated. Various levels of significant multistep connections are then imaged via the neural field theory (NFT) result that these correspond to powers of the deCM; these are shown to be predictable from geometric distances between regions.

COMPARISON WITH EXISTING METHODS

This method gives insight into direct and multistep effective connectivity from fCMs and relating to physiology and brain geometry. This contrasts with other methods, which progressively adjust connections without an overarching physiologically based framework to deal with multistep or poorly estimated connections.

CONCLUSIONS

deCMs can be usefully estimated using this method and the results enable multistep connections to be investigated systematically.

Macroscopic neural mass model constructed from a current-based network model of spiking neurons

Neural mass models (NMMs) are efficient frameworks for describing macroscopic cortical dynamics including electroencephalogram and magnetoencephalogram signals. Originally, these models were formulated on an empirical basis of synaptic dynamics with relatively long time constants. By clarifying the relations between NMMs and the dynamics of microscopic structures such as neurons and synapses, we can better understand cortical and neural mechanisms from a multi-scale perspective. In a previous study, the NMMs were analytically derived by averaging the equations of synaptic dynamics over the neurons in the population and further averaging the equations of the membrane-potential dynamics. However, the averaging of synaptic current assumes that the neuron membrane potentials are nearly time invariant and that they remain at sub-threshold levels to retain the conductance-based model. This approximation limits the NMM to the non-firing state. In the present study, we newly propose a derivation of a NMM by alternatively approximating the synaptic current which is assumed to be independent of the membrane potential, thus adopting a current-based model. Our proposed model releases the constraint of the nearly constant membrane potential. We confirm that the obtained model is reducible to the previous model in the non-firing situation and that it reproduces the temporal mean values and relative power spectrum densities of the average membrane potentials for the spiking neurons. It is further ensured that the existing NMM properly models the averaged dynamics over individual neurons even if they are spiking in the populations.

Mapping how local perturbations influence systems-level brain dynamics

The human brain exhibits a distinct spatiotemporal organization that supports brain function and can be manipulated via local brain stimulation. Such perturbations to local cortical dynamics are globally integrated by distinct neural systems. However, it remains unclear how local changes in neural activity affect large-scale system dynamics. Here, we briefly review empirical and computational studies addressing how localized perturbations affect brain activity. We then systematically analyze a model of large-scale brain dynamics, assessing how localized changes in brain activity at the different sites affect whole-brain dynamics. We find that local stimulation induces changes in brain activity that can be summarized by relatively smooth tuning curves, which relate a region's effectiveness as a stimulation site to its position within the cortical hierarchy. Our results also support the notion that brain hubs, operating in a slower regime, are more resilient to focal perturbations and critically contribute to maintain stability in global brain dynamics. In contrast, perturbations of peripheral regions, characterized by faster activity, have greater impact on functional connectivity. As a parallel with this region-level result, we also find that peripheral systems such as the visual and sensorimotor networks were more affected by local perturbations than high-level systems such as the cingulo-opercular network. Our findings highlight the importance of a periphery-to-core hierarchy to determine the effect of local stimulation on the brain network. This study also provides novel resources to orient empirical work aiming at manipulating functional connectivity using non-invasive brain stimulation.

Eigenmodes of brain activity: Neural field theory predictions and comparison with experiment

Neural field theory of the corticothalamic system is applied to predict and analyze the activity eigenmodes of the bihemispheric brain, focusing particularly on their spatial structure. The eigenmodes of a single brain hemisphere are found to be close analogs of spherical harmonics, which are the natural modes of the sphere. Instead of multiple eigenvalues being equal, as in the spherical case, cortical folding splits them to have distinct values. Inclusion of interhemispheric connections between homologous regions via the corpus callosum leads to further splitting that depends on symmetry or antisymmetry of activity between brain hemispheres, and the strength and sign of the interhemispheric connections. Symmetry properties of the lowest observed eigenmodes strongly constrain the interhemispheric connectivity strengths and unihemispheric mode spectra, and it is predicted that most spontaneous brain activity will be symmetric between hemispheres, consistent with observations. Comparison with the eigenmodes of an experimental anatomical connectivity matrix confirms these results, permits the relative strengths of intrahemispheric and interhemispheric connectivities to be approximately inferred from their eigenvalues, and lays the foundation for further experimental tests. The results are consistent with brain activity being in corticothalamic eigenmodes, rather than discrete "networks" and open the way to new approaches to brain analysis.

Neural field theory of nonlinear wave-wave and wave-neuron processes

Systematic expansion of neural field theory equations in terms of nonlinear response functions is carried out to enable a wide variety of nonlinear wave-wave and wave-neuron processes to be treated systematically in systems involving multiple neural populations. The results are illustrated by analyzing second-harmonic generation, and they can also be applied to wave-wave coalescence, multiharmonic generation, facilitation, depression, refractoriness, and other nonlinear processes.

Determination of effective brain connectivity from functional connectivity using propagator-based interferometry and neural field theory with application to the corticothalamic system

It is shown how to compute both direct and total effective connection matrices (deCMs and teCMs), which embody the strengths of neural connections between regions, from correlation-based functional CMs using propagator-based interferometry, a method that stems from geophysics and acoustics, coupled with the recent identification of deCMs and teCMs with bare and dressed propagators, respectively. The approach incorporates excitatory and inhibitory connections, multiple structures and populations, and measurement effects. The propagator is found for a generalized scalar wave equation derived from neural field theory, and expressed in terms of neural activity correlations and covariances, and wave damping rates. It is then related to correlation matrices that are commonly used to express functional and effective connectivities in the brain. The results are illustrated in analytically tractable test cases.

Stability constraints on large-scale structural brain networks

Stability is an important dynamical property of complex systems and underpins a broad range of coherent self-organized behavior. Based on evidence that some neurological disorders correspond to linear instabilities, we hypothesize that stability constrains the brain's electrical activity and influences its structure and physiology. Using a physiologically-based model of brain electrical activity, we investigated the stability and dispersion solutions of networks of neuronal populations with propagation time delays and dendritic time constants. We find that stability is determined by the spectrum of the network's matrix of connection strengths and is independent of the temporal damping rate of axonal propagation with stability restricting the spectrum to a region in the complex plane. Time delays and dendritic time constants modify the shape of this region but it always contains the unit disk. Instabilities resulting from changes in connection strength initially have frequencies less than a critical frequency. For physiologically plausible parameter values based on the corticothalamic system, this critical frequency is approximately 10 Hz. For excitatory networks and networks with randomly distributed excitatory and inhibitory connections, time delays and non-zero dendritic time constants have no impact on network stability but do effect dispersion frequencies. Random networks with both excitatory and inhibitory connections can have multiple marginally stable modes at low delta frequencies.

Statistical method for detecting phase shifts in alpha rhythm from human electroencephalogram data

We developed a statistical method for detecting discontinuous phase changes (phase shifts) in fluctuating alpha rhythms in the human brain from electroencephalogram (EEG) data obtained in a single trial. This method uses the state space models and the line process technique, which is a Bayesian method for detecting discontinuity in an image. By applying this method to simulated data, we were able to detect the phase and amplitude shifts in a single simulated trial. Further, we demonstrated that this method can detect phase shifts caused by a visual stimulus in the alpha rhythm from experimental EEG data even in a single trial. The results for the experimental data showed that the timings of the phase shifts in the early latency period were similar between many of the trials, and that those in the late latency period were different between the trials. The conventional averaging method can only detect phase shifts that occur at similar timings between many of the trials, and therefore, the phase shifts that occur at differing timings cannot be detected using the conventional method. Consequently, our obtained results indicate the practicality of our method. Thus, we believe that our method will contribute to studies examining the phase dynamics of nonlinear alpha rhythm oscillators.

From oscillatory transcranial current stimulation to scalp EEG changes: a biophysical and physiological modeling study

Both biophysical and neurophysiological aspects need to be considered to assess the impact of electric fields induced by transcranial current stimulation (tCS) on the cerebral cortex and the subsequent effects occurring on scalp EEG. The objective of this work was to elaborate a global model allowing for the simulation of scalp EEG signals under tCS. In our integrated modeling approach, realistic meshes of the head tissues and of the stimulation electrodes were first built to map the generated electric field distribution on the cortical surface. Secondly, source activities at various cortical macro-regions were generated by means of a computational model of neuronal populations. The model parameters were adjusted so that populations generated an oscillating activity around 10 Hz resembling typical EEG alpha activity. In order to account for tCS effects and following current biophysical models, the calculated component of the electric field normal to the cortex was used to locally influence the activity of neuronal populations. Lastly, EEG under both spontaneous and tACS-stimulated (transcranial sinunoidal tCS from 4 to 16 Hz) brain activity was simulated at the level of scalp electrodes by solving the forward problem in the aforementioned realistic head model. Under the 10 Hz-tACS condition, a significant increase in alpha power occurred in simulated scalp EEG signals as compared to the no-stimulation condition. This increase involved most channels bilaterally, was more pronounced on posterior electrodes and was only significant for tACS frequencies from 8 to 12 Hz. The immediate effects of tACS in the model agreed with the post-tACS results previously reported in real subjects. Moreover, additional information was also brought by the model at other electrode positions or stimulation frequency. This suggests that our modeling approach can be used to compare, interpret and predict changes occurring on EEG with respect to parameters used in specific stimulation configurations.

Neural field theory of plasticity in the cerebral cortex

A generalized timing-dependent plasticity rule is incorporated into a recent neural field theory to explore synaptic plasticity in the cerebral cortex, with both excitatory and inhibitory populations included. Analysis in the time and frequency domains reveals that cortical network behavior gives rise to a saddle-node bifurcation and resonant frequencies, including a gamma-band resonance. These system resonances constrain cortical synaptic dynamics and divide it into four classes, which depend on the type of synaptic plasticity window. Depending on the dynamical class, synaptic strengths can either have a stable fixed point, or can diverge in the absence of a separate saturation mechanism. Parameter exploration shows that time-asymmetric plasticity windows, which are signatures of spike-timing dependent plasticity, enable the richest variety of synaptic dynamics to occur. In particular, we predict a zone in parameter space which may allow brains to attain the marginal stability phenomena observed experimentally, although additional regulatory mechanisms may be required to maintain these parameters.

Complementarity of spike- and rate-based dynamics of neural systems

Relationships between spiking-neuron and rate-based approaches to the dynamics of neural assemblies are explored by analyzing a model system that can be treated by both methods, with the rate-based method further averaged over multiple neurons to give a neural-field approach. The system consists of a chain of neurons, each with simple spiking dynamics that has a known rate-based equivalent. The neurons are linked by propagating activity that is described in terms of a spatial interaction strength with temporal delays that reflect distances between neurons; feedback via a separate delay loop is also included because such loops also exist in real brains. These interactions are described using a spatiotemporal coupling function that can carry either spikes or rates to provide coupling between neurons. Numerical simulation of corresponding spike- and rate-based methods with these compatible couplings then allows direct comparison between the dynamics arising from these approaches. The rate-based dynamics can reproduce two different forms of oscillation that are present in the spike-based model: spiking rates of individual neurons and network-induced modulations of spiking rate that occur if network interactions are sufficiently strong. Depending on conditions either mode of oscillation can dominate the spike-based dynamics and in some situations, particularly when the ratio of the frequencies of these two modes is integer or half-integer, the two can both be present and interact with each other.

Neural field theory with variance dynamics

Previous neural field models have mostly been concerned with prediction of mean neural activity and with second order quantities such as its variance, but without feedback of second order quantities on the dynamics. Here the effects of feedback of the variance on the steady states and adiabatic dynamics of neural systems are calculated using linear neural field theory to estimate the neural voltage variance, then including this quantity in the total variance parameter of the nonlinear firing rate-voltage response function, and thus into determination of the fixed points and the variance itself. The general results further clarify the limits of validity of approaches with and without inclusion of variance dynamics. Specific applications show that stability against a saddle-node bifurcation is reduced in a purely cortical system, but can be either increased or decreased in the corticothalamic case, depending on the initial state. Estimates of critical variance scalings near saddle-node bifurcation are also found, including physiologically based normalizations and new scalings for mean firing rate and the position of the bifurcation.

Spike, rate, field, and hybrid methods for treating neuronal dynamics and interactions

Spike-, rate-, and field-based approaches to neural dynamics are adapted and hybridized to provide new methods of analyzing dynamics of single neurons and large neuronal systems, to elucidate the relationships and intermediate forms between these limiting cases, and to enable faster simulations with reduced memory requirements. At the single-neuron level, the new approaches involve reformulation of dynamics in synapses, dendrites, cell bodies, and axons to enable new types of analysis, longer numerical timesteps, and demonstration that rate-based methods can predict spike times. In multineuron systems, hybrids and intermediates between spike-based and field-based coupling between neurons are used to provide stepping stones between descriptions based on pairwise spike-based interactions between neurons and ones based on neural field-based interactions within and between populations, including arbitrary spatial structure and temporal delays in the connections in general. In particular, a new neuron-in-cell approach is introduced that is a hybrid between neural field theory and spiking-neuron models in analogy to particle-in-cell methods in plasma physics. This approach enables large speedups in computations while preserving spike shapes and times. Various approaches are illustrated numerically for specific cases.

Interrelating anatomical, effective, and functional brain connectivity using propagators and neural field theory

It is shown how to compute effective and functional connection matrices (eCMs and fCMs) from anatomical CMs (aCMs) and corresponding strength-of-connection matrices (sCMs) using propagator methods in which neural interactions play the role of scatterings. This analysis demonstrates how network effects dress the bare propagators (the sCMs) to yield effective propagators (the eCMs) that can be used to compute the covariances customarily used to define fCMs. The results incorporate excitatory and inhibitory connections, multiple structures and populations, asymmetries, time delays, and measurement effects. They can also be postprocessed in the same manner as experimental measurements for direct comparison with data and thereby give insights into the role of coarse-graining, thresholding, and other effects in determining the structure of CMs. The spatiotemporal results show how to generalize CMs to include time delays and how natural network modes give rise to long-range coherence at resonant frequencies. The results are demonstrated using tractable analytic cases via neural field theory of cortical and corticothalamic systems. These also demonstrate close connections between the structure of CMs and proximity to critical points of the system, highlight the importance of indirect links between brain regions and raise the possibility of imaging specific levels of indirect connectivity. Aside from the results presented explicitly here, the expression of the connections among aCMs, sCMs, eCMs, and fCMs in terms of propagators opens the way for propagator theory to be further applied to analysis of connectivity.

Quantitative modelling of sleep dynamics

Arousal is largely controlled by the ascending arousal system of the hypothalamus and brainstem, which projects to the corticothalamic system responsible for electroencephalographic (EEG) signatures of sleep. Quantitative physiologically based modelling of brainstem dynamics theory is described here, using realistic parameters, and links to EEG are outlined. Verification against a wide range of experimental data is described, including arousal dynamics under normal conditions, sleep deprivation, stimuli, stimulants and jetlag, plus key features of wake and sleep EEGs.

Neural field theory of synaptic plasticity

Plasticity is crucial to neural development, learning, and memory. In the common in vivo situation where postsynaptic neural activity results from multiple presynaptic inputs, it is shown that a widely used class of correlation-dependent and spike-timing dependent plasticity rules can be written in a form that can be incorporated into neural field theory, which enables their system-level dynamics to be investigated. It is shown that the resulting plasticity dynamics depends strongly on the stimulus spectrum via overall system frequency responses. In the case of perturbations that are approximately linear, explicit formulas are found for the dynamics in terms of stimulus spectra via system transfer functions. The resulting theory is applied to a simple model system to reveal how collective effects, especially resonances, can drastically modify system-level plasticity dynamics from that implied by single-neuron analyses. The simplified model illustrates the potential relevance of these effects in applications to brain stimulation, synaptic homeostasis, and epilepsy.

Firing responses of bursting neurons with delayed feedback

Thalamic neurons, which play important roles in the genesis of rhythmic activities of the brain, show various bursting behaviors, particularly modulated by complex thalamocortical feedback via cortical neurons. As a first step to explore this complex neural system and focus on the effects of the feedback on the bursting behavior, a simple loop structure delayed in time and scaled by a coupling strength is added to a recent mean-field model of bursting neurons. Depending on the coupling strength and delay time, the modeled neurons show two distinct response patterns: one entrained to the unperturbed bursting frequency of the neurons and one entrained to the resonant frequency of the loop structure. Transitions between these two patterns are explored in the model's parameter space via extensive numerical simulations. It is found that at a fixed loop delay, there is a critical coupling strength at which the dominant response frequency switches from the unperturbed bursting frequency to the loop-induced one. Furthermore, alternating occurrence of these two response frequencies is observed when the delay varies at fixed coupling strength. The results demonstrate that bursting is coupled with feedback to yield new dynamics, which will provide insights into such effects in more complex neural systems.

Influence of coherence between multiple cortical columns on alpha rhythm: a computational modeling study

In electroencephalographic (EEG) and magnetoencephalographic (MEG) signals, stimulus-induced amplitude increase and decrease in the alpha rhythm, known as event-related synchronization and desynchronization (ERS/ERD), emerge after a task onset. ERS/ERD is assumed to reflect neural processes relevant to cognitive tasks. Previous studies suggest that several sources of alpha rhythm, each of which can serve as an alpha rhythm generator, exist in the cortex. Since EEG/MEG signals represent spatially summed neural activities, ERS/ERD of the alpha rhythm may reflect the consequence of the interactions between multiple alpha rhythm generators. Two candidates modulate the magnitude of ERS/ERD: (1) coherence between the activities of the alpha rhythm generators and (2) mean amplitude of the activities of the alpha rhythm generators. In this study, we use a computational model of multiple alpha rhythm generators to determine the factor that dominantly causes ERS/ERD. Each alpha rhythm generator is modeled based on local column circuits in the primary visual cortex and made to interact with the neighboring generators through excitatory connections. We observe that the model consistently reproduces spontaneous alpha rhythms, event-related potentials, phase-locked alpha rhythms, and ERS/ERD in a specific range of connectivity coefficients. Independent analyses of the coherence and amplitude of multiple alpha rhythm generators reveal that the ERS/ERD in the simulated data is dominantly caused by stimulus-induced changes in the coherence between multiple alpha rhythm generators. Nonlinear phenomena such as phase-resetting and entrainment of the alpha rhythm are related to the neural mechanism underlying ERS/ERD.

Mean-field modeling of the basal ganglia-thalamocortical system. II Dynamics of parkinsonian oscillations

Neuronal correlates of Parkinson's disease (PD) include a shift to lower frequencies in the electroencephalogram (EEG) and enhanced synchronized oscillations at 3-7 and 7-30 Hz in the basal ganglia, thalamus, and cortex. This study describes the dynamics of a recent physiologically based mean-field model of the basal ganglia-thalamocortical system, and shows how it accounts for many key electrophysiological correlates of PD. Its detailed functional connectivity comprises partially segregated direct and indirect pathways through two populations of striatal neurons, a hyperdirect pathway involving a corticosubthalamic projection, thalamostriatal feedback, and local inhibition in striatum and external pallidum (GPe). In a companion paper, realistic steady-state firing rates were obtained for the healthy state, and after dopamine loss modeled by weaker direct and stronger indirect pathways, reduced intrapallidal inhibition, lower firing thresholds of the GPe and subthalamic nucleus (STN), a stronger projection from striatum to GPe, and weaker cortical interactions. Here it is shown that oscillations around 5 and 20 Hz can arise with a strong indirect pathway, which also causes increased synchronization throughout the basal ganglia. Furthermore, increased theta power with progressive nigrostriatal degeneration is correlated with reduced alpha power and peak frequency, in agreement with empirical results. Unlike the hyperdirect pathway, the indirect pathway sustains oscillations with phase relationships that coincide with those found experimentally. Alterations in the responses of basal ganglia to transient stimuli accord with experimental observations. Reduced cortical gains due to both nigrostriatal and mesocortical dopamine loss lead to slower changes in cortical activity and may be related to bradykinesia. Finally, increased EEG power found in some studies may be partly explained by a lower effective GPe firing threshold, reduced GPe-GPe inhibition, and/or weaker intracortical connections in parkinsonian patients. Strict separation of the direct and indirect pathways is not necessary to obtain these results.

Modeling distributed axonal delays in mean-field brain dynamics

The range of conduction delays between connected neuronal populations is often modeled as a single discrete delay, assumed to be an effective value averaging over all fiber velocities. This paper shows the effects of distributed delays on signal propagation. A distribution acts as a linear filter, imposing an upper frequency cutoff that is inversely proportional to the delay width. Distributed thalamocortical and corticothalamic delays are incorporated into a physiologically based mean-field model of the cortex and thalamus to illustrate their effects on the electroencephalogram (EEG). The power spectrum is acutely sensitive to the width of the thalamocortical delay distribution, and more so than the corticothalamic distribution, because all input signals must travel along the thalamocortical pathway. This imposes a cutoff frequency above which the spectrum is overly damped. The positions of spectral peaks in the resting EEG depend primarily on the distribution mean, with only weak dependences on distribution width. Increasing distribution width increases the stability of fixed point solutions. A single discrete delay successfully approximates a distribution for frequencies below a cutoff that is inversely proportional to the delay width, provided that other model parameters are moderately adjusted. A pair of discrete delays together having the same mean, variance, and skewness as the distribution approximates the distribution over the same frequency range without needing parameter adjustment. Delay distributions with large fractional widths are well approximated by low-order differential equations.

The dynamic brain: from spiking neurons to neural masses and cortical fields

The cortex is a complex system, characterized by its dynamics and architecture, which underlie many functions such as action, perception, learning, language, and cognition. Its structural architecture has been studied for more than a hundred years; however, its dynamics have been addressed much less thoroughly. 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. Computational models at different space-time scales help us understand the fundamental mechanisms that underpin neural processes and relate these processes to neuroscience data. 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. Mesoscopic models tell us how neural elements interact to yield emergent behavior at the level of microcolumns and cortical columns. 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. 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). Models of the cortex can establish which types of large-scale neuronal networks can perform computations and characterize their emergent properties. Mean-field and related formulations of dynamics also play an essential and complementary role as forward models that can be inverted given empirical data. This makes dynamic models critical in integrating theory and experiments. 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.

Bayesian estimation of synaptic physiology from the spectral responses of neural masses

We describe a Bayesian inference scheme for quantifying the active physiology of neuronal ensembles using local field recordings of synaptic potentials. This entails the inversion of a generative neural mass model of steady-state spectral activity. The inversion uses Expectation Maximization (EM) to furnish the posterior probability of key synaptic parameters and the marginal likelihood of the model itself. The neural mass model embeds prior knowledge pertaining to both the anatomical [synaptic] circuitry and plausible trajectories of neuronal dynamics. This model comprises a population of excitatory pyramidal cells, under local interneuron inhibition and driving excitation from layer IV stellate cells. Under quasi-stationary assumptions, the model can predict the spectral profile of local field potentials (LFP). This means model parameters can be optimised given real electrophysiological observations. The validity of inferences about synaptic parameters is demonstrated using simulated data and experimental recordings from the medial prefrontal cortex of control and isolation-reared Wistar rats. Specifically, we examined the maximum a posteriori estimates of parameters describing synaptic function in the two groups and tested predictions derived from concomitant microdialysis measures. The modelling of the LFP recordings revealed (i) a sensitization of post-synaptic excitatory responses, particularly marked in pyramidal cells, in the medial prefrontal cortex of socially isolated rats and (ii) increased neuronal adaptation. These inferences were consistent with predictions derived from experimental microdialysis measures of extracellular glutamate levels.

Population dynamics: variance and the sigmoid activation function

This paper demonstrates how the sigmoid activation function of neural-mass models can be understood in terms of the variance or dispersion of neuronal states. We use this relationship to estimate the probability density on hidden neuronal states, using non-invasive electrophysiological (EEG) measures and dynamic casual modelling. The importance of implicit variance in neuronal states for neural-mass models of cortical dynamics is illustrated using both synthetic data and real EEG measurements of sensory evoked responses.

Physiologically based calculation of steady-state evoked potentials and cortical wave velocities

Steady-state evoked potentials (SSEPs) elicited by sinusoidal stimuli are predicted from a physiologically-based model, including bielectrode and volume conduction effects. Comparison with visual SSEPs yields constraints on phase and latency of the retinothalamic transfer function that are consistent with experiment. Predictions of phase velocities measured as SSEPs cross the cortex are consistent with low values measured for slow waves in sleep, while resonant behavior induced by corticothalamic loops, especially near the alpha peak, contributes to wide scatter in waking-state phase velocity measurements comparable to effects from volume conduction. The common use of bielectrode derivations to compensate for volume conduction effects is examined and shown to be incomplete, tending to lead to underestimates of phase velocity, especially at low frequencies and near the alpha peak, due to incorrect elimination of true long-wavelength contributions to the SSEP.

Neural rate equations for bursting dynamics derived from conductance-based equations

A method of obtaining rate equations from conductance-based equations is developed and applied to fast-spiking and bursting neocortical neurons. It involves splitting systems of conductance-based equations into fast and slow subsystems, and averaging the effects of fast terms that drive the slowly varying quantities by showing that their average is closely proportional to the firing rate. The dependence of the firing rate on the injected current is then approximated in the analysis. The resulting behavior of the slow variables is then substituted back into the fast equations, with the further approximation of replacing the fast voltages in these terms by effective values. For bursting neurons the method yields two coupled limit-cycle oscillators: a self-exciting oscillator for the slow variables that commences limit-cycle oscillations at a critical current and modulates a fast spike-generating oscillator, thereby leading to slowly modulated bursts with a group of spikes in each burst. The dynamics of these coupled oscillators are then verified against those of the conductance-based equations. Finally, it is shown how to place the results in a form suitable for use in mean-field equations for neural population dynamics.

Visual gamma oscillations: waves, correlations, and other phenomena, including comparison with experimental data

Mean-field theory of brain dynamics is applied to explain the properties of gamma (> or approximately 30 Hz) oscillations of cortical activity often seen during vision experiments. It is shown that mm-scale patchy connections in the primary visual cortex can support collective gamma oscillations with the correct frequencies and spatial structure, even when driven by uncorrelated inputs. This occurs via resonances associated with the the periodic modulation of the network connections, rather than being due to single-cell properties alone. Near-resonant gamma waves are shown to obey the Schrödinger equation, which enables techniques and insights from quantum theory to be used in exploring these classical oscillations. Resulting predictions for gamma responses to stimuli account in a unified way for a wide range of experimental results, including why oscillations and zero-lag synchrony are associated, and variations in correlation functions with time delay, intercellular distance, and stimulus features. They also imply that gamma oscillations may enable a form of frequency multiplexing of neural signals. Most importantly, it is shown that correlations reproduce experimental results that show maximal correlations between cells that respond to related features, but little correlation with other cells, an effect that has been argued to be associated with segmentation of a scene into separate objects. Consistency with infill of missing contours and increase in response with length of bar-shaped stimuli are discussed. Background correlations expected in the absence of stimulation are also calculated and shown to be consistent in form with experimental measurements and similar to stimulus-induced correlations in structure. Finally, possible links of gamma instabilities to certain classes of photically induced seizures and visual hallucinations are discussed.

A neural mass model of spectral responses in electrophysiology

We present a neural mass model of steady-state membrane potentials measured with local field potentials or electroencephalography in the frequency domain. This model is an extended version of previous dynamic causal models for investigating event-related potentials in the time-domain. In this paper, we augment the previous formulation with parameters that mediate spike-rate adaptation and recurrent intrinsic inhibitory connections. We then use linear systems analysis to show how the model's spectral response changes with its neurophysiological parameters. We demonstrate that much of the interesting behaviour depends on the non-linearity which couples mean membrane potential to mean spiking rate. This non-linearity is analogous, at the population level, to the firing rate–input curves often used to characterize single-cell responses. This function depends on the model's gain and adaptation currents which, neurobiologically, are influenced by the activity of modulatory neurotransmitters. The key contribution of this paper is to show how neuromodulatory effects can be modelled by adding adaptation currents to a simple phenomenological model of EEG. Critically, we show that these effects are expressed in a systematic way in the spectral density of EEG recordings. Inversion of the model, given such non-invasive recordings, should allow one to quantify pharmacologically induced changes in adaptation currents. In short, this work establishes a forward or generative model of electrophysiological recordings for psychopharmacological studies.

Compact dynamical model of brain activity

A compact physiologically based mean-field formulation of brain dynamics is proposed to model observed brain activity and electroencephalographic (EEG) signals. In contrast to existing formulations, which are more detailed and complicated, our model is described by a single second-order delay differential equation that encapsulates salient aspects of the physiology. The model captures essential features of activity mediated by fast corticocortical connections and delayed feedbacks via extracortical pathways and external stimuli. In the linear regime, these features can be simply expressed by three coefficients derived from the properties of these physiological pathways and explicit nonlinear approximations are also derived. This compact model successfully reproduces the main features of experimental EEG's and the predictions of previous models, including resonance peaks in EEG spectra and nonlinear dynamics. As an illustration, key features of the dynamics of epileptic seizures are shown to be reproduced by the model. Due to its compact form, the model will facilitate insight into nonlinear brain dynamics via standard nonlinear techniques and will guide analysis and investigation of more complex models. It is thus a useful tool for analyzing complex brain activity, especially when it exhibits low-dimensional dynamics.