Dynamics of large-scale brain activity in normal arousal states and epileptic seizures

Transition to absence seizures and the role of GABA(A) receptors

Absence seizures appear to be initiated in a putative cortical ‘initiation site’ by the expression of medium-amplitude 5–9 Hz oscillations, which may in part be due to a decreased phasic GABA_A receptor function. These oscillations rapidly spread to other cortical areas and to the thalamus, leading to fully developed generalized spike and wave discharges. In thalamocortical neurons of genetic models, phasic GABA_A inhibition is either unchanged or increased, whereas tonic GABA_A inhibition is increased both in genetic and pharmacological models. This enhanced tonic inhibition is required for absence seizure generation, and in genetic models it results from a malfunction in the astrocytic GABA transporter GAT-1. Contradictory results from inbred and transgenic animals still do not allow us to draw firm conclusions on changes in phasic GABA_A inhibition in the GABAergic neurons of the nucleus reticularis thalami. Mathematical modelling may enhance our understanding of these competing hypotheses, by permitting investigations of their mechanistic aspects, hence enabling a greater understanding of the processes underlying seizure generation and evolution.

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.

Biophysical mechanisms of multistability in resting-state cortical rhythms

The human alpha (8-12 Hz) rhythm is one of the most prominent, robust, and widely studied attributes of ongoing cortical activity. Contrary to the prevalent notion that it simply "waxes and wanes," spontaneous alpha activity bursts erratically between two distinct modes of activity. We now establish a mechanism for this multistable phenomenon in resting-state cortical recordings by characterizing the complex dynamics of a biophysical model of macroscopic corticothalamic activity. This is achieved by studying the predicted activity of cortical and thalamic neuronal populations in this model as a function of its dynamic stability and the role of nonspecific synaptic noise. We hence find that fluctuating noisy inputs into thalamic neurons elicit spontaneous bursts between low- and high-amplitude alpha oscillations when the system is near a particular type of dynamical instability, namely a subcritical Hopf bifurcation. When the postsynaptic potentials associated with these noisy inputs are modulated by cortical feedback, the SD of power within each of these modes scale in proportion to their mean, showing remarkable concordance with empirical data. Our state-dependent corticothalamic model hence exhibits multistability and scale-invariant fluctuations-key features of resting-state cortical activity and indeed, of human perception, cognition, and behavior-thus providing a unified account of these apparently divergent phenomena.

Synchronization in interdependent networks

We explore the synchronization behavior in interdependent systems, where the one-dimensional (1D) network (the intranetwork coupling strength J(I)) is ferromagnetically intercoupled (the strength J) to the Watts-Strogatz (WS) small-world network (the intranetwork coupling strength J(II)). In the absence of the internetwork coupling (J=0), the former network is well known not to exhibit the synchronized phase at any finite coupling strength, whereas the latter displays the mean-field transition. Through an analytic approach based on the mean-field approximation, it is found that for the weakly coupled 1D network (J(I)≪1) the increase of J suppresses synchrony, because the nonsynchronized 1D network becomes a heavier burden for the synchronization process of the WS network. As the coupling in the 1D network becomes stronger, it is revealed by the renormalization group (RG) argument that the synchronization is enhanced as J(I) is increased, implying that the more enhanced partial synchronization in the 1D network makes the burden lighter. Extensive numerical simulations confirm these expected behaviors, while exhibiting a reentrant behavior in the intermediate range of J(I). The nonmonotonic change of the critical value of J(II) is also compared with the result from the numerical RG calculation.

A spatially extended model for macroscopic spike-wave discharges

Spike-wave discharges are a distinctive feature of epileptic seizures. So far, they have not been reported in spatially extended neural field models. We study a space-independent version of the Amari neural field model with two competing inhibitory populations. We show that this competition leads to robust spike-wave dynamics if the inhibitory populations operate on different time-scales. The spike-wave oscillations present a fold/homoclinic type bursting. From this result we predict parameters of the extended Amari system where spike-wave oscillations produce a spatially homogeneous pattern. We propose this mechanism as a prototype of macroscopic epileptic spike-wave discharges. To our knowledge this is the first example of robust spike-wave patterns in a spatially extended neural field model.

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.

Experimental observation of increased fluctuations in an order parameter before epochs of extended brain synchronization

The identification of epileptic seizure precursors has potential clinical relevance. It is conjectured that seizures may be represented by dynamical bifurcations and that an adequate order parameter to characterize brain dynamics is the phase difference in the oscillatory activity of neural systems. In this study, the critical point hypothesis that seizures, or more generally periods of widespread high synchronization, represent bifurcations is empirically tested by monitoring the growth of fluctuations in the putative order parameter of phase differences between magnetoencephalographic and electroencephalographic signals in nearby brain regions in patients with epilepsy and normal subjects during hyperventilation. Implications of the results with regard to epileptic phenomena are discussed.

Dynamics of coupled thalamocortical modules

We develop a model of thalamocortical dynamics using a shared population of thalamic neurons to couple distant cortical regions. Behavior of the model is determined as a function of the connection strengths with shared and unshared populations in the thalamus, either within a relay nucleus or the reticular nucleus. When the coupling is via the reticular nucleus, we locate solutions of the model where distant cortical regions maintain the same activity level, and regions where one region maintains an elevated activity level, suppressing activity in the other. We locate and investigate a region where both types of solutions exist and are stable, yielding a mechanism for spontaneous changes in global activity patterns. Power spectra and coherence are computed, and marked differences in the coherence are found between the two kinds of modes. When, on the other hand, the coupling is via a shared relay nuclei, the features seen with the reticular coupling are absent. These considerations suggest a role for the reticular nucleus in modulating long distance cortical communication.

A method for detecting false bifurcations in dynamical systems: application to neural-field models

In this article, we present a method for tracking changes in curvature of limit cycle solutions that arise due to inflection points. In keeping with previous literature, we term these changes false bifurcations, as they appear to be bifurcations when considering a Poincaré section that is tangent to the solution, but in actual fact the deformation of the solution occurs smoothly as a parameter is varied. These types of solutions arise commonly in electroencephalogram models of absence seizures and correspond to the formation of spikes in these models. Tracking these transitions in parameter space allows regions to be defined corresponding to different types of spike and wave dynamics, that may be of use in clinical neuroscience as a means to classify different subtypes of the more general syndrome.

Axonal velocity distributions in neural field equations

By modelling the average activity of large neuronal populations, continuum mean field models (MFMs) have become an increasingly important theoretical tool for understanding the emergent activity of cortical tissue. In order to be computationally tractable, long-range propagation of activity in MFMs is often approximated with partial differential equations (PDEs). However, PDE approximations in current use correspond to underlying axonal velocity distributions incompatible with experimental measurements. In order to rectify this deficiency, we here introduce novel propagation PDEs that give rise to smooth unimodal distributions of axonal conduction velocities. We also argue that velocities estimated from fibre diameters in slice and from latency measurements, respectively, relate quite differently to such distributions, a significant point for any phenomenological description. Our PDEs are then successfully fit to fibre diameter data from human corpus callosum and rat subcortical white matter. This allows for the first time to simulate long-range conduction in the mammalian brain with realistic, convenient PDEs. Furthermore, the obtained results suggest that the propagation of activity in rat and human differs significantly beyond mere scaling. The dynamical consequences of our new formulation are investigated in the context of a well known neural field model. On the basis of Turing instability analyses, we conclude that pattern formation is more easily initiated using our more realistic propagator. By increasing characteristic conduction velocities, a smooth transition can occur from self-sustaining bulk oscillations to travelling waves of various wavelengths, which may influence axonal growth during development. Our analytic results are also corroborated numerically using simulations on a large spatial grid. Thus we provide here a comprehensive analysis of empirically constrained activity propagation in the context of MFMs, which will allow more realistic studies of mammalian brain activity in the future.

Neurophysiological changes with age probed by inverse modeling of EEG spectra

OBJECTIVE

To investigate age-associated changes in physiologically-based EEG spectral parameters in the healthy population.

METHODS

Eyes-closed EEG spectra of 1498 healthy subjects aged 6-86 years were fitted to a mean-field model of thalamocortical dynamics in a cross-sectional study. Parameters were synaptodendritic rates, cortical wave decay rates, connection strengths (gains), axonal delays for thalamocortical loops, and power normalizations. Age trends were approximated using smooth asymptotically linear functions with a single turning point. We also considered sex differences and relationships between model parameters and traditional quantitative EEG measures.

RESULTS

The cross-sectional data suggest that changes tend to be most rapid in childhood, generally leveling off at age 15-20 years. Most gains decrease in magnitude with age, as does power normalization. Axonal and dendritic delays decrease in childhood and then increase. Axonal delays and gains show small but significant sex differences.

CONCLUSIONS

Mean-field brain modeling allows interpretation of age-associated EEG trends in terms of physiological processes, including the growth and regression of white matter, influencing axonal delays, and the establishment and pruning of synaptic connections, influencing gains.

SIGNIFICANCE

This study demonstrates the feasibility of inverse modeling of EEG spectra as a noninvasive method for investigating large-scale corticothalamic dynamics, and provides a basis for future comparisons.

Comparative study of nonlinear properties of EEG signals of normal persons and epileptic patients

BACKGROUND

Investigation of the functioning of the brain in living systems has been a major effort amongst scientists and medical practitioners. Amongst the various disorder of the brain, epilepsy has drawn the most attention because this disorder can affect the quality of life of a person. In this paper we have reinvestigated the EEGs for normal and epileptic patients using surrogate analysis, probability distribution function and Hurst exponent.

RESULTS

Using random shuffled surrogate analysis, we have obtained some of the nonlinear features that was obtained by Andrzejak et al. [Phys Rev E 2001, 64:061907], for the epileptic patients during seizure. Probability distribution function shows that the activity of an epileptic brain is nongaussian in nature. Hurst exponent has been shown to be useful to characterize a normal and an epileptic brain and it shows that the epileptic brain is long term anticorrelated whereas, the normal brain is more or less stochastic. Among all the techniques, used here, Hurst exponent is found very useful for characterization different cases.

CONCLUSION

In this article, differences in characteristics for normal subjects with eyes open and closed, epileptic subjects during seizure and seizure free intervals have been shown mainly using Hurst exponent. The H shows that the brain activity of a normal man is uncorrelated in nature whereas, epileptic brain activity shows long range anticorrelation.

Bistability and non-Gaussian fluctuations in spontaneous cortical activity

The brain is widely assumed to be a paradigmatic example of a complex, self-organizing system. As such, it should exhibit the classic hallmarks of nonlinearity, multistability, and "nondiffusivity" (large coherent fluctuations). Surprisingly, at least at the very large scale of neocortical dynamics, there is little empirical evidence to support this, and hence most computational and methodological frameworks for healthy brain activity have proceeded very reasonably from a purely linear and diffusive perspective. By studying the temporal fluctuations of power in human resting-state electroencephalograms, we show that, although these simple properties may hold true at some temporal scales, there is strong evidence for bistability and nondiffusivity in key brain rhythms. Bistability is manifest as nonclassic bursting between high- and low-amplitude modes in the alpha rhythm. Nondiffusivity is expressed through the irregular appearance of high amplitude "extremal" events in beta rhythm power fluctuations. The statistical robustness of these observations was confirmed through comparison with Gaussian-rendered phase-randomized surrogate data. Although there is a good conceptual framework for understanding bistability in cortical dynamics, the implications of the extremal events challenge existing frameworks for understanding large-scale brain systems.

Characterisation of cortical activity in response to deep brain stimulation of ventral-lateral nucleus: modelling and experiment

Motivated by its success as a therapeutic treatment in other neurological disorders, most notably Parkinson's disease, Deep Brain Stimulation (DBS) is currently being trialled in a number of patients with drug unresponsive epilepsies. However, the mechanisms by which DBS interferes with neuronal activity linked to the disorder are not well understood. Furthermore, there is a need to identify optimized values of parameters (for example in amplitude/frequency space) of the stimulation protocol with which one aims to achieve the desired outcome. In this paper we characterise the system response to stimulation, to gain an understanding of the role different brain regions play in generating the output observed in EEG. We perform a number of experiments in healthy rats, where the ventral-lateral thalamic nucleus is stimulated using a train of square-waves with different frequency and amplitudes. The response to stimulation in the motor cortex is recorded and the drive-response relationship over frequency/amplitude space is considered. Subsequently, we compare the experimental data with simulations of a mean-field model, finding good agreement between the output of the model and the experimental data--both in the time and frequency domains--when considering a transition to oscillatory activity in the cortex as the frequency of stimulation is increased. Overall, our study suggests that mean-field models can appropriately characterise the stimulus-response relationship of DBS in healthy animals. In this way, it constitutes a first step towards the goal of developing a closed-loop feedback control protocol for suppressing epileptic activity, by adaptively adjusting the stimulation protocol in response to EEG activity.

Transitions to spike-wave oscillations and epileptic dynamics in a human cortico-thalamic mean-field model

In this paper we present a detailed theoretical analysis of the onset of spike-wave activity in a model of human electroencephalogram (EEG) activity, relating this to clinical recordings from patients with absence seizures. We present a complete explanation of the transition from inter-ictal activity to spike and wave using a combination of bifurcation theory, numerical continuation and techniques for detecting the occurrence of inflection points in systems of delay differential equations (DDEs). We demonstrate that the initial transition to oscillatory behaviour occurs as a result of a Hopf bifurcation, whereas the addition of spikes arises as a result of an inflection point of the vector field. Strikingly these findings are consistent with EEG data recorded from patients with absence seizures and we present a discussion of the clinical significance of these results, suggesting potential new techniques for detection and anticipation of seizures.

Two-point heterogeneous connections in a continuum neural field model

We examine a novel heterogeneous connection scheme in a 1D continuum neural field model. Multiple two-point connections are added to a local connection function in order to model the "patchy" connections seen in, for example visual cortex. We use a numerical approach to solve the equations, choosing the locations of the two-point connections stochastically. We observe self-sustained persistent fluctuations of activity which can be classified into two types (one of which is similar to that seen in network models of discrete excitable neurons, the other being particular to this model). We study the effect of parameters such as system size and the range, number and strength of connections, on the probability that a particular realisation of the connections is able to exhibit persistent fluctuations.

Onset of polyspike complexes in a mean-field model of human electroencephalography and its application to absence epilepsy

In this paper, we introduce a modification of a mean-field model used to describe the brain's electrical activity as recorded via electroencephalography (EEG). The focus of the present study is to understand the mechanisms giving rise to the dynamics observed during absence epilepsy, one of the classical generalized syndromes. A systematic study of the data from a number of different subjects with absence epilepsy demonstrates a wide variety of dynamical phenomena in the recorded EEG. In addition to the classical spike and wave activity, there may be polyspike and wave, wave spike or even no discernible spike-wave onset during seizure events. The model we introduce is able to capture all of these different phenomena and we describe the bifurcations giving rise to these different types of seizure activity. We argue that such a model may provide a useful clinical tool for classifying different subclasses of absence epilepsy.

From working memory to epilepsy: dynamics of facilitation and inhibition in a cortical network

Persistent states are believed to be the correlate for short-term or working memory. Using a previously derived model for working memory, we show that disruption of the lateral inhibition can lead to a variety of pathological states. These states are analogs of reflex or pattern-sensitive epilepsy. Simulations, numerical bifurcation analysis, and fast-slow decomposition are used to explore the dynamics of this network.

Derivation and analysis of an ordinary differential equation mean-field model for studying clinically recorded epilepsy dynamics

In this paper we describe how an ordinary differential equation model of corticothalamic interactions may be obtained from a more general system of delay differential equations. We demonstrate that transitions to epileptic dynamics via changes in system parameters are qualitatively the same as in the original model with delay, as well as demonstrating that the onset of epileptic activity may arise due to regions of bistability. Hence, the model presents in one unique framework, two competing theories for the genesis of epileptiform activity. Similarities between model transitions and clinical data are presented and we argue that statistics obtained from, and a parameter estimation of this model may be a potential means of classifying and predicting the onset and offset of seizure activity.

Mean-field modeling of the basal ganglia-thalamocortical system. I Firing rates in healthy and parkinsonian states

Parkinsonism leads to various electrophysiological changes in the basal ganglia-thalamocortical system (BGTCS), often including elevated discharge rates of the subthalamic nucleus (STN) and the output nuclei, and reduced activity of the globus pallidus external (GPe) segment. These rate changes have been explained qualitatively in terms of the direct/indirect pathway model, involving projections of distinct striatal populations to the output nuclei and GPe. Although these populations partly overlap, evidence suggests dopamine depletion differentially affects cortico-striato-pallidal connection strengths to the two pallidal segments. Dopamine loss may also decrease the striatal signal-to-noise ratio, reducing both corticostriatal coupling and striatal firing thresholds. Additionally, nigrostriatal degeneration may cause secondary changes including weakened lateral inhibition in the GPe, and mesocortical dopamine loss may decrease intracortical excitation and especially inhibition. Here a mean-field model of the BGTCS is presented with structure and parameter estimates closely based on physiology and anatomy. Changes in model rates due to the possible effects of dopamine loss listed above are compared with experiment. Our results suggest that a stronger indirect pathway, possibly combined with a weakened direct pathway, is compatible with empirical evidence. However, altered corticostriatal connection strengths are probably not solely responsible for substantially increased STN activity often found. A lower STN firing threshold, weaker intracortical inhibition, and stronger striato-GPe inhibition help explain the relatively large increase in STN rate. Reduced GPe-GPe inhibition and a lower GPe firing threshold can account for the comparatively small decrease in GPe rate frequently observed. Changes in cortex, GPe, and STN help normalize the cortical rate, also in accord with experiments. The model integrates the basal ganglia into a unified framework along with an existing thalamocortical model that already accounts for a wide range of electrophysiological phenomena. A companion paper discusses the dynamics and oscillations of this combined system.

Cortical local and long-range synchronization interplay in human absence seizure initiation

Brain activity relies on transient, fluctuating interactions between segregated neuronal populations. Synchronization within a single and between distributed neuronal clusters reflects the dynamics of these cooperative patterns. Thus absence epilepsy can be used as a model for integrated, large-scale investigation of the emergence of pathological collective dynamics in the brain. Indeed, spike-wave discharges (SWD) of an absence seizure are thought to reflect abnormal cortical hypersynchronization. In this paper, we address two questions: how and where do SWD arise in the human brain? Therefore, we explored the spatio-temporal dynamics of interactions within and between widely distributed cortical sites using magneto-encephalographic recordings of spontaneous absence seizures. We then extracted, from their time-frequency analysis, local synchronization of cortical sources and long-range synchronization linking distant sites. Our analyses revealed a reproducible sequence of 1) long-range desynchronization, 2) increased local synchronization and 3) increased long-range synchronization. Although both local and long-range synchronization displayed different spatio-temporal profiles, their cortical projection within an initiation time window overlap and reveal a multifocal fronto-central network. These observations contradict the classical view of sudden generalized synchronous activities in absence epilepsy. Furthermore, they suggest that brain states transition may rely on multi-scale processes involving both local and distant interactions.

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 brain activation patterns for the default and cognitive states

We argue that spatial patterns of cortical activation observed with EEG, MEG and fMRI might arise from spontaneous self-organisation of interacting populations of excitatory and inhibitory neurons. We examine the dynamical behavior of a mean-field cortical model that includes chemical and electrical (gap-junction) synapses, focusing on two limiting cases: the "slow-soma" limit with slow voltage feedback from soma to dendrite, and the "fast-soma" limit in which the feedback action of soma voltage onto dendrite reversal potentials is instantaneous. For slow soma-dendrite feedback, we find a low-frequency (approximately 1 Hz) dynamic Hopf instability, and a stationary Turing instability that catalyzes formation of patterned distributions of cortical firing-rate activity with pattern wavelength approximately 2 cm. Turing instability can only be triggered when gap-junction diffusion between inhibitory neurons is strong, but patterning is destroyed if the tonic level of subcortical excitation is raised sufficiently. Interaction between the Hopf and Turing instabilities may describe the non-cognitive background or "default" state of the brain, as observed by BOLD imaging. In the fast-soma limit, the model predicts a high-frequency Hopf (approximately 35 Hz) instability, and a traveling-wave gamma-band instability that manifests as a 2-D standing-wave pattern oscillating in place at approximately 30 Hz. Small levels of inhibitory diffusion enhance and broaden the definition of the gamma antinodal regions by suppressing higher-frequency spatial modes, but gamma emergence is not contingent on the presence of inhibitory gap junctions; higher levels of diffusion suppress gamma activity. Fast-soma instabilities are enhanced by increased subcortical stimulation. Prompt soma-dendrite feedback may be an essential component of the genesis and large-scale cortical synchrony of gamma activity observed at the point of cognition.

Modeling brain dynamics using computational neurogenetic approach

The paper introduces a novel computational approach to brain dynamics modeling that integrates dynamic gene-protein regulatory networks with a neural network model. Interaction of genes and proteins in neurons affects the dynamics of the whole neural network. Through tuning the gene-protein interaction network and the initial gene/protein expression values, different states of the neural network dynamics can be achieved. A generic computational neurogenetic model is introduced that implements this approach. It is illustrated by means of a simple neurogenetic model of a spiking neural network of the generation of local field potential. Our approach allows for investigation of how deleted or mutated genes can alter the dynamics of a model neural network. We conclude with the proposal how to extend this approach to model cognitive neurodynamics.

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.

A study on the possible usefulness of detrended fluctuation analysis of the electroencephalogram background activity in Alzheimer's disease

We studied the EEG background activity of Alzheimer's disease (AD) patients with detrended fluctuation analysis (DFA). DFA provides an estimation of the scaling information and long-range correlations in time series. We recorded the EEG in 11 AD patients and 11 age-matched controls. Our results showed two scaling regions in all subjects' channels (for limited time scales from 0.01 to 0.04 s and from 0.08 to 0.43 s, respectively), with a clear bend when their corresponding slopes (alpha(1) and alpha(2)) were different. No significant differences between groups were found with alpha(1). However, alpha(2) values were significantly lower in control subjects at electrodes T5, T6, and O1 (p < 0.01, Student's t-test). These findings suggest that the scaling behavior of the EEG is sensitive to AD. Although alpha(2) values allowed us to separate AD patients and controls, accuracies were lower than with spectral analysis. However, a forward stepwise linear discriminant analysis with a leave-one-out cross-validation procedure showed that the combined use of DFA and spectral analysis could improve the diagnostic accuracy of each individual technique. Thus, although spectral analysis outperforms DFA, the combined use of both techniques may increase the insight into brain dysfunction in AD.

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.

Mean field model of acetylcholine mediated dynamics in the thalamocortical system

A recent continuum model of the large scale electrical activity of the thalamocortical system is generalized to include cholinergic modulation. The model is examined analytically and numerically to determine the effect of acetylcholine (ACh) on its steady states, linear stability, spectrum, and temporal responses. Changing the ACh concentration moves the system between zones of one, three, and five steady states, showing that neuromodulation of synaptic strength is a possible mechanism by which multiple steady states emerge in the brain. The lowest firing rate steady state is always stable, and subsequent fixed points alternate between stable and unstable. Increasing ACh concentration changes the form of the spectrum. Increasing the tonic level of ACh concentration increases the magnitudes of the N100 and P200 in the evoked response potential (ERP), without changing the timing of these peaks. Driving the system with a pulse of cholinergic activity results in a transient increase in the firing rate of cortical neurons that lasts over 10s. Step-like increases in cortical ACh concentration cause increases in the firing rate of cortical neurons, with rapid responses due to fast acting nicotinic receptors and slower responses due to muscarinic receptor suppression of intracortical connections.

Narrow-band oscillations in probabilistic cellular automata

Dynamical properties of neural populations are studied using probabilistic cellular automata. Previous work demonstrated the emergence of critical behavior as the function of system noise and density of long-range axonal connections. Finite-size scaling theory identified critical properties, which were consistent with properties of a weak Ising universality class. The present work extends the studies to neural populations with excitatory and inhibitory interactions. It is shown that the populations can exhibit narrow-band oscillations when confined to a range of inhibition levels, with clear boundaries marking the parameter region of prominent oscillations. Phase diagrams have been constructed to characterize unimodal, bimodal, and quadromodal oscillatory states. The significance of these findings is discussed in the context of large-scale narrow-band oscillations in neural tissues, as observed in electroencephalographic and magnetoencephalographic measurements.

Computer modelling of epilepsy

Epilepsy is a complex set of disorders that can involve many areas of the cortex, as well as underlying deep-brain systems. The myriad manifestations of seizures, which can be as varied as déjà vu and olfactory hallucination, can therefore give researchers insights into regional functions and relations. Epilepsy is also complex genetically and pathophysiologically: it involves microscopic (on the scale of ion channels and synaptic proteins), macroscopic (on the scale of brain trauma and rewiring) and intermediate changes in a complex interplay of causality. It has long been recognized that computer modelling will be required to disentangle causality, to better understand seizure spread and to understand and eventually predict treatment efficacy. Over the past few years, substantial progress has been made in modelling epilepsy at levels ranging from the molecular to the socioeconomic. We review these efforts and connect them to the medical goals of understanding and treating the disorder.

Computational models of epileptic activity: a bridge between observation and pathophysiological interpretation

Epilepsy is a neurological disorder characterized by the recurrence of seizures. It affects 50 million people worldwide. Although a considerable number of new antiepileptic drugs with reduced side effects and toxicity have been introduced since the 1950s, 30% of patients remain pharmacoresistant. Although epilepsy research is making progress, advances in understanding drug resistance have been hampered by the complexity of the underlying neuronal systems responsible for epileptic activity. In such systems where short- or long-term plasticity plays a role, pathophysiological alterations may take place at subcellular (i.e., membrane ion channels and neurotransmitter receptors), cellular (neurons), tissular (networks of neurons) and regional (networks of networks of neurons) scales. In such a context, the demand for integrative approaches is high and neurocomputational models become recognized tools for tackling the complexity of epileptic phenomena. The purpose of this report is to provide an overview on computational modeling as a way of structuring and interpreting multimodal data recorded from the epileptic brain. Some examples are briefly described, which illustrate how computational models closely related with either experimental or clinical data can markedly advance our understanding of essential issues in epilepsy such as the transition from background to seizure activity. A commentary is also made on the potential use of such models in the study of therapeutic strategies such as rational drug design or electrical stimulations.

Characteristics of temporal fluctuations in the hyperpolarized state of the cortical slow oscillation

We present evidence for the hypothesis that transitions between the low- and high-firing states of the cortical slow oscillation correspond to neuronal phase transitions. By analyzing intracellular recordings of the membrane potential during the cortical slow oscillation in rats, we quantify the temporal fluctuations in power and the frequency centroid of the power spectrum in the period of time before "down" to "up" transitions. By taking appropriate averages over such events, we present these statistics as a function of time before transition. The results demonstrate an increase in fluctuation power and time scale broadly consistent with the slowing of systems close to phase transitions. The analysis is complicated and limited by the difficulty in identifying when transitions begin, and removing dc trends in membrane potential.

A dynamic extraversion model. The brain's response to a single dose of a stimulant drug

The aim of this paper is to present a mathematical dynamic modelling of the effect a stimulant drug has on different people which, at the same time, can be a useful tool for future brain studies. To this end, a dynamic model of the evolution of extraversion (considering its tonic and phasic aspects) has been constructed taking into account the unique personality trait theory and the general modelling methodology. This model consists of a delayed differential equation which, on one hand, considers that the active stimulus, a consequence of a single intake, is not constant; on the other hand, it contemplates that the state variable representing the phasic extraversion also represents the brain activation. The derivative of this state variable is calculated as the sum of the homeostatic control flow, the excitatory effect flow and the inhibitor effect flow. The solutions of this equation relate the tonic activation of an individual (that characterizes his or her personality) with his or her phasic activation level, whose evolution over time describes the organism's response to a single drug intake. These solutions quantitatively reproduce the predictions of current personality theories and anticipate vulnerability to drug misuse and addiction development.

Controlling limit-cycle behaviors of brain activity

The limit cycles of brain activity are studied using a compact continuum model that reproduces the main features of electroencephalographic signals, including bifurcations of fixed points and limit cycles in seizures. Frequencies and amplitudes are predicted analytically and related to physiology. Gaussian stimuli yield two distinct evoked responses in the linearly stable zone, consistent with experiment. Limit cycles can be initiated or suppressed by control signals or stimuli.

Physiology-based modeling of cortical auditory evoked potentials

Evoked potentials are the transient electrical responses caused by changes in the brain following stimuli. This work uses a physiology-based continuum model of neuronal activity in the human brain to calculate theoretical cortical auditory evoked potentials (CAEPs) from the model's linearized response. These are fitted to experimental data, allowing the fitted parameters to be related to brain physiology. This approach yields excellent fits to CAEP data, which can then be compared to fits of EEG spectra. It is shown that the differences between resting eyes-open EEG and standard CAEPs can be explained by changes in the physiology of populations of neurons in corticothalamic pathways, with notable similarities to certain aspects of slow-wave sleep. This pilot study demonstrates the ability of our model-based fitting method to provide information on the underlying physiology of the brain that is not available using standard methods.

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.

Mean field model of acetylcholine mediated dynamics in the cerebral cortex

A recent continuum model of the large scale electrical activity of the cerebral cortex is generalized to include cholinergic modulation. In this model, dynamic modulation of synaptic strength acts over the time scales of nicotinic and muscarinic receptor action. The cortical model is analyzed to determine the effect of acetylcholine (ACh) on its steady states, linear stability, spectrum, and temporal responses to changes in subcortical input. ACh increases the firing rate in steady states of the system. Changing ACh concentration does not introduce oscillatory behavior into the system, but increases the overall spectral power. Model responses to pulses in subcortical input are affected by the tonic level of ACh concentration, with higher levels of ACh increasing the magnitude firing rate response of excitatory cortical neurons to pulses of subcortical input. Numerical simulations are used to explore the temporal dynamics of the model in response to changes in ACh concentration. Evidence is seen of a transition from a state in which intracortical inputs are emphasized to a state where thalamic afferents have enhanced influence. Perturbations in ACh concentration cause changes in the firing rate of cortical neurons, with rapid responses due to fast acting facilitatory effects of nicotinic receptors on subcortical afferents, and slower responses due to muscarinic suppression of intracortical connections. Together, these numerical simulations demonstrate that the actions of ACh could be a significant factor modulating early components of evoked response potentials.