Prediction of electroencephalographic spectra from neurophysiology

Waves, bumps, and patterns in neural field theories

Neural field models of firing rate activity have had a major impact in helping to develop an understanding of the dynamics seen in brain slice preparations. These models typically take the form of integro-differential equations. Their non-local nature has led to the development of a set of analytical and numerical tools for the study of waves, bumps and patterns, based around natural extensions of those used for local differential equation models. In this paper we present a review of such techniques and show how recent advances have opened the way for future studies of neural fields in both one and two dimensions that can incorporate realistic forms of axo-dendritic interactions and the slow intrinsic currents that underlie bursting behaviour in single neurons.

Propagator theory of brain dynamics

A physiologically based continuum model of brain dynamics is extended to incorporate arbitrary numbers of structures and neural populations, multiple outgoing fields of activity from a single population of neurons to various targets, improved treatment of converging or diverging projections and mesoscopic structure, and generalized connections to quantities observable via electroencephalography and other methods. The results are applied to study the corticothalamic system, predicting an intracortical resonance that leads to enhancements of electroencephalographic activity in the gamma (>30 Hz) range. This resonance involves feedback loops incorporating slow, short-range inhibitory fibers.

Dynamics of a neural system with a multiscale architecture

The architecture of the brain is characterized by a modular organization repeated across a hierarchy of spatial scales-neurons, minicolumns, cortical columns, functional brain regions, and so on. It is important to consider that the processes governing neural dynamics at any given scale are not only determined by the behaviour of other neural structures at that scale, but also by the emergent behaviour of smaller scales, and the constraining influence of activity at larger scales. In this paper, we introduce a theoretical framework for neural systems in which the dynamics are nested within a multiscale architecture. In essence, the dynamics at each scale are determined by a coupled ensemble of nonlinear oscillators, which embody the principle scale-specific neurobiological processes. The dynamics at larger scales are 'slaved' to the emergent behaviour of smaller scales through a coupling function that depends on a multiscale wavelet decomposition. The approach is first explicated mathematically. Numerical examples are then given to illustrate phenomena such as between-scale bifurcations, and how synchronization in small-scale structures influences the dynamics in larger structures in an intuitive manner that cannot be captured by existing modelling approaches. A framework for relating the dynamical behaviour of the system to measured observables is presented and further extensions to capture wave phenomena and mode coupling are suggested.

Stimulant drug action in attention deficit hyperactivity disorder (ADHD): inference of neurophysiological mechanisms via quantitative modelling

OBJECTIVE

To infer the neural mechanisms underlying tonic transitions in the electroencephalogram (EEG) in 11 adolescents diagnosed with attention deficit hyperactivity disorder (ADHD) before and after treatment with stimulant medication.

METHODS

A biophysical model was used to analyse electroencephalographic (EEG) measures of tonic brain activity at multiple scalp sites before and after treatment with medication.

RESULTS

It was observed that stimulants had the affect of significantly reducing the parameter controlling activation in the intrathalamic pathway involving the thalamic reticular nucleus (TRN) and the parameter controlling excitatory cortical activity. The effect of stimulant medication was also found to be preferentially localized within subcortical nuclei projecting towards frontal and central scalp sites.

CONCLUSIONS

It is suggested that the action of stimulant medication occurs via suppression of the locus coeruleus, which in turn reduces stimulation of the TRN, and improves cortical arousal. The effects localized to frontal and central sites are consistent with the occurrence of frontal delta-theta EEG abnormalities in ADHD, and existing theories of hypoarousal.

SIGNIFICANCE

To our knowledge, this is the first study where a detailed biophysical model of the brain has been used to estimate changes in neurophysiological parameters underlying the effects of stimulant medication in ADHD.

On the role of general system theory for functional neuroimaging

One of the most important goals of neuroscience is to establish precise structure-function relationships in the brain. Since the 19th century, a major scientific endeavour has been to associate structurally distinct cortical regions with specific cognitive functions. This was traditionally accomplished by correlating microstructurally defined areas with lesion sites found in patients with specific neuropsychological symptoms. Modern neuroimaging techniques with high spatial resolution have promised an alternative approach, enabling non-invasive measurements of regionally specific changes of brain activity that are correlated with certain components of a cognitive process. Reviewing classic approaches towards brain structure-function relationships that are based on correlational approaches, this article argues that these approaches are not sufficient to provide an understanding of the operational principles of a dynamic system such as the brain but must be complemented by models based on general system theory. These models reflect the connectional structure of the system under investigation and emphasize context-dependent couplings between the system elements in terms of effective connectivity. The usefulness of system models whose parameters are fitted to measured functional imaging data for testing hypotheses about structure-function relationships in the brain and their potential for clinical applications is demonstrated by several empirical examples.

Analysis of the electroencephalographic activity associated with thalamic tumors

A physiologically based model of corticothalamic dynamics is used to investigate the electroencephalographic (EEG) activity associated with tumors of the thalamus. Tumor activity is modeled by introducing localized two-dimensional spatial non-uniformities into the model parameters, and calculating the resulting activity via the coupling of spatial eigenmodes. The model is able to reproduce various qualitative features typical of waking eyes-closed EEGs in the presence of a thalamic tumor, such as the appearance of abnormal peaks at theta ( approximately 3Hz) and spindle ( approximately 12Hz) frequencies, the attenuation of normal eyes-closed background rhythms, and the onset of epileptic activity, as well as the relatively normal EEGs often observed. The results indicate that the abnormal activity at theta and spindle frequencies arises when a small portion of the brain is forced into an over-inhibited state due to the tumor, in which there is an increase in the firing of (inhibitory) thalamic reticular neurons. The effect is heightened when there is a concurrent decrease in the firing of (excitatory) thalamic relay neurons, which are in any case inhibited by the reticular ones. This is likely due to a decrease in the responsiveness of the peritumoral region to cholinergic inputs from the brainstem, and a corresponding depolarization of thalamic reticular neurons, and hyperpolarization of thalamic relay neurons, similar to the mechanism active during slow-wave sleep. The results indicate that disruption of normal thalamic activity is essential to generate these spectral peaks. Furthermore, the present work indicates that high-voltage and epileptiform EEGs are caused by a tumor-induced local over-excitation of the thalamus, which propagates to the cortex. Experimental findings relating to local over-inhibition and over-excitation are discussed. It is also confirmed that increasing the size of the tumor leads to greater abnormalities in the observable EEG. The usefulness of EEG for localizing the tumor is investigated.

Unifying and interpreting the spectral wavenumber content of EEGs, ECoGs, and ERPs

A biological model of corticothalamic dynamics is used to investigate the spatial power spectrum (wavenumber spectrum) of electrical activity in the brain. The model provides a single framework for unifying different aspects of activity. Comparisons of the predicted spectra with published electrocorticographic, electroencephalographic, and evoked response potential data enable physiology and anatomy to be inferred, producing results which are complementary to those obtained from comparisons in the frequency domain; the inferred quantities are consistent with, and complementary to, direct physiological and anatomical measurements. We also use the model to quantify the interdependence of the wavenumber and frequency domains, and deduce that further experiments that cover large wavenumber and frequency ranges simultaneously would greatly increase our knowledge of brain function. We conclude that both the frequency and wavenumber domains should be studied in order to build the fullest picture of brain dynamics: the two domains are both complementary and interdependent.

Estimation of multiscale neurophysiologic parameters by electroencephalographic means

It is shown that new model-based electroencephalographic (EEG) methods can quantify neurophysiologic parameters that underlie EEG generation in ways that are complementary to and consistent with standard physiologic techniques. This is done by isolating parameter ranges that give good matches between model predictions and a variety of experimental EEG-related phenomena simultaneously. Resulting constraints range from the submicrometer synaptic level to length scales of tens of centimeters, and from timescales of around 1 ms to 1 s or more, and are found to be consistent with independent physiologic and anatomic measures. In the process, a new method of obtaining model parameters from the data is developed, including a Monte Carlo implementation for use when not all input data are available. Overall, the approaches used are complementary to other methods, constraining allowable parameter ranges in different ways and leading to much tighter constraints overall. EEG methods often provide the most restrictive individual constraints. This approach opens a new, noninvasive window on quantitative brain analysis, with the ability to monitor temporal changes, and the potential to map spatial variations. Unlike traditional phenomenologic quantitative EEG measures, the methods proposed here are based explicitly on physiology and anatomy.

"Dynamic" connectivity in neural systems: theoretical and empirical considerations

The study of functional interdependences between brain regions is a rapidly growing focus of neuroscience research. This endeavor has been greatly facilitated by the appearance of a number of innovative methodologies for the examination of neurophysiological and neuroimaging data. The aim of this article is to present an overview of dynamical measures of interdependence and contrast these with statistical measures that have been more widely employed. We first review the motivation, conceptual basis, and experimental approach of dynamical measures of interdependence and their application to the study of neural systems. A consideration of boot-strap "surrogate data" techniques, which facilitate hypothesis testing of dynamical measures, is then used to clarify the difference between dynamical and statistical measures of interdependence. An overview of some of the most active research areas such as the study of the "synchronization manifold," dynamical interdependence in neurophysiology data and the putative role of nonlinear desynchronization is then given. We conclude by suggesting that techniques based on dynamical interdependence--or "dynamical connectivity"--show significant potential for extracting meaningful information from functional neuroimaging data.

A stochastic limit cycle oscillator model of the EEG

We present an empirical model of the electroencephalogram (EEG) signal based on the construction of a stochastic limit cycle oscillator using Ito calculus. This formulation, where the noise influences actually interact with the dynamics, is substantially different from the usual definition of measurement noise. Analysis of model data is compared with actual EEG data using both traditional methods and modern techniques from nonlinear time series analysis. The model demonstrates visually displayed patterns and statistics that are similar to actual EEG data. In addition, the nonlinear mechanisms underlying the dynamics of the model do not manifest themselves in nonlinear time series analysis, paralleling the situation with real, non-pathological EEG data. This modeling exercise suggests that the EEG is optimally described by stochastic limit cycle behavior.

Spatially uniform and nonuniform analyses of electroencephalographic dynamics,with application to the topography of the alpha rhythm

Corticothalamic dynamics are investigated using a model in which spatial nonuniformities are incorporated via the coupling of spatial eigenmodes. Comparison of spectra generated using the nonuniform analysis with those generated using a uniform one demonstrates that, for most frequencies, local activity is only weakly dependent on activity elsewhere in the cortex; however, dispersion of low-wave-number activity ensures that distant dynamics influence local dynamics at low frequencies (below approximately 2 Hz ), and at the alpha frequency (approximately 10 Hz ), where propagating signals are inherently weakly damped, and wavelengths are large. When certain model parameters have similar spatial profiles, as is expected from physiology, the low-frequency discrepancies tend to cancel, and the uniform analysis with local parameter values is an adequate approximation to the full nonuniform one across the whole spectrum, at least for large-scale nonuniformities. After comparing the uniform and nonuniform analyses, we consider one possible application of the nonuniform analysis: studying the phenomenon of occipital alpha dominance, whereby the alpha frequency and power are greater at the back of the head (occipitally) than at the front. In order to infer realistic nonuniformities in the model parameters, the uniform version of the model is first fitted to data recorded from 98 normal subjects in a waking, eyes-closed state. This yields a set of parameters at each of five electrode sites along the midline. The inferred parameter nonuniformities are consistent with anatomical and physiological constraints. Introducing these spatial profiles into the full nonuniform model then quantitatively reproduces observed site-dependent variations in the alpha power and frequency. The results confirm that the frequency shift is mainly due to a decrease in the corticothalamic propagation delay, but indicate that the delay nonuniformity cannot account for the observed occipital increase in alpha power; the occipital alpha dominance is due to decreased cortical gains and increased thalamic gains in occipital regions compared to frontal ones.

A novel method for the topographic analysis of neural activity reveals formation and dissolution of 'Dynamic Cell Assemblies'

The study of synchronous oscillations in neural systems is a very active area of research. However, cognitive function may depend more crucially upon a dynamic alternation between synchronous and desynchronous activity rather than synchronous behaviour per se. The principle aim of this study is to develop and validate a novel method of quantifying this complex process. The method permits a direct mapping of phase synchronous dynamics and desynchronizing bursts in the spatial and temporal domains. Two data sets are analyzed: Numeric data from a model of a sparsely coupled neural cell assembly and experimental data consisting of scalp-recorded EEG from 40 human subjects. In the numeric data, the approach enables the demonstration of complex relationships between cluster size and temporal duration that cannot be detected with other methods. Dynamic patterns of phase-clustering and desynchronization are also demonstrated in the experimental data. It is further shown that in a significant proportion of the recordings, the pattern of dynamics exhibits nonlinear structure. We argue that this procedure provides a 'natural partitioning' of ongoing brain dynamics into topographically distinct synchronous epochs which may be integral to the brain's adaptive function. In particular, the character of transitions between consecutive synchronous epochs may reflect important aspects of information processing and cognitive flexibility.

A neural mass model for MEG/EEG: coupling and neuronal dynamics

Although MEG/EEG signals are highly variable, systematic changes in distinct frequency bands are commonly encountered. These frequency-specific changes represent robust neural correlates of cognitive or perceptual processes (for example, alpha rhythms emerge on closing the eyes). However, their functional significance remains a matter of debate. Some of the mechanisms that generate these signals are known at the cellular level and rest on a balance of excitatory and inhibitory interactions within and between populations of neurons. The kinetics of the ensuing population dynamics determine the frequency of oscillations. In this work we extended the classical nonlinear lumped-parameter model of alpha rhythms, initially developed by Lopes da Silva and colleagues [Kybernetik 15 (1974) 27], to generate more complex dynamics. We show that the whole spectrum of MEG/EEG signals can be reproduced within the oscillatory regime of this model by simply changing the population kinetics. We used the model to examine the influence of coupling strength and propagation delay on the rhythms generated by coupled cortical areas. The main findings were that (1) coupling induces phase-locked activity, with a phase shift of 0 or pi when the coupling is bidirectional, and (2) both coupling and propagation delay are critical determinants of the MEG/EEG spectrum. In forthcoming articles, we will use this model to (1) estimate how neuronal interactions are expressed in MEG/EEG oscillations and establish the construct validity of various indices of nonlinear coupling, and (2) generate event-related transients to derive physiologically informed basis functions for statistical modelling of average evoked responses.

A disturbance of nonlinear interdependence in scalp EEG of subjects with first episode schizophrenia

It has been proposed that schizophrenia arises through a disturbance of coupling between large-scale cortical systems. This "disconnection hypothesis" is tested by applying a measure of dynamical interdependence to scalp EEG data. EEG data were collected from 40 subjects with a first episode of schizophrenia and 40 matched healthy controls. An algorithm for the detection of dynamical interdependence was applied to six pairs of bipolar electrodes in each subject. The topographic organization of the interdependence was calculated and served as the principle measure of cortical integration. The rate of occurrence of dynamical interdependence did not statistically differ between subject groups at any of the sites. However, the topography across the scalp was significantly different between the two groups. Specifically, nonlinear interdependence tended to occur in larger concurrent "clusters" across the scalp in schizophrenia than in the healthy subjects. This disturbance was reflected most strongly in left intrahemispheric coupling and did not differ significantly according to symptomatology. Medication dose and subject arousal were not observed to be confounding factors. The study of dynamical interdependence in scalp EEG data does not support a straightforward interpretation of the disconnection hypothesis-that there is a decrease in the strength of functional coupling between adjacent cortical regions. Rather, it suggests a dysregulation in the organization of dynamical interactions across supraregional brain systems.

Drug-induced modification of the system properties associated with spontaneous human electroencephalographic activity

The benzodiazepine (BZ) class of minor tranquilizers are important modulators of the gamma-amino butyric acid (GABA(A))/BZ receptor complex that are well known to affect the spectral properties of spontaneous electroencephalographic activity. While it is experimentally well established that the BZs reduce total alpha band (8-13 Hz) power and increase total beta band (13-30 Hz) power, it is unclear what the physiological basis for this effect is. Based on a detailed theory of cortical electrorhythmogenesis it is conjectured that such an effect is explicable in terms of the modulation of GABAergic neurotransmission within locally connected populations of excitatory and inhibitory cortical neurons. Motivated by this theory, fixed order autoregressive moving average (ARMA) models were fitted to spontaneous eyes-closed electroencephalograms recorded from subjects before and approximately 2 h after the oral administration of a single 1 mg dose of the BZ alprazolam. Subsequent pole-zero analysis revealed that BZs significantly transform the dominant system pole such that its frequency and damping increase. Comparisons of ARMA derived power spectra with fast Fourier transform derived spectra indicate an enhanced ability to identify benzodiazepine induced electroencephalographic changes. This experimental result is in accord with the theoretical predictions implying that alprazolam enhances inhibition acting on inhibitory neurons more than inhibition acting on excitatory neurons. Further such a result is consistent with reported cortical neuronal distributions of the various GABA(A) receptor pharmacological subtypes. Therefore physiologically specified fixed order ARMA modeling is expected to become an important tool for the systematic investigation and modeling of a wide range of cortically acting compounds.

Nonuniform corticothalamic continuum model of electroencephalographic spectra with application to split-alpha peaks

Recent theoretical work has successfully predicted electroencephalographic spectra from physiology using a model corticothalamic system with spatially uniform parameters. The present work incorporates parameter nonuniformities into this model via the coupling they induce between spatial eigenmodes. Splitting of the spectral alpha peak, an effect seen in a small percentage of the normal population, is investigated as an illustrative special case. It is confirmed that weak splitting can arise from mode structure if the peak is sufficiently sharp, even for uniform parameters. However, it is further demonstrated that greater splitting can result from nonuniformities, and it is argued that this mechanism for split alpha is better able to account quantitatively for this effect than previously suggested alternatives of pacemakers or purely cortical resonances. On introducing nonuniformities in corticothalamic loop time delays, we find that the alpha frequency also varies as one moves from the front to the back of the head, in accord with observations, and that analogous (but less distinct) variations are seen in the beta peak. Analysis shows realistic variations of around +/-10 ms relative to the mean loop delay of approximately 80 ms can account for observed splittings of about 1 Hz. It is also suggested that subjects who display clear alpha splitting form the tail of a distribution of magnitude of cortical inhomogeneity, rather than a separate population.

Simulated electrocortical activity at microscopic, mesoscopic, and global scales

Simulation of electrocortical activity requires (a) determination of the most crucial features to be modelled, (b) specification of state equations with parameters that can be determined against independent measurements, and (c) explanation of electrical events in the brain at several scales. We report our attempts to address these problems, and show that mutually consistent explanations, and simulation of experimental data can be achieved for cortical gamma activity, synchronous oscillation, and the main features of the EEG power spectrum including the cerebral rhythms and evoked potentials. These simulations include consideration of dendritic and synaptic dynamics, AMPA, NMDA, and GABA receptors, and intracortical and cortical/subcortical interactions. We speculate on the way in which Hebbian learning and intrinsic reinforcement processes might complement the brain dynamics thus explained, to produce elementary cognitive operations.

Integrative neuroscience

A fundamental impediment to an "Integrative Neuroscience" is the sense that scientists building models at one particular scale often see that scale as the epicentre of all brain function. This fragmentation has begun to change in a very distinctive way. Multidisciplinary efforts have provided the impetus to break down the boundaries and encourage a freer exchange of information across disciplines and scales. Despite huge deficits of knowledge, sufficient facts about the brain already exist, for an Integrative Neuroscience to begin to lift us clear of the jungle of detail, and shed light upon the workings of the brain as a system. Integrations of brain theory can be tested using judicious paradigm designs and measurement of temporospatial activity reflected in brain imaging technologies. However, to test realistically these new hypotheses requires consistent findings of the normative variability in very large numbers of control subjects, coupled with high sensitivity and specificity of findings in psychiatric disorders. Most importantly, these findings need to be analyzed and modeled with respect to the fundamental mechanisms underlying these measures. Without this convergence of theory, databases, and methodology (including across scale physiologically realistic numerical models), the clinical utility of brain imaging technologies in psychiatry will be significantly impeded. The examples provided in this paper of integration of theory, temporospatial integration of neuroimaging technologies, and a numerical simulation of brain function, bear testimony to the ongoing conversion of an Integrative Neuroscience from an exemplar status into reality.

Wave-number spectrum of electrocorticographic signals

A physiologically based continuum model of corticothalamic electrodynamics is generalized and used to derive the theoretical form of the electrocorticographic (ECoG) wave-number spectrum. A one-dimensional projection of the spectrum is derived, as is the azimuthally averaged two-dimensional spectrum for isotropic and anisotropic cortices. The predicted spectra are found to consist of a low-k plateau followed by three regions of power-law decrease, which result from filtering of the electrical activity through physical structures at different scales in the cortex. The magnitude of the maximum theoretical power-law exponent is larger for the two-dimensional (2D) spectrum than for its 1D counterpart. The predicted spectra agree well with experimental data obtained from 1D and 2D recording arrays on the cortical surface, enabling the structures in the brain that are important in determining spatial cortical dynamics to be identified. The cortical dispersion relation predicted by our model is also investigated, providing insight into the relationships between temporal and spatial brain dynamics.

Interpretation of scaling properties of electroencephalographic fluctuations via spectral analysis and underlying physiology

Detrended fluctuation analysis has recently demonstrated the existence of two approximate temporal scaling regimes in locally detrended human electroencephalographic (EEG) fluctuations, and has suggested a connection between the location of the breakpoint between regimes and the alpha resonance near 10 Hz. It is shown here that these scalings can be explained in terms of the filtering of the underlying power spectrum implied by the detrending process. Using a recent physiologically based model of EEG generation, the main features of the scalings, and deviations from them, are related to the underlying physiology of dendritic propagation and muscle electrical activity, and it is concluded that the effects of such physiological features are usually clearer in spectra.

Wave-number spectrum of electroencephalographic signals

A recently developed, physiologically based continuum model of corticothalamic electrodynamics is used to derive the theoretical form of the electroencephalographic wave-number spectrum and its projection onto a one-dimensional recording array. The projected spectrum is found to consist of a plateau followed by regions of power-law decrease with various exponents, which are dependent on both model parameters and temporal frequency. The theoretical spectrum is compared with experimental results obtained in other studies, showing good agreement. The model provides a framework for understanding the nature of the spatial power spectrum by linking the underlying physiology with the large-scale dynamics of the brain.

Topographic organization of nonlinear interdependence in multichannel human EEG

This paper investigates the spatial organization of nonlinear interactions between different brain regions in healthy human subjects. This is achieved by studying the topography of nonlinear interdependence in multichannel EEG data, acquired from 40 healthy human subjects at rest. An algorithm for the detection and quantification of nonlinear interdependence is applied to four pairs of bipolar electrode derivations to detect posterior and anterior interhemispheric and left and right intrahemispheric interdependences. Multivariate surrogate data sets are constructed to control for linear coherence and finite sample size. Nonlinear interdependence is shown to occur in a small but statistically robust number of epochs. The occurrence of nonlinear interdependence in any region is correlated with the concurrent presence of nonlinear interdependence in other regions at high levels of significance. The strength, direction and topography of the interdependences are also correlated. For example, posterior interhemispheric interdependence from right-to-left is strongly correlated with right intrahemispheric interdependence from back-to-front. There is a subtle change in these correlations when subjects open their eyes. These results suggest that nonlinear interdependence in the human brain has a specific topographic organization which reflects simple cognitive changes. It sometimes occurs as an isolated phenomenon between two brain regions, but often involves concurrent interdependences between multiple brain regions.

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

Links between electroencephalograms (EEGs) and underlying aspects of neurophysiology and anatomy are poorly understood. Here a nonlinear continuum model of large-scale brain electrical activity is used to analyze arousal states and their stability and nonlinear dynamics for physiologically realistic parameters. A simple ordered arousal sequence in a reduced parameter space is inferred and found to be consistent with experimentally determined parameters of waking states. Instabilities arise at spectral peaks of the major clinically observed EEG rhythms-mainly slow wave, delta, theta, alpha, and sleep spindle-with each instability zone lying near its most common experimental precursor arousal states in the reduced space. Theta, alpha, and spindle instabilities evolve toward low-dimensional nonlinear limit cycles that correspond closely to EEGs of petit mal seizures for theta instability, and grand mal seizures for the other types. Nonlinear stimulus-induced entrainment and seizures are also seen, EEG spectra and potentials evoked by stimuli are reproduced, and numerous other points of experimental agreement are found. Inverse modeling enables physiological parameters underlying observed EEGs to be determined by a new, noninvasive route. This model thus provides a single, powerful framework for quantitative understanding of a wide variety of brain phenomena.

Nonlinear phase desynchronization in human electroencephalographic data

Ensembles of coupled nonlinear systems represent natural candidates for the modeling of brain dynamics. The objective of this study is to examine the complex signal produced by coupled chaotic attractors, to discuss their potential relevance to distributed processes in the brain, and to illustrate a method of detecting their contribution to human EEG morphology. Two measures of quantifying the behavior of coupled nonlinear systems are presented: a measure of phase synchrony and a novel measure of intermittent phase desynchronization. These are used to quantify the behavior of numerical examples of coupled chaotic attractors. Experimental evidence of their contribution to the morphology of the human alpha rhythm is then illustrated in a study of EEG recordings from 40 healthy human subjects. Amplitude-adjusted phase-randomized surrogate data is used to test the null hypothesis that the observed patterns of phase coherence can be described by purely linear methods. Statistical analysis reveals that this null hypothesis can be robustly rejected in a small number (approximately 4%) of EEG epochs. These findings are discussed with reference to the adaptive function and complex dynamics of the brain.

Toward an integrated continuum model of cerebral dynamics: the cerebral rhythms, synchronous oscillation and cortical stability

Continuum models of cerebral cortex with parameters derived from physiological data, provide explanations of the cerebral rhythms, synchronous oscillation, and autonomous cortical activity in the gamma frequency range, and suggest possible mechanisms for dynamic self-organization in the brain. Dispersion relations and derivations of power spectral response for the models, show that a low frequency resonant mode and associated travelling wave solutions of the models' equations of state can account for the predominant 1/f spectral content of the electroencephalogram (EEG). Large scale activity in the alpha, beta, and gamma bands, is accounted for by thalamocortical interaction, under regulation by diffuse cortical excitation. System impulse responses can be used to model Event-Related Potentials. Further classes of local resonance may be generated by rapid negative feedbacks at active synapses. Activity in the gamma band around 40 Hz, associated with large amplitude oscillations of pulse density, appears at higher levels of cortical activation, and is unstable unless compensated by synaptic feedbacks. Control of cortical stability by synaptic feedbacks offers a partial account of the regulation of autonomous activity within the cortex. Synchronous oscillation occurs between concurrently excited cortical sites, and can be explained by analysis of wave motion radiating from each of the co-active sites. These models are suitable for the introduction of learning rules-most notably the coherent infomax rule.