EEG spike and wave modelled by a stochastic limit cycle

Stochastic non-linear oscillator models of EEG: the Alzheimer's disease case

In this article, the Electroencephalography (EEG) signal of the human brain is modeled as the output of stochastic non-linear coupled oscillator networks. It is shown that EEG signals recorded under different brain states in healthy as well as Alzheimer's disease (AD) patients may be understood as distinct, statistically significant realizations of the model. EEG signals recorded during resting eyes-open (EO) and eyes-closed (EC) resting conditions in a pilot study with AD patients and age-matched healthy control subjects (CTL) are employed. An optimization scheme is then utilized to match the output of the stochastic Duffing—van der Pol double oscillator network with EEG signals recorded during each condition for AD and CTL subjects by selecting the model physical parameters and noise intensity. The selected signal characteristics are power spectral densities in major brain frequency bands Shannon and sample entropies. These measures allow matching of linear time varying frequency content as well as non-linear signal information content and complexity. The main finding of the work is that statistically significant unique models represent the EC and EO conditions for both CTL and AD subjects. However, it is also shown that the inclusion of sample entropy in the optimization process, to match the complexity of the EEG signal, enhances the stochastic non-linear oscillator model performance.

Stochastic coupled oscillator model of EEG for Alzheimer's disease

Coupled nonlinear oscillator models of EEG signals during resting eyes-closed and eyes-open conditions are presented based on Duffing-van der Pol oscillator dynamics. The frequency and information entropy contents of the output of the nonlinear model and the actual EEG signal is matched through an optimization algorithm. The framework is used to model and compare EEG signals recorded from Alzheimer's disease (AD) patients and age-matched healthy controls (CTL) subjects. The results show that 1) the generated model signal can capture the frequency and information entropy contents of the EEG signal with very similar power spectral distribution and non-periodic time history; 2) the EEG and the generated signal from the eyes-closed model are α band dominant for CTL subjects and θ band dominant for AD patients; and 3) statistically distinct models represent the EEG signals from AD patients and CTL subject during resting eyes-closed condition.

Effective connectivity: influence, causality and biophysical modeling

This is the final paper in a Comments and Controversies series dedicated to "The identification of interacting networks in the brain using fMRI: Model selection, causality and deconvolution". We argue that discovering effective connectivity depends critically on state-space models with biophysically informed observation and state equations. These models have to be endowed with priors on unknown parameters and afford checks for model Identifiability. We consider the similarities and differences among Dynamic Causal Modeling, Granger Causal Modeling and other approaches. We establish links between past and current statistical causal modeling, in terms of Bayesian dependency graphs and Wiener-Akaike-Granger-Schweder influence measures. We show that some of the challenges faced in this field have promising solutions and speculate on future developments.

Estimating short-run and long-run interaction mechanisms in interictal state

We address the issue of analyzing electroencephalogram (EEG) from seizure patients in order to test, model and determine the statistical properties that distinguish between EEG states (interictal, pre-ictal, ictal) by introducing a new class of time series analysis methods. In the present study: firstly, we employ statistical methods to determine the non-stationary behavior of focal interictal epileptiform series within very short time intervals; secondly, for such intervals that are deemed non-stationary we suggest the concept of Autoregressive Integrated Moving Average (ARIMA) process modelling, well known in time series analysis. We finally address the queries of causal relationships between epileptic states and between brain areas during epileptiform activity. We estimate the interaction between different EEG series (channels) in short time intervals by performing Granger-causality analysis and also estimate such interaction in long time intervals by employing Cointegration analysis, both analysis methods are well-known in econometrics. Here we find: first, that the causal relationship between neuronal assemblies can be identified according to the duration and the direction of their possible mutual influences; second, that although the estimated bidirectional causality in short time intervals yields that the neuronal ensembles positively affect each other, in long time intervals neither of them is affected (increasing amplitudes) from this relationship. Moreover, Cointegration analysis of the EEG series enables us to identify whether there is a causal link from the interictal state to ictal state.

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.

Phase response curves in the characterization of epileptiform activity

Coordinated cellular activity is a major characteristic of nervous system function. Coupled oscillator theory offers unique avenues to address cellular coordination phenomena. In this study, we focus on the characterization of the dynamics of epileptiform activity, based on some seizures that manifest themselves with very periodic rhythmic activity, termed absence seizures. Our approach consists in obtaining experimentally the phase response curves (PRCs) in the neocortex and thalamus, and incorporating these PRCs into a model of coupled oscillators. Phase preferences of the stationary states and their stability are determined, and these results from the model are compared with the experimental recordings, and interpreted in physiological terms.

Nonlinear dynamical analysis of EEG and MEG: review of an emerging field

Many complex and interesting phenomena in nature are due to nonlinear phenomena. The theory of nonlinear dynamical systems, also called 'chaos theory', has now progressed to a stage, where it becomes possible to study self-organization and pattern formation in the complex neuronal networks of the brain. One approach to nonlinear time series analysis consists of reconstructing, from time series of EEG or MEG, an attractor of the underlying dynamical system, and characterizing it in terms of its dimension (an estimate of the degrees of freedom of the system), or its Lyapunov exponents and entropy (reflecting unpredictability of the dynamics due to the sensitive dependence on initial conditions). More recently developed nonlinear measures characterize other features of local brain dynamics (forecasting, time asymmetry, determinism) or the nonlinear synchronization between recordings from different brain regions. Nonlinear time series has been applied to EEG and MEG of healthy subjects during no-task resting states, perceptual processing, performance of cognitive tasks and different sleep stages. Many pathologic states have been examined as well, ranging from toxic states, seizures, and psychiatric disorders to Alzheimer's, Parkinson's and Cre1utzfeldt-Jakob's disease. Interpretation of these results in terms of 'functional sources' and 'functional networks' allows the identification of three basic patterns of brain dynamics: (i) normal, ongoing dynamics during a no-task, resting state in healthy subjects; this state is characterized by a high dimensional complexity and a relatively low and fluctuating level of synchronization of the neuronal networks; (ii) hypersynchronous, highly nonlinear dynamics of epileptic seizures; (iii) dynamics of degenerative encephalopathies with an abnormally low level of between area synchronization. Only intermediate levels of rapidly fluctuating synchronization, possibly due to critical dynamics near a phase transition, are associated with normal information processing, whereas both hyper-as well as hyposynchronous states result in impaired information processing and disturbed consciousness.

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.

Impulse response function based on multivariate AR model can differentiate focal hemisphere in temporal lobe epilepsy

The purpose of this study is to propose and investigate a new approach for discriminating between focal and non-focal hemispheres in intractable temporal lobe epilepsy, based on applying multivariate time series analysis to the discharge-free background brain activity observed in nocturnal electrocorticogram (ECoG) time series. Five unilateral focal patients and one bilateral focal patient were studied. In order to detect the location of epileptic foci, linear multivariate autoregressive (MAR) models were fitted to the ECoG data; as a new approach for the purpose of summarizing these models in a single relevant parameter, the behavior of the corresponding impulse response functions was studied and described by an attenuation coefficient. In the majority of unilateral focal patients, the averaged attenuation coefficient was found to be almost always significantly larger in the focal hemisphere, as compared to the non-focal hemisphere. Also the amplitude of the fluctuations of the attenuation coefficient was significantly larger in the focal hemisphere. Moreover, in one patient showing a typical regular sleep cycle, the attenuation coefficient in the focal hemisphere tended to be larger during REM sleep and smaller during Non-REM sleep. In the bilateral focal patient, no statistically significant distinction between the hemispheres was found. This study provides encouraging results for new investigations of brain dynamics by multivariate parametric modeling. It opens up the possibility of relating diseases like epilepsy to the properties of inconspicuous background brain dynamics, without the need to record and analyze epileptic seizures or other evidently pathological waveforms.

Exploring the nonlinear dynamics of the brain

The growing need for a better understanding of large-scale brain dynamics has stimulated in the last decade the development of new and more advanced data analysis techniques. Progress in this domain has greatly benefited from developments in nonlinear time series analysis. This review gives a short overview of some of the nonlinear properties one may wish to infer from brain recordings and presents some examples and recent applications.

Toward a neurodynamical understanding of ictogenesis

Although considerable information on cellular and network mechanisms of epilepsy exists, it is still not understood why, how, and when the transition from interictal to ictal state takes place. The authors review their work on nonlinear EEG analysis and provide consistent evidences that dynamical changes in the neural activity allows the characterization of a preictal state several minutes before seizure onset. This new neurodynamical approach of ictogenesis opens new perspectives for studying the basic mechanisms in epilepsy as well as for possible therapeutic interventions.

Estimating Lyapunov exponents in biomedical time series

Among nonlinear dynamical invariants, determination of the largest Lyapunov exponent is well suited to positive identification of chaos in observed time series. When analyzing the dynamics of biomedical series, such as an electro-encephalogram (EEG), model-based methods should be used. Moreover, in the absence of any well founded theoretical model, and because of unexplained variability in the data, candidate models must provide for a stochastic component. Here we use nonlinear autoregressive stochastic modeling to estimate the dominant Lyapunov exponent in an EEG series and compute confidence intervals from surrogate data. The results are found to differ from those of approaches which aim at deleting noise prior to analysis.

Characterizing neurodynamic changes before seizures

The study of dynamic changes in neural activity preceding epileptic seizure allows the characterization of a preictal state several minutes before seizure onset. This opens up new perspectives for studying the mechanisms of epileptogenesis as well as for possible therapeutic interventions, which represent a major breakthrough. In this review the authors present and discuss the results from their group in this domain using nonlinear analysis of brain signals, as well as the limitations of this topic and current questions.

A comparison of non-linear non-parametric models for epilepsy data

EEG spike and wave (SW) activity has been described through a non-parametric stochastic model estimated by the Nadaraya-Watson (NW) method. In this paper the performance of the NW, the local linear polynomial regression and support vector machines (SVM) methods were compared. The noise-free realizations obtained by the NW and SVM methods reproduced SW better than as reported in previous works. The tuning parameters had to be estimated manually. Adding dynamical noise, only the NW method was capable of generating SW similar to training data. The standard deviation of the dynamical noise was estimated by means of the correlation dimension.

Testing non-linearity and directedness of interactions between neural groups in the macaque inferotemporal cortex

Information processing in the visual cortex depends on complex and context sensitive patterns of interactions between neuronal groups in many different cortical areas. Methods used to date for disentangling this functional connectivity presuppose either linearity or instantaneous interactions, assumptions that are not necessarily valid. In this paper a general framework that encompasses both linear and non-linear modelling of neurophysiological time series data by means of Local Linear Non-linear Autoregressive models (LLNAR) is described. Within this framework a new test for non-linearity of time series and for non-linearity of directedness of neural interactions based on LLNAR is presented. These tests assess the relative goodness of fit of linear versus non-linear models via the bootstrap technique. Additionally, a generalised definition of Granger causality is presented based on LLNAR that is valid for both linear and non-linear systems. Finally, the use of LLNAR for measuring non-linearity and directional influences is illustrated using artificial data, reference data as well as local field potentials (LFPs) from macaque area TE. LFP data is well described by the linear variant of LLNAR. Models of this sort, including lagged values of the preceding 25 to 60 ms, revealed the existence of both uni- and bi-directional influences between recording sites.