Robert Savit

NON-LINEARITY IN INVASIVE EEG RECORDINGS FROM PATIENTS WITH TEMPORAL LOBE EPILEPSY

Electrographic recordings from depth and subdural electrodes, performed in two patients with seizures of mesial temporal origin, were analyzed for the presence of non-linearities in the signal. The correlation integral, a measure sensitive to a wide variety of non-linearities, was used for detection. Statistical significance was determined by comparison of the original signal to surrogate datasets. Statistically significant non-linearities were present in signals generated by the epileptogenic hippocampus and interictal spike foci in the temporal neocortex. Less prominent non-linearities were found in EEG signals generated by more normal areas of the brain. These results indicate that techniques developed for the study of non-linear systems can be used to characterize the epileptogenic regions of the brain during the interictal period and can elucidate the dynamical mechanisms of the epileptic transition.

Stationarity and nonstationarity in time series analysis

Abstract In this paper we introduce a new class of methods to test, model and describe nonstationary processes. To frame these methods, we generalize the dynamical description of autonomous systems to the case of nonautonomous systems. Of particular interest are systems for which the driving force is recurrent. For these systems we describe a method to find recurrences and to improve the statistics in reconstructing the tine series and, consequently, to improve the predictability. Another objective is a proper description of the nonstationarity. All these methods are applied to four examples.

Time series and dependent variables

We present a new method for analyzing time series which is designed to extract inherent deterministic dependencies in the series. The method is particularly suited to series with broad-band spectra such as chaotic series with or without noise. We derive quantities, ~j(e), based on conditional probabilities, whose magnitude, roughly speaking, is an indicator of the extent to which the kth element in the series is a deterministic function of the (k -j)th element to within a measurement uncertainty, e. We apply our method to a number of deterministic time series generated by chaotic processes such as the tent, logistic and H~non maps, as well as to sequences of quasi-random numbers. In all cases the 6j correctly indicate the expected dependencies. We also show that the ~j are robust to the addition of substantial noise in a deterministic process. In addition, we derive a predictability index which is a measure of the extent to which a time series is predictable given some tolerance, e. Finally, we discuss the behavior of the 6i as e approaches zero.

Time dependencies in the occurrences of epileptic seizures.

A new method of analysis, developed within the framework of nonlinear dynamics, is applied to patient recorded time series of the occurrence of epileptic seizures. These data exhibit broad band spectra and generally have no obvious structure. The goal is to detect hidden internal dependencies in the data without making any restrictive assumptions, such as linearity, about the structure of the underlying system. The basis of our approach is a conditional probabilistic analysis in a phase space reconstructed from the original data. The data, recorded from patients with intractable epilepsy over a period of 1-3 years, consist of the times of occurrences of hundreds of partial complex seizures. Although the epileptic events appear to occur independently, we show that the epileptic process is not consistent with the rules of a homogeneous Poisson process or generally with a random (IID) process. More specifically, our analysis reveals dependencies of the occurrence of seizures on the occurrence of preceding seizures. These dependencies can be detected in the interseizure interval data sets as well as in the rate of seizures per time period. We modeled patients inaccuracy in recording seizure events by the addition of uniform white noise and found that the detected dependencies are persistent after addition of noise with standard deviation as great as 1/3 of the standard deviation of the original data set. A linear autoregressive analysis fails to capture these dependencies or produces spurious ones in most of the cases.

Characterizing nonlinearity in invasive EEG recordings from temporal lobe epilepsy

Abstract Invasive electroencephalographic (EEG) recordings from depth and subdural electrodes, performed in eight patients with temporal lobe epilepsy, are analyzed using a variety of nonlinear techniques. A surrogate data technique is used to find strong evidence for nonlinearities in epileptogenic regions of the brain. Most of these nonlinearities are characterized as “spiking” by a wavelet analysis. A small fraction of the nonlinearities are characterized as “recurrent” by a nonlinear prediction algorithm. Recurrent activity is found to occur in spatio-temporal patterns related to the location of the epileptogenic focus. Residual delay maps, used to characterize “lag-one nonlinearity”, are remarkably stationary for a given electrode, and exhibit striking variations among electrodes. The clinical and theoretical implications of these results are discussed.

The structure of adaptive competition in minority games

In this paper we present results and analyses of a class of games in which heterogeneous agents are rewarded for being in a minority group. Each agent possesses a number of fixed strategies each of which are predictors of the next minority group. The strategies use a set of aggregate, publicly available information (reflecting the agents’ collective previous decisions) to make their predictions. An agent chooses which group to join at a given moment by using one of his strategies. These games are adaptive in that agents can choose, at different points of the game, to exercise different strategies in making their choice of which group to join. The games are not evolutionary in that the agents’ strategies are fixed at the beginning of the game. We find, rather generally, that such systems evidence a phase change from a maladaptive, informationally efficient phase in which the system performs poorly at generating resources, to an inefficient phase in which there is an emergent cooperation among the agents, and the system more effectively generates resources. The best emergent coordination is achieved in a transition region between these two phases. This transition occurs when the dimension of the strategy space is of the order of the number of agents playing the game. We present explanations for this general behavior, based in part on an information theoretic analysis of the system and its publicly available information. We also propose a mean-field-like model of the game which is most accurate in the maladaptive, efficient phase. In addition, we show that the best individual agent performance in the two different phases is achieved by sets of strategies with markedly different characteristics. We discuss implications of our results for various aspects of the study of complex adaptive systems.

Linear and Nonlinear Measures and Seizure Anticipation in Temporal Lobe Epilepsy

In a recent paper, we showed that the value of a nonlinear quantity computed from scalp electrode data was correlated with the time to a seizure in patients with temporal lobe epilepsy. In this paper we study the relationship between the linear and nonlinear content and analyses of the scalp data. We do this in two ways. First, using surrogate data methods, we show that there is important nonlinear structure in the scalp electrode data to which our methods are sensitive. Second, we study the behavior of some simple linear metrics on the same set of scalp data to see whether the nonlinear metrics contain additional information not carried by the linear measures. We find that, while the nonlinear measures are correlated with time to seizure, the linear measures are not, over the time scales we have defined. The linear and nonlinear measures are themselves apparently linearly correlated, but that correlation can be ascribed to the influence of a small set of outliers, associated with muscle artifact. A remaining, more subtle relation between the variance of the values of a nonlinear measure and the expectation value of a linear measure persists. Implications of our observations are discussed.

Nonstationarity in epileptic EEG and implications for neural dynamics

In this paper, we use a recently developed method to analyze the nonstationarity in time series from intracranial depth and subdural recordings of patients with temporal lobe epilepsy. We show that the nonstationarity in the signal can be accounted for by the variation of a single parameter. We then show that the various dominant nonlinear waveforms observed in different electrodes can be explained by a simple stochastic model in which the mesoscopic collection of neurons, whose potential the electrodes measure, can be on one of two states. The nonstationarity observed in our analysis is a consequence of a time-dependent transition probability between these two states. In general, this transition probability increases as a seizure is approached. The model that we propose incorporates this bistability. We find good agreement between real data and simulated data generated by our model. We understand that this mesoscopic bistability may be associated with the existence of excitation waves traversing the brain in these patients.

Evolution in minority games. (II). Games with variable strategy spaces

We continue our study of evolution in minority games by examining games in which agents with poorly performing strategies can trade in their strategies for new ones from a different strategy space. In the context of the games discussed in this paper, this means allowing for strategies that use information from different numbers of time lags, m. We find, in all the games we study, that after evolution, wealth per agent is high for agents with strategies drawn from small strategy spaces (small m), and low for agents with strategies drawn from large strategy spaces (large m). In the game played with N agents, wealth per agent as a function of m is very nearly a step function. The transition is at m=mt, where mt≈mc−1. Here mc is the critical value of m at which N agents playing the game with a fixed strategy space (fixed m) have the best emergent coordination and the best utilization of resources. We also find that overall system-wide utilization of resources is independent of N. Furthermore, although overall system-wide utilization of resources after evolution varies somewhat depending on some other aspects of the evolutionary dynamics, in the best cases, utilization of resources is on the order of the best results achieved in evolutionary games with fixed strategy spaces. Simple explanations are presented for some of our main results.

The minority game with variable payoffs

In the standard minority game, each agent in the minority group receives the same payoff regardless of the size of the minority group. Of great interest for real social and biological systems are cases in which the payoffs to members of the minority group depend on the size of the minority group. This latter includes the fixed sum game. We find, remarkably, that the phase structure and general scaling behavior of the standard minority game persists when the payoff function depends on the size of the minority group. There is still a phase transition at the same value of z, the ratio of the dimension of the strategy space to the number of agents playing the game. We explain the persistence of the phase structure and argue that it is due to the absence of temporal cooperation in the dynamics of the minority game. We also discuss the behavior of average agent wealth and the wealth distribution in these variable payoff games.