Symbolic dynamics of event-related brain potentials

Abstract

We apply symbolic dynamics techniques such as word statistics and measures of complexity to nonstationary and noisy multivariate time series of electroencephalograms (EEG) in order to estimate event-related brain potentials (ERP). Their significance against surrogate data as well as between different experimental conditions is tested. These methods are validated by simulations using stochastic dynamical systems with time-dependent control parameters and compared with traditional ERP-analysis techniques. Continuous EEG data are cut into epochs according to stimuli events presented to the subjects. These ensembles of time series can be considered as ensembles of trajectories given by some dynamical systems. We employ a statistical mechanics approach motivated by the Frobenius-Perron equation and apply it to coarse-grained symbolic descriptions of the dynamics. We develop time-dependent measures of complexity founded on running cylinder sets and show that these quantities are able to distinguish simulated data obtained by different control parameters as well as experimental data between different experimental conditions. As a first finding, our approach restores the well-known ERP components and it reveals additionally qualitative changes in the EEG that cannot be detected by means of the traditional techniques. We criticize the prerequisites of the traditional approach to ERP analysis and propose to consider ERP instead in terms of dynamical system theory and information theory.

Finding needles in haystacks: Symbolic resonance analysis of event-related potentials unveils different processing demands

“…The common overlap of their time ranges across trials yields again a measure of the intertrial coherence. However, these coarse-graining techniques were lacking a theoretical foundation that has recently been provided by beim Graben et al [6] (cf. [3,5,16]) in the framework of symbolic dynamics [18,23].…”

“…2 After collecting all the sequences corresponding to one experimental condition in an ensemble, the number of b+Q or bÀQ symbols across all trials yields the polarity histogram [8]. beim Graben et al [6] have shown that these polarity histograms can be formally captured by probability measures of cylinder sets, which are subsets drawn from an ensemble of sequences having a common building block, which is called a word. beim Graben et al [6] have further argued that ERP components are characterized by large cylinder sets corresponding to a particular word.…”

“…beim Graben et al [6] have shown that these polarity histograms can be formally captured by probability measures of cylinder sets, which are subsets drawn from an ensemble of sequences having a common building block, which is called a word. beim Graben et al [6] have further argued that ERP components are characterized by large cylinder sets corresponding to a particular word. The likelihood of these cylinders is then assessed by generalized polarity histograms, the word statistics [5,16], and by information theoretic measures such as the Shannon entropy or measures of complexity [2,31], which serve as indicators of the intertrial coherence [24].…”

“…From the word statistics, we then derive the running cylinder entropies [6] that are presented in Table 6 for the example data.…”

“…This might not be the case for other symbolic encoding procedures, such as the median threshold encoding (cf. [6]). Table 6 Running cylinder entropies of the mean-field transformed example data In order to achieve the aim of the symbolic resonance analysis, we have to compute the estimator of the signal-tonoise ratio (SNR) [4] depending on the noise strength of the EEG.…”