Boris P. Bezruchko

The causal relationship between subcortical local field potential oscillations and Parkinsonian resting tremor

To study the dynamical mechanism which generates Parkinsonian resting tremor, we apply coupling directionality analysis to local field potentials (LFP) and accelerometer signals recorded in an ensemble of 48 tremor epochs in four Parkinsonian patients with depth electrodes implanted in the ventro-intermediate nucleus of the thalamus (VIM) or the subthalmic nucleus (STN). Apart from the traditional linear Granger causality method we use two nonlinear techniques: phase dynamics modelling and nonlinear Granger causality. We detect a bidirectional coupling between the subcortical (VIM or STN) oscillation and the tremor, in the theta range (around 5 Hz) as well as broadband (>2 Hz). In particular, we show that the theta band LFP oscillations definitely play an efferent role in tremor generation, while beta band LFP oscillations might additionally contribute. The brain-->tremor driving is a complex, nonlinear mechanism, which is reliably detected with the two nonlinear techniques only. In contrast, the tremor-->brain driving is detected with any of the techniques including the linear one, though the latter is less sensitive. The phase dynamics modelling (applied to theta band oscillations) consistently reveals a long delay in the order of 1-2 mean tremor periods for the brain-->tremor driving and a small delay, compatible with the neural transmission time, for the proprioceptive feedback. Granger causality estimation (applied to broadband signals) does not provide reliable estimates of the delay times, but is even more sensitive to detect the brain-->tremor influence than the phase dynamics modelling.

Granger causality: Cortico-thalamic interdependencies during absence seizures in WAG/Rij rats

Linear Granger causality was used to identify the coupling strength and directionality of information transport between frontal cortex and thalamus during spontaneous absence seizures in a genetic model, the WAG/Rij rats. Electroencephalograms were recorded at the cortical surface and from the specific thalamus. Granger coupling strength was measured before, during and after the occurrence of spike-wave discharges (SWD). Before the onset of SWD, coupling strength was low, but associations from thalamus-to-cortex were stronger than vice versa. The onset of SWD was associated with a rapid and significant increase of coupling strength in both directions. There were no changes in Granger causalities before the onset of SWD. The strength of thalamus-to-cortex coupling remained constantly high during the seizures. The strength of cortex-to-thalamus coupling gradually diminished shortly after the onset of SWD and returned to the pre-SWD level when SWD stopped. In contrast, the strength of thalamus-to-cortex coupling remained elevated even after cessation of SWD. The strong and sustained influence of thalamus-to-cortex may facilitate propagation and maintenance of seizure activity, while rapid reduction of cortex-to-thalamus coupling strength may prompt the cessation of SWD. However, the linear estimation of Granger coupling strength does not seem to be sufficient for predicting episodes with absence epilepsy.

The generation of Parkinsonian tremor as revealed by directional coupling analysis

To reveal the dynamic mechanism underlying Parkinsonian resting tremor, we applied a phase dynamics modelling technique to local field potentials and accelerometer signals recorded in three Parkinsonian patients with implanted depth electrodes. We detect a bidirectional coupling between the subcortical oscillation and the tremor. The tremor → brain driving is a linear effect with a small delay corresponding to the neural transmission time. In contrast, the brain → tremor driving is a nonlinear effect with a long delay in the order of 1–2 mean tremor periods. Our results are well reproduced for an ensemble of 41 tremor epochs in three Parkinsonian patients and confirmed by surrogate data tests and model simulations. The uncovered mechanism of tremor generation suggests to specifically counteract tremor by desynchronizing the subcortical oscillatory neural activity.

Synchronization of low-frequency oscillations in the human cardiovascular system.

We investigate synchronization between the low-frequency oscillations of heart rate and blood pressure having in humans a basic frequency close to 0.1 Hz. A method is proposed for quantitative estimation of synchronization between these oscillating processes based on calculation of relative time of phase synchronization of oscillations. It is shown that healthy subjects exhibit on average substantially longer epochs of internal synchronization between the low-frequency oscillations in heart rate and blood pressure than patients after acute myocardial infarction.

EEG nonstationarity during intracranially recorded seizures : statistical and dynamical analysis

OBJECTIVE The investigation of nonstationarity in complex, multivariable signals, such as electroencephalographic (EEG) recordings, requires the application of different and novel approaches to analysis. In this study, we have divided the EEG recordings during epileptic seizures into sequential stages using spectral and statistical analysis, and have as well reconstructed discrete-time models (maps) that reflect dynamical (deterministic) properties of the EEG voltage time series. METHODS Intracranial human EEG recordings with epileptic seizures from three different subjects with medically intractable temporal lobe epilepsy were studied. The methods of statistical (power spectra, wavelet spectra, and one-dimensional probability distribution functions) and dynamical (comparison of dynamical models) nonstationarity analysis were applied. RESULTS Dynamical nonstationarity analysis revealed more detailed inner structure within the seizures than the statistical analysis. Three or four stages with different dynamics are typically present within seizures. The difference between interictal activity and seizure events was also more evident through dynamical analysis. CONCLUSIONS Nonstationarity analysis can reveal temporal structure within an epileptic seizure, which could further understanding of how seizures evolve. The method could also be used for identification of seizure onset. SIGNIFICANCE Our approach reveals new information about the temporal structure of seizures, which is inaccessible using conventional methods.

Characterizing direction of coupling from experimental observations

We demonstrate that the direction of coupling of two interacting self-sustained electronic oscillators can be determined from the realizations of their signals. In our experiments, two electronic generators, operating in a periodic or a chaotic state, were subject to symmetrical or unidirectional coupling. In data processing, first the phases have been extracted from the observed signals and then the directionality of coupling was quantitatively estimated from the analysis of mutual dependence of the phase dynamics.

Spurious causalities due to low temporal resolution: Towards detection of bidirectional coupling from time series

The detection of causal influences is a topical problem in time series analysis. A traditional approach is based on Granger causality and increasingly often used in very diverse fields. However, a principal possibility of spurious detection of a bidirectional coupling due to low sampling rate, noted by statisticians and econometricians, remains overlooked in physical research. With models widely used in physics, including linear oscillators and nonlinear chaotic maps, we show that spurious coupling characteristics can be rather large and one may even incorrectly identify directionality of a unidirectional coupling if a sampling interval is not small enough. To avoid erroneous conclusions, we suggest a practical test to distinguish between uni- and bi-directional couplings and illustrate it with mathematical systems and climatic data.

Estimation of coupling between oscillators from short time series via phase dynamics modeling: Limitations and application to EEG data

We demonstrate in numerical experiments that estimators of strength and directionality of coupling between oscillators based on modeling of their phase dynamics [D. A. Smirnov and B. P. Bezruchko, Phys. Rev. E 68, 046209 (2003)] are widely applicable. Namely, although the expressions for the estimators and their confidence bands are derived for linear uncoupled oscillators under the influence of independent sources of Gaussian white noise, they turn out to allow reliable characterization of coupling from relatively short time series for different properties of noise, significant phase nonlinearity of the oscillators, and nonvanishing coupling between them. We apply the estimators to analyze a two-channel human intracranial epileptic electroencephalogram (EEG) recording with the purpose of epileptic focus localization.

Application of adaptive nonlinear Granger causality: disclosing network changes before and after absence seizure onset in a genetic rat model.

BACKGROUND Advanced methods of signal analysis of the preictal and ictal activity dynamics characterizing absence epilepsy in humans with absences and in genetic animal models have revealed new and unknown electroencephalographic characteristics, that has led to new insights and theories. NEW METHOD Taking into account that some network associations can be considered as nonlinear, an adaptive nonlinear Granger causality approach was developed and applied to analyze cortico-cortical, cortico-thalamic and intrathalamic network interactions from local field potentials (LFPs). The outcomes of adaptive nonlinear models, constructed based on the properties of electroencephalographic signal and on statistical criteria to optimize the number of coefficients in the models, were compared with the outcomes of linear Granger causality. RESULTS The nonlinear adaptive method showed statistically significant preictal changes in Granger causality in almost all pairs of channels, as well as ictal changes in cortico-cortical, cortico-thalamic and intrathalamic networks. Current results suggest rearrangement of interactions in the thalamo-cortical network accompanied the transition from preictal to ictal phase. COMPARISON WITH EXISTING METHOD(S) The linear method revealed no preictal and less ictal changes in causality. CONCLUSIONS Achieved results suggest that this proposed adaptive nonlinear method is more sensitive than the linear one to dynamics of network properties. Since changes in coupling were found before the seizure-related increase of LFP signal amplitude and also based on some additional tests it seems likely that they were not spurious and could not result from signal to noise ratio change.

Oscillation types, multistability, and basins of attractors in symmetrically coupled period-doubling systems

Abstract Symmetrically coupled nonlinear oscillator systems demonstrating transition to chaos via a sequence of period-doubling bifurcations under variation of the control parameter exhibit various types of mutual synchronization. For these coupled systems, with dissipatively coupled logistic maps, we consider a hierarchy of possible oscillation types using the value of the time shift between oscillations of the subsystems as a basis for the classification of multistable states. For oscillation states and their basins of attraction the ways of evolution are studied under variation of the parameters of nonlinearity and coupling. The obtained results are compared with those of physical experiment with a system of coupled, periodically driven nonlinear resonators.