D. A. Smirnov

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.

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.

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.

Assessments of the relationship of changes of the global surface air temperature with different natural and anthropogenic factors based on observations

We obtained estimates of the relationship of changes in the global surface air temperature (GSAT) with different natural and anthropogenic factors based on empirical data beginning from the middle of the 19th century using the Granger causality test estima� tion and application of cross wavelet analysis. Along with the solar and volcanic activity and changes of the carbon dioxide concentration in the atmosphere, we estimated the role of quasicyclic processes in the Earths climatic system. We analyzed the climatic vari� ations detected by the index of the Atlantic multidec� adal oscillation (AMO) with a characteristic period of approximately 60-70 years and the variations in the angular velocity of the Earth. We made a conclusion on the basis of the empirical regression models based on data beginning from the middle of the 19th century that the changes of the CO2 concentration in the atmosphere have a determining influence on the longterm (secular) GSAT trends. The natural climatic cycles with periods of a few decades influence significantly only on the relatively fast GSAT variations. The influence of natural factors related to solar and volcanic activity on the longterm trends appeared to be much less significant. One of the modern key problems is estimating the role of natural and anthropogenic factors of global cli� mate changes. The natural climatic variability not related to external forcing is characterized by a wide spectrum of temporal and spatial scales and the effects of an anthropogenic character can hardly be distin� guished against the background of the natural variabil� ity. The problem of distinguishing the anthropogenic influence is strongly complicated by the effects of non� linearity and stochasticity in the climatic system under

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.

Diagnostics of a cause-effect relation between solar activity and the Earth’s global surface temperature

The influence of solar activity on the Earth’s global surface temperature (GST) was quantified. The method for estimation of the Granger causality was used, with analysis of the improvement of the prediction of one process by using data from another process as compared to autoprediction. Two versions of reconstructions of the solar flux variations associated with solar activity were used, according to Hoyt et al. [1997] for 1680–1992 (data H) and according to Lean et al. [2005] for 1610–2005 (data L). In general, the estimation results for the two reconstructions are reasonably well consistent. A significant influence of solar activity on GST with a positive sign was found for two periods, from the late 19th century to the late 1930s and from the latter half of the 1940s to the early 1990s, with no inertia or time delay. In these periods, up to 8 and 25% of the variance of the GST change, respectively, can be attributed to solar activity variations. The solar influence increased in the 1980s to the early 1990s according to data H and began to decrease in the latter half of the 1980s according to data L.

Study of the Mutual Influence of the El Niño-Southern Oscillation Processes and the North Atlantic and Arctic Oscillations

On the basis of the nonlinear techniques for the estimation of coupling between oscillatory systems from time series, we investigate the dynamics of climatic modes characterizing global and Northern Hemisphere (NH) processes. The North Atlantic Oscillation (NAO) and Arctic Oscillation indices and the El Niño-Southern Oscillation (ENSO) indices are analyzed in terms of the most reliable data from 1950 through 2004 and earlier data since the 19th century. These indices characterize changes in NH atmospheric pressure (specifically, sea-level pressure in the North Atlantic and NH extratropical latitudes as a whole) and in equatorial Pacific sea-surface temperature and sea-level pressure to which the strongest variations of global surface temperature and global climate on interannual time scales and of regional climatic anomalies in the NH are linked. The methods used are based on phase-dynamics modeling and nonlinear prediction models (a nonlinear version of Granger causality). From both methods and various ENSO indices, the inference about the ENSO effect on the NAO during the latter half of the 20th century and in the early 21st century is made with confidence probability of at least 0.95. The influence is characterized by a time delay of about two years. No inverse influence is found with a similar degree of reliability. Results of estimating the coupling between the ENSO and the NAO depend on the type of index that is used to describe the NAO. The ENSO effect on the NAO is detected with sufficient confidence when the NAO index is chosen to be a larger scale characteristic. However, when a more local index of the NAO is used, no statistically significant coupling to the ENSO is found. Increasing the length of the analyzed ENSO and NAO series (over more than 100 yr) does not lead to any more reliable detection of coupling. Analysis of the data for different time intervals during 1950–2004 has revealed a strengthening of the ENSO effect on the NAO, although this inference is not reliable.

A generalization of the van-der-Pol oscillator underlies active signal amplification in Drosophila hearing

The antennal hearing organs of the fruit fly Drosophila melanogaster boost their sensitivity by an active mechanical process that, analogous to the cochlear amplifier of vertebrates, resides in the motility of mechanosensory cells. This process nonlinearly improves the sensitivity of hearing and occasionally gives rise to self-sustained oscillations in the absence of sound. Time series analysis of self-sustained oscillations now unveils that the underlying dynamical system is well described by a generalization of the van-der-Pol oscillator. From the dynamic equations, the underlying amplification dynamics can explicitly be derived. According to the model, oscillations emerge from a combination of negative damping, which reflects active amplification, and a nonlinear restoring force that dictates the amplitude of the oscillations. Hence, active amplification in fly hearing seems to rely on the negative damping mechanism initially proposed for the cochlear amplifier of vertebrates.