An important theme of computational neuroscience is to develop analysis techniques for complex electrophysiological signals. In my PhD thesis work with Agnessa Babloyantz and Grégoire Nicolis (University of Brussels, Belgium), methods from nonlinear dynamics were designed and applied to analyze human electroencephalogram (EEG) data. We showed that the complex dynamics of EEG oscillations have similar statistical and dynamical properties as low-dimensional chaotic systems [1,2]. The design of methods based on symbolic dynamics [3] suggested that patterns of oscillations are not random but exhibit temporal correlations. These results were modeled by simple networks where the thalamus was represented by a pacemaker [4] (see Section 3.2). An explanation for the presence of chaotic states in neural systems was proposed based on the maximization of information transport [5], similar to the onset of turbulent states which maximize transport properties in fluids.
We also investigated the spatiotemporal dynamics of oscillations in cat cerebral cortex and thalamus based on multi-electrode recordings (in collaboration with Diego Contreras and Mircea Steriade). We combined multisite recordings in vivo with computer analyses to investigate the role of corticothalamic feedback connections on the synchrony and spatiotemporal coherence of thalamic oscillations [6,7,8]. Here, relatively simple analysis techniques such as spatiotemporal maps, local Fourier analysis and spatial correlations, were used to characterize the spatiotemporal coherence. These methods helped to establish that the spatiotemporal coherence of thalamic oscillations is destroyed by removal of the cortex [6]. These experimental measurements of spatiotemporal properties were successfully modeled by thalamocortical networks [9,10] (see Section 3.2).
More recently, still in collaboration with Diego Contreras and Mircea Steriade, we characterized the spatiotemporal distribution of oscillatory activity from multisite field potential recordings in cat cerebral cortex during natural wake and sleep states [11]. The spatiotemporal pattern of activity was markedly different for slow-wave events and fast oscillations: slow waves were characterized by a generalized silence in all cell types and were of remarkable spatiotemporal coherence, whereas during fast oscillations the activity was less coherent and correlations were local within a millimeter range [11]. Although fast oscillations are present during wake and REM sleep, brief periods of fast oscillations with identical spatiotemporal characteristics were also present during slow-wave sleep. These results suggest that slow-wave sleep in cats consists in brief periods of activity ("up states") with low spatiotemporal coherence, similar to wakefulness, interleaved with slow-wave complexes coherent over large cortical territories. We recently wrote a review article that summarizes the evidence that these "up" states represents fragments of wakefulness that are replayed during sleep [12].
In a recent study [18], we compared the above evidence for stochastic dynamics of single neurons with the earlier evidence for low-dimensional dynamics in the EEG (see above). To attempt reconcile these results, we investigated models of randomly-connected networks of integrate-and-fire neurons, and also contrast global (averaged) variables, with neuronal activity. The network displays different states, such as "synchronous regular" (SR) or "asynchronous irregular" (AI) states. In SR states, the global variables display coherent behavior with low dimensionality, while in AI states, the global activity is high-dimensionally chaotic with exponentially distributed neuronal discharges, similar to awake cats. Scale-dependent Lyapunov exponents and epsilon-entropies show that the seemingly stochastic nature at small scales (neurons) can coexist with more coherent behavior at larger scales (averages). Thus, we suggest that brain activity obeys similar scheme, with seemingly stochastic dynamics at small scales (neurons), while large scales (EEG) display more coherent behavior or high-dimensional chaos [18].
More recently, with Claude Bedard and Helmut Kröger, we investigated the presence of self-organized critical (SOC) states in cerebral cortex in vivo [13]. Many complex systems display SOC states characterized by 1/f frequency scaling of power spectra. Global variables such as the electroencephalogram, scale as 1/f, which could be the sign of SOC states in neuronal activity. By analyzing simultaneous recordings of global and neuronal activities, we confirmed the 1/f scaling of global variables (such as LFPs) for selected frequency bands. However, by analyzing neuronal activities, we did not find the typical power-law scaling of SOC states ("avalanche analysis"), which suggests that neuronal activity does not stem from critical states. The 1/f scaling of LFPs can be explained by a model which does not rely on critical states, but is rather due to a filtering process from extracellular space. This latter hypothesis was recently shown to be plausible [14]. The predictions of this model are testable experimentally, and are presently under investigation (see also Section 1.3).
To check if other models than SOC can account for unit activity, we examined Ising type models to predict the occurrence of population patterns in distributed spiking activity [15]. Using a maximum entropy principle with a Markovian assumption, we elaborated a model that accounts for both spatial and temporal pairwise correlations among neurons. This model was tested on model data as well as on experimental data, and it was shown that this approach correctly predicts the occurrence probabilities of spatio-temporal patterns of spikes, significantly better than Ising type models only based on pairwise correlations. The same approach can be used to generate surrogates that reproduce the spatial and temporal correlations of a given data set.
Another approach to power-law relations (in collaboration with Sami El Boustani, Olivier Marre and Yves Fregnac at the UNIC) consists of extracting correlations from the power spectral density (PSD) of the Vm recorded in vivo. Vm recordings with different stimulus types (from moving gratings to natural images) revealed a different frequency scaling as a function of the stimulus. To explain this, we are developing a model where the "effective connectivity'' leads to special correlations in the Vm activity, which may appear as different slopes in the Vm PSD. This approach could lead to methods to estimate the effective connectivity from the Vm, and is still under study. We also plan to test this method experimentally in dynamic-clamp (in collaboration with Sebastien Behuret and Thierry Bal at the UNIC), where the amount of correlation will be controlled within synthetic inputs, and related to the PSD of the Vm obtained experimentally (see details in [16]).
Recently, with Jonathan Touboul, we investigated whether the LFP signal can show evidence for SOC states [17], as found by other authors in awake monkeys. By using the same techniques, we could show that indeed, the statistics of negative LFP peaks (which are related to increase of firing), can show power-law scaling which could be taken as evidence for SOC. However, we did the same analysis for positive LFP peaks, which are unrelated to firing activity, and found the same results. Moreover, shuffled peaks also demonstrated apparent power-law scaling, suggesting that power-law scaling may be a generic property of thresholded stochastic processes. We next showed that, indeed, spurious power-law scaling can appear from stochastic processes without the presence of underlying self-organized criticality. However, this power-law is only apparent in logarithmic representations, but does not resist to more severe analysis such as the Kolmogorov-Smirnoff test. We conclude that logarithmic representations can lead to spurious power-law scaling induced by the stochastic nature of the phenomenon, and should be demonstrated by more stringent statistical tests (see details in [17]).
In conclusion, these studies suggest that SOC does not seem to be a satisfactory model to explain the statistics of electrical brain activity, both at the level of the LFP and at the level of unit activity. Simple models based on pairwise correlations (such as the Ising model or its variants) seem to account for much of the statistics measured experimentally.
[1] Babloyantz, A. and Destexhe, A., Low dimensional chaos in an instance of epileptic seizure. Proc. Natl. Acad. Sc. USA 83: 3513-3517 , 1986. (see abstract)
[2] Destexhe, A., Sepulchre, J.A. and Babloyantz, A. A comparative study of the experimental quantification of deterministic Chaos. Phys. Lett. A 132: 101-106, 1988. (see abstract)
[3] Destexhe, A. Symbolic dynamics from biological time series. Phys. Lett. A 143: 373-378, 1990. (see abstract)
[4] Destexhe, A. and Babloyantz, A. Pacemaker-induced coherence in cortical networks. Neural Computation 3: 145-154, 1991. (see abstract)
[5] Destexhe, A. Oscillations, complex spatiotemporal behavior and information transport in networks of excitatory and inhibitory neurons. Physical Review E50: 1594-1606, 1994. (see abstract)
[6] Contreras, D., Destexhe, A., Sejnowski, T.J. and Steriade, M. Control of spatiotemporal coherence of a thalamic oscillation by corticothalamic feedback. Science 274: 771-774, 1996. (see abstract)
[7] Contreras, D., Destexhe, A., Sejnowski, T.J. and Steriade, M. Spatiotemporal patterns of spindle oscillations in cortex and thalamus. J. Neurosci. 17: 1179-1196, 1997. (see abstract)
[8] Contreras, D., Destexhe, A. and Steriade, M. Spindle oscillations during cortical spreading depression in naturally sleeping cats. Neuroscience 77: 933-936, 1997. (see abstract)
[9] Destexhe, A., Contreras, D. and Steriade, M. Mechanisms underlying the synchronizing action of corticothalamic feedback through inhibition of thalamic relay cells. J. Neurophysiol. 79: 999-1016, 1998. (see abstract)
[10] Destexhe, A., Contreras, D. and Steriade, M. Cortically-induced coherence of a thalamic-generated oscillation. Neuroscience 92: 427-443, 1999 (see abstract)
[11] Destexhe, A., Contreras, D. and Steriade, M. Spatiotemporal analysis of local field potentials and unit discharges in cat cerebral cortex during natural wake and sleep states. J. Neurosci. 19: 4595-4608, 1999 (see abstract)
[12] Destexhe, A., Hughes, S.W., Rudolph, M. and Crunelli, V. Are corticothalamic `up' states fragments of wakefulness? Trends Neurosci. 30: 334-342, 2007 (see abstract)
[13] Bedard, C., Kröger, H. and Destexhe, A. Does the 1/f frequency-scaling of brain signals reflect self-organized critical states ? Physical Review Letters 97: 118102, 2006 (see abstract).
[14] Bedard, C. and Destexhe, A. Macroscopic models of local field potentials the apparent 1/f noise in brain activity. Biophysical Journal 96: 2589-2603, 2009 (see abstract).
[15] Marre, O., El Boustani, S., Frégnac, Y. and Destexhe, A. Prediction of spatio-temporal patterns of neural activity from pairwise correlations. Physical Review Letters 102: 138101, 2009 (see abstract).
[16] El Boustani, S., Marre, O., Behuret, S., Baudot, P., Yger, P., Bal, T., Destexhe, A. and Fr\'egnac, Y. Network-state modulation of power-law frequency-scaling in visual cortical neurons. PLoS Computational Biology 5: e1000519, 2009 (see abstract).
[17] Touboul, J. and Destexhe, A. Can power-law scaling and neuronal avalanches arise from stochastic dynamics? PLoS-One 5: e8982, 2010 (see abstract).
[18] El Boustani, S. and Destexhe, A. Brain dynamics at multiple scales: can one reconcile the apparent low-dimensional chaos of macroscopic variables with the seemingly stochastic behavior of single neurons? International J. Bifurcation & Chaos 20: 1687-1702, 2010 (see abstract)
Unité de Neurosciences, Information & Complexité
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