In collaboration with Sami ElBoustani (PhD student in my laboratory), we have recently designed a new approach to model cortical network activity at macroscopic scales. The goal of this study is to describe the activity of large-scale networks, at a level of description adapted to macroscopic measurements such as LFPs or optical imaging. In this case, the "unit" of the system is not the neuron, but a network of neurons, which would correspond to a "pixel" in imaging experiments. To obtain such a description, we have used a mean-field approach to describe the mean activity level of a network of neuron, but our study had two particularities: (1) we considered self-sustained irregular activity states (asynchronous irregular or AI states); (2) we considered a two-dimensional approach where not only the mean activity but also its variance (and correlations) are described. This was realized by deriving a Master equation for the mean and variance of the activity of the network. This approach successfully reproduced the complex state diagrams calculated numerically in networks of excitatory and inhibitory neurons (Fig. 1; see details in [1]). The approach is pursued presently towards obtaining a macroscopic description of cortical activity in relation to optical imaging experiments.
A macroscopic description was also developed to model extracellular field potentials in the brain [2]. Starting from first principles (Maxwell equations), a macroscopic formalism was developed, in which macroscopic measurements of permittivity and conductivity are naturally incorporated. The study evidences that ionic diffusion must be taken into account to match the frequency dependence of electric parameters observed experimentally (in addition to electric field effects). The same mechanisms also reproduce the typical 1/f frequency dependence of local field potentials from plausible physical causes. The predictions of this model are testable experimentally, and are presently under investigation.
Recently, we extended this macroscopic analysis to brain signals at multiple scales [3]. Macroscopic variables, such as the EEG, can display low dimensionality for some brain states, such as slow-wave sleep or pathological states like epilepsy. In awake and attentive subjects, however, there is not such low dimensionality, and the EEG is more similar to a stochastic variable. In contrast, "microscopic" recordings with microelectrodes inserted in cortex show that global variables such as local field potentials (local EEG) are similar to the human EEG. However, in all cases, neuronal discharges are highly irregular and exponentially distributed, similar to Poisson stochastic processes. 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 [3].
[1] El Boustani, S. and Destexhe, A. A master equation formalism for macroscopic modeling of asynchronous irregular activity states. Neural Computation 21: 46-100, 2009 (see abstract)
[2] 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)
[3] 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)
Figure 1: Macroscopic model of activated states in networks of neurons. A. Networks of randomly-connected excitatory and inhibitory IF neurons with conductance-based synaptic interactions display asynchronous irregular (AI) states. The raster (red=excitatory cells, blue=inhibitory) shows that spike discharges are irregular, so is the instantaneous activity (firing rate, bottom). B. Decay of the auto-correlation function (dashed line = exponential fit) and activity distribution (dashed line = Gaussian fit) during AI states. C. Results of a Master Equation model which can be used to predict the state diagrams of such networks. The colorized region corresponds to AI states. The firing rate and its standard deviation (as well as cross-correlations) are well predicted by the formalism. Similar results have been obtained for locally-connected networks. Modified from El Boustani and Destexhe, Neural Computation, 2009 (see abstract).
Unité de Neurosciences, Information & Complexité
(UNIC)
CNRS
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