Macroscopic models of cortical networks
In collaboration with Sami ElBoustani (PhD student in my laboratory), we have 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 ). The approach is pursued presently towards obtaining a macroscopic description of cortical activity in relation to optical imaging experiments.
This macroscopic analysis was extended to brain signals at multiple scales . 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 .
A macroscopic description is also needed to correctly model (and understand) the measurements of extracellular field potentials, which are "macroscopic" variables that represent the summated activity of many thousands of neurons. 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.
This macroscopic approach to Maxwell equations was further developed to study macroscopic current sources, using the concept of "current source density" (CSD) . The CSD analysis is a well known method to estimate the CSD from LFP recordings. We showed that the classic CSD method is invalid if the extracellular medium is frequency dependent or non-Ohmic. The macroscopic approach (mean-field) was used to generalize the CSD method to more realistic extracellular media with frequency dependent properties . It was also shown that the power spectrum of the signal contains the signature of the nature of current sources and medium, which provides a direct way to identify the presence of frequency-dependent properties from experimental data. These concepts were the subject of a recent short review paper .
A significant advance was realized recently by proposing a semi-analytical approach  for the calculation of the TF of AdEx neurons, which could be potentially applicable to any neuron model and, more interestingly, to biological neurons. This approach permits to build a mean-field model of networks of complex neurons, such as done recently using AdEx networks . This model provides a good prediction of the level of spontaneous activity of the network, with excitatory and inhibitory cells firing at different rates. But more importantly, it can also predict the dynamics of the response of the network to an external input. Because it is conductance-based, this mean-field model accounts for shunting effects, such as shunting inhibition for example. The response to oscillatory inputs, however, is not well captured because it would require to take into account spike-frequency adaptation (work in progress).
Using the semi-analytic approach, it is also possible to measure the transfer function from real neurons. However, it is non trivial because the inputs must be conductance-based, so the dynamic-clamp method should be used. A first study of this kind was done recently, where we measured the transfer function of Layer V cortical neurons in mice visual cortex by using perforated patch recordings . This study revealed that it is possible to obtain a compact description of the transfer function of individual pyramidal neurons, which opens the perspective of building ``realistic'' mean-field models. The study also evidenced a strong cell to cell diversity of firing responses. It suggests that appropriate mean-field formalisms have to be designed in order to integrate this diversity.
This approach was more recently extended to neurons with dendrites and dendritic inputs . This constitutes a step towards the design of mean-field models of population of neurons with dendrites (work in progress).
Finally, it is worth noting that mean-field models not only can model a single population of neurons, but it can also form the basis for very large scale models of many interacting populations. This was done recently , where the mean-field model was used to model the entire V1 area as seen by voltage-sensitive dye (VSD) recordings. Here, each ``pixel'' of the imaging is modeled by a neural population, and has its own mean-fied model. An array of pixels is built, and the different populations are interconnected. This model reproduced macroscopic features of brain activity, such as propagating waves in awake monkey V1 . It is presently extended to more complex paradigms involving multiple inputs (work in progress, in collaboration with F. Chavane).
 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)
 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)
 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)
 Bedard, C. and Destexhe, A. A generalized theory for current-source density analysis in brain tissue. Physical Review E 84: 041909, 2011 (see abstract).
 Bedard, C. and Destexhe, A. Mean-field formulation of Maxwell equations to model electrically inhomogeneous and isotropic media. J. Electromagnetic Analysis and Applications 6: 296-302, 2014 (see abstract)
 Zerlaut, Y., Telenczuk, B., Deleuze, C., Bal, T., Ouanounou, G. and Destexhe, A. Heterogeneous firing rate response of mice layer V pyramidal neurons in the fluctuation-driven regime. J. Physiol. 594: 3791-3808, 2016 (see abstract)
 Zerlaut, Y. and Destexhe, A. Heterogeneous firing responses predict diverse couplings to presynaptic activity in mice Layer V pyramidal neurons. PLOS Comp. Biol. 13: e1005452, 2017 (see abstract)
 Zerlaut, Y., Chemla, S., Chavane, F. and Destexhe, A. Modeling mesoscopic cortical dynamics using a mean-field model of conductance-based networks of adaptive exponential integrate-and-fire neurons. J. Computational Neurosci. 44: 45-61, 2018 (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).
Department of Integrative and Computational Neuroscience (ICN),