|
Macroscopic models of neuronal activityIn collaboration with Sami ElBoustani (PhD student in my laboratory), we have designed a mean-field 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 to describe the time evolution 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 (see Mean-field models of neuronal populations).This macroscopic analysis was extended to brain signals at multiple scales [2]. 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 [2]. 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 [3], 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) [4]. 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 [4]. 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 [5]. A significant advance was realized recently by proposing a semi-analytical approach [6] for the calculation of the transfer function of complex 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 [8]. 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 (see below). 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 [6]. 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. Finally, it is worth noting that mean-field were extended in many directions, such as neurons with dendrites [7], large-scale models of propagating waves in primary visual cortex as seen from voltage-sensitive dyes measurements [8]. In the latter case, 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 [8]. Another recent extension was to integrate spike-frequency adaptation in the mean field [9], yielding a very realistic mean-field model capable of precisely reproducing the network responses to external inputs. Finally, the mean-field approach was extended to model the spread of activit in the entire human brain [10]. These different extensions are described in more detail in Section Mean-field models of neuronal populations.  Alain Destexhe |