Noise-Induced Behaviors in Neural Mean Field Dynamics

Abstract

Abstract. The collective behavior of cortical neurons is strongly affected by the presence of noise at the level of individual cells. In order to study these phenomena in large-scale assemblies of neurons, we consider networks of firing-rate neurons with linear intrinsic dynamics and nonlinear coupling, belonging to a few types of cell populations and receiving noisy currents. Asymptotic equations as the number of neurons tends to infinity (mean field equations) are rigorously derived based on a probabilistic approach. These equations are implicit on the probability distribution of the solutions which generally makes their direct analysis difficult. However, in our case, the solutions are Gaussian, and their moments satisfy a closed system of nonlinear ordinary differential equations (ODEs), which are much easier to study than the original stochastic network equations, and the statistics of the empirical process uniformly converge towards the solutions of these ODEs. Based on this description, we analytically and numerically study the influence of noise on the collective behaviors, and compare these asymptotic regimes to simulations of the network. We observe that the mean field equations provide an accurate description of the solutions of the network equations for network sizes as small as a few hundreds of neurons. In particular, we observe that the level of noise in the system qualitatively modifies its collective behavior, producing for instance synchronized oscillations of the whole network, desynchronization of oscillating regimes, and stabilization or destabilization of stationary solutions. These results shed a new light on the role of noise in shaping collective dynamics of neurons, and gives us clues for understanding similar phenomena observed in biological networks. Key words. mean field equations, neural mass models, bifurcations, noise, dynamical systems AMS subject classifications. 34C15, 34C23, 60F99, 60B10, 34C25Introduction. The brain is composed of a very large number of neurons interacting in a complex nonlinear fashion and subject to noise. Because of these interactions, stimuli tend to produce coherent global responses, often with high reliability. At the scale of single neurons the presence of noise and nonlinearities often results in highly intricate behaviors. However, at larger scales, neurons form large ensembles that share the same input and are strongly connected, and at these scales, reliable responses to specific stimuli may arise. Such population assemblies (cortical columns or cortical areas) feature a very large number of neurons. Understanding the global behavior of these large-scale neural assemblies has been a focus of many investigations in the past decades. One of the main interests of large-scale modeling is to characterize brain function at the scale most non-invasive imaging techniques operate. Its relevance is also connected to the fact that anatomical data recorded in the cortex reveal its columnar organization at scales ranging from about 50µm to 1mm. These so-called ...

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