Linear response in neuronal networks: from neurons dynamics to collective response

Description

We review two examples where the linear response of a neuronal network submitted to an external stimulus can be derived explicitly, including network parameters dependence. This is done in a statistical physics-like approach where one associates to the spontaneous dynamics of the model a natural notion of Gibbs distribution inherited from ergodic theory or stochastic processes. These two examples are the Amari-Wilson-Cowan model 3,131 and a conductance based Integrate and Fire model 101,102. One of the contemporary challenges in neuroscience is to understand how our brain processes external world information. For example, our retina receives the light coming from a visual scene and efficiently converts it into trains of impulses (action potentials) sent to the brain via the optic nerve. The visual cortex is then able to decode this flow of information in a fast and efficient way. How does a neu-ronal network, like the retina, adapts its internal dynamics to stimuli, yet providing a response that can be successfully deciphered by another neuronal network ? Even if this question is far from being resolved, there exist successful methods and strategies providing partial answers. To some extent, as developed in this paper, this question can be addressed from the point of non equilibrium statistical physics and linear response theory. Although neuronal networks are outside the classical scope of non equilibrium statistical physics-interactions (synapses) are not symmetric , equilibrium evolution is not time-reversible, there is no known conserved quantity, no Lyapunov function-an extended notion of Gibbs distribution can be proposed, directly constructed from the dynamics where the linear response can be derived explicitly, including network parameters dependence.

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