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On a voltage-conductance kinetic system for integrate & fire neural networks. (English) Zbl 1284.35103

Summary: The voltage-conductance kinetic equation for integrate and fire neurons has been used in neurosciences since a decade and describes the probability density of neurons in a network. It is used when slow conductance receptors are activated and noticeable applications to the visual cortex have been worked-out. In the simplest case, the derivation also uses the assumption of fully excitatory and moderately all-to-all coupled networks; this is the situation we consider here. { } We study properties of solutions of the kinetic equation for steady states and time evolution and we prove several global a priori bounds both on the probability density and the firing rate of the network. The main difficulties are related to the degeneracy of the diffusion resulting from noise and to the quadratic aspect of the nonlinearity. { } This result constitutes a paradox; the solutions of the kinetic model, of partially hyperbolic nature, are globally bounded but it has been proved that the fully parabolic integrate and fire equation (some kind of diffusion limit of the former) blows-up in finite time.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35Q84 Fokker-Planck equations
62M45 Neural nets and related approaches to inference from stochastic processes
82C32 Neural nets applied to problems in time-dependent statistical mechanics
92B20 Neural networks for/in biological studies, artificial life and related topics
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