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Persistent neural states: stationary localized activity patterns in nonlinear continuous $n$-population, $q$-dimensional neural networks

Olivier Faugeras 1, * Romain Veltz 1 François Grimbert 1
* Corresponding author
1 ODYSSEE - Computer and biological vision
DI-ENS - Département d'informatique de l'École normale supérieure, CRISAM - Inria Sophia Antipolis - Méditerranée , ENS Paris - École normale supérieure - Paris, Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech
Abstract : Neural continuum networks are an important aspect of the modeling of macroscopic parts of the cortex. Two classes of such networks are considered: voltage- and activity-based. In both cases our networks contain an arbitrary number, $n$, of interacting neuron populations. Spatial non-symmetric connectivity functions represent cortico-cortical, local, connections, external inputs represent non-local connections. Sigmoidal nonlinearities model the relationship between (average) membrane potential and activity. Departing from most of the previous work in this area we do not assume the nonlinearity to be singular, i.e., represented by the discontinuous Heaviside function. Another important difference with previous work is our relaxing of the assumption that the domain of definition where we study these networks is infinite, i.e. equal to $\R$ or $\R^2$. We explicitely consider the biologically more relevant case of a bounded subset $\Omega$ of $\R^q,\,q=1,\,2,\,3$, a better model of a piece of cortex. The time behaviour of these networks is described by systems of integro-differential equations. Using methods of functional analysis, we study the existence and uniqueness of a stationary, i.e., time-independent, solution of these equations in the case of a stationary input. These solutions can be seen as ``persistent'', they are also sometimes called ``bumps''. We show that under very mild assumptions on the connectivity functions and because we do not use the Heaviside function for the nonlinearities, such solutions always exist. We also give sufficient conditions on the connectivity functions for the solution to be absolutely stable, that is to say independent of the initial state of the network. We then study the sensitivity of the solution(s) to variations of such parameters as the connectivity functions, the sigmoids, the external inputs, and, last but not least, the shape of the domain of existence $\Omega$ of the neural continuum networks. These theoretical results are illustrated and corroborated by a large number of numerical experiments in most of the cases $2\leq n \leq 3,\,2\leq q \leq 3$.
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Olivier Faugeras, Romain Veltz, François Grimbert. Persistent neural states: stationary localized activity patterns in nonlinear continuous $n$-population, $q$-dimensional neural networks. [Research Report] RR-6393, INRIA. 2007, pp.37. ⟨inria-00198808v3⟩

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