Analysis of a neural network model

In this paper we study a stochastic model of a neural network of Purkinje cells proposed by Axelrad et al. 2], 5]. In this model only the inhibitory interaction between the cells is considered. We analyze the stability properties of networks whose graph is complete N-partite. It is shown that if the parameters of the model are below some critical values, then the network converges to unique equilibrium. In this case, the invariant measure of the inhibition states is explicited as well as the distribution of the duration of the interspikes of a given cell. If this stability condition is not satissed, the state of the network converges to some asymptotic state depending on its initial state. In this case, the set of possible asymptotic states is given. Nous etudions un mod ele probabiliste d'un r eseau de cellules de Purkinje propos e par Axelrad et al. 2], 5]. Dans ce mod ele, nous consid erons unique-ment l'interaction inhibitrice entre les cellules. Si le graphe du r eseau est N-partite complet et si les param etres de celui ci sont inf erieurs a certaines constantes, nous montrons que l' etat du r eseau converge vers un unique etat d' equilibre. Dans ce cas, la mesure invariante de l' etat du r eseau est explicit ee, de m^ eme que la loi de la dur ee entre deux excitations cons ecutives d'une m^ eme cellule. Si cette condition de stabilit e n'est pas v erii ee, nous donnons les dii erents etats asymptotiques possibles, l' etat asymptotique du r eseau d epend alors des conditions initiales. Abstract In this paper we study a stochastic model of a neural network of Purkinje cells proposed by Axelrad et al. 2], 5]. In this model only the inhibitory interaction between the cells is considered. We analyze the stability properties of networks whose graph is complete N-partite. It is shown that if the parameters of the model are below some critical values, then the network converges to unique equilibrium. In this case, the invariant measure of the inhibition states is explicited as well as the distribution of the duration of the interspikes of a given cell. If this stability condition is not satissed, the state of the network converges to some asymptotic state depending on its initial state. In this case, the set of possible asymptotic states is given.