Bifurcation analysis of a neural network model

Abstract

This paper describes the analysis of the well known neural network model by Wilson and Cowan. The neural network is modeled by a system of two ordinary differential equations that describe the evolution of average activities of excitatory and inhibitory populations of neurons. We analyze the dependence of the model's behavior on two parameters. The parameter plane is partitioned into regions of equivalent behavior bounded by bifurcation curves, and the representative phase diagram is constructed for each region. This allows us to describe qualitatively the behavior of the model in each region and to predict changes in the model dynamics as parameters are varied. In particular, we show that for some parameter values the system can exhibit long-period oscillations. A new type of dynamical behavior is also found when the system settles down either to a stationary state or to a limit cycle depending on the initial point.

Finite mixture distribution models of simple discrimination learning

“…Here, unless the gradual change in parameter value is extremely rapid, this does not result in a two-component mixture distribution of the behavioral variable. However, in special cases, a gradual variation of a model parameter will induce a discontinuous change in behavior, which is called a bifurcation (e.g., Borisyuk & Kirillov, 1992;Pollack, 1991;Raijmakers, van der Maas, & Mohenaar, 1996). Such a discontinuous change will result in a two-component mixture distribution of the behavioral variable.…”

Patterns, Memory and Periodicity in Two-Neuron Delayed Recurrent Inhibitory Loops

“…Such twoneuron recurrent inhibitory loops with delay display similar complex dynamic behaviors as larger networks and many techniques developed to deal with two-neuron networks can carry over to networks of large size. Moreover, two-neuron networks are sometimes thought of as systems of two modules, where each module represents the mean activity of a spatially localized neural population [3,25].…”

Attractive Periodic Sets in Discrete-Time Recurrent Networks (with Emphasis on Fixed-Point Stability and Bifurcations in Two-Neuron Networks)

“…There is a considerable amount of literature on two-unit recurrent networks (Beer, 1995;Borisyuk & Kirillov, 1992;Botelho, 1999;Klotz & Brauer, 1999;Pakdamann, Grotta-Ragazzo, Malta, Arino, & Vibert, 1998;Pasemann, 1993;Wang, 1991;Zhou, 1996). This is partly due to the lack of mathematical tools for a detailed analysis of higher-dimensional dynamical systems and partly due to the high expressive power of such simple networks, in which many generic properties of larger networks are already present (Botelho, 1999).…”

“…This is partly due to the lack of mathematical tools for a detailed analysis of higher-dimensional dynamical systems and partly due to the high expressive power of such simple networks, in which many generic properties of larger networks are already present (Botelho, 1999). Moreover, two-neuron networks are sometimes thought of as systems of two modules, where each module represents the mean activity of a spatially localized neural population (Borisyuk & Kirillov, 1992;Pasemann, 1995a;Tonnelier, Meignen, Bosh, & Demongeot, 1999).…”