Estudi de les equacions de Wilson i Cowan, un model de xarxa neuronal

Abstract

The theoretical knowledge about dynamical systems has important applications to the study of those biological models that are described by means of differential equations. This work is a study of some basic aspects about the theory of dynamical systems and its application to the analysis of a statistical neural model: the Wilson-Cowan model. First of all, we have explored some important results about bifurcations of equilibria in systems of ordinary differential equations that depend on parameters. Later on, we have presented the Wilson-Cowan model and its mathematical formulation. The model describes, by means of a system of two ordinary differential equations, the temporal evolution of the mean activity in two neuronal populations of a network, one of them being excitatory and the other, inhibitory. The equations include constants that depend on intrinsic properties of the network and two real parameters that represent the external stimuli received by the network. We have applied some of the learnt theoretical results to the analysis of the model and its dynamical regimes. The results show that, for some chosen values in the constant parameters of the model, the system exhibits rich dynamics, with possible state changes between oscillatory and stationary regimes when the external stimuli vary in certain regions in the parameter space. These phenomena may represent basic mechanisms underlying more complex processes as sleep oscillatory rhythms or the establishment of memories.