Different Cases of Singular Domains in Two Population Synaptic Drive Model

Abstract

In this work we have investigated the dynamics of synaptic drive two-population neural firing rate model for the case of steep firing rate function by applying the theory of switching dynamical systems. The switching dynamic theory simplifies the dynamical system and receive the approximation of the solution for the system with certain modes of parameters. In such type of outlook, the phase space is divided into regular as well as singular domains. These singular domains are characterized into black, white and transparent on the basis of the behavior of trajectories on the sides of singular domain.

Moreover, there has been investigation of the existence and uniqueness solutions with corresponding singular and regular stationary points. It has been observed that regular stationary point, which are located within regular domain, are always constant and this characteristic is secured for smooth and sufficiently steep firing rate function. Yousaf.et al investigated the dynamics for the same model. They investigated the dynamics only for some particular cases. We have explored some more cases of singular domains and investigated the dynamics in regular and singular domains. Moreover, general formulae for the stationary points of the system have been calculated and the existence of singular stationary points which belong to the singular domains have been investigated. Furthermore, these singular domains in new dynamical system have also been transformed. Besides, the stability of singular stationary points has been investigated. In the wake of outcomes, the stationary points are attended by implementing the singular perturbation analysis and Thikhonov theorem.