Slow manifold analysis of modified burst model in the saccadic system

Abstract

The saccade is one of the eye movements that resulted in the creation of the saccadic model. This work is grounded in the basic principles of the saccadic system, which are burst neurons and a resettable integrator model. Considering the possibility of strengthening the saccadic model based on its fundamental model, we introduce a replacement function for use in the burster equation that explains the preservation of the on response’s form and also considers the off response. The new model is a two-dimensional map containing slow and fast variables with a new burster function, which solves the lack of differentiability of the primary function at the equilibrium point. By applying time series approaches and phase portraits, the mechanisms underlying the generation of spikes and spike bursts in the behavior of the new model are revealed. The present research’s other main focus is to determine the geometry of the slow manifold for the newly developed system. Specifically, we examine the dynamics around an equilibrium point and the geometry of a slow manifold by using Fenichel’s theorem. In addition, we use the center manifold theory to describe some dynamical characteristics of the center manifold that the slow manifold matches. Finally, this study aims to figure out the effects of geometric singular perturbations on this fast-slow burster equation, which finds dynamical behaviors such as being uniformly asymptotically stable and locally attractive.