One of the eye movements is the saccade, which has led to the introduction of the saccadic model. This study is based on the part of the saccadic model, which means the burst neurons and resettable integrator model. The principal limitation of the original model is the lack of differentiability at the equilibrium point. By using the Naka–Rushton function, we introduce a new model in place of the original one, so that the equilibrium point of the system becomes a differentiable point in the modified model. Our focus in this work is to investigate the fundamental properties of the discrete model of our novel system. We apply the forward Euler method to transform the new model to a discrete model. With the utilization of the center manifold theory, we describe some of its dynamical features, such as stability, instability, and bifurcation at a fixed point. Finally, both analytical and numerical simulations are used to continue investigating the period-doubling bifurcation according to the numerical parameters in the saccadic model.
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. We intend to substitute a new model for the original one so that the equilibrium point of the system becomes a differentiable point in the changed model to eliminate the main shortcoming of the original model at the equilibrium point. Our principal objective in this study is to determine the geometry of the slow manifold for the innovative system that has one fast and one slow variable. Specifically, we examine the dynamics around an equilibrium point and the geometry of a slow manifold by using Fenichel's theorem. The current study is to ascertain the effects of geometric singular perturbations on this fast-slow system. To conclude the debate, the stability or instability at the equilibrium point is found by using the center manifold theory.
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