Saccadic model and stability of equilibrium point with different sigmoidal functions
Department of Mathematical Sciences, Yazd University, Yazd, Iran
Accepted: 31 January 2023
Published online: 17 February 2023
Saccades are defined as rapid eye movements that occur between gaze fixation and the desired object and can be either reflexive or voluntary. These voluntary gaze movements are made of a coordinated series of head and eye movements that have been modeled by a dynamical system and are quick or slow. In particular, one of the purposes of modeling saccadic eye movements based on mathematical dynamical models is to anticipate the experimental data behavior of this system. Considering how it is possible to enhance the saccadic model based on its basic model, namely the burst neurons and resettable integrator model, we introduce alternative functions that can more effectively represent the response of the bursters to an error signal. So, we utilize a special sigmoidal function, that is, the logistic function, and the Naka-Rushton function. These functions are exceedingly close to the experimental data and contribute to solving the absence of a derivative at the equilibrium point in the original saccadic model. Employing new functions, we also investigate the dynamical behavior of the modified saccadic model, in which the role of omnipause cells is considered, and evaluated the stability, instability, and existence or lack of bifurcation. Since the saccadic model has a lot of parameters that can change, we use simulation results to compare the changes of the phase portrait and time series in special and fundamental conditions. Another focus of the numerical simulations is to compare the changes in the model’s main factors against each other, i.e., left burst neuron, right burst neuron, motor error, and fringe rates.
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