Analysis of Neuronal Oscillations of Fractional-order Morris-Lecar Model

Abstract

Fractional calculus is a new approach for modeling biological and physical phenomena with memory effects. Fractional calculus uses differential and integral operators including non-integer orders to study the non-linear behavior of physical and biological systems with some degrees of fractionality or fractality. Since the long memory properties of neuronal responses can be better explained using fractional derivative, in this study we generalize the integer-order Morris-Lecar model in the fractionalorder domain to better modeling of neuron dynamics. To investigate the complex spiking patterns of fractional-order Morris-Lecar neural system the fractional calculus has been applied to build this new mathematical model. We compare the results with integer-order Morris-Lecar model. The analytical solutions of these equations cannot explicitly be obtained. Therefore, to find the dynamical behaviors of solutions, we used approximation and numerical schemes. Depending on the different parame-ters values for 0 < \(\eta\) \(\le\) 1, the fractional-order Morris-Lecar reproduces quiescent, spiking and bursting activities the same as its original model but for higher input current. We numerically discover the hopf bifurcation, saddle node bifurcation of limit cycle and homoclinic bifurcation for this model for different input current and derivative orders. Taking the advantages of the fractional order derivative, for a variety of orders, we define different classes of this model which helps to better extract all the complicated dynamics of this single neuron model.