Fractional-order neuronal maps: dynamics, control and stability analysis

Excitable systems have important biological applications, such as impulse transmission in neurons. The Chialvo map and Rulkov map are popular and important for studying the nonlinear dynamics of neurons. Both belong to the family of two-dimensional maps used to describe excitable systems in general and neural dynamics in particular. They show rich dynamics ranging from oscillatory, bursting to chaotic bursting behaviour. Neurons have memory and extension of the above maps to fractional order is an efficient way to model the effects of memory. This generalisation provides a wider scope of understanding the neuron systems. We study the dynamics of the above maps for fractional orders. We observe that whether the chaotic range is suppressed, enhanced or remains unchanged depends on the details of the eigenvalues of Jacobian obtained by the linearisation around the fixed point. We show that the three cases studied give three different results. The impact of fractional order on the efficacy of feedback control as well as the frequency and amplitude of bursts in the chaotic region is different for different maps. Thus, the impact of memory on the stability of neuronal maps depends on the details of the underlying system.