Abstract
Fractional-order dynamics of excitable systems can be physically described as a memory dependent phenomenon. It can produce diverse and fascinating oscillatory patterns for certain types of neuron models. To address these characteristics, we consider a nonlinear fast-slow FitzHugh-Rinzel (FH-R) model that exhibits elliptic bursting at a fixed set of parameters with a constant input current. The generalization of this classical order model provides a wide range of neuronal responses (regular spiking, fast-spiking, bursting, mixed-mode oscillations, etc.) in understanding the single neuron dynamics. So far, it is not completely understood to what extent the fractional-order dynamics may redesign the firing properties of excitable systems. We investigate how the classical order system changes its complex dynamics and how the bursting changes to different oscillations with stability and bifurcation analysis depending on the fractional exponent (0 < α ≤ 1). This occurs due to the memory trace of the fractional-order dynamics. The firing frequency of the fractional-order FH-R model is less than the classical order model, although the first spike latency exists there. Further, we investigate the responses of coupled FH-R neurons with small coupling strengths that synchronize at specific fractional-orders. The interesting dynamical characteristics suggest various neurocomputational features that can be induced in this fractional-order system which enriches the functional neuronal mechanisms.
Highlights
Collective oscillatory dynamics and synchronous activity are the fundamental phenomena in dynamical systems[1,2]
The characteristics of a fast-slow FH-R model has been introduced in this article by using a commensurate fractional-order derivative
It has been examined that how fractional exponents influence the dynamics of the system and make it different from classical-order FH-R model that exhibits elliptic bursting
Summary
The fractional-order system has a Hopf bifurcation at α = 0.80828 and the system goes to quiescent state when α < 0.80828 i.e., it converges to the stable fixed point (v⁎ = −0.885098) at α = 0.79 (see Fig. 3(d)). When the fractional-order is decreased, the spike frequency is decreased and the period of small amplitude oscillations increases, i.e., it is growing with larger time duration with α = 0.98 (see Fig. 3(p)) It goes to the complete quiescent phase at α = 0.95 i.e.; the system converges to the stable fixed point (v⁎ = −0.948702) (see Fig. 3(q)). We characterize these diverse neuronal responses with stability and bifurcation analysis. This verified the coupling scheme and effectiveness of the method for synchronization
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