In this chapter, the dynamical behavior of the incommensurate fractional-order FitzHugh-Nagumo model of neuron is explored in details from local stability analysis. First of all, considering that the FitzHugh-Nagumo model is a mathematical simplification of the Hodgkin-Huxley model, the considered model is derived from the fractional-order Hodgkin-Huxley model obtained taking advantage of the powerfulness of fractional derivatives in modeling certain biophysical phenomena as the dielectrics losses in cell membranes, and the anomalous diffusion of particles in ion channels. Then, it is shown that the fractional-order FitzHugh-Nagumo model can be simulated by a simple electrical circuit where the capacitor and the inductor are replaced by corresponding fractional-order electrical elements. Then, the local stability of the model is studied using the Theorem on the stability of incommensurate fractional-order systems combined with the Cauchy’s argument Principle. At last, the dynamical behavior of the model are investigated, which confirms the results of local stability analysis. It is found that the simple model can exhibit, among others, complex mixed mode oscillations, phasic spiking, first spike latency, and spike timing adaptation. As the dynamical richness of a neuron expands its computational capacity, it is thus obvious that the fractional-order FitzHugh-Nagumo model is more computationally efficient than its integer-order counterpart.
We study networks of coupled oscillators whose local dynamics are governed by the fractional-order versions of the paradigmatic van der Pol and Rayleigh oscillators. We show that the networks exhibit diverse amplitude chimeras and oscillation death patterns. The occurrence of amplitude chimeras in a network of van der Pol oscillators is observed for the first time. A form of amplitude chimera, namely, “damped amplitude chimera” is observed and characterized, where the size of the incoherent region(s) increases continuously in the course of time, and the oscillations of drifting units are damped continuously until they are quenched to steady state. It is found that as the order of the fractional derivative decreases, the lifetime of classical amplitude chimeras increases, and there is a critical point at which there is a transition to damped amplitude chimeras. Overall, a decrease in the order of fractional derivatives reduces the propensity to synchronization and promotes oscillation death phenomena including solitary oscillation death and chimera death patterns that were unobserved in networks of integer-order oscillators. This effect of the fractional derivatives is verified by the stability analysis based on the properties of the master stability function of some collective dynamical states calculated from the block-diagonalized variational equations of the coupled systems. The present study generalizes the results of our recently studied network of fractional-order Stuart–Landau oscillators.
This paper presents a report on the microcontroller implementation of an autonomous three-dimensional oscillator with five terms (ATDOFT) and performance analysis based on partial and total amplitude controls. ATDOFT displays periodic spiking behaviors, period-tripling bifurcation to chaos, chaotic spiking attractors, coexisting attractors and bistable attractors. ATDOFT is divided into two subsystems; namely the fast and slow subsystems to investigate the mechanism of the spiking dynamics. Relying on the stability analysis based on the fast subsystem with respect to the slow variable, it is shown that the spiking oscillations present in the ATDOFT arise from the system switching between the unstable state and the stable state of the lone equilibrium point of the fast subsystem. By inserting two controller parameters into the rate equations of the ATDOFT, total and partial amplitude controls are achieved. Finally, the dynamical behaviors found in ATDOFT are validated by the microcontroller implementation.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.