A FRACTIONAL DIFFERENCE EQUATION MODEL OF A SIMPLE NEURON MAP

In this work, we present the dynamics of the one dimension fractional-order Rulkov map of biological neurons. The one-dimensional neuron map shows all the dynamical behaviors observed in the real-time experiment. The integer order one-dimensional Rulkov map exhibits chaotic dynamics in the presence of time-dependent external stimuli like periodic sinusoidal force or random Gaussian process. When we construct a large complex network of neurons, the higher system dimension, as well as the external forcing, is always an obstacle. Interestingly, our study shows even with constant external stimuli, the fractional-order one-dimensional neuron shows a rich variety of complex dynamics including chaotic dynamics. We present our results based on the Lyapunov exponent of the fractional-order systems.

Automated summary

In this study, the dynamics of one-dimensional fractional-order Rulkov map in biological neurons are presented, showing various dynamical behaviors and the influence of external stimuli on both integer and fractional-order maps. The results are based on the Lyapunov exponent of the fractional-order systems.