Complex transitions between spiking, bursting and silent regimes in a new memristive Rulkov neuronal model

Abstract

The Rulkov model, which simulates the behavior of biological neurons, is modified by replacing one of its control parameters with a memristive, sigmoid-type function of finite memory. This modification causes the parameter to vary according to the system's history throughout the simulation. Previous works usually modify the Rulkov model by introducing additional parameters altering its behavior. Here, by contrast, we retain the original equations and allow the control parameters to vary in time, thereby preserving the model's fundamental properties. In this sense, the proposed model is locally equivalent in time to the original one. However, unlike the original model, which reproduces a single neuronal regime per simulation, the new memristive version exhibits both uniform and chaotic transitions among multiple neuronal activity regimes. Its dynamics are examined with respect to the rate at which the memristive function changes and the number of internal states it stores. Three distinct scenarios emerge around a bifurcation point. Before the bifurcation, the system undergoes uniform transitions toward a stable bursting regime. After the bifurcation, it shows uniform transitions toward a final spiking or silent regime. At the bifurcation point, highly complex transitions arise. As examples, we present trajectories in which the neuron chaotically switches between regimes without ever settling, and trajectories for which it requires around 140000 map iterations to reach a stationary regime.