Bifurcation and Chaos Control in the Hindmarsh–Rose Neuronal Model with Distributed Time Delay

In this paper, we analyze the effect of distributed time delay on the dynamics of the Hindmarsh–Rose (HR) neuronal model, particularly in the context of chaos control by generalizing the Pyragas delayed feedback method (LDFC). We model the distributed time delay through a weak exponential memory kernel, which better captures realistic neuronal processes. Using distributed time delay as a bifurcation parameter, we identify the conditions leading to Hopf bifurcation and the emergence of periodic oscillations. These analytical conditions are then derived and expressed in an explicit formula. This framework allows us to reformulate the chaos control problem as a bifurcation analysis problem in the parameter plane defined by the feedback gain and distributed time delay. Our results show that the distributed time delay plays a stabilizing role in the chaotic system dynamics, that is, with appropriate values of the feedback gain and the distributed time delay, it can stabilize the chaotic regime by transforming it into a periodic regime. Numerical simulations confirm the theoretical predictions and demonstrate that a proper choice of parameters values enables effective control of neuronal chaos. In the end, a comparative numerical study between discrete and distributed delay cases and from performance of the LDFC and performance indicators, we deduce that our approach provides a general framework for analyzing and stabilizing chaotic regimes in excitable neuronal systems.