Uncovering various neuronal responses in a fractional-order generalized HR system

This study investigates neuronal electrical activities in a fractional-order generalized Hindmarsh-Rose (HR) system and explores an extended model incorporating an induced electric field. Stability and bifurcation analyses examine the impact of external electrical stimulation on neuronal dynamics. The results show how electric field parameters, including amplitude and frequency, modulate neuronal excitability and stability. The H-R model is a mathematical representation that captures diverse neuronal activities, and the introduction of fractional-order derivatives allows us to explore non-local dynamics in greater depth. We analyze the effects of fractional-order derivatives on the system's behavior, including the generation of action potential dynamics. We discuss some biophysical aspects of the different firing patterns that we encounter. In addition, the study employs both analytical and numerical methods to investigate the stability of bursting and spiking patterns, using linear stability analysis to examine the transitions between stable and unstable states. Simulations reveal significant memory effects even with a slight decrease in fractional order. This underscores the versatility of fractional-order models in bridging mathematical theory with biologically plausible phenomena. The findings of this study demonstrate the potential of fractional-order systems in capturing the intricacies of neuronal responses, highlighting the need for further exploration of these phenomena in excitable biophysical systems.