Bifurcation mechanisms and FPGA implementation of comb-shaped chaos and coexisting behaviors in a modified Hindmarsh-Rose model

The edge of chaos is regarded as a critical regime for neuronal complexity, learning, and adaptability. In Hindmarsh-Rose type models, this regime manifests as comb-shaped chaotic regions in two-dimensional parameter space. Although the phenomenon has been widely observed, the specific types of bifurcation points and structures that underlie it remain unclear. This paper investigates the bifurcation mechanisms underlying comb-shaped chaos and the dynamics of different coexisting behaviors related to bifurcations in a modified HR model. The results indicate that, as a single parameter varies, the neuron exhibits an alternation between periodic and chaotic firings. This can be explained by the special bifurcation structure to an alternation between fold bifurcations of limit cycles (Flc) and period-doubling bifurcations (PD), where Flc leads to periodic firing and PD leads to chaotic firing, in a repeating sequence. In the two-dimensional parameter space, comb-shaped chaos can be well explained by the alternation of periodic and chaotic regions formed by codimension-1 bifurcation curves of Flc and PD. Additionally, three types of coexisting behaviors are identified: two spiking states associated with supercritical Hopf and homoclinic bifurcations, chaos and resting associated with subcritical Hopf bifurcation, and a hidden coexistence of two bursting states related to PD bifurcations. The coexisting behaviors are explained by fast-slow variable dissection. These findings are validated through hardware implementation using a Field-Programmable Gate Array (FPGA), which successfully observes the alternation of periodic and chaotic firing and the three coexistence behaviors. This study elucidates the bifurcation mechanisms of comb-shaped chaos and coexistence dynamics, and demonstrates their physical realizability, laying a foundation for biologically realistic neuromorphic computing.