A Hénon-like map inspired by the generalized discrete-time FitzHugh–Nagumo model

The patterns of discharge activity are closely associated with the physiological manifestation of neurons, and hence, it is particularly significant to understand the dynamic behavior of the patterns. In this paper, a Hénon-like map (generalized map) system is reported, which is derived from the discrete-time FitzHugh–Nagumo model that is a simplification of the classical Hodgkin–Huxley neuron model. Our main contributions are as follows: (i) proposing the Hénon-like map; (ii) discovering the fractal structure and the chaos of the map; and (iii) presenting the horseshoe structure of the subsequence of the map. Specifically, it is observed that bifurcation structure in the process of period-doubling cascade to chaos is compatible with the largest Lyapunov exponent spectrum. Moreover, some basic properties of the generalized map are also analyzed with repeated numerical simulation, such as phase plane, bifurcation diagram, power spectrum and chaotic attractor. In addition, it is confirmed that the hidden horseshoe structure also exists in the generalized map. The potential connection between decomposition of chaotic behavior and period-3 is logically prospected. The behavior of period-doubling cascade to chaos and Smale horseshoe coexists in the map. Furthermore, the relationship between the internal mechanisms of the map and external performance may be discovered. Finally, the effects of forced external factors—delayed feedback, noise and electromagnetic field on the system—are evaluated. It may be the first time to study discrete neuron models by using Simulink simulation in the field of neurodynamics. In the above process, a circuit simulation is built in the Simulink package, which opens up a new path for the detection of discrete neuron network.