Nivea D. Bosco

We uncover the global organization of resonance structures in a periodically driven FitzHugh–Nagumo neuron model. High-resolution Lyapunov diagrams with local-maxima (LM) classification expose a hierarchical network of Arnold tongues in parameter space. Frequency-locking regions emerge from quasiperiodic dynamics and extend into chaotic domains, following a period-adding organization consistent with Farey-tree ordering. Successive magnifications reveal multiscale embeddings of stable periodic structures containing quint points, where five periodic domains coalesce and satisfy local LM-adding relations. Basin-of-attraction analysis confirms multistability, including coexistence of periodic and chaotic attractors. In a neurostimulation context, these results show that periodic forcing can induce multiple coexisting frequency-locked firing regimes with distinct LM counts, providing a dynamical basis for stimulus-dependent neural encoding and selective response modulation.

In this paper, we investigate a seven-parameter, five-dimensional dynamical system, specifically a unidirectional coupling of two FitzHugh–Nagumo neuron models, with one neuron being sinusoidally driven. This master–slave configuration features neuron N1 as the master, subjected to an external sinusoidal electrical current, and neuron N2 as the slave, interacting with N1 through an electrical force. We report numerical results for three distinct scenarios where N1 operates in (i) periodic, (ii) quasiperiodic, and (iii) chaotic regimes. The primary objective is to explore how the dynamics of the master neuron N1 influence the coupled system’s behavior. To achieve this, we generated cross sections of the seven-dimensional parameter space, known as parameter planes. Our findings reveal that in the periodic regime of N1, the coupled system exhibits period-adding sequences of Arnold tongue-like structures in the parameter planes. Furthermore, regions of multistability can also be identified in these parameter planes of the coupled system. In the quasiperiodic regime, regions of periodic motion are absent, with only regions of quasiperiodic and chaotic dynamics present. In the chaotic regime of N1, the parameter planes display regions of chaos, hyperchaos, and transient hyperchaos.