Modulating Neuronal Entrainment via Distinct Fractional Derivative Orders in the FitzHugh–Nagumo Model

The synchronization of neural activity to external rhythms is fundamental to information processing and network coordination. However, classical integer-order models often neglect the power-law adaptation and memory effects inherent in biological tissues, such as ion channel gating and membrane impedance. This study investigates the synchronization dynamics of a periodically forced FitzHugh–Nagumo system, incorporating fractional-order calculus to explicitly model these history-dependent behaviors. We also introduce an asymmetry into the model by assigning distinct fractional orders to the fast membrane potential and the slow recovery variable to isolate their specific biophysical contributions.

The system is numerically investigated across a broad two-dimensional parameter space of forcing amplitude and frequency. We employ rotation numbers and inter-spike interval statistics to systematically map phase-locking zones—known as Arnold tongues—and to classify complex dynamical regimes versus stable periodic firing.

The investigation reveals that the two fractional orders play fundamentally different, non-interchangeable roles in shaping the entrainment landscape. Introducing memory into the voltage dynamics acts primarily as a stabilizing damping mechanism, significantly suppressing chaos and higher-order locking in favor of robust 1:1 synchronization. In contrast, increasing memory in the recovery variable functions as a timescale modulator, systematically shifting the system's resonant frequency and entrainment windows toward lower bands. Furthermore, we observe that these effects can compete; recovery memory can partially restore dynamical complexity to a system otherwise simplified by voltage memory.

We conclude that fractional-order asymmetry serves as a flexible biophysical mechanism for tuning neuronal response. By independently adjusting stability and frequency selectivity, neurons can optimize their entrainment to rhythmic stimuli, offering new theoretical insights into the modulation of neural circuits and the potential for targeted neuromodulation strategies.