Asymmetric effects of fractional orders on synchronization in a periodically forced FitzHugh–Nagumo system

Synchronization of a neuron with a periodic external input is a fundamental process in neuroscience, critical for information processing, network coordination, and therapeutic neuromodulation. In this work, we use the periodically forced fractional-order FitzHugh–Nagumo system to investigate how this phenomenon is affected by memory asymmetry, modeled by distinct fractional orders, α and β, of the Caputo derivatives governing the voltage and recovery variables. The system is numerically explored across a biologically plausible range of forcing parameters to identify a spectrum of dynamical responses, including phase-locking, complex dynamics, and quiescence. Phase-locking structures (Arnold tongues) are systematically mapped using the rotation number and inter-spike interval statistics, revealing how the asymmetry in memory reshapes the entrainment landscape. We find that memory in the fast voltage variable has a simplifying effect, suppressing complex firing patterns and promoting 1:1 synchronization. In contrast, memory in the slow recovery variable regulates the system’s frequency preference, shifting the entire entrainment window. These results demonstrate that the fractional orders have distinct, non-interchangeable roles, suggesting that memory asymmetry could serve as a biophysical mechanism for modulating the complexity and frequency selectivity of neuronal response to rhythmic input, with potential relevance to neural processing and neuromodulation strategies.