Frequency-input-timescale relations in two-dimensional Hindmarsh-Rose model

Stability analysis and control of two-timescale systems are of importance for understanding oscillation emergence mechanisms. In this work, we study theoretically frequency-input-timescale (called the f-z-μ) relation in the two-dimensional Hindmarsh-Rose (2DHR) oscillator. All bifurcations of saddle-node, Andronov-Hopf and saddle-node on invariant cycle are also computed with the respective parameter sets, giving the respective frequency gradient equation in terms of z and μ derived by approaches of their small perturbation on the oscillation. The core target in this analysis is to understand dynamical property of the f-μ cleft on the 2DHR undergoing the SNIC bifurcation, compared to oscillatory dynamics of the system exhibiting the other bifurcations.