A nullcline-guided discrete-time map for neurons with subcritical Hopf bifurcation dynamics

Neurons near a subcritical Hopf bifurcation are distinguished by their ability to produce subthreshold oscillations, maintain bistability between rest and repetitive spiking, and fire preferentially in response to inputs near their intrinsic resonant frequency. Simulating these properties in large neural networks using continuous-time models such as the Hodgkin-Huxley formalism is computationally prohibitive, motivating the search for reduced representations that retain the essential dynamics. Here we derive a two-dimensional discrete-time map from the continuous I _{Na, p} +I _K model by exploiting the time-scale separation between the membrane potential and the potassium gating variable, and using nullcline geometry to approximate the slow drift along each branch of the cubic v-nullcline. The resulting map consists of an outer piecewise-linear loop that produces realistic action potential waveforms and an inner switching region that reproduces focus-like subthreshold oscillations, with a slow recovery variable governing the transition between the two. The injected current modulates the size of the inner region, naturally encoding the bifurcation structure of the original system. Numerical simulations confirm that the discrete model reproduces the main dynamical signatures of bistability, hysteresis under a slowly varying current ramp, and frequency-selective firing in response to periodic burst stimulation-all without solving a single differential equation. Crucially, the map retains selected physically interpretable parameters of the continuous model while replacing continuous-time integration with a lightweight algebraic update. Benchmarking against explicit Euler and fourth-order Runge-Kutta integration shows that the optimized map has a per-update cost comparable to Euler and substantially lower than Runge-Kutta, supporting its use as a compact, interpretable discrete representation of subcritical-Hopf dynamics rather than as a direct numerical integrator of the continuous model.