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

Area of Science:

• Computational Neuroscience
• Mathematical Biology
• Neural Dynamics Modeling

Background:

• Neurons near subcritical Hopf bifurcations exhibit complex dynamics: subthreshold oscillations, bistability, and resonant firing.
• Simulating these dynamics in large neural networks using continuous-time models (e.g., Hodgkin-Huxley) is computationally intensive.
• Reduced models are needed to efficiently capture essential neuronal behaviors.

Purpose of the Study:

• To derive a computationally tractable discrete-time map from a continuous-time model of neuronal dynamics.
• To retain key dynamical properties like subthreshold oscillations and bistability in the reduced model.
• To provide a compact and interpretable representation of subcritical Hopf dynamics.

Main Methods:

• Derived a two-dimensional discrete-time map from the continuous I_Na,p + I_K model.
• Exploited time-scale separation and nullcline geometry for model reduction.
• Approximated slow drift along the cubic v-nullcline.

Main Results:

• The discrete map features a piecewise-linear loop for action potentials and an inner region for subthreshold oscillations.
• Injected current naturally encodes the bifurcation structure, modulating the inner region size.
• Simulations confirmed accurate reproduction of bistability, hysteresis, and frequency-selective firing.

Conclusions:

• The discrete map offers a computationally efficient and interpretable alternative to continuous-time models for subcritical Hopf dynamics.
• It retains key neuronal properties while significantly reducing computational cost compared to numerical integration methods.
• This reduced model facilitates simulations of large neural networks exhibiting complex firing patterns.