Analysis of a Hopf Bifurcation in a Neural System by Impedance Spectroscopy

A neuromorphic system characterized by a system of two differential equations, similar to the Morris-Lecar model, is considered. The system can be formally represented as an equivalent electrical circuit composed of elementary components: a capacitor, a resistor, and an inductor. This analogy allows the derivation of an analytical impedance function, which describes the system's linear response to periodic external perturbations. The impedance, expressed in terms of these circuit elements, provides insight into how the system processes input signals across frequencies. Numerical simulations show that, as a control parameter is varied, the system undergoes a Hopf bifurcation that marks the transition from a stable fixed point to a regime of sustained oscillations. Across this transition, the impedance spectrum experiences qualitative changes, including the emergence of resonance peaks and shifts in phase behavior, reflecting the underlying reorganization of the system's dynamics. These results illustrate how impedance measurements can reveal critical transitions in nonlinear excitable systems and draw analogies with resonance phenomena in both biological and electrochemical contexts.