Revisiting the modeling of cell electrophysiology using a delay-based equation

Abstract Voltage-gated ion channels respond to the transmembrane voltage by transitioning among conduction states, thereby shaping macroscopic ionic currents and the voltage itself. Although microscopic gating is usually modeled as a memoryless Markov process, the resulting population-level conductances exhibit an effective dependence on the past transmembrane-voltage trajectory and can therefore be represented using discrete delays (Dirac kernels) or, more generally, Volterra memory kernels. In this study, we propose a simple mathematical model that describes the action potential using a single delay differential equation based on the conductances of sodium and potassium channels. Our model accurately reproduces the behavior of a standard Hodgkin–Huxley action potential, including the characteristic sodium and potassium conductance profiles. This framework captures essential electrophysiological features such as subthreshold and suprathreshold responses, and the all-or-nothing principle. Through this delay formulation, we derive explicit analytical approximations for strength-interval relations, expressing excitability recovery directly in terms of a small set of interpretable parameters. We also provide analytical expressions for the relation between spike frequency and the amplitude of a prolonged stimulus. Additionally, by changing the memory kernel, we connect the discrete-delay model to other representations of macroscopic channel memory, including linear-chain (Erlang) ODE approximations, Markov (CTMC) models, and fractional calculus formulations. This approach highlights the potential of delayed differential equations and memory kernels to capture essential neuronal dynamics.