Hill-Based Reformulation of the Hodgkin-Huxley Model for Interpretable Neuronal Excitability

Conductance-based models of neuronal excitability depend critically on the mathematical form used to describe voltage-dependent ion channel gating. The classical Hodgkin-Huxley (HH) formalism employs empirically derived rate expressions fitted to squid giant axon data that are not readily transferable across cell types or interpretable in terms of measurable gating properties. Here we introduce a Hill-based reformulation of the HH model in which steady-state sodium and potassium activation curves and the sodium inactivation rate are recast using Hill-type sigmoidal functions, a biologically motivated family widely used to describe cooperative and saturating processes in enzyme kinetics, gene regulation, and receptor binding. Systematic benchmarking against four compact sigmoid alternatives demonstrates that Hill functions provide superior fits to the original HH-derived gating data across all three targets. The resulting hybrid model reproduced canonical spike waveforms and frequency-current behavior, preserving the broad input-output organization of the original model. Importantly, the reformulation linked specific gating parameters to firing regimes and spike features, revealing how shifts in activation, inactivation, and steepness can systematically reshape excitability phenotypes. By making the relationship between channel kinetics and neuronal output more transparent, this framework provides an interpretable route for adapting conductance-based models to cell-specific excitability and channel-dependent changes in neural function.