Hodgkin–Huxley Excitability Model

• The excitability model is a quantitative framework that employs voltage-dependent ion channel gating to simulate action potential generation and propagation in neurons.
• It integrates classical Hodgkin–Huxley kinetics with bifurcation and energy-based threshold analyses to distinguish spiking regimes.
• Hybrid and stochastic extensions enrich the model by incorporating data-driven, mesoscale, and homeostatic dynamics to capture realistic neuronal variability.

The excitability model of Hodgkin–Huxley (HH) defines a quantitative and biophysically interpretable framework for understanding how voltage-dependent ion channel dynamics underlie the generation and propagation of action potentials in excitable cells. The standard HH formulation treats individual channel populations—predominantly Na⁺ and K⁺—as sets of macroscopic, voltage- and time-dependent conductances gated by empirically defined kinetic variables. Recent work has both expanded and hybridized the classical paradigm, embedding it within stochastic, mesoscale, electromechanical, homeostatic, and data-driven architectures.

1. Classical Hodgkin–Huxley Formalism: Core Equations and Kinetics

The HH model specifies the temporal evolution of the membrane potential $V(t)$ in response to an externally injected current $I_\mathrm{ext}(t)$ through the equation

$$C_m \frac{dV}{dt} = - \left[ \bar{g}_{\mathrm{Na}} m^3 h (V - E_{\mathrm{Na}}) + \bar{g}_K n^4 (V - E_K) + g_L (V - E_L) \right] + I_\mathrm{ext}(t)$$

where $C_m$ is membrane capacitance, $\bar{g}_{\mathrm{Na}}$, $\bar{g}_K$, $g_L$ are maximal conductances for sodium, potassium, and leakage, $E_{\mathrm{Na}}, E_K, E_L$ are their Nernst potentials, and $m$, $h$, $n$ are gating variables for sodium activation, sodium inactivation, and potassium activation, respectively.

The gating variables evolve according to

$\frac{dx}{dt} = \alpha_x(V)(1-x) - \beta_x(V)x,$

for $x \in \{m, h, n\}$, with rate functions empirically derived from voltage-clamp data, e.g. (units: $V$ in mV): $$\begin{aligned} \alpha_m(V) &= 0.1 (25 - V)/(\exp((25-V)/10) - 1), &\beta_m(V) = 4 \exp(-V/18) \ \alpha_h(V) &= 0.07 \exp(-V/20), &\beta_h(V) = 1/(\exp((30-V)/10)+1) \ \alpha_n(V) &= 0.01 (10 - V)/(\exp((10-V)/10) - 1), &\beta_n(V) = 0.125 \exp(-V/80) \end{aligned}$$ This deterministic, nine-parameter system robustly reproduces all-or-none action potentials, threshold phenomena, refractoriness, and repetitive firing (Estienne, 2023).

2. Modern Bifurcation, Threshold, and Excitability Mechanics

2.1 Type I/II Excitability, Bifurcations, and Periodic Orbits

Periodic firing and underlying bifurcations are explicit in the HH model. The saddle–node of limit cycles at $I_{5} \approx 6.265\,\mu\mathrm{A}/\mathrm{cm}^2$ underlies type I excitability: arbitrarily low-frequency oscillations near threshold. Type II excitability emerges via a subcritical Hopf bifurcation at $I_{2} \approx 9.74\,\mu\mathrm{A}/\mathrm{cm}^2$, where oscillations appear at nonzero frequency. Beyond the Hopf point, unstable periodic orbits and additional bifurcations (period-doubling, homoclinic) structure the transition from quiescence to repetitive spiking (Balti et al., 2015).

2.2 Energy-Based Threshold: Dissipativity and Required Supply

A recent approach formalizes threshold as a local maximum of the required supply $S_r(x^*)$, defined as the minimal external energy required to drive the system from rest to a target state $x^*$ (Sepulchre et al., 2 Apr 2025): $$S_r(x^*) = \inf_{i_{(-\infty,0]}\colon\,x(0)=x^*} \int_{-\infty}^0 i(t) v(t) dt$$ In the HH framework, the energy-based threshold unambiguously separates subthreshold and suprathreshold responses and generalizes classical definitions, offering robust, input–output-based criteria independent of specific voltage, current, or channel state (Sepulchre et al., 2 Apr 2025).

3. Dimensional Reduction, Robustness, and Homeostasis

Although sensitive to parameter variations, the functional output of the HH model is largely determined by two dimensions: the instantaneous “structural” conductance ratio ($S$) and the “kinetic” recovery rate ($K$) (Ori et al., 2018). Specifically,

$$S = \frac{\bar{g}_{\mathrm{Na}}}{\bar{g}_{\mathrm{Na}} + \bar{g}_K}$$

$$K = \frac{\alpha_n(\bar{V}) + \beta_m(\bar{V})}{\alpha_n(\bar{V}) + \beta_m(\bar{V}) + \alpha_m(\bar{V}) + \beta_n(\bar{V})}$$

Models collapse, in $(S, K)$ space, into three regimes: non-excitable, single-spike excitable, and oscillatory (pacemaking). Slow inactivation of Na⁺ channels is modeled as a dynamic reduction of $S$, providing automatic, local homeostatic stabilization of excitability (Ori et al., 2018).

4. Extended and Hybrid Models

4.1 Data-Driven Hybrid Hodgkin–Huxley Models

Estienne’s hybrid HH-ANN model replaces each empirical rate function $\alpha_x(V), \beta_x(V)$ with a trainable two-layer neural network (ANN), optimized via backpropagation to match experimentally observed voltage traces under known stimuli:

• Each rate function net has two layers, each with a single neuron.
• First layer activation: (log-)sigmoid (monotonicity); second: ReLU (non-negativity).
• Only six nets (one for each $\alpha_x$, $\beta_x$).
• Training on only two suprathreshold pulses (with augmentation) suffices to recover thresholds, spike shape, refractoriness, and frequency–current curves with fidelity (errors: waveform amplitude/duration $<5\%$, f–I curve error $\sim$20%) (Estienne, 2023).

4.2 Mesoscopic and Stochastic Extensions

Recent mesoscale HH reductions describe collective excitability of fields, with state variables such as sodium kinetic-energy density $J(x,t)$ and excitability $H(x,t)$, and treat firing rate as a dynamic redistribution of energy. These models reproduce wave propagation, oscillation spectra, and damped temporal/spatial responses in neural tissue (Qin et al., 2022).

Stochastic HH models introduce rigorously validated multiplicative noise into gate kinetics: $dg_i(V, x_i) = \sigma_i x_i(1 - x_i) dW_i(t)$ ensuring invariance of gating variables in $[0,1]$ (both Itô and Stratonovich sense), and reproducing channel-noise-induced firing, spike-time jitter, and physiologically realistic subthreshold fluctuations (Cresson et al., 2012).

5. Biophysical, Homeostatic, and Multiscale Generalizations

Subsequent models have incorporated additional physical mechanisms:

• Electrodiffusion: A “primitive” model treats Na⁺ spike activation/deactivation as an electrodiffusive phenomenon without gating variables, reproducing HH-like spikes solely via Nernst–Planck and Poisson equations in a growing hemispherical volume (Patel et al., 12 Jul 2024).
• Ionic Homeostasis: Extensions with dynamic intra/extracellular ion concentrations and pumps (e.g., Na⁺/K⁺-ATPase), as well as bath or glial buffer coupling, reveal new classes of bistable (pathological free-energy-starvation, FES) and excitable (spreading depression, seizure-like) behaviors via low-dimensional bifurcation structures, e.g., in the “potassium gain/loss” parameter (Hübel et al., 2014, Hübel et al., 2013).
• Quantum Biophysics: Selectivity-filter gating models (BS model) introduce quantum corrections to HH sodium conductance by modulating $$g_{\mathrm{Na}} m^3 \to g_{\mathrm{Na}} \delta(m,k) m^3$$, where $\delta(m,k)$ encodes filter entanglement; these corrections slightly sharpen spike onset and preserve all standard excitability features (Moradi et al., 2014).

6. Mechanistic, Spatio-Temporal, and Network-Scale Formulations

Alternative mechanistic models replace empirical gating by ODEs grounded in first-principles conductance kinetics, predicting saddle-node thresholds and all-or-nothing responses with reduced parameter sets (Deng, 2015). Memristive circuit models generalize HH–type excitability to spatio-temporal domains, unifying positive/negative feedback and threshold mechanisms from single neurons to population-level neural fields, and reproducing both SNIC bifurcations and Amari-type spatial bumps (Burger et al., 28 May 2025).

7. Integrative and Extended Theoretical Frameworks

Modern theoretical developments place HH excitability within input–output dissipativity theory and multivariable singular perturbation; the mesoscopic, energy-based, and multiscale approaches enable quantitative, robust, and physiologically meaningful descriptions of neuronal excitability. These frameworks offer interpretability, data-driven adaptability, and extensibility to include engineering, network, and clinical perspectives, supporting future research in channelopathy, neural coding, and complex systems neuroscience (Estienne, 2023, Sepulchre et al., 2 Apr 2025, Ori et al., 2018, Qin et al., 2022).