Voltage-Gated Ion Channel Isoforms

• Voltage-gated ion channel isoforms are distinct molecular variants that modulate neuronal excitability via unique kinetic and conductance properties.
• Conductance-based models using six-state Markov representations and first-order dynamics elucidate stability, Hopf bifurcations, and oscillatory thresholds.
• Isoform-specific excitability landscapes guide synthetic system design by aligning channel profiles with desired oscillatory and stability characteristics.

Voltage-gated ion channel isoforms comprise distinct molecular variants of sodium and potassium channels that govern the electrical excitability of neurons. Each isoform exhibits unique kinetic and conductance properties, exerting differential effects on action potential generation. A conductance-based neuron model featuring a six-state Markov representation of human NaV sodium channel isoforms and a first-order KV3.1 potassium channel model reveals how the biophysical fingerprint of each isoform sculpts neuronal excitability, stability, and the capacity for repetitive firing (Abdullah et al., 30 Nov 2025).

1. Mathematical Framework for Voltage-Gated Channel Dynamics

A biophysical neuron model is composed of voltage-dependent sodium (NaV) and potassium (KV3.1) channels. Sodium channel isoforms are described by a six-state Markov model, including two closed ($C_1, C_2$), two open ($O_1, O_2$), and two inactivated ($I_1, I_2$) states. State transitions occur at voltage-dependent rates: $$r_{ij}(V) = B^{\mathrm{hyp}_{ij}}\left[1+e^{(V - V^{\mathrm{hyp}_{ij}})/K^{\mathrm{hyp}_{ij}}}\right]^{-1} + B^{\mathrm{dep}_{ij}}\left[1+e^{(V - V^{\mathrm{dep}_{ij}})/K^{\mathrm{dep}_{ij}}}\right]^{-1}$$ The occupation probability vector $s(t) = [C_1, C_2, O_1, O_2, I_1, I_2]^\top$ evolves according to master equations: $$\frac{d s_i}{d t} = \sum_{j=1}^6 Q_{ij}(V)\,s_j, \quad i=1\ldots 6$$ with $Q_{ii}(V) = -\sum_{j\neq i} Q_{ji}(V)$ and normalization $\sum_i s_i(t) = 1$. The macroscopic sodium current is given by: $$I_{Na} = \bar g_{Na}(s_{O_1} + s_{O_2})(V_m - E_{Na})$$ The KV3.1 potassium current uses a single gating variable $m(t)$: $$\frac{d m}{d t} = [m_\infty(V_m) - m] / \tau_m(V_m)$$ where

$$m_\infty(V) = \frac{1}{1 + e^{(18.7-V)/9.7}}, \quad \tau_m(V) = \frac{1}{1 + e^{-(V+46.56)/44.14}}$$

and the current is

$I_K = \bar g_K m (V_m - E_K)$

The complete system state is $$x = [V_m, m, s_1, \dots, s_6]^\top \in \mathbb{R}^8$$.

2. Stability and Bifurcation Analysis

Action potential generation depends critically on stability in the high-dimensional conductance-kinetics space. Equilibrium points $x^*$ are computed by setting all time derivatives to zero and solving: $$\frac{d V_m}{dt} = (1/C_m)[I_{ext} - I_{Na} - I_K] = 0$$

$\frac{d m}{dt} = 0, \quad Q(V_m^*) s^* = 0$

The Jacobian matrix $J$ at equilibrium is block-partitioned: $$J = \begin{bmatrix} A & B \ C & Q(V_m^*) \end{bmatrix}$$ where $A$ (2×2) corresponds to $(V_m, m)$, $B$ (2×6) and $C$ (6×2) couple sodium channel states, and $Q$ (6×6) contains transition rates. Hopf bifurcation occurs when a complex conjugate pair of Jacobian eigenvalues, $\lambda_{1,2}$, crosses the imaginary axis: $$\det(J - \lambda I) = 0 \qquad \operatorname{Re}(\lambda_{1,2}) = 0,\quad \operatorname{Im}(\lambda_{1,2}) = \pm \omega_0 \neq 0$$ Nondegeneracy requires: $$\frac{\partial}{\partial \mu}\operatorname{Re}(\lambda_{1,2})\bigg|_{\mu=\mu_H} \neq 0,$$ where $\mu = (\bar g_{Na}, \bar g_K, I_{ext})$.

3. Isoform-Specific Excitability Landscapes

Excitability landscapes, constructed via heatmaps, systematically compare how maximal sodium and potassium conductances ($\bar g_{Na}$, $\bar g_K$) shape the minimal stimulus ($I^*_{ext}$) required for sustained oscillatory firing. Each isoform yields a distinct landscape, revealing both quantitative and qualitative variations:

NaV Isoform	Excitable Regime Width	$I^*_{ext}$ Threshold	Oscillatory Capability
NaV1.3/1.4/1.6	Very broad	Low (5–15 mA/cm²)	Robust repetitive firing
NaV1.1/1.2	Moderate	Moderate at high $\bar g_{Na}$	Sustained, higher threshold
NaV1.5/1.8	Narrow	High	Limited to select conditions
NaV1.7	Minimal/None	Not achievable	No stable oscillations
NaV1.9	Very limited	High	Rare oscillatory episodes

Plots of $I^*_{ext}$ as heatmaps clarify regions of high (bright) versus low (dark/black) excitability. Superimposed Hopf bifurcation boundaries define where sustained oscillations are theoretically attainable.

4. Comparative Isoform Dynamics and Physiological Implications

NaV1.3, NaV1.4, and NaV1.6 isoforms support extensive oscillatory domains: they enable action potential trains at relatively low stimulus, with stability preserved over a broad span of $\bar g_{Na}/\bar g_K$ ratios. These properties designate them as optimal candidates where robust, rhythmic firing is required. In contrast, NaV1.7 and NaV1.9 isoforms display minimal or negligible oscillatory regimes, necessitating high stimuli for rare firing or none at all, making them suitable for contexts demanding cellular quiescence or single-spike precision. Isoforms such as NaV1.5 and NaV1.8 occupy intermediate positions, with oscillations constrained to higher conductance ratios or current stimuli (Abdullah et al., 30 Nov 2025).

5. Engineering Synthetic Excitable Systems

These findings deliver actionable parameters for synthetic excitable system design. Selecting NaV isoforms with broad oscillatory regimes, especially NaV1.3, NaV1.4, or NaV1.6, in combination with moderate overexpression of $\bar g_{Na}$ and $\bar g_K$, ensures robust limit-cycle firing in engineered cells. Safe-operating regions in $(\bar g_{Na}, \bar g_K, I_{ext})$ parameter space are delineated by heatmap excitability zones and Hopf bifurcation curves, enabling reliable action potential trains under physiological and synthetic variability. Isoforms with narrow or absent excitable regions, such as NaV1.7 and NaV1.9, may be selectively expressed when it is necessary to suppress spontaneous firing or allow only high-threshold responses.

6. Key Bifurcation and Stability Formulas

Critical formulas for analyzing channel-mediated instability and excitability include:

• Characteristic equation: $\det(J - \lambda I) = 0$
• Hopf bifurcation condition: $\operatorname{Re}(\lambda_{1,2}) = 0$, $$\operatorname{Im}(\lambda_{1,2}) = \pm \omega_0 \neq 0$$
• Nondegeneracy (transversality): $$\frac{\partial}{\partial \mu} \operatorname{Re}(\lambda_{1,2})|_{\mu=\mu_H} \neq 0$$, $\mu = (\bar g_{Na}, \bar g_K, I_{ext})$
• For reduced $2\times2$ Jacobian cases: $\mathrm{tr}(J) = 0$, $\det(J) > 0$

7. Channel Heterogeneity, Excitability, and Design Guidelines

The heterogeneity among NaV isoforms directly shapes the landscape of neuronal excitability. The rigorous mapping of isoform-dependent action potential generation provides quantifiable guidelines—low-threshold repetitive firing requires choice of NaV1.3, 1.4, or 1.6 and suitable adjustment of conductance parameters. These insights facilitate not only the analysis of physiological neural dynamics but also the rational engineering of synthetic excitable cells with tailored firing characteristics (Abdullah et al., 30 Nov 2025).