Stability analysis of action potential generation using Markov models of voltage-gated sodium channel isoforms

TL;DR

The paper addresses how voltage-gated sodium channel isoforms influence action potential generation by integrating a six-state Markov model of NaV channels with a single KV3.1 potassium current in a minimal, single-compartment neuron. It employs bifurcation analysis and excitability heatmaps to map where stable oscillations occur as Na and K conductances are varied, and validates predictions with time-domain simulations. Key findings show that certain isoforms (e.g., NaV1.3, NaV1.4, NaV1.6) support broad excitable regimes at relatively low external stimuli, while others (NaV1.7, NaV1.9) exhibit minimal or highly restricted oscillatory behavior; NaV1.5 also requires higher stimuli for firing. The work provides a mechanistic framework for isoform-aware control of neuronal excitability, with implications for synthetic excitable systems and understanding channel heterogeneity in neural dynamics.

Abstract

We investigate a conductance-based neuron model to explore how voltage-gated ion channel isoforms influence action-potential generation. The model combines a six-state Markov representation of NaV channels with a first-order KV3.1 model, allowing us to vary maximal sodium and potassium conductances and compare nine NaV isoforms. Using bifurcation theory and local stability analysis, we map regions of stable limit cycles and visualize excitability landscapes via heatmap-based diagrams. These analyses show that isoforms NaV1.3, NaV1.4 and NaV1.6 support broad excitable regimes, while isoforms NaV1.7 and NaV1.9 exhibit minimal oscillatory behavior. Our findings provide insights into the role of channel heterogeneity in neuronal dynamics and may help to guide the design of synthetic excitable systems by narrowing the parameter space needed for robust action-potential trains.

Figures (6)

• Figure 1: A simple equivalent electrical circuit for a neuronal membrane
• Figure 1: Minimum external stimulus current $I_{\mathrm{ext}}$ required to induce stable oscillations across the parameter space defined by maximal sodium and potassium conductances $(\bar{g}_{\mathrm{Na}}, \bar{g}_{\mathrm{K}})$ for the NaV1.1– KV3.1 channel combination. Color intensity indicates the threshold stimulus required for a stable limit cycle, with darker regions indicating higher thresholds or stationary regimes. Black regions denote parameter combinations where no stable oscillations are observed for any tested $I_{\mathrm{ext}}$. Red dots mark representative points A, B, and C used for time-domain validation (see Figure \ref{['fig:time-series']}). The $\dagger$ point is sodium and potassium conductance in a typical neuron of a human that uses these specific potassium and sodium channel isoforms.
• Figure 2: Generalized six-state Markov model for voltage-gated sodium channels. The kinetic scheme includes: Two closed states (C1, C2) Two open states (O1, O2) Two inactivated states (I1, I2)
• Figure 2: Time series simulations of membrane potential $V_m(t)$ at three representative points (A, B, and C) in the conductance parameter space. The behavior of each point (oscillatory or stationary) confirms the predictions from eigenvalue-based stability analysis.
• Figure 3: Bifurcation structure of the Markov model of VGSC isoform NaV1.1 in the $(\bar{g}_{\mathrm{K}}, \bar{g}_{\mathrm{Na}})$ conductivity space. The heatmap shows the minimum external current $I_{\mathrm{ext}}$ [nA] required to sustain a stable limit cycle. The red curve marks the Hopf bifurcation boundary separating excitable (oscillatory) and quiescent (stationary) regimes.