INa,p+IK Model for Excitable Membranes

• INa,p+IK model is a reduced-dimensional framework that defines the interplay between persistent sodium and delayed potassium currents in excitable membranes.
• It employs biophysical reduction techniques and quasi-steady-state approximations to simplify complex Hodgkin–Huxley dynamics while preserving key spiking behaviors.
• The model’s computational efficiency supports neuromorphic hardware designs, enabling minimal-component, biologically plausible circuits for neural and cardiac simulations.

The INa,p + IK model is a reduced dimensional model for excitable membranes that captures the essential dynamical interactions between a persistent sodium current (INa,p) and a potassium current (IK), serving as a tractable alternative to the full Hodgkin–Huxley system. This model has become increasingly relevant in both computational neuroscience and neuromorphic hardware design, owing to its computational efficiency and its ability to faithfully represent neuronal spiking behaviors such as resonance, adaptation, and plateau oscillations. The INa,p + IK model describes the interplay between rapid sodium activation with (often) persistent current and slower potassium activation, under simplified gating kinetics, and is supported by biophysical reduction techniques, experimental validation, and recent circuit realizations.

1. Biophysical and Mathematical Foundations

The INa,p + IK model arises from the drive to simplify the complex gating kinetics of voltage-gated sodium and potassium channels while retaining their physiological roles in action potential generation and propagation. Formally, the model describes membrane voltage dynamics as:

$$\frac{dV}{dt} = \frac{I - I_L - I_{Na,p} - I_K}{C},$$

where $C$ is the membrane capacitance, $I$ the injected current, $I_L$ the leak current, $I_{Na,p}$ the persistent sodium current, and $I_K$ the potassium current. The sodium current is treated as having instantaneous or steady-state activation:

$I_{Na,p} = g_{Na} m_\infty(V) (V - E_{Na}),$

$m_\infty(V) = \frac{1}{1 + e^{(V_{1/2} - V)/k}},$

with $g_{Na}$ the sodium conductance, $E_{Na}$ the reversal potential, and parameters $V_{1/2}$, $k$ specifying half-activation and slope. Potassium channel dynamics are governed by a first-order kinetics:

$I_K = g_K n (V - E_K),$

$\frac{dn}{dt} = \frac{n_\infty(V) - n}{\tau(V)},$

$n_\infty(V) = \frac{1}{1 + e^{(V^*_n - V)/k_n}},$

mirroring the slow delayed rectifier activation (Nabil et al., 3 Jun 2025).

This two-dimensional reduction retains the crucial feedbacks of the full Hodgkin–Huxley scheme while drastically simplifying both analysis and simulation.

2. Reduction from Detailed Gating Kinetics

The INa,p + IK model is rigorously justified by systematic reduction of high-dimensional kinetic models. Detailed schemes, such as the fifteen-state sodium channel Markov model, include multiple closed/open states, multi-stage fast inactivation, and nested slow inactivation transitions (Vaccaro, 2018). By leveraging methods of multiple scales and quasi-steady-state (QSS) approximations, rates are partitioned by their timescales:

• Rapid transitions (activation sensor movements, sub-states within fast-/slow-inactivation) are averaged using QSS.
• Independence of activation sensors allows representing the aggregate activation by a binomial distribution of open/closed channels.
• State aggregation: Fast inactivated and slow inactivated channel clusters are lumped into single effective states.

The reduced system expresses closed, open, fast inactivated, and slow inactivated macro-states using three key variables: the activation variable $m$, a fast inactivation variable $h$, and a slow inactivation variable $s$. The coupled gating kinetics are captured by:

$\frac{dm}{dt} = \alpha_m(V) [1 - m] - \beta_m(V) m$

$\frac{dh}{dt} = \alpha_h(V) [1 - h] - \beta_h(V) h$

$\frac{ds}{dt} = \alpha_s(V) [1 - s] - \beta_s(V) s$

Subtle modulation of rate coefficients by activation variables (for instance, the effective inactivation rate $p(t) \approx B_h(m)$) ensures the preserved dynamics of fast/slow inactivation and adaptation. The resulting current is

$I_{Na} = g_{Na}\, m^3\, h\, s\, (V_{Na} - V),$

and can be paired with a standard $I_K = g_K\, n^4\, (V - V_K)$ term. This approach captures spike frequency adaptation and plateau behaviors in neural and cardiac membranes while remaining computationally accessible (Vaccaro, 2018).

3. Channel-Level Biophysics and Model Calibration

Contemporary approaches ground INa,p + IK models in microscopic electrodiffusion dynamics using Poisson–Nernst–Planck (PNP) theory (Werneck et al., 2023). At the single-channel level, channel geometry (radius, length), ion concentration profiles $c(z)$, and electric field $\phi(z)$ are computed from:

$c''(z) + \frac{ZF}{RT} (c(z) \phi'(z))' = 0$

$\varepsilon \phi''(z) + ZF c(z) = 0$

The channel flux density $H_i$ for species $i$ and membrane-spanning current $I_c^i$ are given by:

$$H_i = -D_i \left( \nabla c_i + \frac{Z_i F}{RT} c_i \nabla \phi \right)$$

$$I_c^i = Z_i F \int_A \left[ \frac{dc_i}{dz} + \frac{Z_i F}{RT} c_i \frac{d\phi}{dz} \right]_{z=z_0} dA$$

The population-level current is then modeled as a sum over channel types weighted by empirically calibrated, voltage-dependent activation functions:

$$I_{tot}(V) = b_{Na}(V) I_c^{Na} + b_K(V) I_c^K + \dots$$

with $b_i(V)$ approximated by sigmoidal/Boltzmann fits:

$$b_i(V) \approx \frac{b_i^0}{1 + \exp \left( -\frac{V - V_i^*}{k_i} \right)}$$

Model predictions closely match patch-clamp data for human neural cells, confirming the sufficiency of geometric and gating representations in capturing INa,p and IK behavior. Effects such as negative $b_K(V)$ at certain voltages are attributable to local double-layer physics at the membrane (Werneck et al., 2023).

4. Model Reduction and Gating Variable Interdependence

The model dimensionality can be further reduced by leveraging empirical and theoretical relationships between gating variables. In the full Hodgkin–Huxley model, the sodium inactivation variable $h$ closely parallels the potassium activation $n$. This manifests in the approximation:

$h \simeq c(I) - n,$

where $c(I)$ depends upon the applied input current. This coupling reflects physiological interdependence possibly influenced by Na$^+$/K$^+$-ATPase activity and permits a reduction of the original 4D system to 3D and 2D systems:

• 3D Model: Replace $h$ by $c(I)-n$; retain dynamic $m$ and $n$.
• 2D Model: Further set $m \approx m_\infty(V)$, reducing ODEs to $(V, n)$.

These reduced systems preserve Hopf and Bautin bifurcations and the spiking dynamics of the parent model, providing a practical means to simulate and analyze membrane excitability with persistent sodium and potassium currents (Branco et al., 26 Feb 2024).

Cable equations derived from this framework reveal that the velocity of action potential propagation scales as

$v(R, C_m) = \frac{\alpha}{C_m R^\beta},$

with $\alpha > 0$, $0 < \beta < 1$, linking macroscopic conduction to biophysical membrane properties and axon geometry. This facilitates quantitative predictions about conduction velocity in response to parameter changes.

5. Hardware Implementation: Minimal Neuron Circuits

The INa,p + IK model's low dimensionality makes it ideal for efficient neuromorphic hardware realization. Circuits emulate the model with a minimal number of components (e.g., three MOSFETs, two capacitors, one resistor) (Nabil et al., 3 Jun 2025). The methodology is as follows:

• Membrane capacitance is implemented directly as a physical capacitor.
• Potassium conductance (slow activation) uses a MOSFET plus an RC network, capturing the delayed rectification and afterhyperpolarization.
• Sodium conductance (instantaneous activation) is implemented with a device exhibiting type-N negative differential resistance (NNDR), such as a JFET-based element or similar. The N-shaped I–V curve provides voltage-controlled positive feedback: as the membrane voltage rises, the NNDR device generates a rapid upstroke, mimicking biological sodium inflow.

The circuit equations for hardware instantiation are structurally analogous to the model equations:

$$\frac{dV_{out}}{dt} = \frac{1}{C_1} [I - i_{ps_{Q1}} - i_{DS_{Q3}}],$$

$$\frac{dV_{GS}}{dt} = \frac{V_{out} - V_{GS}}{R_1 C_2},$$

where $i_{ps_{Q1}}$ and $i_{DS_{Q3}}$ are potassium and sodium analog currents, respectively.

A distinctive classification is made between "resonator" and "integrator" neuron circuits. Resonators, realized using the INa,p + IK paradigm and NNDR elements, exhibit subthreshold oscillations and bifurcations characteristic of Hopf transitions, whereas integrators (detailed in subsequent work) display sharp spike thresholds and binary response. Use of INa,p + IK circuits enables efficient scaling to neuromorphic systems while preserving biological plausibility and a spectrum of firing behaviors (Nabil et al., 3 Jun 2025).

6. Functional Significance and Model Implications

The INa,p + IK model, both in its computational and circuit instantiations, offers advantages for simulating and analyzing neural and cardiac excitability:

• Spike frequency adaptation arises from cumulative modulation of inactivated sodium channels (slow inactivation), prolonging interspike intervals and yielding neuronal adaptation.
• Plateau phase in cardiac action potentials can be generated by appropriate adjustment of recovery rates from fast inactivation, extending the persistence of sodium currents and supporting oscillatory plateau phenomena when combined with potassium current dynamics.
• Computational efficiency is greatly improved over the full Markov or Hodgkin–Huxley representations, without sacrificing the emergent behaviors relevant for system-level modeling or hardware emulation.
• Biophysically plausible parameters can be calibrated from single-channel measurements and aggregate current-voltage relationships, making the model directly applicable and interpretive in experimental contexts.

A plausible implication is that alterations in sodium channel recovery and coupling (e.g., due to pharmacological modulation or disease) will be directly and sensitively manifest in action potential dynamics within the INa,p + IK framework.

7. Summary Table: Principal Components of the INa,p + IK Model

Construct	Mathematical Formulation	Interpretation
Sodium current	$I_{Na,p} = g_{Na} m_\infty(V)(V-E_{Na})$	Persistent/instantaneous activation
Potassium current	$I_K = g_K n (V - E_K)$	Delayed rectifier, slow activation
Activation function	$m_\infty(V)$ (Boltzmann/sigmoidal)	Voltage-gated sodium opening
Kinetics (n)	$\frac{dn}{dt} = (n_\infty(V) - n)/\tau(V)$	First-order potassium gating dynamics
Hardware emulation	NNDR device + RC network	Minimal, biologically plausible circuit

The INa,p + IK model therefore represents a significant reduction in complexity, enabling both high-fidelity modeling of excitable membranes and scalable, biologically grounded hardware realization. It is well suited for both research and engineering contexts that require efficient yet mechanistically interpretable descriptions of sodium/potassium channel interplay in neural and cardiac systems.