Hodgkin-Huxley Conductance Calculator
Find the instantaneous Na+ and K+ membrane conductance and ionic current from Hodgkin-Huxley gating variables m, h, and n.
⚡ What is the Hodgkin-Huxley Conductance Calculator?
The Hodgkin-Huxley conductance calculator finds the instantaneous Na+ and K+ membrane conductance, and the resulting ionic current, from the classic Hodgkin-Huxley gating-variable formulation gNa = gNaBar x m^3 x h and gK = gKBar x n^4. This is the mathematical core of the Nobel Prize winning Hodgkin-Huxley model, which was the first quantitative description of how voltage-gated ion channels generate the action potential.
Neuroscience and computational biology courses use this calculator to build intuition for how the gating variables m, h, and n combine to set conductance at any instant, before tackling the full system of differential equations that describes how those variables evolve over time. Students verify by-hand homework calculations of conductance and current at specific points along an action potential, such as at rest, at the depolarization peak, or during repolarization. Computational neuroscientists use the same gNa/gK formulas as building blocks inside larger conductance-based neuron simulations.
A common misconception is that m, h, and n are simple fixed constants for a given neuron. They are not, each one is itself a time- and voltage-dependent variable that relaxes toward a voltage-dependent steady-state value with its own time constant, this calculator computes conductance and current for whatever instantaneous m, h, n values you provide, it does not simulate how those values change over time.
This calculator is useful because manually computing m cubed, n to the fourth power, and then two Ohm's-law current calculations is tedious and error-prone by hand, especially when comparing several points along an action potential's time course side by side.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Near Resting Potential
ḡNa=120, ḡK=36 mS/cm², m=0.05, h=0.6, n=0.32, V=−65 mV, ENa=50 mV, EK=−77 mV
Example 2 — Peak of the Action Potential Upstroke
Same neuron, Na+ channels mostly open and inactivating, K+ channels beginning to open: m=0.95, h=0.1, n=0.5, V=+40 mV
Example 3 — Repolarization Phase
Na+ channels inactivated, K+ conductance dominant and falling voltage: m=0.3, h=0.02, n=0.75, V=−20 mV
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Hodgkin-Huxley conductance model?
The Hodgkin-Huxley model describes a neuron's Na+ and K+ membrane conductance as gNa = gNaBar x m^3 x h and gK = gKBar x n^4, where gNaBar and gKBar are maximum conductances and m, h, n are voltage- and time-dependent gating variables between 0 and 1 that were fit to real squid giant axon voltage-clamp recordings.
What is the formula for Na+ and K+ conductance?
gNa = gNaBar x m^3 x h for sodium and gK = gKBar x n^4 for potassium, where gNaBar and gKBar are the maximum possible conductances (all channels open) and m, h, n are dimensionless gating variables ranging from 0 to 1.
What do the gating variables m, h, and n represent?
m is the Na+ activation variable (rises with depolarization, opens the channel), h is the Na+ inactivation variable (falls with depolarization, closes the channel after a delay), and n is the K+ activation variable (rises with depolarization, more slowly than m). All three evolve toward voltage-dependent steady-state values over time.
Why is m cubed but n raised to the fourth power?
Hodgkin and Huxley modeled each Na+ channel as needing 3 independent activation subunits in the permissive state to conduct, and each K+ channel as needing 4, so the probability that all subunits are simultaneously permissive is m^3 for Na+ and n^4 for K+. These specific exponents were chosen because they best fit the shape of their experimental voltage-clamp current traces.
How do I find the ionic current from conductance?
Ohm's law for ion channels: I = g x (V - E), where g is the channel's conductance, V is the current membrane voltage, and E is that ion's reversal (equilibrium) potential, typically obtained from the Nernst equation. A negative I is an inward (depolarizing) current; a positive I is an outward (repolarizing) current.
What are typical gNaBar and gKBar values?
Hodgkin and Huxley's original squid giant axon measurements used gNaBar around 120 mS/cm² and gKBar around 36 mS/cm², the defaults used on this page. Different neuron types and species have different maximum conductance densities depending on their ion channel expression.
Why is gNa so small near resting potential?
At rest, the Na+ activation variable m is small (most Na+ channels are closed) while the inactivation variable h is close to 1 (channels are available but not yet opened), so the product m^3 x h stays small, keeping resting gNa low even though gNaBar itself is large.
How does this relate to the actual shape of an action potential?
During the action potential upstroke, m rises rapidly (Na+ channels open) while h has not yet fallen much, producing a large transient gNa and a large inward INa that drives rapid depolarization. As h falls (inactivation) and n rises (delayed K+ activation), gK becomes dominant and an outward IK repolarizes the membrane back toward rest.
Is this a full dynamic simulation of an action potential?
No, this calculator computes conductance and current at a single instant for gating variable values you supply directly. The full Hodgkin-Huxley model additionally specifies differential equations for how m, h, and n evolve continuously over time as functions of voltage, producing the complete time course of an action potential.
What units does this calculator use?
Conductances gNaBar, gKBar, gNa, and gK are in millisiemens per square centimeter (mS/cm²), the standard Hodgkin-Huxley unit for membrane area-normalized conductance. Currents INa and IK are in microamps per square centimeter (µA/cm²), and voltages are in millivolts (mV).