Explainer16 min

Neuron biophysics

Nernst and Goldman, the cable equation, dendrites, myelin, and conduction velocity. The physics underneath the four Hodgkin–Huxley equations you just integrated.

AssumesProject A: Neurons from the equations up SpineComputational neuroscienceElectrophysiology / wet lab

On this page16 min

Where the reversal potentials come from
The membrane as capacitor
The cable equation
Propagation and myelin
Dendrites compute

You integrated Hodgkin–Huxley and a spike came out. The parameters were given: E_Na = 50 mV, E_K = −77, a membrane capacitance of 1 µF/cm². Each of those is a piece of physics, and once you see where they come from, the model stops being a curve fit and becomes a description of a cell. Let’s derive the numbers, then extend the point neuron into the long, branching thing a real neuron is.

Hodgkin–Huxley

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Figure 1. The model again, for reference while reading.

Where the reversal potentials come from

An ion is pulled two ways across a membrane: down its concentration gradient by diffusion, and along the electric field by its charge. The voltage at which these balance for one ion is its Nernst potential: $E = \frac{RT}{zF}\ln\frac{C_{out}}{C_{in}}$ , about 61 mV times the log10 of the ratio for a monovalent ion at body temperature. Potassium is roughly 140 mM inside and 5 outside, so $E_K \approx 61 \times \log_{10}(5/140) \approx -88$ mV (squid values give −77). Sodium is about 15 inside and 145 outside: $E_{Na} \approx +60$ mV. Chloride, more outside than in, sits near the resting potential. Every “battery” in the membrane circuit is a concentration ratio.

At body temperature the constants collapse to one number:

E \approx 61 \log_{10}\frac{C_{out}}{C_{in}}\ \text{mV}

Here $R$ is the gas constant, $T$ the absolute temperature, $z$ the ion’s valence, $F$ Faraday’s constant, and $C_{out}/C_{in}$ the concentration ratio across the membrane.

The resting potential is not any single Nernst potential; it is the weighted average, weighted by permeability, given by the Goldman–Hodgkin–Katz equation. At rest, potassium permeability dominates, so the cell sits near E_K. During the spike, sodium permeability dominates for a moment and the voltage heads toward E_Na. The action potential is a competition between two batteries, decided by which channels are open.

Goldman, Hodgkin, and Katz extended Nernst to several ions at once, weighting each by its permeability:

V_{rest} = \frac{RT}{F}\ln\frac{P_K[K^+]

Here $P$ is each ion’s permeability. Whichever ion’s path is most open pulls the voltage toward its own battery.

The membrane as capacitor

A lipid bilayer is about 5 nanometres thick with a dielectric constant around 2. That gives a capacitance close to 1 µF/cm² for every cell membrane ever measured, which is why that number is a constant in the model. It matters because the membrane time constant, τ = R_m C_m, sets how fast a neuron can change its voltage: with a membrane resistance of 10,000 Ω·cm², τ is 10 ms, and a neuron cannot respond to inputs faster than that unless the input is enormous.

Resistance times capacitance is the time constant:

\tau = R_m C_m

With $R_m = 10000\ \Omega\cdot\text{cm}^2$ and $C_m = 1\ \mu\text{F/cm}^2$ , $\tau = 10\ \text{ms}$ : the voltage cannot follow anything much faster unless the input is enormous.

Predict before you look

A neuron's dendrite is a long thin cable. A synaptic input arrives at its far end. What happens to that signal by the time it reaches the cell body?

Attenuated and slowed. A dendrite is a leaky cable: current flows along its interior and leaks out through the membrane as it goes. The voltage falls off exponentially with a length constant λ = √(r_m / r_i), typically a few hundred micrometres, and the membrane capacitance smears the signal in time. A synapse at the end of a long thin dendrite has a small, slow effect at the soma unless the dendrite has active channels to boost it, which many do.

The cable equation

Treat a dendrite or axon as a cylinder with internal resistance per length r_i, membrane resistance per length r_m, and membrane capacitance per length c_m. Current conservation gives λ² ∂²V/∂x² = τ ∂V/∂t + V, the cable equation, with λ the length constant and τ the time constant. Its solutions say how far and how fast a passive signal spreads. Every simulation of a neuron with morphology (NEURON, the software, is the standard) is this equation on a branching tree with Hodgkin–Huxley channels sprinkled along it.

The length constant sets the spatial scale, and the voltage falls exponentially along the cable:

\lambda = \sqrt{r_m/r_i}, \quad V(x) = V_0 e^{-x/\lambda}

At one length constant the signal is down to $1/e$ of where it started. Current conservation on the cylinder gives the cable equation itself:

\lambda^2 \frac{\partial^2 V}{\partial x^2} = \tau \frac{\partial V}{\partial t} + V

Its solutions say how far a passive signal spreads before it leaks away, and how slowly.

Propagation and myelin

An action potential propagates because the depolarized patch of membrane pushes current into the neighbouring patch, bringing it to threshold. Speed scales with the square root of axon diameter in unmyelinated fibres; the squid’s giant axon is a millimetre across to reach 25 m/s. Vertebrates found a better trick: wrap the axon in myelin, an insulator that raises r_m and lowers c_m so the signal jumps between exposed nodes. A 20 µm myelinated fibre conducts at 100 m/s. This is why peripheral nerve stimulation recruits large myelinated fibres first, why demyelinating diseases slow conduction, and why the compound action potential you might record from a nerve has separate peaks for fibre classes.

Speed grows with the square root of diameter in unmyelinated fibres:

v \propto \sqrt{d}

The squid buys speed with width; vertebrates buy it with myelin, which changes the constant by raising $r_m$ and lowering $c_m$ along the insulated stretches.

Dendrites compute

Real dendrites are not passive. They carry voltage-gated channels that produce local spikes, NMDA receptors whose conductance depends on voltage, and branch geometries that make some inputs sum supralinearly and others not. A single pyramidal neuron does the work of a small network. The point neuron in your model is a caricature, and knowing exactly what it leaves out is what lets you use it well.

Stop here if you like. Nernst, GHK, capacitance, the cable equation, myelin. That is the physics of one cell.

Deep dive The sodium-potassium pump and why the batteries do not run down 2 min

Every spike lets some sodium in and some potassium out, which would eventually erase the gradients. The Na⁺/K⁺-ATPase pumps three sodium out and two potassium in per ATP, continuously, consuming a large fraction of the brain’s energy budget. It is slow and it does not participate in the spike itself; it maintains the batteries in the background. In a Hodgkin–Huxley simulation the gradients are fixed constants, which is a fine approximation over seconds and a poor one over minutes of high activity.

Deep dive Channel diversity 3 min

The squid had two voltage-gated channel types. Mammalian neurons express dozens: several sodium types with different kinetics, potassium channels that activate fast or slow or only after calcium enters, calcium channels that produce bursts, hyperpolarization-activated channels that produce rhythms. The Izhikevich model’s twenty firing patterns are a phenomenological summary of this diversity, and the thalamic pacemaker behind alpha and sleep spindles is a specific combination (a low-threshold calcium current and a hyperpolarization-activated current) that the oscillations explainer describes.

Recall

Where does the sodium reversal potential of about +60 mV come from?

The Nernst equation for sodium's concentration ratio (roughly 145 mM outside, 15 inside): E ≈ 61 mV × log10(145/15) ≈ +60 mV.

Recall

What does the length constant of a dendrite tell you?

How far a passive voltage signal spreads before decaying to 1/e of its value: λ = √(r_m/r_i), typically a few hundred micrometres. Distant synapses have small, slow effects at the soma unless boosted by active channels.

Recall

Why does myelin speed conduction?

It insulates the axon, raising membrane resistance and lowering capacitance, so the depolarization jumps between exposed nodes instead of charging the whole membrane continuously.

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