Neuron biophysics
Nernst and Goldman, the cable equation, dendrites, myelin, and conduction velocity. The physics underneath the four Hodgkin–Huxley equations you just integrated.
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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.
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 = (RT/zF) ln(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 ≈ 61 × log10(5/140) ≈ −88 mV (squid values give −77). Sodium is about 15 inside and 145 outside: E_Na ≈ +60 mV. Chloride, more outside than in, sits near the resting potential. Every “battery” in the membrane circuit is a concentration ratio.
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.
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.
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.
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.
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.
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.
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