Stochastic Hodgkin-Huxley

A neuron fires because thousands of microscopic gates — its ion channels — pop open to let charged atoms rush across the membrane. Each gate is like a tiny door on a faulty spring: it flips open and shut at random, on its own schedule. Watch one door and you can never predict the exact instant it opens. But when millions of doors share the job, their random flickers mostly cancel and the cell behaves smoothly. The fewer doors there are, the more each one's private coin-flip shows through — that leftover randomness is channel noise.

Call the number of channels $N$. Because each opens or closes like an independent coin toss, the relative size of the jitter shrinks as $1/\sqrt{N}$. With $N=100$ channels the current wobbles by roughly 10%; with $N=10{,}000$ it wobbles by only about 1%. So a fat axon carrying billions of channels is almost noiseless, while a thin dendrite or a tiny soma — with only a few hundred — is genuinely jittery. Channel noise is fundamentally a small-cell problem.

Rather than track every door, Fox & Lu (1994) keep the smooth Hodgkin–Huxley gating variables $m,h,n$ and add a random kick: $\frac{dm}{dt}=\alpha_m(1-m)-\beta_m m+\xi_m(t)$. The kick's strength comes straight from binomial channel statistics, $\sigma_m^2=\frac{\alpha_m\beta_m}{(\alpha_m+\beta_m)^2 N_{Na}}$ — the same $1/N$ rule in disguise. The physiological consequences are real: spike times jitter by about $0.1$–$1\,\text{ms}$, the firing threshold softens into a fuzzy band instead of a sharp line, and the cell can fire spontaneously even when the input current $I$ sits just below the deterministic threshold (rheobase).

① Set Input I just below the firing threshold and lengthen T sim — with noise on, watch the occasional unprompted spike a noiseless neuron could never produce. ② Treat Parameter 1 as the channel count: lower it (fewer doors) and spike timing scatters more; raise it and the train sharpens back up. ③ Hold Input I fixed and sweep Parameter 2 to compare a jittery small cell against a near-deterministic large one.

N_Na ≈ 60/μm², N_K ≈ 18/μm². Conductance variance: σ²_gNa = N_Na × A × m³h(1-m³h) × γ_Na². For soma area A=100 μm²: σ_gNa ≈ 0.1 nS → voltage fluctuation ≈ 0.5 mV. Significant for small neurons.

Fox & Lu (1994): replace 8-state Markov chain with Langevin SDEs for m,h,n. Noise amplitude from binomial statistics. Valid when N>>1. For small neurons (N<100): exact Markov simulation required. Computational cost: deterministic HH + 4 random numbers per step.

Channel noise: spike-timing jitter ~0.1–1 ms; spontaneous spikes at sub-rheobase I; soft effective threshold. For large neurons: negligible. For small neurons (soma < 100 μm²): physiologically significant — constrains the minimum coding precision.