FR42 · Neural statistical physics

Stochastic Ion Channels: Why Small Neurons Fire Unreliably

When a patch holds only a few thousand channels, does the same shock always fire a spike? Not in a small patch. Each channel opens and closes at random, and with only a few thousand of them the noise decides the outcome near threshold: a patch of 100 square micrometres fires on about half of identical shocks. Larger patches return to the reliable Hodgkin–Huxley spike, and tiny ones fire with no stimulus at all.

Shrink a membrane patch and repeat the same shock. With fewer channels the spike becomes a coin flip, its timing jitters and noise alone starts to fire it.

Channels in the patch $N_{\mathrm{Na}}=6000,\ N_{\mathrm{K}}=1800$

Trials that fire a spike $\hat p_{\mathrm{spike}}=0.527\pm0.041\ (n=150)$

Spread of first-spike times $\sigma_t=1.57\pm0.13\ \mathrm{ms}$

Deterministic Hodgkin–Huxley latency $t_{\mathrm{HH}}=6.63\ \mathrm{ms}$

Shrink the patch, repeat the shock

Patch area

100\ \mu\mathrm{m^2}

\log_{10}(A/\mu\mathrm{m^2})

Shock charge, nC per cm²

6.5

q

Temperature, °C

6.3

T

Drag the rim of the patch to shrink or grow it. Every new trial gets the same shock, so any difference between trials comes from the channels.

Recording, seed and playback

Model ms shown per second

3

\dot t

Random seed of the channels

42

k

Model, method and sources

$\langle f_{\mathrm{Na}}\rangle=m^3h,\quad \operatorname{Var}f_{\mathrm{Na}}=\frac{m^3h\,(1-m^3h)}{N_{\mathrm{Na}}},\qquad \sigma_t\propto N^{-1/2}\ \ (N\to\infty)$

Each gate flips between closed and open with the Hodgkin–Huxley rates at the present voltage. Over each 0.01 ms step the voltage is held, the number of gates that flip is drawn from the exact two-state probability, and the voltage then advances with the conductance of the open channels. At a fixed voltage the open fraction is binomial with the mean and variance shown; the benchmarks check that with 1000 trials. The $N^{-1/2}$ scaling of latency jitter is a small-noise limit. In the benchmark it holds above about $1000\ \mu\mathrm{m^2}$ , and smaller patches jitter more than it predicts.

Goldwyn & Shea-Brown (2011), arXiv:1104.4823v1 · Hodgkin & Huxley (1952), J. Physiol. 117, 500–544

Read why channel noise matters

Model assumptions

Every sodium channel has three activation gates and one inactivation gate, every potassium channel four gates, and each gate opens and closes at random with the Hodgkin–Huxley rates. A channel conducts only when all its gates are open. The patch holds 60 sodium and 18 potassium channels per square micrometre, each 20 pS, as in Goldwyn and Shea-Brown (2011). Each trial settles for 10 ms without being recorded, then receives the same brief shock. The dashed reference is the deterministic Hodgkin–Huxley membrane, the limit of infinitely many channels. Every trial is simulated; no patch-clamp data are shown.

Recent stochastic trials and the deterministic membrane

First-spike latency over all trials

Open sodium channels in the trial shown, against the binomial expectation

More numbers and how they are computed

Patch area: $0$
Mean first-spike latency: $0$
All spikes per second of trial time: $0$
Trials run: $0$
Deterministic shock threshold: $0$
Channel densities and conductance: $0$
Time step of the channel chains: $0$

Each trial uses its own seed and settles for 10 ms before the record starts. A trial counts as firing when the depolarization crosses 50 mV upward after the shock. Standard errors are binomial for the fraction and large-sample for the spread. Estimates stop changing after 1500 trials.

Physics tutorial

Fewer channels, less reliable spikes

BackgroundThe Hodgkin–Huxley gate variables are averages over many channels. Each real channel is a protein that snaps open and shut at random, so a patch with a finite number of them carries a current that fluctuates even when nothing outside changes.

Why it mattersThin axons, dendrites and small neurons have far fewer channels than a squid giant axon. Whether their spikes are reliable is a question about counting statistics, and the answer changes near threshold first.

Start with the essentials

Focus question: When a patch holds only a few thousand channels, does the same shock always fire a spike?
One-sentence intuition: At a fixed voltage the number of open channels is binomial, so its relative spread falls as $N^{-1/2}$ . Near threshold the deterministic membrane balances on a knife edge, and that small spread decides the outcome: a $100\ \mu\mathrm{m^2}$ patch fires on about half of identical shocks.

Core mathematical model

Open channels at a fixed voltage

N_{\mathrm{open}}\sim\mathrm{Bin}(N_{\mathrm{Na}},\,m^3h),\qquad \frac{\sqrt{\operatorname{Var}N_{\mathrm{open}}}}{\langle N_{\mathrm{open}}\rangle}=\sqrt{\frac{1-m^3h}{N_{\mathrm{Na}}\,m^3h}}

Channels are independent and each is open with the Hodgkin–Huxley probability, so the count is binomial. The benchmark checks the mean and variance at six times after a voltage step.

Channel numbers in the patch

N_{\mathrm{Na}}=60\,A,\qquad N_{\mathrm K}=18\,A,\qquad A\ \text{in}\ \mu\mathrm{m^2}

With 20 pS per channel these densities give the 1952 maximum conductances, $120$ and $36\ \mathrm{mS\,cm^{-2}}$ .

Jitter in the small-noise limit

\sigma_t\propto N^{-1/2}\qquad (N\to\infty)

Valid only when the noise is small. Between $100$ and $1000\ \mu\mathrm{m^2}$ the simulated jitter falls faster than this, so do not extrapolate it to small patches.

Common difficulties

The noise is in the membrane, not the stimulus

Typical misconceptionTrials differ because the shock is not exactly repeated.

Better mental modelEvery trial gets an identical shock. The only random numbers in the model decide which gates open and close.

Large patches are not noise-free

Typical misconceptionAbove some size the patch behaves exactly like the equations.

Better mental modelThe spread shrinks but never vanishes. At $1000\ \mu\mathrm{m^2}$ , sixty thousand sodium channels still give a latency spread of about 0.06 ms.

Run the experiment

01
Shock at threshold
Leave the coin-flip preset running for twenty seconds.
What to observe: About half the trials fire. The deterministic membrane with the same shock fires only once, after about 6.6 ms, because $6.5\ \mathrm{nC\,cm^{-2}}$ is just above its threshold.
02
Shock hard and watch the timing
Choose Spike times jitter.
What to observe: Nearly every trial now fires, but the first spike times have a spread of about 0.28 ms. The deterministic membrane fires after 0.87 ms every time.
03
Add channels
Choose Many channels: back to Hodgkin–Huxley, then drag the rim inward.
What to observe: With ten times the channels the spread falls to about 0.06 ms. Shrinking the patch brings the jitter back. Small patches also fire with no shock at all, about 38 times per second at ten square micrometres.

Stochastic Ion Channels: Why Small Neurons Fire Unreliably

Fewer channels, less reliable spikes

Core mathematical model

Open channels at a fixed voltage

Channel numbers in the patch

Jitter in the small-noise limit

Common difficulties

The noise is in the membrane, not the stimulus

Large patches are not noise-free

Run the experiment

Shock at threshold

Shock hard and watch the timing

Add channels

Related experiments and articles

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