Stochastic Resonance in Neurons

Try to hear a friend whisper in a quiet room when the whisper is just a hair too soft to make out. Surprisingly, adding a bit of background hiss can help: every so often the hiss happens to land on top of the whisper and pushes it just loud enough to notice — and it tends to do that right when the whisper is at its loudest. Too little hiss and you still hear nothing; too much hiss and it drowns everything out. There is a "just right" amount in between. That sweet spot is stochastic resonance: the right amount of randomness makes a weak signal easier to detect, not harder.

A neuron only fires when its input crosses a threshold. Call the smallest steady current that makes it fire the rheobase, $I_{rheo}$. Feed it a weak rhythmic signal $A\sin(\Omega_0 t)$ whose peak $A$ is below threshold — on its own it never fires. Now add random noise of strength $\sigma$ (its typical size). When the signal is near a peak, a chance noise bump can carry the total over threshold, producing a spike timed to the signal. Example: if threshold sits $4$ units above where the signal peaks, almost no noise ($\sigma\!\approx\!1$) rarely reaches it, while a moderate $\sigma\!\approx\!4$ pushes it over mostly near the peaks — spikes that echo the rhythm.

Detection quality is the output signal-to-noise ratio $\text{SNR}_{out}(\sigma)$. As $\sigma$ grows it rises, peaks, then falls — the inverted-U "SR curve" — so the best noise level $\sigma^*$ satisfies $\frac{d\,\text{SNR}}{d\sigma}\big|_{\sigma^*}=0$. The noise-driven threshold-crossing rate follows a Kramers/Arrhenius form $r\propto\exp(-\Delta U/\sigma^2)$, where $\Delta U$ is the "barrier" between rest and threshold; resonance is loudest when this rate matches the signal frequency $\Omega_0$. With $N$ independent noisy neurons the population SNR grows like $\sqrt{N}$ (array SR), so a crowd of imperfect detectors beats any single one. On the panel, the parameter sliders set the gain and barrier, Input I sets the signal amplitude $A$, and T sim sets how long you average (longer = cleaner SNR estimate).

(1) Set Input I low so the signal stays subthreshold, then slowly raise the noise parameter: detection should improve, peak, then degrade — watch for the inverted-U. (2) Open the Param Sweep tab to see SNR plotted against noise directly and read off $\sigma^*$. (3) Push the noise parameter very high and notice the output becomes random spikes with no rhythm — proof that more noise is not always better.

Double-well potential + subthreshold periodic input + noise: transitions between wells synchronise with signal. SNR peaks at intermediate noise σ* where Kramers rate = signal frequency. Benzi et al. (1981) introduced SR to explain ice-age periodicity.

Subthreshold I(t) = A sin(Ω₀t), A < I_rheobase. Without noise: no spikes. With optimal noise σ*: spikes phase-lock to input, SNR peaks. Demonstrated in mechanoreceptors (Douglass 1993), auditory nerve (Jaramillo 1998), human muscles (Collins 1996).

N independent neurons with uncorrelated noise: average output SNR ∝ √N (CLT). Optimal σ* ∝ 1/√N. Array SR: a population of noisy neurons outperforms any single optimal-noise neuron for detection. Explains why sensory populations with heterogeneous thresholds outperform uniform arrays.