Mean-Field Theory for Spiking Networks

You could never track every neuron in a network one at a time — even a small cortical patch has thousands, each firing erratically. But you don't have to. Picture a packed stadium doing "the wave": nobody watches every single fan, yet everyone sees the average motion sweep around the bowl. Mean-field theory plays the same trick on a spiking network. Instead of following each noisy cell, it describes the whole population by a few averages — above all, how fast the typical neuron fires. Track the herd, not the sheep.

Two numbers set a neuron's pace. Every cell is pelted by thousands of tiny synaptic taps; sum them and, by the central limit theorem, the bombardment looks like a steady push of mean size $\mu$ riding on Gaussian jitter of size $\sigma$. The neuron charges like a leaky bucket until it reaches threshold, fires, and resets. More mean drive $\mu$ refills the bucket faster, so it fires more often: the population firing rate is some function of the input, $\nu = F(\mu,\sigma)$. A worked number: if $\mu$ alone leaves the bucket just shy of the rim, then with no noise it never fires — but add a little $\sigma$ and random tips push it over a handful of times each second, say $\nu \approx 5$ Hz. Noise, not just the mean, makes the rate.

Precisely, the transfer function for a leaky integrate-and-fire neuron is the Siegert formula, $\nu = \left[\tau_{ref} + \tau_m\sqrt{\pi}\int_{(V_r-\mu)/\sigma}^{(V_{th}-\mu)/\sigma} e^{u^2}(1+\text{erf}(u))\,du\right]^{-1}$. The twist for a recurrent network is self-consistency: the population's own spikes are part of the input, so $\mu$ and $\sigma$ themselves depend on $\nu$. You solve the fixed-point equation $\nu = F(\mu(\nu),\sigma(\nu))$. In a balanced network, excitation and inhibition nearly cancel, leaving $\mu$ below threshold so firing is driven by the fluctuations $\sigma$ — the cortex's "fluctuation-driven" regime. When the fixed-point curve crosses itself more than once, the network holds two stable rates at once: the UP and DOWN states of slow-wave sleep.

Try this in the sim above: (1) Open the Param Sweep tab and raise Input I — watch the output rate ride up the S-shaped transfer curve, steep in the middle and flat at the ends. (2) Push Parameter 2 (the noise $\sigma$) up while keeping the mean low: the rate stays above zero even when the mean alone would be silent — that is fluctuation-driven firing. (3) Switch between the Default and Variant presets and use the Compare tab to hunt for a regime with two coexisting fixed points (bistability).

μ = mean input, σ_V = input noise. Self-consistency: μ and σ_V depend on ν → solve iteratively.

Siegert (1951); Brunel (2000) J Comput Neurosci; Dayan & Abbott Ch.7

Gerstner et al. — Neuronal Dynamics (neuronaldynamics.epfl.ch, free) · Dayan & Abbott — Theoretical Neuroscience (MIT Press) · Izhikevich — DSN (dynamicalsystems.org, free)