The integrate-and-fire neuron

In 1907 the sodium and potassium currents behind a nerve impulse were more than forty years from being separated. Louis Lapicque modelled nerve excitation anyway, as a circuit: a capacitor charged through a resistance, a leak across it, and a threshold at which the nerve responds. Rebuilt in the 1960s as a neuron that fires and resets, the circuit became the theorist’s tool of choice, and the neuron that engineers put into silicon.

A capacitor with a leak

The question was the one a stimulator designer still asks: how strong, and for how long? Following Nernst, Lapicque took a current’s effect on a nerve to be polarisation — ions piling up against a semipermeable membrane — and treated the membrane as electrochemists treated a polarised metal electrode: to a first approximation, a capacitor. “One should just not forget that this is an approximation.”

His circuit is a capacitor $K$ with a leak $\rho$ across it, charged through a series resistance $R$; excitation happens when the capacitor reaches a fixed potential. The threshold voltage for a pulse of duration $t$ then goes as $1/(1 - e^{-t/\beta})$, with $\beta = KR\rho/(R + \rho)$. It fitted the shortest pulses worse than Weiss’s empirical law of 1901 and the longest better, and its constants meant something physical.

The argument that mattered was about impedance. In the usual stimulating circuit, he reckoned, $R$ ran to hundreds of kilohms and $\rho$ to under ten, so the source drove a nearly constant current and $\beta \approx K\rho$: the membrane’s capacitance times its own resistance. Both depend on how much membrane the electrode touches, in opposite directions, so their product does not. Change the electrodes and the threshold voltage changes, but the time course of excitation should not — and in his and Mme Lapicque’s experiments it had not. For frog sciatic nerve $\beta$ came out near a millisecond, with pulses timed by a bullet cutting two wires: 27 cm of flight to the millisecond.

What the paper does not contain

Worth being clear about what is missing: no spike, no reset, no firing rate, no “integrate-and-fire”. Abbott’s 1999 appreciation credits Lapicque with a reset and a computed firing frequency; Brunel and van Rossum, translating the paper for its centenary, say it proposes no spike mechanism and no reset, and the text bears them out.

The spiking model came nearly sixty years later. Gerstein and Mandelbrot analysed an integrator without a leak in 1964. Richard Stein’s 1965 neuron has the rest: random excitatory and inhibitory inputs that decay exponentially and sum to a threshold, a firing, a reset and a refractory period — with Lapicque’s strength–duration curve, Stein noted, as a special case. The name is Bruce Knight’s, first found in print in his 1972 paper; “leaky integrate-and-fire”, the name that stuck, Brunel and van Rossum credit to Rick Purple.

In modern form the voltage $V$, measured from rest, relaxes towards $RI$ with time constant $\tau = RC$; at a threshold $\theta$ the neuron spikes and $V$ returns to rest for a refractory period $t_{\text{ref}}$. Under a constant current $I$ it fires at a rate Stein gave in 1965:

$$f = \frac{1}{t_{\text{ref}} + \tau \ln\!\left(\dfrac{RI}{RI - \theta}\right)}$$

Below the rheobase current $\theta/R$ it never fires; far above, it tends to one spike per $C\theta$ of charge, until the refractory period caps it. A current-to-frequency converter with a dead zone and a ceiling: an engineer’s reading of the stretch receptor behind rate coding.

Knight’s question was about timing: how faithfully a population’s firing follows a changing stimulus. In his “simple” model, without a leak, the pooled rate copies it exactly; his “forgetful” one, with a leak, over-responds at some frequencies and its neurons tend to fall into step, until noise in the individual neurons restores the copy.

Beside the Hodgkin–Huxley model this is abstraction, not mechanism: there four state variables compute the spike; here one crosses a threshold and the spike is declared. Abbott’s defence is that the spike is fast and stereotyped, so skipping its trajectory costs little. Izhikevich counted four floating-point operations and a comparison per millisecond of model time, against about 1,200 for Hodgkin–Huxley.

Back into silicon

Lapicque took his circuit from the polarised electrode; neuromorphic engineering took the neuron back into circuits. The abstraction swaps the sodium channel’s regenerative positive feedback for a comparison, and in silicon the comparison has to be built. Carver Mead’s axon-hillock circuit (1989) uses an amplifier, usually two inverters in series. Input current charges a membrane capacitor to the switching point; the output flips, kicks the membrane further past threshold through a feedback capacitor — positive feedback again, as a capacitive divider — and switches on a reset current that drains the membrane until the output flips back. With no leak, the interval between spikes is inversely proportional to the input current: Knight’s simple model. Later circuits added the leak, an explicit threshold and a refractory period.

The circuit also shows what a threshold costs in CMOS: it sits wherever the inverter happens to switch, set by transistor geometry and process, and while a slow input creeps through that point the inverter conducts from supply to ground.

The large digital chips kept the model and dropped the analog. IBM’s TrueNorth (2014) holds a million neurons, digital integrate-and-fire units with leaks, and 256 million synapses in 4,096 cores; on 400 × 240 video at 30 frames a second it drew 63 mW. Intel’s Loihi (2018) runs “a variation of the well-known CUBA leaky-integrate-and-fire model” in discrete time steps, on 128 cores of 1,024 compartments each.

It suits a chip for the reason it suits a simulation: a neuron is a number or two and an update ending in a comparison, and its output is an event, not a waveform — the event-driven style that neuromorphic computing credits for the brain’s energy efficiency. The RC circuit Lapicque drew for a frog nerve came back as the neuron these chips build.

What one variable leaves out

Lapicque named the first failure himself: slowly rising currents do not excite a real nerve, and his equation could not say why — “a hiatus, not an objection.” Neurons accommodate: during a slow ramp, inward currents inactivate and outward ones activate, and excitability falls. A fixed threshold cannot move; in the model, any current that ends above rheobase fires eventually.

Izhikevich’s list is longer. With one variable the model cannot burst in any form, fire only at an input’s onset, rebound after inhibition, or vary its threshold — “one of the worst models to use in simulations, unless one wants to prove analytical results.” Each can be bought back with more variables — an outward current stepped up at every spike buys adaptation — at the price of the simplicity that was the point.

It is also a point neuron, blind to everything cable theory says about where on a dendrite an input lands. Loihi added some of that back: a neuron there can be a tree of compartments, of which only the root spikes.

The fixed threshold shows in stimulation too, Lapicque’s own field. TMS studies have fitted his first-order membrane to motor thresholds at several pulse widths, finding time constants of 150–200 µs. But when a 2023 study shaped monophasic-equivalent pulses for lower coil heating — the work behind programmable stimulation waveforms — the linear leaky integrate-and-fire membrane demanded substantially larger amplitude corrections than the nonlinear models: it cannot represent a hyperpolarising leading phase lowering the threshold by relieving sodium-channel inactivation.