Leaky integrate-and-fire · Lobeworks/17

The leaky integrate-and-fire model is a mathematical model of a neuron that reduces the cell to one variable, the voltage across its membrane, which adds up its inputs, leaks back toward a resting value, and produces a spike and a reset whenever it crosses a threshold.

Leaky integrate-and-fire. The leaky integrate-and-fire model is a mathematical model of a neuron that reduces the cell to one variable, the voltage across its membrane, which adds up its inputs, leaks back toward a resting value, and produces a spike and a reset whenever it crosses a threshold.

The model is built from the physics of the membrane. The membrane is a thin insulating film between two salt solutions, so it stores charge like a capacitor, and the stored charge is the voltage; channels that are always slightly open let the charge seep away like a resistor. With τm\tau_mτm​ the membrane time constant, the voltage obeys τm dV/dt=(Vrest−V)+R I(t)\tau_m,dV/dt = (V_{\text{rest}} - V) + R,I(t)τm​dV/dt=(Vrest​−V)+RI(t): the first term pulls it back toward rest, faster the further it has strayed, and the second adds the input. A threshold and a reset are added by hand, because the equation contains no spike. Louis Lapicque used a model of this kind in 1907 to describe the excitation of nerve, long before the ion channels behind it were known.

It is cheap and has few parameters. A time constant, a resting voltage, a threshold, a reset and a refractory pause describe a cell, which is why it can run a whole brain when the real parameters of each cell are unknown.

It leaves most of the cell out. The shape of the action potential, the sodium and potassium channels that make it, the branching dendrites, adaptation, vesicle release, neuromodulation and plasticity are all absent.

It ran the first whole-brain fly model. Shiu and colleagues gave all 127,400 proofread neurons of FlyWire the same parameters (time constant 11 ms, threshold −45 mV, reset −52 mV), so every difference between cells came from the connectome, and 91% of the 164 predictions they could test held in real flies.

A bucket with a hole in the bottom.

Pour slowly and the hole empties it as fast as it fills; pour hard and it reaches the brim, tips over and starts empty again. Inhibition is a cup taking water out.

Questions: How much can a model with identical neurons predict from wiring alone? More than expected, in the circuits tried so far: a great deal. Shiu and colleagues gave all 127,400 proofread neurons of the fly brain the same parameters and let only the connectome differ, then activated taste neurons in the simulation. Of 164 predictions they could test in real flies, 91% held, and ten of eleven cell types predicted to drive the proboscis did so when switched on with light. Circuits run by neuromodulation, they warned, would be modelled poorly. Why do some connectome models train their parameters and others do not? Because they ask different questions. A model with every parameter fixed and equal, as in Shiu and colleagues' whole fly brain, tests how much the wiring explains by itself. A model that trains the parameters the wiring leaves open, as Lappalainen and colleagues did for the fly's motion vision by asking the network to estimate motion in video, tests whether wiring plus a task is enough to recover how the cells respond; their trained networks agreed with measurements from 26 studies. What does a leaky integrate-and-fire model leave out of a spike? It leaves out the spike itself. The model tracks only the voltage below threshold and replaces the action potential with a rule: when the voltage crosses the threshold, record a spike and set the voltage back to a reset value. The opening and closing of sodium and potassium channels, the shape and width of the spike, adaptation and bursting are all absent, which keeps the model cheap enough to run every neuron of a fly brain. What did the virtual fly of March 2026 actually run? The FlyWire brain as a leaky integrate-and-fire model with uniform neurons, whose descending neurons were read out and mapped onto commands for walking, turning, grooming and feeding, executed by the controllers of NeuroMechFly, a fly body in the MuJoCo physics engine. The brain chose the behaviour; the body model produced the movement, because the brain map stops at the neck and the nerve cord that drives the legs was not in it. Why can't a whole-brain model of a fly be hungry? Because hunger reaches the fly brain as chemistry the model does not contain. In a starved fly, dopamine released onto the sugar-sensing neurons makes them respond more strongly to the same sugar, and hormones from the body shift other circuits. The connectome holds the wires from taste to proboscis but not the chemical knob that sets their gain, nor the body that turns it, so a simulated hungry fly and a full one respond identically. Why do whole-brain fly models read out descending neurons? Because they are the brain's only output to the body. Every command to the legs and wings leaves the brain through about 1,300 descending neurons, so a simulation of the brain alone can do nothing with its activity except read those neurons and translate their firing into commands for a body model, which is what the virtual flies of 2026 did.