Leaky integrate-and-fire neuron

The leaky integrate-and-fire model turns input current into a changing voltage and a sequence of spike events. It captures integration and leak with a simple equation, then adds a threshold rule.

Intuition: accumulation with leakage

Constant input initially pushes voltage away from rest. Leakage grows as the voltage moves away, so the rise slows. If the limiting voltage is below threshold, no spike occurs. If it is above threshold, the model eventually emits an event and resets.

Two ingredients, not one equation

Between events,

$$\tau_m\dot V=-(V-E_L)+RI.$$

At threshold, register a spike, reset to $V_r$, and hold there for the specified refractory time. The differential equation alone does not specify the spike mechanism. Reset and refractory conventions are part of the model definition.

For a constant current, the equilibrium is $V_\infty=E_L+RI$, with an exponential approach from initial voltage $V_0$:

$$V(t)=V_\infty+(V_0-V_\infty)e^{-t/\tau_m}.$$

This expression provides a direct way to check a simulator. At zero current and initial resting voltage, voltage should remain at rest. Below threshold it should approach the equilibrium without generating events.

Calculate before simulating

With illustrative values $E_L=V_r=-65$ mV, $V_{th}=-50$ mV and $R=10$ MΩ, the limiting threshold current is $1.5$ nA. At larger currents, the charging time from reset is

$$T_c=\tau_m\log\frac{V_\infty-V_r}{V_\infty-V_{th}}.$$

With a refractory hold $T_{ref}$, the sustained constant-current interspike interval is $T_c+T_{ref}$. Its reciprocal is the steady firing rate, after converting milliseconds to seconds. This formula assumes a constant current and identical reset on every event; arbitrary time-varying input needs a different solution strategy.

What it cannot explain

Threshold/reset is a convenient event rule, not an ion-channel model of the action potential. The basic LIF model omits adaptation and many response patterns. Its simplicity is useful when the question concerns timing under stated assumptions, but cannot establish biological realism or energy efficiency.

Where you will see this

Change the current in the LIF explorable, and compare the trace with its spike train. This material reflects current learning rather than applicant research results.

References

• Neuronal Dynamics: integrate-and-fire models.
• Neuronal Dynamics: limitations.