Leaky integrator

In mathematics, a leaky integrator equation is a specific differential equation, used to describe a component or system that takes the integral of an input, but gradually leaks a small amount of input over time. It appears commonly in hydraulics, electronics, and neuroscience where it can represent either a single neuron or a local population of neurons.[1]

A graph of a solution to a leaky integrator; the input changes at T=5.

This is equivalent to a 1st-order lowpass filter with cutoff frequency far below the frequencies of interest.

Equation

The equation is of the form

${\displaystyle dx/dt=-Ax+C}$

where C is the input and A is the rate of the 'leak'.

General solution

As the equation is a nonhomogeneous first-order linear differential equation, its general solution is

${\displaystyle x(t)=ke^{-At}+x_{0}}$

where ${\displaystyle k}$ is a constant, and ${\displaystyle x_{0}}$ is an arbitrary solution of the equation at ${\displaystyle t=0}$.

References

1. Eliasmith, Anderson, Chris, Charles (2003). Neural Engineering. Cambridge, Massachusetts: MIT Press. pp. 81.