Leaky integrator

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

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.

Equation

The equation is of the form

$dx/dt=-Ax+C$

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

General solution

The equation is a nonhomogeneous first-order linear differential equation. For constant C its solution is

$x(t)=k{e}^{-At}+\frac{C}{A}$

where $k$ is a constant encoding the initial condition.