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Leaky integrator

Type of differential equation

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Source updated September 17, 2026

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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.

01Equation

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

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

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

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Sources and credits

This article is adapted from the Wikipedia article “Leaky integrator”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

Leakyintegrator.png by Unknown author, PD

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