mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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. This is equivalent to a 1st-order highpass filter with cutoff frequency far below the frequencies of interest.

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) = ke^ + \frac

where

k

is a constant encoding the initial condition.

References

Differential equations {{mathapplied-stub