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

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

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

where $k$ is a constant, and $x_{0}$ is an arbitrary solution of the equation.

gollark: I don't think I can type that fast.

gollark: Hmm, 23 LoC/s, impressive.

gollark: Idea: video compression?

gollark: n is all quaternionic values satisfying n³ = 7, yes.

gollark: Not really. You can run arbitrary shell commands. They exist in a container or something.

References

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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

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