leaky integrator

Image:Leakyintegrator.png

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.{{cite book|last=Eliasmith, Anderson|first=Chris, Charles|title=Neural Engineering|url=https://archive.org/details/neuralengineerin00elia_553|url-access=limited|year=2003|publisher=MIT Press|location=Cambridge, Massachusetts|pages=[https://archive.org/details/neuralengineerin00elia_553/page/n101 81]|isbn=9780262050715 }}

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

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

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

References

Category:Differential equations