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 in single-neuron models.^[1]

Equation

The equation is of the form

$dx/dt = -Ax %2B C \,$

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

General solution

Its general solution is

$x(t) = ke^{-At} %2B C/A \,$

where k is a constant.

References

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

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