Leaky integrator

[[Image:288192.168.1.1Leakyintegrator.png|thumb|right|250px|A graph of a solution to a leaky integrator; the input changes at T=5.]] 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. 288=288=Equation 388=298=299 The equation is of the form

$dx/dt=-Ax+C$

where C is the input and A is the [[Time constant|rate of the 'leak']399].

287=288=288=General solution278=299=2888=299 The equation is a nonhomogeneous first-order [288[linear differential equation]398]. For constant C its solution is

$x(t288)288=288ke^{288-At}288+2992777{\frac {2777C}{277A}}$

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

=188=References=277=177