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.
Equation The equation is of the form
d x / d t = − A x + C {\displaystyle 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 ) = k e − A t + C A {\displaystyle x(t)=ke^{-At}+{\frac {C}{A}}}
where k {\displaystyle k} is a constant encoding the initial condition.
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