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[[Image:Leakyintegrator.png|thumb|right|250px|A graph of 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.<ref>{{cite book|last=Eliasmith, Anderson|first=Chris, Charles|title=Neural Engineering|year=2003|publisher=MIT Press|location=Cambridge, Massachusetts|pages=81}}</ref> {{Clarify|date=July 2009}}


==Equation==
The equation is of the form


:<math>dx/dt = -Ax + C \,</math>
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where C is the input and A is the [[Time constant|rate of the 'leak']].
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===General solution===
Its general solution is
 
:<math>x(t) = ke^{-At} + Ct \,</math>
 
where k is a constant.
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Leaky Integrator}}
[[Category:Differential equations]]
 
{{mathanalysis-stub}}
{{mathapplied-stub}}

Latest revision as of 21:01, 30 December 2014


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