In
mathematics, a leaky integrator equation is a specific
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
, used to describe a component or system that takes the
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
of an input, but gradually leaks a small amount of input over time. It appears commonly in
hydraulics
Hydraulics (from Greek: Υδραυλική) is a technology and applied science using engineering, chemistry, and other sciences involving the mechanical properties and use of liquids. At a very basic level, hydraulics is the liquid counter ...
,
electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification a ...
, and
neuroscience
Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a multidisciplinary science that combines physiology, anatomy, molecular biology, developm ...
where it can represent either a single neuron or a local population of neurons.
This is equivalent to a 1st-order
highpass filter with cutoff frequency far below the frequencies of interest.
Equation
The equation is of the form
:
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
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...
. For constant C its solution is
:
where
is a constant encoding the initial condition.
References
Differential equations
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