In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a leaky integrator equation is a specific differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

, used to describe a component or system that takes the integral

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

 of an input, but gradually leaks a small amount of input over time. It appears commonly in hydraulics

Hydraulics

Hydraulics is a topic in applied science and engineering dealing with the mechanical properties of liquids. Fluid mechanics provides the theoretical foundation for hydraulics, which focuses on the engineering uses of fluid properties. In fluid power, hydraulics is used for the generation, control,...

, electronics

Electronics

Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...

, and neuroscience

Neuroscience

Neuroscience is the scientific study of the nervous system. Traditionally, neuroscience has been seen as a branch of biology. However, it is currently an interdisciplinary science that collaborates with other fields such as chemistry, computer science, engineering, linguistics, mathematics,...

in single-neuron models.

Equation

The equation is of the form


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

General solution

Its general solution is


where k is a constant.