A Hodgkin-Huxley model algorithm for the numerical simulation of noise in neurons is contracted from a master equation description (cellular automoton) into a Langevin description. This reduction reduces the time required for a simulation by about two orders of magnitude. Earlier work is summarized, condensed, and made explicit to make the algorithm transparent and facilitate applications. Two approximate treatments are reported. An extension of this approach is presented that includes spatial dependence and the propagation of a noisy action potential along an axon.
The velocity of a particle in Brownian motion as described by the Langevin equation is a stationary Gaussian–Markov process. Similarly, in the formulation of the laws of non-equilibrium thermodynamics by Onsager and Machlup, the macroscopic variables describing the state of a system lead to an n-component stationary Gaussian–Markov process, which, in addition, these authors assumed to be even in time. By dropping this assumption, the most general stationary Gaussian–Markov process is discussed. As a consequence, the theory becomes applicable to the linearized hydrodynamical equations and suggests that the Navier–Stokes equations require additional fluctuation terms which were first proposed by Landau and Lifshitz. Such terms and their correlation properties are presented, and these equations are then used to derive the Langevin equation for the Brownian motion of a particle of arbitrary shape.
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