The Master Equation for Neural Interaction

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Abstract
Based on neural interaction equations a random walk model for the stochastic dynamics of a single neuron is introduced. In this model the somatic potential corresponds to a state in the state space and action potentials provide the mechanism causing transitions. Time is made discrete, consisting of small finite increments δ t ; assumptions are made about the transitions within such an increment and the associated probabilities are formulated. These quantities depend on δ t and on parameters derived from neural interaction equations. Moreover the model is chosen so that the sequence of somatic potentials is a Markov chain. By appropriately scaling the parameters, in the limit as δ t → 0, a master equation for the probability in continuous time is obtained. Depending on the parameters, the master equation describes the evolution of a deterministic, a diffusion, or a discrete process. An interpretation for the diffusion and discrete processes is outlined. The conclusion is that the stochastic equations for neural interaction lead to a master equation representing a diffusion or a discrete process depending on the number, size of synaptic connectivity coefficients, and probability distribution of neural activity. An example is included describing how a master equation may be used to derive properties of the single neuron's output process.

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