Each spike arrival at a synapse causes an excursion of the membrane potential of the postsynaptic neuron. If spikes arrive stochastically the membrane potential exhibits fluctuations around a mean trajectory. If the fluctuations stay in the sub-threshold regime where the membrane properties can be approximated by a linear equation, the mean and the standard deviation of the trajectory can be calculated analytically, given the parameters of the stochastic process that characterize spike arrivals. In the presence of a firing threshold, the fluctuations in the membrane potential caused by stochastic spike arrivals can make the neuron fire even if the mean trajectory would never reach the firing threshold.

In the limit that the rate of spike arrival at excitatory and inhibitory synapses is high while each spike causes only a small jump of the membrane potential, synaptic bombardment can be approximated by the sum of two terms: A mean input current and a Gaussian white noise input. The white noise leads to a ’diffusion’ of the membrane potential trajectory around the mean trajectory. The evolution of the probability distribution $p(u,t)$ of the membrane potential over time is described by a Fokker-Planck equation. For the leaky integrate-and-fire model and stationary input, the Fokker-Planck equation can be solved analytically. For nonlinear integrate-and-fire neurons and time-dependent input numerical solutions are possible. We will return to the Fokker-Planck equations in Ch. 13 where further results will be derived.

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