Figure 4.
A, Particle diffusing in periodic potential well Equation 3 is an idealized model of the bump diffusing over the network with spatially heterogeneous synapses, given by the stochastic process Equation 2. B, Probability density p(θ, t) of particle position θ spreads diffusively in time (n = 8; h = 1; noise variance σ2 = 0.16). C, Profile of p(θ, t) computed from 10,000 realizations (red); effective diffusion theory Equation 6 (black); and effective diffusion theory with periodic correction Equation 7 (blue). D, Effective diffusion theory using Deff in Equation 8 matches variance scaling from simulations of stochastic equation. Variance increases monotonically with the well frequency n. E, An effective diffusion coefficient Deff can be computed by treating well hopping as a jump Markov process (Lifson and Jackson, 1962; Lindner et al., 2001), yielding formula Equation 8.