Figure 2

Example of microscopic and mesoscopic synaptic dynamics for 200 presynaptic stationary Poisson neurons. (a1) Raster plot of \(N = 200\) presynaptic stationary Poisson neurons with rate 10 Hz. (a2–4) Trajectories of variables \(u_{j}(t)\) and \(x_{j}(t)\) and the resulting modulation factor \(R_{j}(t) \equiv u_{j}(t)x_{j}(t)\) for two example neurons (gray lines). The black line shows the population averages \(u(t)\), \(x(t)\) and \(R(t)\) calculated from Eqs. (15a)–(15c). (b1) Population activity \(A^{N}(t)\) corresponding to the 200 spike trains shown in (a1). (a2–4) Trajectories of the mesoscopic variables \(u(t)\), \(x(t)\) and \(R(t)\) predicted by the first- and second-order MF (blue and red, respectively) compared to the microscopic simulation (black) which correspond to the population averages shown on the left. Note that the y-axis scale is different in (a2–4) and (b2–4). In (b4), we see that, while finite-size fluctuations in R for the population average are reproduced by both first- and second-order MF, the first-order MF makes an error in predicting the mean. (c1–4) is the same as (b1–4) except that we force \(A^{N}\) to be constant: while the \(s_{j}\) are still Poisson spike trains with rate 10 Hz, they are generated such that \(A^{N}\) is constant over time. This removes the effect of the finite-size fluctuations of \(A^{N}\) on the finite-size fluctuations of the mesoscopic STP u, x and R. (b2–4) In contrast with the first-order MF, the second-order MF reproduces the residual finite-size fluctuations observed in the microscopic simulation. Synaptic parameters: \(\tau _{\mathrm {D}}= 0.15\text{ s}\), \(\tau _{\mathrm {F}}= 0.15\text{ s}\), \(U = U_{0} = 0.2\). (b1) and (c1) are binned with bin size 0.005 ms