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Figure 4 | The Journal of Mathematical Neuroscience

Figure 4

From: Mesoscopic population equations for spiking neural networks with synaptic short-term plasticity

Figure 4

First- and second-order statistics of the TPSI (y) for a presynaptic population of 100 stationary Poisson neurons. (A), (B) Relative error of the mean TPSI over time \(\langle y \rangle _{t}\) (A) and the coefficient of variation of the TPSI over time (\(\text{CV}(y)_{t}\)) (B) predicted by the first- and second-order MF (left and right column respectively) with respect to microscopic simulation, as a function of the synaptic parameters \(\tau _{\mathrm {D}}\), \(\tau _{\mathrm {F}}\) and for two values of U (with \(U=U_{0}\)); U is set to 0.5 on the upper row and 0.2 on the lower row. On the x- and y-axes, \(\tau _{\mathrm {F}}\cdot r\) is a unitless quantity. In (A), the maximum relative error is 4.7% for the first-order MF and 0.3% for the second-order MF. In (B), the maximum relative error is 28.6% for the first-order MF and 4.0% for the second-order MF. As scaling \(\tau _{\mathrm {F}}\) and \(\tau _{\mathrm {D}}\) is equivalent to scaling the firing rate r, the relative error at different firing rates can be read moving along the diagonal (white line). (C) Mean modulating factor \(\langle R \rangle _{t}\) over time predicted by the first- and second-order MF (dotted blue and solid red lines respectively) compared to microscopic simulations (dashed black line) as a function of the firing rate r for a specific set of synaptic parameters. (D) TPSI variance over time (\(\operatorname{Var} (y )_{t}\)) predicted by the first- and second-order MF compared to microscopic simulations as a function of the firing rate r for a specific set of synaptic parameters. Synaptic parameters used in (C)–(D) correspond to the white line in (A)–(B) and are: \(\tau _{\mathrm {D}}= 0.3\text{ s}\), \(\tau _{\mathrm {F}}= 0.3\text{ s}\) and \(U =U_{0}=0.2\). Simulation time step is 0.5 ms

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