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

Valentin Schmutz,Tilo Schwalger,Wulfram Gerstner

doi:10.1186/s13408-020-00082-z

Abstract

Coarse-graining microscopic models of biological neural networks to obtain mesoscopic models of neural activities is an essential step towards multi-scale models of the brain. Here, we extend a recent theory for mesoscopic population dynamics with static synapses to the case of dynamic synapses exhibiting short-term plasticity (STP). The extended theory offers an approximate mean-field dynamics for the synaptic input currents arising from populations of spiking neurons and synapses undergoing Tsodyks–Markram STP. The approximate mean-field dynamics accounts for both finite number of synapses and correlation between the two synaptic variables of the model (utilization and available resources) and its numerical implementation is simple. Comparisons with Monte Carlo simulations of the microscopic model show that in both feedforward and recurrent networks, the mesoscopic mean-field model accurately reproduces the first- and second-order statistics of the total synaptic input into a postsynaptic neuron and accounts for stochastic switches between Up and Down states and for population spikes. The extended mesoscopic population theory of spiking neural networks with STP may be useful for a systematic reduction of detailed biophysical models of cortical microcircuits to numerically efficient and mathematically tractable mean-field models.

Highlights

One of the primary goals in computational neuroscience is to understand how brain functions arise from the interactions of billions of nerve cells and their underlying biophysical processes at the microscopic scale
4 Discussion We have derived stochastic mean-field (MF) equations that capture the effect of synaptic short-term plasticity (STP) at the level of populations
The mesoscopic STP dynamics is compatible with a recent mesoscopic population model [12], which has been originally derived for static synapses

Summary

Introduction

One of the primary goals in computational neuroscience is to understand how brain functions arise from the interactions of billions of nerve cells and their underlying biophysical processes at the microscopic scale. We compare numerically the effective input given by the mesoscopic mean-field dynamics with simulations of the full microscopic population, in the case where the presynaptic population consists of N Poisson neurons. We present our main result, which provides a mean-field approximation y(t) that is determined by the history of the mesoscopic presynaptic population activity AN (t) rather than individual presynaptic spike trains sj(t) or synaptic variables uj and xj. “constant” means that the time-discretized population activity AN (t) = n(t)/(N t) in the numerical simulation with time step t is constant, i.e. The deviations of the first-order MF become more pronounced during non-stationary transients caused by stepwise increases of the rate of the Poisson process (Fig. 3). While mean responses for stationary cases are well captured by the firstorder MF, the second-order MF gives a significantly better description of transient responses (Fig. 3) and fluctuations (Figs. 4(B), (D) and 5)

Microscopic model

Mesoscopic mean-field model

Discussion

U0u – τF