Mean Field Analysis of a Stochastic STDP model

Biological neural network models with synaptic plasticity pose challenges for both theoretical and numerical analyses. The numerical bottleneck arises from the $N^2$ scaling of synapses. Additionally, the intricate coupling between neuron and synapse dynamics, along with plasticity-induced heterogeneity, hinders the use of classical tools from theory. One approach to address these challenges has been to assume plasticity to be very slow compared to neural activity and leverage slow-fast theory [1]. However, this slowness is not universally applicable in the brain, leaving unclear how to analyse such complex systems without relying on the slow-fast assumption. Tools recently developed in statistical physics [2] enabled going beyond such an assumption in a rate-based model [3]. In this work we tackle the problem on a spiking model by performing a mean-field analysis on a spiking neural network model with plastic interactions governed by a stochastic STDP. The generality of our results could then help deriving previously inaccessible mean-field limits for other models with jumping particles with plastic interaction.

Here we study a STDP model implemented within a probabilistic Wilson-Cowan neural network model (binary neural activity). The neural network is then described by $N$ triplets composed of the neuron potential, $V_t^{i,N} \in \{0,1\}$, the time since its last spike, $S_t^{i,N} \in \bf{R}^+$, and its $N$ incoming synaptic weights, $(W_t^{ij,N})_j \in \bf{Z}$.

In this model, the definition of a $typical$ neuron highly non trivial. A possible choice is to use $new$ variables that ease performing a mean field approximation, namely the $empirical\ distributions$, $\xi_t^{i,N}$, of the state of the pre-synaptic neurons (neuron state, time since last spike and outcoming synaptic weight to neuron $i$), $\xi_t^{i,N} = \frac1N \sum_j \delta_{ ( V_t^{j,N}, S_t^{j,N}, W_t^{ij,N} ) }.$
Considering this new system $X_t^{i,N}=(V_t^{i,N}, S_t^{i,N}, \xi_t^{i,N})$, we are able to derive the dynamics of any limit point $(V_t^*,S_t^*,\xi_t^*)$. We illustrated in Fig. 1 this limit dynamics with simulations by comparing the finite size neural network to the mean field limit system. Importantly, our approach significantly reduces simulation costs, crucial for models involving synaptic plasticity. A timeline example is the consequence of deep brain stimulation (DBS) on the weights linking the neurons stimulated, especially in an adaptive setting.