Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mean-field analysis of a neural network with stochastic STDP

Pascal Helson, Etienne Tanré, Romain Veltz

TL;DR

This work derives a McKean-Vlasov mean-field (MKV-MF) limit for a stochastic spike-timing-dependent plasticity (STDP) network of interacting spiking neurons. By introducing a typical-neuron description and an empirical measure over the extended state, the authors obtain a piecewise deterministic Markov process of McKean-Vlasov type that captures both the binary spiking dynamics and heterogeneous, time-evolving synaptic weights. The main contributions include (i) a rigorous mean-field limit framework that handles plasticity without requiring symmetric STDP or slow-fast reductions, (ii) a coupled MKV-SDE/PDMP system for the typical neuron with a PDE for the distribution of the triplet state, and (iii) numerical validation showing close agreement with the full $N$-neuron network and substantial computational savings. The MKV-MF model paves the way for analytical and computational tools to study transient dynamics, weight divergence prevention, and control problems in networks with plastic synapses, with potential applications to deep brain stimulation and other neuromodulation contexts.

Abstract

Analysing biological spiking neural network models with synaptic plasticity has proven to be challenging both theoretically and numerically. In a network with N all-to-all connected neurons, the number of synaptic connections is on the order of $N^2$, making these models computationally demanding. Furthermore, the intricate coupling between neuron and synapse dynamics, along with the heterogeneity generated by plasticity, hinder the use of classic theoretical tools such as mean-field or slow-fast analyses. To address these challenges, we introduce a new variable which we term a typical neuron X. Viewed as a post-synaptic neuron, X is composed of the activity state V , the time since its last spike S, and the empirical distribution $ξ$ of the triplet V , S and W (incoming weight) associated to the pre-synaptic neurons. In particular, we study a stochastic spike-timing-dependent plasticity (STDP) model of connection in a probabilistic Wilson-Cowan spiking neural network model, which features binary neural activity. Taking the large N limit, we obtain from the empirical distribution of the typical neuron a simplified yet accurate representation of the original spiking network. This mean-field limit is a piecewise deterministic Markov process (PDMP) of McKean-Vlasov type, where the typical neuron dynamics depends on its own distribution. We term this analysis McKean-Vlasov mean-field (MKV-MF). Our approach not only reduces computational complexity but also provides insights into the dynamics of this spiking neural network with plasticity. The model obtained is mathematically exact and capable of tracking transient changes. This analysis marks the first exploration of MKV-MF dynamics in a network of spiking neurons interacting with STDP.

Mean-field analysis of a neural network with stochastic STDP

TL;DR

This work derives a McKean-Vlasov mean-field (MKV-MF) limit for a stochastic spike-timing-dependent plasticity (STDP) network of interacting spiking neurons. By introducing a typical-neuron description and an empirical measure over the extended state, the authors obtain a piecewise deterministic Markov process of McKean-Vlasov type that captures both the binary spiking dynamics and heterogeneous, time-evolving synaptic weights. The main contributions include (i) a rigorous mean-field limit framework that handles plasticity without requiring symmetric STDP or slow-fast reductions, (ii) a coupled MKV-SDE/PDMP system for the typical neuron with a PDE for the distribution of the triplet state, and (iii) numerical validation showing close agreement with the full -neuron network and substantial computational savings. The MKV-MF model paves the way for analytical and computational tools to study transient dynamics, weight divergence prevention, and control problems in networks with plastic synapses, with potential applications to deep brain stimulation and other neuromodulation contexts.

Abstract

Analysing biological spiking neural network models with synaptic plasticity has proven to be challenging both theoretically and numerically. In a network with N all-to-all connected neurons, the number of synaptic connections is on the order of , making these models computationally demanding. Furthermore, the intricate coupling between neuron and synapse dynamics, along with the heterogeneity generated by plasticity, hinder the use of classic theoretical tools such as mean-field or slow-fast analyses. To address these challenges, we introduce a new variable which we term a typical neuron X. Viewed as a post-synaptic neuron, X is composed of the activity state V , the time since its last spike S, and the empirical distribution of the triplet V , S and W (incoming weight) associated to the pre-synaptic neurons. In particular, we study a stochastic spike-timing-dependent plasticity (STDP) model of connection in a probabilistic Wilson-Cowan spiking neural network model, which features binary neural activity. Taking the large N limit, we obtain from the empirical distribution of the typical neuron a simplified yet accurate representation of the original spiking network. This mean-field limit is a piecewise deterministic Markov process (PDMP) of McKean-Vlasov type, where the typical neuron dynamics depends on its own distribution. We term this analysis McKean-Vlasov mean-field (MKV-MF). Our approach not only reduces computational complexity but also provides insights into the dynamics of this spiking neural network with plasticity. The model obtained is mathematically exact and capable of tracking transient changes. This analysis marks the first exploration of MKV-MF dynamics in a network of spiking neurons interacting with STDP.

Paper Structure

This paper contains 27 sections, 7 theorems, 121 equations, 2 figures.

Table of Contents

  1. Presentation of the model
  2. The microscopic model
  3. Towards the macroscopic model
  4. Main results
  5. Numerical comparison with the neural network
  6. Discussion
  7. Conclusion
  8. Acknowledgements
  9. Proofs
  10. Density and Markovianity of the empirical distribution
  11. Properties of TEXT
  12. TEXT is Markov
  13. Preliminaries for the derivation of the main result
  14. The case without plasticity
  15. Generator of TEXT
  16. ...and 12 more sections

Key Result

Lemma 8

Grant Assumption hyp:cdt-init. Consider the process $X_t^N$ solution of the microscopic model described in Section sec:micro_model. Then, for any $t\geq 0$ and for all $i\neq j \in \llbracket 1,N\rrbracket$, we have almost surely $S_t^{i,N}\neq S_t^{j,N}$.

Figures (2)

  • Figure 1: Comparisons between the MKV-MF model and the initial model. (\ref{['sub:esperance-V-final-500ms']}-c): Empirical mean (see Definition \ref{['def:empirical_dist_mean']} and below) of the potentials \ref{['sub:esperance-V-final-500ms']}, time since last spike \ref{['sub:esperance-S-final-500ms']}, and the weights \ref{['sub:esperance-W-final-500ms']} across time. \ref{['sub:distribution-I-final-500ms']}: Empirical distributions (see Definition \ref{['def:empirical_dist_mean']}) at time $t=500ms$ of the synaptic currents onto the neurons. (\ref{['sub:distribution-S-final-V0-500ms']}-f): Distributions (see Definition \ref{['def:empirical_dist_mean']} and below) at time $t=500ms$ of the time since last spike for neurons in state $V=0$\ref{['sub:distribution-S-final-V0-500ms']} and $V=1$\ref{['sub:distribution-S-final-V1-500ms']}.
  • Figure S1: Distributions of the time from last spikes for neurons in state $V=0$ in \ref{['sub:distribution-S-V0-initial']} and for neurons in state $V=1$ in \ref{['sub:distribution-S-V1-initial']}, at the initial time $t=0\, ms$. Parameters' values of Section \ref{['sec:Simulations']} are used.

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Remark 3
  • Remark 4
  • Remark 6
  • Definition 7
  • Lemma 8
  • proof
  • Lemma 9
  • proof
  • ...and 16 more