Stochastic Models of Neural Plasticity: A Scaling Approach

TL;DR

This paper develops a scaling approach for stochastic models of synaptic plasticity, demonstrating how long-term plasticity evolves slowly compared to cellular activity, and provides limit theorems for various STDP models.

Contribution

It introduces a novel scaling method for stochastic neural plasticity models and proves averaging principles for their long-term behavior.

Findings

Long-term synaptic plasticity evolves on a slower timescale.

Averaging principles are established for a broad class of models.

Comparison with neuroscience models shows good approximation.

Abstract

In neuroscience, synaptic plasticity refers to the set of mechanisms driving the dynamics of neuronal connections, called synapses and represented by a scalar value, the synaptic weight. A Spike-Timing Dependent Plasticity (STDP) rule is a biologically-based model representing the time evolution of the synaptic weight as a functional of the past spiking activity of adjacent neurons. A general mathematical framework has been introduced in~arXiv:2010.08195. In this paper we develop and investigate a scaling approach of these models based on several biological assumptions. Experiments show that long-term synaptic plasticity evolves on a much slower timescale than the cellular mechanisms driving the activity of neuronal cells, like their spiking activity or the concentration of various chemical components created/suppressed by this spiking activity. For this reason, a scaled version of the stochastic model of~arXiv:2010.08195 is introduced and a limit theorem, an averaging principle, is stated for a large class of plasticity kernels. A companion paper~arXiv:2010.08790 is entirely devoted to the tightness properties used to prove these convergence results. These averaging principles are used to study two important STDP models: pair-based rules and calcium-based rules. Our results are compared with the approximations of neuroscience STDP models. A class of discrete models of STDP rules is also investigated for the analytical tractability of its limiting dynamical system.