[Paper Review] A new stochastic STDP Rule in a neural Network Model

Pascal Helson|arXiv (Cornell University)|Jun 1, 2017

Advanced Memory and Neural Computing4 references3 citations

TL;DR

This paper proposes a novel stochastic Spike-Timing Dependent Plasticity (STDP) rule with discrete synaptic weights in a Wilson-Cowan neural network model, leveraging slow-fast time-scale separation to derive a deterministic equation for weight dynamics. The key contribution is a simple, analytically tractable condition ensuring ergodicity of the weight process without requiring bounds or metaplasticity, revealing counterintuitive results such as weight divergence despite a negative integral over the learning window.

ABSTRACT

Thought to be responsible for memory, synaptic plasticity has been widely studied in the past few decades. One example of plasticity models is the popular Spike Timing Dependent Plasticity (STDP). The huge litterature of STDP models are mainly based deterministic rules whereas the biological mechanisms involved are mainly stochastic ones. Moreover, there exist only few mathematical studies on plasticity taking into account the precise spikes timings. In this article, we aim at proposing a new stochastic STDP rule with discrete synaptic weights which allows a mathematical analysis of the full network dynamics under the hypothesis of separation of timescales. This model attempts to answer the need for understanding the interplay between the weights dynamics and the neurons ones.

Motivation & Objective

• To develop a mathematically tractable stochastic STDP model with discrete synaptic weights to enable analytical study of plasticity dynamics.
• To bridge the timescale gap between fast neural spiking and slow synaptic plasticity using a slow-fast analysis framework.
• To derive a reduced equation for weight dynamics by averaging over the stationary distribution of neural activity.
• To identify sufficient conditions for ergodicity of the weight process without imposing artificial constraints like bounds or metaplasticity.
• To reveal non-intuitive dynamical behaviors, such as divergence under negative learning window integrals, through rigorous probabilistic analysis.

Proposed method

• Models the neural network using the stochastic Wilson-Cowan framework, where synaptic weights evolve via a piecewise deterministic Markov process.
• Applies a slow-fast time-scale separation, assuming plasticity evolves infinitely slowly compared to neural spiking dynamics.
• Derives the effective weight dynamics by replacing fast neural processes with their invariant measures, obtained via Lyapunov functions and Laplace transforms.
• Uses matrix inversion and diagonal dominance arguments to prove uniqueness of the invariant measure for the joint neuron-weight process.
• Employs Laplace transforms of the invariant measure to characterize the long-term behavior of the weight process.
• Analyzes recurrence and transience of the weight process using spectral properties of transition rate matrices derived from the model.

Experimental results

Research questions

• RQ1 Under what conditions does the synaptic weight process remain ergodic in the absence of artificial bounds or metaplasticity?
• RQ2 Can a stochastic STDP rule with discrete weight updates lead to stable, analytically tractable weight dynamics?
• RQ3 Why does the weight process diverge even when the integral of the STDP learning window is negative?
• RQ4 How can the slow dynamics of synaptic weights be rigorously derived from the fast neural spiking dynamics using probabilistic methods?
• RQ5 What is the role of intrinsic noise in the STDP rule, and how does it affect long-term stability and ergodicity?

Key findings

• The weight process remains positive recurrent and ergodic if the parameters satisfy a simple condition derived from the spectral properties of the transition rate matrix.
• Counterintuitively, the weight process can diverge even when the integral of the STDP learning window is negative, due to the interplay between stochasticity and discrete weight updates.
• The invariant measure for the joint neuron-weight process is unique, proven via Laplace transform and diagonal dominance of the associated matrices.
• The slow dynamics of synaptic weights are governed by an effective equation derived by replacing neural activity with its stationary distribution, enabling analytical tractability.
• The model avoids the need for numerical estimation of cross-correlation matrices by using probabilistic tools instead of Taylor or Fourier expansions.
• The analysis confirms that intrinsic noise in the STDP rule can lead to stable, long-term behavior under a simple, explicit condition on the model parameters.