Emergent Dynamical Properties of the BCM Learning Rule

Lawrence C Udeigwe,G Bard Ermentrout,Paul W Munro

doi:10.1186/s13408-017-0044-6

Lawrence C Udeigwe, G Bard Ermentrout + Show 1 more

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https://doi.org/10.1186/s13408-017-0044-6

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Abstract

The Bienenstock–Cooper–Munro (BCM) learning rule provides a simple setup for synaptic modification that combines a Hebbian product rule with a homeostatic mechanism that keeps the weights bounded. The homeostatic part of the learning rule depends on the time average of the post-synaptic activity and provides a sliding threshold that distinguishes between increasing or decreasing weights. There are, thus, two essential time scales in the BCM rule: a homeostatic time scale, and a synaptic modification time scale. When the dynamics of the stimulus is rapid enough, it is possible to reduce the BCM rule to a simple averaged set of differential equations. In previous analyses of this model, the time scale of the sliding threshold is usually faster than that of the synaptic modification. In this paper, we study the dynamical properties of these averaged equations when the homeostatic time scale is close to the synaptic modification time scale. We show that instabilities arise leading to oscillations and in some cases chaos and other complex dynamics. We consider three cases: one neuron with two weights and two stimuli, one neuron with two weights and three stimuli, and finally a weakly interacting network of neurons.

Highlights

IntroductionThe topic of synaptic plasticity has remained relevant. A pioneering theory on this topic is the Hebbian theory of synaptic modification [1, 2], in which Donald Hebb proposed that when neuron A repeatedly participates in firing neuron B, the strength of the action of A onto B increases
For several decades the topic of synaptic plasticity has remained relevant
If the weights and the threshold change slowly compared to the change in the stimulus presentation, the differential equations for the BCM rule can be averaged over the inputs: m τwwj = pkxkj vk(vk − θ ), k=1 m τθ θ = pkvk2 − θ, k=1 where vk is given in Eq (10)

Summary

Introduction

The topic of synaptic plasticity has remained relevant. A pioneering theory on this topic is the Hebbian theory of synaptic modification [1, 2], in which Donald Hebb proposed that when neuron A repeatedly participates in firing neuron B, the strength of the action of A onto B increases. A few decades later, Nass and Cooper [3] developed a Hebbian synaptic modification theory for the synapses of the visual cortex, which was later extended to a threshold dependent setup by Cooper et al [4]. In this setup, the sign of a weight modification is based on whether the post-synaptic response is below or above a static threshold. A response above the threshold is meant to strengthen the active synapse, and a response below the threshold should lead to a weakening of the active synapse

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