An analysis of synaptic normalization in a general class of Hebbian models

In standard Hebbian models of developmental synaptic plasticity, synaptic normalization must be introduced in order to constrain synaptic growth and ensure the presence of activity-dependent, competitive dynamics. In such models, multiplicative normalization cannot segregate afferents whose patterns of electrical activity are positively correlated, while subtractive normalization can. It is now widely believed that multiplicative normalization cannot segregate positively correlated afferents in any Hebbian model. However, we recently provided a counterexample to this belief by demonstrating that our own neurotrophic model of synaptic plasticity, which can segregate positively correlated afferents, can be reformulated as a nonlinear Hebbian model with competition implemented through multiplicative normalization. We now perform an analysis of a general class of Hebbian models under general forms of synaptic normalization. In particular, we extract conditions on the forms of these rules that guarantee that such models possess a fixed-point structure permitting the segregation of all but perfectly correlated afferents. We find that the failure of multiplicative normalization to segregate positively correlated afferents in a standard Hebbian model is quite atypical.