Synaptic normalization is used to enforce competitive dynamics in many models of developmental synaptic plasticity. In linear and semilinear Hebbian models, multiplicative synaptic normalization fails to segregate afferents whose activity patterns are positively correlated. To achieve this, the biologically problematic device of subtractive synaptic normalization must be used instead. Our own model of competition for neurotrophic support, which can segregate positively correlated afferents, was developed in part in an attempt to overcome these problems by removing the need for synaptic normalization altogether. However, we now show that the dynamics of our model decompose into two decoupled subspaces, with competitive dynamics being implemented in one of them through a nonlinear Hebb rule and multiplicative synaptic normalization. This normalization is “emergent” rather than imposed. We argue that these observations permit biologically plausible forms of synaptic normalization to be viewed as abstract and general descriptions of the underlying biology in certain scaleless models of synaptic plasticity.

Neurotrophic factors, particularly the neurotrophin gene family of neurotrophic factors, are implicated in activity-dependent anatomical plasticity in the visual cortex and at the neuromuscular junction. Accumulating evidence implicates neurotrophic factors as possible mediators of activity-dependent competition between afferents, leading to the segregation of afferents' arbors on the target space. We present a biologically plausible mathematical model of competition for neurotrophic factors. We show that the model leads to anatomical segregation, provided that the levels of neurotrophic factors released in an activity-independent manner, or the levels available by exogenous infusion, are below a critical value, which we derive. Above this critical value, afferent segregation breaks down. We also show that the model segregates afferents even in the presence of very highly correlated patterns of afferent activity. The model is therefore ideally suited for application to the development of ocular dominance columns in the kitten visual cortex.

We examine the claim that a class of sprouting-and-retraction models is mathematically equivalent to a fixed-anatomy model. We accept, subject to important caveats, a narrow mathematical equivalence of the energy functions in both classes of model. We argue that this narrow equivalence of energy functions does not, however, entail equivalence of the models. Indeed, the claim of complete model equivalence hides significant dynamical differences between the approaches, which we discuss. We also disagree that our work demonstrates that subtractive constraint enforcement is natural in fixed-anatomy models.