We propose a novel rigorous approach for the analysis of the Linsker's unsupervised Hebbian learning network. The behavior of this model is determined by the underl~ingn onlinear dynamics that are parameterized by a set of parameters originating from the Hebbian rule and the arbor density of the synapses. These parameters determine the presence or absence of a specilic receptive field (also referred to as a 'connection pattern') as a saturated fixed point attractor of the model. In this paper, we perform a qualitative analysis of the underlying nonlinear dynamics over the parameter space, determine the effects of the system parameters on the emergence of various receptive fields, and provide a rigorous criterion for the parameter regime in which the network will have the potential to develop a specially designated connection pattern. In particular, this approach analytically demonstrates, for the first time, the crucial role played by the synaptic arbor density. For example, our analytic predictions indicate that no structured connection pattern can emerge in a Linsker's network that is fully feedforward connected without localized synaptic arbor density. On the other hand, we also show that if the synaptic density functions are appropriately chosen, then any kind of connection pattern may emerge. Our general theorems lead to a complete and precise picture of the parameter space that defines the relationships among the different receptive fields, and yielcl a method to predict whether a given connection pattern will emerge under a given set of parameters without running a numerical simulation of the model. The theoretical results are corroborated by our examples (including center-surround and certain oriented receptive fields), and match key observations reported in Linsker's numerical simulation. The rigorous approach resented here provides a unified treatment of many diverse problems about the dynamical mechanism of the model, and applies not only to the Linsker's network but also to other related self-organization models about neural development.
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