Topographic maps are brain structures which connect two regions [1]. These maps are essential features of primary sensory signal processing. A prototypical animal model of such a system is the mouse retinotopic map [2]. Topography is developed using three distinct mechanisms: chemotaxis, competition, and activity based refinement [3]. Chemotaxis establishes a coarse topography with broad dendritic arbors which is followed by three stages of spontaneously generated waves of electrical activity in the retina: first at E16-P0, then from P0-P11, and finally from P11-P14 [4]. These three periods of have distinct spatio-temporal characteristics and likely perform different functions in the development of the retinotopic system. They are concurrent with electrical activity in the SC and the correlations between these signals guide Hebbian plasticity to make the refinement. Unified models of activity and genetics have found success in predicting the effects of chemical perturbations, but not activity-based perturbations [5]. The activity mechanism in these models condenses the activity into a purely spatial and radially symmetric isotropic form.
A good model of electrical activity in brain regions with lateral connectivity and dense homogenous cell types such as those in the SC is neural field theory (NFT) [6]. A theoretical framework of Hebbian-based plasticity that can incorporate time-signatures of activity has been developed for NFT [7]. This framework allows for the incorporation of a more accurate and complete description of spatio-temporally varying waves. In this paper we shall demonstrate that NFT can support the refinement and establishment of precise topography via waves of propagating activity and biologically reasonable Hebbian learning rules and therefore establish it as a useful model to study the development of topographic systems.
We develop an analytical solution to the field equation by first linearizing the sigmoid activation function. We then proceed with computational analysis of three key parameters: the width of the wave stimulus, wave-speed, and the width of the lateral connections. Finally, we discuss the limitations of the model, implications of these results in the context of the β2 knock-out (an activity perturbation), and future directions.
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