Mathematical models of cortical development

We extend classical activity-based developmental models of ocular dominance column (ODC) formation in primary visual cortex (V1) to include cortical growth and cortex's laminar structure. We show that with cortical growth the OD pattern exhibits a sequence of pattern forming instabilities as the size of the cortex increases. Each instability results in the insertion of an additional OD column such that over the course of development, the mean width of an ODC is approximately preserved, consistent with recent experimental observations of postnatal growth in cat. The other biologically motivated extension we make is to consider a multilayer representation of V1 with thalamic and vertical connections taken to be modifiable by activity. By including the layer-specific thalamic input to V1 along with the interlaminar projections to a correlation-based Hebbian learning rule, our model allows for the joint development of OD columns and cytochrome oxidase (CO) blobs in primate V1. The developed OD map in layer 4C is inherited by layer 2/3 via the vertical projections. Competition between these projections and the direct thalamic input to layer 2/3 then results in the formation of CO blobs superimposed upon the OD map. The spacing of the OD columns is determined by the spatial profile of the intralaminar connections within layer 4, while the spacing of CO blobs depends both on the width of the ODCs inherited and the spatial distribution of intralaminar connections within the superficial layer. These mathematical models of cortical development demonstrate that simple models with key aspects of the biological system give us insight into the underlying mechanisms for observed patterns in V1.