Barrels are patterned groups of neurons in rodent somatosensory cortex that correspond one to one with the animal's facial whiskers. Dirichiet domains are a class of convex polygon found frequently in nature, often arising by nucleation from center points. Analytic and graphical methods were devised to verify the hypothesis that Dirichlet domains accurately describe the adult barrel fields of normal mice. We found that normal barrel fields and abnormal barrel fields caused by supernumerary whiskers or lesions to the whisker pad are closely approximated by this mathematical formalism. This implies that each developing cortical barrel organizes about a center point. Experiments in neonatal animals (Senft and Woolsey, 1991a) demonstrate foci in the thalamocortical afferent (TCA) distributions. These results support an hypothesis in which TCAs are the nucleating agents causing barrels to organize as Dirichlet domains. This is made possible because TCA terminals from each barreloid (a whisker-related group of cells in the ventrobasal complex of the thalamus) initially colonize somatosensory cortex with an approximately “Gaussian” distribution. These peaked groups of related TCAs behave as Dirichlet domain centers. They generate barrel structures competitively, in animals with normal or with perturbed whisker patterns, via statistical epigenetic interactions within and between distinct TCA Gaussians associated with separate whiskers. This leads to selective axon outgrowth and pruning of single TCA branches, regulated by the TCA population, and creates beneath each Gaussian the dense knot of related TCA arbors typical of the barrel cortex. Similar parcellation of neuronal processes into contending subgroups having spatially coherent actions could lead to nucleation of other geometric patterns as Dirichlet domains elsewhere in the brain.

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