Turing models of cortical folding on exponentially and logistically growing domains

The brain's cerebral cortex is folded into intricate patterns of gyri (hills) and sulci (valleys). The complete biological mechanism underlying the cortical folding process continues to remain elusive to this day. One proposed biological mechanism of cortical folding is the Intermediate Progenitor Model, which correlates patterns of regional self-amplification of neural intermediate progenitor cells with cortical folds. We construct mathematical models of cortical folding utilizing a Turing reaction-diffusion system on an exponentially or logistically growing prolate spheroidal domain. Patterns generated by our Turing system models represent prepatterns for regional activation of self-amplification of intermediate progenitor cells, which may be correlated with cortical folding patterns according to the Intermediate Progenitor Model. We observe that the presence of exponential domain growth generates a Turing pattern that is continually transient and evolves from one pattern to another. This differs from patterns generated by a static domain Turing system, which converge to a final pattern. We also observe that a logistically growing domain Turing system generates a pattern that is transient during the period of rapid domain growth, but then converges to a final pattern similar to a static domain Turing system. Our Turing models of cortical folding can be used to explain certain diseases of cortical folding.