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The brain is one of nature's greatest mysteries. The mechanism by which the folds of the brain's cerebral cortex, called gyri (hills) and sulci (valleys), are formed remains unknown. Existing biological hypotheses that attempt to explain the underlying mechanism of cortical folding conflict. Some hypotheses, such as the Intermediate Progenitor Model, emphasize genetic chemical factor control. Others, such as the Axonal Tension Hypothesis, emphasize the influence of physical tension due to axonal connections. To bring mathematics into this debate, this dissertation presents two biomathematical models of cortical folding that utilize a Turing reaction-diffusion system on an exponentially or logistically growing prolate spheroidal domain. These models are used to investigate the validity of the Intermediate Progenitor Model, thereby investigating the role of genetic chemical factor control of the development of cortical folding patterns. We observe that the presence of domain growth drives the patterns generated by our growing prolate spheroidal Turing systems to become transient. An exponentially growing prolate spheroidal domain generates a pattern that continually evolves, while a logistically growing prolate spheroidal domain generates a pattern that evolves while the domain is growing but then converges to a final pattern once the domain growth asymptotically stops. Patterns generated by the model systems represent genetic chemical prepatterns for self-amplification of intermediate progenitor cells, which may be correlated to cortical folding patterns according to the Intermediate Progenitor Model. By altering system parameters, we are able to model diseases of cortical folding such as polymicrogyria where the cortex possesses too many folds as well as diseases where the cortex has too few cortical folds such as Norman-Roberts Syndrome (microcephalic lissencephaly) and normocephalic lissencephaly. Our ability to model such a variety of diseases lends support to the role of genetic control of cortical folding pattern development and therefore to the Intermediate Progenitor Model.
cortical folding, growing domain, pattern formation, reaction-diffusion system, Turing system
Date of Defense
March 18, 2013.
A Dissertation submitted to the Department of Mathematics in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy.
Includes bibliographical references.
Monica K. Hurdal, Professor Directing Dissertation; Oliver Steinbock, University Representative; Richard Bertram, Committee Member; Nick Cogan, Committee Member; Brian Ewald, Committee Member.
Florida State University
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