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Case Western Reserve University DEPARTMENT OF MATHEMATICS |
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DEPARTMENT OF MATHEMATICS SEMINARS and COLLOQUIUMMATHEMATICAL LIFE SCIENCES SEMINARS |
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Deena SCHMIDT ABSTRACT Sleep and wake states are each maintained by activity in a corresponding neuronal network, with mutually inhibitory connections between the networks. In infant mammals, the durations of both states are exponentially distributed, whereas in adults, the wake states yield a heavy-tailed distribution. What drives this transformation of the wake distribution? Is it the altered network structure or a change in neuronal dynamics? What properties of the network are necessary for maintenance of neural activity on the network and what mechanisms are involved in transitioning between sleep and wake states? We explore these issues using random graph theory, specifically looking at stohastic processes occurring on random graphs, and also by investigating the accuracy of predictions made by deterministic approximations of stochastic processes on networks.
Professor Clyde F. MARTIN ABSTRACT This informal talk will discuss some of the modeling efforts as well as some of the clinical trials that we are trying to establish. We have worked extensively on the modeling of the interaction between glucose and insulin in the context of pharmacokinetic models based on the original dissertation of Sorenson at MIT. It would be interesting to discuss the incorporation of these models into the planning of a clinical trial.
Professor Wojbor WOYCZYNSKI ABSTRACT (Joint work with B. Jourdain and S. Meleard) We are interested in modeling Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by each individual’s trait values, and interactions between individuals. An offspring usually inherits the trait values of her progenitor, except when a random mutation causes the offspring to take an instantaneous mutation step at birth to new trait values. In the case we are interested in, the probability distribution of mutations has a heavy tail and belongs to the domain of attraction of a stable law and the cor responding diffusion admits jumps. This could be seen as an alternative to Gould and Eldredge’s model of evolutionary punctuated equilibria.We investigate the large population limit with allometric demographies: larger populations made up of smaller individuals which reproduce and die faster, as is typical for micro-organisms. We show that depending on the allometry coefficient the limit behavior of the population process can be approximated by nonlinear Lévy flights of different nature: either determin16 istic, in the form of non-local fractional reaction–diffusion equations, or stochastic, as nonlinear super-processes with the underlying reaction and a fractional diffusion operator. These approximation results demonstrate the existence of such non-trivial fractional objects; their uniqueness is also proved
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