Diffusion in pipes and thin channel surfaces
Eligibility: Undergraduate students enrolled in the Fields Undergraduate Summer Research Program (FUSRP 2022) [http://www.fields.utoronto.ca/activities/22-23/2022-FUSRP]
Neuron-Glial Interaction Laboratory
Brain Division, Krembil Research Institute
University Health Network, Toronto, Ontario
Maurizio De Pitta
Department of Physiology, Faculty of Medicine
University of Toronto
applied mathematics, differential geometry, diffusion theory, complex variable calculus, elliptic integrals, computational neuroscience, cell segmentation, computer-aided design
An essential challenge in fundamental neuroscience is cell reconstruction. This reconstruction is based on a geometrical component that accounts for the cell anatomy and a dynamical part, which instead describes the cell's physiology, that is, the ensemble of molecular signals that go on inside the cell, and together, account for the cell's functions. The majority of those signals are chemical and depend on the intracellular diffusion of molecules and ions. While it is possible to describe intracellular signaling in brain cells by diffusion-like equations, the boundary conditions set by the complicated anatomy of these cells make the problem analytically intractable and computationally inefficient. For this reason, we choose to compromise between anatomical resolution and signal dynamics, approximating cellular anatomy by ensembles of connected compartments of defined geometries, like generalized cylinders (or pipes) and shells thereof, and infer simplified yet realistic boundary conditions amenable to mathematical tractability.
We aim to derive simplified equations for diffusion in pipes (or channels) and thin shells built around these pipes. These geometries are chosen because they can be realistically adapted to segment brain cells. At the same time, the type of equations that we want to derive aim to mimic essential features of intracellular molecular signals in a mathematically tractable framework. This is a challenging but fun project that looks for students interested in applied math and multidisciplinary research. The student should have a background in differential geometry and complex variable calculus.
Future Perspectives and contacts
Depending on the student's engagement and performance, I will happily discuss future directions in the student's
training and consider the opportunity to have her/him/they join my group for Master's or Ph.D. projects.
Interested students are encouraged to reach out to Dr. De Pitta at the Krembil Research Institute (firstname.lastname@example.org) upon coordination with their supervisor in Fields Undergraduate Research Program.