The course proposes an introduction to the mathematical modeling and simulation of physiological systems in biological sciences (cellular physiology dynamics of excitable cells) providing the main analytical and numerical methods for the investigation of the mathematical models and the interpretation of the simulated results
Course Prerequisites
Basic mathematical courses of the "laurea triennale" (undergraduate) + the course Dynamical systems: theory and numerical methods
Teaching Methods
Lectures + Matlab laboratory
Assessment Methods
Learning will be assessed by periodic tests during the Matlab labs and by a final written exam consisting in the solution of 4 complex problems in 3 hours, where the students will be allowed to use notes, slides, books, and the codes writtens during the course.
Texts
F. Britton. Essential Mathematical Biology. Springer-Verlag, Heidelberg, 2003.
J.P. Keneer, J. Sneyd. Mathematical Physiology. Springer-Verlag, New York, 1998.
J.P. Keneer, J. Sneyd. Mathematical Physiology I: Cellular Physiology. Springer-Verlag, New York, 2009.
J.P. Keneer, J. Sneyd. Mathematical Physiology II: System Physiology. Springer-Verlag, New York, 2009.
P. Colli Franzone, L. F. Pavarino, S. Scacchi. Mathematical Cardiac Electrophysiology. Springer, 2014
Contents
The course proposes an introduction to the mathematical modeling and simulation of some physiological systems: enzyme kinetics, dynamics of excitable cells, reaction-diffusion systems, bioelectric cardiac processes. - Models of cellular physiology. - Mass action law, biochemical and enzymatic reactions, enzyme kinetics and quasi-steady approximation, cooperative and inhibition phenomena. - Cellular electrophysiology, Nernst potential, electro-diffusion models, approximate current-voltage relationships. - Ionic currents, ion channels with multiple subunits, voltage-clamping, Hodgkin-Huxley formalism. - Approximate two variable excitable models, FitzHugh-Nagumo model: threshold effect and limit cycles. - Hodgkin - Huxley model for the action potential, threshold effects, refractoriness, bifurcation diagrams. - Introduction to reaction - diffusion models, balance laws, diffusion equation. Reaction and transport terms. Initial and boundary conditions. - Numerical approximation of evolution problems. - Introduction to propagation in excitable media. - Cable equation, bistable equation, traveling waves. - Computational Electrocardiology. Anisotropic bidomain model, excitation wavefront propagation, reentry phenomena.
