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Mathematics 8200.001

Applied Mathematics in Neuroscience

TR 2:00–3:20

Wachman Hall 527

Wachman Hall 514

TR 1:00–2:00 & TR 3:20–3:50

The course is intended to bring together fundamentals of biochemical and electrical signaling in brain cells, development of mathematical models, and numerical methods for solving biophysical models. Computational frameworks for simulating processes in brain cells will be introduced and used in practice. Topics will include modeling of brain cell signals as purely time-dependent processes (integrate-and-fire models), one-dimensional compartment models (Hodgkin-Huxley models), all the way to a full three-dimensional structure-function models. These modeling approaches motivate mathematical and computational topics, including numerical methods for ordinary differential equations (ODEs) and partial differential equations (PDEs). The course will give a practical approach to time-stepping, Finite Difference and Finite Element methods for solving ODEs as well as elliptic and parabolic PDEs. Introduction of computational methods for reconstructing realistic cellular morphologies and computational grid generation (domain discretization) will allow the treatment of research-relevant problems. Finally, methods for solving large linear equation systems will complete the cycle from modeling to simulation. 


Find below material relevant for the course

Course Notes

Week 1 - Biological Foundation: Download notes

Week 2 - Physiology Basics: Download notes

Week 3 - Integrate and Fire models: Download notes

Week 4 - Modeling synapses: Download notes

Week 5 - Cable equation: Download notes

Week 6 - Multicompartment models: Download notes

Week 7 - NeuroBox application and

Week 8 - Calcium models: Download notes

Week 9 - Image processing: Download notes

Week 10 - Solving the filter equation: Download notes

Week 11 - Functional analysis: Download notes

Week 12 - Variational problem: Download notes

Week 13 - Galerkin method: Download notes

Week 14 - Courant example: Download notes

Week 15 - Finite Elements (1D): Download notes

Week 16 - Finite Elements (2D): Download notes

Student Presentations

Presentation 1 - Hodgkin and Huxley model: Download papers

Presentation 2 - Leaky integrate and fire models: Download paper

Presentation 3 - Basics of ODE solvers: Download paper

Presentation 4 - Synapse models: Download paper

Presentation 5 - The NEURON simulator: Download paper

Presentation 6 - Cellular calcium dynamics: Download paper

Presentation 7 - Image processing: Download paper

Presentation 8 - Orders of finite difference methods

Presentation 9 - Inverse Modeling: Download paper

Presentation 10 - TMS: Download paper