Advanced Topics in Numerical Analysis: Computational Neuroscience

Aaditya Rangan

Computer Science

Prerequisites: Students are expected to be familiar with ordinary and partial differential equations, as used in applications. Some familiarity with a programming language (such as Matlab, Java, C, etc.) is expected. The necessary background in biology will be explained within the course, and so there is no biology prerequisite.
This course focuses on the computational techniques used to model neuronal networks. Along the way, we will discuss and comment on the biological phenomena which have been successfully modeled using these techniques. We will also try and highlight the features of these models which enable them to capture different types of network dynamics.
The syllabus for this course will include:
1. the analytical and numerical consequences of various approximate neuron-models (e.g., Cable Equations, Hodgkin-Huxley and Integrate-and-fire models, and reductions from deterministic point-neuron models to stochastic processes).
2. large-scale models of the early olfactory system (e.g., the antennal-lobe models of Bazhenov, Laurent, Kopell and others), and some numerical details necessary for the fast simulation of stiff neuron-neuron interactions.
3. large-scale models of the early visual system (e.g., the "NYU" model of McLaughlin, Shelley and Shapley, as well as models of primary visual cortex developed by Miller, Roque and others) and numerical details necessary for fast simulation of spatially extended systems.
4. numerical methods for the coarse-grained population-dynamics models introduced by Knight, Abbott, van Vreeswijk, Sirovich, Tranchina and others --- focusing on the numerical details required to resolve the discontinuous solutions which arise from these population-dynamics equations.
Grading: This course will be graded as a regular course. The course grade will involve homework assignments, some of which will involve computation, as well as a computing project. There will not be an exam. Students are encouraged to collaborate and work in teams, both for the homework and the project.