Advanced Topics in Numerical Analysis: Computational Neuroscience
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
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
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
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.