Mathematical Neuroscience

Department: 
MATH
Course Number: 
8803-CHO
Hours - Lecture: 
3
Hours - Total Credit: 
3
Typical Scheduling: 
Not Regularly Scheduled

This course will fill the gap between the computational and engineering aspects of neuroscience covered in BMED 7610 and ECE/BMED6790 by covering mathematical analysis and simulation of neural systems  across single cells, networks, and populations, employing methods from dynamical systems, network  science, and stochastic processes. The topics will include single-neuron excitability and bifurcation,  network structure and synchrony, and statistical dynamics of large neural populations. 

Grades will be based on problem sets and a term project. 

Prerequisites: 

The prerequisite of the course will be familiarity with dynamical systems (at the level of MATH 4541, although it can be flexible) and either MATLAB or Python.

Course Text: 

There will be no textbook requirement, but topics and materials will be adapted from

1) “Neuronal Dynamics” by Gerstner, Kistler, Naud, and Paninski;

2) “Dynamical Systems in Neuroscience” by Izhikevich;

3) Some key classical papers in the field. 

Topic Outline: 

1. Nonlinear dynamics of single neurons

     a. Bifurcations in single neurons

     b. Timescale separation in generalized integrate and fire models
     c. Data-driven, reduced neuronal models
2. Spiking dynamics
     a. Spike time correlations
     b. Lyapunov exponents, attractors
     c. Information measures in spike trains
3. Population dynamics
     a. Diffusion approximations and stochastic-differential equations
     b. Mean-field models
     c. Balanced networks
4. Learning and neural networks
     a. Spike-time-dependent plasticity
     b. Recurrent neural networks
5. (If time permits) Understanding brain networks
     a. Local & global network measures 
     b. Generative models of brain networks