Instructors: Professor Jérémie Lefebvre, University of Ottawa & Professor John Griffiths, University of Toronto

Course Description

This course is in preparation for the Fall 2025 Thematic Program on the Mathematics of Neuroscience at the Fields Institute.

The goal of this special topics course is to provide an overview of models, methods and techniques commonly used in the discipline, to better understand how they are used to probe brain function at various spatial and temporal scales. The course will review important models of both single neurons and networks and discuss current challenges in the field. Topics will include (but not limited to): Hodgkin-Huxley model, Fitzhugh-Nagumo model, Perceptrons, Hopfield networks, Neural Fields, and Wilson-Cowan equations. We will also try to discuss Bayesian networks and probabilistic computations. We will also discuss issues of neural spiking analysis, pattern formation, feedback and synchrony in neural networks. In order to appreciate the value of these models, some preliminary neurobiological facts and principles will also be presented in the lectures. Emphasis will be placed on connecting mathematical insight with neurobiological mechanisms, and how this can be achieved using a combination of theoretical and numerical approaches. We will use elements of dynamical systems (especially bifurcation theory), probability, signal processing and numerical methods. This course is also meant provide an interdisciplinary experience, in which the complexity of neurobiology oftentimes inspires new mathematical challenges.

Format: 3 hours per week

Evaluation: 3 assignments (20% each), project (40%)

Prerequisites: Calculus, Statistics, Dynamical Systems and Introductory Biology/Neuroscience-type courses.

Primary Reading: Research papers & custom learning materials to be provided by lecturers

Supplementary reading:

• “Foundations of Mathematical Neuroscience”, Bard Ermentrout, David Terman, Springer (ISBN-13: 978-0387877075)
• “Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting”, Eugene M. Izhikevich, MIT Press (ISBN-13: 978-0262514200)
• “Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems”, Peter Dayan, Laurence F. Abbott, MIT Press (ISBN-13: 978-0262541855)
• “Neuronal Dynamics: From single neurons to networks and models of cognition.” Wulfram Gerstner, Werner M. Kistler, Richard Naud and Liam Paninski (online: http://neuronaldynamics.epfl.ch/)

Approximate Schedule

• Week #1: Single Neuron Models
• Week #2: Neural Networks (Hopfield, Perceptron, Machine learning)
• Week #3: Wilson Cowan model
• Week #4: Neural fields
• Mid-Term Exam
• Week #5: Oscillations and synchrony
• Week #6: Large Scale Simulations
• Week #7: Probabilistic Brain
• Week #8: Effect of Noise in Neural systems
• Week #9: Anesthesia & plasticity
• Week #10: Presentations

(More logistical details are coming soon.)