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

Course Dates: February 5th - April 16th, 2025
Mid-Semester Break: February 24th-28th, 2025
Lecture Times: Wednesdays | 2:00 PM - 5:00 PM (ET)
Office Hours: TBA

Registration Fee: PSU Students - Free | Other Students - CAD$500

Capacity Limit: 35 students

Format: Hybrid synchronous delivery

• In-Person: Room 210, Fields Institute (for students in Toronto)
• Online: via Zoom (for students not in Toronto)

There will be one 3 hour session per week, each consisting of a 1.5 hour lecture, followed by a 1.5 hour coding-based practical tutorial. Lectures and tutorials will be given in-person at the Fields Institute, located at the corner of UofT St George campus. These will be streamed live on zoom and recorded for non-local attendees, and interaction from the online audience will be encouraged and facilitated.

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 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 to provide an interdisciplinary experience, in which the complexity of neurobiology oftentimes inspires new mathematical challenges. The practical aspects of the course (assignments and final project) will involve hands-on tutorial content, implementing and exploring computational neuroscience, data science, data visualization, and scientific computing concepts in Python using jupyter/google colab notebooks.

Course Texts:

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/)

Prerequisites: Basic Calculus, Basic Neurobiology, Basic Python.

Evaluation:

• Assignment 1: 30%
• Assignment 2: 30%
• Presentation: 20% 
• Engagement: 10%

There will be 2 coding-based assignments (30% each), covering two major topics from the lecture series. For each of these, students will submit a short jupyter notebook-based report, in which they will engage with a number of concepts introduced in the lectures and practical tutorials. The assignments will include answering some specific mathematical and neuroscientific questions posed, and some more open-ended exploration of the topic. This will be done with (short) simulations and data/results visualizations written in Python, plus additional narrative commentary, that build on examples provided in the tutorials. A final oral presentation on a subset of the work done in the two assignments will receive 20% of the mark, with the final 10% allocated to engagement demonstrated throughout the course. 

Schedule:

All lectures are on Wednesdays, 2:00 PM - 5:00 PM (ET), in-person at the Fields Institute and online via Zoom

• Feb 05 - Lecture 01: Single Neuron Models
• Feb 12 - Lecture 02: Neural Networks (Hopfield, Perceptron, Machine learning)
• Feb 19 - Lecture 03: Wilson Cowan Model
• Feb 26 - No Lecture
• Mar 05 - Lecture 04: Neural fields - Assignment #1 released
• Mar 12 - Lecture 05: Oscillations and synchrony (fully online class)
• Mar 19 - Lecture 06: Large Scale Simulations - Assignment #2 released
• Mar 26 - Lecture 07: Probabilistic Brain
• Apr 02 - Lecture 08: Effect of Noise in Neural Systems
• Apr 09 - Lecture 09: Anaesthesia & Plasticity
• Apr 16 - Presentations & Conclusions