University of Illinois Urbana-Champaign · Fall 2026

ECE/CS 598 RE: Dynamical Systems and Neural Networks

Use dynamical systems to understand computation, learning, and chaos in artificial and biological neural networks through theory, simulation, and a hands-on research project that culminates in a NeurIPS/PRL-style report.

Teaching Team

Prof. Rainer Engelken
engelken@illinois.edu
Research Website
Teaching Assistant: Sophia Wilhelm

Lectures

Tuesday & Thursday
11:00 am–12:20 pm
ECEB 4070

Course

4 graduate credit hours
No written midterm or final exam
Official course listing

About This Course

Dynamical systems theory gives us a quantitative language for opening the black box of neural networks.

We ask how network states evolve, why perturbations grow or decay, how collective activity emerges, and how those dynamics shape computation and learning. The course connects mathematical analysis with simulations and current research in machine learning and computational neuroscience.

Scientific Content

The common toolkit includes discrete- and continuous-time dynamics, phase space, fixed points and limit cycles, linearization and stability, attractors, perturbation growth, chaos, Lyapunov spectra, recurrent gradients, and optimization dynamics.

Mean-field theory describes the complex dynamics of large networks with a few effective variables. We apply these ideas to rate-based models and to spiking neural network models, in which neuron models communicate through discrete events called spikes. Numerical work will primarily use Python or Julia.

Six phase-space maps showing basins of attraction as a model parameter changes
Basins of attraction of a neural circuit model change as a biophysical parameter changes.
Two-dimensional Poincaré section colored by local Lyapunov exponents
Two-dimensional Poincaré section of a three-dimensional dynamical system, colored by local Lyapunov exponents.

Hands-On Research Project

The project begins early and may be completed individually or in a small team. Many projects start from provided code or a validated baseline and connect closely to current research. Students move from a precise question to a checked analytical or computational result through a project pitch and plan, a preliminary report, and peer review at each major written milestone.

The final deliverables are a concise NeurIPS/PRL-style report and a presentation.

Prerequisites

Required: Multivariable calculus and linear algebra (for example, UIUC MATH 415 and MATH 416 or equivalent), plus basic proficiency in Python, Julia, or a similar language.

Recommended: Familiarity with ordinary differential equations and foundational machine-learning concepts.

Prior coursework in neuroscience is not required.

Course Format

Two 80-minute meetings per week combine theory, worked examples, discussion, and computational analysis. The semester is organized around developing the tools needed for the research project.

Short analytical and computational project toolkits develop techniques used in the projects. Brief in-class mini-quizzes are based on those toolkits.

Course Communication

This website provides the course overview and protected course materials. Zulip is used for announcements and discussion; Gradescope is used for submissions, feedback, and grades.

Grading

There will be no written midterm or final exam. Assessment is based on project toolkits, mini-quizzes, staged project milestones, the final project, and peer review.

Resources & Reading Material

Primary Texts (Excerpts)

  • Pikovsky & Politi (2016). Lyapunov Exponents: A Tool to Explore Complex Dynamics.
  • Izhikevich, E. M. (2007). Dynamical Systems in Neuroscience.
  • Gerstner, W., et al. (2014). Neuronal Dynamics. [Online Version]
  • Strogatz, S. H. (2018). Nonlinear Dynamics and Chaos.

Software

We will primarily use Python or Julia for computational assignments. Familiarity with libraries such as NumPy, SciPy, and Matplotlib is expected. We may also use machine-learning libraries such as PyTorch, Flux, or JAX.

Relevant Papers