Syllabus
CNeuro 501 · Fall 2026
Syllabus
BIO4502e.01 · Computational Neuroscience · Fall 2026
Course Description
The emergence of intelligence and behavior from the complex interactions within the brain remains one of the most significant and unsolved mysteries in modern science. This is an exciting era. In the last decade, we have witnessed rapid advancements in experimental tools that now enable us to monitor and manipulate brain circuits with unprecedented precision. However, it is also a perplexing time. Neuroscientists are navigating the intricate landscapes of brain structures and dynamics. Mathematical theory has become crucial for integrating seemingly unrelated evidence, providing new insights, guiding new experiments, and identifying concepts and principles of brain function.
In this course, we will explore how physics, engineering, and mathematics have shaped our understanding of the brain. In particular, we will investigate the relationship between structure, dynamics, representation, and behavior. Special topics may include wiring optimization in neural circuit, attractor and chaotic dynamics in neural network, sensory and motor representation, biological learning rules, Hopfield network, and hierarchical control of behaviors. We will also discuss connections with modern machine learning methods.
What You Will Learn
Connect neurons to perform computation: How a network of simple, noisy, nonlinear units performs operations — amplification, integration, selection, gating — that no single neuron can perform alone. How wiring optimization principles can be used to explain various structures in the brain circuits. We also introduce the basic circuit motifs (feedforward, recurrent, lateral inhibition, gain modulation) and the rate-model formalism used throughout the rest of the course.
Recurrent network and attractor dynamics: Connectivity structure creates fixed points, line attractors, and ring attractors that hold information without input. This gives physics students a familiar entry point — energy landscapes, stability analysis, symmetry — and shows how memory and integration arise from wiring rather than from single-cell properties.
E/I balance and chaotic dynamics: Random connectivity with balanced excitation and inhibition produces irregular, high-dimensional activity with a sharp transition to chaos. This is in contrast with attractor dynamics that are low dimensional. Random matrix theory and mean-field methods here are the same tools used in disordered systems, and they explain why cortical spiking looks stochastic despite deterministic dynamics.
Geometry and dimensionality of population neural activity: Population activity occupies low-dimensional manifolds embedded in the space of all neurons, and the shape of that manifold constrains what downstream areas can read out. This shifts the unit of analysis from the single neuron to the collective state.
Efficient coding principle: Given limited spikes, noise, and metabolic cost, the optimal code adapts to input statistics — predicting receptive fields, adaptation, and contrast response from the statistics of natural signals. This is the first normative argument in the course: structure is derived rather than described.
Stimulus discrimination, Fisher information and information-limited correlations: Fisher information quantifies how well a population distinguishes nearby stimuli, and correlated noise aligned with the signal direction sets a hard ceiling that adding neurons cannot lift. This makes precise why population geometry, not neuron count, determines coding fidelity.
Normative approach to neural computation and the emergence of neural representation: Instead of postulating representations, one specifies an objective and a constraint and asks what representation emerges — in trained recurrent networks, in Bayesian inference schemes, in optimization-derived circuits. Learning appears here as the mechanism that turns a task into a representation.
Reward and reinforcement learning: Temporal-difference learning, value functions, and policy optimization, together with the dopamine prediction-error evidence that made this the clearest link between a normative algorithm and a measured neural signal. This introduces learning driven by scalar feedback.
How does an animal move — an integrated approach: Closing the loop: sensory input, internal state, motor output, and the body’s mechanics form one dynamical system, illustrated with whole-brain imaging in C. elegans and larval zebrafish. Every earlier concept reappears here as a component of behavior in a real animal.
Grading
| Component | Weight |
|---|---|
| Homework | 70% |
| Final | 30% |
| Bonus | Depend |
If a solution has novelty or answers a question tagged as bonus, you can get a bonus. Posting a good question or offering a good answer on the issue page will also earn you a bonus.
Score = 70% × (average homework) + 30% × (final exam) + bonus
Submission: USTC bb platform / offline
Tip: Although we have not yet encountered such a situation, please do not abuse the GitHub issue page for bonus. The TA reserves the right of final interpretation.
Recommended Textbooks
- Abbott & Dayan, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems — MIT Press
- Xiao-Jing Wang, Theoretical Neuroscience: Understanding Cognition — Taylor & Francis
- Sterling & Laughlin, Principles of Neural Design — MIT Press
- Grace Lindsay, Models of the Mind: How Physics, Engineering and Mathematics Have Shaped Our Understanding of the Brain — Amazon
Tip: You don’t have to read them all. Learning to reduce redundant information is also an important skill in this course.
Resources of Previous Semesters
The course repository provides course resources of different semesters in distinct branches. If you are interested in how senior students learned this course in the good old days, or want a preview of topics we will consider, you can browse these branches.
Policies
Late homework: Homework submitted after the deadline will incur a penalty of 2 points per day, unless you have emailed the TA in advance to arrange an extension.
Collaboration: Collaboration is allowed, but you must state it in your submission, cite all sources properly, and clearly declare any use of AI tools. In this era, we will unavoidably collaborate with AI tools, and sometimes feel overwhelmed or discouraged. Thus, as a researcher, you should try to preserve your subjectivity.
…if the authors cannot convincingly demonstrate that they can give a clear, expert-level talk on their results, that is correct and properly attributed, then the result should not be published.
— Terence Tao, the commentary of the Leiden Declaration on Artificial Intelligence and Mathematics, in “Mathematics in the Age of AI” (ICM 2026 invited lecture; preprint)