AMATH 534 A: Dynamics of Neurons and Networks

Winter 2022
TTh 11:30am - 12:50pm / SMI 309
Section Type:
Joint Sections:
Syllabus Description (from Canvas):

Course Information

Introduction to Computational Models in Biology
AMATH 422/522

Prof. Eric Shea-Brown


Office Hours:  Tu 8:30-9:20


About the Course

Course Overview

Here, you will learn about models that arise in the life sciences and how they're analyzed using mathematical and computational techniques. We will cover statistical models, discrete- and continuous- time dynamical models, and stochastic models. Applications will sample a wide range of scales, from biomolecules to population dynamics, with an emphasis on common mathematical concepts and computational techniques. Throughout, our themes will include interpretation of existing data and predictions for new experiments.

In class, MATLAB will mostly be used for numerical computation, visualization, and data analysis -- and mathematical tools taught in parallel with their computational implementation. No prior programming experience is assumed. We may introduce and use PYTHON toward the end of the quarter in class. Students are welcome to complete HW assignments in PYTHON at any point, too.

A major component of the course is a mentored, team-based project in a computational biology area of your choosing that makes real and vibrant contact with the current literature.  

Learning Goals

For students with mostly biosciences backgrounds:  

  • Fundamental mathematical and scientific programming techniques for dynamic modeling biological systems across the scales, including matrix models, Markov chains, dynamical systems, and stochastic differential equations. 

For students with mostly mathematical backgrounds:  

  • Fundamental biology concepts from ecology, cell biology, and neuroscience.  Learning some new mathematical or programing techniques from class, and more  from project.

For all students:

  • Application of dynamic modeling techniques to biology:  connection to biology, model formulation, critical analysis of questions that can and cannot be answered.
  • Identifying and implementing rich and plausible mathematical models in the biology literature
  • Asking an open, interesting, and plausible questions based on the research literature, and attempting to answer them 
  • Interdisciplinary team building and collaboration


Required Texts

Required text: "Dynamic Models in Biology,"  by Stephen Ellner and John Guckenhiemer (called EG below).  First chapter provided at this link.

Readings from the literature will be provided with each module as well.


Course topics

(1) Course overview, introduction to programming, and introduction to mathematical models in the life sciences. (0.5 weeks, EG Chapter 1)

  • Modeling objectives: prediction and theory development
  • Rate equations, inflow-outflow models
  • Michaelis-Mentin Kinetics: enzyme-mediated chemical reactions
  • Complex models -- pharmacokinetics
  • Exponential and chaotic population growth in a simple system


(2) Matrix models -- discrete time, linear maps  and Introduction to population biology (1.5 weeks, EG Chapter 2)

  • Euler-Lotka formula and root-finding for age-class models: "Leslie Matrices"
  • Matrix multiplication and eigenvalues
  • The Perron-Frobenius Theorem, dominant eigenvalues and population growth
  • Eigenvalue sensitivity formulas and applications in ecology


(3) Stochastic models (2.5 weeks) and Introduction to neuroscience (2.5 Weeks, EG Chapter 3)

  • Introduction to electrical membranes and neurons
  • Random variables and probability
  • Channel statistics: the binomial distribution
  • Transition probabilities and Markov chains
  • Equilibrium states -- dominant eigenvalues and Perron-Frobenius return!
  • Central limit theorem and deterministic limits
  • Applications to ion channels and reverse-engineering molecular configurations


(4) Continuous time models, systems biology, and biological networks and Introduction to systems biology and genetic networks (3 weeks, EG Chapters 4, 5 , 6.1-6.3)

  • Ordinary differential equations and vector "arrow" fields -- visualizing flows in MATLAB
  • Nullclines, equilibria, and numerical root finding
  • Numerical solution methods
  • Stability and oscillations
  • Lyapunov exponents and predictability, and numerical algorithms
  • Genetic oscillators
  • Genetic switches
  • Stochastic differential equations and numerical methods


(5) Computing networks in biology (1-2 weeks)

  • Biological memory models and attractors
  • Intro to random connectivity matrices, girko's circle rule for eigenvalues
  • Nonlinear dynamics on complex networks: the transition to chaos
  • Complex networks and motifs



Course grades are composed of: Problem Sets 40%, Case study presentation 10%, Project presentation 10%, Project paper 40%.  Problem sets will be due on select Wednesdays at the start of class.

Important formatting instructions for problem sets: in your writeup please present all material for a given problem together -- e.g. under "Problem II" you'd have any and all code that you used for that problem, a written answer (i.e., "the dominant eigenvalue is 0.921"), plots that explain and back up your findings and answers, and any analysis. Then we'd go to the next problem. (Not stapling all code for all problems together as an appendix at the end.) The "live script" format in MATLAB and the similar Jupyter notebook in PYTHON will be very helpful here.  Handwritten work is welcome; just scan and insert in your writeup.  

Academic Integrity

Working together on problem sets is encouraged.  The work you turn in on problem sets should be your own understanding and calculations.  Please do not refer to previous years' solutions.



University practices and policies


Access and Accommodations

Your experience in this class is important to me. If you have already established accommodations with Disability Resources for Students (DRS), please communicate your approved accommodations to me at your earliest convenience so we can discuss your needs in this course.  If you have not yet established services through DRS, but have a temporary health condition or permanent disability that requires accommodations, you are welcome to contact DRS at 206-543-8924 or or DRS offers resources and coordinates reasonable accommodations for students with disabilities and/or temporary health conditions.  Reasonable accommodations are established through an interactive process between you, your instructor(s) and DRS.  It is the policy and practice of the University of Washington to create inclusive and accessible learning environments consistent with federal and state law.

Religious Accommodations

Required Syllabus Language: “Washington state law requires that UW develop a policy for accommodation of student absences or significant hardship due to reasons of faith or conscience, or for organized religious activities. The UW’s policy, including more information about how to request an accommodation, is available at Religious Accommodations Policy.  Accommodations must be requested within the first two weeks of this course using the Religious Accommodations Request form.

Notice to Students - Use of Plagiarism Detection Software

Notice: The University has a license agreement with SimCheck, an educational tool that helps prevent or identify plagiarism from Internet resources. Your instructor may use the service in this class by requiring that assignments are submitted electronically to be checked by SimCheck. The SimCheck Report will indicate the amount of original text in your work and whether all material that you quoted, paraphrased, summarized, or used from another source is appropriately referenced.

Catalog Description:
Covers mathematical analysis and simulation of neural systems - singles cells, networks, and populations - via tolls of dynamical systems, stochastic processes, and signal processing. Topics include single-neuron excitability and oscillations; network structure and synchrony; and stochastic and statistical dynamics of large cell populations. Prerequisite: either familiarity with dynamical systems and probability, or permission of instructor.
Last updated:
May 24, 2024 - 8:27 pm