# Nonlinear Dynamics in Biological Systems

## General Announcements

Final Project Presentation Schedule

Mon 11/19

• Neuro Team #4 Davis, Meehan, Paydar, Pierce, Dadarlat

Mon 11/26

• Cell differentiation: noble, critser, siddavatam, chen.

Wed 11/28

• glucose, calcium team
• cardiac rhyth. team 1: gore, johannes, hou, ranganath, zakaeifar

Fri 11/30

• biomechanics team
• Rock Paper Scissors

Mon 12/3

• cardiac team 6: croxall, floyd, halsey, hatfield, shah
• The Bowman Group

Wed 12/5

• Higbee, Burton, MacBroom, Lee
• neuronal firing: gallien, morby, scheidt, wilson, green

Friday 12/7

• schultz, fohl, kennedy, lubkert
• HH: beier, brandner, casey, hodgman

## Coures Outline and Syllabus

This course is an introduction to nonlinear dynamics with applications to biology targeted to junior/senior engineering students and 1st year graduate students in engineering and quantitative life sciences.

We will use Strogatz as our main text and supplement with outside biological examples.

## Homework Assignments

homework 4: in Strogatz: 5.2.1 ,5.2.2a, 5.2.11, 5.3.2

homework 5: in strogatz: 6.1.5, 6.3.10, 6.4.4, 6.8.1 a, 6.8.1 d, 6.8.7, 6.8.8

homework 6 (Due: Wed Oct 24): 7.1.1, 7.1.5, 7.3.3, 8.1.1b, 8.2.3

homework 7 (Due: Monday Nov 12): 9.2.1 .. finding the critical parameter value, rH, for the hopf bifurcation around C+ in hte Lorenz equations

## Questions for Professor Rickus or TA Mike

9/7/07

Q: I was wondering how to show that the origin is stable in a super critical pitchfork with r=0 since the slope test fails. Also, what is the stability at the origin when r=0 for the subcritical pitchfork case and by what reasoning?

A: When the linear stability analysis (slope test) fails, the way to determine the stability definitively is to look at the phase portrait. Plot dx/dt versus x for that value of your parameter, r. Is the fixed point of interest stable or unstable? You should be able to tell by the direction of the vector fields on either side of the fixed point. Try this yourself for the supercritical and subcritical and see what you get. After doing this. See me if you are still confused.

## Organize Team Groups and Topics

Team Cardiac Rhythm (5) - Athurva Gore, Iecun Johanes, Yi (Gary) Hou, Harsha Ranganath, Hamid Zakaeifar

Calcium Homeostasis (6) - Jeffrey Kras, Andy Deeds, Brian Kaluf, Dan Song, Keith Rennier (added late), Kyuwan Lee

Glucose-Insulin Model (5)- Zach Featherstone, Ian Thorson, Tracy Liu, Lauren Hamamoto, Kyle Amick

neurological signals (5): Brandon Davis, Nicole Meehan, Omeed Paydar, Andrew Pierce, Christina Dadarlat

neuronal firing:(5) Timu Gallien, Julie Morby, Michelle Scheidt, Mark Wilson, Mandy Green

cadiac modeling(5), Matt Croxall, Meghan Floyd, Erica Halsey, Shari Hatfield, Rohit Shah

bacterial rock paper scissors(3): Team Grad Minority. Alex DiMauro, Trisha Eustaquio, and Nick Snead

circadian rhythms(5): Jeremy Schaeffer, Arun Mohan , Drew Lengerich, Shaunak A Kothari, Iunia Dadarlat

cell differentiation(4): Sarah Noble, Paul Critser, Prasad Siddavatam, Jiji Chen

team name: The Bowman Group, members: Chris Fancher, Todd Shuba, and Ben Zajeski and area of interest: Fermentation

Hospital sustainability(4): Steve Higbee, Halle Burton, Tyler MacBroom, Steven Lee

The Metabolites (4), Members: Brooke Beier, Eric Brandner, Elizabeth Casey, Eric Hodgman, Areas of interest: Metabolism and neuron cells, Potential Project Area: Metabolic flux of neuron cells during firing

human Immune Response - Will Schultz, Ezra Fohl, Eric Kennedy, and Jon Lubkert

## Lecture Notes and Topics

Monday August 20 Lecture 1 powerpoint

Wed Aug 22 in class covered: projects, email list, class wiki, state space, existence and uniqueness, trajectory, dimensionality, possible behavior of 1,2,3 D systems, coverting higher order and time dependent equations to state space, intro to stability, intro to vector fields, autocatalysis example Chapter 1 notes

Fri Aug 24th in class covered: projects, review stability of fixed points, look at linear examples, linear stability analysis, classic May problem: cows in the field

Mon Aug 27th in class covered: identify the bifurcations in the May cow problem, what is a bifurcation, critical parameter values, saddle node bifurcations, introduce bifurcation diagrams

Wed Aug 29th in classe covered: transcritical bifurcations, pitchfork (super and subcritical)

Fri Aug 31st tumor problem - 2 parameter, 1D system with multiple bifurcations. creating stability diagrams. the tangency condition of saddle node bifurcations

Mon Sept 3 - no class labor day holiday

Fri Sept 7 - Nova Chaos Video

Mon Sept 10 - Strogatz Chapter 4, Flow on a Circle, Nonlinear Oscillator, Excitable Cells , Basics of Neuron Physiology

Wed Sept 13 - work on projects in teams

Fri Sept 15 - finish basics of neuron physiology, return to nonlinear oscillator example of a simple excitable cell

Mon Sept 17 - chapter 5 linear 2D systems, phase plane, vector fields in 2D

Wed Sept 19 - 2D stability intro, eigenvalues eigenvectors

Fri Sept 21 - chapter 6 non-linear 2D systems, Example Lotka-Volterra problem. Nullclines, eigenvector analysis.

Mon Sept 24

Wed Sept 26 EXAM TONIGHT EE270

Fri Sept 28 - Chapter 6 continued.

Mon Oct 1 - index theory, introduction to limit cycles

Wed Oct 3 - pass back exam. review problem 8 ... gene switch problem on exam, review curvature of trajectories near stable and unstasble nodes in nonlinear systems

Mon Oct 15 - begin Chap 8 ... 2D bifurcations .... bifurcations of fixed points in 2D ... 2D transcritical bifurication and summary slide

Wed Oct 17 - Bifurcations of Cycles - super and subcritical Hopf bifurcations

Fri Oct 19 - Bifurcations of Cycles: infinite period, saddle node of cycles, homoclinic bifurcations

Mon Oct 22 - Circadian Rhythms & Biological Oscillations

Wed Oct 24 - Prof Rickus Tips for Writing .... Details of Project Requirements

Fri Oct 26 - Scaling Laws of Cycles

Mon Oct 29 - Review Session for Exam

Wed Oct 31 - NO class due to exam on Thursday night

Thur Nov 1 - EXAM II EE270 7 - 9 pm : covers chapters 6, 7, 8. nullclines, index theory, stability of limit cycles, poincare-bendixson theorm, 2D bifurcations (saddle node, transcritical, supercritical pitchfork, subcritical pitchfork, bifurcations of cycles including supercritical Hopf, subcritical Hopf, saddle node bifurcations of cycles, homoclinic bifurcations, infinite period bifurcations

Fri Nov 2 - start 3D systems and the Lorenz equations

Mon Nov 3 - Lorenz equations class powerpoint

## Software Tools

Mathematica

1. Tip: Do the 5 minute (or 10 minute in older versions) tutorial which can be found in the Help Menu
2. Purdue CS hosts a site with an intro to mathematica basics
3. A list of many hosted tutorials can be found here
4. How Purdue students can get a copy of Mathematica Students may purchase an annual "student edition" license by visiting the BoilerCopyMaker facility on the main floor of the Purdue Memorial Union. The cost is \$45 per license (multiple licenses may be purchased - e.g., one each for a desktop and laptop or one for use on your Windows computer and another for your Linux system). These licenses expire at the end of each academic year (in mid-late August). The fee is not pro rated.

XPPAUT XPP/AUTO is designed to solve differential equations with an emphasis on a phase plane and bifurcation graphing. You may find this useful for creating particularly hairly bifurcation diagrams. the software can be downloaded from here

## Background Math to Brush Up On

these following things should be 2nd nature to you. if they are hazing from the summer fun, it would be best to brush up now.

1. sketching of common functions: exponentials exp(ax), sin(x), cos(x), x / x + 1, more generally axn / (xn + b), polynomials
2. taking derivatives of common functions
3. solving simple linear ODEs dx / dt = kx
4. finding eigenvalues and eigenvectors
5. Taylor series expansion
6. solving polynomials
7. complex numbers