# Nonlinear Dynamics in Biological Systems

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## Contents |

## 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.
- circadian rhyth.

**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

**Wed Sept 5** - finish stability diagram and phase portraits of tumor treatment problem. Mathematica File Used in Class

**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

**Fri Oct 12** - glycolysis oscillations example using mathematica

**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

**Wed Nov 5** - Lorenz Equations

**Fri Nov**

**Mon Nov 12** - class cancelled. prof rickus in boston

**Wed Nov 14** - TA Mike lecture.

**Fri Nov 16** - Chapter 10 - 1D maps

**Mon Nov 19** - Group Presentation: polynomial neuron model

**Wed Nov 21 / Fri Nov 23** - THANKSGIVING BREAK

**Mon Nov 26** - Group Presentations

## Software Tools

**Mathematica**

- Tip: Do the 5 minute (or 10 minute in older versions) tutorial which can be found in the Help Menu
- Purdue CS hosts a site with an intro to mathematica basics
- A list of many hosted tutorials can be found here
- 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.

- sketching of common functions: exponentials
*e**x**p*(*a**x*),*s**i**n*(*x*),*c**o**s*(*x*),*x*/*x*+ 1, more generally*a**x*^{n}/ (*x*^{n}+*b*), polynomials - taking derivatives of common functions
- solving simple linear ODEs
*d**x*/*d**t*=*k**x* - finding eigenvalues and eigenvectors
- Taylor series expansion
- solving polynomials
- complex numbers