SimulationsPDF

# SimulationsPDF - BME 580.223 Models and Simulation Part 2 Introduction to Nonlinear Dynamical Systems Modeling Raimond L Winslow Department of

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BME 580.223 Models and Simulation Part 2 Introduction to Nonlinear Dynamical Systems Modeling Raimond L. Winslow Department of Biomedical Engineering The Johns Hopkins University School of Medicine Instructor Version – Do Not Distribute

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Table of Contents Part 2 . .................................................................................................................... 1 Introduction to Nonlinear Dynamical . .................................................................. 1 Systems Modeling . ................................................................................................ 1 Raimond L. Winslow . ........................................................................................... 1 Department of Biomedical Engineering . .............................................................. 1 The Johns Hopkins University School of Medicine . ............................................ 1 Lecture 1: Introduction to Iterated Maps . ............................................................. 5 1. General Iterated Map . ........................................................................................................ 5 1.1 Definition of an Iterated Map . ..................................................................................... 5 2. Linear Maps . ...................................................................................................................... 5 3. Orbits of Iterated Maps . ..................................................................................................... 6 4. Some Questions We Can Ask About Iterated Maps . ......................................................... 6 5. Asymptotic Behavior of Orbits . ......................................................................................... 6 5.1 Fixed Points of a Scalar Map . ...................................................................................... 6 5.2 Stability of Fixed Points. .............................................................................................. 7 5.3 Periodic Solutions . ..................................................................................................... 10 6. What You Need to Know From Lecture 1 . ...................................................................... 13 Lecture 2: Bifurcation and Chaos in Iterated Maps . ........................................... 14 1. The Logistic Map . ............................................................................................................ 14 2. Biological Interpretation of the Logistic Map . ............................................................... 14 3. Fixed Points of the Logistic Map: . ................................................................................... 15 3.1 Stability of Fixed Points of the Quadratic Map . ........................................................ 17 3.2 Bifurcations: . .............................................................................................................. 18 4. Liapunov Exponents . ....................................................................................................... 22 5. What You Need to Know From Lecture 2 . ...................................................................... 26 Lecture 3: Review of Linear Systems . ................................................................ 27 1. Scalar Linear System – An Important Biophysical Example . ......................................... 27 2. Multi-Variate (Coupled) Linear Systems. ........................................................................ 28 3. Eigenvalues and Solution Behavior . ................................................................................ 35 Fig. 3.63. What You Need to Know From Lecture 3 . ......................................................... 38 3. What You Need to Know From Lecture 3 . ...................................................................... 39 Lecture 4: Basics of Continuous Time Nonlinear Systems . ............................... 41 1. The Initial Value Problem . ............................................................................................... 41 2. Existence and Uniqueness . ............................................................................................... 41 3. Autonomous and Non-Autonomous Systems . ................................................................. 41 We will only study autonomous systems. . ....................................................................... 44 4. A Continuous Time Population Model Similar to the Logistic Equation . ....................... 44 5.General Consideration of Fixed Points and Stability . ....................................................... 46 5.1 Fixed Points . .............................................................................................................. 46 5.2 Stability . ..................................................................................................................... 46 5.3 Asymptotic Stability . ................................................................................................. 47 5.4 Exponential Stability . ................................................................................................. 48
6. Stability Analysis of Scalar Nonlinear System . ............................................................... 49 7. Determining Stability Without Solving for Trajectories - Multivariate Case . ................. 50 8. What You Need to Know From Lecture 4 . ...................................................................... 52 Lecture 5A: Background for Biological Applications – Nonlinear Models of Excitable Cells . ................................................................................................... 53 1. The Generic Cell Model (Fig. 5A.1) . ............................................................................... 53 2. Simple Models of Membrane Permeability: . ................................................................... 53 3. Ion Channels . ................................................................................................................... 56 4. Simple Cell Model . .......................................................................................................... 56 Equivalent Circuit Model . .................................................................................................... 57 5. Hodgkin-Huxley Theory . ................................................................................................. 58 5.1 Experimental Approach: . ........................................................................................... 58 5.2 Modeling: . .................................................................................................................. 60 6. What You Need to Know From Lecture 5A . ................................................................... 63 Lecture 5B: Introductory Applications of Nonlinear Systems Theory to the Analysis of Cell Models - Horizontal Cell Models . .......................................... 64 1. Retinal Horizontal Cells . .................................................................................................. 64 2. The Model . ....................................................................................................................... 65 2.1 The System is Bi-Stable When Gsyn is Small . .......................................................... 68 2.2 The System Has a Single Real Fixed Point When Gsyn is Large: . ........................... 69 2.3 Termination (Recovery Phase) of the AP . ................................................................. 69 3. What You Need to Know From Lecture 5B . ................................................................... 72 Lecture 5C: The Moris-LeCar Oscillator Model . ............................................... 73 1. Figure 5C.1 illustrates the model: . ................................................................................... 73 1.1 Key Assumption: . ...................................................................................................... 73 2 Currents: . ........................................................................................................................... 73 2.1 Ca Current . ................................................................................................................. 73 2.2 K Current: . ................................................................................................................. 74 2.3 Leak Current: . ............................................................................................................ 75 3 Simulations: . ..................................................................................................................... 75 3.1 Fixed Points: . ............................................................................................................. 76 3.2 Stability of Fixed Points: . .......................................................................................... 76 4. What You Need to Know From Lecture 5C . ................................................................... 78 Lecture 6A: Nonlinear Oscillators - Van Der Pol’s Oscillator . .......................... 79 1. Van der Pol’s Oscillator: . ................................................................................................. 79 1.1 The X-Nullcline: . ....................................................................................................... 79 1.2 The Y-Nullcline: .

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## This note was uploaded on 05/27/2008 for the course BME 213 taught by Professor Winslow during the Spring '08 term at Johns Hopkins.

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SimulationsPDF - BME 580.223 Models and Simulation Part 2 Introduction to Nonlinear Dynamical Systems Modeling Raimond L Winslow Department of

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