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- Permanent Link:
- http://digital.auraria.edu/AA00004016/00001
## Material Information- Title:
- Coupled biological oscillators and animal gaits
- Creator:
- Saffari-Parizi, Mozhdeh
- Publication Date:
- 2004
- Language:
- English
- Physical Description:
- xix, 148 leaves : ; 28 cm
## Subjects- Subjects / Keywords:
- Harmonic oscillators ( lcsh )
Nonlinear oscillators ( lcsh ) Van der Pol oscillators (Physics) ( lcsh ) Gait in animals -- Mathematical models ( lcsh ) Bifurcation theory ( lcsh ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references (leaves 145-148).
- General Note:
- Integrated Sciences Program
- Statement of Responsibility:
- by Mozhdeh Saffari-Parizi.
## Record Information- Source Institution:
- |University of Colorado Denver
- Holding Location:
- |Auraria Library
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 57662666 ( OCLC )
ocm57662666 - Classification:
- LD1190.L584 2004m S33 ( lcc )
## Auraria Membership |

Full Text |

COUPLED BIOLOGICAL OSCILLATORS AND ANIMAL GAITS
by Mozhdeh Saffari-Parizi B.S., University of Colorado at Denver, 1993 A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Master of Integrated Science Physics and Mathematics This thesis for the Master of Integrated Science degree by Mozhdeh Saffari-Parizi has been approved by / Date Saffari-Parizi, Mozhdeh (M.I.S., Master of Integrated Science) Coupled Biological Oscillators and Animal Gaits Thesis directed by Associate Professor Randall P. Tagg ABSTRACT Nature is full of rhythmic phenomena. Mathematical models based on cou- pled oscillators successfully describe many of these behaviors. In this thesis the fundamental concepts of linear and nonlinear oscillators are first discussed. Af- ter a review of the physics of oscillators like the simple harmonic oscillator, the Duffing oscillator and the Van der Pol oscillator, we consider electrical oscillator circuits. Then examples of biological oscillators are explored. Finally coupled identical oscillator models for animal gaits are discussed. The dynamics of indi- vidual oscillators (internal dynamics) in the network is nonlinear and must be at least two-dimensional to have a Hopf bifurcation. One example of this type of dynamics is a network of Van der Pol oscillator which is used to produce primary gaits such as walk, trot, pace and bound. When coupled together, these oscilla- tors produce patterns of relative phases that correspond to the different animal gaits. The patterns emerge through bifurcations as parameters describing the coupling are varied. We will see that at least eight cells are required to model the quadruped gaits. We conclude with suggestions for practical demonstration of such coupled oscillator systems. m This abstract accurately represents the content of the candidates thesis. I recommend its publication. Signed Randall P. Tagg IV DEDICATION To my parents, my husband and all of my family. ACKNOWLEDGMENT I would like to thank my husband, Masoud Asadi-Zeydabadi, who helped me with mathematics and computer programming in this thesis. This thesis would not have been possible without his help. CONTENTS Figures .................................................................... x Tables.................................................................... xix Chapter 1. Introduction............................................................. 1 1.1 Coupled Oscillators and Biological Synchronization..................... 1 2. Fundamental Oscillators.................................................. 9 2.1 Simple Harmonic Oscillator............................................. 9 2.1.1 Simple Harmonic Oscillator Without Damping......................... 9 2.1.2 Simple Harmonic Oscillator with Damping........................... 15 2.1.3 Simple Harmonic Oscillator with Periodic Applied Force............ 17 2.2 Duffing Oscillator.................................................... 23 2.3 Van der Pol Oscillator................................................ 33 2.3.1 Hopf Bifurcation and the Limit Cycle............................... 37 2.3.1.1 Limit Cycle in Van der Pol Oscillator .......................... 43 2.3.1.2 Hopf Bifurcation in Van der Pol Oscillator......................... 45 2.3.2 Relaxation Oscillator.............................................. 47 3. Electrical Oscillators.................................................. 51 3.1 Resistor Capacitor (RC) Circuits...................................... 51 3.1.1 Charging the Capacitor............................................. 52 3.1.2 Discharging the Capacitor.......................................... 54 vii 3.2 Relaxation Oscillator 55 3.2.1 Neon Bulb........................................................ 55 3.2.2 Neon Tube Relaxation Circuit..................................... 56 3.2.3 The 555 Relaxation Oscillator.................................... 59 3.3 Oscillation by Gain and Positive Feedback......................... 60 3.3.1 The Wien Bridge Oscillator....................................... 64 4. Oscillators in Biology............................................ 67 4.1 Excitable Neurons ................................................ 67 4.1.1 Hodgkin-Huxley Model............................................. 69 4.1.2 Fitzhugh-Nagumo Model............................................ 73 4.2 Heart and Pacemaker............................................... 86 4.3 Synchronization of Coupled Biological Oscillator (Fireflies)....... 92 5. Coupled Oscillators and Animal Gaits.............................. 95 5.1 Bipedal Gaits...................................................... 96 5.2 Quadrupedal Gaits.................................................. 96 5.3 Central Pattern Generators ........................................ 98 5.4 Symmetries of Animal Gaits........................................ 100 5.5 Two Coupled Oscillators........................................... 101 5.6 Four Coupled Oscillators......................................... 102 5.7 Mathematical Models of Animal Gaits......................... 103 5.7.1 Four Coupled Van Der Pol Oscillators............................ 103 5.7.2 Eight Coupled Oscillators....................................... 105 5.7.3 Eight Coupled FitzHugh-Nagumo Oscillators for Modelling of Pri- mary Gaits............................................................ 112 viii 5.7.4 Eight Coupled Van Der Pol Oscillators for Modelling of Primary Gaits ........................................................... 113 5.7.5 Mathematical Model of Primary and Secondary Gaits by Morris- Lecar Equations ................................................. 121 5.7.5.1 Primary Gait................................................... 129 5.7.5.2 Secondary Gait................................................. 130 6. Conclusion, Suggestions and Applications........................... 139 6.1 Simulation of Pulse Coupling ...................................... 139 6.2 Model of Electronic Pulse Coupling................................. 139 6.3 Application of the Gait Simulation in Animals with Birth Defects . 142 6.4 Conclusion......................................................... 143 References.............................................................. 145 IX FIGURES Figure 2.1 Mass and spring diagram............................................. 10 2.2 Preservation of phase space area.................................... 13 2.3 Time series, phase trajectory and phase portrait of simple harmonic oscillator: u = l,o;(0) = 1, (^f)t=o = y(0) = 0; (a) and (b) time series for x and for y, (c) trajectory in phase space, (d) a phase portrait consisting of several orbits and the fixed point at the origin. 13 2.4 Simple pendulum diagram.............................................. 14 2.5 RLC circuit diagram.................................................. 15 2.6 Simple oscillator with damping: over damped( 72 > u2), critically damped (y2 u2) and under damped ('y2 < ui2)...................... 17 2.7 Mass and spring with driving force diagram. Force is f = k(x xo) where xq = Ad cos (fit)............................................ 18 2.8 Simple oscillator with damping and driving force: (a) under damped (y2 < lo2), (b) critically damped fy2 = u>2) and (c) over damped (y2 > lu2) with driving force for F = 1............................ 19 2.9 Time series and phase trajectory for simple oscillator with damping and driving force (under damped)................................... 20 2.10 The ratio of long-time response amplitude to driving amplitude, A/(^)2, versus driving frequency, Q (top), phase shift, (ip), versus driving frequency, 0 (bottom), for simple harmonic oscillator with damping. For to = 1,F = 1,7 = (0.1,0.2)...................... 22 2.11 Time series and phase trajectory for Duffing oscillator with damping: to = 1,7 = 0.1, e = 0.01 and x(0) = l,x'(0) = v(0) = 0; (a) time series and (b) phase trajectory.......................................... 25 2.12 Phase trajectory of Duffing oscillator with driving force: u = 1,7 = 0.1, e = 0.01, 0 = a; + e, F = 1 and x(O) = 1,2/(0) = f(0) = 0. ... 26 2.13 Numerical solution of equation (2.46) and the approximate solutions from equations (2-49) and (2.51), for Duffing oscillator with driving force: (u = 1,7 = 0.1, e = 0.01,0 = + e, F = 1 and x(0) = 1,^(0) =u(0) = 0).................................................... 28 2.14 Duffing oscillator (weak forcing): a = versus (A = Â£Lzmj for Ad = 0.5,u = 1,7 = 0.1, e = 0.01..................................... 30 2.15 Duffing oscillator (Moderate forcing): a = ^ versus (A = Q=^) for Ad = 1, cu = l,7 = 0.1,e = 0.01.................................. 31 2.16 Duffing oscillator (strong forcing): a = ^ versus (A = ^=sl) for Ad = 5, to = 1,7 = 0.1, e = 0.01,0 = u + e........................... 32 2.17 Time series (top) and phase trajectory (bottom), critical control pa- rameter, ec = 0: (left) e = 0.2 < 0, stable spiral and (right) e = 0.2 > 0, unstable spiral......................................... 41 xi 2.18 The approximate solution by averaging method (equation (2.103)) and numerical solution by Runge-Kutta method for Van der Pol os- cillator: (e = 0.2 and x(0) = 1, ^'(O) = i'(O) = 0,)...................... 42 2.19 Limit cycle of Van der Pol oscillator: (e = 0.2 and top: x(0) = l,a;'(0) = i/(0) = 0, bottom: x(0) = ^'(O) = 'u(O) = 2)................. 44 2.20 Hopf bifurcation in Van der Pol equation (time series (top) and phase trajectory (bottom), critical control parameter, ec = 0: (Left) e = 0.2 < 0, stable spiral and (right) c = 0.2 > 0, stable limit cycle. . 46 2.21 Time series for relaxation oscillator: (a) p = l,a;(0) = 1, ^'(O) = v(0) = 0, (b) p = 5, x(0) = l,a/(0) = v(0) = 0 and (c) p = 10, x(0) = 1, x'(0) = v(0) = 0............................................ 48 2.22 Limit cycle of relaxation oscillator (initial condition inside the limit cycle), (a) p = 1, x(0) = lj-X^O) = f/(0) = 0, (b) p == 5, x(0) = ljX^O) = 1/(0) = 0 and (c) p = 10, a;(0) = 1,Â£'(0) = i>(0) = 0. ... 49 2.23 Limit cycle of relaxation oscillator (initial condition outside the limit cycle), (a) p = l,x(0) = l,a/(0) = u(0) = 2, (b) p = 5, x(0) 1. a/'(0) = v(0) = 3 and (c) p 10, x(0) = 1, x'(0) = t/(0) 5. . 50 3.1 Resistor capacitor (RC) circuit............................................ 51 3.2 Charge and current versus time for charging resistor capacitor (RC) circuit...................................................................... 53 3.3 Charge and current versus time for discharging resistor capacitor (RC) circuit................................................................. 54 3.4 Neon tube circuit............................................................. 56 3.5 Voltage V versus current I for neon bulb...................................... 57 xii 3.6 Voltage V versus time t for neon bulb.............................. 58 3.7 The 555 oscillator circuit......................................... 60 3.8 Voltage V versus time, t for 555 oscillator........................ 61 3.9 General configuration of a system with gain and feedback........... 62 3.10 A system capable of self-sustained oscillation at frequency to0 where G{uiq)H{u)q) = 1.................................................. 63 3.11 Wien Bridge oscillator. For comparison to figure (3.10) we note that y' = y+-y_........................................................ 64 4.1 Schematic diagram of Hodgkin-Huxley model.......................... 70 4.2 x and y Nullclines for Fitzhugh-Nagumo Model: (a = 0.7, b = 0.8, c = 3 and z = 0)...................................................... 75 4.3 Time series and phase trajectory for Fitzhugh-Nagumo model: (a = 0.7, b = 0.8, c = 3 and z = 0).................................... 76 4.4 Time series and phase trajectory for Fitzhugh-Nagumo model: (a = 0.7, b = 0.8, c = 3 and z 0A).................................. 77 4.5 Time series and phase trajectory for Fitzhugh-Nagumo model: (a = 0.7, b = 0.8,c = 3, z = 0.336,) and initial condition (0.25, 0.1). . 78 4.6 Time series and phase trajectory for Fitzhugh-Nagumo model: (a = 0.7, b 0.8,c = 3,z = 0.34) and initial condition (0.6,1.4). .. . 80 4.7 Time series and phase trajectory for Fitzhugh-Nagumo model: (a = 0.7, b = 0.8, c = 3, z = 0.34j and initial condition (0.6, 0.2). ... 81 4.8 Time series and phase trajectory for Fitzhugh-Nagumo model: (a = 0.7, b = 0.8, c = 3, z = 0.34,) and initial condition (0.4,0.7). 82 xiii 4.9 Time series and phase trajectory for Fitzhugh-Nagumo Mmodel: (a = 0.7, b = 0.8, c 3,z = 0.34,) and initial condition (0.6, 0.1). 83 4.10 Time series and phase trajectory for Fitzhugh-Nagumo model: (a = 0.7, b = 0.8, c = 3, z = 0.34,) and initial condition (0.6, 0.15). . 84 4.11 Phase portrait for Fitzhugh-Nagumo model: (a = 0.7,6 = 0.8, c = 3,z 0.34:) and initial conditions (0.6,1.4), (0.6, 0.2), (0.4,0.7) and (0.6, 0.1)................................................. 85 4.12 Time series (top), phase trajectory (middle) and an enlargement of the phase trajectory (Bottom) for Fitzhugh-Nagumo model: (a = 0.7,6 = 0.8, c = 3, ^ = -0.25;.................................. 87 4.13 Fitzhugh-Nagumo model with excitation term: (a = 0.7,6 = 0.8, c = 3, z = 0.2b) and E = 6 at time t = 12,16,25,33,38, initial condi- tion (1,0) and time step dt = 0.02........................... 88 4.14 A typical electrocardiogram (ECG) configuration................. 90 4.15 Fitzhugh-Nagumo model with periodical sinoatrial node signal term: (a = 0.7,6 = 0.8, c = 3, z = 0.33) (a) without sinoatrial signal, (b)with sinoatrial signal and (c) Sn(t) = 0.35sin(0.4t)....... 91 5.1 Relative phase for quadrupedal gaits............................. 97 5.2 Two identical coupled oscillator (in phase and out of phase patterns), (a) in pase pattern, (b) out of phase patterns.................. 101 5.3 Five possible networks for four coupled oscillators. Front is shown by arrow........................................................ 102 5.4 Four coupled cell network: (a) bidirectional and unidirectional, (b) bidirectional. Figure (a) and (b) are equivalent................ 104 xiv 5.5 Cartoon horse-camel and camel-horse by Hossein Moradi-Khalaj and Yahya Movahedi-Parizi (top right) horse-camel trot, (top left) horse- camel pace, (bottom right) Camel-horse trot, (bottom left) camel- horse pace.................................................... 5.6 Zig-zag network for eight coupled cell oscillators............. 5.7 Criss-cross network for eight coupled cells oscillators........ 5.8 Model ofpronk by FitzHugh-Nagumo equation: a = 0.02, b = 0.2, c = 0.5, a = 0.01, Â£ = 0.014,7 = 0.025,5 = 0.02, (27(0), ^(0)) = (0.06,0.04) and (27(0), y;(0)) = (0,0) for 2 < i < 8.......... 5.9 Model of pace by FitzHugh-Nagumo equation: a = 0.02,5 = 0.2, c = 0.44, a = 0.025, (3 = 0.02,7 = -0.01,5 = 0.012, (0), yi(0)) = (0.06,0.04) and (27(0), ^(0)) = (0,0) for 2 < i < 8........... 5.10 Model of bound by FitzHugh-Nagumo: a = 0.02,5 = 0.2, c 0.44, a = -0.01, P = -0.0102,7 = 0.025,5 = 0.02, (27(0), jft(O)) = (0.06,0.04) and (27(0), j/i(0)) = (0,0) for 2 < i < 8.............. 5.11 Model of trot by FitzHugh-Nagumo equation: a = 0.02,5 = 0.2, c = 0.44, a = -0.02, (3 = -0.002,7 = -0.025,5 = -0.015, (27(0), 2/1(0)) = (0.06,0.04) and (27(0), y,(0)) = (0,0) for 2 < i < 8............ 117 5.12 Model of jump by FitzHugh-Nagumo equation: a = 0.02,5 = 0.2, c = 0.44, a = 0.02, /? = 0.01,7 = 0.025,5 = 0.015, (27(0), yx(0)) = (0.06,0.04) and (27(0), 2/i(0)) = (0,0) for 2 < i < 8........... 118 5.13 Model of walk by FitzHugh-Nagumo equation: a = 0.02,5 = 0.2, c = 0.44, a -0.01,/? = 0.0102,7 = -0.025,5 = 0.02, (27(0),yi(0)) = (0.06,0.04) and (27(0), ^(0)) = (0,0) for 2 < i < 8............. 119 106 108 108 114 115 116 xv 5.14 Model of pronk by Van der Pol equation: p = 0.5, a = 0.02, (3 = 0.02,7 = 0.02,5 = 0.02,Â£ = r) = 0, (x1(0),y1(0)) = (0.06,-0.04) and (Xi(0)^(0)) = (0,0) for 2 < i < 8........................................ 122 5.15 Model of pace by Van der Pol equation: p = 0.5, a = 0.02, /3 = 0.02,7 = 0.02,8 = -0.02, Â£ = p = 0, (rn(0),yi(0)) = (0.06,-0.04) and (xi(0),yi(0)) = (0,0) for 2 < i < 8.................................. 123 5.16 Model of bound by Van der Pol equation: p = 0.5, ct = 0.02,/? = -0.02,7 = 0.02, S = 0.02,Â£ = 77 = 0, (xi(0),j/i(0)) = (0.06,-0.04) and (Xi(0),yi(0)) = (0,0) for 2 < i < 8.................................. 124 5.17 Model of trot by Van der Pol equation: p = 0.5, a = 0.02,/? = -0.02,7 = -0.02, <*= -0.02, f = ?7 = 0, (zi(0), 2/i(0)) = (0.06,-0.04) and (xi(0),yi(0)) = (0,0) for 2 < i < 8. .......................... 125 5.18 Model of jump by Van der Pol equation: p = 0.5, a = 0.02,/? = -0.02,7 = 0.02,5 = 0.02, Â£ = r) = 0, (zi(0),2/i(0)) = (0.06,-0.04) and (rci(0),2/i(0)) = (0,0) for 2 < i < 8................................ 126 5.19 Model of walk by Van der Pol equation: p = 0.5, a = 0.02,/? = -0.02,7 = -0.02, 5 = -0.02, Â£ = 7 = 0, (m(0), yi(0)) = (0.06, -0.04) and (xi(0),yi(0)) = (0,0) for 2 < i < 8.............................. 127 5.20 Model of pronk by Van der Pol equation (phase trajectory): p = 0.5, a = 0.02,/? = 0.02,7 = 0.02,5 = 0.02, Â£ = r] = 0, (zi(0),2/i(0)) = (0.06,-0.04) and (xi(0),yi(0)) = (0,0) for 2 < i < 8................. 128 xvi 5.21 Model of jump by Van der Pol equation for reference foot of left hind limb and left fore limb; left panel: (p = 0.5, a = 0.02, (3 = 0.02,7 = 0.02,8 = 0.02, Â£ = rj = 0) and right panel: (p = 0.5, a = 0.02, f3 0.02,7 = 0.02,5 = 0.02, Â£ = 7/ = 0). For both left and right panels: (07(0), 2/1(0)) = (0.06,-0.04) and (2^(0), 2/* (0)) = (0,0) for2
5.22 Model of canter by Morris-Lecar equation: a = 0.17,/? = 0.2,7 =-0.9,5 = 1, Â£ = 0,7/ = 0(27(0), 2/i(0)) = (0.4,0.3) and (27(0), 2/*(0)) = (0,0) for 2 < i < 8............................................... 133 5.23 Phase trajectory for model of canter by Morris-Lecar equation: a = 0.17, (3 = -0.2,7 = -0.9,5 = -1,Â£ = 0,7/ = 0(27 (0), 2/1 (0)) = (0.4,0.3) and (27(0), 2/i(0)) = (0,0) for 2 < i < 8. ............... 134 5.24 Model of transverse gallop by Morris-Lecar equation: a = 0.09,/? = -0.8,7 = -0.3,5 = -0.08, Â£ = 0,t/ = 0(27(0), 2/1(0)) = (0.061,0.047) and (21^(0), 2/i(0)) = (0,0) for 2 < i < 8.......................... 135 5.25 Phase trajectory for model of transverse gallop by Morris-Lecar equa- tion: a = 0,09,13 = 0.8,7 = 0.3,5 = 0.08, Â£ = 0,7/ = 0 (27(0), 2/i(0)) = (0.061, 0.047) and (xi(0),yi(0)) = (0, 0)for2
8................................................................... 1365.26 Model of rotary gallop by Morris-Lecar equation: a = 0.02,/? = 0-23,7 = 0.4,5 = 0.7,Â£ = 0,7/ = 0, (27(0),2/i(0)) = (-0.79, O.48), (22(0), 2/2(0)) = (0, 0.56) and (2^(0), 2/i(0)) = (0,0) for 3 < i < 8. . 137 XVII 5.27 Phase trajectory for model of rotary gallop by Morris-Lecar equation: a = 0.02,0 = 0.23,7 = -0.4,5 = -0.7, f = 0, rj = 0, (i(0), yi(0)) = (-0.79,0.48), (Â£2(0), 2/2(0)) = (0,0.56) and (xi(0), yi{0)) = (0,0) for 3 < i < 8.................................................................... 138 6.1 555 circuit as unit cell in the model of pulse coupled oscillator. . 140 6.2 The model of pulse coupled oscillators. Each square represents a circuit like the one shown in figure (6.1)............................. 141 6.3 Two views of an animal with three legs: Museum of Natural History and Technology, Shiraz University, Iran, (by Samira Moradi-Khalaj). 142 XVlll TABLES Table 5.1 Possible assignments of cell to legs in zig-zag network....... 109 5.2 Possible assignments of cell to legs in criss-cross network... 109 5.3 The phase shift for primary gaits in term of xi...................... 110 5.4 Symmetry of primary gaits............................................ 110 5.5 Parameter values of primary gaits in Fitzhugh-Nagumo model. ... 113 5.6 Parameter values of primary gaits in Van der Pol model............... 121 5.7 Parameter in the case of linear synaptic coupling leading to stable Hopf bifurcations................................................... 130 5.8 Parameters of primary gait for synaptic coupling Morris-Lecar model: Â£ = ?7 = 0................................................... 131 5.9 Secondary gaits with linear diffusive coupling in Morris-Lecar. . . . 131 5.10 Secondary gaits with linear synaptic coupling in Morris-Lecar. . . . 132 5.11 Nonzero initial conditions in Morris-Lecar........................... 132 xix 1. Introduction 1.1 Coupled Oscillators and Biological Synchronization The purpose of this thesis is to guide the selection of experimental oscil- lators that will be used to create arrays of coupled dynamical systems whose behavior is governed by symmetries in the couplings. In particular, the thesis is motivated by recent work on animal gaits. Transitions to different patterns of footfalls can be understood on broad grounds by the effect of symmetry on an oscillator network modelling the neurophysiology of the animals. An experimen- tal robotic horse is under construction whose leg motions will be orchestrated by coupled electronic or fluidic oscillators. The choice of oscillators for this robot will be governed by several considerations: experimental realizability, precision of control of the couplings, the availability of matching mathematical models, and potential for comparison to biological oscillators. These critical factors will be examined in detail for various oscillator models in this thesis. Following this introduction, chapter 2 concentrates on oscillator fundamen- tals and mathematical prototypes in order to develop the language for describing and evaluating the dynamics of individual oscillators. Chapter 3 then investi- gates specific electronic circuits that might conveniently be used in coupled- oscillator experiments. In chapter 4, there is a shift of emphasis in order to describe some of the more basic theoretical models for biological oscillators. This gives a perspective for experiments, even if a specific circuit implemen- tation does not attempt to mimic most of the biological details. Indeed, one 1 expectation to come from the study of nonlinear dynamics is that the overall behavior is independent of many of the details and depends instead on such broad features as symmetry and the number of elements interacting with one another. Chapter 5 continues with the biological orientation by describing the animal gait problem and some corresponding oscillator models. Concludes by outlining principal features one would like to mimic in an electronic coupled- oscillator model of central pattern generators. First, however, we give an overview of the coupled oscillator problem as it arises in a broad variety of physical and biological systems. In the seventeenth century, Huygens invented the pendulum clock [31]. He watched two clocks that were hanging side by side. He noticed that the two pendulums were swinging in perfect synchrony. They swung for hours without any broken steps. Then he tried disturbing them and within half an hour they regained synchrony. Huygens believed that the clocks must somehow influence each other by air movement or vibrations in their common support. In other observations he moved the clocks to opposite sides of the room. The clocks gradually fell out of step, one losing five seconds a day relative to the other. His observations initiated the mathematics of the theory of coupled oscillators [38]. Recently this investigation has been reproduced using modern laboratory equipment [4]. Coupled oscillators can be found in nature. Examples are the pacemaker cells in the heart, insulin-secreting cells in the pancreas, and the neural networks in the brain and spinal cord that control the rhythmic behaviors of breathing, running, and chewing. Another example is the synchronization of congregations of fireflies [38]. In each case, an element isolated from the system can continue 2 to oscillate in its own rhythm. This rhythm is entirely dependent on proper- ties of the elemental subsystems itself. Typically, the elemental oscillators are autonomous nonlinear dynamical systems, meaning that their behavior is not determined by an external time-dependent forcing. In fact, the systems are ca- pable of self-sustained oscillation. For example one can isolate a firefly from other insects, put it into a place with a constant temperature and illuminance and observe that the lone firefly produces rhythmic flashes. The systems that generate these rhythms are dissipative and thus need some energy source to sus- tain their oscillations. By contrast, ah example of non self-sustained oscillation is the tidal flow caused by the daily variation of gravitational forces due to the Moon. These oscillations would not happened if there were no Moon to provide external forcing [31]. One class of natural oscillator is called a relaxation oscillator. In rhythmic fashion, some quantity slowly grows to a threshold level and then suddenly resets to a lower value. Then the process repeats. This behavior occurs, for example, in spontaneously firing neurons. If a current is injected into the cell, the electric potential slowly changes until the cell potential reaches a threshold of about 50mV. At this point the electric potential suddenly spikes for a duration of about two milliseconds. After discharging, the cell resets to about 70rnV. Such integrate and fire oscillators are very important in various problems of neuroscience [31]. Another example is an experimental situation in which an isolated heart preserves its ability to rhythmically contract in vitro, i.e. outside the body and decoupled from the bodys nervous system. If the heart is arterially perfused 3 with a physiologic oxygenated solution, the activity of the primary pacemaker, the sinoatrial node, continues to trigger the beats. The pacemaker is composed of a population of rhythm-generating cells that synchronize their action potentials. This system is different from the heart in vivo because the frequency of the heart in vitro is higher and the variability of the interbeat intervals is lower than those of the heart in vivo. Thus the heart is a highly complex system that can behave as a self-sustained oscillator. In the body, the normal heartbeat is influenced by the the whole cardiovascular system, including all the control loops of the autonomic neural system and the brain centers involved in the regulation of the heart rate. This system is also affected by other physiological rhythms, such as respiration. Nonetheless, we see that the heart is capable of producing its rhythm by itself in spite of these perturbations [31]. Peskin worked on a schematic model of the hearts pacemaker (10,000 cells of the sinoatrial node) in order to understand how they synchronize their indi- vidual electrical rhythms to generate a heartbeat [30]. He suggested that the pacemaker is similar to a large number of coupled identical oscillators In the model each oscillator is an electrical circuit with a capacitor in parallel with a resistor. A constant current causes the voltage across the capacitor to increase steadily. The voltage reaches a threshold where the capacitor discharges, and the voltage drops to zero (firing and discharging of the pacemaker). In the model, each oscillator affects the others only when it fires by causing a fixed jump in the voltage of every oscillator coupled to the one that fired. Peskin stated that the system would always eventually become synchronized, and it would synchronize even if the oscillators were not quite identical. Peskin mod- 4 elled only two coupled oscillators. He used a variation of Poincares concept of a mathematical equivalent to stroboscopic photography. Using the pulse of one oscillator (A) as a strobe, he took the other oscillators (Bs) voltage every time A fired. Peskin found a formula for the change in Bs voltage between snapshots. Based on the formula, if the voltage is less than a certain critical value, it will decrease until it reaches zero, whereas if it is larger, it will increase. Eventually B will synchronize with A. If Bs voltage is almost equal to the critical voltage, then it stays poised at criticality and the oscillators fire about half a cycle out of phase from each other. This equilibrium is unstable however. Note that a general observation seen in both Peskins model and in Huygens experiments is the following: When two identical oscillators are coupled, there are two possi- bilities: synchrony with a phase difference Of zero, and anti-synchrony with a phase difference of one half period [38]. Natural oscillators may be coupled only to a few immediate neighbors or they may be coupled to all the oscillators in a larger community. In such systems of coupled oscillators, synchrony is the most familiar mode of organization. For ex- ample, thousands of male fireflies gather in trees at night and flash on and off in unison to attract females. Fireflies are a paradigm of a pulse coupled oscillator system. They interact only when one sees the sudden flash of another and shifts its rhythm accordingly. In models of such behavior Strogatz proved that oscil- lators started at different times will always become synchronized [25],[38],[36]. Inspired by Peskins work, Strogatz wrote a computer program to do nu- merical experiments with a large number of pulse-coupled oscillators. Mirollo and Strogatz proved mathematically that the system always becomes synchro- 5 nized for any number of oscillators and for almost all initial conditions. If one oscillator kicks another over threshold, they will remain synchronized forever [25], In contrast to the firefly behavior bioluminescent algae show desynchroniza- tion when removed from daily light variations. J. Woodland Hastings and his colleagues at Harvard University have found that when a tank full of Gonyaulax is kept in constant dim light, a circadian glow rhythm is observed with a period close to 23 hours. As time goes by, however, the rhythm gradually damps out. The individual cells continue to oscillate, but they drift out of phase because of differences in their natural frequencies [38]. Do we account for synchronization of some systems and desynchronization in others? Consider each element in a system to be an oscillator with a preferred amplitude and frequency of oscillation. This preferred behavior, to which a lone oscillator will return even if it is momentarily perturbed, is called a limit cycle. Winfree pointed out that if oscillators are weakly coupled, they remain close to their limit cycles at all times. Thus amplitude variations could be neglected and only the spread or decrease of relative phases need be considered. He found that the systems behavior depends on the width of the frequency distribution. If the spread of frequencies is large compared with coupling, the system always lapses into incoherence, as if it were not coupled at all. But if the frequency spread decreases below a critical value, part of the system spontaneously freezes into synchrony [44]. Finally, a brief mention is made of the animal gait problem here, with more detail to come in chapter 5. Quadruped gaits follow patterns of four oscillator 6 systems. They illustrate how the symmetry of a system oscillating in phase can be broken. For example when a rabbit bounds, it moves its front legs together and then its back legs. There is a phase difference of zero between the two front legs and of one half of a cycle between the front and back legs. In a giraffes pace the front and rear legs on each side move together. In a horses trot the diagonal legs move in synchrony. When an elephant ambles it lifts each food in turn, with phase differences of one quarter at each stage. When gazelles pronk, all legs move in synchrony. Biologists believe that the nervous system must produce the different patterns of locomotion. They hypothesize the existence of networks of coupled neurons called central pattern generators, which act as oscillators that control the gait more or less independently from signals from the brain [34], [19]. Most animals have several gaits. For example horses walk, trot, canter and gallop. Mathematical models show that the same central pattern generator circuit can produce all of an animals gaits. Only the strength of the couplings among neural oscillators must vary. An interesting question concerns 3-legged gaits. When three identical os- cillators are coupled in a ring they can be phase-locked in four basic patterns. (1) three in synchrony, (2) each one third out of phase with the others, (3) two in synchrony and one with random phase shift and (4) two half a cycle out of phase and one twice as fast. An example of this last case is a person using a walking stick [38]. The variety of coupled oscillators systems discussed above suggests that a rich phenomenology should be achieved with laboratory coupled oscillator cir- 7 cuits. By incorporating significant features of the models of various biological systems, one hopes to mimic the spatial and temporal patterns discussed above. This may lead to new insights into these models and to the corresponding biolog- ical systems. Laboratory systems may also lead to novel methods of preparing and controlling coupled networks and to the discovery of dynamics whose bio- logical counterparts are yet to be identified. 8 2. Fundamental Oscillators In this chapter we will study the mathematics of several fundamental types of oscillators. We will start with the simple harmonic oscillator as the basis of several key ideas. Most notable about the simple harmonic oscillator is its linearity: if x (t) and x2(t) represent solutions that describe the oscillators po- sition as a function of time, then an arbitrary combination AiX]_{t) + A2x2(t) is also a solution. While suitable for modelling many types of physical systems, especially if their oscillations are small, the linear simple harmonic oscillator model must be extended to nonlinear models in order to describe a much wider range of phenomena. Fundamental mathematical models for nonlinear oscilla- tors include the Duffing and Van der Pol oscillators, whose behavior will also be described in this chapter. 2.1 Simple Harmonic Oscillator The motion of a mass on spring, figure (2.1), is one example of the simple harmonic oscillator. The motion of the simple pendulum in the approximation of small angle is another example. 2.1.1 Simple Harmonic Oscillator Without Damping The force on a mass attached to a spring is proportional to the displacement x from the equilibrium point but acts in the opposite direction. We have fa = ~kx, (2.1) where k is the spring constant. From Newtons second law we have d?x m = kx dt2 (2.2) 9 5S ^W> X m 3 ^ x m m x = 0 Figure 2.1: Mass and spring diagram. where m is the mass of the object. Define the angular frequency as k uj = \l m Then we have the equation of a simple harmonic oscillator d2x dt2 + U! X = 0. The solution of this equation is sinusoidal with a period of 2?r T = u in the general form of x = A sin tot + B cos u>t = C sin (not + ip). (2.3) (2.4) (2.5) (2.6) The constants (A and B) or (C and ip) are determined by initial conditions. 10 Further insight comes from re-examining the dynamics from a geometric viewpoint. The second order equation (2.4) is equivalent to the system of two first order equations dx dt dy dt y (2.7) UJ2X (2.8) These describe motion of a state (x,y) in an abstract plane called the phase plane. Plotting y versus x for any initial condition in the phase plane makes a curve that is called a trajectory. An accumulation of trajectories for a variety of initial conditions provides a comprehensive view of the dynamics that is called a phase portrait. Note that (^, describes a vector that is tangent to the trajectory at point (x,y). An accumulation of such vectors is similar to the velocity field of a moving fluid: here the fluid consists of points in the phase plane. Linking the vectors gives streamlines that correspond to trajectories. The mapping of points of the phase plane onto other points in the same plane is called the flow of this dynamical system. The fixed (equilibrium) point of this system can be determined by putting the right side of the equation (2.7) and (2.8) equal zero. The only fixed point for this system is the origin (0,0). All other points belong to trajectories that form closed curves This can be seen as follows. From equation (2.6) and (2.7) we have Eliminating time between (2.6) and (2.9) by summing suitable multiples of the squares of each side of the equations gives x2 + y c2 (Cu>y = sin 2(ut + ip) + cos2(a)t + (p) = 1. (2.10) Thus the phase space (trajectory) for a simple harmonic oscillator is a closed orbit which is an ellipse. Its equation given above can be rewritten as ^4-^ = 1 a2 b2 (2.11) where a = C and b = Cu. The center of the ellipse is at the fixed point which in this case is the origin and the length of its axes along the x and y axes are 2a and 2b respectively. Each orbit represents a constant total energy. This means the system of a mass and spring without damping is a conservative system. One of the important properties of a phase portrait in a conservative system is area preservation. This means that all of the points in an area, at time t\ occupy an equal area, A2, at time t2 (figure (2.2)). Another property of phase portraits of any kind of system (not just conservative systems) is that any trajectory never crosses itself. This is the result of uniqueness of the trajectory in a deterministic dynamical system. Suppose the initial condition is (:r(0), y(0)) = (1,0). Then the constants in equation (2.6) can be determined to be (A = 0,B = 1) or (C l,(p = 7r/2). Let the natural frequency lj = 1. Then the trajectory is a circle with radius of 1 and its center is located at the origin. Figure (2.3) shows the time series and phase space trajectory of this example. The dynamics of motion of a pendulum is another example of the funda- 12 y Figure 2.2: Preservation of phase space area. Figure 2.3: Time series, phase trajectory and phase portrait of simple har- monic oscillator: u> = l,o;(0) = 1, (fjr)t=o = y(0) = 0; (a) and (b) time series for x and for y, (c) trajectory in phase space, (d) a phase portrait consisting of several orbits ana the fixed point at the origin. 13 \^x\ Figure 2.4: Simple pendulum diagram. mental oscillators. The equation of a simple pendulum is given by d29 o. - + ur. sm 6 = 0 dt2 where 0 is the angular position and the angular frequency is (2.12) CJ 9 (2.13) Here g is the acceleration of gravity and l is the length of the pendulum (figure (2.4)). For small angles we have sin0 9. (2.14) Thus in the small angle approximation the equation of a pendulum becomes a simple harmonic equation j2 a ^+u29 = 0. (2.15) A key point for both types of linear oscillators is that the frequency does not depend on the amplitude of the oscillation. 14 2.1.2 Simple Harmonic Oscillator with Damping Consider a third prototype for the harmonic oscillator: an RLC circuit composed of a resistor R, capacitor C, and inductor L, arranged in a loop figure (2.5). The charge on the capacitor is q and the current flowing in the circuit is R Figure 2.5: RLC circuit diagram. i. Applying Kirchoffs Law of voltages to the loop, we have Q di iR L = 0. C dt (2.16) Now the current i around the loop is related to the charge q on the capacitor by -dq i dr (2.17) Thus, differentiating equation (2.16) once gives i % _ _ L*i = o. C dt dt dt2 Substituting for ^ from equation (2.17) gives (2.18) -1. C dt dt2 (2.19) 15 Rearranging terms gives d2i Rdi 1 . dt? + TJt + lc1 = Let the variable i be replaced with x, define 2y = ^, and identify u2 = to get d2x dx 9 dF + 2T*+1JI = 0' (2'21) This is the equation for a simple harmonic oscillator with damping. The angular frequency u> is the natural frequency of the simple harmonic oscillator. The second term, 2y^|, is a damping term. In the solution of equation (2.21) we have three cases [32]. The first case is known as over-damped motion (y2 > u2) and its solution is x = e-7* (Aeat + Be~at) where a = a/t2 ^2> (2.22) and where A and B are constants that can be determined by initial conditions. The second case is critically damped motion (y2 = uj2). The solution of equation (2.21) in this case is x = e~'rt(A + Bt), (2.23) where A and B can found from initial conditions. The last case is called under-damped or damped oscillatory motion (y2 < to2). In this case equation (2.21) has the solution of x = e7t (A sin pt + B cos fit) with fi = \/uj2 y2. (2.24) Again, the constants A and B are determined by initial conditions. Equation (2.24) can be written as x = Ce~jt cos (fit ip) with fi = \/u2 ~ y2, (2.25) 16 where C = y/A2 + B2 is the amplitude and
Figure 2.6: Simple oscillator with damping: over damped(72 > oj2), critically
ing frequency, Â£1 (bottom), for simple harmonic oscillator with damping. For |