Coupled Oscillators: Joggers, Fireflies, and Finger Coordination

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Coupled Oscillators Joggers, Fireflies, and
Finger Coordination

• Tanya Leise
• Amherst College
• tleise_at_amherst.edu

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Jogger on a Circular Track
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Jogger on a Circular Track
We assume the jogger runs on a circular track and
represent the joggers position as an angle ? and
the joggers speed as an angular rate of change
d?/dt.
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Two Joggers on a Track
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Two Joggers Trying to Jog Together

• Look at the problem from the point of view of the
  red jogger
• If the blue jogger is behind me, I should slow
  down.
• If the blue jogger is ahead of me, I should
  speed up.
• To model this, we want to adjust the red joggers
  speed using an appropriate function of their
  relative positions

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Two Joggers Trying to Jog Together
If the blue jogger is behind me, I should slow
down.
If the blue jogger is ahead of me, I should
speed up.
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Two Joggers Matching Speeds on a Track
C0.1
C0.4
C0.8
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Two Joggers Matching Speeds
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Fireflies

• Certain species of fireflies try to flash
  simultaneously synchronized flashing.
• Model synchronization of a population of N
  identical oscillators, n1,2,,N

(Kuramoto model with mean field coupling)
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Synchronization of Fireflies
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Synchronized Fireflies
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Synchronization Index R
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Synchronized Fireflies
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Single Finger Oscillation
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Single Finger Oscillation
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Coupled Oscillations
Left hand Right hand
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Bimanual Oscillations

• Increasing frequency of motion

In-phase
Out-of-phase
Transition
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Developing a Model

• Goals
• To develop a minimal model that can reproduce the
  qualitative features of this experiment
• To gain insight into underlying neuromuscular
  system (how both flexibility and stability can be
  achieved)
• Nature uses only the longest threads to weave
  her pattern, so each small piece of the fabric
  reveals the organization of the entire tapestry.
• ?R.P. Feynman

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A Minimal Model
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Analysis of Minimal Model
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Analysis of Minimal Model
p
0
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An Energy Well Model
Slow twiddling frequency
Fast twiddling frequency
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Explanation for Dynamics

• Increasing frequency w of motion

In-phase
Out-of-phase
As w increases, A(w) decreases. Once A(w) falls
below ¼, out-of-phase motion becomes unstable so
system switches to in-phase motion.
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Further Reading

• H. Haken, J.A.S. Kelso, and H. Bunz. A
  theoretical model of phase transitions in human
  hand movements. Biol. Cybern. 51347-356, 1985.
• T. Leise and A. Cohen. Nonlinear oscillators at
  our fingertips. American Mathematical Monthly
  114(1)14-28, 2007.
• S. Strogatz. Human sleep and circadian rhythms a
  simple model based on two coupled oscillators.
  J. Math. Biol. 25327-347, 1987.
• S. Strogatz. Nonlinear dynamics and chaos, 1994.
• S. Strogatz. From Kuramoto to Crawford exploring
  the onset of synchronization in populations of
  coupled oscillators. Physica D 1431-20, 2000.
• S. Strogatz. SYNC The emerging science of
  spontaneous order, 2003.
• Synchronous fireflies, www.nps.gov/grsm/naturescie
  nce/fireflies.htm