1. Jun-ichi Yamanishi, Mitsuo Kawato, Ryoji Suzuki Emilie Dolan
Two Coupled Oscillators as a Model for the Coordinated Finger Tapping by Both Hands
Jun-ichi Yamanishi, Mitsuo Kawato, Ryoji Suzuki
Emilie Dolan

2. Background Oscillations Phase Difference Milliseconds – seconds – days
Difference between two waves with the same frequency

3. Rhythmic movement is probably controlled by oscillatory neural networks
Invertebrates
Verberates
Previous Paper (1979)
Finger tapping in L/R hand controlled by separate oscillatory neural networks
Assumption that coordinated finger tapping by both hands is controlled by two interacting neural networks

4. Method 2 Groups (all R handed) Learning
4 unskilled subjects, normal motor function
5 skilled subjects, studying piano
Learning
Tap the key in synchrony with L/R pacing signals
Learn a tapping interval
Delays between L and R signal chosen out of 10 steps
Phase differences of 0.0, 0.1, 0.2…
Called “standard phase differences”

5. Methods Cont’d Test Phase differences measured
Pacing signals presented 10 times
Subjects continue without signals until end (20 taps)
Phase differences measured
Accuracy found using SD of measured and standard delay

6. Results
Systematic errors in SD during self-paced finger tapping

7. Phase difference 0  Synchronous rhythm
Phase difference 0.5  Alternate rhythm
Tendency to approach these rhythms
Ex: phase difference of .1 has a systematic error of -30ms, phase difference comes to 70ms, shows a tendency to resemble synchronous rhythm.

8. Results Steady states: 0.0 and 0.5 Systematic error pattern
Negative slope
Includes 0.2 and 0.75 for skilled participants
Systematic error pattern
4-5 subjects gather around avg at 0.0 and 0.5
Data splits in two for other points, bimodal distribution
Why bimodal?
Drawn towards 0.5, negative error
Drawn towards 1.0, positive error

9. Results
Systematic errors in SD during self-paced finger tapping

10. One Hand vs. Two

11. Model for Two Coupled Oscillators
Suppose there is a neural circuit producing periodic outputs
Not clarified physiologically
1979 Paper
Find that network controlling R hand finger tapping has the same characteristics as the L

12. Relation between networks
Is there a relation between the neural networks controlling both hands versus one hand?
Assume timed oscillations in coordinated tapping
Find points of equilibrium via equations
In stable equilibrium, curve is closer to the horizontal

14. Distribution of stable and unstable equilibrium points

15. Symmetry between R and L phase transition curves
Skilled are more symmetrical
Most equilibrium points at 0.0 and 0.5
Finger tapping tends to be synchronous or alternate
Not indicative of stability of equilibrium points

16. Degree of Stability Degree of stability (C)
Negative is more stable
Phase difference of high stability can be performed more accurately

18. Discussion
Prediction of two coupled oscillators is in good agreement with data
Interaction between two oscillators of unskilled subjects is stronger
Weakened by learning
There must be a higher motor center controlling the coupled system of two neural oscillators

19. Conclusion Greatest performance in synchronous and alternate rhythm
Stable rhythm patterns
Data support
Neural mechanism controlling coordinated finger tapping may very well be composed of the coupled system of two neural oscillators, controlling the left and right hands respectively.

21. Equilibrium Equ’s ∆ø(ø) curve
phase advance (∆ø>0) and delay (∆ø<0) plotted
ø’(ø) is defined as follows from p-response curve: ø’(ø)= ø+∆ø(ø).
Ø­2= 1-f(ø1)- ø1 = 1-( ø1+f(ø1))=F(ø1) (1)
Ø’1= 1-g(ø2)- ø2 = 1-( ø2+g(ø2)) = G(ø2) (2)
Ø­2= 1 - ( ø1+f(ø1))=F(ø1) (3)
Ø’1= 1-( ø2+g(ø2)) = G(ø2) (4)
|G’(ø2e)•F’(ø1e)| < (5)

22. Degree of stability Equs
ø1= ø1e + (G’•F’) δ
Ci = 10 (G’(ø2e) •F’(ø1e)| - 1)