Rhythmic Interlimb Coordination Lyapunov exponent (slope at  = 0° and 180°) The Variability of Rhythmic Interlimb Coordination * * * * ( 1, Q 1 ) ( 1, Q 2 ) | 1 | = | 1 | 1 1 Q 1 << Q 2 Q1Q1 Q2Q2 Q1Q1 Q1Q1 Q 1 = Q | 1 | >> | 2 | * * * * ( 1, Q 1 )( 2, Q 1 ) The same SD  can result for attractors of the same strength, but that contain different magnitudes Q of noise. The greater Q the more  is perturbed away from  * and the greater the SD . Intrapersonal Interpersonal Predicts two stable phase modes: Interlimb coordination dynamics  dynamics of coupled oscillators (Schmidt et al., 1990). (Haken, Kelso, & Bunz, 1985)  Qba  2sin2 Rate of Change of Relative Phase Angle Detuning Gaussian Noise Coupling Strength Coefficients  = 0° or Inphase  = 180° or Antiphase Two coordination modes: Neuromuscular Coupling of Visual Coupling of SD  = Q 2 Inphase Antiphase IntrapersonalInterpersonal (SD  ) = d  d   =  0 SD  confounds λ and Q. Cross-recurrence analysis can measure λ and Q independently (Richardson et al., 2007). Index λ and Q using Cross-Recurrence Quantification Analysis Cross-recurrence quantification analysis (CRQA) statistics of Percent Recurrence (%REC) and Maxline (L max ) can index the effects of Q and λ respectively. L max (Longest diagonal line) Sensitive to deterministic processes. Provides a reliable index of attractor strength ( ) independent of ( Q ) (Marwan, 2003; Richardson et al., 2007). %REC (Density of recurrent points) Sensitive to stochastic processes. Provides a reliable index of Q independent of ( ) (Kudo et al., 2006; Pellecchia et al, 2005; Zbilut & Webber, 1992). Cross-Recurrence Analysis Determines recurrent states in reconstructed phase space (Marwan, 2003; Zbilut & Webber, 1992). Above, the measured scalar sequences x(t) and y(t) are embedded (unfolded) into a phase space of 3-dimensions using time-delayed (  ) copies of x(t) [x(t+  ), x(t+2  )) and y(t) (y(t+  ), y(t+2  )] as the surrogate 2 nd and 3 rd dimensions. In the corresponding cross-recurrence plot, a point of the trajectory at y j is considered to be recurrent with a point x i, when y j falls within a sphere of radius r about x i. 1,2,3……N points Right wrist 1,2,3……N points y Left wrist x & y (t) x & y (t +2  ) x & y (t +  ) Threshol d radius (r) Reconstructed Phase Space xixi yjyj Cross-Recurrence Plot x Method Current Research Question: Are differences in variability (SD  ) between intra- and interpersonal coordination, as well as between inphase and antiphase, due to differences in attractor strength ( ) or differences in magnitude ( Q ) of noise? 8 Pairs of University of Connecticut Graduate Students Intrapersonal Coordination (Without Vision) Interpersonal Coordination (With Vision) Always 1 st Block 2 trials per condition = 8 trials per pair. Inphase Antiphase Experimental Procedures Conclusions: Haken, H., Kelso, J. A. S., & Bunz, H. (1985). A theoretical model of phase transitions in human hand movements. Biological Cybernetics, 51, Kudo, K., Park, H., Kay, B. A., & Turvey, M. T. (2006). Environmental coupling modulates the attractors of rhythmic coordination. In press. Marwan, N. (2003). Encounters with neighbors: Current developments of concepts based on recurrence plots and their applications. Doctoral Thesis, University of Potsdam. Pellecchia, G. L., Shockley, K., & Turvey, M. T. (2005). Concurrent cognitive task modulates coordination dynamics. Cognitive Science, 29, Richardson, M. J., Schmidt, R. C. & Kay, B. (2007). Distinguishing the noise and attractor strength of coordinated limb movements using recurrence analysis. Biological Cybernetics, 96, Schmidt, R. C., Carello, C. & Turvey, M. T. (1990). Phase transitions and critical fluctuations in the visual coordination of rhythmic movements between people.. JEP: HPP, 16, Zbilut, J. P., & Webber, C. L., Jr. (1992). Embeddings and delays as derived from quantification of recurrence plots. Physics Letters A, 171, Hypotheses: SD  (inphase) < SD  (antiphase) L max (inphase) > L max (antiphase) %REC (inphase) ≈ %REC (antiphase) SD  (intrapersonal) < SD  (interpersonal) L max (intrapersonal) > L max (interpersonal) %REC (intrapersonal) ≈ %REC (interpersonal) Results: Eigenperiod = 1.1s 1 Beep/Cycle Metronome Set pace 1 st 15 s then stopped. 45 s continued to swing at same pace. Wrist movements recorded at 50 Hz with electrogoniometers. Right wrist left person, left wrist right person for interpersonal trials. Phase: Coordination: SD  5 InphaseAntiphase Intra- Inter L max InphaseAntiphase Intra- Inter- %REC Intra- Inter- InphaseAntiphase SD  (inphase) < SD  (antiphase) L max (inphase) > L max (antiphase) %REC (inphase) > %REC (antiphase) SD  (intrapersonal) < SD  (interpersonal) L max (intrapersonal) > L max (interpersonal) %REC (intrapersonal) ≈ %REC (interpersonal) Phase: Coordination: Differences in variability (SD  ) between intra- and interpersonal coordination, as well as between in- and antiphase, are due to differences in attractor strength ( ) as indexed by L max, not magnitude ( Q ) of noise, as indexed by %REC? References: Stacy Lopresti-Goodman, Marisa C. Mancini 1,2, Richard C. Schmidt 1,3, Bruce Kay 1, & Michael J. Richardson 14 1 Center for the Ecological Study of Perception and Action, University of Connecticut; 2 Federal University of Minas Gerais, Brazil; 3 College of the Holy Cross; 4 Colby College Acknowledgments. This research was supported by grants from the National Science Foundation (BCS and BCS ) and a CAPES Award from the Brazilian Ministry of Education (BEX 0330/05-1). Comparing the Attractor Strength of Intra- and Interpersonal Interlimb Coordination