Exit from Synchrony in Joint Improvised Motion

Abstract

Motion synchrony correlates with effective and well-rated human interaction. However, people do not remain locked in synchrony; Instead, they repeatedly enter and exit synchrony. In many important interactions, such as therapy, marriage and parent-infant communication, it is the ability to exit and then re-enter synchrony that is thought to build strong relationship. The phenomenon of entry into zero-phase synchrony is well-studied experimentally and in terms of mathematical modeling. In contrast, exit-from-synchrony is under-studied. Here, we focus on human motion coordination, and examine the exit-from-synchrony phenomenon using experimental data from the mirror game paradigm, in which people perform joint improvised motion, and from human tracking of computer-generated stimuli. We present a mathematical mechanism that captures aspects of exit-from-synchrony in human motion. The mechanism adds a random motion component when the accumulated velocity error between the players is small. We introduce this mechanism to several models for human coordinated motion, including the widely studied HKB model, and the predictor-corrector model of Noy, Dekel and Alon. In all models, the new mechanism produces realistic simulated behavior when compared to experimental data from the mirror game and from tracking of computer generated stimuli, including repeated entry and exit from zero-phase synchrony that generates a complexity of motion similar to that of human players. We hope that these results can inform future research on exit-from-synchrony, to better understand the dynamics of coordinated action of people and to enhance human-computer and human-robot interaction.

Fig 1. Human tracking (blue) of computer-generated stimuli (red) shows jitter, brief periods of synchrony and transient decay when the stimulus ends.

(a) Jittery motion which weaves around the stimulus. In this interval of time, the tracker did not synchronize to the signal according to the criteria described in Methods (CC motion). (b) Velocity trace of a subject which showed two intervals of high zero-phase synchrony with the stimulus (gray boxes), and some jitter when not synchronized. (c) An example of a full round in which the subject entered and exited synchrony with the stimulus. (d, e) When the stimulus suddenly stopped (velocity went to zero discontinuously), subjects showed decaying overshoots and undershoots lasting a few periods. Discontinuities in the stimulus velocity occur at transitions between pieces of constant amplitude and frequency in the stimulus position signal.

Fig 2. The PC model shows unrealistically high jitter when tracking stimuli.

(a) Tracking a stimulus with f(0) = 0.1, A₂₁(0) = −1, initial velocity v₁(0) = −1, v₂(0) = −0.5. Jitter amplitude is about 5-fold higher than observed in Fig 1a. (b) at high stimulus frequency (not obeying $\omega \ll \sqrt{k}$), f(0) = 2, A₂₁(0) = 1.5, v₁(0) = 0.5, v₂(0) = −0.5, jitter increases to an amplitude to be almost 4 times higher than the stimulus and 20-fold higher than observed in Fig 1a. Such phenomena occur for a wide range of parameters. Here we used k = 8, g = 0.3 and initial amplitudes A_1n(0) = 0. (c) When stimulus suddenly stops, PC model does not settle to zero velocity. Here we use three periods in the predictor, and f(0) = 0.1, A₂₁(0) = −1, v₁(0) = 0.5, v₂(0) = −0.5 and all other initial amplitudes A_in(0) = 0.

Fig 3. Model with friction and exit-from-synchrony (PCFE) tracks an input signal with realistic jitter and shows entry and exit from synchrony.

(a) The PCFE model shows dynamics qualitatively similar to human players, with the best fit parameter set. (b) PCFE model with parameters that produce high-error tracking with jitter similar to the human player in Fig 1a. Parameters are same as in (a) except for a lower corrector damping parameter, α = 1. (c) High precision tracking to a changing stimulus with frequent entry and exit-from-synchrony, similar to the subjects in Fig 1b, is obtained with PCFE model with α = 5. (d) When stimulus suddenly stops, PCFE model converges to zero velocity after a few periods, as in Fig 1d. Model parameters: k = 8, g = 0.3, τ = 1, E₀ = 0.04, β = 3, α = 4, v₁(0) = 1, v₂(0) = −1, A₁₁(0) = 0.2, and A₁₂(0) = A₁₃(0) = 0, except as noted.

Fig 4. Mirrored PCFE models capture key aspects of human joint improvised motion in the mirror game.

(a, b) Examples of JI rounds by human players from the data of Ref. [59]. Periods of co-confident motion (CC) are in gray boxes. (c) PC model simulation converges to synchrony and does not exit it. Here f₁(0) = f₂(0) = 0, k₁ = k₂ = 8, g₁ = g₂ = 0.5, v₁(0) = 1 v₂(0) = −1, A₁₁(0) = 0.2, and all other initial amplitudes A_ji(0) = 0. (d, e) Mirrored PCFE model dynamics show more complexity and realistic entry and exit from synchrony. Parameters of both mirrored PCFE equations are identical and equal to the best fit parameters found in the tracking experiment, except for (e) where β = 1 is used to describe a player dyad with lower exit-from-synchrony amplitude.

Fig 5. HKB model with an added exit-from-synchrony term shows entry and exit-from-synchrony.

(a) Original HKB model of Ref [26] converges to synchrony and stays locked in a periodic synchronized motion. ω₁ = 1.5, ω₂ = 1, ϵ = −1, β = 0.5, δ = 1, α = 4, γ = 1, v₁(0) = 2, v₂(0) = 1, x₁(0) = x₂(0) = 0. (b) Modified HKB model according to Eqs 13 and 14, with exit-from-synchrony term, shows entry and exit-from-synchrony and complex non-periodic motion. Parameters are the same as in a), with exit-from-synchrony parameters β₁ = β₂ = 3, τ = 1, E₀ = 0.04, ϵ₀ = 0.001, η = 2.

Fig 6. Bingham model with an added exit-from-synchrony term shows entry and exit-from-synchrony.

(a) Original Bingham model of Ref [29,30] with out-of-phase initial conditions converges to synchrony and stays locked in a periodic synchronized motion. Parameters: c = 0.7, b = 0.35, k = 0.3, α = 0.3, x10 = 2, x20 = −2, v10 = 1, v20 = −1, (b) Modified Bingham model with exit-from-synchrony term shows entry and exit-from-synchrony and complex non-periodic motion. Parameters are c = 0.7, b = 0.35, k = 0.3, α = 0.3, x10 = 2, x20 = 2, v10 = 2, v20 = 1, β = 3, τ = 12, E0 = 0.001, ϵ = 0.001.