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S4 Appendix
The Circle Map model
The Farey tree provides a general structure for understanding the dynamics of the mode-locking behavior of a coordinative system as predicted by a nonlinear dynamics model known as the Circle Map model. In short, the Circle Map model defines the structural stability of the rational modes to which the coordinative system can be attracted depending on its initial condition, i.e., whether the ratio of the eigenfrequencies of the coupled rhythmic components falls within the resonance region, also called the attraction or stability region, of a mode (Kelso, 1995; Pikovsky, Rosenblum and Kurths, 2003; Treffner and Turvey, 1993; Jensen, Bak, and Bohr, 1984; Kelso, 1991). Whether the system stays attracted or not to a particular mode depends on systems fluctuations and how wide the resonance region of the mode is: the wider the resonance region, the more likely the mode resists the system fluctuations. Lower-order modes, i.e., modes with small integers p:q, benefit from wider resonance regions than higher-order modes. Accordingly lower-order ratios have been classically observed as being more frequently produced and as being more stable in human coordinated behaviors (Treffner and Turvey, 1993; Peper, Beek and van Wieringen, 1995; Kelso and deGuzman, 1988). One advantage of the Farey tree is to reveal the transition routes that occur when the coordinative system loses attraction to one mode to the benefit of another. This likely happens when a decrease in coupling strength occurs, leading to a narrowing of the resonance region of the mode of coordination that may then become not wide enough to resist systems fluctuations. In this case, the system is generally attracted to the nearest lower-order mode following a transition route along the branches of the Farey tree (Peper, Beek and van Wieringen, 1995).
In more detail, the circle map is a nonlinear dynamics model that provides a general mathematical framework for understanding the dynamics of the mode-locking behavior of two oscillating components coupled within a coordinated system. The mode-locking behavior is captured as a function of two parameters: the m:n ratio of the eigen-frequencies of both oscillating components, i.e., the ratio of their uncoupled frequency, and the strength of their coupling. One advantage of the circle map approach is that it reduces the analysis of the system’s dynamics to a one-dimensional discrete map defined by:
(15) –
with (mod 1), where represents the phase of one oscillating component (e.g., A) measured at periodic time intervals , using the observed frequency of the other oscillating component (i.e., B) as a clock (Jensen, Bak and Bohr, 1984). Accordingly the phase shift represents the evolution of the phase of A for a full rotation of B. represents the frequency ratio of the components in the absence of coupling g, that is where and are the respective eigen-frequency of the oscillators A and B. is the coupling function introducing a periodic nonlinear influence to the iteration of the circle map. The resulting mode-locking behavior can be predicted by the average phase shift after a great number of iterations, i.e., the average number of rotations completed by A per the completed rotation of B. This rotation number, also called the ‘dressed’ winding number, is written as:
(16) –
where represents the initial phase of A and k the number of iterations.
Two types of dynamics can be considered depending on the rationality of the dressed winding number. When is rational, i.e., when it can be expressed by the ratio of two integers, the oscillations are mode-locked or frequency synchronised, and is periodic. In contrast when is irrational, the oscillations are desynchronised and results in a quasiperiodic dynamic (Kelso, 1995; Pikovsky, Rosenblum and Kurths, 2003).
Computational implementations of the circle map usually are written as:
(17) –
where represents the coupling strength (also called ‘sine circle map’). The structural stability of the frequency-locking behavior, i.e., its resistance to any given random perturbation (e.g., biological noise), can be understood from the parameter space. This space predicts W from an initial ratio , and the strength of their coupling .
In the absence of coupling (), it is clear that . In that case, W is called the ‘bare’ winding number, i.e., the observed mode matches the ratio of the uncoupled frequencies of the oscillating components. In contrast for , the system can be attracted toward a specific rational mode :, if its initial conditions falls within the resonance region of this mode. The regions of attraction, or resonance regions, are represented by the so-called Arnold tongues, i.e., wedges emanating from the horizontal axis and widening as increases. The width of the tongue characterises the stability of the mode, i.e., wider resonances will remain more stable than narrower resonances under random perturbation (Treffner and Turvey, 1993). Regardless of the strength of the coupling , lower-order modes, i.e., modes with small integers p:q, benefit from wider resonance regions than higher-order modes. Accordingly lower-order ratios have been classically observed as more stable in human coordinated behaviors (Treffner and Turvey, 1993; Peper, Beek and van Wieringen, 1995; Kelso and deGuzman, 1988).