Cross-frequency coupling explains the preference for simple ratios in rhythmic behaviour and the relative stability across non-synchronous patterns

Rhythms are important for understanding coordinated behaviours in ecological systems. The repetitive nature of rhythms affords prediction, planning of movements and coordination of processes within and between individuals. A major challenge is to understand complex forms of coordination when they differ from complete synchronization. By expressing phase as ratio of a cycle, we adapted levels of the Farey tree as a metric of complexity mapped to the range between in-phase and anti-phase synchronization. In a bimanual tapping task, this revealed an increase of variability with ratio complexity, a range of hidden and unstable yet measurable modes, and a rank-frequency scaling law across these modes. We use the phase-attractive circle map to propose an interpretation of these findings in terms of hierarchical cross-frequency coupling (CFC). We also consider the tendency for small-integer attractors in the single-hand repeated tapping of three-interval rhythms reported in the literature. The phase-attractive circle map has wider basins of attractions for such ratios. This work motivates the question whether CFC intrinsic to neural dynamics implements low-level priors for timing and coordination and thus becomes involved in phenomena as diverse as attractor states in bimanual coordination and the cross-cultural tendency for musical rhythms to have simple interval ratios. This article is part of the theme issue ‘Synchrony and rhythm interaction: from the brain to behavioural ecology’.

STRANGE NONCHAOTIC ATTRACTORS IN A QUASIPERIODICALLY-FORCED PIECEWISE SMOOTH SYSTEM WITH FAREY TREE

The existence of strange nonchaotic attractors (SNAs) is verified in a simple quasiperiodically-forced piecewise smooth system with Farey tree. It can be seen that more and more jumping discontinuities appear on the smooth torus and the torus becomes extremely fragmented with the change of control parameter. Finally, the torus becomes an SNA with fractal property. In order to confirm the existence of SNAs in this system, we preliminarily use the estimation of the phase sensitivity exponent, estimation of the largest Lyapunov exponent and rational approximation. SNAs are further characterized by power spectra, recurrence plots, the largest Lyapunov exponents and their variance, the distribution of the finite-time Lyapunov exponents, the spectral distribution function and scaling laws.

Groups Acting on Trees

This chapter considers groups acting on trees. It examines which groups act on which spaces and, if a group does act on a space, what it says about the group. These spaces are called trees—that is, connected graphs without cycles. A group action on a tree is free if no nontrivial element of the group preserves any vertex or any edge of the tree. The chapter first presents the theorem stating that: If a group G acts freely on a tree, then G is a free group. The condition that G is free is equivalent to the condition that G acts freely on a tree. The discussion then turns to the Farey tree and shows how to construct the Farey complex using the Farey graph. The chapter concludes by describing free and non-free actions on trees. Exercises and research projects are included.

AMBIGUITY IN THE DETERMINATION OF THE FREE ENERGY ASSOCIATED WITH THE CRITICAL CIRCLE MAP

We consider a simple model to describe the widths of the mode-locked intervals for the critical circle map. By using two different partitions of the rational numbers based on Farey series and Farey tree levels, respectively, we calculate the free energy analytically at selected points for each partition. It emerges that the result of the calculation depends on the method of partition. An implication of this finding is that the generalized dimensions Dq are different for the two types of partition except when q=0; that is, only the Hausdorff dimension is the same in both cases.