Multistability and metastability: understanding dynamic coordination in the brain

Abstract

Multistable coordination dynamics exists at many levels, from multifunctional neural circuits in vertebrates and invertebrates to large-scale neural circuitry in humans. Moreover, multistability spans (at least) the domains of action and perception, and has been found to place constraints upon, even dictating the nature of, intentional change and the skill-learning process. This paper reviews some of the key evidence for multistability in the aforementioned areas, and illustrates how it has been measured, modelled and theoretically understood. It then suggests how multistability--when combined with essential aspects of coordination dynamics such as instability, transitions and (especially) metastability--provides a platform for understanding coupling and the creative dynamics of complex goal-directed systems, including the brain and the brain-behaviour relation.

Figure 1.

(a,b) Hallmarks of non-equilibrium phase transitions illustrated in the extended HKB model of coordination dynamics, i.e. the potential, V, in equation (3.4) with δω = 0 (solid curves in (a)). Enhancement of fluctuations is indicated by the widening of the probability distribution, p, of relative phase around Φ = π (dashed and dotted lines in (a) respectively) as the control parameter k = b/a decreases. Critical slowing down is revealed by an increase in the time it takes for the system to recover from a small perturbation (b). Note how this relaxation time increases as the system approaches instability. At the critical point (k = 0.25) and beyond, the system does not return to its former (non-equilibrium) state.

Figure 2.

Potential plots (equations (3.4) and (3.5)) of the intrinsic (dotted grey) and intentional dynamics scaled according to basal ganglia activity prior to (black) and during pattern switching (grey; see text). (a) In-phase initial condition. (b) Anti-phase initial condition.

Figure 3.

Two routes to learning. RMSE (in degrees) of the performed relative phase during scanning probes that specify a required phase before (a,c) and after (b,d) learning a new pattern. Arrows indicate minimum error corresponding to wells in potential functions (shown for illustration purposes only). (a,b) The bifurcation route in initially bistable participants: with learning two minima shift to three, a qualitative change. (c,d) The shift route in initially tristable participants: the number of minima do not change with learning (see text). Data and figure courtesy Viviane Kostrubiec (adapted from Zanone et al. [60]).

Figure 4.

(a–d) Trajectories of the extended HKB model of coordination dynamics (equation (3.2)) illustrating how metastability (c) emerges from multistability (a) as a control parameter (b/a) changes. For (a–c) the coupling is fixed and δω is varied. The case of no coupling (d) is shown for comparison. (e) The phase diagram or parameter space of the extended HKB model showing the range of behaviours possible as a function of coupling strength (k = b/a) and broken symmetry (δω). Possible paths in parameter space are illustrated by arrows. See text for discussion.