Phase reduction. The solid circular line shows the state space of a system on a limit cycle. The limit cycle is assumed stable so that after a small perturbation, the system returns to . Although may be high-dimensional the state will be uniquely determined by its position around the orbit, or the ‘phase’, . The dynamics of perturbed solutions are constrained to the space X shown by the torus. The solid disc corresponds to an ‘isochron’, meaning that all points on this disc have the same asymptotic phase. Using this notion, as we show in the main text, the high-dimensional state equation can be reduced to the one-dimensional system . This is known as a phase reduction.

Bivariate sine interactions. The left column shows the network structure used to generate the data in each row. The middle column shows the corresponding bivariate time series for two oscillators, (red) and (blue). The right column shows the corresponding phase diagrams on the unit circle with initial phases marked as a red cross for the first oscillator, and as a blue circle for the second. Subsequent phase evolutions are shown using dots. These data were generated from bivariate WCO models with sine interaction functions and . Different rows correspond to data generated using different model parameters and/or initial phases. The first row was produced using , , the second row , and the third , . In all cases negative a values move the system towards zero lag synchronization, the absolute value of a indicating the speed of convergence. For the first row oscillator 2 slows down. In the second row, due to the different initial conditions oscillator 2 speeds up. In the third row oscillator 2 speeds up and oscillator 1 slows down. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

DCM versus EMA. The figure plots the log of the parameter estimate error (mean and 1SD error bars), versus the observation noise level, , for the DCM (red) and EMA (blue) estimation methods. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

Experimental paradigm. MEG data was acquired during a working memory task using pictures of visual scenes. After a 1 s inter-trial-interval, a visual scene was presented for 3 s (encoding). This was followed by a blank screen with a fixation cross for 5 s (delay) and then by two test stimuli for 3 s (probe). For ‘memory’ trials subjects were required to press a button at probe indicating which of the two test pictures was presented during encoding. For ‘control’ trials, the button-press at probe indicated whether the probe images were the same or different.

Model comparison. This bar graph plots the log model evidence (relative to the worst model, model 4) for each model structure in Fig. 8. It shows that model 3, in which occipital cortex enslaves activity in IFG and MTL is the most likely cause of synchronized theta activity during maintenance.

Control. This figure presents a state-space diagram of the estimated phase dynamics for the ‘control’ MEG data. The blue arrows show the flow vector and the background grey scale maps the magnitude . The red dots show the fitted trajectories of the ten control trials, with initial values marked with open red circles. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

Exemplar trial. This figure shows the fitted time series for the th trial (the 5th memory trial), plotted as , for data in the MTL, VIS and IFG regions during the first second of the delay period. This trial is also shown in state-space format in Fig. 12. One can see how activity in the different regions becomes synchronized.

Pair of weakly coupled oscillators. The figure shows two oscillators that are weakly coupled via the perturbation function . By assuming that the phase difference changes on a slower time scale than the period of oscillation , the right hand side of the above equations can be rewritten as a function solely of phase differences , where is referred to as the phase interaction function.

Bimanual finger movement. Potential functions and phase interaction functions for (a) low frequency and (b) high frequency bimanual finger movement. The phase difference changes by following the gradient of the potential function (see filled circles and arrows). At low frequency, both in-phase () and anti-phase () minima are stable, whereas at high frequency only the in-phase minimum is stable.

Multiple trials. The figure plots the total number of correct model comparisons, , out of 100, versus the number of trials used in making each comparison, , for two different sets of initial conditions. For the red curve the initial phase difference was drawn from a uniform distribution between 0 and , and for the blue curve from a uniform distribution between and . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

Hypothesized model structures. Theta activity observed using MEG during the delay period of a working memory task is hypothesized to arise from master-slave, partial mutual entrainment or total-mutual entrainment mechanisms.

Network parameters. The numbers next to the arrows indicate estimated values of the intrinsic connections ( in Eq. (13)). The lines ending in filled circles indicate modulatory connections, and the numbers at the end of them show the estimated values ( in Eq. (13)). This follows the usual DCM network diagram semantics. A larger connection value denotes that the receiving region changes its phase more quickly.

Memory. This figure presents a state-space diagram of the estimated phase dynamics for the ‘memory’ MEG data. The blue arrows show the flow vector and the background grey scale maps the magnitude . The red dots show the fitted trajectories of the 10 memory trials, with initial values marked with open red circles. One can see that the FPs have moved, as compared to Fig. 11. The number 15 marks the start of the trajectory of the th trial (the 5th memory trial), which is also shown in time series format in Fig. 13. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)