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In practice

These two equations are how we create our quasi-quantum mechanical analogues. The second equation is an extension of Ehrenfest’s theorem, relating the average momenta of a particle to the time derivative of its average position. Where we have assumed a Hamiltonian with only a spatially dependent potential. Note that as the positions are fixed in space (positions of the electrodes) only the probability changes in time. Throughout this paper the mass m has been taking to be unity for both the $x$ and $y$ momenta. Each of the 92 electrodes were projected onto the horizontal plane, thus the $j$th electrode was described by one unique $(x_{j},y_{j})$ point.

We first examined this model by grouping the 92 electrodes into eight regions on the scalp: Anterior L/R, Posterior L/R, Parietal L/R, Occipital L/R and the probabilities of each electrode in the region were summed to give a region-level probability. Figure 1A shows the $(x_{j},y_{j})$ locations of each electrode, with different colours representing each of the eight groups. Figure 1B displays the frequency of entering each region, grouped by the four task conditions and two resting conditions. This reflects the normalized count of regional probabilities integrated in time. We found that each anterior region was entered more frequently while at rest than when subjects were engaged in either movie. Specifically, the anterior left and right regions had significant within stimulus change, with $P<0.001$ (Tukey adjusted) for the Taken Rest—Taken, Taken Rest—Taken Scrambled, BYD Rest—BYD and BYD Rest—BYD Scrambled. This is in line with Axelrod and colleagues’ findings which showed activation in the frontal region was associated with mind wandering. We found frequency suppression in posterior regions, and an increase in anterior frequency in rest compared to the stimulated conditions, consistent with fMRI studies showing increased activation in the posterior cingulate cortex, and the medial prefrontal cortex during rest . Thus, suggesting our model captures the frontal tendency associated with the brain activity while at rest.

Figure 1: (A) Electrode locations for each of the 92 electrodes on the Electrical Geodesics Inc. headcap. Electrodes were projected onto a horizontal plane with the nose in the positive y direction. Electrodes have been colour-coded to display the constituent parts of the 8 groups for the frequency analysis, namely, occipital left (blue)/right (orange), parietal left (green)/right (red), posterior left (purple)/right (brown) and anterior left (pink)/right (grey). (B) Histograms representing the frequency of entering each region fG are displayed for the six conditions tested. Significant within stimulus change is present between each of the Anterior Left and Right regions when comparing the pre-stimulus rest and the respective stimulated condition (P < 0.001, Tukey adjusted.). Error bars display the 1 standard deviation confidence interval.

Phase space

We also explored the average-valued phase space of this system. The phase space for each subject was plotted as the average position and momentum along the $x$ direction $(\langle x(t)\rangle ,\langle p_{x}(t)\rangle )$ or as the average position and momentum along the $x$ direction $(\langle y(t)\rangle ,\langle p_{y}(t)\rangle )$. Figure 2 shows the centroids of the phase space scatter plots for each subject with an ellipse representing the one standard deviation confidence interval. Note that values are only reported for the intact stimuli as an analysis of variance shows the scrambled and intact movies are indistinguishable in phase space (P$P<0.85$, Tukey adjusted). Figure 2A and B show the projection of the phase space centroid onto the plane spanned by $x$ and $p_{x}$ for “Bang! You’re Dead” and “Taken” respectively, and Fig. 2C and D ($y,p_{y}$) plane. The average position along the $y$ axis $(\langle y\rangle )$ for the intact stimulus (“BYD” and “Taken”) and their scrambled forms are significantly different from the pre-stimulus rest counterparts with $P<0.001$ (Tukey adjusted) whereas the task-positive and resting centroids are indistinguishable in the $x$ plane ($P<0.05$, Tukey adjusted). The averages of the group are reported in Table ​Table11 along with their standard deviations. These values are the averaged value of the centroids (average of the within stimuli centre points in Fig. 2) for the respective position/momenta within each stimulus level. As also seen in Fig. 2C and D, there is a striking difference of one order of magnitude for $\langle y\rangle$ between the resting and task conditions, yet no marked differences in $x$,$\langle p_{x}\rangle$ , or $\langle p_{y}\rangle$.

Figure 2: Mean phase space centroids for each subject. Ellipses represent the 1 standard deviation confidence interval. Centroids for the scrambled stimuli were omitted as they are indistinguishable from intact stimuli (P > 0.85) (A) Centroids for “Bang! You’re Dead” along the x direction. (B) Centroids for “Taken” along the x direction. (C) Centroids for “Bang! You’re Dead” along the y direction. (D) Centroids for “Taken” along the y direction. Differences are only apparent in the y direction (P < 0.001, Tukey adjusted) indicative of the higher level of anterior activation as noted in Fig. 1.