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| ===== Phase space =====
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| 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 <math>x</math> direction <math>(\langle x(t)\rangle,\langle p_x(t)\rangle)</math> or as the average position and momentum along the <math>x</math> direction <math>(\langle y(t)\rangle,\langle p_y(t)\rangle)</math>. 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<math>P<0.85</math>, Tukey adjusted). Figure 2A and B show the projection of the phase space centroid onto the plane spanned by <math>x</math> and <math>p_x</math> for “''Bang! You’re Dead''” and “''Taken''” respectively, and Fig. 2C and D (<math>y,p_y</math>) plane. The average position along the <math>y</math> axis <math>(\langle y\rangle)</math> for the intact stimulus (“BYD” and “Taken”) and their scrambled forms are significantly different from the pre-stimulus rest counterparts with <math>P<0.001</math> (Tukey adjusted) whereas the task-positive and resting centroids are indistinguishable in the <math>x</math> plane (<math>P<0.05</math>, 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 <math>\langle y\rangle</math> between the resting and task conditions, yet no marked differences in <math>x</math>,<math>\langle p_x\rangle</math> , or <math>\langle p_y\rangle</math>.
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| [[File:Figure 2.jpeg|thumb|<small>'''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.</small>|alt=|center|500x500px]]
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| === Table 1 ===
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| Group averages of the centroids.
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| {| class="wikitable" | |
| |+
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| !Stimulus
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| !<math>\langle x\rangle</math>
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| !<math>\langle y\rangle</math>
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| !<math>\langle p_x\rangle</math>
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| !<math>\langle p_y\rangle</math>
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| |-
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| |Taken
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| | colspan="1" rowspan="1" |<small><math>(-1.4\pm5.8)\times10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math>(2.4\pm8.0)\times10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math>(-5.8\pm27.0)\times10^{-2}</math></small>
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| | colspan="1" rowspan="1" |<small><math>(-1.0\pm4.1)\times10^{-1}</math></small>
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| |-
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| |Taken Scrambled
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| | colspan="1" rowspan="1" |<small><math>(-7.7\pm35.0)\times10^{-2}</math></small>
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| | colspan="1" rowspan="1" |<small><math>(1.1\pm9.3)\times10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math>(4.1\pm13.0)\times10^{-2}</math></small>
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| | colspan="1" rowspan="1" |<small><math>(6.3\pm35.0)\times10^{-2}</math></small>
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| |-
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| | colspan="1" rowspan="1" |Bang! You’re Dead
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| | colspan="1" rowspan="1" |<small><math>(1.2\pm4.7)\times10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math>(3.5\pm74.0)\times10^{-2}</math></small>
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| | colspan="1" rowspan="1" |<small><math>(2.6\pm33.0)\times10^{-2}</math></small>
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| | colspan="1" rowspan="1" |<small><math>(-3.0\pm42.0)\times10^{-1}</math></small>
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| |-
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| | colspan="1" rowspan="1" |Bang! You’re Dead Scrambled
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| | colspan="1" rowspan="1" |<small><math>(1.4\pm5.7)\times10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math>(-2.6\pm7.5)\times10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math>(-1.5\pm2.8)\times10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math>(-5.5\pm53.0)\times10^{-2}</math></small>
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| |-
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| | colspan="1" rowspan="1" |Rest (Pre-Taken)
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| | colspan="1" rowspan="1" |<small><math>(-1.3\pm4.6.0)\times10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math>(2.0\pm1.4)\times10^{0}</math></small>
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| | colspan="1" rowspan="1" |<small><math>(9.1\pm19.0)\times10^{-2}</math></small>
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| | colspan="1" rowspan="1" |<small><math>(-6.3\pm7.3)\times10^{-1}</math></small>
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| |-
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| | colspan="1" rowspan="1" |Rest (Pre-BYD)
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| | colspan="1" rowspan="1" |<small><math>(1.1\pm66.0)\times10^{-3}</math></small>
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| | colspan="1" rowspan="1" |<small><math>(1.9\pm1.2)\times10^{0}</math></small>
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| | colspan="1" rowspan="1" |<small><math>(1.0\pm26.0)\times10^{-2}</math></small>
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| | colspan="1" rowspan="1" |<small><math>(-4.3\pm7.5)\times10^{-1}</math></small>
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| |}
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| Significant differences are only noted for the rest acquired before Taken and Bang! You’re Dead when comparing the average y location to either of their task counterparts (scrambled and intact stimulus).
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| Group averages of the centroids.
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| This analysis revealed two notable findings. First, there was a lack of significant differences in the momenta of the brain along the x and y direction. Second, the averages in momenta were not significantly different from 0 at the group level. The positive or negative momenta come from the competing time derivative of the probability and location of the electrode. Since the momenta average to 0, there is an equal number of anterior and posterior electrodes with both increases and decreases in probability.
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| Further, we examined changes in the probability values in both resting and active states. Animations of the probability distributions are present in Supplementary Material 1. In these animations, the differences in rest and task are apparent through the evolution of probability in time.
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| ==== Uncertainty principle ====
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| Despite the confirmation of previous neuroscientific results, and the apparent success of our quasi-quantum model, our research question as posed above remains only half answered. Using this model, we noted differences in the probability distributions and the phase space centroids in rest when compared to task. However, we still sought a parameter from the model that would remain the same in rest and task. To this end, we defined an analogous Heisenberg uncertainty principle of the form,
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| <center>
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| {| width="80%" |
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| | width="33%" |
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| | width="33%" |{{CD1}}<math>\bigtriangleup x(t)\bigtriangleup p_x(t)\geq K_{brain}</math>{{CD2}}
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| | width="33%" align="right" |<math>(5)</math>
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| |}<blockquote></blockquote>
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| </center>
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| Table 2 displays the values of this constant (<math>K_{brain}</math>) acquired in all conditions, as well as the maximum value, mean value, and standard deviation. We found that this quasi-quantum model leads to a constant minimum value across <math>\Delta x(t)\Delta p_x(t)</math> and <math>\Delta y(t)\Delta p_y(t)</math> of <math>0,78\pm0,41\tfrac{cm^2}{4ms}</math> with <math>T=0, P=1</math>. Note the unit of <math>\tfrac{cm^2}{4ms}</math> is a result of the EEG being sampled at 250 Hz and the mass being taken to be unity. Furthermore, the average value and standard deviation of these quantities remains consistent across conditions with an average value of <math>9,3\pm4,4\tfrac{cm^2}{4ms}</math> (<math>T=0, P=1</math>) and a standard deviation of <math>18\pm29\tfrac{cm^2}{4ms}</math> (<math>T=0, P=1</math>).
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| Notably, the maximum value does vary between conditions, with the largest value occurring while subjects watched the intact clip from Bang! You’re Dead. Despite the average position of the signal along the y direction being different in rest than during a task (<math>P<0.001</math>), the quasi-quantum mathematical methodology leads to a constant uncertainty value. Quite remarkably, the values in the table display that the average uncertainty and minimum uncertainty is the same across different conditions, despite maxima varying by over two orders of magnitude. Thus, giving further credence to the idea that this uncertainty relation captures the similarities of the brain across the vastly different conditions. Figure 3 displays the probability distribution at the time corresponding to the minimum in uncertainty for both <math>x</math> and <math>y</math>.
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| === Table 2===
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| {| class="wikitable"
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| !Stimulus
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| !<math>\Delta x\Delta p_x</math>
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| !<math>\Delta y\Delta p_y</math>
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| !<math>\Delta x\Delta p_x</math>
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| !<math>\Delta y\Delta p_y</math>
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| !<math>\Delta x\Delta p_x</math>
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| !<math>\Delta y\Delta p_y</math>
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| !<math>\Delta x\Delta p_x</math>
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| !<math>\Delta y\Delta p_y</math>
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| |-
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| | colspan="1" rowspan="1" |Taken
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| | colspan="1" rowspan="1" |<small><math display="inline">(7\pm2.1)10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(7.2\pm1.8)10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(1.9\pm1.0)10^{3}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(1.4\pm0.8)10^{3}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(8.2\pm2.2)10^{0}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(8.2\pm2.2)10^{0}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(1.4\pm0.4)10^{1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(1.3\pm0.4)10^{1}</math></small>
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| |-
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| | colspan="1" rowspan="1" |Taken Scrambled
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| | colspan="1" rowspan="1" |<small><math display="inline">(6.4\pm2.6)10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(6.8\pm2.1)10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(1.7\pm1.2)10^{3}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(2.1\pm2.2)10^{3}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(8.1\pm1.9)10^{0}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(7.8\pm2.0)10^{0}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(1.4\pm0.4)10^{1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(1.4\pm0.7)10^{1}</math></small>
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| |-
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| | colspan="1" rowspan="1" |Bang! You’re Dead
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| | colspan="1" rowspan="1" |<small><math display="inline">(7.6\pm4.9)10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(7.5\pm3.1)10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(0.1\pm3.1)10^{5}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(0.7\pm1.4)10^{4}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(9.4\pm6.7)10^{0}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(8.3\pm3.6)10^{0}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(4.1\pm8.9)10^{1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(2.7\pm3.9)10^{1}</math></small>
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| |-
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| |colspan="1" rowspan="1" |Bang! You’re Dead Scrambled
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| | colspan="1" rowspan="1" |<small><math display="inline">(7.4\pm3.2)10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(7.1\pm2.9)10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(2.5\pm1.2)10^{3}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(2.5\pm1.6)10^{3}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(9.3\pm5.1)10^{0}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(8.6\pm4.4)10^{0}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(1.6\pm0.7)10^{1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(1.5\pm0.8)10^{1}</math></small>
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| |-
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| |colspan="1" rowspan="1" |Rest (Pre-Taken)
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| | colspan="1" rowspan="1" |<small><math display="inline">(9.7\pm4.2)10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(1.1\pm0.6)10^{0}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(3.5\pm3.1)10^{2}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(3.5\pm1.7)10^{2}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(9.6\pm2.1)10^{0}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(1.3\pm0.4)10^{1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(1.5\pm0.8)10^{1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(1.9\pm0.7)10^{1}</math></small>
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| |-
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| |Rest (Pre-BYD)
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| | colspan="1" rowspan="1" |<small><math display="inline">(6.3\pm3.7)10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(8.6\pm6.1)10^{-1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(3.7\pm2.0)10^{2}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(4.3\pm2.5)10^{2}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(8.7\pm3.3)10^{0}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(1.2\pm0.6)10^{1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(1.4\pm0.5)10^{1}</math></small>
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| | colspan="1" rowspan="1" |<small><math display="inline">(1.9\pm0.8)10^{1}</math></small>
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| |}
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| Various values extracted from the time courses of the products
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| <math>\Delta x(t)\Delta p_x(t)</math> and <math>\Delta y(t)\Delta p_y(t)</math>.
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| [[File:Figure 3.jpeg|center|thumb|788x788px|<small>'''Figura 3:''' Probability maps corresponding to the least uncertain time point for each of the six experimental conditions. ('''A''') The probabilities which lead to the minimum uncertainty as defined by the minimum of . ('''B''') The probabilities which lead to the minimum uncertainty as defined by the minimum of . One subject is displayed for all Taken stimuli, and another for all Bang! You’re Dead stimuli.</small>]]
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| ==== Discussion ====
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| In the current study, we investigated the spatial-extent and the associated transitional properties of neural activity in the brain during active and resting conditions, and whether similar underlying network properties exist. We found that applying the Hilbert transformation to the EEG data and normalizing it (Eq. 2) imposes a probabilistic structure to the EEG signal across the brain (Eq. 3), which we used to identify probability of spatial patterns of activity along with transitions in activity across the scalp. We found more anterior activity during rest relative to the movie watching, in both amplitude and phase space. This finding is in line with previous results showing increased activation in anterior region during rest <ref name=":1" /><ref name=":1" /><ref name=":2" /><ref name=":4" /><ref name=":5" /><ref>Christoff K, Gordon AM, Smallwood J, Smith R, Schooler JW. Experience sampling during fMRI reveals default network and executive system contributions to mind wandering. Proc. Natl. Acad. Sci. U. S. A. 2009;106:8719–8724. doi: 10.1073/pnas.0900234106.[PMC free article] [PubMed] [CrossRef] [Google Scholar]</ref>. Moreover, by normalizing the Hilbert transformed EEG signals and extracting average values akin to those of the wavefunction formulation of quantum mechanics, we were able to compute uncertainty in the ‘position’ and ‘momentum’ during rest and movie-watching, which is set by the new constant
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| <math>K_{brain}=0,78\pm0,41\tfrac{cm^2}{4ms}</math>.
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| It is alluring to associate the constant related to the ‘position’ and ‘momentum’ of neural activity to a fundamental principle, such as, the Heisenberg uncertainty principle. However, it is still unclear what this uncertainty means. It could imply limits to the degree to which the brain is accessible; increasing information about the precise location of the brain state (as described by our quasi-quantum ‘wavefunctions’) will produce a bigger uncertainty about where it will be at a subsequent time. These results offer an interesting perspective on the link between neural function and cognitive processes. For instance, as the ‘wavefunction’ becomes localized in space along a train of thoughts, we become distracted to increase the uncertainty, which may explain why minds wander and thoughts are fleeting?
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| Is the <math> K_b </math> value we found constant across different stimulus conditions, and independent of the number of electrodes used to acquire the data? To test this, we down sampled the EEG electrodes from 92 to 20 and performed the same analysis as in the main text. In line with 92 channels, we found the anterior tendency in rest, but we found reducing the electrodes to 20 resulted in a different constant <math>K_b= 0,03\pm0,02\tfrac{cm^2}{4ms}</math> (See Supplementary Material). This demonstrates that the model is able to capture the differences of rest/task, but a montage-dependent normalisation condition may need to be introduced.
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| It is important to note that uncertainty values of this form are inherent to any Fourier conjugate variables, as a value spreads out in one variable, it localizes in the other. This suggests that after defining the square of the Hilbert transformed EEG electrode time course to be the probability and imposing the properties of a Hilbert space onto the electrode signals, an uncertainty values can be extracted. In quantum mechanics, this uncertainty sets the limit for the scales that cannot be observed. This approach was inspired from the need in neuroscience for novel models to help interpret neuroimaging data. While this is an interesting methodological step forward, we still must determine if the observed uncertainty in the EEG data is supported by a new fundamental principle like in quantum mechanics, or if it is just the outcome of having built two new Fourier conjugate variables from the EEG signal.
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| Further work must be done to explore this constant with respect to the rich taxonomy of tasks and stimuli and varying states of consciousness that are routinely used in cognitive neuroscience. This methodology could be extended into fMRI, where the BOLD time courses could be Hilbert transformed creating a three-dimensional analogue of the EEG model presented in this paper.
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| Ultimately, this paper presented a novel methodology for analysing EEG data. Normalizing the data and treating it as a probability amplitude led to parameters that changed with the presence or lack of stimulus, while simultaneously establishing a constant value independent of stimulus. We have successfully applied a mathematical framework based on the formalisms of quantum mechanics to the resting and task paradigm in EEG (without claiming the brain is a quantum object). As neuroscience continues to evolve, the analytic tools at its disposal must also progress accordingly. We hope that this analytical tool, along with the advances in modelling and machine learning will aid in our understanding of the nature of consciousness.
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| == Methods== | | == Methods== |