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Difference between revisions of "Exploring electroencephalography with a model inspired by quantum mechanics"

Exploring electroencephalography with a model inspired by quantum mechanics (view source)

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=== Table 1 ===
Group averages of the centroids.


{| class="wikitable"
|+
!Stimulus
!<math>\langle x\rangle</math>
!<math>\langle y\rangle</math>
!<math>\langle p_x\rangle</math>
!<math>\langle p_y\rangle</math>
|-
|Taken
| colspan="1" rowspan="1" |<small><math>(-1.4\pm5.8)\times10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math>(2.4\pm8.0)\times10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math>(-5.8\pm27.0)\times10^{-2}</math></small>
| colspan="1" rowspan="1" |<small><math>(-1.0\pm4.1)\times10^{-1}</math></small>
|-
|Taken Scrambled
| colspan="1" rowspan="1" |<small><math>(-7.7\pm35.0)\times10^{-2}</math></small>
| colspan="1" rowspan="1" |<small><math>(1.1\pm9.3)\times10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math>(4.1\pm13.0)\times10^{-2}</math></small>
| colspan="1" rowspan="1" |<small><math>(6.3\pm35.0)\times10^{-2}</math></small>
|-
| colspan="1" rowspan="1" |Bang! You’re Dead
| colspan="1" rowspan="1" |<small><math>(1.2\pm4.7)\times10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math>(3.5\pm74.0)\times10^{-2}</math></small>
| colspan="1" rowspan="1" |<small><math>(2.6\pm33.0)\times10^{-2}</math></small>
| colspan="1" rowspan="1" |<small><math>(-3.0\pm42.0)\times10^{-1}</math></small>
|-
| colspan="1" rowspan="1" |Bang! You’re Dead Scrambled
| colspan="1" rowspan="1" |<small><math>(1.4\pm5.7)\times10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math>(-2.6\pm7.5)\times10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math>(-1.5\pm2.8)\times10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math>(-5.5\pm53.0)\times10^{-2}</math></small>
|-
| colspan="1" rowspan="1" |Rest (Pre-Taken)
| colspan="1" rowspan="1" |<small><math>(-1.3\pm4.6.0)\times10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math>(2.0\pm1.4)\times10^{0}</math></small>
| colspan="1" rowspan="1" |<small><math>(9.1\pm19.0)\times10^{-2}</math></small>
| colspan="1" rowspan="1" |<small><math>(-6.3\pm7.3)\times10^{-1}</math></small>
|-
| colspan="1" rowspan="1" |Rest (Pre-BYD)
| colspan="1" rowspan="1" |<small><math>(1.1\pm66.0)\times10^{-3}</math></small>
| colspan="1" rowspan="1" |<small><math>(1.9\pm1.2)\times10^{0}</math></small>
| colspan="1" rowspan="1" |<small><math>(1.0\pm26.0)\times10^{-2}</math></small>
| colspan="1" rowspan="1" |<small><math>(-4.3\pm7.5)\times10^{-1}</math></small>
|}




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).


Group averages of the centroids.


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.


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.
==== Uncertainty principle ====
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,
<center>
{| width="80%" |
|-
| width="33%" |&nbsp;
| width="33%" |<math>\bigtriangleup x(t)\bigtriangleup p_x(t)\geq K_{brain}</math>
| width="33%" align="right" |<math>(5)</math>
|}<blockquote></blockquote>
</center>
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>).
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>.
=== Table 2===
{| class="wikitable"
!Stimulus
!<math>\Delta x\Delta p_x</math>
!<math>\Delta y\Delta p_y</math>
!<math>\Delta x\Delta p_x</math>
!<math>\Delta y\Delta p_y</math>
!<math>\Delta x\Delta p_x</math>
!<math>\Delta y\Delta p_y</math>
!<math>\Delta x\Delta p_x</math>
!<math>\Delta y\Delta p_y</math>
|-
| colspan="1" rowspan="1" |Taken
| colspan="1" rowspan="1" |<small><math display="inline">(7\pm2.1)10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(7.2\pm1.8)10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(1.9\pm1.0)10^{3}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(1.4\pm0.8)10^{3}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(8.2\pm2.2)10^{0}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(8.2\pm2.2)10^{0}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(1.4\pm0.4)10^{1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(1.3\pm0.4)10^{1}</math></small>
|-
| colspan="1" rowspan="1" |Taken Scrambled
| colspan="1" rowspan="1" |<small><math display="inline">(6.4\pm2.6)10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(6.8\pm2.1)10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(1.7\pm1.2)10^{3}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(2.1\pm2.2)10^{3}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(8.1\pm1.9)10^{0}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(7.8\pm2.0)10^{0}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(1.4\pm0.4)10^{1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(1.4\pm0.7)10^{1}</math></small>
|-
| colspan="1" rowspan="1" |Bang! You’re Dead
| colspan="1" rowspan="1" |<small><math display="inline">(7.6\pm4.9)10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(7.5\pm3.1)10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(0.1\pm3.1)10^{5}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(0.7\pm1.4)10^{4}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(9.4\pm6.7)10^{0}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(8.3\pm3.6)10^{0}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(4.1\pm8.9)10^{1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(2.7\pm3.9)10^{1}</math></small>
|-
|colspan="1" rowspan="1" |Bang! You’re Dead Scrambled
| colspan="1" rowspan="1" |<small><math display="inline">(7.4\pm3.2)10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(7.1\pm2.9)10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(2.5\pm1.2)10^{3}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(2.5\pm1.6)10^{3}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(9.3\pm5.1)10^{0}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(8.6\pm4.4)10^{0}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(1.6\pm0.7)10^{1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(1.5\pm0.8)10^{1}</math></small>
|-
|colspan="1" rowspan="1" |Rest (Pre-Taken)
| colspan="1" rowspan="1" |<small><math display="inline">(9.7\pm4.2)10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(1.1\pm0.6)10^{0}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(3.5\pm3.1)10^{2}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(3.5\pm1.7)10^{2}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(9.6\pm2.1)10^{0}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(1.3\pm0.4)10^{1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(1.5\pm0.8)10^{1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(1.9\pm0.7)10^{1}</math></small>
|-
|Rest (Pre-BYD)
| colspan="1" rowspan="1" |<small><math display="inline">(6.3\pm3.7)10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(8.6\pm6.1)10^{-1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(3.7\pm2.0)10^{2}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(4.3\pm2.5)10^{2}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(8.7\pm3.3)10^{0}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(1.2\pm0.6)10^{1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(1.4\pm0.5)10^{1}</math></small>
| colspan="1" rowspan="1" |<small><math display="inline">(1.9\pm0.8)10^{1}</math></small>
|}
Various values extracted from the time courses of the products
<math>\Delta x(t)\Delta p_x(t)</math> and  <math>\Delta y(t)\Delta p_y(t)</math>.
[[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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