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| == Model == | | == Model == |
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| Each of the j electrodes is described by an ordered pair (<math>x_j,y_j,z_j</math>) in 3-dimensional space. To complete this analysis, the electrodes were first projected onto the (<math>x,y</math>) plane, removing the depth of the head. Figure 1A shows the locations of each electrode in this 2d-space. Following this projection, the time courses for each of the 92 electrodes were Hilbert transformed and then normalized following the procedure listed using Eq. (2). A probability was defined in this electrode-position space as the square of the Hilbert transformed time course (Eq. 3), analogous to the wavefunctions of quantum mechanics. Eight regions Anterior L/R, Posterior L/R, Parietal L/R, Occipital L/R) were then defined by grouping the 92 electrodes, and the frequencies of entering each region fG were obtained by summing the probabilities electrodes within the group, then integrating in time.
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| <center>
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| {| width="80%" | | |
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| | width="33%" |
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| | width="33%" |<math>Prob_G(t)=\sum_{j=1}^{92} \Psi_k^* (t)\times\Psi_k (t); f_G=\tfrac{1}{T}\sum_{j=1}^{T}Prob_G(t)</math>
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| | width="33%" align="right" |<math>(6)</math>
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| |}
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| </center>
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| where each of the eight groups denoted by the subscript have a different number of constituent electrodes N. In the occipital left and right there are 10 electrodes each, in the parietal left and right there are 17 electrodes each, in the posterior left and right there are 10 and 11 electrodes respectively, and in the anterior left and right there are 8 and 9 electrodes respectively.
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| Upon getting the group level frequencies average values for position and momentum were calculated using Eqs. (4) and (5) (with identical expressions for y). Finally, to ascertain our analogous uncertainty principle, we sought expressions of the form
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| {| width="80%" |
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| | width="33%" |
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| | width="33%" |<math>\bigtriangleup x=\sqrt{\langle x^2(t)\rangle-\langle x(t)\rangle^2};
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| \bigtriangleup p_x=\sqrt{\langle p_x^2(t)\rangle-\langle p_x(t)\rangle^2}</math>
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| | width="33%" align="right" |<math>(7)</math>
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| |}
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| The expression for can be readily applied to the probabilities and positions as defined above, resulting in the first term given by
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| <center>
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| {| width="80%" |
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| | width="33%" |
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| | width="33%" |<math>\langle x^2(t)\rangle=\sum_{j=1}^{92}P_j(t) x_j^2</math>
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| | width="33%" align="right" |<math>(8)</math>
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| And the second term given by the square of Eq. (4). The second term of <math>\Delta P_x</math> is given by the square of Eq. (5), but the first term is more nuanced. This is owing to the complex number returned when acting the derivative operator twice on the probability. To overcome this, Fourier transforms were used to change Eq. (5) into the momentum basis which then allowed for the efficient calculation of <math>P_x^2(t)</math>.
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| Denoting <math>\tilde{P}_j(t)</math> as the momentum-space probability obtained through a 2-dimensional, non-uniform Fourier transform of the position space pseudo-wavefunction, Eq. (5) can be rewritten as,
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| <center>
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| {| width="80%" |
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| | width="33%" |
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| | width="33%" |<math>\langle p_x(t)\rangle=\sum_{j=1}^{92}\tilde{P}_j(t)p_j</math>
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| | width="33%" align="right" |<math>(9)</math>
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| </center>
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| Leading to the first term in the expression to be written as,
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| <center>
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| {| width="80%" |
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| | width="33%" |
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| | width="33%" |<math>\langle p_x^2(t)\rangle=m^2\sum_{j=1}^{92}\tfrac{x_j^2}{\tilde{p}_j(t)}[{d \over dt}P_j(t)]^2</math>
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| | width="33%" align="right" |<math>(10)</math>
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| |}
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| </center>
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| The FINUFFT python wrapper was used to take the Fourier transform using a type 3, 2d non-uniform FFT<ref>Barnett AH, Magland J, Klinteberg LAF. A parallel nonuniform fast Fourier transform library based on an “Exponential of semicircle” kernel. SIAM J. Sci. Comput. 2019;41:C479–C504. doi: 10.1137/18M120885X. [CrossRef] [Google Scholar]</ref><ref>Barnett, A. H. Aliasing error of the kernel in the nonuniform fast Fourier transform. arXiv:2001.09405 [math.NA] (2020).</ref>, and the minimum value in time of the uncertainty relation was found. Points in momentum space were sampled on <math>p_x\in[-4,4]</math> and <math>p_y\in[-4,5]</math> along with the two additional points (<math>[-5,-4]</math>) and (<math>[-4,-5]</math>).
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| Figure 4 shows the position and momentum probabilities respectively in their own basis. An animation showing how these evolve in time for the different conditions is presented in Supplementary Material 2.
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| [[File:Figure 4.jpeg|center|thumb|500x500px|<small>'''Figure 4: (A''') Probability distribution for a single subject in the position basis. ('''B''') Momentum basis probability distribution for a single subject. The momentum values used for the Fourier transform are indicated by the point locations. Points are colour-/size-coded to represent the probability value at that location.</small>]]
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| To compute the values reported in Table 2, the corresponding value was found for each subject, and these were used to calculate the group average reported here.
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| == Supplementary Information == | | == Supplementary Information == |