View source for Store:Exel05

With  <math>i=\sqrt{-1}</math>. We then imposed the normalization condition,
<center>
{|  width="80%"  |
|-
| width="33%" |&nbsp;
| width="33%" |{{CD1}}<math>\hat{\Psi}_j(t)= \frac{\Psi_j(t)}{\sqrt{\sum_{j=1}^{92}|\Psi_j|^2}}</math>{{CD2}}
| width="33%" align="right" |<math>(2)</math>
|}
</center>



The summation extends to 92, corresponding to the 92 electrodes selected from the original 129 on the head cap (channels removed from the face and neck for this analysis). This normalization constraint allowed us to define the probability at time <math>t</math> of the <math>j</math> ''j''th electrode as
<center>
{|  width="80%"  |
|-
| width="33%" | &nbsp;
| width="33%" | {{CD1}}<math>P_j(t)=\hat{\Psi}^*_j(t) \times\hat{\Psi}_j(t)</math>{{CD2}}
| width="33%" align="right" |<math>(3)</math><br />
|}
</center>

With the * denoting complex conjugation<ref name=":7" />. We then can describe each moment in time as a ‘brain state’ that is fully described (in the context of this model) through the ‘wavefunction’. This ‘brain state’ uniquely specifies the EEG signal, and hence the dynamics of interest, at each moment in time. Using this definition of probability, we defined two average quantities of interest. The average position and momentum are given explicitly by,

<center>
{|
| width="33%" | &nbsp;
| width="33%" |{{CD1}}<math>\langle x(t)\rangle=\sum_{j=1}^{92}x_jP_j(t)</math> <math>\langle p_x(t)\rangle=m{d \over dt}\langle x(t)\rangle=m\sum_{j=1}^{92}x_j{d \over dt}P_j(t)</math>{{CD1}}
| width="33%" align="right" |<math>(4)</math>
|}
</center>