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=== Results ===
In this paper, we adapted the probability amplitudes of quantum mechanics to define new metrics for examining EEG data—the ‘average position’ and ‘average momentum’ of the EEG signal. These were constructed from our definition of ‘brain states’ based on the quasi-quantum model. This allowed us to ascertain the frequency with which unique brain regions are entered by the pseudo-wavefunction, as well as explore the average-valued phase space. Finally, an analogous uncertainty relationship to that of quantum mechanics was established, with the full mathematical derivation described in the methods.

==== Average values ====
The ‘average position’ of the EEG data was first extracted performing a Hilbert transform of the pre-processed time courses, and then applying a normalization constraint. Typically, the Hilbert transformed data is used to generate a metric of power dispersion or to extract the phase of the signal<ref>Freeman WJ, Vitiello G. Nonlinear brain dynamics as macroscopic manifestation of underlying many-body field dynamics. Phys. Life Rev. 2006;3:93–118. doi: 10.1016/j.plrev.2006.02.001.[CrossRef] [Google Scholar]
</ref><ref>le Van Quyen M, et al. Comparison of Hilbert transform and wavelet methods for the analysis of neuronal synchrony. J. Neurosci. Methods. 2001;111:83–98. doi: 10.1016/S0165-0270(01)00372-7.[PubMed] [CrossRef] [Google Scholar]</ref><ref>Freeman WJ. Deep analysis of perception through dynamic structures that emerge in cortical activity from self-regulated noise. Cogn. Neurodyn. 2009;3:105–116. doi: 10.1007/s11571-009-9075-3.[PMC free article] [PubMed] [CrossRef] [Google Scholar]</ref>. Instead, we imposed a new normalization condition, thereby creating an analogy to the wavefunctions of quantum mechanics. Denoting the Hilbert transformed time course of the <math>j</math>th electrode as <math>\Psi_j</math>, this is equivalent to
<center>
{|  width="80%"  |
|-
| width="33%" | &nbsp;
| width="33%" | <math>\Psi_j(t)=A_j(t)\exp (i\theta_j(t))</math>
| width="33%" align="right" | <math>(1)</math>
|}
</center>

With  <math>i=\sqrt{-1}</math>. We then imposed the normalization condition,
<center>
{|  width="80%"  |
|-
| width="33%" |&nbsp;
| width="33%" |<math>\hat{\Psi}_j(t)= \frac{\Psi_j(t)}{\sqrt{\sum_{j=1}^{92}|\Psi_j|^2}}</math>
| 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%" | <math>P_j(t)=\hat{\Psi}^*_j(t) \times\hat{\Psi}_j(t)</math>
| 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%" |<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>
| width="33%" align="right" |<math>(4)</math>
|}
</center>