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Resulti

In questo documento, abbiamo adattato le ampiezze di probabilità della meccanica quantistica per definire nuove metriche per l'esame dei dati EEG: la "posizione media" e il "momento medio" del segnale EEG. Questi sono stati costruiti dalla nostra definizione di "stati cerebrali" basata sul modello quasi quantistico. Ciò ci ha permesso di accertare la frequenza con cui le regioni cerebrali uniche vengono inserite dalla pseudo-funzione d'onda, nonché di esplorare lo spazio delle fasi di valore medio. Infine, è stata stabilita una relazione di incertezza analoga a quella della meccanica quantistica, con la piena derivazione matematica descritta nei metodi.

Valore medio

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. 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 $j$th electrode as $\Psi _{j}$, this is equivalent to

$\Psi _{j}(t)=A_{j}(t)\exp(i\theta _{j}(t))$

$(1)$

With $i={\sqrt {-1}}$. We then imposed the normalization condition,

${\hat {\Psi }}_{j}(t)={\frac {\Psi _{j}(t)}{\sqrt {\sum _{j=1}^{92}|\Psi _{j}|^{2}}}}$

$(2)$

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 $t$ of the $j$ jth electrode as

$P_{j}(t)={\hat {\Psi }}_{j}^{*}(t)\times {\hat {\Psi }}_{j}(t)$

$(3)$

With the * denoting complex conjugation. 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,

$\langle x(t)\rangle =\sum _{j=1}^{92}x_{j}P_{j}(t)$ $\langle p_{x}(t)\rangle =m{d \over dt}\langle x(t)\rangle =m\sum _{j=1}^{92}x_{j}{d \over dt}P_{j}(t)$

$(4)$

With the same holding true for y.