New results

Average time-frequency maps of energy density of EEG responses (Figures 5, 6) were obtained as follows:

1. Single trials from the data described in chapter 2 were subjected to MP decomposition in stochastic Gabor dictionaries.
2. For each trial, a time-frequency map of signal's energy density was constructed. Time-frequency structures of duration below 0.125 sec. were excluded from the maps--this can be understood as a kind of non-linear denoising, and corresponds to smoothing in time curves from Figure 3 by 16-points window. Parallel procedure (exclusion of structures longer than 4 sec.) was used to remove sharp frequency artifacts.
3. Finally, all the time-frequency maps constructed for single trials were added, to produce an average (across trials) distribution of signal's energy in the time-frequency plane (Figures 5 and 6).

Slices corresponding to the frequency bands used in classical ERD/ERS quantification (section 3.1) were cut from this distribution in Figure 4. Curves presented above those slices are corresponding frequency integrals. Value at each point of a curve is a sum of energies at this time point for all the frequencies in given band, i.e. from the the map below.

Figure 4: Average time-frequency maps of EEG energy density, obtained from the same data as analyzed in Figure 3. Horizontal axis--time in seconds relative to the finger movement. Vertical axes on the right of those maps indicate frequency in Hz. Energy proportional to shades of gray. Scale is arbitrary and different for each map. Curves above the maps, corresponding to figure 3, are frequency integrals of the maps below them. Vertical axes on their left sides indicate percent of change, relative to energy integrated between 3.5 and 4.5 seconds before the movement in given band.

Figure 5: Complete time-frequency map (from 5 to 40 Hz), from which slices presented in Figure 4 were cut. Horizontal scale--seconds relative to the finger movement, vertical--frequency (Hz). Top--the same in 3 dimensions, but energy of the alpha band cut off in 50% of the height to show weaker high-frequency structures. Presentation as in Figure 1.

Figure 6: The same as Fig. 5, calculated for referential data. Alpha band energy in the 3-D plot cut at 25%.

Parametric description of signal structures opens also amazing possibilities of signal analysis in the space of parameters of functions from the signal's expansion (3). As an example, Figure 7 presents estimation of the energy carried by structures with frequency centers lying within certain interval--as opposed to the energy calculated as an integral of the spectral power within the same interval or energy of band-pass filtered signals. Difference between those estimates is explained in Figure 8. Choice of the gamma band between 35 and 38 Hz, based upon the overall picture from Figure 5, combined with such a selective energy estimate, increased the estimated gamma ERS above 4000%--double the result obtained via integration of the overall MP estimate between 36 and 40 Hz (Fig. 4).⁶

Figure 7: Bottom: time-frequency distribution of energy carried by structures with frequency centers lying between 35 and 38 Hz (interval chosen from the full picture of energy changes displayed in Figure 5). Top: gamma ERS based upon this estimate. Presentation as in Figure 4.

Figure 8: Difference between the spectral integral from $f_1$ to $f_2$ and the actual power carried by structures with frequency centers lying within this interval. Black solid line--spectrum resulting from the presence of three structures (spectra plotted with dotted and dashed lines in (a)). (b)--the actual power carried by the structure with frequency center between $f_1$ and $f_2$ (dark gray), (c)--spectral integral from $f_1$ to $f_2$. We observe that part of the energy originating within the interval spreads outside it, while part of energy of structures originating at outside frequencies contributes to the power integrated within the interval. Method proposed in this paper offers estimate corresponding to the dark area in (b). On the contrary, the classical estimate obtained via band-pass filtering is a smeared version of the light gray area in (c), since the band-pass properties of a filter are never perfect.

Footnotes

...fig:local).⁶

These numbers are quoted here only to illustrate the increase of sensitivity; per se they do not provide direct measures of the underlying phenomena. If we assume a complete absence of gamma in the reference period, their theoretical value would be infinite.