70
M. Congedo
If induced components are of interest, instead of using (4.14) we average the
envelope computed on each sweep as
¯
r tf
1
K
k
|z ktf ||
1
K
k
a
2
ktf + b
2
ktf
(4.17)
In this case, the average envelope depends on the amplitude of the coefficients in
each sweep and is not affected by the randomness of the analytic signal phase. Note
that it does not make sense to average phase values ϕ ktf estimated at each sweep, as
we have done with amplitude in (4.17), since the phase is a circular quantity.
1
Measures (4.15), (4.16) and their normalized (non-linear) versions can be modified computing a weighted average of the normalized analytic signal. Note that the
non-normalized average analytic signal is equal to the normalized average analytic
signal weighted by its own envelope. Choosing the weights differently, we obtain
quite different measures of phase consistency. For instance, weights can be given
by experimental or behavioral variables such as reaction time, stimulus luminance,
etc. In this way, we can discover phase consistency effects that are specific to certain
properties of the stimulus or certain behavioral responses [14, 15]. Taking as weight
the envelope of the signal at the frequency under analysis and the analytic signal of
another frequency (that we name here the modulating frequency) we obtain a measure of phase-amplitude coupling named modulation index (MI: [10, 14, p. 413]). If
the distribution of the modulating phase is uniform, high values of MI reveal dependency between the two frequencies. The modulating frequency is usually lower than
the frequency under analysis. Note that by weighting the normalized analytic signal
arbitrarily, the obtained average amplitude is no longer guaranteed to be bounded
superiorly by 1.0. Furthermore, such measures are subjected to several confounding
effects and must be standardized using resampling methods (for details see [10, 14,
pp. 253–257, 413–418]). An alternative to the MI measure that does not require such
standardization is the phase-amplitude coupling (PAC), which is the MI normalized
by the amplitude [72]. Measures such as MI and PAC and other variants, along with
bivariate counterparts (e.g., [94]), are used to study an important class of phenomena
that can be found in the literature under the name of amplitude-amplitude, phaseamplitude and phase-phase nesting (or coupling, interaction, binding…), amplitude
modulation and more [16, 36, 54, 55, 73, 93].
Several measures of amplitude and phase in the time-frequency plane are shown
in the following real-data example. Figure 4.7 shows a time-frequency analysis of
source S7 and S13 of Fig. 4.2. The analysis has been performed on the average of the
80 target sweeps, from −1000 to +1000 ms with respect to the flash (visual stimulus),
indicated on the abscissa as the time “0”. Successively, the first and last 200 ms have
been trimmed at both sides to remove edge effects. See the caption of the figure for
explanations and the interpretation of results.
1 The time of the day is also a circular quantity and provides a good example. The appropriate
average of 22 h and 1 h is 23 h 30, but this is very far from their arithmetic mean. See also Cohen
[14, pp. 214–246].
M. Congedo
If induced components are of interest, instead of using (4.14) we average the
envelope computed on each sweep as
¯
r tf
1
K
k
|z ktf ||
1
K
k
a
2
ktf + b
2
ktf
(4.17)
In this case, the average envelope depends on the amplitude of the coefficients in
each sweep and is not affected by the randomness of the analytic signal phase. Note
that it does not make sense to average phase values ϕ ktf estimated at each sweep, as
we have done with amplitude in (4.17), since the phase is a circular quantity.
1
Measures (4.15), (4.16) and their normalized (non-linear) versions can be modified computing a weighted average of the normalized analytic signal. Note that the
non-normalized average analytic signal is equal to the normalized average analytic
signal weighted by its own envelope. Choosing the weights differently, we obtain
quite different measures of phase consistency. For instance, weights can be given
by experimental or behavioral variables such as reaction time, stimulus luminance,
etc. In this way, we can discover phase consistency effects that are specific to certain
properties of the stimulus or certain behavioral responses [14, 15]. Taking as weight
the envelope of the signal at the frequency under analysis and the analytic signal of
another frequency (that we name here the modulating frequency) we obtain a measure of phase-amplitude coupling named modulation index (MI: [10, 14, p. 413]). If
the distribution of the modulating phase is uniform, high values of MI reveal dependency between the two frequencies. The modulating frequency is usually lower than
the frequency under analysis. Note that by weighting the normalized analytic signal
arbitrarily, the obtained average amplitude is no longer guaranteed to be bounded
superiorly by 1.0. Furthermore, such measures are subjected to several confounding
effects and must be standardized using resampling methods (for details see [10, 14,
pp. 253–257, 413–418]). An alternative to the MI measure that does not require such
standardization is the phase-amplitude coupling (PAC), which is the MI normalized
by the amplitude [72]. Measures such as MI and PAC and other variants, along with
bivariate counterparts (e.g., [94]), are used to study an important class of phenomena
that can be found in the literature under the name of amplitude-amplitude, phaseamplitude and phase-phase nesting (or coupling, interaction, binding…), amplitude
modulation and more [16, 36, 54, 55, 73, 93].
Several measures of amplitude and phase in the time-frequency plane are shown
in the following real-data example. Figure 4.7 shows a time-frequency analysis of
source S7 and S13 of Fig. 4.2. The analysis has been performed on the average of the
80 target sweeps, from −1000 to +1000 ms with respect to the flash (visual stimulus),
indicated on the abscissa as the time “0”. Successively, the first and last 200 ms have
been trimmed at both sides to remove edge effects. See the caption of the figure for
explanations and the interpretation of results.
1 The time of the day is also a circular quantity and provides a good example. The appropriate
average of 22 h and 1 h is 23 h 30, but this is very far from their arithmetic mean. See also Cohen
[14, pp. 214–246].