11 Coupling Functions in Neuroscience
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according to some widely accepted protocols, like the International 10–20 system
[31] (internationally recognized protocol to describe and apply the location of scalp
electrodes).
At first sight the EEG signal looks random-like and complex (see e.g., Fig. 11.1),
however, a detailed spectral analysis reveals that there are number of distinct oscillating intervals—called brainwaves. The most commonly studied brainwaves include
the delta δ, theta θ , alpha α, beta β and gamma γ neural oscillation [7]. The frequency intervals of these brainwaves are also given in Fig. 11.1. Apart from these,
there are also other brainwaves, including the mu μ, faster gamma1 γ 1 and gamma2 γ 2
brainwaves, and other more characteristic oscillations like the sleep spindles, thalamocortical oscillations, subthreshold membrane potential oscillations, cardiac cycle
etc. The brainwaves are often linked to specific brain functions and mechanisms,
though not all of them are known and they are still a very active field of research.
The existence and strength of the brainwave oscillations are usually determined by
spectral Fourier or Wavelet analysis.
The brainwave oscillations emanate from the dynamics of large-scale cell ensembles which oscillate synchronously within characteristic frequency intervals. The
different ensembles communicate with each other to integrate their local information
flows into a common brain network. One of the most appropriate ways of describing
communication of that kind is through cross-frequency coupling, and there has been
a large number of such studies in recent years to elucidate the functional activity of
the brain underlying e.g., cognition, attention, learning and working memory [9, 29,
30, 43, 82]. The different types of cross-frequency coupling depend on the dynamical
properties of the oscillating systems that are coupled, e.g., phase, amplitude/power
and frequency, and different combinations of brainwaves have been investigated,
including often the δ-α, θ -γ and α-γ cross-frequency coupling relation. These types
of investigation are usually based on the statistics of the cross-frequency relationship
e.g., in terms of correlation or phase-locking, or on a quantification of the coupling
amplitude.
Recently, a new type of measure for brain interactions was introduced called neural
cross-frequency coupling functions [73]. This measure is one of the central aspects
in this chapter. The neural cross-frequency coupling functions describe interactions
which are cross-frequency coupling i.e. between brainwaves but now describing not
only the coupling existence and strength but also the form of coupling function. This
functional form acts as another dimension of the coupling with the ability to describe
the mechanisms, or the functional law, of the underlying coupling connection in
question [69]. In simple words, not only if, but also how the neural coupling takes
place.
When studying brainwave interactions the neural cross-frequency coupling functions are very suitable. Namely, the fact that the brainwaves are described by oscillations can be used to model the interacting dynamics with the coupled phase oscillator
model [37]. In this way one can have a direct 1:1 correspondence between the number of observables and the dimensions of the measured signals—having a 1D signal
and 1D model for the phase dynamics for each system i.e. there will be no hidden
181
according to some widely accepted protocols, like the International 10–20 system
[31] (internationally recognized protocol to describe and apply the location of scalp
electrodes).
At first sight the EEG signal looks random-like and complex (see e.g., Fig. 11.1),
however, a detailed spectral analysis reveals that there are number of distinct oscillating intervals—called brainwaves. The most commonly studied brainwaves include
the delta δ, theta θ , alpha α, beta β and gamma γ neural oscillation [7]. The frequency intervals of these brainwaves are also given in Fig. 11.1. Apart from these,
there are also other brainwaves, including the mu μ, faster gamma1 γ 1 and gamma2 γ 2
brainwaves, and other more characteristic oscillations like the sleep spindles, thalamocortical oscillations, subthreshold membrane potential oscillations, cardiac cycle
etc. The brainwaves are often linked to specific brain functions and mechanisms,
though not all of them are known and they are still a very active field of research.
The existence and strength of the brainwave oscillations are usually determined by
spectral Fourier or Wavelet analysis.
The brainwave oscillations emanate from the dynamics of large-scale cell ensembles which oscillate synchronously within characteristic frequency intervals. The
different ensembles communicate with each other to integrate their local information
flows into a common brain network. One of the most appropriate ways of describing
communication of that kind is through cross-frequency coupling, and there has been
a large number of such studies in recent years to elucidate the functional activity of
the brain underlying e.g., cognition, attention, learning and working memory [9, 29,
30, 43, 82]. The different types of cross-frequency coupling depend on the dynamical
properties of the oscillating systems that are coupled, e.g., phase, amplitude/power
and frequency, and different combinations of brainwaves have been investigated,
including often the δ-α, θ -γ and α-γ cross-frequency coupling relation. These types
of investigation are usually based on the statistics of the cross-frequency relationship
e.g., in terms of correlation or phase-locking, or on a quantification of the coupling
amplitude.
Recently, a new type of measure for brain interactions was introduced called neural
cross-frequency coupling functions [73]. This measure is one of the central aspects
in this chapter. The neural cross-frequency coupling functions describe interactions
which are cross-frequency coupling i.e. between brainwaves but now describing not
only the coupling existence and strength but also the form of coupling function. This
functional form acts as another dimension of the coupling with the ability to describe
the mechanisms, or the functional law, of the underlying coupling connection in
question [69]. In simple words, not only if, but also how the neural coupling takes
place.
When studying brainwave interactions the neural cross-frequency coupling functions are very suitable. Namely, the fact that the brainwaves are described by oscillations can be used to model the interacting dynamics with the coupled phase oscillator
model [37]. In this way one can have a direct 1:1 correspondence between the number of observables and the dimensions of the measured signals—having a 1D signal
and 1D model for the phase dynamics for each system i.e. there will be no hidden