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T. Stankovski
dimensions. To illustrate the steps of the analysis an example of δ-to-α phase neural
coupling function is considered:
• First one needs to extract the δ and α oscillation signals—this is done with standard
filtering of the EEG signals.
• After this, one needs to detect the instantaneous phase signals from the oscillations, which can be done by Hilbert transform, and further transforming this with
protophase-to-phase transformation [34].
• Such phases φ δ (t) and φ α (t) are then inputs to a method for dynamical inference
which can infer a model of two coupled phase oscillators where the base functions
are represented by Fourier series (set of sine and cosine functions of the φ δ (t) and
φ α (t) arguments). In our calculations we used the method for dynamical Bayesian
inference [66] and Fourier series as base function up to the second order.
• The resulting inferred model explicitly gives the desired neural coupling functions.
• After reconstructing the neural coupling functions of interest, one can use them
to perform coupling function analysis in order to extract and quantify unique
characteristics.
The phase coupling functions give the precise mechanism of how one oscillation is accelerated or decelerated as an effect of another oscillation. For example,
let us consider the δ-to-α phase neural coupling function. Figure 11.2 presents such
δ-to-α coupling function q α (φ δ (t), φ α (t)) from three studies involving resting state
and anaesthesia, from single electrode or from spatially distributed electrodes [71–
73]. Figure 11.2a shows the coupling existence, strength and significance in respect
of surrogates, while the Fig. 11.2b shows the all-important neural coupling function q α (φ δ (t), φ α (t)). Observing closely the 3D plot in Fig. 11.2 describes that the
q α (φ δ (t), φ α (t)) coupling function which is evaluated in the φ α (t) dynamics changes
mostly along the φ δ (t) axis, meaning it is a predominantly direct coupling from δ
oscillations. Detailed description of the direct form of coupling function, which is
not analytical for non-parametric functional form, are presented elsewhere [72]. The
specific form of the coupling function describes the coupling mechanism that when
the δ oscillations are between 0 and π the coupling function is negative and the α
oscillations are decelerated, while when the δ oscillations are between π and 2π
the coupling function is positive and the α oscillations are accelerated. The rest of
the figures tell similar story—Fig. 11.2c present three cases of q α (φ δ (t), φ α (t)) coupling functions for awake and anaesthetized subjects (with propofol and sevoflurane
anaesthetics, respectively), while Fig. 11.2d, e present the q α (φ δ (t), φ α (t)) in spatial
distribution on the cortex and its average value. The 3D plots present the qualitative description, while for quantitative analysis one can extract two measures—the
coupling strength and the similarity of form of coupling function [69].
T. Stankovski
dimensions. To illustrate the steps of the analysis an example of δ-to-α phase neural
coupling function is considered:
• First one needs to extract the δ and α oscillation signals—this is done with standard
filtering of the EEG signals.
• After this, one needs to detect the instantaneous phase signals from the oscillations, which can be done by Hilbert transform, and further transforming this with
protophase-to-phase transformation [34].
• Such phases φ δ (t) and φ α (t) are then inputs to a method for dynamical inference
which can infer a model of two coupled phase oscillators where the base functions
are represented by Fourier series (set of sine and cosine functions of the φ δ (t) and
φ α (t) arguments). In our calculations we used the method for dynamical Bayesian
inference [66] and Fourier series as base function up to the second order.
• The resulting inferred model explicitly gives the desired neural coupling functions.
• After reconstructing the neural coupling functions of interest, one can use them
to perform coupling function analysis in order to extract and quantify unique
characteristics.
The phase coupling functions give the precise mechanism of how one oscillation is accelerated or decelerated as an effect of another oscillation. For example,
let us consider the δ-to-α phase neural coupling function. Figure 11.2 presents such
δ-to-α coupling function q α (φ δ (t), φ α (t)) from three studies involving resting state
and anaesthesia, from single electrode or from spatially distributed electrodes [71–
73]. Figure 11.2a shows the coupling existence, strength and significance in respect
of surrogates, while the Fig. 11.2b shows the all-important neural coupling function q α (φ δ (t), φ α (t)). Observing closely the 3D plot in Fig. 11.2 describes that the
q α (φ δ (t), φ α (t)) coupling function which is evaluated in the φ α (t) dynamics changes
mostly along the φ δ (t) axis, meaning it is a predominantly direct coupling from δ
oscillations. Detailed description of the direct form of coupling function, which is
not analytical for non-parametric functional form, are presented elsewhere [72]. The
specific form of the coupling function describes the coupling mechanism that when
the δ oscillations are between 0 and π the coupling function is negative and the α
oscillations are decelerated, while when the δ oscillations are between π and 2π
the coupling function is positive and the α oscillations are accelerated. The rest of
the figures tell similar story—Fig. 11.2c present three cases of q α (φ δ (t), φ α (t)) coupling functions for awake and anaesthetized subjects (with propofol and sevoflurane
anaesthetics, respectively), while Fig. 11.2d, e present the q α (φ δ (t), φ α (t)) in spatial
distribution on the cortex and its average value. The 3D plots present the qualitative description, while for quantitative analysis one can extract two measures—the
coupling strength and the similarity of form of coupling function [69].