LEiDA - Leading Eigenvector Dynamics Analysis
LEiDA serves to detect phase-locked oscillatory patterns emerging in large systems of coupled dynamical units. It has been applied to large datasets of brain activity recorded with functional Magnetic Resonance Imaging, revealing meaningful functional subsystems relating to cognitive and emotional processing (Cabral et al, 2017, Figueroa et al., 2019; Lord et al., 2019). These results point to the direction that the macroscopic functional networks emerging recurrently from brain activity during rest are the expression of a repertoire of phase-locked solutions shaping hemodynamic fluctuations on ultra-slow time-scales.
Collective behaviour in coupled dynamical systems can emerge spontaneously in Nature through phase-locking mechanisms (phase-locking is a form of synchrony in systems organized in space and time). In complex systems where more than one phase-locked solutions co-exist, the system may display erratic switches between phase-locked solutions driven by chaos intrinsic to the system.
Given the signals (i.e., BOLD time-series) in different points of the system (be they voxels, regions of interest, or nodes), the first step is to capture the analytic 'phase' of the signals. To do so, we use the Hilbert transform, which expresses a given time-varying signal X(t) as the product between a time-varying amplitude A(t) and the cosine of a time-varying phase Theta(t), as illustrated in the figure below.
Figure 1: This figure illustrates the Hilbert decomposition of a signal into its time-varying amplitude (A) and phase (Theta). In the top row we plot the original time-series X, corresponding to a pre-processed BOLD signal recorded in a given brain area over 2000 seconds in the MRI scanner. In red, we plot the Hilbert amplitude of the signal, obtained as abs(hilbert(X)) in Matlab, and in yellow the Hilbert phase, obtained as angle(hilbert(X)). Since the phase is expressed in the complex plane, the real part of the phase can be captured by its cosine (plotted in dark green), whereas the imaginary part is captured by its sine. In the bottom line, we plot the product of the amplitude and the cosine of the phase, obtained in matlab as abs(hilbert(X))*cos(angle(hilbert(X)), which corresponds exactly to the original signal X. This figure is from the Supplementary Material of Cabral et al., 2017.
To capture phase-locking patterns emerging at the system level, we analyse the signal in the complex domain, i.e. considering both its real component given by A*cos(theta) and its imaginary part given by A*sin(theta). The analytic signal is therefore the result of adding together the real and imaginary parts together as A*cos(theta)+1i*A*sin(theta). It seems that the Hilbert transform of a signal be interpreted in the sense that we 'imagine' that the real signal we see is only one perspective of an oscillatory process occurring together with another hidden dimension.
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