Criticality in a multisignal system using principal component analysis

In systems with dynamical transitions, criticality is usually defined by the behavior of suitable individual variables of the system. In the case of time series, the usual procedure involves the analysis of the statistical properties of the selected variable as a function of a control parameter in both the time and frequency domains. An interesting question, however, is how to identify criticality when multiple simultaneous signals are required to provide a reliable representation of the system, especially when the signals exhibit different dynamics and do not individually display the characteristic signs of criticality. In that situation, a technique that analyzes the collective behavior of the signals is necessary. In this work we show that the eigenvalues and eigenvectors obtained from principal components analysis (PCA) can be used as a way to identify collective criticality. To do this, we construct a multilayer Ising model comprised of coupled two-dimensional Ising lattices that have distinct critical temperatures when isolated. We apply PCA to the collection of magnetization signals for a range of global temperatures and study the resulting eigenvalues. We find that there exists a single global temperature at which the eigenvalue spectrum follows a power law, and identify this as an indicator of “multicriticality” for the system. We then apply the technique to electroencephalographic recordings of brain activity, as this is a prime example of multiple signals with distinct individual dynamics. The analysis reveals a power-law eigenspectrum, adding further evidence to the brain criticality hypothesis. We also show that the eigenvectors can be used to distinguish the recordings in the resting state from those during a cognitive task, and that there is important information contained in all eigenvectors, not just the first few dominant ones, establishing that PCA has great utility beyond dimensionality reduction.

Figure 1

Diagram of the coupled multilayer Ising model. Each horizontal layer is a classic 2D Ising model, where the spin-spin interaction constant $\mathcal{J}$ varies with the “height” of the layer (as shown by the color gradient). The vertical red links show that each layer is coupled to those above and below it through the small coupling constant $\mathcal{C}$.

Figure 2

Average magnetization of the layers vs global temperature. Each layer is given by its height or, equivalently, the value of its spin-spin constant $\mathcal{J}$ as indicated by the color gradient.

Figure 3

Variances of the magnetization time series as a function of global temperature. Each layer is color coded and identified by its value of $\mathcal{J}\left(h\right)$, and the temperature-dependent average magnetization of the array is shown in red. The global variance is seen to peak near a value of $T\approx 3.85$.

Figure 4

Using PCA we can see that at different temperatures, the eigenvalues only form a power law at a temperature of 3.841. Note that the vertical scale is logarithmic.

Figure 5

The eigenspectrum at the critical temperature when the phases of the signals of the individual layers are randomized, thus erasing interlayer correlations.

Figure 6

PCA eigenspectra of ensembles of artificially generated correlated $1/f^{\beta }$ noise signals. The value of $\beta$ used for each ensemble is indicated to the right.

Figure 7

The PCA eigenspectra for the control states and arithmetic task experiment, averaged over all 36 subjects. The vertical bars indicate 1-$\sigma$ spreads over the 36 subjects. The straight line indicates a power law with slope about $-1.8$.

Figure 8

The spatial patterns of the first four PCA eigenvectors interpolated from the coefficients corresponding to the positions of the electrodes.