Complexity or Spectrum? A Surrogate Program for Interpreting Lempel–Ziv Increases in EEG

When you measure "complexity" in a brain signal, you might actually be measuring two very different things — and this paper is about telling them apart.

Lempel-Ziv complexity (LZC) compresses a signal and asks how hard it is to describe. Neuroscientists use it constantly: higher LZC supposedly means richer, more dynamic brain activity. The problem is that LZC can go up for a boring reason — the signal just got spectrally wider (more frequencies present) — or for an interesting reason — the phases of those frequencies became more organized in a nonlinear, non-equilibrium way. A single LZC number cannot tell you which.

The paper makes this precise and then proposes a two-step diagnostic. Step 1: generate "surrogate" versions of the signal using IAAFT (a method that scrambles the phases while exactly preserving the power spectrum and amplitude distribution), then check whether the real signal's LZC sits inside the surrogate cloud. If it does, the LZC is fully explained by the spectrum — nothing deeper is going on. Controlled simulations confirm the intuition: what really drives LZC in a band-limited, median-binarized signal is spectral bandwidth, not mean frequency. Widen the band and LZC shoots up; slide the center frequency and almost nothing happens. Step 2 is only triggered when the real signal stands outside the surrogate cloud — meaning there is genuine phase structure left to explain. Three markers then localize it: bicoherence (are frequencies phase-locked to each other, indicating nonlinear waveform shape?), time-irreversibility (does the signal have a statistical arrow of time, a hallmark of non-equilibrium dynamics?), and INSIDEOUT (is there directed information flow between brain networks?). The connection to Kolmogorov complexity theory is explicit: random-phase "complexity" is just colored noise dressed up as structure, while phase organization is the real thing — compressible in a deep sense, not a trivial one.

The motivating application is chronic low back pain (CLBP), where source-space EEG shows elevated global LZC in patients versus controls, while conventional state-transition metrics show nothing. The paper frames this as exactly the kind of ambiguous finding the program is designed to resolve — is the elevated LZC a spectral artifact or a sign of genuinely altered brain dynamics? The full group-level analysis applying both steps to the CLBP cohort is described as ongoing with collaborators at the University of Otago, so the paper is primarily a methods contribution with the clinical application still in progress.

The practical upshot: before claiming that a brain condition changes neural "complexity," run the surrogate test. If it passes Step 1, you have a spectral story, not a complexity story. Only a surviving residual, characterized through the phase markers, licenses the stronger claim.