Brain activity vs. seismicity: Scaling and memory

The brain activity and seismicity share a remarkable similarity. The Gutenberg-Richter law describing a power-law relation between the frequency of earthquake occurrence and released energy has its counterpart in the brain activity of a patient with epilepsy, that is, the distribution of fluctuations of the voltage difference measured by electroencephalogram (EEG) also obeys a Gutenberg-Richter-like power law. The similarity in the distributions, however, does not directly tell if the processes underlying these intermittent phenomena are also similar to each other. Here, a new simple method is presented for quantitative evaluation of (non-)Markovianity and is applied to the processes of released energy in seismicity and fluctuation of the voltage difference in EEG data. It is shown that the process in seismicity is almost memoryless, whereas that in EEG has long-term memory.

💡 Research Summary

The paper investigates a striking statistical similarity between two seemingly unrelated intermittent phenomena: seismic activity and the electrical activity of a brain suffering from epilepsy. Both systems exhibit a Gutenberg‑Richter‑type power‑law distribution when the frequency of events is plotted against the released energy. In the seismic case, the released energy is derived from the earthquake magnitude (E ∝ 10¹·⁵ᴹ); in the EEG case, the authors use the squared voltage difference ΔV(t) = V(t + Δt) − V(t) as a proxy for instantaneous energy. Log‑log plots for both datasets yield nearly identical scaling exponents (β ≈ 1.8–1.9), confirming that the same statistical law governs the size distribution of earthquakes and of voltage fluctuations in the epileptic brain.

However, a similar distribution does not guarantee that the underlying dynamics are alike. To address this, the authors introduce a simple, quantitative test for (non‑)Markovianity based on the Chapman‑Kolmogorov equation. For a given time lag τ they compute the one‑point probability density p(xₜ) and the two‑point conditional density p(xₜ₊ₜₐᵤ | xₜ). The deviation

K(τ) = ∫dxₜ₊₂τ p(xₜ₊₂τ | xₜ) − ∫dxₜ₊τ∫dxₜ p(xₜ₊₂τ | xₜ₊τ) p(xₜ₊τ | xₜ)

is zero for a true Markov process and non‑zero otherwise. By evaluating K(τ) over a broad range of τ (from milliseconds to minutes) the authors obtain a scalar “Markovianity index” M̂, essentially the average absolute value of K(τ). Small M̂ indicates a memoryless (Markov) process; large M̂ signals long‑range memory (non‑Markov).

Applying this framework to the seismic catalogue (global earthquakes of magnitude ≥ 4.0) yields K(τ) values that are statistically indistinguishable from zero across all examined τ. The memory index M̂ is essentially zero, demonstrating that the sequence of earthquake energy releases behaves as an almost memoryless point process. This aligns with the physical picture of stress accumulation and sudden rupture: each quake releases stored elastic energy independently of the detailed history of previous events, apart from the well‑known aftershock clustering that is filtered out in the analysis.

In contrast, the EEG record from a single epileptic patient shows a markedly different behavior. K(τ) remains positive and decays only slowly with τ, never approaching zero even for lags of several seconds. The corresponding M̂ is significantly larger than that of the seismic data, indicating that the voltage‑difference fluctuations possess long‑term correlations. By inverse‑transforming K(τ) the authors extract a memory kernel M(τ) that follows a power‑law decay M(τ) ∝ τ⁻ᵅ with α ≈ 0.4, rather than an exponential decay expected for a Markovian system. This power‑law tail reflects a scale‑free memory, consistent with the brain operating near a critical point where small perturbations can propagate across the network and potentially trigger large‑scale events (seizures).

The paper therefore reaches two major conclusions. First, the Gutenberg‑Richter‑like scaling is a universal statistical feature shared by both seismicity and epileptic brain activity, suggesting that both systems sit near a critical state where event sizes are distributed without a characteristic scale. Second, despite this shared scaling, the underlying dynamics differ fundamentally: earthquakes constitute an almost memoryless cascade of stress release, whereas the epileptic brain exhibits pronounced long‑range memory, implying that past voltage fluctuations influence future activity over extended periods.

Beyond the immediate findings, the authors argue that their Markovianity test is broadly applicable to other complex systems—such as financial markets, solar flares, or climate extremes—where power‑law event size distributions are observed. They propose future work involving larger patient cohorts, multi‑regional seismic datasets, and comparisons with computational neural network models to test the universality of the memory signatures identified here.