Multifractality in critical neural field dynamics

TL;DR

This study links temporal multifractality to brain criticality by analyzing a Landau-Ginzburg–based neural field model that undergoes a synchronization phase transition. Temporal fluctuations, quantified via wavelet $p$-leader multifractal formalism, reveal that multifractality emerges near the critical point and can differ across phase-transition types. The findings show that both low-frequency activity and the oscillation envelope exhibit multifractal scaling, with peak complexity at criticality, providing a formal basis for interpreting multifractality in brain recordings. The work highlights practical considerations for data analysis, such as finite-size effects and potential spurious multifractality from envelope extraction, and suggests broader applicability to neural-field theories and clinical contexts.

Abstract

The brain criticality hypothesis has largely only characterized brain dynamics in terms of their self-similarity, although experimental evidence suggests that the brain exhibits significant multifractality. To understand how multifractality may emerge in critical-like systems modeling neuronal activity, we used a neural field model exhibiting neural oscillations and a critical phase transition. We find that multifractality emerges near a synchronization phase transition, and that the pattern of variation of multifractality changes when placing the model at a different phase transition. These findings show that multifractality in temporal dynamics emerges near criticality in neural fields, providing a formal basis for interpreting multifractality in brain recordings.

Figures (4)

• Figure 1: Schema of the discretized model (a) in blue, the excitatory activity field $\rho$; in orange, the resource field $R$. Spatial average $\bar{\rho}$ of the activity field from model simulations, observed at the minimal (b) at maximal (c) characteristic temporal scales used in subsequent multifractal analysis. The values of the control parameter $\tau_D$ vary from subcritical (bottom) to supercritical (top) through the critical point around $\tau_D \approx 99$. Time series are offset on the y-axis by increasing multiples of 1.5 for legibility. The black line represents the extracted envelope of the oscillations for a single time series ($\tau_D=116$).
• Figure 2: Cumulant scaling functions $C_1(j)$ (top row) and $C_2(j)$ (bottom row) averaged over 20 simulations, for the low-frequency domain (a-b) and oscillation envelope (c-d), for five $\tau_D$ values introduced in Fig. \ref{['fig:model']}. Shaded area indicates 90% confidence interval over 20 simulations. In blue: range of scales $[j_1, j_2]$ over which the log-cumulants were estimated. The characteristic temporal scale of the oscillation is $j=7.2$.
• Figure 3: Continuous lines: log-cumulants $c_1$ (top row) and $c_2$ (bottom row) across the phase transition for the low frequency domain (a-b) and the envelope (c-d). Error bars show the standard deviation over 20 simulations. Dashed grey lines show model order (Kuramoto synchronization parameter $K$). Vertical dotted lines indicate the true critical point; the shaded area delimits the bistability region.
• Figure 4: Second order non-absorbing phase transition. Multifractal exponents $c_1$ (a) and $c_2$ (b) as a function of the parameter $\xi$, averaged across 10 simulations. The shaded area indicates the range of extreme observations. Power spectral density of the time series, with normalized energy (c); the color bar shows the values of $\xi$ used.