Fractal geometry predicts dynamic differences in structural and functional connectomes

TL;DR

The paper tackles the limitation of static graph metrics in capturing brain network dynamics by introducing a fractal-dynamics framework based on complex quadratic networks and equi-M sets. By mapping the asymptotic behavior of iterated quadratic dynamics onto empirically derived Structural and Functional connectomes from the HCP, the authors show that equi-M shape and size invariants reveal robust distinctions between network types and cognitive states, with positive vs signed interactions playing a central role. Structural connectomes exhibit consistent, cusp-right equi-M geometry and strong correlations between graph and geometric measures, while Functional connectomes (Rest vs Emotion) display leftward cusps and greater boundary variability, and partial-network analyses fail to recapitulate full dynamics. The results suggest fractal-based invariants offer a compact, interpretable descriptor of emergent network dynamics with potential applicability across complex systems and neuroscience contexts, providing a new lens to classify and compare dynamic states beyond traditional connectivity metrics.

Abstract

Understanding the intricate architecture of brain networks and its connection to brain function is essential for deciphering the underlying principles of cognition and disease. While traditional graph-theoretical measures have been widely used to characterize these networks, they often fail to fully capture the emergent properties of large-scale neural dynamics. Here, we introduce an alternative approach to quantify brain networks that is rooted in complex dynamics, fractal geometry, and asymptotic analysis. We apply these concepts to brain connectomes and demonstrate how quadratic iterations and geometric properties of Mandelbrot-like sets can provide novel insights into structural and functional network dynamics. Our findings reveal fundamental distinctions between structural (positive) and functional (signed) connectomes, such as the shift of cusp orientation and the variability in equi-M set geometry. Notably, structural connectomes exhibit more robust, predictable features, while functional connectomes show increased variability for non-trivial tasks. We further demonstrate that traditional graph-theoretical measures, when applied separately to the positive and negative sub-networks of functional connectomes, fail to fully capture their dynamic complexity. Instead, size and shape-based invariants of the equi-M set effectively differentiate between rest and emotional task states, which highlights their potential as superior markers of emergent network dynamics. These results suggest that incorporating fractal-based methods into network neuroscience provides a powerful tool for understanding how information flows in natural systems beyond static connectivity measures, while maintaining their simplicity.

Figures (16)

• Figure 1: Measurable differences in equi-M set shape can be captured by computing a set of geometric measures, illustrated here for two random Structural connectomes in our data set: subject #22 (panel A) and subject #5 (panel B). In the left panel, we have the following landmarks and distances: $C = 0.13$, $T = -0.9$, $L = 0.22 + 0.16 i$, $R = -0.06 + 0.43 i$, $d_{TC} = 1.03$, $d_{TR} = 0.94$, $d_{TL} = 1.14$, $d_{CL} = 0.19$, $d_{R \bar{R}} = 0.41$, $\epsilon = 0.87$. In the right panel: $C = 0.08$, $T = -0.39$, $L = 0.13 + 0.06 i$, $R = -0.03 + 0.2 i$, $d_{TC} = 0.47$, $d_{TR} = 0.42$, $d_{TL} = 0.53$, $d_{CL} = 0.008$, $d_{R \bar{R}} = 0.87$, $\epsilon = 0.84$.
• Figure 2: Measurable differences in equi-M set shape captured by computing geometric measures for sets derived from Functional connectomes. The two panels illustrate the different geometry landmarks and corresponding measures between two Emotion connectomes: subject #28 (panel A) and subject #3 (panel B). In the left panel, we have the following landmarks and distances: $C = -0.01$, $T = 0.024$, $S = 0.025 + 0.001 i$, $L = -0.021 + 0.003 i$, $R = 0.003 + 0.026 i$, $d_{TC} = 0.04$, $d_{SR} = 0.03$, $d_{SL} = 0.04$, $d_{CL} = 0.006$, $d_{R \bar{R}} = 0.05$, $\epsilon = 1.3$. In the right panel: $C = -0.008$, $T = 0.015$, $S = 0.015$, $L = -0.013 + 0.007 i$, $R = -0.0006 + 0.013 i$, $d_{TC} = 0.024$, $d_{SR} = 0.021$, $d_{SL} = 0.03$, $d_{CL} = 0.008$, $d_{R \bar{R}} = 0.02$, $\epsilon=1.1$.
• Figure 3: Fourier mode reconstruction of the Structural connectome from \ref{['fig:structural_landmarks']}a (top) and Functional connectome from \ref{['fig:emotion_landmarks']}b (bottom) with $N + 1$ modes in $\{-N/2, \dots, N/2\}$.
• Figure 4: Different approaches to the Structural connectome data using the equi-M set. (a) Statistical equi-M set: the light to dark gradient shows the fraction of equi-M sets that the respective point in $\mathbb{C}$ belongs to. (b) The equi-M set computed for the prototypical Structural connectome (i.e., the matrix computed as the mean over all Structural connectomes).
• Figure 5: Correlations between the graph-theoretical measures and geometric measures for the Structural connectomes (panel A) and for randomizations of the Structural connectomes that preserve the number of edges into each node (panel B) . The measures are listed along both coordinate axes, in the order found in the tables, and in the order introduced in the text. For $C$, $\operatorname{Re}(R)$ and $\operatorname{Re}(L)$, we used their absolute values, to illustrate the distance to the origin. For the complex modes $m_{-2}$ to $m_2$, we use their moduli (to illustrate their strength). In this scheme, the three diagonal squares show correlations between measures within the same category, and the off-diagonal rectangles show correlations between measures in different categories. For simplicity, we only included the $\rho$ values here.