Fig. 5. Structural eigenmode decomposition generalises the Fourier transform to the network structure of the brain.

a | In traditional Fourier analysis, a signal in the time domain (represented in terms of successive time points) is decomposed into temporal harmonics of different frequencies and thereby rendered in terms of a different set of basis functions. b | High-frequency temporal harmonics correspond to rapidly varying signals, such that data points may have very different values even if they are close in time. In contrast, low-frequency temporal harmonics correspond to signals that change slowly over time, such that temporally contiguous data points have similar values, reflecting a greater time dependence of the signal. c | Harmonic decomposition of the connectome involves decomposing a signal in the spatial domain (represented in terms of fMRI activation at discrete spatial locations over the cortex) into harmonic modes of the structural connectome, resulting in a different set of basis functions in terms of whole-brain distributed patterns of activity propagation distributed throughout the brain at different spatial scales (granularity), from large-scale patterns of smooth variation along geometrical axes (left–right and anterior–posterior being the most prominent) to increasingly fine-grained patterns. Note that here, frequency is not about time, but about spatial scale. d | Low-frequency (coarse-grained) connectome harmonics indicate that the spatial organisation of the functional signal closely matches the underlying organisation of the structural connectome: Nodes that are strongly connected exhibit similar functional signals (indicated by colour). High-frequency (fine-grained) patterns indicate divergence between the spatial organisation of the functional signal and the underlying network structure, where nodes may exhibit different functional signals even if they are closely connected in the structural network²⁵.