Functional harmonics reveal multi-dimensional basis functions underlying cortical organization

The human brain consists of specialized areas that flexibly interact to form a multitude of functional networks. Complementary to this notion of modular organization, brain function has been shown to vary along a smooth continuum across the whole cortex. We demonstrate a mathematical framework that accounts for both of these perspectives: harmonic modes. We calculate the harmonic modes of the brain’s functional connectivity graph, called ‘‘functional harmonics,’’ revealing a multi-dimensional, frequency-ordered set of basis functions. Functional harmonics link characteristics of cortical organization across several spatial scales, capturing aspects of intra-areal organizational features (retinotopy, somatotopy), delineating brain areas, and explaining macroscopic functional networks as well as global cortical gradients. Furthermore, we show how the activity patterns elicited by seven different tasks are reconstructed from a very small subset of functional harmonics. Our results suggest that the principle of harmonicity, ubiquitous in nature, also underlies functional cortical organization in the human brain.K.G. and P.H. are supported by Swiss National Science Foundation (170873). M.L.K. is supported by the Center for Music in the Brain funded by the Danish National Research Foundation (DNRF117), and Centre for Eudaimonia and Human Flourishing funded by the Pettit and Carlsberg Foundations. G.D. is supported Spanish National Research Project (PID2019-105772GB-I00 MCIU AEI) funded by the Spanish Ministry of Science, Innovation and Universities (MCIU), State Research Agency (AEI); the Human Brain Project Specific Grant Agreement 3 (HBP SGA3) (945539) funded by the EU H2020 FET Flagship programme; the SGR Research Support Group (2017 SGR 1545) funded by the Catalan Agency for Management of University and Research Grants (AGAUR); Neurotwin Digital twins for model-driven non-invasive electrical brain stimulation (101017716) funded by the EU H2020 FET Proactive programme; European School of Network Neuroscience (860563) funded by the EU H2020 MSCA-ITN Innovative Training Networks; CECH The Emerging Human Brain Cluster (001-P-001682) within the framework of the European Research Development Fund Operational Program of Catalonia 2014-2020; Brain-Connects: Brain Connectivity during Stroke Recovery and Rehabilitation (201725.33) funded by the Fundacio La Marato TV3; and Corticity, FLAG-ERA JTC 2017 (PCI2018-092891) funded by the Spanish Ministry of Science, Innovation and Universities (MCIU), State Research Agency (AEI). J.P. is supported by Australian NHMRC (APP1024800, APP1046198, and APP1085404), a Career Development Fellowship (APP1049596), and an ARC discovery project (DP140101560). S.A. is supported by the ERC (CAREGIVING 615539)