For over a century, wave equations of physical systems have played a pivotal role in understanding diverse phenomena, ranging from Schrödinger’s model of the atom to the neural field theory of the brain. These mathematical representations predict that the expressed energy of a system is confined into natural modes or ‘eigenstates’, largely determined by the system’s geometry. The implication of these eigenstates transcends the realm of atomic structures, providing a theoretical framework that finds resonance in the dynamics of the human brain.
In a groundbreaking study by Pang et al. (2023), solutions to the Helmholtz equation applied on the cortical geometry have been shown to outperform competing approaches across a wide array of tasks and resting state fMRI data. This comes as a significant development, as the Helmholtz equation, originally formulated in the context of wave propagation and harmonics, finds a novel application in the realm of neuroscience. The promise it holds is to illuminate the complex structure and function of the brain through a fresh lens, one that emphasizes geometric constraints.
This paper shows that solutions of the Helmholtz equation on the cortical geometry outperform competing approaches across a diverse range of task & resting state fMRI data.Michael Breakspear
Remarkably, the application of geometric modes is not confined to the cortical surface alone. They extend into subcortical structures, presenting a striking correspondence with patterns of subcortical functional connectivity. This uncovers a new facet of the brain’s functional network, suggesting that the geometric principles that govern cortical activity may also apply to deeper, subcortical regions. This is a potential game-changer for understanding the intricate dynamics of the human brain.
Neural field theory, an established mathematical framework for modeling large-scale brain activity, predicts that waves of activity propagate across the cortex through the successive excitation of these modes. This perspective of wave propagation provides a unique insight into the dynamism and complexity of brain activity, suggesting that the brain’s activity is not just a product of its functional connections, but also a consequence of its physical structure and geometry.
The predictions of neural field theory and the geometric approach indeed provide a more parsimonious account of evoked and resting-state cortical activity than competing network-based neural mass models. This superiority in explanatory power not only validates the geometric approach but also underscores the need for more refined and holistic models in neuroscience. It suggests that our understanding of the brain can be significantly enriched by considering its physical structure alongside its functional connectivity.
“The anatomy of the brain necessarily constrains its function, but precisely how remains unclear,” state Pang et al. (2023). This statement encapsulates the ongoing conundrum in neuroscience. While traditional neuroscience has often attributed neuronal dynamics to interactions between discrete, functionally specialized cell populations connected by a complex array of axonal fibers, the geometric approach presents a fundamentally different perspective, one that elevates the importance of the brain’s physical form.
Predictions from neural field theory suggest that the geometry of the brain may represent a more fundamental constraint on dynamics than complex interregional connectivity. This groundbreaking idea proposes that the physical shape and structure of the brain itself might play a more decisive role in influencing brain activity than previously thought. It is a shift in perspective that challenges the traditional view of the brain as a network of functionally connected regions.
These theoretical predictions, confirmed by analyses of human magnetic resonance imaging data, have profound implications for neuroimaging studies. The findings indicate the possibility of interpreting fMRI data through the lens of geometric constraints, thus offering a new dimension to understanding brain functionality. This could potentially revolutionize how we process and interpret brain imaging data, opening up new avenues for research and discovery.
The geometric approach challenges the classical paradigm of functional specialization and complex connectivity as the primary drivers of brain dynamics. Instead, it positions the physical geometry of the brain as a fundamental determinant of neuronal dynamics. This shift in perspective not only brings a fresh dimension to our understanding of brain function but also promises a more unified and comprehensive model of brain dynamics.
The geometric approach opens up exciting new directions for future research. It invites researchers to explore how physical and geometric properties of the brain interact with neuronal activity, laying the groundwork for a deeper understanding of the brain’s functioning. This approach could potentially spawn a whole new field of research, opening up unexplored avenues and opportunities in the study of the human brain.
Rooted in the principles of wave propagation and eigenstates, it provides a powerful tool for unraveling the intricacies of the human brain. As we continue to explore this promising direction, we may well be on the cusp of groundbreaking discoveries that will fundamentally reshape our understanding of this complex organ.
The anatomy of the brain necessarily constrains its function, but precisely how remains unclear. The classical and dominant paradigm in neuroscience is that neuronal dynamics are driven by interactions between discrete, functionally specialized cell populations connected by a complex array of axonal fibres1–3. However, predictions from neural field theory, an established mathematical framework for modelling large-scale brain activity4–6, suggest that the geometry of the brain may represent a more fundamental constraint on dynamics than complex interregional connectivity7,8. Here, we confirm these theoretical predictions by analysing human magnetic resonance imaging data acquired under spontaneous and diverse task-evoked conditions. Specifically, we show that cortical and subcortical activity can be parsimoniously understood as resulting from excitations of fundamental, resonant modes of the brain’s geometry (that is, its shape) rather than from modes of complex interregional connectivity, as classically assumed. We then use these geometric modes to show that task-evoked activations across over 10,000 brain maps are not confined to focal areas, as widely believed, but instead excite brain-wide modes with wavelengths spanning over 60 mm. Finally, we confirm predictions that the close link between geometry and function is explained by a dominant role for wave-like activity, showing that wave dynamics can reproduce numerous canonical spatiotemporal properties of spontaneous and evoked recordings. Our findings challenge prevailing views and identify a previously underappreciated role of geometry in shaping function, as predicted by a unifying and physically principled model of brain-wide dynamics.
Geometric constraints on human brain function, James C. Pang, Kevin M. Aquino, Marianne Oldehinkel, Peter A. Robinson, Ben D. Fulcher, Michael Breakspear & Alex Fornito
Published: 31 May 2023