Meet the project - 'Mathematical Challenges in Brain Mechanics'

At CAS this year, the project Mathematical Challenges in Brain Mechanics is bringing mathematicians and neuroscientists together to do exactly that. Their aim is ambitious: to build the mathematical foundations needed to understand how fluids move through the brain and why these movements matter for diseases such as Alzheimer’s and Parkinson’s.

When Professor Jan Martin Nordbotten and Professor Kent-Andre Mardal speak about the brain, they sound as much like explorers as scientists. Their project starts from a simple but striking paradox: despite remarkable advances in neuroscience, some of the brain’s most basic housekeeping functions are still poorly understood. “That rather simple questions like ‘how is nutrition and waste transported through this incredibly complex material with the high precision required for normal function?’ is not resolved in modern day medicine fascinated us deeply,” they explain.

This curiosity sits at the heart of their work at CAS. Over the past decade, a shift has taken place in medicine: disorders such as Alzheimer’s and Parkinson’s are now increasingly described not just in terms of damaged cells, but as problems involving fluid flow and solute transport. These processes do not behave like the systems engineers typically simulate. They unfold simultaneously across microscopic and macroscopic scales, involve multiple physical phenomena, and require mathematical descriptions that connect structures of different dimensions. In short: they are messy, and existing mathematical tools do not fully capture them.

“The human brain seems to require the development of new models and new simulation tools,” they note, and this is precisely the challenge they have set out to tackle. Their main objective, as they put it, is “to develop a mathematical framework that enables simulation of the mechanics of the human brain, in particular the fluid flows within and outside the brain.”

It’s a task that sits at the frontier of numerical mathematics. Enormous progress has been made in the past half-century—faster computation, more accurate algorithms, smarter numerical techniques. “In fact, the speed up and accuracy improvements provided by smart numerics has been greater than the hardware improvements,” they observe. But much of this progress has focused on single-physics, single-scale problems. Extending mathematics to capture processes that are multiscale, multiphysics, and mixed-dimensional remains an open challenge.

This is where CAS becomes crucial. The project’s success hinges on bringing together expertise that normally sits far apart—in mathematics, fluid mechanics, computational science, and neuroscience. The group leaders are very clear-eyed about what this requires: persistent, practical collaboration. “Neuroscience is a vast topic, as is mathematics, and of course there is only a certain overlap,” they say. “However, we believe that the overlap is actually larger than one might think and that the main obstacle is that of communication and also to find interesting topics of mutual scientific benefit.”

To overcome this, the group is organising frequent meetings, inviting neuroscientists for workshops both at CAS and in clinical environments, and hosting long-term visitors who can properly settle into the work. CAS, they say, is uniquely suited for this combination of focus and exchange. “For long term visitors, the calm and peaceful atmosphere and break from teaching is stimulating for thinking ‘long thoughts’.”

A key part of the project is the scholars they have assembled. Their selection criteria were refreshingly straightforward: “We chose the scholars we thought were the best on particular topics of relevance.” This means the team includes leading specialists in mathematical modelling, numerical analysis, and the physics of porous media, alongside researchers working at the forefront of experimental and clinical neuroscience.

The potential impact of this collaboration is wide-ranging. On the medical side, Nordbotten and Mardal describe a long-term hope: “While we hope to understand the cause and develop the cure for diseases like Alzheimer’s and Parkinson’s, a more realistic but still ambitious goal is to develop a mathematical foundation for the study of the particular aspects of brain function that is currently believed to be the causes of these diseases.” On the mathematical side, they aim to push forward the theory needed to work across scales and physical processes—tools that are still largely missing today.

If successful, their framework could become a cornerstone for the next generation of quantitative, physics-based neuroscience. It is precisely the kind of cross-disciplinary, curiosity-driven effort that aligns with the CAS mission. As they put it: “In our opinions, most big question and societal challenges lie between disciplines. For example, it is impossible—in our view—to understand even simple questions regarding the brain’s function without mathematics beyond current state-of-the-art.”

The brain remains one of science’s most enigmatic landscapes. With this project, Nordbotten, Mardal, and their collaborators hope to illuminate part of that terrain—not by peering through a microscope, but by sharpening the mathematics that help us understand how the brain moves, breathes, and sustains itself. And in that work lies the possibility of new insights, both for mathematics and for medicine.