Research led by a Wayne State University Department of Mathematics professor is aiding researchers in Wayne State’s Department of Psychiatry and Behavioral Neurosciences in analyzing fMRI data. fMRI is the preeminent class of signals collected from the brain in vivo and is irreplaceable in the study of brain dysfunction in many medical fields, including psychiatry, neurology and pediatrics.
Andrew Salch, Ph.D., associate professor of mathematics in Wayne State’s College of Liberal Arts and Sciences, is leading the multidisciplinary team that is investigating how concepts of topological data analysis, a subfield of mathematics, can be applied to recovering “hidden” structure in fMRI data.
“We hypothesized that aspects of the fMRI signal are not easily discoverable using many of the standard tools used for fMRI data analysis, which strategically reduce the number of dimensions in the data to be considered. Consequently, these aspects might be uncovered using concepts from the mathematical field of topological data analysis, also called TDA, which is intended for use on high-dimensional data sets,” said Salch. “The high dimensionality that characterizes fMRI data includes the three dimensions of space — that is, where in the brain the signal is being acquired — time — or how the signal varies as brain states change in time — and signal intensity — or how the strength of the fMRI signal changes in response to the task. When related to task-induced changes, the results reflect biologically meaningful aspects of brain function and dysfunction. This is a unique collaborative work focused on the complexities of both TDA and fMRI respectively, show how TDA can be applied to real fMRI data collected, and provide open access computational software we have developed for implementing the analyses.”
The research article, “From mathematics to medicine: A practical primer on topological data analysis and the development of related analytic tools for the functional discovery of latent structure in fMRI data,” appears in the Aug. 12 issue of PLOS ONE.
In it, the team used TDA to discover data structures in the anterior cingulate cortex, a critical control region in the brain. These structures — called non-contractible loops in TDA — appeared in specific conditions of the experiment, and were not identified using conventional techniques for fMRI analyses.
“We expect this work to become a citation classic,” said Vaibhav Diwadkar, Ph.D., professor of psychiatry and behavioral neurosciences and research collaborator. “Instead of merely applying TDA to fMRI, we provide a lucid argument for why medical researchers who use fMRI should consider using TDA, and why topologists should turn their attention to the study of complex fMRI data. Moreover, this important work provides readers with empirical demonstrations of such applications, and we provide potential users with the tools we used so they can in turn apply it to their own data.”
“Our ongoing research utilizing TDA with fMRI will provide a unique and complementary method for assessing brain function, and will give medical researchers greater flexibility in tackling complex properties in their data,” said Salch. “In particular, our work will help fMRI researchers become aware of the significant power of TDA that is designed to address complexity in data, and will enhance the value of using fMRI in neuroscience and medicine.”
In addition to Salch and Diwadkar, co-authors on the paper include Adam Regalski, Wayne State mathematics graduate student; Hassan Abdallah, Wayne State mathematics department alumni and current graduate student at the University of Michigan; and Michael Catanzaro, assistant professor of mathematics at Iowa State University and Wayne State mathematics department alumni.
This work is supported by the National Institutes of Health (MH111177 and MH059299), the Jack Dorsey Endowment, the Cohen Neuroscience Endowment, and the Lycaki-Young Funds from the State of Michigan.
For millennia, humanity has observed the changing phases of the Moon. The rise and fall of sunlight reflected off the Moon, as it presents its different faces to us, is known as a “phase curve.” Measuring phase curves of the Moon and Solar System planets is an ancient branch of astronomy that goes back at least a century. The shapes of these phase curves encode information on the surfaces and atmospheres of these celestial bodies. In modern times, astronomers have measured the phase curves of exoplanets using space telescopes such as Hubble, Spitzer, TESS and CHEOPS. These observations are compared with theoretical predictions. In order to do so, one needs a way of calculating these phase curves. It involves seeking a solution to a difficult mathematical problem concerning the physics of radiation.
Approaches for the calculation of phase curves have existed since the 18th century. The oldest of these solutions goes back to the Swiss mathematician, physicist and astronomer, Johann Heinrich Lambert, who lived in the 18th century. “Lambert’s law of reflection” is attributed to him. The problem of calculating reflected light from Solar System planets was posed by the American astronomer Henry Norris Russell in an influential 1916 paper. Another well-known 1981 solution is attributed to the American lunar scientist Bruce Hapke, who built on the classic work of the Indian-American Nobel laureate Subrahmanyan Chandrasekhar in 1960. Hapke pioneered the study of the Moon using mathematical solutions of phase curves. The Soviet physicist Viktor Sobolev also made important contributions to the study of reflected light from celestial bodies in his influential 1975 textbook. Inspired by the work of these scientists, theoretical astrophysicist Kevin Heng of the Center for Space and Habitability CSH at the University of Bern has discovered an entire family of new mathematical solutions for calculating phase curves. The paper, authored by Kevin Heng in collaboration with Brett Morris from the National Center of Competence in Research NCCR PlanetS — which the University of Bern manages together with the University of Geneva — and Daniel Kitzmann from the CSH, has just been published in Nature Astronomy.
Generally applicable solutions
“I was fortunate that this rich body of work had already been done by these great scientists. Hapke had discovered a simpler way to write down the classic solution of Chandrasekhar, who famously solved the radiative transfer equation for isotropic scattering. Sobolev had realised that one can study the problem in at least two mathematical coordinate systems.” Sara Seager brought the problem to Heng’s attention by her summary of it in her 2010 textbook.
By combining these insights, Heng was able to write down mathematical solutions for the strength of reflection (the albedo) and the shape of the phase curve, both completely on paper and without resorting to a computer. “The ground-breaking aspect of these solutions is that they are valid for any law of reflection, which means they can be used in very general ways. The defining moment came for me when I compared these pen-and-paper calculations to what other researchers had done using computer calculations. I was blown away by how well they matched,” said Heng.
Successful analysis of the phase curve of Jupiter
“What excites me is not just the discovery of new theory, but also its major implications for interpreting data,” says Heng. For example, the Cassini spacecraft measured phase curves of Jupiter in the early 2000s, but an in-depth analysis of the data had not previously been done, probably because the calculations were too computationally expensive. With this new family of solutions, Heng was able to analyze the Cassini phase curves and infer that the atmosphere of Jupiter is filled with clouds made up of large, irregular particles of different sizes. This parallel study has just been published by the Astrophysical Journal Letters, in collaboration with Cassini data expert and planetary scientist Liming Li of Houston University in Texas, U.S.A.
New possibilities for the analysis of data from space telescopes
“The ability to write down mathematical solutions for phase curves of reflected light on paper means that one can use them to analyze data in seconds,” said Heng. It opens up new ways of interpreting data that were previously infeasible. Heng is collaborating with Pierre Auclair-Desrotour (formerly CSH, currently at Paris Observatory) to further generalize these mathematical solutions. “Pierre Auclair-Desrotour is a more talented applied mathematician than I am, and we promise exciting results in the near future,” said Heng.
In the Nature Astronomy paper, Heng and his co-authors demonstrated a novel way of analyzing the phase curve of the exoplanet Kepler-7b from the Kepler space telescope. Brett Morris led the data analysis part of the paper. “Brett Morris leads the data analysis for the CHEOPS mission in my research group, and his modern data science approach was critical for successfully applying the mathematical solutions to real data,” explained Heng. They are currently collaborating with scientists from the American-led TESS space telescope to analyze TESS phase curve data. Heng envisions that these new solutions will lead to novel ways of analyzing phase curve data from the upcoming, 10-billion-dollar James Webb Space Telescope, which is due to launch later in 2021. “What excites me most of all is that these mathematical solutions will remain valid long after I am gone, and will probably make their way into standard textbooks,” said Heng.
Researchers have developed a mathematical model that can predict the optimum exercise regime for building muscle.
The researchers, from the University of Cambridge, used methods of theoretical biophysics to construct the model, which can tell how much a specific amount of exertion will cause a muscle to grow and how long it will take. The model could form the basis of a software product, where users could optimise their exercise regimes by entering a few details of their individual physiology.
The model is based on earlier work by the same team, which found that a component of muscle called titin is responsible for generating the chemical signals which affect muscle growth.
The results, reported in the Biophysical Journal, suggest that there is an optimal weight at which to do resistance training for each person and each muscle growth target. Muscles can only be near their maximal load for a very short time, and it is the load integrated over time which activates the cell signalling pathway that leads to synthesis of new muscle proteins. But below a certain value, the load is insufficient to cause much signalling, and exercise time would have to increase exponentially to compensate. The value of this critical load is likely to depend on the particular physiology of the individual.
We all know that exercise builds muscle. Or do we? “Surprisingly, not very much is known about why or how exercise builds muscles: there’s a lot of anecdotal knowledge and acquired wisdom, but very little in the way of hard or proven data,” said Professor Eugene Terentjev from Cambridge’s Cavendish Laboratory, one of the paper’s authors.
When exercising, the higher the load, the more repetitions or the greater the frequency, then the greater the increase in muscle size. However, even when looking at the whole muscle, why or how much this happens isn’t known. The answers to both questions get even trickier as the focus goes down to a single muscle or its individual fibres.
Muscles are made up of individual filaments, which are only 2 micrometres long and less than a micrometre across, smaller than the size of the muscle cell. “Because of this, part of the explanation for muscle growth must be at the molecular scale,” said co-author Neil Ibata. “The interactions between the main structural molecules in muscle were only pieced together around 50 years ago. How the smaller, accessory proteins fit into the picture is still not fully clear.”
This is because the data is very difficult to obtain: people differ greatly in their physiology and behaviour, making it almost impossible to conduct a controlled experiment on muscle size changes in a real person. “You can extract muscle cells and look at those individually, but that then ignores other problems like oxygen and glucose levels during exercise,” said Terentjev. “It’s very hard to look at it all together.”
Terentjev and his colleagues started looking at the mechanisms of mechanosensing — the ability of cells to sense mechanical cues in their environment — several years ago. The research was noticed by the English Institute of Sport, who were interested in whether it might relate to their observations in muscle rehabilitation. Together, they found that muscle hyper/atrophy was directly linked to the Cambridge work.
In 2018, the Cambridge researchers started a project on how the proteins in muscle filaments change under force. They found that main muscle constituents, actin and myosin, lack binding sites for signalling molecules, so it had to be the third-most abundant muscle component — titin — that was responsible for signalling the changes in applied force.
Whenever part of a molecule is under tension for a sufficiently long time, it toggles into a different state, exposing a previously hidden region. If this region can then bind to a small molecule involved in cell signalling, it activates that molecule, generating a chemical signal chain. Titin is a giant protein, a large part of which is extended when a muscle is stretched, but a small part of the molecule is also under tension during muscle contraction. This part of titin contains the so-called titin kinase domain, which is the one that generates the chemical signal that affects muscle growth.
The molecule will be more likely to open if it is under more force, or when kept under the same force for longer. Both conditions will increase the number of activated signalling molecules. These molecules then induce the synthesis of more messenger RNA, leading to production of new muscle proteins, and the cross-section of the muscle cell increases.
This realisation led to the current work, started by Ibata, himself a keen athlete. “I was excited to gain a better understanding of both the why and how of muscle growth,” he said. “So much time and resources could be saved in avoiding low-productivity exercise regimes, and maximising athletes’ potential with regular higher value sessions, given a specific volume that the athlete is capable of achieving.”
Terentjev and Ibata set out to constrict a mathematical model that could give quantitative predictions on muscle growth. They started with a simple model that kept track of titin molecules opening under force and starting the signalling cascade. They used microscopy data to determine the force-dependent probability that a titin kinase unit would open or close under force and activate a signalling molecule.
They then made the model more complex by including additional information, such as metabolic energy exchange, as well as repetition length and recovery. The model was validated using past long-term studies on muscle hypertrophy.
“Our model offers a physiological basis for the idea that muscle growth mainly occurs at 70% of the maximum load, which is the idea behind resistance training,” said Terentjev. “Below that, the opening rate of titin kinase drops precipitously and precludes mechanosensitive signalling from taking place. Above that, rapid exhaustion prevents a good outcome, which our model has quantitatively predicted.”
“One of the challenges in preparing elite athletes is the common requirement for maximising adaptations while balancing associated trade-offs like energy costs,” said Fionn MacPartlin, Senior Strength & Conditioning Coach at the English Institute of Sport. “This work gives us more insight into the potential mechanisms of how muscles sense and respond to load, which can help us more specifically design interventions to meet these goals.”
The model also addresses the problem of muscle atrophy, which occurs during long periods of bed rest or for astronauts in microgravity, showing both how long can a muscle afford to remain inactive before starting to deteriorate, and what the optimal recovery regime could be.
Eventually, the researchers hope to produce a user-friendly software-based application that could give individualised exercise regimes for specific goals. The researchers also hope to improve their model by extending their analysis with detailed data for both men and women, as many exercise studies are heavily biased towards male athletes.