Applied mathematics is used to solve problems in various fields including economics and business. Dr. Jonathan Touboul’s experiments have been directed towards several areas including on biology and how non conformists end up looking the same. In this interview we discuss these areas and much more with him.
Dr. Jonathan Touboul, would you explain to us in layperson’s terms how your research lies within the boundaries of mathematics and biology?
Mathematicians do not always consider ethereal or abstract questions.
In fact, mathematics is everyday more in contact with other disciplines. While abstraction and rigor characterize any research in mathematics, mathematicians do not always consider ethereal or abstract questions. In fact, there is a branch of mathematics, called applied mathematics, whose purpose is to use mathematical approaches to solve a problem arising in another field, such as physics, economics, society, business, industry, or, in my case, biology.
A natural question to ask is why mathematical research can at all be useful for solving a practical question in biology. In fact, mathematics provides a universal language, in which one can express clearly a question and answer it. It is by devising and analyzing mathematical equations and developing appropriate mathematical theories that scientists have been able to describe and predict to some extent the movement of planets, the oscillation of a pendulum, or the turbulence of air around the wings of a plane.
A new frontier in applied mathematics today is biology. Many applied mathematicians worldwide work on mathematical equations describing living systems, from vegetation to the brain, from bacteria to infectious diseases live COVID-19. My research spans different directions in this domain. One of my main interests is modeling and analysis of questions related to neurosciences (the study of neurons and brain function), to embryonic development (how from a single cell a full organism develops) as well as ecology (e.g., what are the determinants of the boundary between savannas and rainforests, and what is the impact of climate change on these).
In contrast with biologists making experiments, collecting and analyzing data in a lab or in the field, the mathematicians gather from multiple sources experimental information and evidences from reading papers or discussing with colleagues doing experiments. Based on this information, the applied mathematicians will write up equations on their blackboard to best representing the concepts and mechanisms that biologists have identified. These equations are called a mathematical model. Oftentimes, writing up a model can take months. Going back and forth to the experiments, talking to the biologists, correcting the equation to better reflect one aspect of it, add or remove a mechanism, and then simplify the equations as much as possible. Once the model is produced, we study it, sometimes using computers solving our equations, sometimes with pen and paper, most of the times doing both. The outcome expected from this research is twofold: on one hand, the applied mathematician shall provide better understanding of biology, through testable predictions that can validate or contradict a theory; on the other hand, new models of raise interesting new questions in mathematics that are explored on their own right.
You collaborate with experimentalists. Tell us about the more interesting studies you have conducted with such collaborations.
I decided to highlight two projects published in scientific journals in 2020 that, I believe, are illustrative of a crosstalk between experiments and mathematics.
I decided to highlight two projects published in scientific journals in 2020 that, I believe, are illustrative of a crosstalk between experiments and mathematics. In this crosstalk, biological problems provide new questions to be studied mathematically, and mathematical models yield new insight about biology. The first project I would like to highlight was a study in which I contributed with a group of biologists at Harvard Medical School (Pourquié lab) to characterize the mechanisms taking place early in the development of embryos. Early in development, embryos undergo somitogenesis, a process leading to the generation of segments in the human body, and the Pourquié lab has been studying in depth the expression of genes associated, and has shown that genes oscillate at unison during this phase in rodents. The problem at hand was to investigate if similar phenomena take place in human, and, if so, characterize the ingredients leading to this synchronization of gene expression. I participated to this study by developing tools to better analyze microscopy data and used mathematical models to characterize the level of synchronization in gene expression across cells (Dias-Cuadros et al, Nature 2020). This helped characterizing the key factors in the generation of oscillations and the spatial organization of these oscillations.
Another topic I have been studying in past years is the question of the synchronization of neurons in normal and pathological brain behaviors. With a neurologist from Paris (Bertrand Degos) and an experimental lab from Collège de France in Paris (Laurent Venance’s lab), we worked to better understand Parkinson’s disease and its treatment. Parkinson’s disease is a common neurodegenerative disorder characterize by motor symptoms such as tremor or akinesia. In the brain of parkinsonian patients, neurons progressively stop functioning as they used to, and tend to synchronize too much, meaning that they activate together and at regular time intervals. A new treatment of Parkinson’s disease, called Deep Brain Stimulation (DBS), consists in sending high frequency electrical stimulation directly in the brain of patients. While surgeons have ideas about why this works, these remain hypotheses and generally we are still not sure why and how DBS induces a dramatic reduction of symptoms. Understanding this is not only important theoretically, but it can help understanding better Parkinson’s disease, and down the line find new treatments that could be less invasive than DBS and therefore be applied to more patients. We thus set out to investigate this question combining mathematical models, numerical simulations and experiments.
Our theoretical work, published this year (Touboul et al, Phys. Rev. X 2020), identified a possible mechanism by which DBS may interrupt the abnormal oscillations in brain, and restore optimal information processing. In parallel, an interdisciplinary work combining more detailed mathematical models with remarkable experiments from the Venance lab (Vandecasteele et al, Nature Comm. 2020) explored new therapeutic avenues. In that work, the Venance lab was able to test hypotheses and predictions by controlling the activation of specific cells using a technique called optogenetics. Such projects span several years, and I have worked on and off on this problem for the past 8 years. In fact, the hipster effect model was a way to approach this question using a simpler setting.
When non conformists end up looking the same. Tell us about this study?
Mathematicians, pure or applied, rely on metaphors to understand or describe phenomena arising in abstract equations. The hipster model is actually one of those metaphors, that I used to better describe the synchronization or unpredictable cells. To understand how this metaphor came about, let me try and explain some background.
In 2012, I discovered with one of my students that unpredictability (or randomness) can lead groups of neurons to actually synchronize into regular behaviors (Hermann et al, SIADS 2012). There is a paradox here, since one would rather expect the opposite: the more unpredictable the less synchronization one could expect.
To understand why this synchronization happens, I decided to work on simplified models, similar to some models existing in physics. I ended up with a model of interacting elements being in one of 2 states and switching between states depending on the states of the others. Two types of agents were considered: those that switch to be more alike other agents, and those that switch to be distinct from others. The novelty was that the prominent role given by agents that oppose to the majority, and that switching occurred after a delay. I found out that this model was in fact completely solvable mathematically (Touboul, DCDS 2019), which allowed me better understand of the mechanisms leading to synchronization. The system was showing that agents would synchronize by trying to be more different from each other, and when the delay in switching was sufficiently long. This, for some reason, made me think of trends that we can see in all domains, mathematics included, where fashions emerge from individuals pursuing originality. That’s when I subtitled the paper when anticonformists all look the same. This subtitle does not imply that the paper has a sociological content, or used field data on the weight of beards in vintage pubs or statistics on the density of plaid shirts in various areas. The hipster effect is just a handy metaphor, and easier way to understand the concept of synchronization in the presence of randomness.
…Stay with us and read the second part of the interview with Professor Touboul on Friday August 21.
Dr. Jonathan Touboul is Associate Professor of Mathematics at the Brandeis University. He is a known expert in Applied Mathematics and has conducted several studies. His research as he describes, lies at the boundary between mathematics and biology. He collaborates with experimentalists, mathematicians and physicists. His study on non-conformists ending up looking the same drew worldwide attention.
Photos: From the Archive of Professor Jonathan Touboul / Shutterstock
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