Research

Glucose regulation and diabetes

An adult human’s blood contains about four grams of glucose. The brain can burn through this amount in a mere half-hour, and in exercise the muscles can churn through it four times faster than that. A single can of soda contains ten times as much sugar as all of your blood, and you might drink it in ten minutes! Despite these huge fluxes, the body controls blood glucose quite well—after all, if the blood glucose drops by a factor of two, you’ll pass out! Diabetes, the world’s 8th leading cause of (human) death, is a disease where this control fails, and the blood glucose is chronically high (as well as having unusual dynamics).

I bring the quantitative approach of physics to the study of both healthy blood glucose regulation and diabetes. In one strand of ongoing work, I am trying to understand why the system that regulates blood glucose has evolved the way it has. Are there other regulatory systems which would work equally well? Can we learn something new about the “goals” of the body by studying the system it uses to control blood glucose theoretically? This understanding may have practical consequences for treatment of diabetes, both by helping us predict how the body adapts physiologically over the course of Type 2 Diabetes (T2D), and by guiding the design of improved hormonal pumps.

In another line of work, I am working to understand the pathogenesis of diabetes in clinical data. Our work here has focused on Ketosis-prone diabetes (KPD), an atypical form of diabetes. KPD resembles T2D in many ways. KPD symptoms can worsen very rapidly compared to conventional T2D, however, with patients showing up in the hospital with a complete lack of insulin production only a few weeks or months after reporting no diabetes symptoms. These patients, however, often achieve a remission after a few weeks or months of insulin treatment, which is quite rare in similarly severe T2D! I developed a mathematical model which explains these and other facts, and we are now testing whether this model can be fit to data from individual patients, in a clinical study that we initiated here at Emory. This will hopefully lead to personalized treatment protocols for individual KPD patients.

When (and why) does machine learning work?

At Emory, I have been involved in a variety of projects using the tools of statistical physics to explain the behaviour of machine-learning algorithms.

Ard Louis’s group has identified a surprising universality in the distribution of functions produced by a variety of randomized neural networks. These distributions decay with “Zipf’s law”, a particular power-law distribution. This provides a measurement of the “inductive bias” that prevents them from overfitting training data. In a soon-to-be submitted manuscript, I developed two theoretical explanations of this universality. Firstly, I show how this universal prior is produced mechanistically by neural networks in the limit of large width. Secondly, I show that, if learning is done using a power-law prior, the Zipfian prior is optimal: any slower decay of the prior hinders generalization to data outside the training set, while any faster decay causes the training set to be underfit.

In another group of projects with a PhD student in the Nemenman group, we have studied simple toy models to understand the limitations and advantages of different machine learning methods. We identified possible flaws in machine-learning methods for connecting structure and dynamics in disordered systems, as well as ways to improve them. We also used random-matrix theory to analyze techniques for dimensionality reduction, identifying situations where joint reduction of two datasets is likely to be far more data-efficient than traditional analysis methods using PCA.

Understanding supercooled liquids through machine learning–informed theory

Solid ice and liquid water are distinguished by a dramatic structural change: in water, molecular positions look more or less random, but in ice, molecules sit on a periodic lattice. As some liquids are cooled, however, they form “glasses”, which are solid, but still have a liquid-like structure! For decades physicists have struggled to connect the dynamics of very cold glass-forming liquids with their structure. Recently, machine learning methods have emerged as a promising approach, because they can identify local structural variables (e.g., “softness”) which correlate much more strongly with particle motion than structural variables which physicists had proposed by pure thought.

A structural variable which can predict short- or long-time dynamics, however, is just the beginning of a scientific understanding of the connection between structure and dynamics. My goal in this area has been to develop theory which uses this machine-learned variable to understand how dynamics evolve in time and space, as well as their temperature dependence. Since the machine-learned variable is given to us by a black box, we need to develop a theory for its time evolution from scratch using empirical observations, and some care is needed to make sure the resulting theory is consistent with the fundamental laws of physics (in particular, time-reversal symmetry, which the fundamental laws must still obey even if the glass state often breaks the symmetry). In my PhD work, I developed the tools necessary to construct this kind of theory, and showed how it reproduces many features of dynamical heterogeneity.

The jamming transition

Many disordered systems show a transition from a liquid-like phase to a rigid phase as a function of some control parameter. A paradigmatic example is the jamming of frictionless repulsive spheres, which has a jamming transition near which various properties have scalings resembling an equilibrium critical point. This transition has an upper critical dimension of 2, and the mean field theory of glasses (which is exact for infinite spatial dimension) accurately describes the critical exponents of this transition in both 2d and 3d systems.

Two projects in my PhD touched on the finite-dimensional consequences of the mean-field theory of glasses. In one project, we showed that, although the mean-field picture would suggest structure becomes rapidly irrelevant for dynamics as spatial dimension increases, local structure appears to be relevant to dynamics in any spatial dimension, raising a puzzle for the interpretation of the otherwise very successful mean-field results. In another project, I helped to analyze simulation data from Eric Corwin’s group. We considered not just critical exponents, but how the prefactors of scaling relations scale with spatial dimension, and showed that these dimensonal scalings match mean-field theory. Since the mean-field theory is exact for large spatial dimensional, this implies, shockingly, that mean-field theory accurately predicts not just critical exponents, but the actual values of some observables near the jamming transition! (I’m still unsure if anyone has ever bothered to check that this isn’t true in, say, the Ising model in 4d—surely someone would have noticed, right?) I was then able to exactly calculate one of these dimensional scalings without the mean-field assumption, in agreement with the numerical results.

In a final project, we showed how known scalings for jamming at finite temperature could be unified into an unusual type of scaling ansatz, where apparently different exponents on different sides of a transition are generated by a dangerous irrelevant variable.