Not long ago we found a way to describe the large-scale statistical dynamics of neocortical neural activity in terms of (a) the equilibria of the mean-field Wilson-Cowan equations, and (b) the fluctuations about such equilibria due to intrinsic noise, as modeled by a stochastic version of such equations. Major results of this formulation include a role for critical branching, and the demonstration that there exists a nonequilibrium phase transition in the statistical dynamics which is in the same universality class as directed percolation (DP). Here we show how the mean-field dynamics of interacting excitatory and inhibitory neural populations is organized around a Bogdanov-Takens bifurcation, and how this property is related to the DP phase transition in the statistical dynamics. The resulting theory can be used to explain the origins and properties of random bursts of synchronous activity (avalanches), population oscillations(quasi-cycles), synchronous oscillations (limit-cycles)and fluctuation-driven spatial patterns (quasi-patterns). We will also show how such a system of interacting neural populations can be made to self-organize to a state near the Bogdanov-Takens bifurcation, if the coupling constants(synaptic weights) are activity-dependent, and follow approximately, a generalization of the Vogels et al. version of spike-time dependent synaptic plasticity (STDP).
As we move through an environment, we are constantly making assessments, judgments, and decisions about the things we encounter. Some are acted upon immediately, but many more become mental notes or fleeting impressions -- our implicit "labeling" of the world. In this talk I will describe our work using physiological correlates of this labeling to construct a hybrid brain-computer interface (hBCI) system for efficient navigation of a 3D environment. Specifically, we record electroencephalographic (EEG), saccadic, and pupillary data from subjects as they move through a small part of a 3D virtual city under free-viewing conditions. Using machine learning, we integrate the neural and ocular signals evoked by the objects they encounter to infer which ones are of subjective interest. These inferred labels are propagated through a large computer vision graph of objects in the city, using semi-supervised learning to identify other, unseen objects that are visually similar to those that are labelled. Finally, the system plots an efficient route so that subjects visit similar objects of interest. We show that by exploiting the subjects' implicit labeling, the median search precision is increased from 25% to 97%, and the median subject need only travel 40% of the distance to see 84% of the objects of interest. We also find that the neural and ocular signals contribute in a complementary fashion to the classifiers' inference of subjects' implicit labeling. In summary, we show that neural and ocular signals reflecting subjective assessment of objects in a 3D environment can be used to inform a graph-based learning model of that environment, resulting in an hBCI system that improves navigation and information delivery specific to the user's interests.
Soundscape Ecology and the concept of biophonies and geophonies in particular, form the basis of an emerging field that is only about dozen years old. While it is being defined to some extent in cultural, acoustic, and biological terms, translating these phenomena to numbers has been elusive and challenging. While my field of expertise is neither mathematics nor statistics — it is limited primarily to field recording, archiving and preliminary analysis of natural soundscapes — this discussion will provide some historical background and then focus on the ways in which biophonies and geophonies, in particular, might be framed through the lens of various statistical models and lay the groundwork for new ones.
Cells are highly ordered, and much of this organization is controlled by active molecular-motor based transport. At the single-molecule level, biophysical studies have been very effective at determining how motors work. However, in cells, motors do not function by themselves, but rather, act in groups. How these groups function is less well understood. In this talk I will discuss our experimental evidence underlying the statement that motors function in groups, and will then discuss how modeling can be used as a key tool to understand both how groups of motors function, and also which aspects of their single-molecule properties are particularly important for controlling this ensemble function.
Despite remarkable advances in protein structure prediction and design, an accurate predictive model of the thermodynamic effects of even point mutations remains elusive. The post-genomic era is a remarkable time to consider the problem of protein stability from a statistical perspective. How is information distributed in protein sequences? What protein properties are encoded by conserved and co-varying amino acids? And how can we use this information to engineer more stable proteins? We have been analyzing the effects of sequence conservation and correlation on enzyme function and physical properties, by engineering consensus and correlated mutations, or fully consensus variants, of the ubiquitous and well-studied metabolic enzymes triosephosphate isomerse (TIM) and adenylate kinase (ADK). We have established useful methods for calculating and visualizing conservation and correlation. From two consensus variants of TIM, we showed that correlated networks in weakly conserved positions can contribute strongly to protein biophysical properties. We also showed that consensus mutations are more likely to stabilize at more conserved positions, unless those positions strongly co-vary with other positions. I will review these results and discuss further efforts to refine our sequence-based algorithm for protein stabilization, and establish its generality. I will also briefly discuss experiments to explore the effects of correlated mutations directly, including through the use of a TIM-knockout based selection we pioneered in our lab.
Protein folding diseases occur when a specific protein fails to fold into its correct functional state as a consequence of mutation in the protein amino acid sequence. In this talk, I present a model of the folded and misfolded protein expression, processing and their interactions, which we have used to investigate how protein folding disease phenotypes develop from mutated genotypes. Modeling protein processing as a continuous flow reactor, we found that the pathogenesis of protein folding diseases can be modulated by a combination of the transition time of folded and misfolded proteins in the reactor, the ratio of folded and misfolded protein inflow rates in the reactor and a chemical interaction parameter between folded and misfolded proteins. Our analysis reveals therapeutic strategies targeting the modulation of protein folding diseases, which have been recently explored in cellular and animal models of Mutant INS-gene Induced Diabetes of Youth and Congenital Hypothyroidism with deficient thyroglobulin.
I will briefly review a main way in which mathematical modeling has been used to understand and predict evolutionary change. I will then highlight an important shortcoming of such approaches and consider an alternative that attempts to overcome the problem. This alternative encompasses what I refer to as "open-ended" evolution. I will then present a proof, using this approach, that certain evolutionary questions are inherently unanswerable unless the process of evolution has specific properties. The cause of this limitation on evolutionary theory is shown to be fundamentally the same as that underlying the Halting Problem from computability theory and Gödel’s Incompleteness Theorem.
From bird flocks to fish schools, animal groups exhibit a remarkable ability to manage a variety of challenging tasks that individuals could not manage on their own. Despite limitations on individual-level sensing, computation, and actuation, and with no centralized instruction, animal groups make decisions quickly, accurately, robustly and adaptively in an uncertain and changing environment. I will describe recent development of analytically tractable models and methods for studying the mechanisms of collective movement and decision-making dynamics in animal groups. I will focus on the application of bifurcation analysis to systematically elucidate the dependence of the collective dynamics on parameters that model the networked multi-agent system and the environment.
Naomi Ehrich Leonard is the Edwin S. Wilsey Professor of Mechanical and Aerospace Engineering and an associated faculty member of the Program in Applied and Computational Mathematics at Princeton. She is currently Director of Princeton's Council on Science and Technology and an affiliated faculty member of the Princeton Neuroscience Institute and Program on Quantitative and Computational Biology. Her research and teaching are in control and dynamical systems with current interests in coordinated control for multi-agent systems, mobile sensor networks and ocean sampling, collective animal behavior, and human and animal decision dynamics. In 2013 she was elected to the American Academy of Arts and Sciences. She received a John D. and Catherine T. MacArthur Foundation Fellowship in 2004, the Mohammed Dahleh Award in 2005, and an Inaugural Distinguished ECE Alumni Award from the University of Maryland in 2012. She is a Fellow of the IEEE, ASME, SIAM, and IFAC. She received the B.S.E. degree in Mechanical Engineering from Princeton University in 1985 and the M.S. and Ph.D. degrees in Electrical Engineering from the University of Maryland in 1991 and 1994. From 1985 to 1989, she worked as an engineer in the electric power industry.
We present a mathematical model for regulation of beta-cell mass and function based on the pioneering work of Topp et al, J. Theor. Biol. 206:605 2000. Their model added a layer of slow negative feedback to the classic insulin-glucose loop in the form of a glucose-dependent growth-death law for beta-cell mass. We add to that model regulation of beta-cell function on intermediate time scales. The model quantifies the relative contributions of insulin action and insulin secretion defects to type 2 diabetes (T2D) and explains why prevention is easier than cure. The latter is a consequence of bistability, which also underlies the success of bariatric surgery and acute caloric restriction in reversing T2D. With a further enhancement to include the dynamics of exocytosis, the model describes the mechanistic bases of the canonical pathways to T2D, elevated fasting glucose vs. elevated post-prandial glucose, and clarifies their relationship to the early transient and late sustained phases of insulin secretion. The model gives new insight into the significance of the fact that insulin secretion is higher for pre-diabetics and early diabetics than for normal individuals ("Starling's law of the pancreas"), which has led some to question whether impaired insulin secretion is necessary for diabetes or even to propose that excessive insulin secretion triggers the disease. The presentation will serve as an example of how theorists can use relatively elementary mathematics to engage constructively in important debates in the experimental community.
Extensive genetic information and the expanding number of techniques available to manipulate the genome of the mouse have led to its widespread use in studies of brain development and to model human neurodevelopmental diseases. We are developing a combination of ultrasound and magnetic resonance micro-imaging approaches with sufficient resolution and sensitivity to provide noninvasive structural, functional and molecular data on developmental and disease processes in normal and genetically-engineered mice. Our efforts over the past decade have focused on in utero and early postnatal imaging and analysis of the developing brain and cerebral vasculature. The advantages and limitations of both ultrasound and MRI for imaging mouse development will be discussed, and examples provided to illustrate the utility of these approaches for 4D mutant phenotype analysis. Recent advances have also made in the area of molecular imaging, including the generation of novel reporter mice that enable cell-specific imaging with ultrasound and MRI contrast agents. Future directions for molecular imaging of mouse brain development will be discussed.
Models of the global terrestrial biosphere in current Earth system models (climate models with coupled atmosphere, ocean and biosphere) uniformly predict a large current carbon sink caused by CO2 fertilization of terrestrial vegetation that sequesters 1-2 GtC/y. Models with a nitrogen cycle generally predict that a large fraction of the sink will disappear by midcentury because of nitrogen limitation. The models all include some form of Liebig’s Law of the Minimum for nitrogen. All models currently predict that water limited systems will see large and sustained sinks because water use efficiency is increased by elevated CO2. However, FACE experiments and other recent evidence implies that the opposite is true: CO2 fertilization sinks are observed to persist despite N-limitation and the benefits of enhanced water use efficiency have not been observed. We developed a mechanistic version of forest simulation models with competition for light, water and nutrients that can be analyzed mathematically. We used it to compute the most competitive strategies of allocation to foliage, stem wood and fine roots as a function of soils and climate. When fertilized by CO2, these most competitive strategies predict the results of FACE experiments and the opposite of previous global models: sustained CO2 sinks in the face of N-limitation and the absence of sinks in water –limited systems. I explain the cause of these results, the mechanism behind them and how one would test them in the field.
Metastatic melanoma is known to be resistant to standard chemotherapy. During the last few years, targeted therapeutic approaches have emerged as the dominant treatment choice, primarily because they target tumor cells that harbor specific genetic mutations. However, even these targeted drugs have limited long term success in treating metastatic melanoma patients, since eventually resistance emerges. Surprisingly, when these treatments are given in combination much better treatment responses are observed. However, little is known about why the combinations are more successful. Our recent experimental results suggest a possible mechanism, in that these treatments differentially induce autophagy (the process of self-digestion by a cell to temporarily extend its life under stressful conditions) in tumor cells. To better understand how autophagy induction might facilitate better treatment response, we developed a mathematical model comprising of a system of ordinary differential equations that explain the dynamics of melanoma cells under different treatments. Specifically, we incorporated an autophagy cell population and examined how this population affects treatment success. Model parameters, such as the growth and death rates with and without treatments, were estimated by comparing model predictions with in vitro experimental data. Model results show that the combination therapy is effective in controlling tumor population over an extended period of time. The resistance, however, eventually emerges driven in part by the autophagy population. To overcome this resistance, we applied a drug that targets the autophagy population and were able to show that additional administration of this drug inhibited growth of the resistant population. In order to place these results in a more clinically relevant setting (e.g. clinical tumor volume doubling times), we generated a small cohort of virtual patients by varying model parameters to capture the diversity of disease response observed in the clinic. Parameters varied include initial proportion of different cell types, net growth rate, autophagy rate, and cell death rates. We then applied 10 different treatment schedules that were composed of different combinations (order and duration) of AKT inhibitor, Chemotherapy and autophagy inhibitor to this virtual patient cohort. This effectively allowed us to implement a “virtual clinical trial” or phase i trial with our model and select the optimal therapeutic approach across a range of patients.