Subekshya Bidari, Department of Applied Mathematics, Â鶹ӰԺ

Evidence accumulation models of social foraging

Foraging is often modeled as a sequence of patch-leaving decisions. The distribution of food in the environment is idealized as being contained in discrete patches (e.g., trees), and animals must decide when to depart one patch to go forage at another. To account for the learning processes involved in determining resource availability within and across patches, we model foraging as an evidence accumulation process. These models associate evidence for leaving a patch with a deterministic drift term and the stochasticity of food encounters and memory with diffusive noise. Thus, the foraging decisions within a patch can be modeled as a drift diffusion process in which decisions are triggered when the process crosses a threshold.

I extend these individual evidence accumulation models to consider patch foraging decisions of multi-agent systems sharing information. Agents’ beliefs are tethered together via different types of coupling: diffusive coupling continuously pulls one forager’s belief about patch quality toward its neighbors whereas pulsatile coupling instantaneously updates a forager’s belief when another departs the patch. Using the patch residence time distribution for individuals and groups, we compare different forms of coupling to get insights into forms of information sharing that are most effective. Using data from capuchin and spider monkeys foraging in groups, we will parametrize our foraging drift diffusion models using Bayesian inference techniques to get insights into the most likely strategies animals use to make patch-leaving decisions in spatially cohesive groups.

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Anthony Kearsley, Mathematical Analysis and Modeling Group, National Institute of Standards and Technology (NIST)

Control of inward solidification in Cryobiology

For many years, mathematical models that predict a cell’s response to encroaching ice has played an important role in developing cryopreservation protocols. It is clear that information about the cellular state as a function of cooling rate can improve the design of cryopreservation protocols and explain reasons for cell damage during freezing.However, previous work has ignored the interaction between the important solutes, the effects on the state of the cell being frozen and encroaching ice fronts. In this talk, I will survey our work on this problem and examine the cryobiologically relevant setting of a spherically-symmetric model of a biological cell separated by a ternary fluid mixturefrom an encroaching solid–liquid interface and will illustrate our work on a simplified 1-D problem. In particular, I will demonstrate how the thermal and chemical states inside the cell are influenced and can potentially be controlled by altering cooling protocols at the external boundary.

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Erin Ellefsen and Lindsey Wong, Department of Applied Mathematics, Â鶹ӰԺ

Lyndsey’s Title: Mathematical Models of Wealth Distribution Through an Amenities-Based Theory
Erin’s Title: Efficiently finding Equilibrium Solutions of Nonlocal Models in Ecology

Lyndsey’s Abstract: 

The dynamics of wealth are not fully understood. In order to gain insight on these dynamics, we can use mathematical models. One application of modeling wealth distribution is gentrification. Gentrification refers to the influx of income into a community leading to the improvement of an area through renovation or the introduction of local amenities. It is often accompanied by an increase in the cost of living, which displaces lower income populations. Gentrification is a core issue that affects many urban areas. In this talk, we will present an overview of the work done with our amenities-based approach to modeling wealth distribution. 

First, we present a PDE model derived from transport theory and prove the existence and stability of spatially heterogeneous solutions through a global bifurcation result. Next, we improve this PDE model by instead deriving from first principles and then perform a sensitivity analysis to see which parameters create the most change in solutions when perturbed. We then begin to work with data for Baltimore, MD via the Baltimore Neighborhood Indicators Alliance-Jacob France Institute Vital Signs report. To better understand this data, we use machine learning to determine what factors are most important in predicting neighborhood change. Lastly, we discuss some preliminary results of our current project in which we are incorporating this data into metapopulation models which are based on our previous PDE models. 

Erin’s Abstract:

Understanding the factors that drive species to move and develop territorial patterns is at the heart of spatial ecology. We introduce a nonlocal system of reaction-advection-diffusion equations that incorporate the use of nonlocal information to influence the movement of species. One benefit of this model is that groups are able to maintain coherence without the inclusion of artificial dynamics. As incorporating nonlocal mechanisms comes with analytical and computational costs, we explore the potential of using long-wave approximations of the nonlocal model to determine if they are suitable alternatives that are more computationally efficient. We use the gradient flow-structure of the both local and nonlocal models to compute the equilibrium solutions of the mechanistic models via energy minimizers. In an effort to incorporate data into our model, we turn to spectral methods to more efficiently find equilibrium solutions of the nonlocal model and test the model against synthetic data.

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Nick Barendregt, Department of Applied Mathematics, Â鶹ӰԺ

Adaptive Decision Rules are Optimal in Simple Environments

Decision-making in uncertain environments often requires adaptive forms of evidence accumulation, but less is known about the decision rules needed to achieve optimal performance. While recent studies of decision models in stochastic and dynamic environments have resulted in several phenomenological models, such as the monotonically collapsing decision threshold of the “urgency-gating model� (UGM), we lack a general, normative description of decision rules and their relation to human decision-making. In this talk, we will study the prevalence of adaptive decision rules by developing a normative, Bayes-optimal framework for relatively simple two-alternative forced choice tasks. By allowing context variables, such as reward or difficulty of the decision, to vary in time, we find rich non-monotonic decision rules that vary throughout task parameter space. By comparing the performance of these strategies against simple heuristics, we show that these complex normative strategies significantly outperform alternative models such as the UGM. Finally, using subject data from the classic “tokens task�, we perform rigorous model fitting and comparison which suggests humans may use adaptive normative strategies in such tasks. These results provide testable hypotheses for experimentalists to validate in future psychophysics tasks and give insights into the complexities of human decision strategies.

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Sabina Altus, Department of Applied Mathematics, Â鶹ӰԺ

Mobility Informed Regional Modeling of the COVID-19 Pandemic in Colorado

The trajectory of the COVID-19 pandemic has varied widely by region, and an understanding of these divergent timelines is of great import to local public health officials, as well as our broader epidemiological perception of disease spread. In this talk, a regional approach to modeling the COVID-19 pandemic in Colorado (developed along with the Colorado School of Public Health COVID-19 Modeling Group) is presented. The meta-population model we use to describe and predict the spread of COVID-19 regionally is an age-structured SEIR-based model, informed by cell phone mobility data. We will first present the model, it's limitations, and the most valuable metrics of spread it allows us to produce. Then, we will discuss the role of mobility in this model with particular attention to the insights gained in our analysis of the flow of infection throughout the state, and the feedback between urban and rural areas.

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Weak-Form Sparse Identification of Nonlinear Dynamics with Applications to Cell Migration

The weak-form sparse identification of nonlinear dynamics (WSINDy) algorithm for inferring nonlinear governing equations from noisy datasets significantly improves the accuracy and robustness to noise of strong-form methods. Furthermore, the weak formulation allows for identification of dynamics from non-classical (weak) solutions. This is accomplished by discretizing a convolutional weak form of the dynamics and using the Fast Fourier Transform to both expedite computations and identify test functions with implicit noise-filtering properties. We will review the nuts and bolts of the WSINDy algorithm and demonstrate its success on several fundamental PDEs including inviscid Burgers, Kuramoto-Sivashinsky, and the Navier-Stokes equations, before diving into new developments relevant to biological equation discovery. In particular, we will discuss the identification of governing equations for particle systems with nonlocal interactions and apply this framework to cellular time series data from wound healing experiments.

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Overview of Math Biology Research in the Applied Math Department

David Bortz will give an overview of the math bio group here in the Applied Math Department. He will also describe his research group and our investigations into the methodology of data-driven modeling and model selection as applied to biological systems including bacterial community population dynamics and biomechanics, cellular migration, infectious diseases, and designs for an artificial pancreas.  Zack Kilpatrick’s group develops and analyzes mathematical and computational models describing behavioral strategies humans and other animals use to make decisions about what they see and where to forage. The mathematical methods he uses include nonlinear and stochastic analysis for integrodifferential equation models of neuronal networks as well as statistical inference for deriving, fitting, and comparing Bayesian evidence-accumulation models.

Nancy Rodriguez will discuss the connection between partial differential equations and complex systems.  She will focus on a few project her group is currently pursuing, including the study of different movements in Ecology, 

spatiotemporal dynamics of protests, the existence and stability of traveling wave solutions, and periodic epidemic models.    

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Tahra Eissa, Department of Applied Mathematics, Â鶹ӰԺ

Normative decision asymmetries with symmetric priors but asymmetric evidence

Decisions based on rare events are challenging because rare events alone can be both informative and unreliable as evidence. How humans should and do overcome this challenge is not well understood. Here we present results from a preregistered study of 200 on-line participants performing a simple inference task in which the evidence was rare and asymmetric but the priors were symmetric. Consistent with a Bayesian ideal observer, most participants exhibited choice asymmetries that reflected a tendency to rationally interpret a rare event as evidence for the alternative likely to produce slightly more events, even when the two alternatives were equally likely a priori. A subset of participants exhibited additional biases based on an under-weighing of rare events. The results provide new quantitative and theoretically grounded insights into rare-event inference, which is relevant to both real-world problems like predicting stock-market crashes and common laboratory tasks like predicting changes in reward contingencies.

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David Stearns, Department of Applied Mathematics, Â鶹ӰԺ

Dynamics and Analysis of Territorial Animals

The ways animals form territories, interact with members of their own social group, and interact with members of other intraspecies social groups play a vital role in the maintenance of an ecosystem and in the evolution of these species. Understanding these territories is also key to evaluating the human impact on these ecosystems through both direct interaction and the effects of climate change. This project focuses on two possible mathematical models for how these territories form and change; they are a discrete interacting particle model, where individuals attract and repel each other based on their identity and the environment, and a continuous advection-diffusion partial differential equation model, in which the groups are modeled as density functions which attract and repel each other. In particular, the threshold for when the discrete model becomes a good approximation to the advection-diffusion model is of particular importance.  

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John Nardini, Department of Applied Mathematics, Â鶹ӰԺ

Data-driven modeling for noisy biological data and agent-based Models

I will consider the problem of inferring the dynamics underlying biological data using two case studies in equation learning and topological data analysis. The math biology field presents several exciting challenges for these methods. I will first investigate the performance of equation learning methods in the presence of noisy and sparsely sampled data with application to glioblastoma multiforme, a harmful cancer of the brain. We will use our results to propose suitable times to collect data for informative datasets. In the second case study, I will demonstrate how topology can accurately summarize the time-varying persistent homology of swarming data over multiple scales. This topological approach can outperform more traditional swarm summaries in supervised machine learning tasks.

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