The study of the brain and the central nervous system is one of the fastest growing areas of science research. Neuroscience research uses a broad range of recording techniques and experimental paradigms including imaging, neurophysiology, and behavioral measurements. As a consequence, neuroscience is currently giving rise to a broad array of challenging signal processing questions. In this talk we survey some of current neuroscience signal processing questions.
Lee Segel one of the greatest applied mathematicians of our time passed away on January 31, 2005. His obituary (SIAM News, 03-10-2005) read "With his death, the applied mathematics community lost one of its finest practitioners, and the theoretical biology community lost a true pioneer who was still a leader at the cutting edge of so many subjects. And most importantly, the world community lost a true mensch, a compassionate and loving individual who inspired so many with his brilliance, his enthusiasm, his sense of humor, and his concern for others." Lee Segel was an extraordinary mentor of women and his former students include many leaders in the field of mathematical biology. In this talk through a series of examples, I will illustrate my experiences with women mathematicians at the undergraduate, graduate and postdoctoral level as they engaged in research in mathematical biology. I will discuss their research as a prelude to the lecture of one of the most outstanding mathematical biologist, Trachette Jackson who chose to become a mathematician through the encouragement and support of the late Joaquin Bustoz Jr.
We propose an algorithm to select parameter combinations that can be estimated using an ordinary least-squares (OLS) inverse problem formulation with a given data set. First, the algorithm selects the parameter combinations that correspond to full-rank sensitivity matrices. Second, the algorithm involves uncertainty quantification by using the inverse of the Fisher Information Matrix. Nominal parameter values are used to explore the effects of removing certain parameters from those to be estimated using OLS procedures. The algorithm is illustrated with synthetic data of seasonal epidemics, real data of an influenza outbreak in a boarding school, real data of 1918 pandemic in San Francisco, and real data of HIV progression with treatment interruption.
Biological flows, such as those surrounding swimming microorganisms, can be properly modeled using the Stokes equations for fluid motion with external forcing. The organism surfaces can be viewed as flexible interfaces imparting force or torque on the fluid. Interesting flows have been observed when the organism swims near a solid wall due to the hydrodynamic interaction of rotating flagella with a neighboring solid surface. I will present the Method of Regularized Stokeslets and some extensions of it that are used to compute these flows. The method includes the use of regularized rotlets and a system of images that exactly cancels the fluid velocity at the wall. The results show features such as an attraction towards the surface and rotations that generate a drag force that allows the flagellum to roll along the surface. Other computed flows resemble observed features of the flow when organisms are near the bottom of the plate in an experimental setting.
In the 2002 film by Gurinder Chadha, character Jesminder 'Jess' Bhamra states "No one can cross a ball or bend it like Beckham" in a reference to the international soccer star's ability to cause the ball to swerve. French researchers Guillaume Dupeux, Anne Le Goff, David Quere and Christophe Clanet published a paper earlier this year in the New Journal of Physics detailing both experimental and mathematical analyses of a spinning ball in a fluid to show that it must follow a spiral. In this talk, we give an overview of their discussion by reviewing the Navier-Stokes equation in a Serret-Frenet coordinate system. This talk is dedicated to the memory of Angela Grant and her love of mathematics in sports.
Breast cancer is one of the most commonly diagnosed cancer among women in the United States. Cancer treatments can be classified as local or systemic. Endocrine therapy is one form of systemic treatment for breast cancer. In this talk we utilize data from published clinical trial results in a mixed integer nonlinear programming model in order to detect an optimal treatment plan. The objective is to maximize the disease-free survival percentage at the end of the treatment subject to constraints dictated by the risk of several side effects. This joint work with Sera Kahruman, Elif Ulusal, Sergiy Butenko, and Kathleen Diehl.
The Mackey-Glass equation is a seemingly simple delay differential equation (DDE) with one fixed delay which can exhibit the full gamut of dynamics from a trivial stable steady state to fully chaotic dynamics, and has inspired decades of mathematical research into DDEs. However, much of that research has focused on equations with fixed or prescribed delays, whereas many biological delays would be more naturally modelled as state-dependent delays. Before incorporating state-dependent delays in complex biochemical network models, it is desirable to understand the dynamics which result from including state-dependent delays in simpler model problems. Accordingly, in this talk we will consider a simple model problem with multiple state-dependent delays, and show that it can exhibit a wide range of dynamical behaviour, including stable periodic solutions and bi-stable periodic solutions, to stable tori, together with the associated bifurcation structures.
Nonlinear partial differential equations arise in stochastic optimal control via dynamic programming equations. In many cases, solutions of these equations aid in the design of optimal controls. We discuss a class of equations where the associated control processes are "singular" with respect to the time variable. These equations arise in models for spacecraft control, financial models that incorporate transaction costs, and in models of queueing systems.
Motility -- random, directed and collective -- is a fundamental property of cells. Coordinated motility of endothelial cells that reside on the inner surface of blood vessels leads to a critical bifurcation point in cancer progression: tumor angiogenesis. Successful angiogenesis is a consequence of integration across multiple levels of biological organization, and several temporal and spatial scales. A major challenge facing the cancer research community is to integrate known information in a way that improves our understanding of the mechanisms driving tumor angiogenesis and that will advance efforts aimed at the development of new therapies for treating cancer.
In this talk, the evolution of spatio-temporal mathematical models of tumor angiogenesis will be explored and recent advances will be highlighted.
Mathematical ecology has its roots in population ecology, which treats the increase and fluctuations of populations. It was along these lines that Lotka (1924) and Volterra (1926) established their original works on the expression of predator-prey and competing species relations in terms of simultaneous nonlinear differential equations, making the first breakthrough in modern mathematical ecology.
The importance of transmissible disease in an ecological situation is not to be ignored. There are many references in this context (Beltrami and Caroll (1994), Chattopadhyay and Pal (2002)) in such eco-epidemiological system. The viral disease can infect bacteria and even phytoplankton in coastal water.
We have dealt with the problem of a classical predator-prey dynamics in which viral infection spread on prey population and classical predator-prey system is splitted into three groups, namely susceptible prey, infected prey and predator. We have observed the dynamical behaviour of this system around each of the equilibrium and pointed out the "exchange of stability".
Force of infection in any eco-epidemiological system is of great importance. Here we have considered standard incidence as the force of infection. This system is of rich dynamics in zero equilibrium. Moreover, the co-existence of the species is of ecological importance. We find some conditions on this force of infection for which the extinction possibilities of the species may overcome. We investigate the criteria for which the system will persist. We may conclude that infection factor may act as a biological control.
Despite the common expectation that each species in a community occupies its own niche, many species that share similar sets of traits actually coexist in nature. Neutral theory provides an alternative to the long-standing notion that species differences promote coexistence. In neutral communities, differences in birth and death rates among similar species are negligible, and stochasticity is the primary factor shaping the distributions of abundances in communities. We use a stochastic Lotka-Volterra model with species along a niche axis to investigate the differences in species abundance distribution patterns that arise due to different mechanisms of coexistence: neutrality in similar species and stability due to trait differences. Here I show preliminary results distinguishing between these two mechanisms and discuss the direction of this work.
A central problem of Climate Variability is global warming. We are familiar with the controversy regarding its social and environmental implications. Much less so about where these controversies arise.
I will describe, first, one of the scenarios predicted by an increase in global temperatures. I will describe the role played by mathematics in climate research and will argue how mathematics must play a central role in answering the largest technical challenges posed by the Intergovernmental Panel on Climate Change report: How confident are we about predictions of future climate scenarios?
I will describe why it is so difficult to pin down uncertainties in climate variability and will highlight some of the mathematical tools being developed to tackle these questions, including techniques developed by my group, the Uncertainty Quantification Group.
Conformational diseases result from the failure of a specific protein to fold into its correct functional state. The misfolded proteins can lead to the toxic aggregation of proteins. In some cases, misfolded proteins interact with bystanders proteins (unfolded and native folded proteins), eliciting a misfolded phenotype. These bystander polypeptides would follow their normal physiological pathways in absence of misfolded proteins. In some conformational diseases, evidence suggests that bystander protein disappearance occurs through direct or indirect interaction with misfolded proteins, resulting in a transformation into aggregate-prone misfolded protein. Protein aggregation in conformational diseases often displays a threshold phenomenon characterized by a sudden shift between nontoxic and toxic levels of protein aggregates. We propose a general mechanism of bystander and misfolded protein interaction to investigate how the threshold phenomenon in protein aggregation is triggered in conformational diseases. Using a continuous flow reactor model of the endoplasmic reticulum, we derived the conditions necessary to produce threshold phenomena. Our results indicate that slight changes in the ratio of misfolded to bystander basal protein concentrations can trigger the threshold phenomena in protein aggregation. Our model proposes a general mechanism for the loss of function observed in certain conformational diseases. We also identify the conditions necessary to trigger the observed threshold phenomena in protein aggregation. Understanding the conditions necessary for the aggregation threshold phenomena is an important step towards developing therapeutic strategies targeting the modulation of conformational diseases.
The Poisson distribution is a popular probability distribution whose underlying structure is assumed in many classical statistical methods, e.g. regression analysis for count data, control charts, etc. In all of these cases, however, we are limited in our ability to accurately model real data due to the constrained assumptions of the Poisson distribution (namely, that the associated mean and variance must equal). Oftentimes, real data do not conform to this requirement in that the data are actually over- or under-dispersed. Thus, we consider using a generalized form of a Poisson distribution, called the Conway-Maxwell-Poisson (COM-Poisson) distribution, to better address and model such data. This work introduces the COM-Poisson distribution and its applications in a variety of fields, including regression analysis, statistical quality control, and general data analysis.
A new way to model the dynamics of malaria transmission that takes into consideration the demography of the transmitting vector will be presented. Model results indicate the existence of nontrivial disease free and endemic steady state solutions which can be driven to instability via a Hopf bifurcation as a parameter is varied in parameter space. The model therefore captures natural occurring oscillations known to occur in the dynamics of mosquito populations and these oscillations lead to oscillations in the dynamics of malaria transmission without recourse to external seasonal forcing, a way that has been used in the past to obtain such oscillations. Possible reasons why it has been difficult to eradicate malaria will also be discussed. The discovery of these natural occurring oscillatory dynamics present a plausible framework for developing and implementing control strategies. These will be discussed.
Multiple cellular processes, such as DNA replication and transcription, affect the topology of DNA. Controlling these changes is key to ensuring stability inside the cell. Changes in DNA topology are mediated by enzymes, such as topoisomerases and site-specific recombinases. We use techniques from knot theory and low-dimensional topology, aided by computational tools, to analyze the action of such enzymes.
I will here present recent advances in our study of DNA unlinking by XerCD-FtsK.
XerC and XerD are site-specific recombinases of Escherichia coli. Replication of circular chromosomes requires unwinding of the DNA and results in the formation of DNA links. In Escherichia coli, error-free unlinking is required to ensure proper segregation at cell division. The site-specific recombination system XerCD mediates sister chromosome unlinking in TopoIV deficient cells. We provide formal proof that, under the model's assumptions, there is a unique pathway taking any torus link to the unlink.
This is joint work with Koya Shimokawa, Kai Ishihara, Ian Grainge, David J.Sherratt.
The Tacoma Narrows Bridge opened on July 1, 1940 in the state of Washington. The bridge earned the name "Galloping Gertie" because it could be seen oscillating up and down. On November 7, 1940, wind gusts with speeds of only 40 miles per hour caused the bridge to collapse into the Pugent Sound. In 1999, Dr. P. Joseph McKenna presented a model that described the bridge's motion and reasons for its collapse. Since his model negated previous theories, a "genteel professorial catfight" broke out between McKenna and some engineers. This talk will examine theories and controversy that surround the collapse of the Tacoma Narrows Bridge. This debate still persists in many undergraduate textbooks.