Complex Biological Systems Group

Department of Mathematics, University of Pittsburgh

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Complex Biological Systems Group
University of Pittsburgh
Department of Mathematics
301 Thackeray Hall
Pittsburgh, PA 15260

cbsg@math.pitt.edu
Tel: (412) 624-6157
Fax: (412) 624-8397

Projects

Algorithms for Spatio-Temporal Models

Novel process coupling/decoupling strategies

Discretizations/griddings of coupled biological processes

Iterative solvers and preconditioners for solving large coupled systems

Neuroscience

Propagating waves and sustained activity patterns

Synchrony and rhythmicity

Role of noise

Synaptic Plasticity

Inflammation and Immunology

Immune Response to Influenza Infection

Lung model

Inflammation and sepsis model

Models of necrotizing enterocolitis and wound healing



Novel process coupling/decoupling strategies

Time-splitting and iterative coupling techniques allow for reducing the solution of the large coupled nonlinear problem to a series of solutions of smaller decoupled (possibly linear) problems. Time-splitting algorithms lead to modeling errors that may not only accumulate in time, but also violate basic conservation laws. It is critical to control such errors, which can be done through iterative coupling approaches. The choice of coupling/decoupling affects the convergence of the iteration. Riviere and Yotov have developed efficient time-splitting methods for multiphase flow in porous media and incompressible fluid flow, as well as couplings of such flows through appropriate interface conditions.

These techniques are being employed in joint work with Clermont, Ermentrout, Hackam, Rubin, Swigon, and Vodovotz on developing multicompartment mathematical and computational models of the NEC inflammatory response and wound healing.

Discretizations/griddings of Coupled Biological Processes

High fidelity simulations of coupled biological phenomena present challenges for developing accurate and efficient discretization and gridding methods. All discretization errors (process specific and coupled) must be controlled through a priori and a posteriori error analysis. Such analysis is challenging when different types of partial differential equations are coupled. Riviere and Yotov have worked extensively on accurate and stable discretization methods for partial differential equations with applications to fluids. In particular Riviere has worked on discontinuous Galerkin (DG) methods and Yotov has worked on mixed finite element (MFE) and related methods. Both methods conserve mass locally and are suitable for complex multiscale simulations. DG methods are very flexible and support high order local approximation and unstructured grids. MFE methods provide accurate approximation of fluxes and a very efficient on relatively structured grids. The inflammation models currently being developed exhibit many similarities with the models of flow and transport in porous media and incompressible fluid flow. Therefore previously developed discretization technologies will be applicable. A MFE method has already been developed for the multicompartment NEC model.

Iterative solvers and preconditioners for solving large coupled biological systems

Due to the large scale of the resulting computational problems, massively parallel computing must be employed in the simulations. Highly scalable solution algorithms are needed for simulators running on parallel computers with a large number of processors. Domain decomposition algorithms are very suitable for this environment, since they localize computations and reduce inter-processor communication. Developing efficient domain decomposition methods for coupled biological problems is especially challenging, since the convergence of the iterative process is affected by nonlinearities due to complex couplings. Yotov's previous work on domain decomposition algotrithms for linear and nonlinear multiphysics fluid couplings will be extended to modeling of inflammation and cardiovascular flows.

Propagating waves and sustained activity patterns

Tissue slice preparations and in vivo imaging show an increasingly important role for the propagation of activity in networks of interconnected neurons in both normal and pathological (e.g. epileptic) states. Models of these phenomena involve large numbers of coupled differential equations, which in certain limits become coupled integro-differential equations. Starting with Hodgkin-Huxley type equations, methods such as averaging and singular perturbation allow one to reduce the dimension of the models which then makes it possible to simulate one- and two-dimensional spatial networks. Examples of this analysis and reduction are found in the papers referred to in prior results, as well as in collaborative efforts of RTG faculty. Several questions remain, related to recent experiments by Pinto (a former student) and Jian-young Wu.There is often a long delay before after stimulus before a wave begins. What causes this long latency? Similarly, many mechanisms have been suggested for the termination of waves. Do these make different testable predictions? Propagation of waves in heterogeneous media results in various phenomena such as block, reflection, and compression.

Synchrony and Rhythmicity

Synchronization of neural ensembles has been an area of great interest in both the experimental and theoretical neuroscience communities. Synchronous firing of neurons has been shown to be necessary for certain kinds of cognitive processing and the loss of this synchrony has been found in some pathologies such as schizophrenia. Excessive synchrony may underly other pathologies like Parkinson's disease and epilepsy. Synchronization also appears to be important in rhythmic activity patterns, in which different subgroups of neurons fire in synchrony at different phases of a global cycle. Determining conditions under which coupled neurons can synchronize is a mathematical question. Work here at the University of Pittsburgh and other places has shown that the types of bifurcations to rhythmicity are major determinants in whether or not neurons can synchronize. Weak coupling theory, singular perturbation theory, and reduction to low-dimensional maps are all tools that we have used to analyze conditions for synchrony. Group members will, for example, investigate mechanisms responsible for maintaining synchrony, and for allowing transitions between multiple inspiratory and expiratory phases, in respiratory rhythms. Moreover, research will address synchrony in heterogeneous networks, examining the roles of network architecture and of distributions of characteristics of intrinsic dynamics in allowing networks to synchronize despite heterogeneity.

Role of Noise

Noise is seen as playing an increasingly important role in biology and in particular neuroscience. For example, in a combined experimental and theoretical study, we showed how synchrony between uncoupled neurons could be effected by correlated noisy inputs into the same cell. By using methods from stochastic calculus, we have been able to derive expressions for the distribution of phase differences in noisy populations, how this distribution depends on the shape of the oscillation, and how the phase-dependent variance of noise can influence the ability to synchronize neural oscillations. This work has been done in conjunction with experiments in the Urban lab. Several students working on these mathematical questions have spent time in Urban's lab doing experiments. Many questions remain, for example: how do the random effects interact with effects due to coupling? what causes the phase-dependence of the variance? Can spatially correlated noise induce pattern formation?

Synaptic Plasticity

Changes in the effectiveness of synaptic connections between neurons are believed to be crucial for learning and memory. We have studied the weight distributions emerging across a network of synaptic inputs to a postsynaptic cell under particular experimentally-observed forms of plasticity and have developed both phenomenological and mechanistic models for these phenomena. Group members will work to link temporal aspects of signalling underlying plasticity to spatial aspects, including the temporal interaction of spatially distributed calcium sources to drive the induction of plasticity. Research will also use computational modeling to investigate how plasticity in particular areas of basal ganglia, associated with the loss of the neurotransmitter dopamine, alters processing of cortical inputs.

Immune Response to Influenza Infection

Hancioglu, Swigon and Clermont have developed a simplified dynamical model of immune response to uncomplicated influenza virus (IAV) infection, which focuses on the control of the infection by the innate and adaptive immunity. They found that for small initial viral load the disease progresses through an asymptomatic course, for intermediate value it takes a typical course with constant duration and severity of infection but variable onset, and for large initial viral load the disease becomes severe. The modeling efforts done in this context, as well as in related work by all RTG PIs, are now faced with the challenge of estimating the parameters that give predictions in accord with experimental observations. Rather than classical estimation techniques producing a single parameter set, we will turn to ensemble models, identical in form and differing only in parameter values, as a description of parameter variability in the system. Using ensemble models, we will give probabilistic predictions about the efficiency of various treatment scenarios.

Lung model

In order to assess the impact of inflammation on respiratory function, Ermentrout, Clermont and Reynolds have developed a multiple compartment model for a single alveolus. Using the microscopic dynamics of a single alveolus, we are scaling up to a multi-alveolar model capable of providing insight into lung failure during inflammation. Since the time scale of breathing is of the order of seconds and the inflammatory process is on the order of a day, this is an example of using an approximation of the microscopic dynamics to build a bridge to a large scale full organ model.

Inflammation and sepsis model

Ermentrout, Rubin, Day, Reynolds, Clermont, and Vodovotz have developed a simple, experimentally-based model for the interactions of pathogens with the intrinsic and activated immune systems. They found that there are several states in the system: clearing the infection and returning to health; clearing the infection but remaining in an inflamed state; not clearing the infection. They found that the time course of anti-inflammatory cyctokines played an important role in the course of the insult. Current and future efforts center on the use of control theory to develop patient-specific intervention strategies and generalization of results to larger models incorporating additional components of the immune response. Furthermore, work is proceeding on the development of multi-compartmental models, ranging from models of interactions between tissue and blood on the millimeter spatial scale to models of multi-organ interactions, relevant to multiple organ failure in severe inflammation.

Models of necrotizing enterocolitis and wound healing

NEC is characterized by an increase in circulating endotoxin (lipopolysaccharide, LPS) and the development of intestinal inflammation and necrosis. Healing from intestinal injury typically occurs through the process of intestinal restitution, which involves the migration of enterocytes from healthy mucosa to sites of injury. Proposed experimental and modeling studies in collaboration with the Hackam lab will be designed to test the hypothesis that NEC results from the initiation of a systemic stress leading to the activation of LPS receptors, bacterial translocation, and impaired gap junction communication resulting in the persistent mucosal defects that characterize NEC. Further studies will be performed to examine the regulation of TLR4 during development of the intestine, the suppression of TLR4 signaling by immuno-modulatory molecules in the pathogenesis of NEC, and the regulation of intestinal stem cells in the presence of inflammatory stimuli. Parallel studies are being performed to evaluate the role of connexins in the pathogenesis of NEC, through their combined effects on the intestinal epithelial monolayer and on the inflammatory macrophages that migrate to the inflamed mucosa.

Mi, Swigon, Riviere, Cetin, Vodovotz and Hackam have developed a one-dimensional mechanical model of cell migration in which the cell layer is treated as a elastic continuum that undergoes motion, adhesion, stretching and growth. The model accurately reproduces motion of enterocyte layer in vitro and, when appropriate assumptions are made about the dependence of adhesion and driving force on integrin concentration, the model predicts dependence of edge velocity on adhesion strength that are in qualitative accord with published data. The model will be extended to two dimensions so as to enable the study the the effect of wound geometry on recovery and healing time. In addition, the group is currently developing a combined mechanistic-cell signaling model that would explain the effect of LPS, TLR4 and other immunological components on cell migration speed.

Yotov, Clermont, Vodovotz, Riviere have worked on general 3D model of inflammation. Dynamic interactions of pathogens and inflammatory agents are coupled with spatial processes such as diffusion, advection, chemotaxis, and cell migration. This is a multidomain model that allows for coupling processes that occur in lumen, organ tissue, and circulatory system. Since in NEC the ability of lumen bacteria to infiltrate the organ tissue critically depends on the integrity of the epithelial layer, cell migration and strength of tight junctions have also been incorporated into the model. Via computer simulations we have studied the effects of hypoxia-induced epithelial damage on the outcome of the inflammatory response. Current work involves studying the effects of formula feeding through anti-inflammatory cytokines. We are also studying the well-posednes of the mathematical model and plan to carry out numerical analysis of the discrete model. Other future work will involve incorporating random effects in the diffusion of bacteria via stochastic PDEs.