| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | June 7, 2021 |
| Latest Amendment Date: | June 7, 2021 |
| Award Number: | 2106566 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Pedro Embid
DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | July 1, 2021 |
| End Date: | June 30, 2024 (Estimated) |
| Total Intended Award Amount: | $350,000.00 |
| Total Awarded Amount to Date: | $350,000.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
1 PROSPECT ST PROVIDENCE RI US 02912-9100 (401)863-2777 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
182 George Street, Box F Providence RI US 02912-9023 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | APPLIED MATHEMATICS |
| Primary Program Source: |
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| Program Reference Code(s): |
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| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
Many biological, chemical, and physical processes exhibit spiral waves and target patterns. Spiral waves occur, for instance, in autocatalytic chemical reactions, during cellular signaling of amoebae, and in cardiac tissue where they are often the cause of cardiac arrhythmias and the precursor to ventricular fibrillation and cardiac arrest. Similarly, target patterns appear prominently in autocatalytic reactions and are also often found in systems with small spatial inhomogeneities. This project focuses on understanding when spiral waves and target patterns arise, under which conditions they can be observed, and what mechanisms cause them to change their shape or disappear. The investigator will develop analytical, computational, and geometric tools to answer these questions and apply the findings to models of autocatalytic chemical reactions and wave propagation in cardiac tissue. Graduate and undergraduate students will be engaged in these research activities.
This project focuses on understanding existence, multiplicity, stability, and bifurcations of spiral waves and target patterns through the development of spatial-dynamics methodologies that can be applied to a broad range of reaction-diffusion systems. The investigator will establish multiplicity results for one-dimensional spiral waves and target patterns in reaction-diffusion systems using spatial dynamics and apply these results to the Brusselator model. The second project centers on transverse instabilities of spiral waves and target patterns. Transverse instabilities do not contribute to the spectra of planar structures, though they do cause linear instabilities. The aim is to investigate how these instabilities manifest themselves on large bounded disks. The third project focuses on making the numerical computation of spiral spectra more robust by developing preconditioners that rely on appropriate exponential weights to reduce the norm of the resolvent independently of the size of the domain. In the last project, singular perturbation methods will be developed to understand the limits of profiles and spectra of planar spiral waves and target patterns when some of the diffusion coefficients in the model approach zero.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
Many biological, chemical, and physical processes exhibit spiral waves and target patterns. Spiral waves occur, for instance, in autocatalytic chemical reactions, during cAMP signaling of amoebae, as calcium waves in frog oocytes, and in cardiac tissue where they can cause cardiac arrhythmias. Similarly, target patterns appear prominently in autocatalytic reactions and are also often found in systems with small spatial inhomogeneities. Amongst the theoretical results of this project are the prediction of robustness, multiplicity, and stability properties of patterned structures and waves, the discovery and analysis of new synchronization and recruitment mechanisms of processes posed on networks and graphs, and the development of preconditioners, and an explanation of their convergence and numerical stability properties, for numerical solvers that help probe for the temporal stability of spiral waves. Amongst the applied outcomes are the design of algorithms for the inference of gene-regulatory networks, for the data-driven continuation of patterned states in reaction-diffusion models, and for the quantification and classification of patterns emerging in models of heterogeneous cell populations. The project provided training opportunities for undergraduate research students, 5 PhD students, and 3 postdoctoral fellow. The results obtained by undergraduate researchers, graduate students, and postdocs were published in premier mathematical journals and presented at national and international conferences.
Last Modified: 08/26/2024
Modified by: Bjorn Sandstede
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