Periodic Reporting for period 1 - GSPTexp (Model Reduction for Complex Systems with Exponential Nonlinearity via Geometric Singular Perturbation Theory)
Reporting period: 2024-04-02 to 2026-04-01
Important aspects of the dynamics of complex systems ranging from the Earth’s climate to the cell cycle can be mathematically modeled using ordinary differential equations. However, direct analysis or numerical simulation is rarely possible due to the fact that these models are typically highly nonlinear, high dimensional and multi-scale. This leads to the problem of model reduction, in which the modeler is faced with the task of deciding which and how much information should be discarded in order to obtain a reduced model which is more amenable to analysis, while preserving the salient features of the original system. Unfortunately, model reductions are rarely justified in mathematical terms, which is problematic because they are often used in order to make predictions about complex dynamical systems of social and economic significance.
The aim of this project was to develop systematic mathematical theory for model reduction, and to apply it to important problems in combustion theory, gene regulatory dynamics and selected biological and biochemical networks which are, from a mathematical point of view, highly non-trivial because of the presence of 'severe' exponential nonlinearities. The primary innovation was to combine different approaches to model reductions in a single mathematical framework using adaptations of an established analytical tool known to experts as the "geometric blow-up method".
By developing a sound mathematical theory for model reduction, we hoped to identify conditions for the validity or invalidity of commonly used model reductions in applications that are well-known within the modeling community, thereby raising awareness and - hopefully - preventing incorrect predictions about important complex dynamical systems in the future which result from unjustified reliance of the 'wrong' reduced model.
The aim of this project was to develop systematic mathematical theory for model reduction, and to apply it to important problems in combustion theory, gene regulatory dynamics and selected biological and biochemical networks which are, from a mathematical point of view, highly non-trivial because of the presence of 'severe' exponential nonlinearities. The primary innovation was to combine different approaches to model reductions in a single mathematical framework using adaptations of an established analytical tool known to experts as the "geometric blow-up method".
By developing a sound mathematical theory for model reduction, we hoped to identify conditions for the validity or invalidity of commonly used model reductions in applications that are well-known within the modeling community, thereby raising awareness and - hopefully - preventing incorrect predictions about important complex dynamical systems in the future which result from unjustified reliance of the 'wrong' reduced model.
Together with K. U. Kristiansen, a project partner based at the Technical University of Copenhagen, we began by focusing on a classical problem in combustion theory, namely, the propagation of planar laminar premixed flames. In this context, flame propagation is mathematically described as a traveling wave in a reaction-diffusion equation known as the Zeldovich-Kamenetskii equation. We provided a rigorous and geometrically informative characterization of formally correct but non-rigorous results which date back to early pioneering works in mathematical combustion theory in the late 1930's. In addition to providing a modern, geometric and mathematically rigorous lens on this historically significant problem, we contributed novel mathematical methods - based on the so-called geometric blow-up method - which are expected to be transferrable and applicable to a wide range of related problems in combustion theory and beyond. The findings were published in a leading dynamical systems journal.
In subsequent collaborations with the same project partner, we initiated a investigations into the possibility of using similar mathematical methods as a mathematically rigorous approach to model reduction of gene regulatory networks. Due to the early termination of the project after approximately 9 months (due to the fact that the Fellow accepted an offer for a permanent academic position), we were unable to complete our investigations within the reduced time-frame of the project. Nevertheless, we have continued this collaboration and expect to publish key results on the validity and invalidity of 'combined' approaches to model reduction that are commonly used in the modeling of gene regulatory dynamics. In particular, we expect to publish our findings for a simple but representative gene regulatory network by May 2025, and for the larger network classes more generally by early 2026.
In addition to the work and achievements outlined above, the Fellow published two additional articles in leading dynamical systems journals, which focused on (i) the geometric description characterization of an experimentally identified multi-scale oscillation in a model for intracellular calcium dynamics, and (ii) mathematical theory for the study of tipping phenomena. Both of these works leveraged mathematical techniques which closely related to the project aims (particularly via the use of geometric blow-up techniques), and demonstrated the broad applicability of these ideas in the mathematical sciences.
In subsequent collaborations with the same project partner, we initiated a investigations into the possibility of using similar mathematical methods as a mathematically rigorous approach to model reduction of gene regulatory networks. Due to the early termination of the project after approximately 9 months (due to the fact that the Fellow accepted an offer for a permanent academic position), we were unable to complete our investigations within the reduced time-frame of the project. Nevertheless, we have continued this collaboration and expect to publish key results on the validity and invalidity of 'combined' approaches to model reduction that are commonly used in the modeling of gene regulatory dynamics. In particular, we expect to publish our findings for a simple but representative gene regulatory network by May 2025, and for the larger network classes more generally by early 2026.
In addition to the work and achievements outlined above, the Fellow published two additional articles in leading dynamical systems journals, which focused on (i) the geometric description characterization of an experimentally identified multi-scale oscillation in a model for intracellular calcium dynamics, and (ii) mathematical theory for the study of tipping phenomena. Both of these works leveraged mathematical techniques which closely related to the project aims (particularly via the use of geometric blow-up techniques), and demonstrated the broad applicability of these ideas in the mathematical sciences.
Although we derived new results in each of the application areas considered, the primary contributions of the project within the field were methodological in nature. In particular, we contributed rigorous and geometrically informative methods for the study of complicated dynamical phenomena that results from the presence of singular exponential nonlinearities. Beyond the field of dynamical systems, our results and methodology shed new light on the nature of so-called essential singularities, which pose a larger and much more pervasive problem in mathematics quite generally.
The primary impact of this project beyond the mathematical community is expected to stem from its impact on the use of model reduction techniques, particularly in the study of gene regulatory networks, biological and biochemical networks. Modeling experts in many areas use model reductions out of necessity, in order to simplify mathematical models for complex dynamical phenomena so that they can be applied in practical situations, e.g. to predict the weather or the behavior of a gene regulatory system. We (the project participants) are continuing our collaborations in this direction, and intend to continue them for an extended period.
The primary impact of this project beyond the mathematical community is expected to stem from its impact on the use of model reduction techniques, particularly in the study of gene regulatory networks, biological and biochemical networks. Modeling experts in many areas use model reductions out of necessity, in order to simplify mathematical models for complex dynamical phenomena so that they can be applied in practical situations, e.g. to predict the weather or the behavior of a gene regulatory system. We (the project participants) are continuing our collaborations in this direction, and intend to continue them for an extended period.