70 research outputs found
Using machine learning to predict catastrophes in dynamical systems
Nonlinear dynamical systems, which include models of the Earth\u27s climate, financial markets and complex ecosystems, often undergo abrupt transitions that lead to radically different behavior. The ability to predict such qualitative and potentially disruptive changes is an important problem with far-reaching implications. Even with robust mathematical models, predicting such critical transitions prior to their occurrence is extremely difficult. In this work, we propose a machine learning method to study the parameter space of a complex system, where the dynamics is coarsely characterized using topological invariants. We show that by using a nearest neighbor algorithm to sample the parameter space in a specific manner, we are able to predict with high accuracy the locations of critical transitions in parameter space. (C) 2011 Elsevier B.V. All rights reserved
A Nonlinear Delay Model for Metabolic Oscillations in Yeast Cells
We introduce two time-delay models of metabolic oscillations in yeast cells.
Our model tests a hypothesis that the oscillations occur as multiple pathways
share a limited resource which we equate to the number of available ribosomes.
We initially explore a single-protein model with a constraint equation
governing the total resource available to the cell. The model is then extended
to include three proteins that share a resource pool. Three approaches are
considered at constant delay to numerically detect oscillations. First, we use
a spectral element method to approximate the system as a discrete map and
evaluate the stability of the linearized system about its equilibria by
examining its eigenvalues. For the second method, we plot amplitudes of the
simulation trajectories in 2D projections of the parameter space. We use a
history function that is consistent with published experimental results to
obtain metabolic oscillations. Finally, the spectral element method is used to
convert the system to a boundary value problem whose solutions correspond to
approximate periodic solutions of the system. Our results show that certain
combinations of total resource available and the time delay, lead to
oscillations. We observe that an oscillation region in the parameter space is
between regions admitting steady states that correspond to zero and constant
production. Similar behavior is found with the three-protein model where all
proteins require the same production time. However, a shift in the protein
production rates peaks occurs for low available resource suggesting that our
model captures the shared resource pool dynamics.Comment: 22 pages, 14 figures. Added analytical characterization of the
trivial equilibrium point. For the three-protein model, we now only consider
results where the protein production times are equal and added corresponding
linearized stability diagrams for this system. We also changed the history
functions for both systems to be consistent with experimental results for
obtaining metabolic oscillation
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