365 research outputs found
Reduction of dimension for nonlinear dynamical systems
We consider reduction of dimension for nonlinear dynamical systems. We
demonstrate that in some cases, one can reduce a nonlinear system of equations
into a single equation for one of the state variables, and this can be useful
for computing the solution when using a variety of analytical approaches. In
the case where this reduction is possible, we employ differential elimination
to obtain the reduced system. While analytical, the approach is algorithmic,
and is implemented in symbolic software such as {\sc MAPLE} or {\sc SageMath}.
In other cases, the reduction cannot be performed strictly in terms of
differential operators, and one obtains integro-differential operators, which
may still be useful. In either case, one can use the reduced equation to both
approximate solutions for the state variables and perform chaos diagnostics
more efficiently than could be done for the original higher-dimensional system,
as well as to construct Lyapunov functions which help in the large-time study
of the state variables. A number of chaotic and hyperchaotic dynamical systems
are used as examples in order to motivate the approach.Comment: 16 pages, no figure
Persistent homology of time-dependent functional networks constructed from coupled time series
We use topological data analysis to study "functional networks" that we
construct from time-series data from both experimental and synthetic sources.
We use persistent homology with a weight rank clique filtration to gain
insights into these functional networks, and we use persistence landscapes to
interpret our results. Our first example uses time-series output from networks
of coupled Kuramoto oscillators. Our second example consists of biological data
in the form of functional magnetic resonance imaging (fMRI) data that was
acquired from human subjects during a simple motor-learning task in which
subjects were monitored on three days in a five-day period. With these
examples, we demonstrate that (1) using persistent homology to study functional
networks provides fascinating insights into their properties and (2) the
position of the features in a filtration can sometimes play a more vital role
than persistence in the interpretation of topological features, even though
conventionally the latter is used to distinguish between signal and noise. We
find that persistent homology can detect differences in synchronization
patterns in our data sets over time, giving insight both on changes in
community structure in the networks and on increased synchronization between
brain regions that form loops in a functional network during motor learning.
For the motor-learning data, persistence landscapes also reveal that on average
the majority of changes in the network loops take place on the second of the
three days of the learning process.Comment: 17 pages (+3 pages in Supplementary Information), 11 figures in many
text (many with multiple parts) + others in SI, submitte
Graph-Facilitated Resonant Mode Counting in Stochastic Interaction Networks
Oscillations in a stochastic dynamical system, whose deterministic
counterpart has a stable steady state, are a widely reported phenomenon.
Traditional methods of finding parameter regimes for stochastically-driven
resonances are, however, cumbersome for any but the smallest networks. In this
letter we show by example of the Brusselator how to use real root counting
algorithms and graph theoretic tools to efficiently determine the number of
resonant modes and parameter ranges for stochastic oscillations. We argue that
stochastic resonance is a network property by showing that resonant modes only
depend on the squared Jacobian matrix , unlike deterministic oscillations
which are determined by . By using graph theoretic tools, analysis of
stochastic behaviour for larger networks is simplified and chemical reaction
networks with multiple resonant modes can be identified easily.Comment: 5 pages, 4 figure
Algebraic Systems Biology: A Case Study for the Wnt Pathway
Steady state analysis of dynamical systems for biological networks give rise
to algebraic varieties in high-dimensional spaces whose study is of interest in
their own right. We demonstrate this for the shuttle model of the Wnt signaling
pathway. Here the variety is described by a polynomial system in 19 unknowns
and 36 parameters. Current methods from computational algebraic geometry and
combinatorics are applied to analyze this model.Comment: 24 pages, 2 figure
Stratifying multiparameter persistent homology
A fundamental tool in topological data analysis is persistent homology, which
allows extraction of information from complex datasets in a robust way.
Persistent homology assigns a module over a principal ideal domain to a
one-parameter family of spaces obtained from the data. In applications data
often depend on several parameters, and in this case one is interested in
studying the persistent homology of a multiparameter family of spaces
associated to the data. While the theory of persistent homology for
one-parameter families is well-understood, the situation for multiparameter
families is more delicate. Following Carlsson and Zomorodian we recast the
problem in the setting of multigraded algebra, and we propose multigraded
Hilbert series, multigraded associated primes and local cohomology as
invariants for studying multiparameter persistent homology. Multigraded
associated primes provide a stratification of the region where a multigraded
module does not vanish, while multigraded Hilbert series and local cohomology
give a measure of the size of components of the module supported on different
strata. These invariants generalize in a suitable sense the invariant for the
one-parameter case.Comment: Minor improvements throughout. In particular: we extended the
introduction, added Table 1, which gives a dictionary between terms used in
PH and commutative algebra; we streamlined Section 3; we added Proposition
4.49 about the information captured by the cp-rank; we moved the code from
the appendix to github. Final version, to appear in SIAG
Joining and decomposing reaction networks
In systems and synthetic biology, much research has focused on the behavior
and design of single pathways, while, more recently, experimental efforts have
focused on how cross-talk (coupling two or more pathways) or inhibiting
molecular function (isolating one part of the pathway) affects systems-level
behavior. However, the theory for tackling these larger systems in general has
lagged behind. Here, we analyze how joining networks (e.g., cross-talk) or
decomposing networks (e.g., inhibition or knock-outs) affects three properties
that reaction networks may possess---identifiability (recoverability of
parameter values from data), steady-state invariants (relationships among
species concentrations at steady state, used in model selection), and
multistationarity (capacity for multiple steady states, which correspond to
multiple cell decisions). Specifically, we prove results that clarify, for a
network obtained by joining two smaller networks, how properties of the smaller
networks can be inferred from or can imply similar properties of the original
network. Our proofs use techniques from computational algebraic geometry,
including elimination theory and differential algebra.Comment: 44 pages; extensive revision in response to referee comment
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