826 research outputs found
Functional optimization of the arterial network
We build an evolutionary scenario that explains how some crucial
physiological constraints in the arterial network of mammals - i.e. hematocrit,
vessels diameters and arterial pressure drops - could have been selected by
evolution. We propose that the arterial network evolved while being constrained
by its function as an organ. To support this hypothesis, we focus our study on
one of the main function of blood network: oxygen supply to the organs. We
consider an idealized organ with a given oxygen need and we optimize blood
network geometry and hematocrit with the constraint that it must fulfill the
organ oxygen need. Our model accounts for the non-Newtonian behavior of blood,
its maintenance cost and F\aa hr\ae us effects (decrease in average
concentration of red blood cells as the vessel diameters decrease). We show
that the mean shear rates (relative velocities of fluid layers) in the tree
vessels follow a scaling law related to the multi-scale property of the tree
network, and we show that this scaling law drives the behavior of the optimal
hematocrit in the tree. We apply our scenario to physiological data and reach
results fully compatible with the physiology: we found an optimal hematocrit of
0.43 and an optimal ratio for diameter decrease of about 0.79. Moreover our
results show that pressure drops in the arterial network should be regulated in
order for oxygen supply to remain optimal, suggesting that the amplitude of the
arterial pressure drop may have co-evolved with oxygen needs.Comment: Shorter version, misspelling correctio
Spectral identification of networks with inputs
We consider a network of interconnected dynamical systems. Spectral network
identification consists in recovering the eigenvalues of the network Laplacian
from the measurements of a very limited number (possibly one) of signals. These
eigenvalues allow to deduce some global properties of the network, such as
bounds on the node degree.
Having recently introduced this approach for autonomous networks of nonlinear
systems, we extend it here to treat networked systems with external inputs on
the nodes, in the case of linear dynamics. This is more natural in several
applications, and removes the need to sometimes use several independent
trajectories. We illustrate our framework with several examples, where we
estimate the mean, minimum, and maximum node degree in the network. Inferring
some information on the leading Laplacian eigenvectors, we also use our
framework in the context of network clustering.Comment: 8 page
Spectral identification of networks using sparse measurements
We propose a new method to recover global information about a network of
interconnected dynamical systems based on observations made at a small number
(possibly one) of its nodes. In contrast to classical identification of full
graph topology, we focus on the identification of the spectral graph-theoretic
properties of the network, a framework that we call spectral network
identification.
The main theoretical results connect the spectral properties of the network
to the spectral properties of the dynamics, which are well-defined in the
context of the so-called Koopman operator and can be extracted from data
through the Dynamic Mode Decomposition algorithm. These results are obtained
for networks of diffusively-coupled units that admit a stable equilibrium
state. For large networks, a statistical approach is considered, which focuses
on spectral moments of the network and is well-suited to the case of
heterogeneous populations.
Our framework provides efficient numerical methods to infer global
information on the network from sparse local measurements at a few nodes.
Numerical simulations show for instance the possibility of detecting the mean
number of connections or the addition of a new vertex using measurements made
at one single node, that need not be representative of the other nodes'
properties.Comment: 3
Extreme phase sensitivity in systems with fractal isochrons
Sensitivity to initial conditions is usually associated with chaotic dynamics
and strange attractors. However, even systems with (quasi)periodic dynamics can
exhibit it. In this context we report on the fractal properties of the
isochrons of some continuous-time asymptotically periodic systems. We define a
global measure of phase sensitivity that we call the phase sensitivity
coefficient and show that it is an invariant of the system related to the
capacity dimension of the isochrons. Similar results are also obtained with
discrete-time systems. As an illustration of the framework, we compute the
phase sensitivity coefficient for popular models of bursting neurons,
suggesting that some elliptic bursting neurons are characterized by isochrons
of high fractal dimensions and exhibit a very sensitive (unreliable) phase
response.Comment: 32 page
Global analysis of a continuum model for monotone pulse-coupled oscillators
We consider a continuum of phase oscillators on the circle interacting
through an impulsive instantaneous coupling. In contrast with previous studies
on related pulse-coupled models, the stability results obtained in the
continuum limit are global. For the nonlinear transport equation governing the
evolution of the oscillators, we propose (under technical assumptions) a global
Lyapunov function which is induced by a total variation distance between
quantile densities. The monotone time evolution of the Lyapunov function
completely characterizes the dichotomic behavior of the oscillators: either the
oscillators converge in finite time to a synchronous state or they
asymptotically converge to an asynchronous state uniformly spread on the
circle. The results of the present paper apply to popular phase oscillators
models (e.g. the well-known leaky integrate-and-fire model) and draw a strong
parallel between the analysis of finite and infinite populations. In addition,
they provide a novel approach for the (global) analysis of pulse-coupled
oscillators.Comment: 33 page
Linear identification of nonlinear systems: A lifting technique based on the Koopman operator
We exploit the key idea that nonlinear system identification is equivalent to
linear identification of the socalled Koopman operator. Instead of considering
nonlinear system identification in the state space, we obtain a novel linear
identification technique by recasting the problem in the infinite-dimensional
space of observables. This technique can be described in two main steps. In the
first step, similar to the socalled Extended Dynamic Mode Decomposition
algorithm, the data are lifted to the infinite-dimensional space and used for
linear identification of the Koopman operator. In the second step, the obtained
Koopman operator is "projected back" to the finite-dimensional state space, and
identified to the nonlinear vector field through a linear least squares
problem. The proposed technique is efficient to recover (polynomial) vector
fields of different classes of systems, including unstable, chaotic, and open
systems. In addition, it is robust to noise, well-suited to model low sampling
rate datasets, and able to infer network topology and dynamics.Comment: 6 page
Geometric Properties of Isostables and Basins of Attraction of Monotone Systems
In this paper, we study geometric properties of basins of attraction of
monotone systems. Our results are based on a combination of monotone systems
theory and spectral operator theory. We exploit the framework of the Koopman
operator, which provides a linear infinite-dimensional description of nonlinear
dynamical systems and spectral operator-theoretic notions such as eigenvalues
and eigenfunctions. The sublevel sets of the dominant eigenfunction form a
family of nested forward-invariant sets and the basin of attraction is the
largest of these sets. The boundaries of these sets, called isostables, allow
studying temporal properties of the system. Our first observation is that the
dominant eigenfunction is increasing in every variable in the case of monotone
systems. This is a strong geometric property which simplifies the computation
of isostables. We also show how variations in basins of attraction can be
bounded under parametric uncertainty in the vector field of monotone systems.
Finally, we study the properties of the parameter set for which a monotone
system is multistable. Our results are illustrated on several systems of two to
four dimensions.Comment: 12 pages, to appear in IEEE Transaction on Automatic Contro
Global computation of phase-amplitude reduction for limit-cycle dynamics
Recent years have witnessed increasing interest to phase-amplitude reduction
of limit-cycle dynamics. Adding an amplitude coordinate to the phase coordinate
allows to take into account the dynamics transversal to the limit cycle and
thereby overcomes the main limitations of classic phase reduction (strong
convergence to the limit cycle and weak inputs). While previous studies mostly
focus on local quantities such as infinitesimal responses, a major and limiting
challenge of phase-amplitude reduction is to compute amplitude coordinates
globally, in the basin of attraction of the limit cycle.
In this paper, we propose a method to compute the full set of phase-amplitude
coordinates in the large. Our method is based on the so-called Koopman
(composition) operator and aims at computing the eigenfunctions of the operator
through Laplace averages (in combination with the harmonic balance method).
This yields a forward integration method that is not limited to two-dimensional
systems. We illustrate the method by computing the so-called isostables of
limit cycles in two, three, and four-dimensional state spaces, as well as their
responses to strong external inputs.Comment: 26 page
Operator-Theoretic Characterization of Eventually Monotone Systems
Monotone systems are dynamical systems whose solutions preserve a partial
order in the initial condition for all positive times. It stands to reason that
some systems may preserve a partial order only after some initial transient.
These systems are usually called eventually monotone. While monotone systems
have a characterization in terms of their vector fields (i.e. Kamke-Muller
condition), eventually monotone systems have not been characterized in such an
explicit manner. In order to provide a characterization, we drew inspiration
from the results for linear systems, where eventually monotone (positive)
systems are studied using the spectral properties of the system (i.e.
Perron-Frobenius property). In the case of nonlinear systems, this spectral
characterization is not straightforward, a fact that explains why the class of
eventually monotone systems has received little attention to date. In this
paper, we show that a spectral characterization of nonlinear eventually
monotone systems can be obtained through the Koopman operator framework. We
consider a number of biologically inspired examples to illustrate the potential
applicability of eventual monotonicity.Comment: 13 page
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