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SIAM J. on Applied Dynamical Systems

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2012

Volume 11, Issue 1 (partial)


Breathers and Q-Breathers: Two Sides of the Same Coin

T. Penati and S. Paleari

SIAM J. Appl. Dyn. Syst. 11, pp. 1-30 (30 pages)

Online Publication Date: January 10, 2012

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We construct, and approximate from the continuum, two-parameter families of time periodic, small amplitude, localized solutions, for both the focusing and defocusing finite discrete nonlinear Schrödinger models, with Dirichlet boundary conditions. Within such families, depending on the parameters, both real space localization (breathers) and Fourier space localization (Q-breathers) are present. For the former type of solutions, convergence to the ground state of the focusing infinite chain is also proved; for the latter, a description of the localization properties is given, and some numerical results on the difference between the focusing and defocusing cases are explained. The proofs are based on continuation tools, ideas from the finite element methods, and techniques of convergence of variational problems.

An Algebraic Approach to Reverse Engineering Finite Dynamical Systems Arising from Biology

Alan Veliz-Cuba

SIAM J. Appl. Dyn. Syst. 11, pp. 31-48 (18 pages)

Online Publication Date: January 10, 2012

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Finite dynamical systems have been used successfully in modeling biological processes. When certain regulatory mechanisms of a biological system or a model are unknown it is important to be able to identify the best model with the available data. In this context, reverse engineering of finite dynamical systems from partial information is an important problem. While this problem has been studied in the past, there are currently no algorithms that can predict the signs of the interactions. In this paper we propose a framework and algorithms to reverse engineer the possible signed wiring diagrams of a finite dynamical system from data. The algorithm consists of encoding all possible wiring diagrams using ideals and algebraic sets and choosing those that are minimal using the primary decomposition and the irreducible components.

Noise-Induced Behaviors in Neural Mean Field Dynamics

Jonathan Touboul, Geoffroy Hermann, and Olivier Faugeras

SIAM J. Appl. Dyn. Syst. 11, pp. 49-81 (33 pages)

Online Publication Date: January 13, 2012

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The collective behavior of cortical neurons is strongly affected by the presence of noise at the level of individual cells. In order to study these phenomena in large-scale assemblies of neurons, we consider networks of firing-rate neurons with linear intrinsic dynamics and nonlinear coupling, belonging to a few types of cell populations and receiving noisy currents. As the number of neurons tends to infinity, asymptotic equations (mean field equations) are rigorously derived based on a probabilistic approach. These equations are implicit on the probability distribution of the solutions, which generally makes their direct analysis difficult. However, in our case, the solutions are Gaussian, and their moments satisfy a closed system of nonlinear ordinary differential equations (ODEs), which are much easier to study than the original stochastic network equations, and the statistics of the empirical process uniformly converge towards the solutions of these ODEs. Based on this description, we analytically and numerically study the influence of noise on the collective behaviors, and compare these asymptotic regimes to simulations of the network. We observe that the mean field equations provide an accurate description of the solutions of the network equations for network sizes as small as a few hundred neurons. In particular, we observe that the level of noise in the system qualitatively modifies its collective behavior, producing, for instance, synchronized oscillations of the whole network, desynchronization of oscillating regimes, and stabilization or destabilization of stationary solutions. These results shed new light on the role of noise in shaping collective dynamics of neurons, and give us clues for understanding similar phenomena observed in biological networks.

Efficient Automation of Index Pairs in Computational Conley Index Theory

Rafael Frongillo and Rodrigo Treviño

SIAM J. Appl. Dyn. Syst. 11, pp. 82-109 (28 pages)

Online Publication Date: January 24, 2012

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We present new methods of automating the construction of index pairs, essential ingredients of discrete Conley index theory. These new algorithms are further steps in the direction of automating computer-assisted proofs of semiconjugacies from a map on a manifold to a subshift of finite type. We apply these new algorithms to the standard map at different values of the perturbative parameter $\varepsilon$ and obtain rigorous lower bounds for its topological entropy for $\varepsilon\in[.7,2]$.

Switching in Mass Action Networks Based on Linear Inequalities

Carsten Conradi and Dietrich Flockerzi

SIAM J. Appl. Dyn. Syst. 11, pp. 110-134 (25 pages)

Online Publication Date: January 26, 2012

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Many biochemical processes can successfully be described by dynamical systems allowing some form of switching when, depending on their initial conditions, solutions of the dynamical system end up in different regions of state space (associated with different biochemical functions). Switching is often realized by a bistable system (i.e., a dynamical system allowing two stable steady state solutions) and, in the majority of cases, bistability is established numerically. In our view, this approach is too restrictive. On the one hand, due to predominant parameter uncertainty, numerical methods are generally difficult to apply to realistic models originating in systems biology. On the other hand, switching already arises with the occurrence of a saddle-type steady state (characterized by a Jacobian where exactly one eigenvalue is positive and the remaining eigenvalues have negative real part). Consequently we derive conditions based on linear inequalities that allow the analytic computation of states and parameters where the Jacobian derived from a mass action network has a defective zero eigenvalue so that—under certain genericity conditions—a saddle-node bifurcation occurs. Our conditions are applicable to general mass action networks involving at least one conservation relation; however, they are only sufficient (as infeasibility of linear inequalities does not exclude defective zero eigenvalues).

Canard-Like Explosion of Limit Cycles in Two-Dimensional Piecewise-Linear Models of FitzHugh–Nagumo Type

Horacio G. Rotstein, Stephen Coombes, and Ana Maria Gheorghe

SIAM J. Appl. Dyn. Syst. 11, pp. 135-180 (46 pages)

Online Publication Date: January 26, 2012

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We investigate the mechanism of abrupt transition between small- and large amplitude oscillations in fast-slow piecewise-linear (PWL) models of FitzHugh–Nagumo (FHN) type. In the context of neuroscience, these oscillatory regimes correspond to subthreshold oscillations and action potentials (spikes), respectively. The minimal model that shows such phenomena has a cubic-like nullcline (for the fast equation) with two or more linear pieces in the middle branch and one piece in the left and right branches. Simpler models with only one linear piece in the middle branch or a discontinuity between the left and right branches (McKean model) show a single oscillatory mode. As the number of linear pieces increases, PWL models of FHN type approach smooth FHN-type models. For the minimal model we investigate the bifurcation structure; we describe the mechanism that leads to the abrupt, canard-like transition between subthreshold oscillations and spikes; and we provide an analytical way of predicting the amplitude regime of a given limit cycle trajectory which includes the approximation of the canard critical control parameter. We extend our results to models with a larger number of linear pieces. Our results for PWL-FHN-type models are consistent with similar results for smooth FHN-type models. In addition, we develop tools that are amenable for the investigation of a variety of related, and more complex, problems including forced, stochastic, and coupled oscillators.

Canonical Discontinuous Planar Piecewise Linear Systems

Emilio Freire, Enrique Ponce, and Francisco Torres

SIAM J. Appl. Dyn. Syst. 11, pp. 181-211 (31 pages)

Online Publication Date: January 31, 2012

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The family of Filippov systems constituted by planar discontinuous piecewise linear systems with two half-plane linearity zones is considered. Under generic conditions that amount to the boundedness of the sliding set, some changes of variables and parameters are used to obtain a Liénard-like canonical form with seven parameters. This canonical form is topologically equivalent to the original system if one restricts one's attention to orbits with no points in the sliding set. Under the assumption of focus-focus dynamics, a reduced canonical form with only five parameters is obtained. For the case without equilibria in both open half-planes we describe the qualitatively different phase portraits that can occur in the parameter space and the bifurcations connecting them. In particular, we show the possible existence of two limit cycles surrounding the sliding set. Such limit cycles bifurcate at certain parameter curves, organized around different codimension-two Hopf bifurcation points. The proposed canonical form will be a useful tool in the systematic study of planar discontinuous piecewise linear systems, in which this paper is a first step.

A Fast-Slow Analysis of the Dynamics of REM Sleep

Cecilia G. Diniz Behn and Victoria Booth

SIAM J. Appl. Dyn. Syst. 11, pp. 212-242 (31 pages)

Online Publication Date: January 31, 2012

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Waking and sleep states are regulated by the coordinated activity of a number of neuronal populations in the brainstem and hypothalamus whose synaptic interactions compose a sleep-wake regulatory network. Physiologically based mathematical models of the sleep-wake regulatory network contain mechanisms operating on multiple time scales including relatively fast synaptic-based interactions between neuronal populations, and much slower homeostatic and circadian processes that modulate sleep-wake temporal patterning. In this study, we exploit the naturally arising slow time scale of the homeostatic sleep drive in a reduced sleep-wake regulatory network model to utilize fast-slow analysis to investigate the dynamics of rapid eye movement (REM) sleep regulation. The network model consists of a reduced number of wake-, non-REM (NREM) sleep-, and REM sleep-promoting neuronal populations with synaptic interactions reflecting the mutually inhibitory flip-flop conceptual model for sleep-wake regulation and the reciprocal interaction model for REM sleep regulation. Network dynamics regularly alternate between wake and sleep states as governed by the slow homeostatic sleep drive. By varying a parameter associated with the activation of the REM-promoting population, we cause REM dynamics during sleep episodes to vary from suppression to single activations to regular REM-NREM cycling, corresponding to changes in REM patterning induced by circadian modulation and observed in different mammalian species. We also utilize fast-slow analysis to explain complex effects on sleep-wake patterning of simulated experiments in which agonists and antagonists of different neurotransmitters are microinjected into specific neuronal populations participating in the sleep-wake regulatory network.

Selection of Ground States in the Zero Temperature Limit for a One-Parameter Family of Potentials

A. T. Baraviera, R. Leplaideur, and A. O. Lopes

SIAM J. Appl. Dyn. Syst. 11, pp. 243-260 (18 pages)

Online Publication Date: February 02, 2012

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For the subshift of finite type $\Sigma=\{0,1,2\}^{\mathbb{N}}$ we study the convergence and the selection at temperature zero of the Gibbs measure associated to a non–locally constant Hölder potential which admits exactly two maximizing ergodic measures. These measures are Dirac measures at two different fixed points, and the potential is flatter at one of these two fixed points. We prove that there always is convergence but not necessarily to the Dirac measure at the point where the potential is the flattest. This is contrary to what was expected in light of the analogous problem in Aubry-Mather theory [N. Anantharaman et al., Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), pp. 513–528]. This is also contrary to the finite range case where the equilibrium state converges to the equi-barycenter of the two Dirac measures. Moreover, we emphasize the unexpected behavior of the Gibbs measure: the eigenmeasure selects one Dirac measure (at the point where the potential is the flattest), and the eigenfunction selects the other one (at the point where the potential is the sharpest).
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