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

Sensitivity analysis of oscillator models in the space of phase-response curves: Oscillators as open systems

Oscillator models are central to the study of system properties such as entrainment or synchronization. Due to their nonlinear nature, few system-theoretic tools exist to analyze those models. The paper develops a sensitivity analysis for phase-response curves, a fundamental one-dimensional phase reduction of oscillator models. The proposed theoretical and numerical analysis tools are illustrated on several system-theoretic questions and models arising in the biology of cellular rhythms

Global Isochrons and Phase Sensitivity of Bursting Neurons

Phase sensitivity analysis is a powerful method for studying (asymptotically periodic) bursting neuron models. One popular way of capturing phase sensitivity is through the computation of isochrons---subsets of the state space that each converge to the same trajectory on the limit cycle. However, the computation of isochrons is notoriously difficult, especially for bursting neuron models. In [W. E. Sherwood and J. Guckenheimer, SIAM J. Appl. Dyn. Syst., 9 (2010), pp. 659--703], the phase sensitivity of the bursting Hindmarsh--Rose model is studied through the use of singular perturbation theory: cross sections of the isochrons of the full system are approximated by those of fast subsystems. In this paper, we complement the previous study, providing a detailed phase sensitivity analysis of the full (three-dimensional) system, including computations of the full (two-dimensional) isochrons. To our knowledge, this is the first such computation for a bursting neuron model. This was made possible thanks to the numerical method recently proposed in [A. Mauroy and I. Mezić, Chaos, 22 (2012), 033112]---relying on the spectral properties of the so-called Koopman operator---which is complemented with the use of adaptive quadtree and octree grids. The main result of the paper is to highlight the existence of a region of high phase sensitivity called the almost phaseless set and to completely characterize its geometry. In particular, our study reveals the existence of a subset of the almost phaseless set that is not predicted by singular perturbation theory (i.e., by the isochrons of fast subsystems). We also discuss how the almost phaseless set is related to empirically observed phenomena such as addition/deletion of spikes and to extrema of the phase response of the system. Finally, through the same numerical method, we show that an elliptic bursting model is characterized by a very high phase sensitivity and other remarkable properties

Interactions of multiple rhythms in a biophysical network of neurons

Neural oscillations, including rhythms in the beta1 band (12–20 Hz), are important in various cognitive functions. Often neural networks receive rhythmic input at frequencies different from their natural frequency, but very little is known about how such input affects the network’s behavior. We use a simplified, yet biophysical, model of a beta1 rhythm that occurs in the parietal cortex, in order to study its response to oscillatory inputs. We demonstrate that a cell has the ability to respond at the same time to two periodic stimuli of unrelated frequencies, firing in phase with one, but with a mean firing rate equal to that of the other. We show that this is a very general phenomenon, independent of the model used. We next show numerically that the behavior of a different cell, which is modeled as a high-dimensional dynamical system, can be described in a surprisingly simple way, owing to a reset that occurs in the state space when the cell fires. The interaction of the two cells leads to novel combinations of properties for neural dynamics, such as mode-locking to an input without phase-locking to it.Published versio

Optimal Control and Synchronization of Dynamic Ensemble Systems

Ensemble control involves the manipulation of an uncountably infinite collection of structurally identical or similar dynamical systems, which are indexed by a parameter set, by applying a common control without using feedback. This subject is motivated by compelling problems in quantum control, sensorless robotic manipulation, and neural engineering, which involve ensembles of linear, bilinear, or nonlinear oscillating systems, for which analytical control laws are infeasible or absent. The focus of this dissertation is on novel analytical paradigms and constructive control design methods for practical ensemble control problems. The first result is a computational method %based on the singular value decomposition (SVD) for the synthesis of minimum-norm ensemble controls for time-varying linear systems. This method is extended to iterative techniques to accommodate bounds on the control amplitude, and to synthesize ensemble controls for bilinear systems. Example ensemble systems include harmonic oscillators, quantum transport, and quantum spin transfers on the Bloch system. To move towards the control of complex ensembles of nonlinear oscillators, which occur in neuroscience, circadian biology, electrochemistry, and many other fields, ideas from synchronization engineering are incorporated. The focus is placed on the phenomenon of entrainment, which refers to the dynamic synchronization of an oscillating system to a periodic input. Phase coordinate transformation, formal averaging, and the calculus of variations are used to derive minimum energy and minimum mean time controls that entrain ensembles of non-interacting oscillators to a harmonic or subharmonic target frequency. In addition, a novel technique for taking advantage of nonlinearity and heterogeneity to establish desired dynamical structures in collections of inhomogeneous rhythmic systems is derived

Synchronization properties and functional implications of parietal beta1 rhythm

Neural oscillations, including rhythms in the beta1 band (12-20 Hz), are important in various cognitive functions. Often brain networks receive rhythmic input at frequencies different than their natural frequency, so understanding how neural networks process rhythmic input is important for understanding their function in the brain. In the current thesis we study a beta1 rhythm that appears in the parietal cortex, focusing on the way it interacts with other incoming rhythms, and the implications of this interaction for cognition. The main part of the thesis consists of two stand-alone chapters, both using as a basis a biophysical neural network model that has been previously proposed to model the parietal beta1 rhythm and validated with in vitro experiments. In the first chapter we use a reduced version of this model, in order to study its dynamics, applying both analytic and numerical methods from dynamical systems. We show that a cell can respond at the same time to two periodic stimuli of unrelated frequencies, firing in phase with one, but with a mean firing rate equal to the other, a consequence of general properties of the dynamics of the network. We next show numerically that the behavior of a different cell, which is modeled as a high-dimensional dynamical system, can be described in a surprisingly simple way, owing to a reset that occurs in the state space when the cell fires. The interaction of the two cells leads to novel combinations of properties for neural dynamics, such as mode-locking to an input without phase-locking to it. In the second chapter, we study the ability of the beta1 model to support memory functions, in particular working memory. Working memory is a highly distributed component of the brain's memory systems, partially based in the parietal cortex. We show numerically that the parietal beta1 rhythm can provide an anatomical substrate for an episodic buffer of working memory. Specifically, it can support flexible and updatable representations of sensory input which are sensitive to distractors, allow for a read-out mechanism, and can be modulated or terminated by executive input

Study of invariant manifolds in two different problems : the Hopf-zero singularity and neural synchrony

The main object of study of this thesis are invariant manifolds in the field of dynamical systems. We deal with two different and independent topics, namely, the study of exponentially small splitting of invariant manifolds in analytic unfoldings of the Hopf-zero singularity (in Part I) and the applications of dynamical systems in problems inspired by neuroscience (in Part II). In general, this thesis studies both theoretical and applied problems in dynamical systems, using analytical as well as computational tools. In Part I, we consider a certain class of generic unfoldings of the so-called Hopf-zero singularity. One can see that the truncation of the normal form at any finite order of such unfoldings possesses two saddle-focus critical points and, when the parameters lie on a certain curve, they are connected by a one- and a two-dimensional heteroclinic manifolds. However, considering the whole vector field, one expects these heteroclinic connections to be destroyed. This fact can lead to the birth of a homoclinic connection to one of the critical points, producing thus a Shilnikov bifurcation. For the case of $C^\infty$ unfoldings, this was proved by Broer and Vegter during the 80's, but for analytic unfoldings it has remained an open problem. Recently, under some assumptions on the size of the splitting of the heteroclinic connections, Dumortier, Ibáñez, Kokubu and Simó proved the existence of Shilnikov bifurcations in the analytic case. Our study concerns the splitting of the one- and two-dimensional heteroclinic connections. These cannot be detected in the truncation of the normal form at any order, and hence they are exponentially small with respect to one of the perturbation parameters. We give asymptotic formulas of these splittings and, in particular, we prove that under generic conditions the main assumptions made by Dumortier, Ibáñez, Kokubu and Simó hold. In Part II, we deal with tools to provide an accurate prediction of phase variations in an oscillator subject to external stimuli. We construct a method based on the concepts of isochrons, Phase Response Functions (PRF) and Amplitude Response Functions (ARF). In particular, the method can be applied to neurons in a state of repetitive firing. In the special case of a pulse-train periodic stimulus, the application of this theoretical frame leads to a 2D map, one variable controlling phase jumps and the other controlling amplitude jumps. We compare these maps to the classical 1D maps obtained via Phase Response Curves (PRC) and we identify circumstances in which the 2D maps give a more accurate prediction of the synchronization. Moreover, we implement some numerical methods to compute the invariant curves of the 2D maps as well as the dynamics inside these curves. Finally, we compute Arnold tongues of these maps, which allow to determine regions in the parameter space for which the neuron is synchronized to the external input.El principal objecte d'estudi d'aquesta tesi són les varietats invariants en el camp dels sistemes dinàmics. Considerem dos temes diferents i independents, concretament l'estudi de l'escissió exponencialment petita de varietats invariants en desplegaments analí­tics de la singularitat Hopf-zero (a la Part I), i les aplicacions dels sistemes dinàmics en problemes inspirats per la neurociència (a la Part II). A la Part I, considerem una classe de desplegaments genèrics de l'anomenada singularitat Hopf-zero. Es pot veure que el truncament de la forma normal a qualsevol ordre finit d'aquests desplegaments té dos punts crítics de tipus sella-focus i, quan els paràmetres estan sobre una certa corba, estan connectats per dues varietats heteroclí­niques, una d'unidimensional i una de bidimensional. No obstant, si es considera tot el camp vectorial, s'espera que aquestes connexions heteroclí­niques desapareguin. Això pot causar el naixement d'una òrbita homoclí­nica en un dels dos punts crí­tics, produint així­ el que es coneix com una bifurcació de Shilnikov. En el cas de desplegaments $C^\infty$, això va ser provat per Broer i Vegter durant els anys 80, però el cas de desplegaments analí­tics ha quedat obert. Recentment, sota certes hipòtesis sobre la mida de l'escissió de les connexions heteroclí­niques, Dumortier, Ibáñez, Kokubu i Simó han provat l'existència de bifurcacions de Shilnikov en el cas analític. El nostre estudi consisteix en el càlcul del trencament de les connexions heteroclí­niques. Aquests trencaments no es poden detectar en la forma normal a cap ordre i, per tant, són exponencialment petits en un dels paràmetres de pertorbació. Donem fórmules asimptòtiques d'aquests trencaments i, en particular, provem que sota certes condicions genèriques les principals hipòtesis fetes per Dumortier, Ibàñez, Kokubu i Simó són vàlides. A la Part II, considerem eines per proporcionar una predicció acurada de la variació de fase en un oscil·lador subjecte a estímuls externs. Construïm un mètode basat en els conceptes d'isòcrones, Funcions de Resposta de Fase (PRF, per les seves inicials en anglès) i Funcions de Resposta d'Amplitud (ARF). En particular, el mètode es pot aplicar a neurones en un estat de dispar repetitiu. En el cas especial d'un tren de pulsos periòdic, l'aplicació d'aquest mètode teòric dóna lloc a una aplicació 2D, on una variable controla els canvis de la fase i l'altra els canvis en l'amplitud. Comparem aquestes aplicacions amb les aplicacions 1D clàssiques obtingudes a través de la Corbes de Resposta de Fase (PRC) i identifiquem circumstàncies en què les aplicacions 2D donen una millora substancial de la predicció de sincronització. A més, implementem alguns mètodes numèrics per calcular les corbes invariants de les aplicacions 2D així­ com la dinàmica dins aquestes corbes. Finalment, calculem les llengües d'Arnold corresponents a aquestes aplicacions, que permeten determinar regions en l'espai de paràmetres per a les quals la neurona es sincronitza amb l'estí­mul extern