160 research outputs found

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

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    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

    On the role of oscillatory dynamics in neural communication

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    In this Thesis we consider problems concerning brain oscillations generated across the interaction between excitatory (E) and inhibitory (I) cells. We explore how two neuronal groups with underlying oscillatory activity communicate much effectively when they are properly phase-locked as suggested by Communcation Through Coherence Theory. In Chapter 1 we introduce the Wilson-Cowan equations (WC), a mean field model describing the mean activity of a network of a single population of E cells and a single popultation of I cells and review the bifurcations that give rise to oscillatory dynamics. In Chapter 2 we study how the oscillations generated across the E-I interaction are affect by a periodic forcing. We take the WC equations in the oscillatory regime with an external time periodic perturbation. We consider the stroboscopic map for this system and compute the bifurcation diagram for its fixed and periodic points as the amplitude and the frequency of the perturbation are varied. From the bifurcation diagram, we can identify the phase-locked states as well as different areas involving bistablility between two invariant objects. Chapter 3 exploits recent techniques based on phase-amplitude variables to describe the phase dynamics of an oscillator under different perturbations. More precisely, the applications of the parameterization method to compute a change of variables that describes correctly the dynamics near a limit cycle in terms of the phase (a periodic variable) and the amplitude. The computational method uses the Floquet normal form to reduce the computational cost. This change provides two remarkable manifolds used in neuroscience: the sets of constant phase/amplitude (isochrons/isostables). Moreover, we compute the functions describing the phase and amplitude changes caused by a perturbation arriving at different phases of the cycle, known as Phase and Amplitude Response Curves, PRCs and ARCs, respectively. The computed parameterization provides also the extension of these curves outside of the limit cycle, defined as the Phase and Amplitude Response Functions, PRFs and ARFs, respectively. We compute these objects for limits cycles in systems with 2 and 3 dimensions. In Chapter 4 we apply the parameterization method to compute Phase Response Curves (PRCs) for a transient stimulus of arbitrary amplitude and duration. The underlying idea is to construct a particular periodic perturbation consisting of the repetition of the transient stimulus followed by a resting period when no perturbation acts. For this periodic system we consider the corresponding stroboscopic map and we prove that, under certain conditions, it has an invariant curve. We prove that this map has an invariant curve and we provide the relationship between the PRC and the internal dynamics of the curve. Moreover, we link the existence properties of this invariant curve as the amplitude of the perturbation is increased with changes in the PRC waveform and with the geometry of isochrons. Furthermore, we also provide algorithms to obtain numerically the PRC and the ARC. In Chapter 5 we study the dynamics arising when two identical oscillators are coupled near a Hopf bifurcation, where we assume the existence of a parameter uncoupling the system when it is equal to zero. Using a recently derived truncated normal form, we perform a theoretical dynamical analysis and study its bifurcations. Computing the normal form coefficients in the case of 2 coupled Wilson-Cowan oscillators gives an understanding of different types of behaviour that arise in this model of perceptual bistability. Notably, we find bistability between in-phase and anti-phase solutions. Using numerical continuation we confirm our theoretical analysis for small coupling strength and explore the bifurcation diagrams for large coupling strength, where the normal form approximation breaks down. We finally discuss the implications of this dynamical study in models of perceptual bistability.Aquesta Tesi estudia problemes relacionats amb les oscil·lacions de l'activitat cerebral. Explorem com dues poblacions neuronals en activitat oscil·latòria es comuniquen més efectivament quan estan lligades en fase, tal com suggereix la teoria de 'Comunicació a Través de la Coherència'. Al capítol 1 introduïm les equacions de Wilson-Cowan (WC), un model de camp mitjà que descriu l' activitat d'una xarxa de neurones excitatòries (E) i inhibitòries (I) i calculem les bifurcacions que generen cicles límit. Al capítol 2 estudiem com un cicle límit generat a través d'aquesta interacció E-I respon a un forçament periòdic. Considerem el model de WC en règim oscil·latori amb una pertorbació externa periòdica en el temps. Considerem el mapa estroboscòpic d'aquest sistema i calculem el diagrama de bifurcació dels seus punts fixos i òrbites periòdiques en funció de l'amplitud i la freqüència de la pertorbació. El diagrama de bifurcació ens permet identificar les àrees amb lligadura de fase, axí com diferents àrees on tenim coexistència de dos objectes invariants estables. Al capítol 3 utilitzem tècniques recents basades en les variables fase-amplitud per a descriure la dinàmica de fase d'un oscil·lador sota diferents pertorbacions. En particular, utilitzem el mètode de la parametrització per a calcular un canvi de variables que descriu correctament la dinàmica prop del cicle límit en termes de la fase (variable periòdica) i l'amplitud. Aquests càlculs estan basats en la forma normal de Floquet que en redueix el cost computacional. Aquest canvi de variables ens permet calcular dos varietats importants en neurociència: els conjunts de fase/amplitud constant (les isòcrones/isostables). A més a més, calculem les funcions que descriuen els canvis de fase i amplitud causats per una pertorbació que arriba a diferents fases del cicle, les Corbes de Resposta de Fase i Amplitud, (PRCs i ARCs), respectivament. El canvi de variables calculat proporciona també l'extensió d'aquestes corbes fora del cicle límit, definides com les Funcions de Resposta de Fase i Amplitud, (PRFs i ARFs). Calculem tots aquests objectes per a cicles límit en 2 i 3 dimensions. Al capítol 4 ens centrem en les aplicacions del mètode de la parametrització per calcular PRCs per a estímuls de duració i amplitud arbitraria. La idea bàsica del mètode és construir una pertorbació periòdica particular que consisteix en la repetició d'un estímul transitori seguit d'un període de relaxació en el qual no actua cap pertorbació. Per a aquest sistema periòdic considerem el seu corresponent mapa estroboscòpic i demostrem que sota certes condicions, té una corba invariant. Demostrem que aquesta aplicació té una corba invariant i donem la relació entre la PRC i la dinàmica interna d'aquesta corba. A més a més, relacionem les propietats d'existència d'aquesta corba quan l'amplitud de la pertorbació augmenta, amb els canvis a la PRC i a la geometria de les isòcrones. Finalment, presentem algoritmes per obtenir numèricament la PRC i la ARC. Al capítol 5 estudiem la dinàmica emergent quan s'acoblen dos oscil·ladors idèntics prop d'una bifurcació de Hopf, pels quals suposem l'existència d'un paràmetre que desacobla el sistema quan s'anul·la. Utilitzant una forma normal derivada recentment per a 2 sistemes idèntics prop d'una bifurcació de Hopf, fem una anàlisi teòrica i estudiem les seves bifurcacions. Identificant els coeficients de la forma normal per a un model de dos oscil·ladors de tipus WC acoblats, il·lustrem els resultats obtinguts en l'anàlisi teòrica en un model amb moltes aplicacions al camp de la percepció biestable. Un resultat important és la biestabilitat entre solucions en fase i en antifase. Utilitzant mètodes de continuacióPostprint (published version

    A geometric approach to phase response curves and its numerical computation through the parameterization method

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    The final publication is available at link.springer.comThe phase response curve (PRC) is a tool used in neuroscience that measures the phase shift experienced by an oscillator due to a perturbation applied at different phases of the limit cycle. In this paper, we present a new approach to PRCs based on the parameterization method. The underlying idea relies on the construction of a periodic system whose corresponding stroboscopic map has an invariant curve. We demonstrate the relationship between the internal dynamics of this invariant curve and the PRC, which yields a method to numerically compute the PRCs. Moreover, we link the existence properties of this invariant curve as the amplitude of the perturbation is increased with changes in the PRC waveform and with the geometry of isochrons. The invariant curve and its dynamics will be computed by means of the parameterization method consisting of solving an invariance equation. We show that the method to compute the PRC can be extended beyond the breakdown of the curve by means of introducing a modified invariance equation. The method also computes the amplitude response functions (ARCs) which provide information on the displacement away from the oscillator due to the effects of the perturbation. Finally, we apply the method to several classical models in neuroscience to illustrate how the results herein extend the framework of computation and interpretation of the PRC and ARC for perturbations of large amplitude and not necessarily pulsatile.Peer ReviewedPostprint (author's final draft

    Mathematical frameworks for oscillatory network dynamics in neuroscience

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    The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathemat- ical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical frame- work for further successful applications of mathematics to understanding network dynamics in neuroscience

    Entrainment and chaos in a pulse-driven Hodgkin-Huxley oscillator

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    The Hodgkin-Huxley model describes action potential generation in certain types of neurons and is a standard model for conductance-based, excitable cells. Following the early work of Winfree and Best, this paper explores the response of a spontaneously spiking Hodgkin-Huxley neuron model to a periodic pulsatile drive. The response as a function of drive period and amplitude is systematically characterized. A wide range of qualitatively distinct responses are found, including entrainment to the input pulse train and persistent chaos. These observations are consistent with a theory of kicked oscillators developed by Qiudong Wang and Lai-Sang Young. In addition to general features predicted by Wang-Young theory, it is found that most combinations of drive period and amplitude lead to entrainment instead of chaos. This preference for entrainment over chaos is explained by the structure of the Hodgkin-Huxley phase resetting curve.Comment: Minor revisions; modified Fig. 3; added reference
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