The uncoupling limit of identical Hopf bifurcations with an application to perceptual bistability

We study the dynamics arising when two identical oscillators are coupled near a Hopf bifurcation where we assume a parameter $\epsilon$ uncouples the system at $\epsilon=0$. Using a normal form for $N=2$ identical systems undergoing Hopf bifurcation, we explore the dynamical properties. Matching the normal form coefficients to a coupled Wilson-Cowan oscillator network gives an understanding of different types of behaviour that arise in a model of perceptual bistability. Notably, we find bistability between in-phase and anti-phase solutions that demonstrates the feasibility for synchronisation to act as the mechanism by which periodic inputs can be segregated (rather than via strong inhibitory coupling, as in existing models). 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

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

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

Bifurcations in a forced Wilson-Cowan neuron pair

We investigate bifurcations of periodic solutions observed in the forced Wilson-Cowan neuron pair by both the brute-force computation and the shooting method. By superimposing the results given by both methods, a detailed topological classification of periodic solutions is achieved that includes tori and chaos attractors in the parameter space is achieved. We thoroughly explore the parameter space composed of threshold values, amplitude, and angular velocity of an external forcing term. Many bifurcation curves that are invisible when using brute-force method are solved by the shooting method. We find out a typical bifurcation structure including Arnold tongue in the angular velocity and the amplitude of the external force parameter plane, and confirm its fractal structure. In addition, the emergence of periodic bursting responses depending on these patterns is explained

On the role of oscillatory dynamics in neural communication

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

Multiscale Computations on Neural Networks: From the Individual Neuron Interactions to the Macroscopic-Level Analysis

We show how the Equation-Free approach for multi-scale computations can be exploited to systematically study the dynamics of neural interactions on a random regular connected graph under a pairwise representation perspective. Using an individual-based microscopic simulator as a black box coarse-grained timestepper and with the aid of simulated annealing we compute the coarse-grained equilibrium bifurcation diagram and analyze the stability of the stationary states sidestepping the necessity of obtaining explicit closures at the macroscopic level. We also exploit the scheme to perform a rare-events analysis by estimating an effective Fokker-Planck describing the evolving probability density function of the corresponding coarse-grained observables

A Model for the Origin and Properties of Flicker-Induced Geometric Phosphenes

We present a model for flicker phosphenes, the spontaneous appearance of geometric patterns in the visual field when a subject is exposed to diffuse flickering light. We suggest that the phenomenon results from interaction of cortical lateral inhibition with resonant periodic stimuli. We find that the best temporal frequency for eliciting phosphenes is a multiple of intrinsic (damped) oscillatory rhythms in the cortex. We show how both the quantitative and qualitative aspects of the patterns change with frequency of stimulation and provide an explanation for these differences. We use Floquet theory combined with the theory of pattern formation to derive the parameter regimes where the phosphenes occur. We use symmetric bifurcation theory to show why low frequency flicker should produce hexagonal patterns while high frequency produces pinwheels, targets, and spirals

Noise-induced synchronization and anti-resonance in excitable systems; Implications for information processing in Parkinson's Disease and Deep Brain Stimulation

We study the statistical physics of a surprising phenomenon arising in large networks of excitable elements in response to noise: while at low noise, solutions remain in the vicinity of the resting state and large-noise solutions show asynchronous activity, the network displays orderly, perfectly synchronized periodic responses at intermediate level of noise. We show that this phenomenon is fundamentally stochastic and collective in nature. Indeed, for noise and coupling within specific ranges, an asymmetry in the transition rates between a resting and an excited regime progressively builds up, leading to an increase in the fraction of excited neurons eventually triggering a chain reaction associated with a macroscopic synchronized excursion and a collective return to rest where this process starts afresh, thus yielding the observed periodic synchronized oscillations. We further uncover a novel anti-resonance phenomenon: noise-induced synchronized oscillations disappear when the system is driven by periodic stimulation with frequency within a specific range. In that anti-resonance regime, the system is optimal for measures of information capacity. This observation provides a new hypothesis accounting for the efficiency of Deep Brain Stimulation therapies in Parkinson's disease, a neurodegenerative disease characterized by an increased synchronization of brain motor circuits. We further discuss the universality of these phenomena in the class of stochastic networks of excitable elements with confining coupling, and illustrate this universality by analyzing various classical models of neuronal networks. Altogether, these results uncover some universal mechanisms supporting a regularizing impact of noise in excitable systems, reveal a novel anti-resonance phenomenon in these systems, and propose a new hypothesis for the efficiency of high-frequency stimulation in Parkinson's disease