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

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

Ap weights for nondoubling measures in Rn and applications

We study an analogue of the classical theory of Ap(µ) weights in Rn without assuming that the underlying measure µ is doubling. Then, we obtain weighted norm inequalities for the (centered) Hardy-Littlewood maximal function and corresponding weighted estimates for nonclassical Calderón-Zygmund operators. We also consider commutators of those Calderón-Zygmund operators with bounded mean oscillation functions (BMO), extending the main result from R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611–635. Finally, we study self-improving properties of Poincaré-B.M.O. type inequalities within this context; more precisely, we show that if f is a locally integrable function satisfying 1 / µ(Q)R Q |f − fQ|dµ ≤ a(Q) for all cubes Q, then it is possible to deduce a higher Lp integrability result for f, assuming a certain simple geometric condition on the functional a.Consell Interdepartamental de Recerca i Innovació TecnològicaDirección General de Investigación Científica y TécnicaDirección General de Enseñanza Superior e Investigación Científic

New estimates for the maximal singular integral

In this paper we pursue the study of the problem of controlling the maximal singular integral $T^{*}f$ by the singular integral $Tf$. Here $T$ is a smooth homogeneous Calder\'on-Zygmund singular integral of convolution type. We consider two forms of control, namely, in the L^2(\Rn) norm and via pointwise estimates of $T^{*}f$ by $M(Tf)$ or $M^2(Tf)$, where $M$ is the Hardy-Littlewood maximal operator and $M^2=M \circ M$ its iteration. It is known that the parity of the kernel plays an essential role in this question. In a previous article we considered the case of even kernels and here we deal with the odd case. Along the way, the question of estimating composition operators of the type $T^\star \circ T$ arises. It turns out that, again, there is a remarkable difference between even and odd kernels. For even kernels we obtain, quite unexpectedly, weak $(1,1)$ estimates, which are no longer true for odd kernels. For odd kernels we obtain sharp weaker inequalities involving a weak $L^1$ estimate for functions in $L LogL$.Comment: v2: 56 pages, with small changes made after acceptance by International Math. Research Notice

Computation of invariant curves in the analysis of periodically forced neural oscillators

Background oscillations, reflecting the excitability of neurons, are ubiq- uitous in the brain. Some studies have conjectured that when spikes sent by one population reach the other population in the peaks of excitability, then informa- tion transmission between two oscillating neuronal groups is more effective. In this context, phase locking relationships between oscillating neuronal populations may have implications in neuronal communication as they assure synchronous activity between brain areas. To study this relationship, we consider a population rate model and perturb it with a time-dependent input. We use the stroboscopic map and apply powerful computational methods to compute the invariant objects and their bifur- cations as the perturbation parameters (frequency and amplitude) are varied. The analysis performed shows the relationship between the appearance of synchronous and asynchronous regimes and the invariant objects of the stroboscopic map.Peer ReviewedPostprint (author's final draft

New estimates for the maximal singular integral

In this paper we pursue the study of the problem of controlling the maximal singular integral T∗ f by the singular integral T f. Here T is a smooth homogeneous Calder´on-Zygmund singular integral of convolution type. We consider two forms of control, namely, in the L2 (Rn) norm and via pointwise estimates of T∗ f by M(T f) or M2 (T f) , where M is the Hardy-Littlewood maximal operator and M2 = M ◦ M its iteration. It is known that the parity of the kernel plays an essential role in this question. In a previous article we considered the case of even kernels and here we deal with the odd case. Along the way, the question of estimating composition operators of the type e T ◦ T arises.. It turns out that, again, there is a remarkable difference between even and odd kernels. For even kernels we obtain, quite unexpectedly, weak (1, 1) estimates, which are no longer true for odd kernels. For odd kernels we obtain sharp weaker inequalities involving a weak L1 estimate for functions in L LogL.Generalitat de CatalunyaMinisterio de Educación y CienciaJunta de Andalucí

Plan de cuidados estandarizado sobre traqueostomías en pacientes ingresados en Unidad de cuidados intensivos

El uso de la implantación de la traqueostomía como medida para controlar la vía aérea de manera segura en pacientes necesitados de soporte ventilatorio en largos períodos de tiempo, o con problemas relacionados con las secreciones o alteraciones de la conciencia, se ha convertido en una técnica bastante frecuente llevada a cabo en pacientes en estado crítico ingresados en las Unidades de Cuidados Intensivos (UCI). La traqueostomía es una técnica quirúrgica consistente en una comunicación directa de la tráquea con el exterior mediante un orificio denominado estoma con la finalidad de proporcionar y facilitar la entrada y salida del aire a los pulmones. Por lo tanto, es imprescindible que Enfermería posea el conocimiento del proceso, sus convenientes y desventajas, indicaciones, contraindicaciones y posibles complicaciones para obrar de manera correcta y actuar con el resto de compañeros sanitarios en coordinación satisfaciendo las necesidades del paciente.<br /