120 research outputs found
A geometric mechanism of diffusion: Rigorous verification in a priori unstable Hamiltonian systems
In this paper we consider a representative a priori unstable Hamiltonian
system with 2+1/2 degrees of freedom, to which we apply the geometric mechanism
for diffusion introduced in the paper Delshams et al., Mem. Amer. Math. Soc.
2006, and generalized in Delshams and Huguet, Nonlinearity 2009, and provide
explicit, concrete and easily verifiable conditions for the existence of
diffusing orbits.
The simplification of the hypotheses allows us to perform explicitly the
computations along the proof, which contribute to present in an easily
understandable way the geometric mechanism of diffusion. In particular, we
fully describe the construction of the scattering map and the combination of
two types of dynamics on a normally hyperbolic invariant manifol
Phasic firing and coincidence detection by subthreshold negative feedback: divisive or subtractive or, better, both
Phasic neurons typically fire only for a fast-rising input, say at the onset of a step current, but not for steady or slow inputs, a property associated with type III excitability. Phasic neurons can show extraordinary temporal precision for phase locking and coincidence detection. Exemplars are found in the auditory brain stem where precise timing is used in sound localization. Phasicness at the cellular level arises from a dynamic, voltage-gated, negative feedback that can be recruited subthreshold, preventing the neuron from reaching spike threshold if the voltage does not rise fast enough. We consider two mechanisms for phasicness: a low threshold potassium current (subtractive mechanism) and a sodium current with subthreshold inactivation (divisive mechanism). We develop and analyze three reduced models with either divisive or subtractive mechanisms or both to gain insight into the dynamical mechanisms for the potentially high temporal precision of type III-excitable neurons. We compare their firing properties and performance for a range of stimuli. The models have characteristic non-monotonic input-output relations, firing rate vs. input intensity, for either stochastic current injection or Poisson-timed excitatory synaptic conductance trains. We assess performance according to precision of phase-locking and coincidence detection by the models' responses to repetitive packets of unitary excitatory synaptic inputs with more or less temporal coherence. We find that each mechanism contributes features but best performance is attained if both are present. The subtractive mechanism confers extraordinary precision for phase locking and coincidence detection but only within a restricted parameter range when the divisive mechanism of sodium inactivation is inoperative. The divisive mechanism guarantees robustness of phasic properties, without compromising excitability, although with somewhat less precision. Finally, we demonstrate that brief transient inhibition if properly timed can enhance the reliability of firing.Postprint (published version
Fast iteration of cocyles over rotations and Computation of hyperbolic bundles
In this paper, we develop numerical algorithms that use small requirements of
storage and operations for the computation of hyperbolic cocycles over a
rotation. We present fast algorithms for the iteration of the quasi-periodic
cocycles and the computation of the invariant bundles, which is a preliminary
step for the computation of invariant whiskered tori
Una panorĂ mica de la literatura per a infants i joves al PaĂs Valencia per entendre l'actualitat
Una revisiĂł de la literatura en catalĂ per a infants i joves escrita al PaĂs ValenciĂ . L’article s’inicia amb una primera part per conèixer el passat i situar el present i s’hi analitza la producciĂł abans de 1939. DesprĂ©s, el perĂode de 1939 a 1970. La tercera part analitza el perĂode de 1970 a 1982, quan comencen a publicar-se les primeres obres dels que esdevindran en pocs anys els autors mĂ©s importants. El punt d’inflexiĂł ve marcat per una data, el 1983, que inicia el darrer perĂode que arriba fins a l’actualitat. És l’any de publicaciĂł de la Llei d’Ús i Ensenyament del ValenciĂ (LUEV). Finalment, de 1984 a l’actualitat: on es revisen les editorials, els premis, els autors, els tĂtols i alguns èxits editorials, la promociĂł i l’ensenyament de la literatura
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
Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems
In the present paper we consider the case of a general \cont{r+2}
perturbation, for large enough, of an a priori unstable
Hamiltonian system of degrees of freedom, and we provide
explicit conditions on it, which turn out to be \cont{2} generic
and are verifiable in concrete examples, which guarantee the
existence of Arnold diffusion.
This is a generalization of the result in Delshams et al.,
\emph{Mem. Amer. Math. Soc.}, 2006, where it was considered the case
of a perturbation with a finite number of harmonics in the angular
variables.
The method of proof is based on a careful analysis of the geography
of resonances created by a generic perturbation and it contains a
deep quantitative description of the invariant objects generated by
the resonances therein. The scattering map is used as an essential
tool to construct transition chains of objects of different
topology. The combination of quantitative expressions for both the
geography of resonances and the scattering map provides, in a
natural way, explicit computable conditions for instability
A geometric mechanism of diffusion: Rigorous verification in a priori unstable Hamiltonian systems
In this paper we consider a representative a priori unstable Hamiltonian system
with 2 + 1/2 degrees of freedom, to which we apply the geometric mechanism for
diffusion introduced in the paper Delshams et al., Mem. Amer. Math. Soc. 2006,
and generalized in Delshams and Huguet, Nonlinearity 2009, and provide explicit,
concrete and easily verifiable conditions for the existence of diffusing orbits.
The simplification of the hypotheses allows us to perform explicitly the computations
along the proof, which contribute to present in an easily understandable way the
geometric mechanism of diffusion. In particular, we fully describe the construction
of the scattering map and the combination of two types of dynamics on a normally
hyperbolic invariant manifold.Preprin
Gluing and grazing bifurcations in periodically forced 2-dimensional integrate-and-fire models
In this work we consider a general class of 2-dimensional hybrid systems. Assuming that the system possesses an attracting equilibrium point, we show that, when periodically driven with a square-wave pulse, the system possesses a periodic orbit which may undergo smooth and nonsmooth grazing bifurcations. We perform a semi-rigorous study of the existence of periodic orbits for a particular model consisting of a leaky integrate-and-fire model with a dynamic threshold. We use the stroboscopic map, which in this context is a 2-dimensional piecewise-smooth discontinuous map. For some parameter values we are able to show that the map is a quasi-contraction possessing a (locally) unique maximin periodic orbit. We complement our analysis using advanced numerical techniques to provide a complete portrait of the dynamics as parameters are varied. We find that for some regions of the parameter space the model undergoes a cascade of gluing bifurcations, while for others the model shows multistability between orbits of different periodsPeer ReviewedPostprint (author's final draft
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