Synchronization and entrainment of coupled circadian oscillators

Circadian rhythms in mammals are controlled by the neurons located in the suprachiasmatic nucleus of the hypothalamus. In physiological conditions, the system of neurons is very efficiently entrained by the 24-hour light-dark cycle. Most of the studies carried out so far emphasize the crucial role of the periodicity imposed by the light dark cycle in neuronal synchronization. Nevertheless, heterogeneity as a natural and permanent ingredient of these cellular interactions is seemingly to play a major role in these biochemical processes. In this paper we use a model that considers the neurons of the suprachiasmatic nucleus as chemically-coupled modified Goodwin oscillators, and introduce non-negligible heterogeneity in the periods of all neurons in the form of quenched noise. The system response to the light-dark cycle periodicity is studied as a function of the interneuronal coupling strength, external forcing amplitude and neuronal heterogeneity. Our results indicate that the right amount of heterogeneity helps the extended system to respond globally in a more coherent way to the external forcing. Our proposed mechanism for neuronal synchronization under external periodic forcing is based on heterogeneity-induced oscillators death, damped oscillators being more entrainable by the external forcing than the self-oscillating neurons with different periods.Comment: 17 pages, 7 figure

On a conjecture on the integrability of Liénard systems

We consider the Liénard differential systems ̇x=y+F(x), ̇y=x (1), in C2 where F(x) is an analytic function satisfying F(0) = 0 and F'(0) ≠ 0. Then these systems have a strong saddle at the origin of coordinates. It has been conjecture that if such systems have an analytic first integral defined in a neighborhood of the origin, then the function F(x) is linear, i.e. F(x) = ax. Here we prove this conjecture, and show that when F(x) is linear and system (1) has an analytic first integral, this is a polynomial

Flow analysis and first integrals of a family of 3D Lotka-Volterra Systems

Tesis leída en l'Universitat de les Illes Balears el 21 de septiembre de 2009Differential equations appear in many interdisciplinary areas, at the interface between mathematics and a variety of fields, from biology and chemistry to economy and physics. They are mathematically studied from several different perspectives, mostly concerned with their solutions, functions that make the equation hold true. In general, not all differential equations admit solutions in terms of elementary functions. Computer simulations and/or the qualitative theory of differential equations are being used to the analyze differential equations whose explicit solutions are hard to find.Peer reviewe