Mathematical models for sleep-wake dynamics: comparison of the two-process model and a mutual inhibition neuronal model

Sleep is essential for the maintenance of the brain and the body, yet many features of sleep are poorly understood and mathematical models are an important tool for probing proposed biological mechanisms. The most well-known mathematical model of sleep regulation, the two-process model, models the sleep-wake cycle by two oscillators: a circadian oscillator and a homeostatic oscillator. An alternative, more recent, model considers the mutual inhibition of sleep promoting neurons and the ascending arousal system regulated by homeostatic and circadian processes. Here we show there are fundamental similarities between these two models. The implications are illustrated with two important sleep-wake phenomena. Firstly, we show that in the two-process model, transitions between different numbers of daily sleep episodes occur at grazing bifurcations.This provides the theoretical underpinning for numerical results showing that the sleep patterns of many mammals can be explained by the mutual inhibition model. Secondly, we show that when sleep deprivation disrupts the sleep-wake cycle, ostensibly different measures of sleepiness in the two models are closely related. The demonstration of the mathematical similarities of the two models is valuable because not only does it allow some features of the two-process model to be interpreted physiologically but it also means that knowledge gained from study of the two-process model can be used to inform understanding of the mutual inhibition model. This is important because the mutual inhibition model and its extensions are increasingly being used as a tool to understand a diverse range of sleep-wake phenomena such as the design of optimal shift-patterns, yet the values it uses for parameters associated with the circadian and homeostatic processes are very different from those that have been experimentally measured in the context of the two-process model

Perturbations of embedded eigenvalues for the planar bilaplacian

Operators on unbounded domains may acquire eigenvalues that are embedded in the essential spectrum. Determining the fate of these embedded eigenvalues under small perturbations of the underlying operator is a challenging task, and the persistence properties of such eigenvalues is linked intimately to the multiplicity of the essential spectrum. In this paper, we consider the planar bilaplacian with potential and show that the set of potentials for which an embedded eigenvalue persists is locally an infinite-dimensional manifold with infinite codimension in an appropriate space of potentials

A robust numerical method to study oscillatory instability of gap solitary waves

The spectral problem associated with the linearization about solitary waves of spinor systems or optical coupled mode equations supporting gap solitons is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. These problems may exhibit oscillatory instabilities where eigenvalues detach from the edges of the continuous spectrum, so called edge bifurcations. A numerical framework, based on a fast robust shooting algorithm using exterior algebra is described. The complete algorithm is robust in the sense that it does not produce spurious unstable eigenvalues. The algorithm allows to locate exactly where the unstable discrete eigenvalues detach from the continuous spectrum. Moreover, the algorithm allows for stable shooting along multi-dimensional stable and unstable manifolds. The method is illustrated by computing the stability and instability of gap solitary waves of a coupled mode model.Comment: key words: gap solitary wave, numerical Evans function, edge bifurcation, exterior algebra, oscillatory instability, massive Thirring model. accepted for publication in SIAD

A stability criterion for the non-linear wave equation with spatial inhomogeneity

In this paper the non-linear wave equation with a spatial inhomogeneity is considered. The inhomogeneity splits the unbounded spatial domain into three or more intervals, on each of which the non-linear wave equation is homogeneous. In such setting, there often exist multiple stationary fronts. In this paper we present a necessary and sufficient stability criterion in terms of the length of the middle interval(s) and the energy associated with the front in these interval(s). To prove this criterion, it is shown that critical points of the length function and zeros of the linearisation have the same order. Furthermore, the Evans function is used to identify the stable branch. The criterion is illustrated with an example which shows the existence of bi-stability: two stable fronts, one of which is non-monotonic. The Evans function also give a sufficient instability criterion in terms of the derivative of the length function

Existence of stationary fronts in a system of two coupled wave equations with spatial inhomogeneity

We investigate the existence of stationary fronts in a coupled system of two sine-Gordon equations with a smooth, "hat-like" spatial inhomogeneity. The spatial inhomogeneity corresponds to a spatially dependent scaling of the sine-Gordon potential term. The uncoupled inhomogeneous sine-Gordon equation has stable stationary front solutions that persist in the coupled system. Carrying out a numerical investigation it is found that these inhomogeneous sine-Gordon fronts loose stability, provided the coupling between the two inhomogeneous sine-Gordon equations is strong enough, with new stable fronts bifurcating. In order to analytically study the bifurcating fronts, we first approximate the smooth spatial inhomogeneity by a piecewise constant function. With this approximation, we prove analytically the existence of a pitchfork bifurcation. To complete the argument, we prove that transverse fronts for a piecewise constant inhomogeneity persist for the smooth "hat-like" spatial inhomogeneity by introducing a fast-slow structure and using geometric singular perturbation theory

Localized mode interactions in 0-pi Josephson junctions

A long Josephson junction containing regions with a phase shift of pi is considered. By exploiting the defect modes due to the discontinuities present in the system, it is shown that Josephson junctions with phase-shift can be an ideal setting for studying localized mode interactions. A phase-shift configuration acting as a double-well potential is considered and shown to admit mode tunnelings between the wells. When the phase-shift configuration is periodic, it is shown that localized excitations forming bright and dark solitons can be created. Multi-mode approximations are derived confirming the numerical results.Comment: 4 pages, to appear in Phys. Rev.