A Fast-Slow Analysis of the Dynamics of REM Sleep

Waking and sleep states are regulated by the coordinated activity of a number of neuronal population in the brainstem and hypothalamus whose synaptic interactions compose a sleep-wake regulatory network. Physiologically based mathematical models of the sleep-wake regulatory network contain mechanisms operating on multiple time scales including relatively fast synaptic-based interations between neuronal populations, and much slower homeostatic and circadian processes that modulate sleep-wake temporal patterning. In this study, we exploit the naturally arising slow time scale of the homeostatic sleep drive in a reduced sleep-wake regulatory network model to utilize fast-slow analysis to investigate the dynamics of rapid eye movement (REM) sleep regulation. The network model consists of a reduced number of wake-, non-REM (NREM) sleep-, and REM sleep-promoting neuronal populations with the synaptic interactions reflecting the mutually inhibitory flip-flop conceptual model for sleep-wake regulation and the reciprocal interaction model for REM sleep regulation. Network dynamics regularly alternate between wake and sleep states as goverend by the slow homeostatic sleep drive. By varying a parameter associated with the activation of the REM-promoting population, we cause REM dynamics during sleep episodes to vary from supression to single activations to regular REM-NREM cycling, corresponding to changes in REM patterning induced by circadian modulation and observed in different mammalian species. We also utilize fast-slow analysis to explain complex effects on sleep-wake patterning of simulated experiments in which agonists and antagonists of different neurotransmitters are microinjected into specific neuronal populations participating in the sleep-wake regulatory network

Triaxial Galaxies with Cusps

We have constructed fully self-consistent models of triaxial galaxies with central density cusps. The triaxial generalizations of Dehnen's spherical models are presented, which have densities that vary as 1/r^gamma near the center and 1/r^4 at large radii. We computed libraries of about 7000 orbits in each of two triaxial models with gamma=1 (weak cusp) and gamma=2 (strong cusp); these two models have density profiles similar to those of the core and power-law galaxies observed by HST. Both mass models have short-to-long axis ratios of 1:2 and are maximally triaxial. A large fraction of the orbits in both model potentials are stochastic, as evidenced by their non-zero Liapunov exponents. We show that most of the stochastic orbits in the strong- cusp potential diffuse relatively quickly through their allowed phase-space volumes, on time scales of 100 - 1000 dynamical times. Stochastic orbits in the weak-cusp potential diffuse more slowly, often retaining their box-like shapes for 1000 dynamical times or longer. Attempts to construct self- consistent solutions using just the regular orbits failed for both mass models. Quasi-equilibrium solutions that include the stochastic orbits exist for both models; however, real galaxies constructed in this way would evolve near the center due to the continued mixing of the stochastic orbits. We attempted to construct more nearly stationary models in which stochastic phase space was uniformly populated at low energies. These ``fully mixed'' solutions were found to exist only for the weak-cusp potential. No significant fraction of the mass could be placed on fully-mixed stochastic orbits in the strong-cusp model, demonstrating that strong triaxiality can be inconsistent with a high central density.Comment: 58 TEX pages, 14 PostScript figures, uses epsf.st

Diffusion-driven instabilities and emerging spatial patterns in patchy landscapes

Spatial variation in population densities across a landscape is a feature of many ecological systems, from self-organised patterns on mussel beds to spatially restricted insect outbreaks. It occurs as a result of environmental variation in abiotic factors and/or biotic factors structuring the spatial distribution of populations. However the ways in which abiotic and biotic factors interact to determine the existence and nature of spatial patterns in population density remain poorly understood. Here we present a new approach to studying this question by analysing a predator–prey patch-model in a heterogenous landscape. We use analytical and numerical methods originally developed for studying nearest- neighbour (juxtacrine) signalling in epithelia to explore whether and under which conditions patterns emerge. We find that abiotic and biotic factors interact to promote pattern formation. In fact, we find a rich and highly complex array of coexisting stable patterns, located within an enormous number of unstable patterns. Our simulation results indicate that many of the stable patterns have appreciable basins of attraction, making them significant in applications. We are able to identify mechanisms for these patterns based on the classical ideas of long-range inhibition and short-range activation, whereby landscape heterogeneity can modulate the spatial scales at which these processes operate to structure the populations

Asymptotic methods for delay equations.

Asymptotic methods for singularly perturbed delay differential equations are in many ways more challenging to implement than for ordinary differential equations. In this paper, four examples of delayed systems which occur in practical models are considered: the delayed recruitment equation, relaxation oscillations in stem cell control, the delayed logistic equation, and density wave oscillations in boilers, the last of these being a problem of concern in engineering two-phase flows. The ways in which asymptotic methods can be used vary from the straightforward to the perverse, and illustrate the general technical difficulties that delay equations provide for the central technique of the applied mathematician. © Springer 2006

Extinction rates of established spatial populations

This paper deals with extinction of an isolated population caused by intrinsic noise. We model the population dynamics in a "refuge" as a Markov process which involves births and deaths on discrete lattice sites and random migrations between neighboring sites. In extinction scenario I the zero population size is a repelling fixed point of the on-site deterministic dynamics. In extinction scenario II the zero population size is an attracting fixed point, corresponding to what is known in ecology as Allee effect. Assuming a large population size, we develop WKB (Wentzel-Kramers-Brillouin) approximation to the master equation. The resulting Hamilton's equations encode the most probable path of the population toward extinction and the mean time to extinction. In the fast-migration limit these equations coincide, up to a canonical transformation, with those obtained, in a different way, by Elgart and Kamenev (2004). We classify possible regimes of population extinction with and without an Allee effect and for different types of refuge and solve several examples analytically and numerically. For a very strong Allee effect the extinction problem can be mapped into the over-damped limit of theory of homogeneous nucleation due to Langer (1969). In this regime, and for very long systems, we predict an optimal refuge size that maximizes the mean time to extinction.Comment: 26 pages including 3 appendices, 16 figure

Global dynamics of a novel delayed logistic equation arising from cell biology

The delayed logistic equation (also known as Hutchinson's equation or Wright's equation) was originally introduced to explain oscillatory phenomena in ecological dynamics. While it motivated the development of a large number of mathematical tools in the study of nonlinear delay differential equations, it also received criticism from modellers because of the lack of a mechanistic biological derivation and interpretation. Here we propose a new delayed logistic equation, which has clear biological underpinning coming from cell population modelling. This nonlinear differential equation includes terms with discrete and distributed delays. The global dynamics is completely described, and it is proven that all feasible nontrivial solutions converge to the positive equilibrium. The main tools of the proof rely on persistence theory, comparison principles and an $L^2$-perturbation technique. Using local invariant manifolds, a unique heteroclinic orbit is constructed that connects the unstable zero and the stable positive equilibrium, and we show that these three complete orbits constitute the global attractor of the system. Despite global attractivity, the dynamics is not trivial as we can observe long-lasting transient oscillatory patterns of various shapes. We also discuss the biological implications of these findings and their relations to other logistic type models of growth with delays

Homogenization techniques for population dynamics in strongly heterogeneous landscapes

An important problem in spatial ecology is to understand how population-scale patterns emerge from individual-level birth, death, and movement processes. These processes, which depend on local landscape characteristics, vary spatially and may exhibit sharp transitions through behavioural responses to habitat edges, leading to discontinuous population densities. Such systems can be modelled using reaction–diffusion equations with interface conditions that capture local behaviour at patch boundaries. In this work we develop a novel homogenization technique to approximate the large-scale dynamics of the system. We illustrate our approach, which also generalizes to multiple species, with an example of logistic growth within a periodic environment. We find that population persistence and the large-scale population carrying capacity is influenced by patch residence times that depend on patch preference, as well as movement rates in adjacent patches. The forms of the homogenized coefficients yield key theoretical insights into how large-scale dynamics arise from the small-scale features