Shared inputs, entrainment, and desynchrony in elliptic bursters: from slow passage to discontinuous circle maps

What input signals will lead to synchrony vs. desynchrony in a group of biological oscillators? This question connects with both classical dynamical systems analyses of entrainment and phase locking and with emerging studies of stimulation patterns for controlling neural network activity. Here, we focus on the response of a population of uncoupled, elliptically bursting neurons to a common pulsatile input. We extend a phase reduction from the literature to capture inputs of varied strength, leading to a circle map with discontinuities of various orders. In a combined analytical and numerical approach, we apply our results to both a normal form model for elliptic bursting and to a biophysically-based neuron model from the basal ganglia. We find that, depending on the period and amplitude of inputs, the response can either appear chaotic (with provably positive Lyaponov exponent for the associated circle maps), or periodic with a broad range of phase-locked periods. Throughout, we discuss the critical underlying mechanisms, including slow-passage effects through Hopf bifurcation, the role and origin of discontinuities, and the impact of noiseComment: 17 figures, 40 page

Uniqueness of Lagrangian Self-Expanders

We show that zero-Maslov class Lagrangian self-expanders in C^n which are asymptotic to a pair of planes intersecting transversely are locally unique if n>2 and unique if n=2.Comment: 32 page

Simple bounds for queues fed by Markovian sources: a tool for performance evaluation

ATM traffic is complex but only simple statistical models are amenable to mathematical analysis. We discuss a class of queuing models which is wide enough to provide models which can reflect the features of real traffic, but which is simple enough to be analytically tractable, and review the bounds on the queue-length distribution that have been obtained. We use them to obtain bounds on QoS parameters and to give approximations to the effective bandwidth of such sources. We present some numerical techniques for calculating the bounds efficiently and describe an implementation of them in a computer package which can serve as a tool for qualitative investigations of performance in queuing systems

Asymptotic solutions of the nonlinear Boltzmann equation for dissipative systems

Analytic solutions $F(v,t)$ of the nonlinear Boltzmann equation in $d$-dimensions are studied for a new class of dissipative models, called inelastic repulsive scatterers, interacting through pseudo-power law repulsions, characterized by a strength parameter $\nu$, and embedding inelastic hard spheres ($\nu=1$) and inelastic Maxwell models ($\nu=0$). The systems are either freely cooling without energy input or driven by thermostats, e.g. white noise, and approach stable nonequilibrium steady states, or marginally stable homogeneous cooling states, where the data, $v^d_0(t) F(v,t)$ plotted versus $c=v/v_0(t)$, collapse on a scaling or similarity solution $f(c)$, where $v_0(t)$ is the r.m.s. velocity. The dissipative interactions generate overpopulated high energy tails, described generically by stretched Gaussians, $f(c) \sim \exp[-\beta c^b]$ with $0 < b < 2$, where $b=\nu$ with $\nu>0$ in free cooling, and $b=1+{1/2} \nu$ with $\nu \geq 0$ when driven by white noise. Power law tails, $f(c) \sim 1/c^{a+d}$, are only found in marginal cases, where the exponent $a$ is the root of a transcendental equation. The stability threshold depend on the type of thermostat, and is for the case of free cooling located at $\nu=0$. Moreover we analyze an inelastic BGK-type kinetic equation with an energy dependent collision frequency coupled to a thermostat, that captures all qualitative properties of the velocity distribution function in Maxwell models, as predicted by the full nonlinear Boltzmann equation, but fails for harder interactions with $\nu>0$.Comment: Submitted to: "Granular Gas Dynamics", T. Poeschel, N. Brilliantov (eds.), Lecture Notes in Physics, Vol. LNP 624, Springer-Verlag, Berlin-Heidelberg-New York, 200

Computing aspects of problems in non-linear prediction and filtering

Imperial Users onl

Information theory and possible mathematical descriptions of economical and social systems based on real physical phenomena

Recent approaches in informatics to model large complex systems are considered following the ideas from real phenomena explained by physical tools. The econo-physics and sociophysics are considered. In particular, Master Equation approach and Markov chains approaches are discussed. Also the partial differential equations as the tool for modeling economical and social systems are represented. New approaches for modeling systems with memory and with accounting internal properties of system elements are considered and some new research problems are proposed.Рассматриваются современные подходы в информатике к моделированию больших сложных систем, аналогичные используемым в физике. Обсуждаются эконофизика и социофизика. Представлены дифференциальные уравнения в частных производных как инструмент для моделирования экономических и общественных систем. Предложены новые подходы к моделированию систем моделирования с памятью и учетом внутренних свойств элементов системы, а также новые проблемы для исследования.Розглядаються сучасні підходи в інформатиці до моделювання великих складних систем, аналогічні тим, що використовуються у фізиці. Обговорюються еконофізика і соціофізика. Наведено диференційні рівняння у частинних похідних як інструмент для моделювання економічних та суспільних систем. Запропоновано нові підходи до моделювання систем із пам’яттю та з урахуванням внутрішніх властивостей елементів системи, а також нові проблеми для досліджень