Oscillations and traveling waves of calcium: a simplified model

We construct a heuristic model of calcium oscillations in pancreatic acinar cells. The model is based on the two-state model of Sneyd et al. (Sneyd, J., A. LeBeau and D. Yule, 2000, Traveling waves of calcium in pancreatic acinar cells: model construction and bifurcation analysis, Physica D, in press) and is similar in spirit to the FitzHugh reduction of the Hodgkin-Huxley equations. The simpli¯ed model successfully reproduces the oscillatory behavior and wave behaviour of the more complex model. In particular, the simpli¯ed model provides an example of a simple, physiologically relevant model that has a T-point and an associated spiral branch of homoclinic orbits

Spike Oscillations

According to Belinskii, Khalatnikov and Lifshitz (BKL), a generic spacelike singularity is characterized by asymptotic locality: Asymptotically, toward the singularity, each spatial point evolves independently from its neighbors, in an oscillatory manner that is represented by a sequence of Bianchi type I and II vacuum models. Recent investigations support a modified conjecture: The formation of spatial structures (`spikes') breaks asymptotic locality. The complete description of a generic spacelike singularity involves spike oscillations, which are described by sequences of Bianchi type I and certain inhomogeneous vacuum models. In this paper we describe how BKL and spike oscillations arise from concatenations of exact solutions in a Hubble-normalized state space setting, suggesting the existence of hidden symmetries and showing that the results of BKL are part of a greater picture.Comment: 38 pages, 14 figure

Noise-Induced Stabilization of Planar Flows I

We show that the complex-valued ODE \begin{equation*} \dot z_t = a_{n+1} z^{n+1} + a_n z^n+\cdots+a_0, \end{equation*} which necessarily has trajectories along which the dynamics blows up in finite time, can be stabilized by the addition of an arbitrarily small elliptic, additive Brownian stochastic term. We also show that the stochastic perturbation has a unique invariant measure which is heavy-tailed yet is uniformly, exponentially attracting. The methods turn on the construction of Lyapunov functions. The techniques used in the construction are general and can likely be used in other settings where a Lyapunov function is needed. This is a two-part paper. This paper, Part I, focuses on general Lyapunov methods as applied to a special, simplified version of the problem. Part II of this paper extends the main results to the general setting.Comment: Part one of a two part pape

Revealing networks from dynamics: an introduction

What can we learn from the collective dynamics of a complex network about its interaction topology? Taking the perspective from nonlinear dynamics, we briefly review recent progress on how to infer structural connectivity (direct interactions) from accessing the dynamics of the units. Potential applications range from interaction networks in physics, to chemical and metabolic reactions, protein and gene regulatory networks as well as neural circuits in biology and electric power grids or wireless sensor networks in engineering. Moreover, we briefly mention some standard ways of inferring effective or functional connectivity.Comment: Topical review, 48 pages, 7 figure

On Dynamics of Integrate-and-Fire Neural Networks with Conductance Based Synapses

We present a mathematical analysis of a networks with Integrate-and-Fire neurons and adaptive conductances. Taking into account the realistic fact that the spike time is only known within some \textit{finite} precision, we propose a model where spikes are effective at times multiple of a characteristic time scale $\delta$, where $\delta$ can be \textit{arbitrary} small (in particular, well beyond the numerical precision). We make a complete mathematical characterization of the model-dynamics and obtain the following results. The asymptotic dynamics is composed by finitely many stable periodic orbits, whose number and period can be arbitrary large and can diverge in a region of the synaptic weights space, traditionally called the "edge of chaos", a notion mathematically well defined in the present paper. Furthermore, except at the edge of chaos, there is a one-to-one correspondence between the membrane potential trajectories and the raster plot. This shows that the neural code is entirely "in the spikes" in this case. As a key tool, we introduce an order parameter, easy to compute numerically, and closely related to a natural notion of entropy, providing a relevant characterization of the computational capabilities of the network. This allows us to compare the computational capabilities of leaky and Integrate-and-Fire models and conductance based models. The present study considers networks with constant input, and without time-dependent plasticity, but the framework has been designed for both extensions.Comment: 36 pages, 9 figure

On the multiresolution structure of Internet traffic traces

Internet traffic on a network link can be modeled as a stochastic process. After detecting and quantifying the properties of this process, using statistical tools, a series of mathematical models is developed, culminating in one that is able to generate ``traffic'' that exhibits --as a key feature-- the same difference in behavior for different time scales, as observed in real traffic, and is moreover indistinguishable from real traffic by other statistical tests as well. Tools inspired from the models are then used to determine and calibrate the type of activity taking place in each of the time scales. Surprisingly, the above procedure does not require any detailed information originating from either the network dynamics, or the decomposition of the total traffic into its constituent user connections, but rather only the compliance of these connections to very weak conditions.Comment: 57 pages, color figures. Figures are of low quality due to space consideration