Rodica Curtu

Pattern Formation in a Network of Excitatory and Inhibitory Cells with Adaptation

A bifurcation analysis of a simplified model for excitatory and inhibitory dynamics is presented. Excitatory cells are endowed with a slow negative feedback and inhibitory cells are assumed to act instantly. This results in a generalization of the Hansel-Sompolinsky model for orientation selectiv- ity. Normal forms are computed for the Turing-Hopf instability, where a new class of solutions is found. The transition from stationary patterns to traveling waves is analyzed by deriving the normal form for a Takens-Bogdanov bifurcation. Comparisons between the normal forms and numerical solutions of the full model are presented.

Mechanisms for Frequency Control in Neuronal Competition Models

We investigate analytically a firing rate model for a two-population network based on mutual inhibition and slow negative feedback in the form of spike frequency adaptation. Both neuronal populations receive external constant input whose strength determines the systems dynamical state-a steady state of identical activity levels or periodic oscillations or a winner-take-all state of bistability. We prove that oscillations appear in the system through supercritical Hopf bifurcations and that they are antiphase. The period of oscillations depends on the input strength in a nonmonotonic fashion, and we show that the increasing branch of the period versus input curve corresponds to a release mechanism and the decreasing branch to an escape mechanism. In the limiting case of infinitely slow feedback we characterize the conditions for release, escape, and occurrence of the winner-take-all behavior. Some extensions of the model are also discussed.

Interaction of Canard and Singular Hopf Mechanisms in a Neural Model

We consider an ordinary differential equation model for neural competition, presented previously in the study of binocular rivalry, which features two adapting populations of neurons interacting through mutual inhibition. This model is known to exhibit a variety of dynamic regimes, including mixed-mode oscillations (MMOs) featuring alternating small- and large amplitude oscillations, depending on the value of an input parameter. In this work, we use geometric dynamical systems techniques to study the structure of the model in the singular limit as well as the emergence of MMOs in the perturbed system. In particular, exploiting a normal form calculation allows us to numerically compute a way-in/way-out function, which we use to elucidate the interaction of canard and singular Hopf mechanisms for small amplitude oscillations that occur as the input parameter approaches a critical value.

Traveling waves in a one-dimensional integrate-and-fire neural network with finite support connectivity

Abstract We study the existence of traveling waves in a one-dimensional network of integrate-and-fire neurons with finite support coupling. We show that when the reset after spiking is sufficiently low, the interspike intervals (ISIs) for a traveling wave with an explicit first wave front can be computed. Further, we analytically derive a self-consistency equation that generates an explicit dispersion relation between the velocity of periodic traveling waves and their corresponding ISIs. Finally, we prove that the ISIs of non-periodic traveling waves converge to the ISI of the periodic waves with the same speed.

Small-Scale Modeling Approach and Circuit Wiring of the Unfolded Protein Response in Mammalian Cells

The accumulation of unfolded proteins in the endoplasmic reticulum (ER) activates a mechanism whose primary functions are to sense any perturbation in the protein-folding capacity of the cell, and correct the situation to restore homeostasis. This cellular mechanism is called the unfolded protein response (UPR). We propose a biologically plausible computational model for the UPR under ER stress in mammalian cells. The model accounts for the signaling pathways of PERK, ATF6, and IRE1 and has the advantage of simulating the dynamical (timecourse) changes in the relative concentrations of proteins without any a priori steady-state assumption. Several types of ER stress can be assumed as input, including long-term (eventually periodic) stress. Moreover, the model allows for outcomes ranging from cell survival to cell apoptosis.

SIMULATION OF A DISTRIBUTED FLOOD CONTROL SYSTEM USING A PARALLEL ASYNCHRONOUS SOLVER FOR SYSTEMS OF ODES

A recently developed parallel asynchronous solver for systems of ordinary differential equations (ODEs) is used to simulate flows along the channels in a river network. In our model, precipitation is applied over the hillslopes adjacent to the river network links and water movement from hillsope to link and along the river network is represented as a system of ODEs. The numerical solver is based on dense output Runge-Kutta methods that allow for asynchronous integration. A static partition method is used to distribute the workload among different processes, enabling a parallel implementation that capitalizes on a distributed memory system. Communication between processes is performed asynchronously. We illustrate the solver capabilities by integrating flow transport equations for a 32,000 km 2 river basin subdivided into 574,000 sub-watersheds that are interconnected by the river network. We show that the runtime for an eight month-long simulation forced by 1-km resolution NEXRAD rainfall is completed in under 4 minutes using 64 computing nodes. In addition, we include equations to simulate small reservoirs spread throughout the river network and estimate changes in hydrographs at multiple locations. Our results provide a firm theoretical basis for the concept of distributed flood control systems.

THE STATIC BIFURCATION DIAGRAM FOR THE GRAY–SCOTT MODEL

The singularity theorem is used to derive the deformations of the static bifurcation diagram of the Gray–Scott model.

Nonlinear response in runoff magnitude to fluctuating rain patterns.

The runoff coefficient of a hillslope is a reliable measure for changes in the streamflow response at the river link outlet. A high runoff coefficient is a good indicator of the possibility of flash floods. Although the relationship between runoff coefficient and streamflow has been the subject of much study, the physical mechanisms affecting runoff coefficient including the dependence on precipitation pattern remain open topics for investigation. In this paper, we analyze a rainfall-runoff model at the hillslope scale as that hillslope is forced with different rain patterns: constant rain and fluctuating rain with different frequencies and amplitudes. When an oscillatory precipitation pattern is applied, although the same amount of water may enter the system, its response (measured by the runoff coefficient) will be maximum for a certain frequency of precipitation. The significant increase in runoff coefficient after a certain pattern of rainfall can be a potential explanation for the conditions preceding flash-floods.