Ihab Haidar

Effects of spatial structure and diffusion on the performances of the chemostat

Given hydric capacity and nutrient flow of a chemostat-like system, we analyse the influence of a spatial structure on the output concentrations at steady-state. Three configurations are compared: perfectly-mixed, serial and parallel with diffusion rate. We show the existence of a threshold on the input concentration of nutrient for which the benefits of the serial and parallel configurations over the perfectly-mixed one are reversed. In addition, we show that the dependency of the output concentrations on the diffusion rate can be non-monotonic, and give precise conditions for the diffusion effect to be advantageous. The study encompasses dead-zone models.

Analysis of delay-induced basal ganglia oscillations: the role of external excitatory nuclei

Basal ganglia are interconnected deep brain structures involved in movement generation. Their persistent beta-band oscillations (13–30 Hz) are known to be linked to Parkinson’s disease motor symptoms. In this paper, we provide conditions under which these oscillations may occur, by explicitly considering the role of the pedunculopontine nucleus (PPN). We analyse the existence of equilibria in the associated firing-rate dynamics and study their stability by relying on a delayed multiple-input/multiple-output (MIMO) frequency analysis. Our analysis suggests that the PPN has an influence on the generation of pathological beta-band oscillations. These results are illustrated by simulations that confirm numerically the analytic predictions of our two main theorems.

Global dynamics of the buffered chemostat for a general class of response functions

We study how a particular spatial structure with a buffer impacts the number of equilibria and their stability in the chemostat model. We show that the occurrence of a buffer can allow a species to persist or on the opposite to go extinct, depending on the characteristics of the buffer. For non-monotonic response functions, we characterize the buffered configurations that make the chemostat dynamics globally asymptotically stable, while this is not possible with single, serial or parallel vessels of the same total volume and input flow. These results are illustrated with the Haldane kinetic function.

Closed-loop deep brain stimulation based on firing-rate regulation

This paper develops a new closed-loop strategy for deep brain stimulation, derived using a model-based analysis of the basal ganglia. The system is described using a firing-rate model that has been proposed recently in the literature, in order to analyze the generation of beta-band oscillations. On this system, a proportional regulation of the firing-rate compensates the loss of stability of the subthalamo-pallidal loop in the parkinsonian case. Nevertheless, because of actuation and measurement delays in the stimulation device, a filter with a well chosen frequency must be added to such proportional schemes, in order to achieve an adequate stability margin.

Closed-loop firing rate regulation of two interacting excitatory and inhibitory neural populations of the basal ganglia

This paper develops a new closed-loop firing rate regulation strategy for a population of neurons in the subthalamic nucleus, derived using a model-based analysis of the basal ganglia. The system is described using a firing rate model, in order to analyse the generation of beta-band oscillations. On this system, a proportional regulation of the firing rate reduces the gain of the subthalamo-pallidal loop in the parkinsonian case, thus impeding pathological oscillation generation. A filter with a well-chosen frequency is added to this proportional scheme, in order to avoid a potential instability of the feedback loop due to actuation and measurement delays. Our main result is a set of conditions on the parameters of the stimulation strategy that guarantee both its stability and a prescribed delay margin. A discussion on the applicability of the proposed method and a complete set of mathematical proofs is included.

Converse Lyapunov-Krasovskii theorems for uncertain retarded differential equations

In this article we give a collection of converse Lyapunov-Krasovskii theorems for uncertain retarded differential equations. We show that the existence of a weakly-degenerate Lyapunov-Krasovskii functional is a necessary and sufficient condition for the global exponential stability of linear retarded functional differential equations. This is carried out using a switched system representation approach.

Converse Lyapunov–Krasovskii theorems for uncertain time-delay systems

Abstract In this article, we give a collection of converse Lyapunov–Krasovskii theorems for uncertain time-delay systems. We show that the existence of a weakly-degenerate Lyapunov– Krasovskii functional is necessary and sufficient condition for the global exponential stability of the time-delay systems. This is carried out using the switched system transformation approach.

The buffered chemostat with non-monotonic response functions

We show how a particular spatial structure with a buffer globally stabilizes the chemostat dynamics with non-monotonic response function, while this is not possible with single, serial or parallel chemostats of the same total volume and input flow. We give a characterization of the set of such configurations that satisfy this property, as well as the configuration that ensures the best nutrient conversion. Furthermore, we characterize the minimal buffer volume to be added to a single chemostat for obtaining the global stability. These results are illustrated with the Haldane kinetic function.