Multimode dynamics in a network with resource mediated coupling

Abstract

The purpose of this paper is to study the special forms of multimode dynamics that one can observe in systems with resource-mediated coupling, i.e., systems of self-sustained oscillators in which the coupling takes place via the distribution of primary resources that controls the oscillatory state of the individual unit. With this coupling, a spatially inhomogenous state with mixed high and lowamplitude oscillations in the individual units can arise. To examine generic phenomena associated with this type of interaction we consider a chain of resistively coupled electronic oscillators connected to a common power supply. The two-oscillator system displays antiphase synchronization, and it is interesting to note that two-mode oscillations continue to exist outside of the parameter range in which oscillations occur for the individual unit. At low coupling strengths, the multioscillator system shows high dimensional quasiperiodicity with little tendency for synchronization. At higher coupling strengths, one typically observes spatial clustering involving a few oscillating units. We describe three different scenarios according to which the cluster can slide along the chain as the bias voltage changes. © 2008 American Institute of Physics. ͓DOI: 10.1063/1.2805194͔ Systems of interacting nonlinear oscillators play an important role in many areas of science and technology. A rat kidney, for instance, contains of the order of 30 000 nephrons, each of which tends to display large amplitude oscillations in the regulation of its incoming blood flow. The arteriolar blood pressure controls the oscillatory state of the individual nephron. At the same time, the nephrons interact via the displacement of blood from one nephron to its neighbors produced by this oscillatory regulation. We refer to such structures, in which the oscillatory dynamics of the individual unit can no longer be separated from the coupling among the units, as a system with resource-mediated coupling. To examine some of the generic aspects of this type of system, the paper considers a chain of resistively coupled electronic oscillators sharing a common power supply. For low coupling strengths, the system tends to produce multimode dynamics in the form of high-dimensional quasiperiodicity with little tendency for synchronization. At higher coupling strengths the resource distributed chain displays spatial clustering involving a small number of oscillatory units. In all cases the distribution of individual amplitudes is highly inhomogeneous, showing an interesting example of spatiotemporal mixed-mode oscillations. We study the mechanisms by which this type of cluster slides along the chain as the external bias voltage is changed.

Multinephron dynamics on the renal vascular network

“…Branches in the network distribute blood to nephrons and serve as nodes at which electrical signals converge. The periodic constriction and relaxation of afferent arterioles also affect the hydrostatic pressure in the nodes, providing a second mechanism for nephrons to interact (30,39,40).…”

“…Bayram and Pitman (3) used the model of Layton et al (22) in a multinephron configuration and found enhanced stability but did not use a parameter set that produced complex dynamics. Postnov et al (39) assembled a multinephron model to examine blood flow dynamics, viewing the vascular structure as a resource distribution network; they used the single nephron model of Barfred et al (2) but did not include electrotonic messaging, Marsh et al (30) studied a multinephron network, also using the Barfred model, and included both internephron signaling and vascular pressure dynamics. The model lacked a myogenic representation and used a simple vascular network structure; it developed some complex dynamics.…”

“…The argument of is the mean phase angle difference, ⌬ , between the two processes. For blood flows in nephron pairs, in-phase synchronization (⌬ ϭ 0) has been predicted to occur if electrotonic signaling is the predominant or only synchronization mechanism(32); antiphase synchronization has been posited when hydrodynamic synchronization prevails (39,40).…”

Coupling-induced complexity in nephron models of renal blood flow regulation

“…Beach et al (4) concluded that the TGF oscillation is more stable if nephrons interact than it is in single nephrons. The renal vascular structure is an example of a resource distribution network that can be a source of complex dynamics from both signaling and cardiovascular sources (27,38,39).…”

Introduction to Focus Issue: Mixed Mode Oscillations: Experiment, Computation, and Analysis

“…As a simple example of a network of oscillators which share a common resource which controls the individual oscillator, Postnov et al 44 consider a network of electric oscillators with a common power supply. Depending on the coupling strength, the system exhibits a range of different MMOs.…”