Phase resetting, phase locking, and bistability in the periodically driven saline oscillator: Experiment and model

Abstract

The saline oscillator consists of an inner vessel containing salt water partially immersed in an outer vessel of fresh water, with a small orifice in the center of the bottom of the inner vessel. There is a cyclic alternation between salt water flowing downwards out of the inner vessel into the outer vessel through the orifice and fresh water flowing upwards into the inner vessel from the outer vessel through that same orifice. We develop a very stable (i.e., stationary) version of this saline oscillator. We first investigate the response of the oscillator to periodic forcing with a train of stimuli (period=Tp) of large amplitude. Each stimulus is the quick injection of a fixed volume of fresh water into the outer vessel followed immediately by withdrawal of that very same volume. For Tp sufficiently close to the intrinsic period of the oscillator (T0) , there is 1:1 synchronization or phase locking between the stimulus train and the oscillator. As Tp is decreased below T0 , one finds the succession of phase-locking rhythms: 1:1, 2:2, 2:1, 2:2, and 1:1. As Tp is increased beyond T0 , one encounters successively 1:1, 1:2, 2:4, 2:3, 2:4, and 1:2 phase-locking rhythms. We next investigate the phase-resetting response, in which injection of a single stimulus transiently changes the period of the oscillation. By systematically changing the phase of the cycle at which the stimulus is delivered (the old phase), we construct the new-phase--old-phase curve (the phase transition curve), from which we then develop a one-dimensional finite-difference equation ("map") that predicts the response to periodic stimulation. These predicted phase-locking rhythms are close to the experimental findings. In addition, iteration of the map predicts the existence of bistability between two different 1:1 rhythms, which was then searched for and found experimentally. Bistability between 1:1 and 2:2 rhythms is also encountered. Finally, with one exception, numerical modeling with a phenomenologically derived Rayleigh oscillator reproduces all of the experimental behavior.

Teaching cardiac electrophysiology modeling to undergraduate students: laboratory exercises and GPU programming for the study of arrhythmias and spiral wave dynamics

“…1A). The difference in densities, combined with gravity, leads to a Rayleigh-Taylor instability (37,64) that results in a rhythmic change in flow of fluid from one container to the other that can last for many hours (12,39,58). Although the phase transition dynamics are complicated (59), qualitatively the denser fluid has a tendency to fall and the less dense fluid has a tendency to rise.…”

“…In the first week, F. H. Fenton and R. Singh gave a series of lectures and laboratory exercises on excitable systems in biology (41,42,66,72), physics (39,45,55), chemistry (6,8,78), and mathematics (26,43,48). This background provided students with a robust understanding of excitable systems and their relationship to complex systems and chaos (35,71,77).…”

Functional Characterization of Oscillatory and Excitable Media

“…We believe that it would still be of interest to carry out detailed resetting experiments on biological and physical oscillators with a view of understanding in detail the evolution of the resetting curves as a function of stimulus amplitude, and using that data to predict entrainment. Although there has been some work done in which the amplitude of the perturbing stimuli has been constant (González et al 2008), systematic analyses of resetting over a broad range of stimuli amplitudes and entrainment over a broad range of stimuli amplitudes and frequencies are rare.…”

Density Oscillators

“…Fresh water The salt oscillator exhibits fluid oscillations even when the salt and fresh water are replaced with two other miscible fluids of different densities, because the density difference is the most crucial factor for generating oscillations (Martin, 1970;Alfredsson and Lagerstedt, 1981;Yoshikawa et al, 1988;Noyes, 1989;Yoshikawa and Nakata, 1990;Yoshikawa et al, 1991;Steinbock et al, 1998;Nakata et al, 1998;Yamada, 1999, 2001;Aoki, 2000;Okamura and Yoshikawa, 2000;Ueno et al, 2006;Kano and Kinoshita, 2007, 2008, 2009González et al, 2008). Thus, this system is also called a density oscillator (Alfredsson and Lagerstedt, 1981;Steinbock et al, 1998;Ueno et al, 2006;Kano and Kinoshita, 2007, 2008, 2009.…”