Frank Moss

Noise in Nonlinear Dynamical Systems

Abstract Noise is commonly regarded as having a destructive but relatively innocuous effect, blurring our view of a system but having no effect on the underlying processes involved. In this paper we show, using examples from stochastic nonlinear dynamics, that these intuitive ideas about noise can be very misleading. For example, an effect known as stochastic resonance means that the addition of extra noise to a system can actually improve the signal-to-noise ratio.

Low-Dimensional Dynamics in Sensory Biology 1: Thermally Sensitive Electroreceptors of the Catfish

We report the results of a search for evidence of periodic unstableorbits in the electroreceptors of the catfish. The function of thesereceptor organs is to sense weak external electric fields. Inaddition, they respond to the ambient temperature and to the ioniccomposition of the water. These quantities are encoded by receptorsthat make use of an internal oscillator operating at the level of themembrane potential. If such oscillators have three or more degreesof freedom, and at least one of which also exhibits a nonlinearity,they are potentially capable of chaotic dynamics. By detecting theexistence of stable and unstable periodic orbits, we demonstratebifurcations between noisy stable and chaotic behavior using theambient temperature as a parameter. We suggest that the techniquedeveloped herein be regarded as an additional tool for the analysisof data in sensory biology and thus can be potentially useful instudies of functional responses to external stimuli. We speculatethat the appearance of unstable orbits may be indicative of a stateof heightened sensory awareness by the animal.

Low-Dimensional Dynamics in Sensory Biology 2: Facial Cold Receptors of the Rat

We report the results of a search for evidence of unstable periodic orbits in the sensory afferents of the facial cold receptors of the rat. Cold receptors are unique in that they exhibit a diversity of action potential firing patterns as well as pronounced transients in firing rate following rapid temperature changes. These characteristics are the result of an internal oscillator operating at the level of the membrane potential. If such oscillators have three or more degree of freedom, and at least one of which also exhibits a nonlinearity, they are potentially capable of complex activity. By detecting the existence of unstable periodic orbits, we demonstrate low-dimensional dynamical behavior whose characteristics depend on the temperature range, impulse pattern, and temperature transients.

Noise sustained waves in subexcitable media: From chemical waves to brain waves

We discuss a novel type of spatiotemporal pattern that can be observed in subexcitable media when coupled to a thermal environment. These patterns have been recently observed in several different types of systems: a subexcitable photosensitive Belousov-Zhabotinsky reaction, hippocampal slices of rat brains, and astrocyte syncytium. In this paper, we introduce the basic concepts of subexcitable media, describe recent experimental observations in chemistry and neurophysiology, and put these observation into context with computer simulations. (c) 1998 American Institute of Physics.

Ito versus Stratonovich revisited

It is shown that a digital simulation of a noise induced phase transition using an algorithm consistent with the Ito stochastic calculus is in agreement with the predictions of that theory, whereas experiments with an analogue simulator yield measured results in agreement with the predictions of the Stratonovich theory.

Noise-induced impulse pattern modifications at different dynamical period-one situations in a computer model of temperature encoding.

We used a minimal Hodgkin-Huxley type model of cold receptor discharges to examine how noise interferes with the non-linear dynamics of the ionic mechanisms of neuronal stimulus encoding. The model is based on the assumption that spike-generation depends on subthreshold oscillations. With physiologically plausible temperature scaling, it passes through different impulse patterns which, with addition of noise, are in excellent agreement with real experimental data. The interval distributions of purely deterministic simulations, however, exhibit considerable differences compared to the noisy simulations especially at the bifurcations of deterministically period-one discharges. We, therefore, analyzed the effects of noise in different situations of deterministically regular period-one discharges: (1) at high-temperatures near the transition to subthreshold oscillations and to burst discharges, and (2) at low-temperatures close to and more far away from the bifurcations to chaotic dynamics. The data suggest that addition of noise can considerably extend the dynamical behavior of the system with coexistence of different dynamical situations at deterministically fixed parameter constellations. Apart from well-described coexistence of spike-generating and subthreshold oscillations also mixtures of tonic and bursting patterns can be seen and even transitions to unstable period-one orbits seem to appear. The data indicate that cooperative effects between low- and high-dimensional dynamics have to be considered as qualitatively important factors in neuronal encoding.

Interactions between slow and fast conductances in the Huber/Braun model of cold-receptor discharges

Abstract Transitions between different types of impulse patterns, according to experimentally recorded cold-receptor discharges, can successfully be mimicked with a minimal Hodgkin–Huxley-type simulation, here referred to as the Huber/Braun cold-receptor model. The model consists of two sets of simplified de- and repolarizing ionic conductances responsible for spike generation and slow-wave potentials, respectively. Over a broad temperature range, spike patterns are determined by the periodicity of subthreshold oscillations. At low temperatures, however, the periodicity of the pattern is destroyed and then appears again but with different patterns of different rhythmicity. We demonstrate that these complex transitions originate from the interactions between slow-wave and spike-generating currents.

Topological analysis of chaos in neural spike train bursts

We show how a topological model which describes the stretching and squeezing mechanisms responsible for creating chaotic behavior can be extracted from the neural spike train data. The mechanism we have identified is the same one (gateau roule, or jelly-roll) which has previously been identified in the Duffing oscillator [Gilmore and McCallum, Phys. Rev. E 51, 935 (1995)] and in a YAG laser [Boulant et al., Phys. Rev. E 55, 5082 (1997)]. (c) 1999 American Institute of Physics.

The breakdown of superfluidity in liquid 4He III. Nucleation of quantized vortex rings

We have measured the rate v at which negative ions nucleate charged vortex rings in isotopically pure superfluid 4He for pressures, P, temperatures, T, and electric fields, E, within the ranges: 15 < P <25 bar; 0.3 < T < 0.9 K; 5 x 104 < E < 106 V m-1. The measurements were done by a novel electrostatic induction technique specially developed for the purpose, and this is described in some detail. It was found that: at fixed E and P, v increases rapidly with T for T ca. 0.5 K, but approaches a temperature-independent limiting value vs for T < 0.5 K; at fixed P and T, v at first increases rapidly with E but then passes through a maximum at ca. 7 x 105 V m-1 and decreases again for larger values of E ;at fixed E and T, v increases rapidly with decreasing P until, below ca. 15 bar, the signal becomes too small to use. In all cases, v was found to be considerably smaller than had been measured for low E by earlier workers using helium of the natural isotopic ratio (ca. 2 x 10-7). The same signals were also used for measuring ionic drift velocities, v for v < ca. 3 x 1O4 s-1. Values of the matrix element for roton pair emission have been deduced from the v(E) measurements for several pressures in the range 17 < P < 25 bar. The pressure dependence of the Landau critical velocity was measured and is compared with predictions based on accepted values of the roton parameters. Analysis of the nucleation data showed that, at fixed v and P,(v—vs) oc nr,where nr is the thermal roton density, suggesting that v is the sum of contributions from two independent nucleation mechanisms: a spontaneous mechanism responsible for vs and a roton driven mechanism responsible for the increase in v with T above 0.5 K. The existence of a maximum in v(E) appears to be inconsistent with the peeling model of vortex nucleation; but it is entirely to be expected on the basis of the quantum transition model. It is shown that all the nucleation rate measurements reported herein are consistent with the quantum transition model, provided that due account is taken of the possibility that roton absorption may give rise to a critical velocity vr that is smaller than the critical velocity vv characteristic of the spontaneous nucleation mechanism. Values of vv and vr are deduced from the experimental data for several pressures. The fact that exponential decay of the bare ion signal still occurs even when v > vv (or vr) constitutes the first experimental evidence that the microscopic mechanisms responsible for vortex nucleation are probabilistic in nature.

Observation of symmetry breaking, state selection and sensitivity in a noisy electronic system

Abstract Measurements of the time evolution of the statistical density as a control parameter is swept through a bifurcation point in the presence of additive noise are presented. We have used these densities to study branch selectivity, which results from a small, additive interaction. A universal time scale for all such processes is established. The results are in good agreement with a recent theory.