Self-tuning to the Hopf bifurcation in fluctuating systems

The problem of self-tuning a system to the Hopf bifurcation in the presence of noise and periodic external forcing is discussed. We find that the response of the system has a non-monotonic dependence on the noise-strength, and displays an amplified response which is more pronounced for weaker signals. The observed effect is to be distinguished from stochastic resonance. For the feedback we have studied, the unforced self-tuned Hopf oscillator in the presence of fluctuations exhibits sharp peaks in its spectrum. The implications of our general results are briefly discussed in the context of sound detection by the inner ear.Comment: 37 pages, 7 figures (8 figure files

Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations

We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Regarding regularity, we show that the RG map, defined on a suitable space of interactions (= formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension $d \ge 3$, these pathologies occur in a full neighborhood $\{ \beta > \beta_0 ,\, |h| < \epsilon(\beta) \}$ of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension $d \ge 2$, the pathologies occur at low temperatures for arbitrary magnetic-field strength. Pathologies may also occur in the critical region for Ising models in dimension $d \ge 4$. We discuss in detail the distinction between Gibbsian and non-Gibbsian measures, and give a rather complete catalogue of the known examples. Finally, we discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems.Comment: 273 pages including 14 figures, Postscript, See also ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.

Power gain exhibited by motile mechanosensory neurons in Drosophila ears

In insects and vertebrates alike, hearing is assisted by the motility of mechanosensory cells. Much like pushing a swing augments its swing, this cellular motility is thought to actively augment vibrations inside the ear, thus amplifying the ear's mechanical input. Power gain is the hallmark of such active amplification, yet whether and how much energy motile mechanosensory cells contribute within intact auditory systems has remained uncertain. Here, we assess the mechanical energy provided by motile mechanosensory neurons in the antennal hearing organs of Drosophila melanogaster by analyzing the fluctuations of the sound receiver to which these neurons connect. By using dead WT flies and live mutants (tilB(2), btv(5P1), and nompA(2)) with defective neurons as a background, we show that the intact, motile neurons do exhibit power gain. In WT flies, the neurons lift the receiver's mean total energy by 19 zJ, which corresponds to 4.6 times the energy of the receiver's Brownian motion. Larger energy contributions (200 zJ) associate with self-sustained oscillations, suggesting that the neurons adjust their energy expenditure to optimize the receiver's sensitivity to sound. We conclude that motile mechanosensory cells provide active amplification; in Drosophila, mechanical energy contributed by these cells boosts the vibrations that enter the ear