Martin Bier

Remerging Feigenbaum trees in dynamical systems

Abstract Finite sequences of remerging period-doubling bifurcations have been recently observed in a variety of physically interesting dynamical systems. We show here that such remerging Feigenbaum trees are quite common in models with more than one parameter and discuss a number of criteria under which they are generally observed. These criteria are applied to simple mappings as well as the conservative Duffings equation where the formation of a primary “bubble” is seen to lead to higher-order bubbles and hence to remerging Feigenbaum sequences. In the case of Duffings equation, we follow the development of one such sequence, with the aid of the variation of the winding number along a symmetry axis of the problem.

Alteration in sensory nerve function following electrical shock

A study of the effects of electrical shock on peripheral nerve fibres is presented. Strength and duration of the applied shocks were similar to those encountered in a typical industrial electrical accident. The purpose of this study is: (i) to identify the electrophysiological and morphological change in nerve fibres after the application of electrical current shocks; (ii) to examine the ability of the peripheral nerve fibres to spontaneously regain function and; (iii) to demonstrate the usefulness of the sensory refractory spectrum as an additional technique in assessing the damage. Three groups of animals received twelve 4-ms electric field pulses of approximately 37 V/cm (n = 5), 75 V/cm (n = 9) and 150 V/cm (n = 6), respectively. Group 4 was a control group and received a direct application of 2 per cent lidocaine over the sciatic nerve for 30 min. Thermal effects of the shocks were negligible. The sensory refractory spectrum shows that electrical shock damage was mainly to the large, fast myelinated fibres and that higher field strengths do more damage. Also in a histological examination it was found that the more heavily shocked myelinated fibres had sustained more damage.

KINETICS OF SEALING FOR TRANSIENT ELECTROPORES IN ISOLATED MAMMALIAN SKELETAL MUSCLE CELLS

Permeabilization of the plasma membrane by electrical forces (electroporation) can be either transient or stable. Although the exact molecular mechanics have not yet been described, electroporation is believed to initiate primarily in the lipid bilayer. To better understand the kinetics of membrane permeabilization, we sought to determine the time constants for spontaneous transient pore sealing. By using isolated rat flexor digitorum brevis skeletal muscle cells and a two-compartment diffusion model, we found that pore sealing times (tau p) after transient electroporation were approximately 9 min. tau p was not significantly dependent on the imposed transmembrane potential. We also determined the transmembrane potential (delta Vm) thresholds necessary for transient and stable electroporation in the skeletal muscle cells. delta VmS ranging between 340 mV and 480 mV caused a transient influx of magnesium, indicating the existence of spontaneously sealing pores. An imposed delta Vm of 540 mV or greater led to complete equilibration of the intracellular and extracellular magnesium concentrations. This finding suggests that stable pores are created by the larger imposed transmembrane potentials. These results may be useful for understanding nerve and skeletal muscle injury after an electrical shock and for developing optimal strategies for accomplishing transient electroporation, particularly for gene transfection and cell transformation.

Brownian ratchets in physics and biology

Thirty years ago Feynman et al. presented a paradox in the Lectures on Physics: an imagined device could let Brownian motion do work by allowing it in one direction and blocking it in the opposite direction. In the chapter Feynman et al. eventually show that such ratcheting can only be achieved if there is, in compliance with the basic conservation laws, some energy input from an external source. Now that technology is going into ever smaller dimensions, ratcheting Brownian motion seems to be a real possibility in nanotechnological applications. Furthermore, Brownian motion plays an essential role in the action of motor proteins (individual molecules that convert chemical energy into motion).

Control analysis of glycolytic oscillations

The principles involved in the control of the frequency of sustained metabolic oscillations are developed in the context of glycolytic oscillations in Saccharomyces cerevisiae. To this purpose, an existing mathematical model that describes the experimentally obtained oscillations was simplified to a core model. Frequency, relative phase, average concentrations and amplitudes of the oscillations were well approximated by writing the two remaining metabolic variables of the core model (representing [ATP] and [hexose]) as harmonic functions of time and by requiring them to fulfill the differential equations. The extent to which an enzyme (-conglomerate) controls the frequency in a sustained oscillation is defined as the log-log derivative of that frequency with respect to enzyme activity. In both the full model and the core model this control of frequency and the control over the average concentrations proved to be distributed over the enzymes. We identified a summation theorem, stating that the sum of such control coefficients over all processes equals unity for frequency and zero for the average concentrations.

Reversals of noise induced flow

Abstract We consider diffusive motion on a periodic, anisotropic potential. Adding a zero-average force that fluctuates between three values: − F , 0 and + F , can bring about net flow. As the frequency of the fluctuations varies the direction of the flux can change. We discuss a possible application for the construction of a device to separate macromolecules.

First order phase transition and hysteresis in a cell's maintenance of the membrane potential: An essential role for the inward potassium rectifiers

Hysteretic behavior is found experimentally in the transmembrane potential at low extracellular potassium in mouse lumbrical muscle cells. Adding isoprenaline to the external medium eliminates the bistable, hysteretic region. The system can be modeled mathematically and understood analytically with and without isoprenaline. Inward rectifying potassium channels appear to be essential for the bistability. Relations are derived to express the dimensions of the bistable area in terms of system parameters. The selective advantage and evolutionary origin of inward rectifying channels and hysteretic behavior is discussed.

On the singularity analysis of intersecting separatrices in near-integrable dynamical systems

Abstract It has been recently proved by S. L. Ziglin that transversal intersections of separatrices (invariant manifolds) in near-integrable Hamiltonian systems of two degrees of freedom imply the existence of multi-valued solutions with infinitely many Riemann sheets. Ziglins theorem is illustrated here on a simple example and then extended and applied to non-Hamiltonian, analytic flows x =f( x ) + eg( x ,t) , with x ≡(x,y) and g( x ,t)=g( x ,t+2π) , which for e = 0 possess the Painleve property. On the other hand, the theoretically expected logarithmic singularities for e ≠ 0 are obtained explicitly in solutions near the intersecting separatrices. Thus, we conjecture that dynamical systems with the Painleve property, can have no separatrix intersections and hence no strange attractors, etc. These singularities are then numerically located and found to form certain very interesting “chimney” patterns in the complex t-plane, on which they accumulate densely. The upper part of the chimneys (away from the Re t axis) is asymptotically quite insensitive to changes in parameter values or initial conditions. The singularity pattern itself, however, becomes “denser” and each chimney is seen to gradually “collapse” towards the Re t axis, as the amplitude of the driving term increases and the motion becomes more chaotic.

On the integrability of some generalized Lotka-Volterra systems

Abstract Several integrable systems of nonlinear ordinary differential equations of the Lotka-Volterra type are identified by the Painleve property and completely integrated. One such integrable case of N first order odes is found, with N − 2 free parameters and N arbitrary. The concept of integrability of a general dynamical system, not necessarily derived from a hamiltonian, is also discussed.

A bistable membrane potential at low extracellular potassium concentration.

In order to understand the electrochemical behavior of a living cell at a low extracellular potassium concentration, a model is constructed. The model involves only the ATP driven sodium-potassium pump, and the sodium and potassium channels. Predictions of the model fit the N-shape of the current-voltage characteristic at low extracellular potassium. The model can, furthermore, quantitatively account for the experimentally observed bistability of the membrane potential at low extracellular potassium concentration. A crucial role in the control of the transmembrane potential appears to be played by how the permeability of the inward rectifying potassium channels depends on the transmembrane potential and on the extracellular potassium concentration.