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Why homeodynamics, not homeostasis?

Ideas of homeostasis derive from the concept of the organism as an open system. These ideas can be traced back to Heraclitus. Hopkins, Bernard, Hill, Cannon, Weiner and von Bertalanffy developed further the mechanistic basis of turnover of biological components, and Schoenheimer and Rittenberg were pioneers of experimental approaches to the problems of measuring pool sizes and dynamic fluxes. From the second half of the twentieth century, a biophysical theory mainly founded on self-organisation and Dynamic Systems Theory allowed us to approach the quantitative and qualitative analysis of the organised complexity that characterises living systems. This combination of theoretical framework and more refined experimental techniques revealed that feedback control of steady states is a mode of operation that, although providing stability, is only one of many modes and may be the exception rather than the rule. The concept of homeodynamics that we introduce here offers a radically new and all-embracing concept that departs from the classical homeostatic idea that emphasises the stability of the internal milieu toward perturbation. Indeed, biological systems are homeodynamic because of their ability to dynamically self-organise at bifurcation points of their behaviour where they lose stability. Consequently, they exhibit diverse behaviour; in addition to monotonic stationary states, living systems display complex behaviour with all its emergent characteristics, i.e., bistable switches, thresholds, waves, gradients, mutual entrainment, and periodic as well as chaotic behaviour, as evidenced in cellular phenomena such as dynamic (supra)molecular organisation and flux coordination. These processes may proceed on different spatial scales, as well as across time scales, from the very rapid processes within and between molecules in membranes to the slow time scales of evolutionary change. It is dynamic organisation under homeodynamic conditions that make possible the organised complexity of life.

Hemodynamics↗

Noise and periodic modulations in neural excitable media.

We have analyzed the interplay between noise and periodic modulations in a mean field model of a neural excitable medium. For this purpose, we have considered two types of modulations, namely, variations of the resistance and oscillations of the threshold. In both cases, stochastic resonance is present, irrespective of whether the system is monostable or bistable.

Biophysical Phenomena↗

Bi-instability as a precursor to global mixed-mode chaos.

Bi-instability, in contrast to bistability, is shown to generate unstable chaotic saddles prior to the onset of chaos. The theory and numerics are applied to a CO2 laser model with modulated losses where unstable pairs of saddles coexist, form heteroclinic connections, and allow mixing between local chaotic attractors to produce global mixed-mode chaos.

Journal Article↗

Spatial multistability and nonvariational effects.

We investigate the phenomenon of spatial multistability of fronts in thin bistable systems and stress the important role played by the absence of a variational principle. Nonvariational effects allow, for instance, two different immobilized fronts to coexist. The morphological instability of the corresponding nucleating solution can then lead, even in the absence of any diffusive instability, to nontrivial patterns in the depth of one-side-fed reactors.

Journal Article↗

Effect of correlation between additive and multiplicative noises on the activation from a double well.

We study the thermal activation problem of a bistable system driven by correlated additive and multiplicative noises by means of numerical simulation. We find that in the colored noise case the suppression effect of the positive correlation decreases and finally is released as the autocorrelation time tau of the colored noise grows. When tau is large enough, negative correlation becomes more suppressive than the positive correlation. Such phenomena are ascribed to the collaboration between noises as well as the memory effect of the multiplicative colored noise.

Journal Article↗

Binary data corruption due to a Brownian agent. II. Two dimensions, competing agents, and generalized couplings.

This work is a continuation of our previous investigation of binary data corruption due to a Brownian agent [Phys. Rev. E 59, 5172 (1999)]. We extend our study in three main directions which allow us to make closer contact with real bistable systems. These are (i) a detailed analysis of two dimensions, (ii) the case of competing agents, and (iii) the cases of asymmetric and quenched random couplings. Most of our results are obtained by extending our original phenomenological model, and are supported by extensive numerical simulations.

Journal Article↗

Self-similar renormalization approach to barrier crossing processes.

An algebraic self-similar renormalization method developed recently for summation of divergent field-theoretical series is applied to the thermally activated escape of a Brownian particle over an arbitrarily shaped barrier. Based on the Mel'nikov-Meshkov result for the underdamped Brownian motion and the inverse friction expansion of the underlying Fokker-Planck equation for strong friction, an overall rate formula is constructed. This formula agrees in the weak friction regime with the rate obtained from a diffusion equation in energy variables and, in the limiting case of strong friction with the rate following from a Smoluchowski equation. Its validity is tested for Brownian motion in bistable potentials with parabolic, cusped, and quartic barriers of different heights. The proposed formula is found to give a reasonable description of activated rate processes even though the barrier is quite low. Our comparison also includes results from various different crossover theories. In most of the cases considered the present formula is in considerably better agreement with exact numerical rates than the other interpolation formulas.

Journal Article↗

Amplification of weak signals and stochastic resonance via on-off intermittency with symmetry breaking.

Nonlinear dynamical systems possessing reflection symmetry have an invariant subspace in the phase space. The dynamics within the invariant subspace can be random or chaotic. As a system parameter changes, the motion transverse to the invariant subspace can lose stability, leading to on-off intermittency. Under certain conditions, the bursting behavior is symmetry breaking. We demonstrate the possibility of observing multiplicative noise(chaos)-induced amplification of weak signal and stochastic resonance via on-off intermittency with symmetry breaking in a general class of symmetrical systems. Differences of this mechanism of stochastic resonance to that in noisy bistable or threshold systems are discussed.

Journal Article↗

Avalanche of bifurcations and hysteresis in a model of cellular differentiation.

Cellular differentiation in a developing organism is studied via a discrete bistable reaction-diffusion model. A system of undifferentiated cells is allowed to receive an inductive signal emanating from its environment. Depending on the form of the nonlinear reaction kinetics, this signal can trigger a series of bifurcations in the system. Differentiation starts at the surface where the signal is received and either cells change type up to a given distance or, under other conditions, the differentiation process propagates throughout the whole domain. When the signal diminishes, hysteresis is observed.

Animals↗

Control of chaotic spatiotemporal spiking by time-delay autosynchronization.

A global time-delayed feedback control is applied to a globally coupled reaction-diffusion system describing charge transport in a bistable semiconductor. We demonstrate that a variety of spatiotemporal unstable periodic orbits (UPOs) embedded in a chaotic attractor of the spatially extended system can be stabilized using an extended time-delay autosynchronization algorithm. These UPOs correspond to spiking current filaments. We critically evaluate analytical approximations for the limits of control originally developed for low-dimensional temporal chaos and show that the delay time can be extrapolated with high accuracy, while the theoretical limit for the control of UPOs in terms of the product of the period and the largest Lyapunov exponent is not reached. If the global feedback is modified by a spatial filter, we achieve stabilization of different spatial patterns.

Journal Article↗

Multistability and symmetry breaking in the two-dimensional flow around a square cylinder.

We use numerical methods to study two-dimensional flow passing a square cylinder at low Reynolds numbers, and observe period-1 and -3 vortices behind the cylinder at the same Reynolds number Re. When Re increases from a small number to a critical value Re(c) approximately equal to 320, the system could change from bistability, which maintains the spatial symmetry, to tristability, which breaks the spatial symmetry. Our results suggest many interesting problems for further studies.

Journal Article↗

Nematic ordering in a cell with modulated surface anchoring: effects of flexoelectricity.

We have analyzed molecular ordering in a nematic sample sandwiched between two parallel substrates, characterized by a periodically varying anchoring easy axis. If the periodicity lambda is smaller than the Debye screening length l(D) and the nematic material possesses flexoelectric properties, it is necessary to take into account also the electrostatic and flexoelectric contributions in the thermodynamical potential when the actual director field is determined. In this framework, for small deviations from the homeotropic alignment we have derived analytical expressions for the tilt angle (theta) and the electrical potential. To establish a connection with experimentally observable quantities, we have related the theta profile to the average and investigated its behavior for different values of lambda, the flexoelectric coefficient, and the anchoring strength w. Our results indicate that in a nematic with pronounced flexoelectric properties for small enough lambda, a kind of subsurface deformation appears, which substantially decreases . Therefore, effects of flexoelectricity cannot be neglected in treating nematic cells with modulated anchoring which allows bistable ordering.

Journal Article↗

Parameter dependence of stochastic resonance in the stochastic Hodgkin-Huxley neuron.

Recently, the phenomena of stochastic resonance (SR) have attracted much attention in the studies of the excitable systems under inherent noise, in particular, nervous systems. We study SR in a stochastic Hodgkin-Huxley neuron under Ornstein-Uhlenbeck noise and periodic stimulus, focusing on the dependence of properties of SR on stimulus parameters. We find that the dependence of the critical forcing amplitude on the frequency of the periodic stimulus shows a bell-shaped structure with a minimum at the stimulus frequency, which is quite different from the monotonous dependence observed in the bistable system at a small frequency range. The frequency dependence of the critical forcing amplitude is explained in connection with the firing onset bifurcation curve of the Hodgkin-Huxley neuron in the deterministic situation. The optimal noise intensity for maximal amplification is also found to show a similar structure.

Animals↗

Stationary and drifting localized structures near a multiple bifurcation point

Localized states embedded in a patterned background are found in numerical simulations of spontaneous pattern formation in a spin-1/2 atomic system with optical feedback. In the vicinity of a parameter region with bistability between two homogeneous states large amplitude peaks as well as dark holes exist as stable localized states on a hexagonal background. Moreover, resonant interaction between oscillatory and stationary inhomogeneous modes produces a nonstationary background which may force the localized states to drift.

Journal Article↗

Pulse-driven switching in one-dimensional nonlinear photonic band gap materials: a numerical study

We numerically examine the time-dependent properties of nonlinear bistable multilayer structures for constant wave illumination. We find that our system exhibits both steady-state and self-pulsing solutions. In the steady-state regime, we examine the dynamics of driving the system between different transmission states by injecting pulses, and we find optimal pulse parameters. We repeat this work for the case of a linear periodic system with a nonlinear impurity layer.

Journal Article↗

Fault tolerance in noise-enhanced propagation.

As a mock-up of synaptic transmission between neurons, we revisit a problem that has recently risen the interest of several authors: the propagation of a low-frequency periodic signal through a chain of one-way coupled bistable oscillators, subject to uncorrelated additive noise. On a numerical study performed in the optimal range of noise intensity for which essentially undamped transmission along the chain has been reported, we focus on the outcome of feeding with noise all the nodes but the central one. A (moderate) critical value of the coupling between oscillators is found such that whereas below it the propagation can be considered to be interrupted at the "dead neuron," it is reestablished above it. Thus, one of the distinguishing features of synaptic transmission, namely, a fault-tolerant behavior that enhances reliability at the expense of efficiency, arises here as an emergent property of the system.

Models, Neurological↗

Variational theory of activated rate processes for an arbitrary barrier

The thermally activated escape of a Brownian particle over a smooth barrier of arbitrary shape and height is considered as an eigenproblem of the Fokker-Planck equation. For the case of moderate and large friction, the least nonzero eigenvalue of this equation is found via a Rayleigh-quotient-based perturbation method. A comparison with existing variational approaches and from numerical simulations for bistable potentials with parabolic and quartic barriers shows that the proposed expression gives unprecedentedly accurate results at all barrier heights including the limit of vanishingly low barrier.

Journal Article↗

Passage through a barrier with a slowly increasing control parameter

A bistable system subjected to noise and a slow increase of the parameter controlling the instability is studied, with emphasis on the kinetics of the transitions across the barrier separating the stable states. It is shown that the presence of a ramp in the control parameter may considerably affect the distribution of probability mass on the two sides of the barrier as compared to the predictions of the classical Kramers theory.

Journal Article↗