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Notes: We investigate stationary and nonstationary probability densities for a weakly forced nonlinear physical or chemical system that displays self-oscillations in the absence of forcing. The period and amplitude of forcing are taken as adjustable constraints. We consider a homogeneous reaction system described by a master equation. Our method of solution is based on the Wentzel–Kramers–Brillouin (WKB) expansion of the probability density with the system size as the expansion parameter. The first term in this expansion is the stochastic potential (eikonal). In the absence of forcing, the probability density is logarithmically flat on the limit cycle. With periodic forcing, the phenomenon of phase locking can occur whereby a stable cycle, which is close to the unforced cycle, adopts a constant relative phase to the forcing. A saddle cycle also exists and has a different constant relative phase. For such phase-locked solutions, the distribution over the relative phases is peaked on the stable cycle and exhibits a logarithmically flat region (a plateau) that originates on the saddle cycle. This plateau is due to a nonzero relative phase slippage: large fluctuations from the stable cycle over the saddle cycle are overwhelmingly more probable in a certain relative phase direction, which depends upon the location of the parameters within an entrainment region. This distribution of relative phases is logarithmically equivalent to that of a Brownian particle in a periodic potential with a constant external force in the strong damping and weak noise limits. For parameter values outside of an entrainment region (for which a quasiperiodic solution exists), the distribution in relative phase is logarithmically flat. For this regime, we investigate the evolution of an initially localized density and show that the width grows proportionally with the square root of time. The proportionality factor depends upon both the position (phase) on the cross section of the peak of the density and the distance in parameter space from the boundary of the entrainment region. For parameter values that approach the boundary of an entrainment region, this proportionality factor tends to infinity. We also determine an expression for the first order correction to the stochastic potential for both entrained and quasiperiodic solutions. A thermodynamic interpretation of these results is made possible by the equality of the stochastic potential with an excess work function. © 1998 American Institute of Physics.

Notes: The eikonal (WKB) approximation is applied to a stationary one-dimensional master equation describing an arbitrary reaction mechanism. The uniqueness of a nontrivial (fluctuational) eikonal solution is proven. Consistent eikonal and exact analytical solutions are obtained for systems with an arbitrary, but unique step size of stochastic transitions. An analytical eikonal solution for the stationary probability density for systems with mixed step sizes of 1 and 2 is also obtained and found to differ significantly from the systems with a uniform step size, particularly in the case of multiple stationary states. © 1995 American Institute of Physics.

Notes: The eikonal approximation (instanton technique) is applied to the problem of large fluctuations of the number of species in spatially homogeneous chemical reactions with the probability density distribution described by a master equation. For both autocatalytic and nonautocatalytic reactions, the analysis of the distribution about a stable stationary state and of the transitions between coexisting stable states comes, to logarithmic accuracy, to the analysis of Hamiltonian dynamics of an auxiliary dynamical system. The latter can be done explicitly in a few cases, including one-species systems, systems with detailed balance, and systems close to the bifurcation points where the number of the stable states changes. In the last case, the fluctuations display universal features, and, for saddle-node bifurcation points, the logarithm of the probability of escape from the metastable state (per unit time) is proportional to the distance to the bifurcation point (in the parameter space) raised to the power 3/2. We compare the eikonal approximation for the stationary distribution of a master equation to Monte Carlo numerical solutions for two chemical two-variable systems with multiple stationary states, where none of the cited restrictions exists. For one of the systems in the pattern of optimal paths we observe caustics emanating from the saddle point.

Notes: We examine the effect of periodic and discrete perturbations on the phase of an oscillatory chemical reaction system near a Hopf bifurcation. Discrete perturbations reset the phase of the oscillation and periodic perturbations entrain the frequency of the oscillation for perturbation frequencies in a small range about each rational multiple of the natural frequency. These phase responses may be determined from time series of a single essential species. The new phase resulting from discrete perturbations and the relative phase between the oscillation and the forcing of an entrained oscillation are described by the same response function, which is a simple sinusoid. We show that for single species perturbations, the amplitude and phase offset of this response function equal the magnitude and the argument, respectively, of the corresponding component of the adjoint eigenvector of the Jacobi matrix (that corresponds to a pure imaginary eigenvalue). These phase response methods are simpler than quenching studies for determining the adjoint eigenvectors, and in addition yield the local isochrons of the periodic orbit. © 1995 American Institute of Physics.

Notes: A random path integral representation of the Ross–Hunt–Hunt thermodynamic and stochastic theory is given for chemical reactions far from equilibrium in the case of constant-step and one-variable processes. An explicit analytical expression for the chemical Lagrangian is presented. A connection is made between the thermodynamic fluctuation–dissipation regimes characteristic to the process and the chemical Lagrangian. The path integral formalism is used to prove the validity of fluctuation regression hypothesis and to derive two variational principles for the most probable and average paths, respectively. The most probable path corresponds to the absolute maximum of the Lagrangian and the average path corresponds to the minimum value of the information gain obtained by observing a certain average path. For nonlinear regimes these two variational principles generally give distinct results; they are identical only in the vicinity of a stable steady state. An eikonal approximation is suggested for evaluating time-dependent probability distributions which reduces the integration of Master Equations to two quadratures. The suggested eikonal approximation leads to a proportionality between the species-specific free energy of the system and the extremal value of the time integral of the chemical Lagrangian. This relationship is similar to the expression of the mechanical action in terms of the Lagrangian in classical mechanics. Most results derived in this paper for one variable can be extended to multivariable systems. Finally a comparison is made with other stochastic approaches to nonequilibrium thermodynamics.

Notes: The transcephalic DC potential is that voltage recorded across the midline surface of the head between the frontal and occipital emissary vein distributions. Some psychophysiological correlates of this potential are described, and it is noted that little is known about the neuronal or biochemical modulation of it. Experiments are described using mature male cats, rabbits and rats as subjects. The frontal potential is found to shift progressively more positive as the depth of anesthesia increases. Pain causes a brief negative frontal shift and visceral irritation a positive one. Pinch mimics pain responses in etherized animals, but the DC shift is dampened in barbiturate anesthetized ones. Intracarotid injections of histamine and a histamine releaser produce a negative frontal shift. Heparin, serotonin, and nembutal produce positive frontal shifts. Epinephrine produces either a positive or negative shift, and potentiates the effects of histamine and serotonin. Histamine and serotonin combined produce a negative frontal shift. Many of these compounds are effective in 1μg or smaller doses. These findings, plus further analyses of the persistence, magnitude, latency, and dosage relatedness of the shifts are presented.