Formation of antiwaves in gap-junction-coupled chains of neurons

Using network models consisting of gap junction coupled Wang-Buszaki neurons, we demonstrate that it is possible to obtain not only synchronous activity between neurons but also a variety of constant phase shifts between 0 and \pi. We call these phase shifts intermediate stable phaselocked states. These phase shifts can produce a large variety of wave-like activity patterns in one-dimensional chains and two-dimensional arrays of neurons, which can be studied by reducing the system of equations to a phase model. The 2\pi periodic coupling functions of these models are characterized by prominent higher order terms in their Fourier expansion, which can be varied by changing model parameters. We study how the relative contribution of the odd and even terms affect what solutions are possible, the basin of attraction of those solutions and their stability. These models may be applicable to the spinal central pattern generators of the dogfish and also to the developing neocortex of the neonatal rat

The scaling behaviour of screened polyelectrolytes

We present a field-theoretic renormalization group (RG) analysis of a single flexible, screened polyelectrolyte chain (a Debye-H\"uckel chain) in a polar solvent. We point out that the Debye-H\"uckel chain may be mapped onto a local field theory which has the same fixed point as a generalised $n \to 1$ Potts model. Systematic analysis of the field theory shows that the system is one with two interplaying length-scales requiring the calculation of scaling functions as well as exponents to fully describe its physical behaviour. To illustrate this, we solve the RG equation and explicitly calculate the exponents and the mean end-to-end length of the chain.Comment: 6 pages, 1 figure; changed title and slight modification to tex

A Perturbative/Variational Approach to Quantum Lattice Hamiltonians

We propose a method to construct the ground state $\psi(\lambda)$ of local lattice hamiltonians with the generic form $H_0 + \lambda H_1$, where $\lambda$ is a coupling constant and $H_0$ is a hamiltonian with a non degenerate ground state $\psi_0$. The method is based on the choice of an exponential ansatz $\psi(\lambda) = {\rm exp}(U(\lambda)) \psi_0$, which is a sort of generalized lattice version of a Jastrow wave function. We combine perturbative and variational techniques to get succesive approximations of the operator $U(\lambda)$. Perturbation theory is used to set up a variational method which in turn produces non perturbative results. The computation with this kind of ansatzs leads to associate to the original quantum mechanical problem a statistical mechanical system defined in the same spatial dimension. In some cases these statistical mechanical systems turn out to be integrable, which allow us to obtain exact upper bounds to the energy. The general ideas of our method are illustrated in the example of the Ising model in a transverse field.Comment: 27 pages, three .ps figures appended, DFTUZ 94-2

Entanglement Mean Field Theory and the Curie-Weiss Law

The mean field theory, in its different hues, form one of the most useful tools for calculating the single-body physical properties of a many-body system. It provides important information, like critical exponents, of the systems that do not yield to an exact analytical treatment. Here we propose an entanglement mean field theory (EMFT) to obtain the behavior of the two-body physical properties of such systems. We apply this theory to predict the phases in paradigmatic strongly correlated systems, viz. the transverse anisotropic XY, the transverse XX, and the Heisenberg models. We find the critical exponents of different physical quantities in the EMFT limit, and in the case of the Heisenberg model, we obtain the Curie-Weiss law for correlations. While the exemplary models have all been chosen to be quantum ones, classical many-body models also render themselves to such a treatment, at the level of correlations.Comment: 5 pages, 4 figure

Finite Temperature and Dynamical Properties of the Random Transverse-Field Ising Spin Chain

We study numerically the paramagnetic phase of the spin-1/2 random transverse-field Ising chain, using a mapping to non-interacting fermions. We extend our earlier work, Phys. Rev. 53, 8486 (1996), to finite temperatures and to dynamical properties. Our results are consistent with the idea that there are ``Griffiths-McCoy'' singularities in the paramagnetic phase described by a continuously varying exponent $z(\delta)$, where $\delta$ measures the deviation from criticality. There are some discrepancies between the values of $z(\delta)$ obtained from different quantities, but this may be due to corrections to scaling. The average on-site time dependent correlation function decays with a power law in the paramagnetic phase, namely $\tau^{-1/z(\delta)}$, where $\tau$ is imaginary time. However, the typical value decays with a stretched exponential behavior, $\exp(-c\tau^{1/\mu})$, where $\mu$ may be related to $z(\delta)$. We also obtain results for the full probability distribution of time dependent correlation functions at different points in the paramagnetic phase.Comment: 10 pages, 14 postscript files included. The discussion of the typical time dependent correlation function has been greatly expanded. Other papers of APY are available on-line at http://schubert.ucsc.edu/pete

Coupling between feedback loops in autoregulatory networks affects bistability range, open-loop gain and switching times

Biochemical regulatory networks governing diverse cellular processes such as stress-response, differentiation and cell cycle often contain coupled feedback loops. We aim at understanding how features of feedback architecture, such as the number of loops, the sign of the loops and the type of their coupling, affect network dynamical performance. Specifically, we investigate how bistability range, maximum open-loop gain and switching times of a network with transcriptional positive feedback are affected by additive or multiplicative coupling with another positive- or negative-feedback loop. We show that a network's bistability range is positively correlated with its maximum open-loop gain and that both quantities depend on the sign of the feedback loops and the type of feedback coupling. Moreover, we find that the addition of positive feedback could decrease the bistability range if we control the basal level in the signal-response curves of the two systems. Furthermore, the addition of negative feedback has the capacity to increase the bistability range if its dissociation constant is much lower than that of the positive feedback. We also find that the addition of a positive feedback to a bistable network increases the robustness of its bistability range, whereas the addition of a negative feedback decreases it. Finally, we show that the switching time for a transition from a high to a low steady state increases with the effective fold change in gene regulation. In summary, we show that the effect of coupled feedback loops on the bistability range and switching times depends on the underlying mechanistic details

Aperiodic Ising model on the Bethe lattice: Exact results

We consider the Ising model on the Bethe lattice with aperiodic modulation of the couplings, which has been studied numerically in Phys. Rev. E 77, 041113 (2008). Here we present a relevance-irrelevance criterion and solve the critical behavior exactly for marginal aperiodic sequences. We present analytical formulae for the continuously varying critical exponents and discuss a relationship with the (surface) critical behavior of the aperiodic quantum Ising chain.Comment: 7 pages, 3 figures, minor correction

Crossover between aperiodic and homogeneous semi-infinite critical behaviors in multilayered two-dimensional Ising models

We investigate the surface critical behavior of two-dimensional multilayered aperiodic Ising models in the extreme anisotropic limit. The system under consideration is obtained by piling up two types of layers with respectively $p$ and $q$ spin rows coupled via nearest neighbor interactions $\lambda r$ and $\lambda$, where the succession of layers follows an aperiodic sequence. Far away from the critical regime, the correlation length $\xi_\perp$ is smaller than the first layer width and the system exhibits the usual behavior of an ordinary surface transition. In the other limit, in the neighborhood of the critical point, $\xi_\perp$ diverges and the fluctuations are sensitive to the non-periodic structure of the system so that the critical behavior is governed by a new fixed point. We determine the critical exponent associated to the surface magnetization at the aperiodic critical point and show that the expected crossover between the two regimes is well described by a scaling function. From numerical calculations, the parallel correlation length $\xi_\parallel$ is then found to behave with an anisotropy exponent $z$ which depends on the aperiodic modulation and the layer widths.Comment: LaTeX file, 9 pages, 8 eps figures, to appear in Phys. Rev.

Griffiths-McCoy singularities in the transverse field Ising model on the randomly diluted square lattice

The site-diluted transverse field Ising model in two dimensions is studied with Quantum-Monte-Carlo simulations. Its phase diagram is determined in the transverse field (Gamma) and temperature (T) plane for various (fixed) concentrations (p). The nature of the quantum Griffiths phase at zero temperature is investigated by calculating the distribution of the local zero-frequency susceptibility. It is pointed out that the nature of the Griffiths phase is different for small and large Gamma.Comment: 21 LaTeX (JPSJ macros included), 12 eps-figures include

Thermodynamics of histories for the one-dimensional contact process

The dynamical activity K(t) of a stochastic process is the number of times it changes configuration up to time t. It was recently argued that (spin) glasses are at a first order dynamical transition where histories of low and high activity coexist. We study this transition in the one-dimensional contact process by weighting its histories by exp(sK(t)). We determine the phase diagram and the critical exponents of this model using a recently developed approach to the thermodynamics of histories that is based on the density matrix renormalisation group. We find that for every value of the infection rate, there is a phase transition at a critical value of s. Near the absorbing state phase transition of the contact process, the generating function of the activity shows a scaling behavior similar to that of the free energy in an equilibrium system near criticality.Comment: 16 pages, 7 figure