The random transitions of ion channels between conducting and nonconducting states generate a source of internal fluctuations in a neuron, known as channel noise. The standard method for modeling the states of ion channels nonlinearly couples continuous-time Markov chains to a differential equation for voltage. Beginning with the work of R. F. Fox and Y.-N. Lu [Phys. Rev. E 49, 3421 (1994)], there have been attempts to generate simpler models that use stochastic differential equation (SDEs) to approximate the stochastic spiking activity produced by Markov chain models. Recent numerical investigations, however, have raised doubts that SDE models can capture the stochastic dynamics of Markov chain models.

We analyze three SDE models that have been proposed as approximations to the Markov chain model: one that describes the states of the ion channels and two that describe the states of the ion channel subunits. We show that the former channel-based approach can capture the distribution of channel noise and its effects on spiking in a Hodgkin-Huxley neuron model to a degree not previously demonstrated, but the latter two subunit-based approaches cannot. Our analysis provides intuitive and mathematical explanations for why this is the case. The temporal correlation in the channel noise is determined by the combinatorics of bundling subunits into channels, but the subunit-based approaches do not correctly account for this structure. Our study confirms and elucidates the findings of previous numerical investigations of subunit-based SDE models. Moreover, it presents evidence that Markov chain models of the nonlinear, stochastic dynamics of neural membranes can be accurately approximated by SDEs. This finding opens a door to future modeling work using SDE techniques to further illuminate the effects of ion channel fluctuations on electrically active cells.

An important goal for synthetic biology is to build robust and tunable genetic regulatory networks that are capable of performing assigned operations, usually in the presence of noise. In this work, a synthetic gene network derived from the bacteriophage $\lambda$ underpins a reconfigurable logic gate wherein we exploit noise and nonlinearity through the application of the logical stochastic resonance paradigm. This biological logic gate can emulate or “morph” the AND and OR operations through varying internal system parameters in a noisy background. Such genetic circuits can afford intriguing possibilities in the realization of engineered genetic networks in which the actual function of the gate can be changed after the network has been built, via an external control parameter. In this article, the full system characterization is reported, with the logic gate performance studied in the presence of external and internal noise. The robustness of the gate, to noise, is studied and illustrated through numerical simulations.

For the direct cascade of steady two-dimensional (2D) Navier-Stokes turbulence, we derive analytically the probability of strong vorticity fluctuations. When $\varpi$ is the vorticity coarse-grained over a scale $R$, the probability density function (PDF), $\mathcal{P}(\varpi )$, has a universal asymptotic behavior $\ln \mathcal{P}~-\varpi /{\varpi }_{\mathrm{rms}}$ at $\varpi \gg {\varpi }_{\mathrm{rms}}=[H\ln (L/R)]{}^{1/3}$, where $H$ is the enstrophy flux and $L$ is the pumping length. Therefore, the PDF has exponential tails and is self-similar, that is, it can be presented as a function of a single argument, $\varpi /{\varpi }_{\mathrm{rms}}$, in distinction from other known direct cascades.

In this paper, we show how proper assignment of weights to the edges of a complex network can enhance the detection of communities and how it can circumvent the resolution limit and the extreme degeneracy problems associated with modularity. Our general weighting scheme takes advantage of graph theoretic measures and it introduces two heuristics for tuning its parameters. We use this weighting as a preprocessing step for the greedy modularity optimization algorithm of Newman to improve its performance. The result of the experiments of our approach on computer-generated and real-world data networks confirm that the proposed approach not only mitigates the problems of modularity but also improves the modularity optimization.

In driving a vehicle, drivers respond to the changes of both the headway and the relative velocity to the vehicle in front. In this paper a new car-following model including these maneuvers is proposed. The acceleration of the model becomes infinite (has a singularity) when the distance between two vehicles is zero, and the asymmetry between the acceleration and the deceleration is incorporated in a nonlinear way. The model is simple but contains enough features of driving for reproducing real vehicle traffic. From the linear stability analysis, we confirm that the model shows the metastable homogeneous flow around the critical density, beyond which a traffic jam emerges. Moreover, we perform experiments to verify this model. From the data it is shown that the acceleration of a vehicle has a positive correlation with the relative velocity.

A crucially important material parameter for all amorphous solids is the yield stress, which is the value of the stress for which the material yields to plastic flow when it is strained quasistatically at zero temperature. It is difficult in laboratory experiments to determine what parameters of the interparticle potential affect the value of the yield stress. Here we use the versatility of numerical simulations to study the dependence of the yield stress on the parameters of the interparticle potential. We find a very simple dependence on the fundamental scales that characterize the repulsive and attractive parts of the potential, respectively, and offer a scaling theory that collapses the data for widely different potentials and in different space dimensions.

A laminar inviscid liquid (typically water) jet issuing from a circular nozzle into otherwise quiescent air disintegrates into droplets periodically at a distance from the nozzle. The Plateau-Rayleigh instability theory and others cannot determine this breakup length because they do not have any logic that determines the initial amplitude of the unstable wave responsible for the breakup. In this paper, a closed spatial evolution solution is derived for a uniformly issued liquid jet by applying a theory that identifies the origin of the unstable wave. This solution describes the self-destabilizing mechanism of the liquid jet in the steady breakup state, showing that the initial amplitude of the unstable wave is determined by the capillary wave with upstream propagating speed that is created by the tip contraction at every breakup. Finally, the developed theory is extended to allow for the self-destabilizing mechanism of a liquid jet issuing from a long nozzle, which initially has a parabolic velocity profile and results in a long breakup length.

The transport of quantum electrons through hierarchical lattices is of interest because such lattices have some properties of both regular lattices and random systems. We calculate the electron transmission as a function of energy in the tight-binding approximation for two related Hanoi networks. HN3 is a Hanoi network with every site having three bonds. HN5 has additional bonds added to HN3 to make the average number of bonds per site equal to five. We present a renormalization group approach to solve the matrix equation involved in this quantum transport calculation. We observe band gaps in HN3, while no such band gaps are observed in linear networks or in HN5. We provide a detailed scaling analysis near the edges of these band gaps.

We investigate the collective properties of three particles performing both independent and interacting random walks on an infinite line by studying the problem on the corresponding distance graph. In the large times limit, we obtain the asymptotically exact behavior of several probability functions regarding maximum and minimum mutual distances among particles. Finally, we suggest possible applications of our results.

Electroconvection (EC) phenomena have been investigated in the nematic phase of a bent-core oxadiazole material with negative dielectric anisotropy and a frequency dependent conductivity anisotropy. The formation of longitudinal roll (LR) patterns is one of the predominant features observed in the complete frequency and voltage range studied. At voltages much above the LR threshold, various complex patterns such as the “crisscrossed” pattern, bimodal varicose, and turbulence are observed. Unusually, the nonstandard EC (ns-EC) instability in this material, is observed in a regime in which we measure the dielectric and conductivity anisotropies to be negative and positive respectively. A further significant observation is that the EC displays distinct features in the high and low temperature regimes of the nematic phase, supporting an earlier report that EC patterns could distinguish between regions that have been reported as uniaxial and biaxial nematic phases.

We construct a theoretical model of the dielectric properties of a ferroelectric LC nanosuspension (FLCNS), using a generalized Maxwell-Garnett picture. The theory supposes that an FLCNS may as a first approximation be considered as a complex homogeneous dielectric ceramic, thus neglecting positional correlations of the colloidal particles. The FLCNS then consists of an anisotropic matrix with a very low concentration ($<1\%$ by volume) of impurity particles. The impurity particles possess both shape and dielectric anisotropy, as well as a permanent electric polarization and strong liquid-crystal director anchoring on the particle surface. We show that the effective dielectric properties for capacitance properties and for effective liquid-crystal free energies do not coincide. We calculate the effect of doping a liquid crystal with ferroelectric impurities on the Frederiks transition. The theory takes account of inclusion shape, dielectric susceptibility, and local field effects. We neglect the possibility of dielectric particle chaining, which appears experimentally not to occur in general. Our calculations suggest, in qualitative agreement with experiment, that doping a nematic liquid crystal with ferroelectric particles, even at very low particle concentration, can in some cases significantly decrease the electric Frederiks threshold field.

Classical theories of gel swelling employ the mixing free energy, thereby ignoring any effects of the free energy of the pure phases,i.e., the polymer standard free energy. In this paper we present a model for the swelling kinetics of gels that incorporates the free energy, including the polymer standard free energy. We provide a complete analysis of how the swelling kinetics and stable states and sizes of the swelled gel depends on the free energy parameters and show that theories that use only the mixing free energy cannot correctly describe equilibrium states or the swelling kinetics.

A link between dimer-scale processes and microtubule (MT) dynamics at macroscale is studied by comparing simulations obtained using computational dimer-scale model with its mean-field approximation. The novelty of the mean-field model (MFM) is in its explicit representation of inter-protofilament cracks, as well as in the direct incorporation of the dimer-level kinetics. Due to inclusion of both longitudinal and lateral dimer interactions, the MFM is two dimensional, in contrast to previous theoretical models of MTs. It is the first analytical model that predicts and quantifies crucial features of MT dynamics such as (i) existence of a minimal soluble tubulin concentration needed for the polymerization (with concentration represented as a function of model parameters), (ii) existence of steady-state growth and shortening phases (given with their respective velocities), and (iii) existence of an unstable pause state near zero velocity. In addition, the size of the GTP cap of a growing MT is estimated. Theoretical predictions are shown to be in good agreement with the numerical simulations.

One of the central challenges facing modern neuroscience is to explain the ability of the nervous system to coherently integrate information across distinct functional modules in the absence of a central executive. To this end, Tononi et al. [Proc. Natl. Acad. Sci. USA. 91, 5033 (1994)] proposed a measure of neural complexity that purports to capture this property based on mutual information between complementary subsets of a system. Neural complexity, so defined, is one of a family of information theoretic metrics developed to measure the balance between the segregation and integration of a system’s dynamics. One key question arising for such measures involves understanding how they are influenced by network topology. Sporns et al. [Cereb. Cortex 10, 127 (2000)] employed numerical models in order to determine the dependence of neural complexity on the topological features of a network. However, a complete picture has yet to be established. While De Lucia et al. [Phys. Rev. E 71, 016114 (2005)] made the first attempts at an analytical account of this relationship, their work utilized a formulation of neural complexity that, we argue, did not reflect the intuitions of the original work. In this paper we start by describing weighted connection matrices formed by applying a random continuous weight distribution to binary adjacency matrices. This allows us to derive an approximation for neural complexity in terms of the moments of the weight distribution and elementary graph motifs. In particular, we explicitly establish a dependency of neural complexity on cyclic graph motifs.

The naming game (NG) describes the agreement dynamics of a population of agents that interact locally in a pairwise fashion, and in recent years statistical physics tools and techniques have greatly contributed to shed light on its rich phenomenology. Here we investigate in details the role played by the way in which the two agents update their states after an interaction. We show that slightly modifying the NG rules in terms of which agent performs the update in given circumstances (i.e., after a success) can either alter dramatically the overall dynamics or leave it qualitatively unchanged. We understand analytically the first case by casting the model in the broader framework of a generalized NG. As for the second case, on the other hand, we note that the modified rule reproducing the main features of the usual NG corresponds in fact to a simplification of it consisting in the elimination of feedback between the agents. This allows us to introduce and study a very natural broadcasting scheme on networks that can be potentially relevant for different applications, such as the design and implementation of autonomous sensor networks, as pointed out in the recent literature.

Extensive systems have no long scale correlations and behave as a sum of their parts. Various techniques are introduced to determine a characteristic length scale of interaction beyond which spatiotemporal chaos is extensive in reaction-diffusion networks. Information about network size, boundary condition, or abnormalities in network topology gets scrambled in spatiotemporal chaos, and the attenuation of information provides such characteristic length scales. Space-time information flow associated with the recovery of spatiotemporal chaos from finite perturbations, a concept somewhat opposite to the paradigm of Lyapunov exponents, defines another characteristic length scale. High-precision computational studies of asymptotic spatiotemporal chaos in the complex Ginzburg-Landau system and transient spatiotemporal chaos in the Gray-Scott network show that these different length scales are comparable and thus suitable to define a length scale of interaction. Preliminary studies demonstrate the relevance of these length scales for stable chaos.

Combined studies by atomic force microscopy, x-ray reflectivity, and Fourier transform infrared spectroscopy on transition-metal stearate ($M$-St, $M$ $=$ Mn, Co, Zn, and Cd) Langmuir-Blodgett films clearly indicate association of bidentate coordination of the metal-carboxylate head group to layer-by-layer growth as observed in MnSt and CoSt and partially in ZnSt. Crossover to islandlike growth, as observed in CdSt and ZnSt, is associated with the presence of unidentate coordination in the head group. Morphological evolutions as obtained from one, three, and nine monolayers (MLs) of $M$-St films are consistent with Frank van der Merwe, Stranski-Krastanov, and Volmer Weber growth modes for $M=\mathrm{Mn}$/Co, Zn, and Cd, respectively, as previously assigned, and are found to vary with number ($n$) of metal atoms per head group, viz. $n=1$ (Mn/Co), $n=0.75$ (Zn), and $n=0.5$ (Cd). The parameter $n$ is found to decide head-group coordination such that $n=1.0$ corresponds to bidentate and $n=0.5$ corresponds to unidentate coordination; the intermediate value in Zn corresponds to a mixture of both. The dependence of the growth mode on head-group structure is explained by the fact that in bidentate head groups, with the in-plane dipole moment being zero, intermolecular forces between adjacent molecules are absent and hence growth proceeds via layering. On the other hand, in unidentate head groups, the existence of a nonzero in-plane dipole moment results in the development of weak in-plane intermolecular forces between adjacent molecules causing in-plane clustering leading to islandlike growth.

The linear viscoelastic properties of aqueous Pluronic L64 solutions have been investigated at high copolymer concentrations (25–62 wt$\%$) using our modified low-frequency mechanical spectroscopy. The concentration-temperature phase diagram of the L64/H${}_2$O system was constructed by studying the evolution of the loss modulus and loss tangent as temperature is increased at a fixed frequency. A particular attention was focused on the dynamics approaching the beginning and ending points (39$\%$ and 60$\%$) of the fusiform gel region in the phase diagram. The dynamics is found to have a similar viscoelastic behavior at the low and high concentrations, where a frequency scaling expected for a static percolated network is exhibited. Moreover, with increasing temperature, the system above the critical gel concentration undergoes a transition from a viscoelastic liquid to a solid gel through a percolated particle network. Therefore, our results suggest that the formation of the gel is dominated by the percolation of the particle clusters.

A network of filaments formed primarily by the abundant cytoskeletal protein actin gives animal cells their shape and elasticity. The rheological properties of reconstituted actin networks have been studied by tracking micron-sized probe beads embedded within the networks. We investigate how microrheology depends on surface properties of probe particles by varying the stickiness of their surface. For this purpose, we chose carboxylate polystyrene (PS) beads, silica beads, bovine serum albumin (BSA) -coated PS beads, and polyethylene glycol (PEG) -grafted PS beads, which show descending stickiness to actin filaments, characterized by confocal imaging and microrheology. Probe size dependence of microrheology is observed for all four types of beads. For the slippery PEG beads, particle-tracking microrheology detects weaker networks using smaller beads, which tend to diffuse through the network by hopping from one confinement “cage” to another. This trend is reversed for the other three types of beads, for which microrheology measures stiffer networks for smaller beads due to physisorption of nearby filaments to the bead surface. We explain the probe size dependence with two simple models. We also evaluate depletion effect near nonadsorption bead surface using quantitative image analysis and discuss the possible impact of depletion on microrheology. Analysis of these effects is necessary in order to accurately define the actin network rheology both in vitro and in vivo.