In this paper, we study optimal control problems of spiking neurons whose dynamics are described by a phase model. We design minimum-power current stimuli (controls) that lead to targeted spiking times. In particular, we consider bounded control amplitude and characterize the range of possible spiking times determined by the bound, which can be chosen sufficiently small within the range where the phase model is valid. We show that for a given bound the corresponding feasible spiking times are optimally achieved by piecewise continuous controls. We present analytic expressions with numerical simulations of the minimum-power stimuli for several phase models. We demonstrate the applicability of our method with an experimentally determined phase response curve.

Phase-locking behaviors in sinoatrial node (SAN) are closely related to cardiac arrhythmias. An ionic model considering structural heterogeneity of SAN is numerically investigated. The bifurcations between phase-locking zones are interpreted by the map derived from the phase resetting curve. Furthermore, the validity of the circle map in describing phase locking of the actual SAN system is evaluated and explained. We reveal also how the phase-locking behaviors in heterogeneous tissue depend on the location of stimulating site and the coupling strength of the tissue. All these results may be of suggestive uses for understanding and controlling practical SAN dynamics.

The principal constituent of the living cell is water. The role of the hydration shell and bulk ${\mathrm{H}}_2$O solvent is well recognized in the dynamics of isolated proteins, but the role of water in the dynamics of the integrated living cytoskeleton (CSK) remains obscure. Here we report a direct connection of dynamics of water to dynamics of the integrated CSK. The latter are known to be scale-free and to hinge upon a frequency $f_0$ that is roughly invariant across cell types. Although $f_0$ is comparable in magnitude to the rotational relaxation frequency of water (gigahertz range), the physical basis of $f_0$ remains unknown. Using the human airway smooth muscle cell as a model system, we show here that replacing water acutely with deuterium oxide impacts CSK dynamics in major ways, slowing CSK remodeling dynamics appreciably, and lowering $f_0$ by up to four orders of magnitude. Although these observations do not distinguish contributions of bulk solvent versus hydration shell, they suggest a unifying hypothesis, namely, that dynamics of integrated CSK networks are slaved in a direct fashion to fluctuations arising in intracellular water.

Thin solid sheets and fluid membranes exhibit buckling under lateral compression. Here it is revealed that buckled fluid membranes have anisotropic mechanical surface tension contrary to solid sheets. Surprisingly, the surface tension perpendicular to the buckling direction shows stronger dependence on the projected area than that parallel to it. Our theoretical predictions are supported by numerical simulations of a meshless membrane model. This anisotropic tension can be used to measure the membrane bending rigidity. It is also found that phase synchronization occurs between multilayered buckled membranes.

The collective migration of cells in tissue pervades many important biological processes, such as wound healing, organism development, and cancer metastasis. Recent experiments on wound healing show that the collective migratory behavior of cells can be quite complex, including transient vortices and long-range correlations. Here, we explore cellular flows in epithelial tissues using a model that considers the force distribution and polarity of a single cell along with cell-cell adhesion. We show that the dipole nature of a crawling cell’s force distribution destabilizes steady cellular motion. We determine the values of the physical parameters that are necessary to produce these complex motions and use numerical simulation to verify the linear analysis and to demonstrate the complex flows. We find that the tendency for cells to align is the dominant physical parameter that determines the stability of steady flows in the epithelium.

We propose a mesoscopic particle-exchange heat engine realized via coupled quantum dots. The engine is able to extract work from a bosonic reservoir and can be used as a refrigerator. The electric current is derived from a master-equation method and is readdressed under a thermodynamic context. The quantum-mechanical result is found to be consistent with thermodynamic analysis.

The exact time-dependent solution for the stochastic equations governing the behavior of a binary self-regulating gene is presented. Using the generating function technique to rephrase the master equations in terms of partial differential equations, we show that the model is totally integrable and the analytical solutions are the celebrated confluent Heun functions. Self-regulation plays a major role in the control of gene expression, and it is remarkable that such a microscopic model is completely integrable in terms of well-known complex functions.

When an initial failure of nodes occurs in interdependent networks, a cascade of failure between the networks occurs. Earlier studies focused on random initial failures. Here we study the robustness of interdependent networks under targeted attack on high or low degree nodes. We introduce a general technique which maps the targeted-attack problem in interdependent networks to the random-attack problem in a transformed pair of interdependent networks. We find that when the highly connected nodes are protected and have lower probability to fail, in contrast to single scale-free (SF) networks where the percolation threshold $p_c=0$, coupled SF networks are significantly more vulnerable with $p_c$ significantly larger than zero. The result implies that interdependent networks are difficult to defend by strategies such as protecting the high degree nodes that have been found useful to significantly improve robustness of single networks.

This paper shows the connection between three previously observed but seemingly unrelated phenomena in hydrodynamic (HD) and magnetohydrodynamic (MHD) turbulent flows, involving the emergence of fluctuations occurring on very long time scales: the low-frequency $1/f$ noise in the power frequency spectrum, the delayed ergodicity of complex valued amplitude fluctuations in wave number space, and the spontaneous flippings or reversals of large-scale fields. Direct numerical simulations of ideal MHD and HD are employed in three space dimensions, at low resolution, for long periods of time, and with high accuracy to study several cases: different geometries, presence of rotation and/or a uniform magnetic field, and different values of the associated conserved global quantities. It is conjectured that the origin of all these long-time phenomena is rooted in the interaction of the longest wavelength fluctuations available to the system, with fluctuations at much smaller scales. The strength of this nonlocal interaction is controlled either by the existence of conserved global quantities with a back-transfer in Fourier space or by the presence of a slow manifold in the dynamics.

Dynamics of a single vesicle under shear flow between two parallel plates is studied in two-dimensions using lattice-Boltzmann simulations. We first present how we adapted the lattice-Boltzmann method to simulate vesicle dynamics, using an approach known from the immersed boundary method. The fluid flow is computed on an Eulerian regular fixed mesh while the location of the vesicle membrane is tracked by a Lagrangian moving mesh. As benchmarking tests, the known vesicle equilibrium shapes in a fluid at rest are found and the dynamical behavior of a vesicle under simple shear flow is being reproduced. Further, we focus on investigating the effect of the confinement on the dynamics, a question that has received little attention so far. In particular, we study how the vesicle steady inclination angle in the tank-treading regime depends on the degree of confinement. The influence of the confinement on the effective viscosity of the composite fluid is also analyzed. At a given reduced volume (the swelling degree) of a vesicle we find that both the inclination angle, and the membrane tank-treading velocity decrease with increasing confinement. At sufficiently large degree of confinement the tank-treading velocity exhibits a nonmonotonous dependence on the reduced volume and the effective viscosity shows a nonlinear behavior.

We describe a number of strategies for minimizing and calculating accurately the statistical uncertainty in quantum Monte Carlo calculations. We investigate the impact of the sampling algorithm on the efficiency of the variational Monte Carlo method. We then propose a technique to maximize the efficiency of the linear extrapolation of diffusion Monte Carlo results to zero time step, finding that a relative time-step ratio of 1:4 is optimal. Finally, we discuss the removal of serial correlation from data sets by reblocking, setting out criteria for the choice of block length and quantifying the effects of the uncertainty in the estimated correlation length.

We use perturbation methods to analyze a two-dimensional random intermittent search process, in which a searcher alternates between a diffusive search phase and a ballistic movement phase whose velocity direction is random. A hidden target is introduced within a rectangular domain with reflecting boundaries. If the searcher moves within range of the target and is in the search phase, it has a chance of detecting the target. A quasi-steady-state analysis is applied to the corresponding Chapman-Kolmogorov equation. This generates a reduced Fokker-Planck description of the search process involving a nonzero drift term and an anisotropic diffusion tensor. In the case of a uniform direction distribution, for which there is zero drift, and isotropic diffusion, we use the method of matched asymptotics to compute the mean first passage time (MFPT) to the target, under the assumption that the detection range of the target is much smaller than the size of the domain. We show that an optimal search strategy exists, consistent with previous studies of intermittent search in a radially symmetric domain that were based on a decoupling or moment closure approximation. We also show how the decoupling approximation can break down in the case of biased search processes. Finally, we analyze the MFPT in the case of anisotropic diffusion and find that anisotropy can be useful when the searcher starts from a fixed location.

Ergodic behavior of the class of $G$ processes $G(t)={\int }_{t_m}^t\mathit{du}\hspace{0.5em}K(t,u)\xi (u)-{\int }_{t_m}^0\mathit{du}\hspace{0.5em}K(0,u)\xi (u)$, $\langle \xi (t)\rangle =0$, $\langle \xi (t)\xi (s)\rangle =\varphi (|t-s|)$ is examined. Ergodic properties are only $G$ extensions of normal diffusion ($K=1$) and of Mandelbrot-Van Ness fractional diffusion [$K(t,u)=K(t-u)$, $t_m\to -\infty$]. Any deviation from these two types results in weak ergodicity breaking which thus is neither exceptional nor limited to some specific events but is typical for much wider class of processes. $G$ processes driven by $\xi (t)$ with nonvanishing correlations are important for describing transport in strongly nonequilibrium systems and may be responsible for peculiarities of diffusion found in biological, glassy, and nanoscale systems.

Fractional Brownian motion is a Gaussian process $x(t)$ with zero mean and two-time correlations $\langle x(t_1)x(t_2)\rangle =D(t_1^{2H}+t_2^{2H}-|t_1-t_2\left|{}^{2H}\right)$, where $H$, with $0<H<1$, is called the Hurst exponent. For $H=1/2$, $x(t)$ is a Brownian motion, while for $H\ne 1/2$, $x(t)$ is a non-Markovian process. Here we study $x(t)$ in presence of an absorbing boundary at the origin and focus on the probability density $P_{+}(x,t)$ for the process to arrive at $x$ at time $t$, starting near the origin at time $0$, given that it has never crossed the origin. It has a scaling form $P_{+}(x,t)~t^{-H}{R}_{+}(x/t^H)$. Our objective is to compute the scaling function $R_{+}(y)$, which up to now was only known for the Markov case $H=1/2$. We develop a systematic perturbation theory around this limit, setting $H=1/2+\epsilon$, to calculate the scaling function $R_{+}(y)$ to first order in $\epsilon$. We find that $R_{+}(y)$ behaves as $R_{+}(y)~y^{\varphi }$ as $y\to 0$ (near the absorbing boundary), while $R_{+}(y)~y^{\gamma }\exp (-y^2/2)$ as $y\to \infty$, with $\varphi =1-4\epsilon +O(\epsilon {}^2)$ and $\gamma =1-2\epsilon +O(\epsilon {}^2)$. Our $\epsilon$-expansion result confirms the scaling relation $\varphi =(1-H)/H$ proposed in Zoia, Rosso, and Majumdar [Phys. Rev. Lett. 102, 120602 (2009)]. We verify our findings via numerical simulations for $H=2/3$. The tools developed here are versatile, powerful, and adaptable to different situations.

We study a decision making model in a condition where it is equivalent to the two-dimensional Ising model, and we show that at the onset of phase transition it generates temporal complexity, namely, nonstationary and nonergodic fluctuations. We argue that this is a general property of criticality, thereby opening the door to the application of the recently discovered phenomenon of complexity matching: For an efficient transfer of information to occur, a perturbing complex network must share the same temporal complexity as the perturbed complex network.

We describe a new, microscopic model for diffusion that captures diffusion induced fluctuations at scales where the concept of concentration gives way to discrete particles. We show that in the limit as the number of particles $N\to \infty$, our model is equivalent to the classical stochastic diffusion equation (SDE). We test our new model and the SDE against Langevin dynamics in numerical simulations, and show that our model successfully reproduces the correct ensemble statistics, while the classical model fails.

We discuss stochastic thermodynamics (ST) for delayed Langevin systems in this paper. By using the general principles of ST, the first-law-like energy balance and trajectory-dependent entropy $s(t)$ can be well defined in a way that is similar to that in a system without delay. Because the presence of time delay brings an additional entropy flux into the system, the conventional second law $\langle \Delta s_{\mathrm{tot}}\rangle ⩾0$ no longer holds true, where $\Delta s_{\mathrm{tot}}$ denotes the total entropy change along a stochastic path and $\langle ·\rangle$ stands for the average over the path ensemble. With the help of a Fokker-Planck description, we introduce a delay-averaged trajectory-dependent dissipation functional $\eta [\chi (t)]$ which involves the work done by a delay-averaged force $\mathrm{F̄}(x,t)$ along the path $\chi (t)$ and equals the medium entropy change $\Delta s_m[x(t)]$ in the absence of delay. We show that the total dissipation functional $R=\Delta s+\eta$, where $\Delta s$ denotes the system entropy change along a path, obeys $\langle R\rangle ⩾0$, which could be viewed as the second law in the delayed system. In addition, the integral fluctuation theorem $\langle e^{-R}\rangle =1$ also holds true. We apply these concepts to a linear Langevin system with time delay and periodic external force. Numerical results demonstrate that the total entropy change $\langle \Delta s_{\mathrm{tot}}\rangle$ could indeed be negative when the delay feedback is positive. By using an inversing-mapping approach, we are able to obtain the delay-averaged force $\mathrm{F̄}(x,t)$ from the stationary distribution and then calculate the functional $R$ as well as its distribution. The second law $\langle R\rangle ⩾0$ and the fluctuation theorem are successfully validated.

We present the theoretical study on nonequilibrium (NEQ) fluctuations for diffusion dynamics in high dimensions driven by a linear drift force. We consider a general situation in which NEQ is caused by two conditions: (i) drift force not derivable from a potential function, and (ii) diffusion matrix not proportional to the unit matrix, implying nonidentical and correlated multidimensional noise. The former is a well-known NEQ source and the latter can be realized in the presence of multiple heat reservoirs or multiple noise sources. We develop a statistical mechanical theory based on generalized thermodynamic quantities such as energy, work, and heat. The NEQ fluctuation theorems are reproduced successfully. We also find the time-dependent probability distribution function exactly as well as the NEQ work production distribution $P(\mathcal{W})$ in terms of solutions of nonlinear differential equations. In addition, we compute low-order cumulants of the NEQ work production explicitly. In two dimensions, we carry out numerical simulations to check out our analytic results and also to get $P(\mathcal{W})$. We find an interesting dynamic phase transition in the exponential tail shape of $P(\mathcal{W})$, associated with a singularity found in solutions of the nonlinear differential equation. Finally, we discuss possible realizations in experiments.

We study the extreme statistics of $N$ nonintersecting Brownian motions (vicious walkers) over a unit time interval in one dimension. Using path-integral techniques we compute exactly the joint distribution of the maximum $M$ and of the time ${\tau }_M$ at which this maximum is reached. We focus in particular on nonintersecting Brownian bridges (“watermelons without wall”) and nonintersecting Brownian excursions (“watermelons with a wall”). We discuss in detail the relationships between such vicious walkers models in watermelon configurations and stochastic growth models in curved geometry on the one hand and the directed polymer in a disordered medium (DPRM) with one free end point on the other hand. We also check our results using numerical simulations of Dyson’s Brownian motion and confront them with numerical simulations of the polynuclear growth model (PNG) and of a model of DPRM on a discrete lattice. Some of the results presented here were announced in a recent letter [J. Rambeau and G. Schehr, Europhys. Lett. 91, 60006 (2010)].

We use Brownian dynamics computer simulations of a moderately dense colloidal system undergoing steady shear flow to investigate the uniqueness of the so-called effective temperature. We compare effective temperatures calculated from the fluctuation-dissipation ratios and from the linear response to a static, long-wavelength, external perturbation along two directions: the shear gradient direction and the vorticity direction. At high shear rates, when the system is strongly anisotropic, the fluctuation-dissipation-ratio-derived effective temperatures are approximately wave-vector independent, but the temperatures along the gradient direction are somewhat higher than those along the vorticity direction. The temperatures derived from the static linear response show the same dependence on the direction as those derived from the fluctuation-dissipation ratio. However, the former and the latter temperatures are different. Our results suggest that the presently used formulas for effective temperatures may not be applicable for strongly anisotropic, driven systems.