Large-scale neural networks can be described in the spatial continuous limit by neural field equations. For large-scale brain networks, the connectivity is typically translationally variant and imposes a large computational burden upon simulations. To reduce this burden, we take a semiquantitative approach and study the dynamics of neural fields described by a delayed integrodifferential equation. We decompose the connectivity into spatially variant and invariant contributions, which typically comprise the short- and long-range fiber systems, respectively. The neural fields are mapped on the two-dimensional spherical surface, which is choice consistent with routine mappings of cortical surfaces. Then, we perform mathematically a mode decomposition of the neural field equation into spherical harmonic basis functions. A spatial truncation of the leading orders at low wave number is consistent with the spatially coherent pattern formation of large-scale patterns observed in simulations and empirical brain imaging data and leads to a low-dimensional representation of the dynamics of the neural fields, bearing promise for an acceleration of the numerical simulations by orders of magnitude.

Single-cell experiments have revealed cell-to-cell variability in generation times and growth rates for genetically identical cells. Theoretical models relating the fluctuating generation times of single cells to the population growth rate are usually based on the assumption that the generation times of mother and daughter cells are uncorrelated. This assumption, however, is inconsistent with the exponential growth of cell volume in time observed for many cell types. Here we develop a more general and biologically relevant model in which cells grow exponentially and generation times are correlated in a manner which controls cell size. In addition to the fluctuating generation times, we also allow the single-cell growth rates to fluctuate and account for their correlations across the lineage tree. Surprisingly, we find that the population growth rate only depends on the distribution of single-cell growth rates and their correlations.

We present a framework for performing input-output system identification near a Hopf bifurcation using data from only the fixed-point branch, prior to the Hopf point itself. The framework models the system with a van der Pol–type equation perturbed by additive noise, and identifies the system parameters via the corresponding Fokker-Planck equation. We demonstrate the framework on a prototypical thermoacoustic oscillator (a flame-driven Rijke tube) undergoing a supercritical Hopf bifurcation. We find that the framework can accurately predict the properties of the Hopf bifurcation and the limit cycle beyond it. This study constitutes an experimental demonstration of system identification on a reacting flow using only prebifurcation data, opening up pathways to the development of early warning indicators for nonlinear dynamical systems near a Hopf bifurcation.

Discrete eigenmodes of the filamentation instability in a weakly ionized current-driven plasma in the presence of a $q$-nonextensive electron velocity distribution is investigated. Considering the kinetic theory, Bhatnagar-Gross-Krook collision model, and Lorentz transformation relations, the generalized longitudinal and transverse dielectric permittivities are obtained. Taking into account the long-wavelength limit and diffusion frequency limit, the dispersion relations are obtained. Using the approximation of geometrical optics and linear inhomogeneity of the plasma, the real and imaginary parts of the frequency are discussed in these limits. It is shown that in the long-wavelength limit, when the normalized electron velocity is increased the growth rate of the instability increases. However, when the collision frequency is increased the growth rate of the filamentation instability decreases. In the diffusion frequency limit, results indicate that the effects of the electron velocity and $q$-nonextensive parameter on the growth rate of the instability are similar. Finally, it is found that when the collision frequency is increased the growth rate of the instability increases in the presence of a $q$-nonextensive distribution.

A quasi-two-dimensional system of hard spheres strongly confined between two parallel plates is considered. The attention is focused on the macroscopic self-diffusion process observed when the system is seen from above or from below. The transport equation, and the associated self-diffusion coefficient, are derived from a Boltzmann-Lorentz kinetic equation, valid in the dilute limit. Since the equilibrium state of the system is inhomogeneous, this requires the use of a modified Chapman-Enskog expansion that distinguishes between equilibrium and nonequilibrium gradients of the density of labeled particles. The self-diffusion coefficient is obtained as a function of the separation between the two confining plates. The theoretical predictions are compared with molecular dynamics simulation results and a good agreement is found.

The magnetoelectric effect of a spin-1/2 Heisenberg-Ising ladder in the presence of external electric and magnetic fields is rigorously examined by taking into account the Katsura-Nagaosa-Balatsky mechanism. It is shown that an applied electric field may control the quantum phase transition between a Néel (stripy) ordered phase and a disordered paramagnetic phase. The staggered magnetization vanishes according to a power law with an Ising-type critical exponent 1/8, the electric polarization exhibits a weak singularity, and the dielectric susceptibility shows a logarithmic divergence at this particular quantum phase transition. The external electric field may alternatively invoke a discontinuous phase transition accompanied with abrupt jumps of the dielectric polarization and susceptibility on the assumption that the external magnetic field becomes nonzero.

We calculate the fractal dimension $d_{\mathrm{f}}$ of critical curves in the $O\left(n\right)$-symmetric ${\left({\vec{\varphi }}^2\right)}^2$ theory in $d=4-\epsilon$ dimensions at 6-loop order. This gives the fractal dimension of loop-erased random walks at $n=-2$, self-avoiding walks ($n=0$), Ising lines $(n=1)$, and $XY$ lines ($n=2$), in agreement with numerical simulations. It can be compared to the fractal dimension $d_{\mathrm{f}}^{\mathrm{tot}}$ of all lines, i.e., backbone plus the surrounding loops, identical to $d_{\mathrm{f}}^{\mathrm{tot}}=1/\nu$. The combination ${\varphi }_c=d_{\mathrm{f}}/d_{\mathrm{f}}^{\mathrm{tot}}=\nu d_{\mathrm{f}}$ is the crossover exponent, describing a system with mass anisotropy. Introducing a self-consistent resummation procedure and combining it with analytic results in $d=2$ allows us to give improved estimates in $d=3$ for all relevant exponents at 6-loop order.

Dynamical stability is a prerequisite for control and functioning of desired nanomachines. We utilize the Caldeira-Leggett master equation to investigate dynamical stability of molecular cogwheels modeled as a rigid, propeller-shaped planar rotator. To match certain expected realistic physical situations, we consider a weakly nonharmonic external potential for the rotator. Two methods for investigating stability are used. First, we employ a quantum-mechanical counterpart of the so-called “first passage time” method. Second, we investigate time dependence of the standard deviation of the rotator for both the angle and angular momentum quantum observables. A perturbationlike procedure is introduced and implemented to provide the closed set of differential equations for the moments. Extensive analysis is performed for different combinations of the values of system parameters. The two methods are, in a sense, mutually complementary. Appropriate for the short time behavior, the first passage time exhibits a numerically relevant dependence only on the damping factor as well as on the rotator size. However, the standard deviations for both the angle and angular momentum observables exhibit strong dependence on the parameter values for both short and long time intervals. Contrary to our expectations, the time decrease of the standard deviations is found for certain parameter regimes. In addition, for certain parameter regimes nonmonotonic dependence on the rotator size is observed for the standard deviations and for the damping of the oscillation amplitude. Hence, nonfulfillment of the classical expectation that the size of the rotator can be reduced to the inertia of the rotator. In effect, the task of designing the desired protocols for the proper control of the molecular rotations becomes an optimization problem that requires further technical elaboration.

The aging process is a common phenomenon in engineering, biological, and physical systems. The hazard rate function, which characterizes the aging process, is a fundamental quantity in the disciplines of reliability, failure, and risk analysis. However, it is difficult to determine the entire hazard function accurately with limited observation data when the degradation mechanism is not fully understood. Inspired by the seminal work pioneered by Jaynes [Phys. Rev. 106, 620 (1956)], this study develops an approach based on the principle of maximum entropy. In particular, the time-dependent hazard rate function can be established using limited observation data in a rational manner. It is shown that the developed approach is capable of constructing and interpreting many typical hazard rate curves observed in practice, such as the bathtub curve, the upside down bathtub, and so on. The developed approach is applied to model a classical single function system and a numerical example is used to demonstrate the method. In addition its extension to a more general multifunction system is presented. Depending on the interaction between different functions of the system, two cases, namely reducible and irreducible, are discussed in detail. A multifunction electrical system is used for demonstration.

Effects of mechanical coupling on cardiac dynamics are studied by monitoring the beating dynamics of a cardiac tissue which is being pulled periodically at a pace slower than its intrinsic beating rate. The tissue is taken from the heart of a bullfrog that includes pacemaker cells. The cardiac tissue beats spontaneously with an almost constant interbeat interval (IBI) when there is no external forcing. On the other hand, the IBI is observed to vary significantly under an external periodic drive. Interestingly, when the period of the external drive is about two times the intrinsic IBI of the tissue without pulling, the IBI as a function of time exhibits a wave packet structure. Our experimental results can be understood theoretically by a phase-coupled model under external driving. In particular, the theoretical prediction of the wave-packet period as a function of the normalized driving period agrees excellently with the observations. Furthermore, the cardiac mechanical coupling constant can be extracted from the experimental data from our model and is found to be insensitive to the external driving period. Implications of our results on cardiac physiology are also discussed.

The twist-bend nematic ($N_{\mathrm{TB}}$) phase is a doubly degenerated heliconical structure with nanometric pitch and spontaneous bend and twist deformations. It is favored by symmetry-breaking molecular structures, such as bent dimers and bent-core molecules, and it is currently one of the burgeoning fields of liquid-crystal research. Although tremendous advances have been reported in the past five years, especially in molecular synthesis, most of its potential applications are held back by the lack of a proper and definitive elastic model to describe its behavior under various situations such as confinement and applied field. In this work we use a recently proposed stable state elastic model and the fact that the mesophase behaves as a lamellar structure to propose a mesoscopic or coarse-grained model for the $N_{\mathrm{TB}}$ phase. By means of standard procedures used for smectic and cholesteric liquid crystals, we arrive at a closed-form energy for the phase and apply it to a few situations of interest. The predicted compressibility for several values of the cone angle and the critical field for field-induced deformation agree well with recent experimental data.

A numerical procedure based on the Schwarz-Christoffel map suitable for the study of the Laplacian growth of thin two-dimensional protrusions is presented. The protrusions take the form of either straight needles or curved fingers satisfying Loewner's equation, and are represented by slits in the complex plane. Particular use is made of Driscoll's numerical procedure, the SC Toolbox, for computing the Schwarz-Christoffel map from a half plane to a slit half plane. Since the Schwarz-Christoffel map applies only to polygonal regions, the growth of curved fingers is approximated by an increasing number of short straight line segments. The growth rate is given by a fixed power $\eta$ of the harmonic measure at the finger or needle tips and so includes the possibility of “screening” as the needles of fingers interact with themselves and with boundaries. The method is illustrated with examples of multiple needle and finger growth in half-plane and channel geometries. The effect of $\eta$ on the trajectories of asymmetric bifurcating fingers is also studied.

We consider a differentially rotating flow of an incompressible electrically conducting and viscous fluid subject to an external axial magnetic field and to an azimuthal magnetic field that is allowed to be generated by a combination of an axial electric current external to the fluid and electrical currents in the fluid itself. In this setting we derive an extended version of the celebrated Hain-Lüst differential equation for the radial Lagrangian displacement that incorporates the effects of the axial and azimuthal magnetic fields, differential rotation, viscosity, and electrical resistivity. We apply the Wentzel-Kramers-Brillouin method to the extended Hain-Lüst equation and derive a comprehensive dispersion relation for the local stability analysis of the flow to three-dimensional disturbances. We confirm that in the limit of low magnetic Prandtl numbers, in which the ratio of the viscosity to the magnetic diffusivity is vanishing, the rotating flows with radial distributions of the angular velocity beyond the Liu limit, become unstable subject to a wide variety of the azimuthal magnetic fields, and so is the Keplerian flow. In the analysis of the dispersion relation we find evidence of a new long-wavelength instability which is caught also by the numerical solution of the boundary value problem for a magnetized Taylor-Couette flow.

An angular momentum conservative pure bulk viscosity term for smoothed particle hydrodynamics (SPH) is proposed in the present paper. This formulation permits independent modeling of shear and bulk viscosities, which is of paramount importance for fluids with large bulk viscosity in situations where sound waves or large Mach numbers are expected. With this aim a dissipative term proportional to the rate of change of the volume is considered at the particle level. The equations of motion are derived from the minimization of a Lagrangian combined with an appropriate dissipation function that depends on this rate of change of particle volume, in analogy with the corresponding entropy production contribution in fluids. Due to the Galilean invariance of the formulation, the new term is shown to exactly conserve linear momentum. Moreover, its invariance under solid-body rotations also ensures the conservation of angular momentum. Two verification cases are proposed: the one-dimensional propagation of a sound pulse and a two-dimensional case, modeling the time decay of an accelerating-decelerating pipe flow. The SPH solutions are compared to exact ones, showing that the newly proposed term behaves indeed as a viscosity associated only with the local expansion-compression of the fluid. In view of these considerations, we conclude that the method presented in this paper allows for setting up a bulk viscosity independently of the shear one and as large as any particular problem may require. At the same time, together with the prescribed momentum conservation to reproduce the Navier-Stokes equation, the new term also keeps the angular momentum conservation required to properly model free interfaces or overall rotations of the bulk fluid.

We introduce and study a simple model for the dynamics of voting intention in a population of agents that have to choose between two candidates. The level of indecision of a given agent is modeled by its propensity to vote for one of the two alternatives, represented by a variable $p\in [0,1]$. When an agent $i$ interacts with another agent $j$ with propensity $p_j$, then $i$ either increases its propensity $p_i$ by $h$ with probability $P_{ij}=\omega p_i+(1-\omega )p_j$, or decreases $p_i$ by $h$ with probability $1-P_{ij}$, where $h$ is a fixed step. We assume that the interactions form a complete graph, where each agent can interact with any other agent. We analyze the system by a rate equation approach and contrast the results with Monte Carlo simulations. We find that the dynamics of propensities depends on the weight $\omega$ that an agent assigns to its own propensity. When all the weight is assigned to the interacting partner ($\omega =0$), agents' propensities are quickly driven to one of the extreme values $p=0$ or $p=1$, until an extremist absorbing consensus is achieved. However, for $\omega >0$ the system first reaches a quasistationary state of symmetric polarization where the distribution of propensities has the shape of an inverted Gaussian with a minimum at the center $p=1/2$ and two maxima at the extreme values $p=0,1$, until the symmetry is broken and the system is driven to an extremist consensus. A linear stability analysis shows that the lifetime of the polarized state, estimated by the mean consensus time $\tau$, diverges as $\tau \sim {(1-\omega )}^{-2}lnN$ when $\omega$ approaches 1, where $N$ is the system size. Finally, a continuous approximation allows us to derive a transport equation whose convection term is compatible with a drift of particles from the center toward the extremes.

We report experimental studies on an azo-substituted compound consisting of bent-core hockey-stick-shaped molecules. The experimental results establish two pseudopolar tilted smectic phases, which are characterized by an in-plane axial-vector order parameter in addition to tilt order in the smectic layers. Electro-optical measurements in the mesophases indicate that the birefringence of the sample strongly depends on the applied electric field. We develop a theoretical model to account for this observation. The change in the birefringence of the sample arises from the field-induced reorientation of the tilt plane of the molecules in the layer above a threshold field. The effect is analogous to the field-induced Freedericksz transition which is quadratic in the applied electric field.

We present a Monte Carlo algorithm based on the stochastic approximation Monte Carlo (SAMC) algorithm for directly calculating the density of states. The proposed method is stochastic approximation with a dynamic update factor (SAD), which dynamically adjusts the update factor ${\gamma }_t$ during the course of the simulation. We test this method on a square-well fluid and a 31-atom Lennard-Jones cluster and compare the convergence behavior of several related Monte Carlo methods. We find that both the SAD and $1/t$-Wang-Landau ($1/t$-WL) methods rapidly converge to the correct density of states without the need for the user to specify an arbitrary tunable parameter $t_0$ as in the case of SAMC. SAD requires as input the temperature range of interest, in contrast with $1/t$-WL, which requires that the user identify the interesting range of energies. The convergence of the $1/t$-WL method is very sensitive to the energy range chosen for the low-temperature heat capacity of the Lennard-Jones cluster. Thus, SAD is more powerful in the common case in which the range of energies is not known in advance.

In a classic paper, Purcell [Proc. Natl. Acad. Sci. U. S. A. 94, 11307 (1997)] analyzed the dynamics of flagellated bacterial swimmers and derived a geometrical relationship which maximizes the propulsion efficiency. Experimental measurements for wild-type bacterial species E. coli have revealed that they closely satisfy this geometric optimality. However, dependence of the flagellar motor speed on the load and more generally the role of the torque-speed characteristics of the flagellar motor are not considered in Purcell's original analysis. Here we derive a tuned condition representing a match between the flagella geometry and the torque-speed characteristics of the flagellar motor to maximize the bacterial swimming speed for a given load. This condition is independent of the geometric optimality condition derived by Purcell. Interestingly, this condition is not satisfied by wild-type E. coli which swims 2–3 times slower than the maximum possible speed given the amount of available motor torque. Finally, we present experimental data on swimming dynamics of a cargo laden bacterial system which follows our analytical model. Our analysis also reveals the existence of an anomalous propulsion regime where the swim speed increases with increasing load (drag).