Large systems of linear equations are ubiquitous in science. Quite often, e.g., when considering population dynamics or chemical networks, the solutions must be nonnegative. Recently, it has been shown that large systems of random linear equations exhibit a sharp transition from a phase, where a nonnegative solution exists with probability one, to one where typically no such solution may be found. The critical line separating the two phases was determined by combining Farkas' lemma with the replica method. Here we show that the same methods remain viable to characterize the two phases away from criticality. To this end we analytically determine the residual norm of the system in the unsolvable phase and a suitable measure of robustness of solutions in the solvable one. Our results are in very good agreement with numerical simulations.

The social amoeba Dictyostelium discoideum performs chemotaxis under starvation conditions, aggregating towards clusters of cells following waves of the signaling molecule cAMP. Cells sense extracellular cAMP and produce internal caches of cAMP to be released, relaying the signal. These events lead to traveling waves of cAMP washing over the population of cells. While much research has been performed to understand the functioning of the chemotaxis network in D. discoideum, limited work has been done to link the operation of the signal relay network with the chemotaxis network to provide a holistic view of the system. We take inspiration from D. discoideum and propose a model that directly links the relaying of a chemical message to the directional sensing of that signal. Utilizing an excitable dynamical systems model that has been previously validated experimentally, we show that it is possible to have both signal amplification and perfect adaptation in a single module. We show that noise plays a vital role in chemotaxing to static gradients, where stochastic tunneling of transient bursts biases the system towards accurate gradient sensing. Moreover, this model also automatically matches its internal time scale of adaptation to the naturally occurring periodicity of the traveling chemical waves generated in the population. Numerical simulations were performed to study the qualitative phenomenology of the system and explore how the system responds to diverse dynamic spatiotemporal stimuli. Finally, we address dynamical instabilities that impede chemotactic ability in a continuum version of the model.

We study continuous-time random walks (CTRW) with power-law distribution of waiting times under resetting which brings the walker back to the origin, with a power-law distribution of times between the resetting events. Two situations are considered. Under complete resetting, the CTRW after the resetting event starts anew, with a new waiting time, independent of the prehistory. Under incomplete resetting, the resetting of the coordinate does not influence the waiting time until the next jump. We focus on the behavior of the mean-squared displacement (MSD) of the walker from its initial position, on the conditions under which the probability density functions of the walker's displacement show universal behavior, and on this universal behavior itself. We show, that the behavior of the MSD is the same as in the scaled Brownian motion (SBM), being the mean-field model of the CTRW. The intermediate asymptotics of the probability density functions (PDF) for CTRW under complete resetting (provided they exist) are also the same as in the corresponding case for SBM. For incomplete resetting, however, the behavior of the PDF for CTRW and SBM is vastly different.

We study the interaction between epidemic spreading and a vaccination process. We assume that, similar to the disease spreading, the vaccination process also occurs through direct contact, i.e., it follows the standard susceptible-infected-susceptible (SIS) dynamics. The two competing processes are asymmetrically coupled as vaccinated nodes can directly become infected at a reduced rate with respect to susceptible ones. We study analytically the model in the framework of mean-field theory finding a rich phase diagram. When vaccination provides little protection toward infection, two continuous transitions separate a disease-free immunized state from vaccinated-free epidemic state, with an intermediate mixed state where susceptible, infected, and vaccinated individuals coexist. As vaccine efficiency increases, a tricritical point leads to a bistable regime, and discontinuous phase transitions emerge. Numerical simulations for homogeneous random networks agree very well with analytical predictions.

We explore the firing-rate model of excitatory neurons with dendritic adaptation (the Feldman–Del Negro model [J. L. Feldman and C. A. Del Negro, Nat. Rev. Neurosci. 7, 232 (2006); D. J. Schwab et al., Phys. Rev. E 82, 051911 (2010)] interacting on a fixed, directed Erdős-Rényi network. This model is applied to the dynamics of the pre-Bötzinger complex, the mammalian central pattern generator with $N\sim {10}^3$ neurons, which produces a collective metronomic signal that times inspiration. In the all-to-all coupled variant of the model, there is spontaneous symmetry breaking in which some fraction of the neurons becomes stuck in a high-firing-rate state, while others become quiescent. This separation into firing and nonfiring clusters persists into more sparsely connected networks. In these sparser networks, the clustering is influenced by $k$ cores of the underlying network. The model has a number of features of the dynamical phase diagram that violate the predictions of mean-field analysis. In particular, we observe in the simulated networks that stable oscillations do not persist in the high-sensitivity limit, in contradiction to the predictions of mean-field theory. Moreover, we observe that the oscillations in these sparse networks are remarkably robust in response to killing neurons, surviving until only approximately $20\%$ of the network remains. This robustness is consistent with experiment.

Cellular movement is a complex dynamic process, resulting from the interaction of multiple elements at the intra- and extracellular levels. This epiphenomenon presents a variety of behaviors, which can include normal and anomalous diffusion or collective migration. In some cases, cells can get neighborhood information through chemical or mechanical cues. A unified understanding about how such information can influence the dynamics of cell movement is still lacking. In order to improve our comprehension of cell migration we have considered a cellular Potts model where cells move actively in the direction of a driving field. The intensity of this driving field is constant, while its orientation can evolve according to two alternative dynamics based on the Ornstein-Uhlenbeck process. In one case, the next orientation of the driving field depends on the previous direction of the field. In the other case, the direction update considers the mean orientation performed by the cell in previous steps. Thus, the latter update rule mimics the ability of cells to perceive the environment, avoiding obstacles and thus increasing the cellular displacement. Different cell densities are considered to reveal the effect of cell-cell interactions. Our results indicate that both dynamics introduce temporal and spatial correlations in cell velocity in a friction-coefficient and cell-density-dependent manner. Furthermore, we observe alternating regimes in the mean-square displacement, with normal and anomalous diffusion. The crossovers between diffusive and directed motion regimes are strongly affected by both the driving field dynamics and cell-cell interactions. In this sense, when cell polarization update grants information about the previous cellular displacement, the duration of the diffusive regime decreases, particularly in high-density cultures.

Chromatin can adopt multiple stable, heritable states with distinct histone modifications and varying levels of gene expression. Insight on the stability and maintenance of such epigenetic states can be gained by mathematical modeling of stochastic reaction networks for histone modifications. Analytical results for the kinetic networks are particularly valuable. Compared to computationally demanding numerical simulations, they often are more convenient at evaluating the robustness of conclusions with respect to model parameters. In this communication, we developed a second-quantization-based approach that can be used to analyze discrete stochastic models with a fixed, finite number of particles using a representation of the $\text{SU}\left(2\right)$ algebra. We applied the approach to a kinetic model of chromatin states that captures the feedback between nucleosomes and the enzymes conferring histone modifications. Using a path-integral expression for the transition probability, we computed the epigenetic landscape that helps to identify the emergence of bistability and the most probable path connecting the two steady states. We anticipate the generalizability of the approach will make it useful for studying more complicated models that couple epigenetic modifications with transcription factors and chromatin structure.

In nonequilibrium active matter systems, a spatial variation in activity can lead to a spatial variation in concentration of active particles satisfying, at steady state, the condition $nU=\text{const}$ [Schnitzer, Phys. Rev. E 48, 2553 (1993); Tailleur and Cates, Phys. Rev. Lett. 100, 218103 (2008)], where $n$ is the number density and $U$ is the active (swim) speed. We show that this condition holds even when the variation is abrupt and when thermal Brownian motion is present provided that the Péclet number is large. This spatial variation in swim speed and concentration produces a fluid pressure distribution that drives a reverse osmotic flow—fluid flows from regions of high concentration to low.

Hiding an object in a chemical gradient requires one to suppress the distortions it would naturally cause on it. To do so, we propose a strategy based on coating the object with a chemical reaction-diffusion network which can act as an active cloaking device. By controlling the concentration of some species in its immediate surrounding, the chemical reactions redirect the gradient as if the object was not there. We also show that a substantial fraction of the energy required to cloak can be extracted from the chemical gradient itself.

We revisit the Born-Markov approximation for an open quantum system by considering a microscopic model of the bath, namely, the Bose-Hubbard chain in the parameter region where it is chaotic in the sense of quantum chaos. It is shown that strong ergodic properties of the bath justify all approximations required for deriving the Markovian master equation from the first principles.

For many networks, it is useful to think of their nodes as being embedded in a latent space, and such embeddings can affect the probabilities for nodes to be adjacent to each other. In this paper, we extend existing models of synthetic networks to spatial network models by first embedding nodes in Euclidean space and then modifying the models so that progressively longer edges occur with progressively smaller probabilities. We start by extending a geographical fitness model by employing Gaussian-distributed fitnesses, and we then develop spatial versions of preferential attachment and configuration models. We define a notion of “spatial strength centrality” to help characterize how strongly a spatial embedding affects network structure, and we examine spatial strength centrality on a variety of real and synthetic networks.

Collective behaviors displayed by groups of social animals are observed frequently in nature. Understanding and predicting the behavior of complex biological systems is dependent on developing effective descriptions and models. While collective animal systems are characteristically nonequilibrium, we can employ concepts from equilibrium statistical mechanics to motivate the measurement of material-like properties in laboratory animal aggregates. Here, we present results from a new set of experiments that utilize high speed footage of two-dimensional schooling events, particle tracking, and projected static and dynamic light fields to observe and control the behavior of negatively phototaxic fish schools (Hemigrammus bleheri). First, we use static light fields consisting of dark circular regions to produce visual stimuli that confine the schools to a range of areas. We find that schools have a maximum density which is independent of group size, and that a swim pressurelike quantity, $\mathrm{\Pi }$ increases linearly with number density, suggesting that unperturbed schools exist on an isotherm. Next, we use dynamic light fields where the radius of the dark region shrinks linearly with time to compress the schools. We find that an effective temperature parameter depends on the compression time and our results are thus consistent with the school having a constant heat flux. These findings further evidence the utility of effective thermodynamic descriptions of nonequilibrium systems in collective animal behavior.

A formulation of Langevin dynamics for discrete systems is derived as a class of generic stochastic processes. The dynamics simplify for a two-state system and suggest a network architecture which is implemented by the Langevin machine. The Langevin machine represents a promising approach to compute successfully quantitative exact results of Boltzmann distributed systems by LIF neurons. Besides a detailed introduction of the dynamics, different simplified models of a neuromorphic hardware system are studied with respect to a control of emerging sources of errors.

We study implications of the assumption of power-law dependence of moments of energy dissipation in turbulence on the Reynolds number $\mathrm{Re}$, holding due to intermittency. We demonstrate that at $\mathrm{Re}\to \infty$ the dissipation's logarithm divided by $ln\mathrm{Re}$ converges with probability one to a negative constant. This implies that the dissipation is singular in the limit, as is known phenomenologically. The proof uses a large deviations function, whose existence is implied by the power-law assumption, and which provides the general asymptotic form of the dissipation's distribution. A similar function exists for vorticity and for velocity differences where it proves the moments representation of the multifractal model (MF). Then we observe that derivative of the scaling exponents of the dissipation, considered as a function of the order of the moment, is small at the origin. Thus the variation with the order is slow and can be described by a quadratic function. Indeed, the quadratic function, which corresponds to log-normal statistics, fits the data. Moreover, combining the lognormal scaling with the MF we derive a formula for the anomalous scaling exponents of turbulence which also fits the data. Thus lognormality, not to be confused with the Kolmogorov (1962) assumption of lognormal dissipation in the inertial range, when used in conjunction with the MF provides a concise way to get all scaling exponents of turbulence available at present.

A fully operational loss-free quantum battery requires an inherent control over the energy transfer process, with the ability of keeping the energy retained with no leakage. Moreover, it also requires a stable discharge mechanism, which entails that no energy revivals occur as the device starts its energy distribution. Here we provide a scalable solution for both requirements. To this aim, we propose a general design for a quantum battery based on an energy current (EC) observable quantifying the energy transfer rate to a consumption hub. More specifically, we introduce an instantaneous EC operator describing the energy transfer process driven by an arbitrary interaction Hamiltonian. The EC observable is shown to be the root for two main applications: (1) a trapping energy mechanism based on a common eigenstate between the EC operator and the interaction Hamiltonian, in which the battery can indefinitely retain its energy even if it is coupled to the consumption hub, and (2) an asymptotically stable discharge mechanism, which is achieved through an adiabatic evolution eventually yielding vanishing EC. These two independent but complementary applications are illustrated in quantum spin chains, where the trapping energy control is realized through Bell pairwise entanglement and the stability arises as a general consequence of the adiabatic spin dynamics.

For the fermionic or bosonic oscillator fully coupled to several heat baths with mixed statistics, the analytical expressions for the occupation numbers are derived within the non-Markovian quantum Langevin approach. Employing two or three heat baths and the Ohmic dissipation with Lorenzian cutoffs, the role of statistics of the system and heat baths in the dynamics of the system is studied.

Network science is a rapidly expanding field, with a large and growing body of work on network-based dynamical processes. Most theoretical results in this area rely on the so-called locally treelike approximation. This is, however, usually an “uncontrolled” approximation, in the sense that the magnitudes of the error are typically unknown, although numerical results show that this error is often surprisingly small. In this paper we place this approximation on more rigorous footing by calculating the magnitude of deviations away from tree-based theories in the context of discrete-time critical network cascades with re-excitable nodes. We discuss the conditions under which tree-like approximations give good results for calculating network criticality, and also explain the reasons for deviation from this approximation, in terms of the density of certain kinds of network motifs. Using this understanding, we derive results for network criticality that apply to general networks that explicitly do not satisfy the locally treelike approximation. In particular, we focus on the biparallel motif, the smallest motif relevant to the failure of a tree-based theory in this context, and we derive the corrections due to such motifs on the conditions for criticality. We verify our claims on computer-generated networks, and we confirm that our theory accurately predicts the observed deviations from criticality. Using our theory, we explain why numerical simulations often show that deviations from a tree-based theory are surprisingly small. More specifically, we show that these deviations are negligible for networks whose average degree is even modestly large compared to one, justifying why tree-based theories appear to work well for most real-world networks.

Electroporation of concentric compound spherical and confocal spheroidal as well as eccentric compound spherical vesicles, considered to be good models for corresponding nucleate cells, are investigated with an emphasis on their response to nanosecond pulse electric field (nsPEF). Analytical models are developed for the estimation of the transmembrane potential (TMP) across the bilayers of the inner and the outer vesicles and finite-element simulations are also carried out for the eccentric case. Our calculations show that with an increase in the aspect ratio, while the TMP decreases when nsPEF is used, it increases for confocal spheroids when the pulse width is greater than the membrane charging time, leading to fully charged vesicles. Bipolar pulses are shown to effectively control the TMP for a desired time period in the nsPEF regime, and a fast decay of the TMP to zero can be achieved by judicious use of pulse polarity. The external conductivity is found to significantly influence the TMP in nsPEF, unlike millisecond pulses where its effect is insignificant. Additionally the critical electric field required to induce a TMP of 1 V at the inner vesicle is presented for different pulse widths, rise time, as well as membrane capacitance, and the TMP of the outer vesicle is found to be within limits of reversible poration. It is found that the maximum TMP has a roughly linear dependence on the outer aspect ratio of the vesicle. We also introduce a new method to obtain the particular solution to the Laplace equation for bispherical system, and it is validated with finite-element simulations. Our study on nsPEF electroporation of bispherical vesicles shows that the north pole TMP is typically greater than the south pole, thereby suggesting the typical pathway a charged species might take inside an eccentric nucleate cell under electroporation.

The single-particle dynamics of sodium ions in molten sodium bromide has been investigated with quasielastic neutron scattering. A detailed and rather extensive data analysis procedure allowed determination of the pure sodium ion dynamics with increasing wave vector. Two different evaluation procedures agree perfectly on the resulting diffusion coefficient of sodium ions on long distances. A simple kinetic theory based on binary collisions of hard spheres is not able to reproduce the sodium diffusion coefficient. The derived reduced linewidth from modeling with a Lorentzian spectral function decreases with increasing wave vector towards the first structure factor maximum. That deviation from the hydrodynamic behavior signals the hindrance of the microscopic diffusion process due to the so-called cage effect when microscopic length scales are probed in a dense fluid. The observed quadratic wave-number-dependent decrease might be evidence for a coupling to density fluctuations as the source of the changes in the diffusion process. The results indicate that in the molten salt NaBr near the melting point the self-diffusion process might be governed by similar processes as already observed in dense metallic liquids.