Eigenmodes of a broad class of “sparse” random matrices, with interactions concentrated near the diagonal, exponentially localize in space, as initially discovered in 1957 by Anderson for quantum systems. Anderson localization plays ubiquitous roles in varieties of problems from electrons in solids to mechanical and optical systems. However, its implications in neuroscience (where the connections can be strongly asymmetric) have been largely unexplored, mainly because synaptic connectivity matrices of neural systems are often “dense,” which makes the eigenmodes spatially extended. Here we explore roles that Anderson localization could be playing in neural networks by focusing on “spatially structured” disorder in synaptic connectivity matrices. Recently neuroscientists have experimentally confirmed that the local excitation and global inhibition (LEGI) ring attractor model can functionally represent head direction cells in Drosophila melanogaster central brain. We first study a non-Hermitian (i.e., asymmetric) tight-binding model with disorder and then establish a connection to the LEGI ring attractor model. We discover that (1) principal eigenvectors of the LEGI ring attractor networks with structured nearest-neighbor disorder are “quasilocalized,” even with fully dense inhibitory connections; and (2) the quasilocalized eigenvectors play dominant roles in the early time neural dynamics, and the location of the principal quasilocalized eigenvectors predicts an initial location of the “bump of activity” representing, for example, a head direction of an insect. Our investigations open up venues for explorations at the intersection between the theory of Anderson localization and neural networks with spatially structured disorder.

High-resolution x-ray imaging was used in combination with differential pressure measurements to measure relative permeability and capillary pressure simultaneously during a steady-state waterflood experiment on a sample of Bentheimer sandstone 51.6 mm long and 6.1 mm in diameter. After prolonged contact with crude oil to alter the surface wettability, a refined oil and formation brine were injected through the sample at a fixed total flow rate but in a sequence of increasing brine fractional flows. When the pressure across the system stabilized, x-ray tomographic images were taken. The images were used to compute saturation, interfacial area, curvature, and contact angle. From this information relative permeability and capillary pressure were determined as functions of saturation. We compare our results with a previously published experiment under water-wet conditions. The oil relative permeability was lower than in the water-wet case, although a smaller residual oil saturation, of approximately 0.11, was obtained, since the oil remained connected in layers in the altered wettability rock. The capillary pressure was slightly negative and 10 times smaller in magnitude than for the water-wet rock, and approximately constant over a wide range of intermediate saturation. The oil-brine interfacial area was also largely constant in this saturation range. The measured static contact angles had an average of ${80}^{\circ }$ with a standard deviation of ${17}^{\circ }$. We observed that the oil-brine interfaces were not flat, as may be expected for a very low mean curvature, but had two approximately equal, but opposite, curvatures in orthogonal directions. These interfaces were approximately minimal surfaces, which implies well-connected phases. Saddle-shaped menisci swept through the pore space at a constant capillary pressure and with an almost fixed area, removing most of the oil.

Mechanical strain and stress play a major role in biological processes such as wound healing or morphogenesis. To assess this role quantitatively, fixed or live images of tissues are acquired at a cellular precision in large fields of views. To exploit these data, large numbers of cells have to be analyzed to extract cell shape anisotropy and cell size. Most frequently, this is performed through detailed individual cell contour determination, using so-called segmentation computer programs, complemented if necessary by manual detection and error corrections. However, a coarse-grained and faster technique can be recommended in at least three situations: first, when detailed information on individual cell contours is not required; for instance, in studies which require only coarse-grained average information on cell anisotropy. Second, as an exploratory step to determine whether full segmentation can be potentially useful. Third, when segmentation is too difficult, for instance due to poor image quality or too large a cell number. We developed a user-friendly, Fourier-transform-based image analysis pipeline. It is fast (typically ${10}^4$ cells per minute with a current laptop computer) and suitable for time, space, or ensemble averages. We validate it on one set of artificial images and on two sets of fully segmented images, one from a Drosophila pupa and the other from a chicken embryo; the pipeline results are robust. Perspectives include in vitro tissues, nonbiological cellular patterns such as foams and $xyz$ stacks.

In large neuronal networks, it is believed that functions emerge through the collective behavior of many interconnected neurons. Recently, the development of experimental techniques that allow simultaneous recording of calcium concentration from a large fraction of all neurons in Caenorhabditis elegans—a nematode with 302 neurons—creates the opportunity to ask whether such emergence is universal, reaching down to even the smallest brains. Here, we measure the activity of 50+ neurons in C. elegans, and analyze the data by building the maximum entropy model that matches the mean activity and pairwise correlations among these neurons. To capture the graded nature of the cells' responses, we assign each cell multiple states. These models, which are equivalent to a family of Potts glasses, successfully predict higher statistical structure in the network. In addition, these models exhibit signatures of collective behavior: the state of single cells can be predicted from the state of the rest of the network; the network, despite being sparse in a way similar to the structural connectome, distributes its response globally when locally perturbed; the distribution over network states has multiple local maxima, as in models of memory; and the parameters that describe the real network are close to a critical surface in this family of models.

We develop the elastic theory for inversion-asymmetric tethered membranes and use it to identify and study their possible phases. Asymmetry in a tethered membrane causes spontaneous curvature, which in general depends on the local in-plane dilation of the tethered network. This in turn leads to long-range interactions between the local mean and Gaussian curvatures, which are not present in symmetric tethered membranes. This interplay between asymmetry and Gaussian curvature leads to a double-spiral phase not found in symmetric tethered membranes. At temperature $T=0$, tethered membranes of arbitrarily large size are always rolled up tightly into a conjoined pair of Archimedes' spirals. At finite $T$ this spiral structure swells up significantly into algebraic spirals characterized by universal exponents, which we calculate. These spirals have long-range orientational order, and are the asymmetric analogs of statistically flat symmetric tethered membranes. We also find that sufficiently strong asymmetry can trigger a structural instability leading to crumpling of these membranes as well. This provides a mechanism for crumpling of asymmetric tethered membranes which is not present for symmetric membranes. We calculate the maximum linear extent $L_c$ beyond which the membrane crumples, and calculate the universal dependence of $L_c$ on the membrane parameters. By tuning the asymmetry parameter, $L_c$ can be continuously varied, implying a scale-dependent crumpling. Our theory can be tested in controlled experiments on lipids with artificial deposits of spectrin filaments, in in vitro experiments on red blood cell membrane extracts, and on graphene coated on one side.

Mammalian cells are crowded with macromolecules, supramolecular complexes, and organelles, all of which equip intracellular fluids, e.g., the cytoplasm, with a dynamic and spatially heterogeneous occupied volume fraction. Diffusion in such fluids has been reported to be heterogeneous, i.e., even individual single-particle trajectories feature spatiotemporally varying transport characteristics. Complementing diffusion-based experiments, we have used here an imaging approach to assess the spatial heterogeneity of the nucleoplasm and the cytoplasm in living interphase cells. As a result, we find that the cytoplasm is more crowded and more heterogeneous than the nucleoplasm on several length scales. This phenomenon even persists in dividing cells, where the mitotic spindle region and its periphery form a contiguous fluid but remain nucleoplasmlike and cytoplasmlike, respectively.

The most popular modeling approach for dielectrophoresis (DEP) is the effective multipole (EM) method. It approximates the polarization-induced charge distribution in an object of interest by a set of multipolar moments. The Coulombic interaction of these moments with the external polarizing electric field then gives the DEP force and torque acting on the object. The multipolar moments for objects placed in arbitrary harmonic electric fields are, however, known only for spherical objects. This shape restriction significantly limits the use of the EM method. We present an approach for online (in real time) computation of multipolar moments for objects of arbitrary shapes having even arbitrary internal composition (inhomogeneous objects, more different materials, etc.). We exploit orthonormality of spherical harmonics to extract the multipolar moments from a numerical simulation of the polarized object. This can be done in advance (offline) for a set of external electric fields forming a basis so that the superposition principle can then be used for online operation. DEP force and torque can thus be computed in fractions of a second, which is needed, for example, in model-based control applications. We validate the proposed model against reference numerical solutions obtained using Maxwell stress tensor. We also analyze the importance of the higher-order multipolar moments using a sample case of a Tetris-shaped micro-object placed inside a quadrupolar microelectrode array and exposed to electrorotation. The implementation of the model in Matlab and Comsol is offered for free download.

Many approaches to modeling reaction-diffusion systems with anomalous transport rely on deterministic equations which ignore fluctuations arising due to finite particle numbers. Starting from an individual-based model we use a generating-functional approach to derive a Gaussian approximation for this intrinsic noise in subdiffusive systems. This results in corrections to the deterministic fractional reaction-diffusion equations. Using this analytical approach, we study the onset of noise-driven quasipatterns in reaction-subdiffusion systems. We find that subdiffusion can be conducive to the formation of both deterministic and stochastic patterns. Our analysis shows that the combination of subdiffusion and intrinsic stochasticity can reduce the threshold ratio of the effective diffusion coefficients required for pattern formation to a greater degree than either effect on its own.

Cellular checkerboard patterns are observed at many stages of embryonic development. We study an analytically tractable model for lateral inhibition and show that the steady states are analogous to optical phonons at the $\mathrm{\Gamma }$ point, which have the wave number $k=0$. We study the cases of cells arranged in linear and hexagonal lattices. To determine how the final pattern is selected it is necessary to take into account the granularity of the pattern and, analogously to solid-state physics, to redefine the basis and lattice sites in terms of a periodic crystal. The sites and basis are determined by looking at the symmetries of inhibitory interactions between cells. The redefined basis for cells placed in a linear lattice is composed by two cells which are embedded in another linear lattice, while for cells placed in a hexagonal lattice the redefined basis consists of three cells embedded in another hexagonal lattice. The pattern in hexagonal lattices can be driven into three different states: two of those states are periodic checkerboards and a third in which both periodic states coexist. These observations provides new predictions for experiments.

Bacteria communicate with each other to coordinate macroscale behaviors including pathogenesis, biofilm formation, and antibiotic production. Empirical evidence suggests that bacteria are capable of communicating at length scales far exceeding the size of individual cells. Several mechanisms of signal interference have been observed in nature, and how interference influences macroscale activity within microbial populations is unclear. Here we examined the exchange of quorum sensing signals to coordinate microbial activity over long distances in the presence of a variable amount of interference through a neighboring signal-degrading strain. As the level of interference increased, communication over large distances was disrupted and at a critical amount of interference, large-scale communication was suppressed. We explored this transition in experiments and reaction-diffusion models, and confirmed that this transition is a two-dimensional percolation transition. These results demonstrate the utility of applying physical models to emergence in complex biological networks to probe robustness and universal quantitative features.

Roots provide basic functions to plants such as water and nutrient uptake and anchoring in soil. The growth and development of root systems contribute to colonizing the surrounding soil and optimizing the access to resources. It is generally known that the variability of plant root architecture results from the combination of genetic, physiological, and environmental factors, in particular soil mechanical resistance. However, this last factor has never been investigated at the soil grain scale for roots. In this paper, we are interested in the effect of the disordered texture of granular soils on the evolution of forces experienced by the root cap during its growth. We introduce a numerical model in which the root is modeled as a flexible self-elongating tube that probes a soil composed of solid particles. By means of extensive simulations, we show that the forces exerted on the root cap reflect interparticle force chains. Our simulations also show that the mean force declines exponentially with root flexibility, the highest force corresponding to the soil hardness. Furthermore, we find that this functional dependence is characterized by a single dimensionless parameter that combines granular structure and root bending stiffness. This finding will be useful to further address the biological issues of mechanosensing and thigmomorphogenesis in plant roots.

The explosion of activity in finding interactions in complex systems is driven by availability of copious observations of complex natural systems. However, such systems, e.g., the human brain, are rarely completely observable. Interaction network inference must then contend with hidden variables affecting the behavior of the observed parts of the system. We present an effective approach for model inference with hidden variables. From configurations of observed variables, we identify the observed-to-observed, hidden-to-observed, observed-to-hidden, and hidden-to-hidden interactions, the configurations of hidden variables, and the number of hidden variables. We demonstrate the performance of our method by simulating a kinetic Ising model, and show that our method outperforms existing methods. Turning to real data, we infer the hidden nodes in a neuronal network in the salamander retina and a stock market network. We show that predictive modeling with hidden variables is significantly more accurate than that without hidden variables. Finally, an important hidden variable problem is to find the number of clusters in a dataset. We apply our method to classify MNIST handwritten digits. We find that there are about 60 clusters which are roughly equally distributed among the digits.

Reinforcement learning in multiagent systems has been studied in the fields of economic game theory, artificial intelligence, and statistical physics by developing an analytical understanding of the learning dynamics (often in relation to the replicator dynamics of evolutionary game theory). However, the majority of these analytical studies focuses on repeated normal form games, which only have a single environmental state. Environmental dynamics, i.e., changes in the state of an environment affecting the agents' payoffs has received less attention, lacking a universal method to obtain deterministic equations from established multistate reinforcement learning algorithms. In this work we present a methodological extension, separating the interaction from the adaptation timescale, to derive the deterministic limit of a general class of reinforcement learning algorithms, called temporal difference learning. This form of learning is equipped to function in more realistic multistate environments by using the estimated value of future environmental states to adapt the agent's behavior. We demonstrate the potential of our method with the three well-established learning algorithms Q learning, SARSA learning, and actor-critic learning. Illustrations of their dynamics on two multiagent, multistate environments reveal a wide range of different dynamical regimes, such as convergence to fixed points, limit cycles, and even deterministic chaos.

The uplift of an initially flat elastic membrane by an upward viscous flow is investigated experimentally. The deformed shape of the membrane results from a balance between the flow pressure, the elastic response of the membrane, and the fluid weight. This last effect becomes non-negligible for a large enough deformed area. The usual theoretical approach supposes the presence of a prewetting film regularizing the viscous stresses according to Lister et al. [Phys. Rev. Lett. 111, 154501 (2013)]. Nevertheless, in our experiments without prewetting films, the measurements are correctly described with this theory in the elastic regime. Microscale roughness of membranes could introduce an equivalent characteristic scale in the problem. An alternative explanation could be provided by the appearance of a fluid lag filled with gas, for which a new theoretical framework has been recently proposed by Ball and Neufeld [Phys. Rev. Fluids 3, 074101 (2018)]. We compare the two approaches and find that both describe reasonably our experiments. However, consistency tests of both models show that the prewetting film model is more appropriate to describe our experimental data.

We present a bottom-up derivation of fluctuation analysis with detrending for the detection of long-range correlations in the presence of additive trends or intrinsic nonstationarities. While the well-known detrended fluctuation analysis (DFA) and detrending moving average (DMA) were introduced ad hoc, we claim basic principles for such methods where DFA and DMA are then shown to be specific realizations. The mean-squared displacement of the summed time series contains the same information about long-range correlations as the autocorrelation function but has much better statistical properties for large time lags. However, the scaling exponent of its estimator on a single time series is affected not only by trends on the data but also by intrinsic nonstationarities. We therefore define the fluctuation function as mean-squared displacement with weighting kernel. We require that its estimator be unbiased and exhibit the correct scaling behavior for the random component of a signal, which is only achieved if the weighting kernel implies detrending. We show how DFA and DMA satisfy these requirements and we extract their kernel weights.

We propose an experiment combining fluid dynamics and strong magnetic field physics to simulate cosmological scenarios. Our proposed system consists of two immiscible, weakly magnetized fluids moved through a strong gradient magnetic field. The diamagnetic and paramagnetic forces thus generated amount to a time-dependent effective gravity, which allows us to precisely control the propagation speed of interface waves. Perturbations on the interface therefore experience a nonstationary effective metric. In what follows, we demonstrate that our proposed system is capable of simulating a variety of cosmological models. We then present a readily realizable experimental setup which will allow us to capture the essential dynamics of standard inflation, wherein interface perturbations experience a shrinking effective horizon and are shown to transition from oscillatory to frozen and squeezed regimes at horizon crossing.

We study interactions between biological cells that apply anisotropic active mechanical forces on an elastic substrate. We model the cells as thin disks that along their perimeters apply radial, but angle-dependent forces on the substrate. We obtain analytical expressions for the elastic energy stored in the substrate as a function of the distance between the cells, the Fourier modes of applied forces, and their phase angles. We show how the relative phases of the forces applied by the cells can switch the interaction between attractive and repulsive, and relate our results to those for linear force dipoles. For long enough distances, the interaction energy decays in magnitude as a power law of the cell-cell distance with an integer exponent that generally increases with the Fourier modes of the applied forces.

We study pattern formation of skin cancers by means of numerical simulation of a binary system consisting of cancer and healthy cells. We extend the conventional model H for macrophase separations by considering a logistic growth of cancer cells and also a mechanical friction between dermis and epidermis. Importantly, our model exhibits a microphase separation due to the proliferation of cancer cells. By numerically solving the time evolution equations of the cancer composition and its velocity, we show that the phase separation kinetics strongly depends on the cell proliferation rate as well as on the strength of hydrodynamic interactions. A steady-state diagram of cancer patterns is established in terms of these two dynamical parameters and some of the patterns correspond to clinically observed cancer patterns. Furthermore, we examine in detail the time evolution of the average composition of cancer cells and the characteristic length of the microstructures. Our results demonstrate that different sequence of cancer patterns can be obtained by changing the proliferation rate and/or hydrodynamic interactions.

Both stimuli-responsive gels and growing biological tissue can undergo pronounced morphological transitions from two-dimensional (2D) layers into 3D geometries. We derive an analytical model that allows us to quantitatively predict the features of 2D-to-3D shape changes in polymer gels that encompasses different degrees of swelling within the sample. We analyze a particular configuration that emerges from a flat rectangular gel that is divided into two strips (bistrips), where each strip is swollen to a different extent in solution. The final configuration yields double rolls that display a narrow transition layer between two cylinders of constant radii. To characterize the rolls' shapes, we modify the theory of thin incompatible elastic sheets to account for the Flory-Huggins interaction between the gel and the solvent. This modification allows us to derive analytical expressions for the radii, the amplitudes, and the length of the transition layer within a given roll. Our predictions agree quantitatively with available experimental data. In addition, we carry out numerical simulations that account for the complete nonlinear behavior of the gel and show good agreement between the analytical predictions and the numerical results. Our solution sheds light on a stress focusing pattern that forms at the border between two dissimilar soft materials. Moreover, models that provide quantitative predictions on the final morphology in such heterogeneously swelling hydrogels are useful for understanding growth patterns in biology as well as accurately tailoring the structure of gels for various technological applications.

In this work we extend the Caspar-Klug construction to the archaeal viruses, which in recent years have captured the attention of many researchers for their ability to thrive in extreme environments. We assume that the shells of archaeal viruses are composed of hexamers and pentamers—as is true for icosahedral viruses—together with heptamers, necessary to introduce negative Gauss curvature. Following the original work of Caspar and Klug, we first construct models capable of reproducing the shape observed in electron microscopy images of archaeal viruses. Next, using the technique of kirigami, we present a systematic way to formulate archaeal virus templates from regular hexagonal lattices. Finally, we utilize the presented techniques to build finite element models of archaeal virus geometries and investigate their shapes as a function of material properties. In particular, using thin-shell elasticity theory, we describe a buckling transition as a function of a modified Föppl–von Kármán number ${\gamma }^{★}$ and we show how changes in ${\gamma }^{★}$ may initiate the tail formation in the Acidianus two-tailed archaeal virus.