The fluidic memristor as a collective phenomenon in elastohydrodynamic networks.

Fluid flow networks are ubiquitous and can be found in a broad range of contexts, from human-made systems such as water supply networks to living systems like animal and plant vasculature. In many cases, the elements forming these networks exhibit a highly non-linear pressure-flow relationship. Although we understand how these elements work individually, their collective behavior remains poorly understood. In this work, we combine experiments, theory, and numerical simulations to understand the main mechanisms underlying the collective behavior of soft flow networks with elements that exhibit negative differential resistance. Strikingly, our theoretical analysis and experiments reveal that a minimal network of nonlinear resistors, which we have termed a 'fluidic memristor', displays history-dependent resistance. This new class of element can be understood as a collection of hysteresis loops that allows this fluidic system to store information, and it can be directly used as a tunable resistor in fluidic setups. Our results provide insights that can inform other applications of fluid flow networks in soft materials science, biomedical settings, and soft robotics, and may also motivate new understanding of the flow networks involved in animal and plant physiology.

From localized to well mixed: How commuter interactions shape disease spread.

Interactions between commuting individuals can lead to large-scale spreading of rumors, ideas, or disease, even though the commuters have no net displacement. The emergent dynamics depend crucially on the commuting distribution of a population, that is how the probability to travel to a destination decays with distance from home. Applying this idea to epidemics, we will demonstrate the qualitatively different infection dynamics emerging from populations with different commuting distributions. If the commuting distribution is exponentially localized, then we recover a reaction-diffusion system and observe Fisher waves traveling at a speed proportional to the characteristic commuting distance. If the commuting distribution has a long tail, then no finite-velocity waves can form, but we show that, in some regimes, there is nontrivial spatial dependence that the well-mixed approximation neglects. We discuss how, in all cases, an initial dispersal-dominated regime can allow the disease to go undetected for a finite amount of time before exponential growth takes over. This "offset time" is a quantity of huge importance for epidemic surveillance and yet largely ignored in the literature.

Flow-network-controlled shape transformation of a thin membrane through differential fluid storage and surface expansion.

The mechanical properties of a thin, planar material, perfused by an embedded flow network, have been suggested to be potentially changeable locally and globally by fluid transport and storage, which can result in both small- and large-scale deformations such as out-of-plane buckling. In these processes, fluid absorption and storage eventually cause the material to locally swell. Different parts can hydrate and swell unevenly, prompting a differential expansion of the surface. In order to computationally study the hydraulically induced differential swelling and buckling of such a membrane, we develop a network model that describes both the membrane shape and fluid movement, coupling mechanics with hydrodynamics. We simulate the time-dependent fluid distribution in the flow network based on a spatially explicit resistor network model with local fluid-storage capacitance. The shape of the surface is modeled by a spring network produced by a tethered mesh discretization, in which local bond rest lengths are adjusted instantaneously according to associated local fluid content in the capacitors in a quasistatic way. We investigate the effects of various designs of the flow network, including overall hydraulic traits (resistance and capacitance) and hierarchical architecture (arrangement of major and minor veins), on the specific dynamics of membrane shape transformation. To quantify these effects, we explore the correlation between local Gaussian curvature and relative stored fluid content in each hierarchy by using linear regression, which reveals that stronger correlations could be induced by less densely connected major veins. This flow-controlled mechanism of shape transformation was inspired by the blooming of flowers through the unfolding of petals. It can potentially offer insights for other reversible motions observed in plants induced by differential turgor and water transport through the xylem vessels, as well as engineering applications.

Leaf Water Storage and Robustness to Intermittent Drought: A Spatially Explicit Capacitive Model for Leaf Hydraulics.

Leaf hydraulic networks play an important role not only in fluid transport but also in maintaining whole-plant water status through transient environmental changes in soil-based water supply or air humidity. Both water potential and hydraulic resistance vary spatially throughout the leaf transport network, consisting of xylem, stomata and water-storage cells, and portions of the leaf areas far from the leaf base can be disproportionately disadvantaged under water stress. Besides the suppression of transpiration and reduction of water loss caused by stomatal closure, the leaf capacitance of water storage, which can also vary locally, is thought to be crucial for the maintenance of leaf water status. In order to study the fluid dynamics in these networks, we develop a spatially explicit, capacitive model which is able to capture the local spatiotemporal changes of water potential and flow rate in monocotyledonous and dicotyledonous leaves. In electrical-circuit analogs described by Ohm's law, we implement linear capacitors imitating water storage, and we present both analytical calculations of a uniform one-dimensional model and numerical simulation methods for general spatially explicit network models, and their relation to conventional lumped-element models. Calculation and simulation results are shown for the uniform model, which mimics key properties of a monocotyledonous grass leaf. We illustrate water status of a well-watered leaf, and the lowering of water potential and transpiration rate caused by excised water source or reduced air humidity. We show that the time scales of these changes under water stress are hugely affected by leaf capacitance and resistances to capacitors, in addition to stomatal resistance. Through this modeling of a grass leaf, we confirm the presence of uneven water distribution over leaf area, and also discuss the importance of considering the spatial variation of leaf hydraulic traits in plant biology.

Distribution Networks Achieve Uniform Perfusion through Geometric Self-Organization.

A generic flow distribution network typically does not deliver its load at a uniform rate across a service area, instead oversupplying regions near the nutrient source while leaving downstream regions undersupplied. In this Letter we demonstrate how a local adaptive rule coupling tissue growth with nutrient density results in a flow network that self-organizes to deliver nutrients uniformly. This geometric adaptive rule can be generalized and imported to mechanics-based adaptive models to address the effects of spatial gradients in nutrients or growth factors in tissues.

Emergent dynamics in excitable flow systems.

Flow networks can describe many natural and artificial systems. We present a model for a flow system that allows for volume accumulation, includes conduits with a nonlinear relation between current and pressure difference, and can be applied to networks of arbitrary topology. The model displays complex dynamics, including self-sustained oscillations in the absence of any dynamics in the inputs and outputs. In this work we analytically show the origin of self-sustained oscillations for the one-dimensional case. We numerically study the behavior of systems of arbitrary topology under different conditions: we discuss their excitability, the effect of different boundary conditions, and wave propagation when the network has regions of conduits with linear conductance.

Hidden Topological Structure of Flow Network Functionality.

The ability to reroute and control flow is vital to the function of venation networks across a wide range of organisms. By modifying individual edges in these networks, either by adjusting edge conductances or creating and destroying edges, organisms robustly control the propagation of inputs to perform specific tasks. However, a fundamental disconnect exists between the structure and function: networks with different local architectures can perform the same functions. Here, we answer the question of how changes at the level of individual edges collectively create functionality at the scale of an entire network. Using persistent homology, we analyze networks tuned to perform complex tasks. We find that the responses of such networks encode a hidden topological structure composed of sectors of nearly uniform pressure. Although these sectors are not apparent in the underlying network structure, they correlate strongly with the tuned function. The connectivity of these sectors, rather than that of individual nodes, provides a quantitative relationship between structure and function in flow networks.

Auxin biosynthesis and cellular efflux act together to regulate leaf vein patterning.

Our current understanding of vein development in leaves is based on canalization of the plant hormone auxin into self-reinforcing streams which determine the sites of vascular cell differentiation. By comparison, how auxin biosynthesis affects leaf vein patterning is less well understood. Here, after observing that inhibiting polar auxin transport rescues the sparse leaf vein phenotype in auxin biosynthesis mutants, we propose that the processes of auxin biosynthesis and cellular auxin efflux work in concert during vein development. By using computational modeling, we show that localized auxin maxima are able to interact with mechanical forces generated by the morphological constraints which are imposed during early primordium development. This interaction is able to explain four fundamental characteristics of midvein morphology in a growing leaf: (i) distal cell division; (ii) coordinated cell elongation; (iii) a midvein positioned in the center of the primordium; and (iv) a midvein which is distally branched. Domains of auxin biosynthetic enzyme expression are not positioned by auxin canalization, as they are observed before auxin efflux proteins polarize. This suggests that the site-specific accumulation of auxin, as regulated by the balanced action of cellular auxin efflux and local auxin biosynthesis, is crucial for leaf vein formation.

Tuning and jamming reduced to their minima.

Inspired by protein folding, we smooth out the complex cost function landscapes of two processes: the tuning of networks and the jamming of ideal spheres. In both processes, geometrical frustration plays a role-tuning pressure differences between pairs of target nodes far from the source in a flow network impedes tuning of nearby pairs more than the reverse process, while unjamming the system in one region can make it more difficult to unjam elsewhere. By modifying the cost functions to control the order in which functions are tuned or regions unjam, we smooth out local minima while leaving global minima unaffected, increasing the success rate for reaching global minima.

Threading the Spindle: A Geometric Study of Chiral Liquid Crystal Polymer Microparticles.

Polymeric particles are strong candidates for designing artificial materials capable of emulating the complex twisting-based functionality observed in biological systems. In this Letter, we provide the first detailed investigation of the swelling behavior of bipolar polymer liquid crystalline microparticles. Deswelling from the spherical bipolar configuration causes the microparticles to contract anisotropically and twist in the process, resulting in a twisted spindle-shaped structure. We propose a model to describe the observed spiral patterns and twisting behavior.

Resilience in hierarchical fluid flow networks.

The structure of flow networks determines their function under normal conditions as well as their response to perturbative damage. Brain vasculature often experiences transient or permanent occlusions in the finest vessels, but it is not clear how these microclots affect the large-scale blood flow or to what extent they decrease functionality. Motivated by this, we investigate how flow is rerouted after the occlusion of a single edge in networks with a hierarchy in edge conductances. We find that in two-dimensional networks, vessels formed by highly conductive edges serve as barriers to contain the displacement of flow due to a localized perturbation. In this way, the vein provides shielding from damage to surrounding edges. We show that once the conductance of the vein surpasses an initial minimal value, further increasing the conductance can no longer extend the shielding provided by the vein. Rather, the length scale of the shielding is set by the network topology. Upon understanding the effects of a single vein, we investigate the global resilience of networks with complex hierarchical order. We find that a system of veins arranged in a grid is able to modestly increase the overall network resilience, outperforming a parallel vein pattern.

Limits of multifunctionality in tunable networks.

Nature is rife with networks that are functionally optimized to propagate inputs to perform specific tasks. Whether via genetic evolution or dynamic adaptation, many networks create functionality by locally tuning interactions between nodes. Here we explore this behavior in two contexts: strain propagation in mechanical networks and pressure redistribution in flow networks. By adding and removing links, we are able to optimize both types of networks to perform specific functions. We define a single function as a tuned response of a single "target" link when another, predetermined part of the network is activated. Using network structures generated via such optimization, we investigate how many simultaneous functions such networks can be programed to fulfill. We find that both flow and mechanical networks display qualitatively similar phase transitions in the number of targets that can be tuned, along with the same robust finite-size scaling behavior. We discuss how these properties can be understood in the context of constraint-satisfaction problems.

Phenotypes of Vascular Flow Networks.

Complex distribution networks are pervasive in biology. Examples include nutrient transport in the slime mold Physarum polycephalum as well as mammalian and plant venation. Adaptive rules are believed to guide development of these networks and lead to a reticulate, hierarchically nested topology that is both efficient and resilient against perturbations. However, as of yet, no mechanism is known that can generate such networks on all scales. We show how hierarchically organized reticulation can be constructed and maintained through spatially correlated load fluctuations on a particular length scale. We demonstrate that the network topologies generated represent a trade-off between optimizing transport efficiency, construction cost, and damage robustness and identify the Pareto-efficient front that evolution is expected to favor and select for. We show that the typical fluctuation length scale controls the position of the networks on the Pareto front and thus on the spectrum of venation phenotypes.

Comparing two classes of biological distribution systems using network analysis.

Distribution networks-from vasculature to urban transportation pathways-are spatially embedded networks that must route resources efficiently in the face of pressures induced by the costs of building and maintaining network infrastructure. Such requirements are thought to constrain the topological and spatial organization of these systems, but at the same time, different kinds of distribution networks may exhibit variable architectural features within those general constraints. In this study, we use methods from network science to compare and contrast two classes of biological transport networks: mycelial fungi and vasculature from the surface of rodent brains. These systems differ in terms of their growth and transport mechanisms, as well as the environments in which they typically exist. Though both types of networks have been studied independently, the goal of this study is to quantify similarities and differences in their network designs. We begin by characterizing the structural backbone of these systems with a collection of measures that assess various kinds of network organization across topological and spatial scales, ranging from measures of loop density, to those that quantify connected pathways between different network regions, and hierarchical organization. Most importantly, we next carry out a network analysis that directly considers the spatial embedding and properties especially relevant to the function of distribution systems. We find that although both the vasculature and mycelia are highly constrained planar networks, there are clear distinctions in how they balance tradeoffs in network measures of wiring length, efficiency, and robustness. While the vasculature appears well organized for low cost, but relatively high efficiency, the mycelia tend to form more expensive but in turn more robust networks. As a whole, this work demonstrates the utility of network-based methods to identify both common features and variations in the network structure of different classes of biological transport systems.

Trophallaxis-inspired model for distributed transport between randomly interacting agents.

Trophallaxis, the regurgitation and mouth to mouth transfer of liquid food between members of eusocial insect societies, is an important process that allows the fast and efficient dissemination of food in the colony. Trophallactic systems are typically treated as a network of agent interactions. This approach, though valuable, does not easily lend itself to analytic predictions. In this work we consider a simple trophallactic system of randomly interacting agents with finite carrying capacity, and calculate analytically and via a series of simulations the global food intake rate for the whole colony as well as observables describing how uniformly the food is distributed within the nest. Our model and predictions provide a useful benchmark to assess to what level the observed food uptake rates and efficiency in food distribution is due to stochastic effects or specific trophallactic strategies by the ant colony. Our work also serves as a stepping stone to describing the collective properties of more complex trophallactic systems, such as those including division of labor between foragers and workers.

The smectic order of wrinkles.

A thin elastic sheet lying on a soft substrate develops wrinkled patterns when subject to an external forcing or as a result of geometric incompatibility. Thin sheet elasticity and substrate response equip such wrinkles with a global preferred wrinkle spacing length and with resistance to wrinkle curvature. These features are responsible for the liquid crystalline smectic-like behaviour of such systems at intermediate length scales. This insight allows better understanding of the wrinkling patterns seen in such systems, with which we explain pattern breaking into domains, the properties of domain walls and wrinkle undulation. We compare our predictions with numerical simulations and with experimental observations.

NET: a new framework for the vectorization and examination of network data.

BACKGROUND: The analysis of complex networks both in general and in particular as pertaining to real biological systems has been the focus of intense scientific attention in the past and present. In this paper we introduce two tools that provide fast and efficient means for the processing and quantification of biological networks like Drosophila tracheoles or leaf venation patterns: the Network Extraction Tool (NET) to extract data and the Graph-edit-GUI (GeGUI) to visualize and modify networks. RESULTS: NET is especially designed for high-throughput semi-automated analysis of biological datasets containing digital images of networks. The framework starts with the segmentation of the image and then proceeds to vectorization using methodologies from optical character recognition. After a series of steps to clean and improve the quality of the extracted data the framework produces a graph in which the network is represented only by its nodes and neighborhood-relations. The final output contains information about the adjacency matrix of the graph, the width of the edges and the positions of the nodes in space. NET also provides tools for statistical analysis of the network properties, such as the number of nodes or total network length. Other, more complex metrics can be calculated by importing the vectorized network to specialized network analysis packages. GeGUI is designed to facilitate manual correction of non-planar networks as these may contain artifacts or spurious junctions due to branches crossing each other. It is tailored for but not limited to the processing of networks from microscopy images of Drosophila tracheoles. CONCLUSION: The networks extracted by NET closely approximate the network depicted in the original image. NET is fast, yields reproducible results and is able to capture the full geometry of the network, including curved branches. Additionally GeGUI allows easy handling and visualization of the networks.

Global Optimization, Local Adaptation, and the Role of Growth in Distribution Networks.

Highly optimized complex transport networks serve crucial functions in many man-made and natural systems such as power grids and plant or animal vasculature. Often, the relevant optimization functional is nonconvex and characterized by many local extrema. In general, finding the global, or nearly global optimum is difficult. In biological systems, it is believed that such an optimal state is slowly achieved through natural selection. However, general coarse grained models for flow networks with local positive feedback rules for the vessel conductivity typically get trapped in low efficiency, local minima. In this work we show how the growth of the underlying tissue, coupled to the dynamical equations for network development, can drive the system to a dramatically improved optimal state. This general model provides a surprisingly simple explanation for the appearance of highly optimized transport networks in biology such as leaf and animal vasculature.

Topological Phenotypes Constitute a New Dimension in the Phenotypic Space of Leaf Venation Networks.

The leaves of angiosperms contain highly complex venation networks consisting of recursively nested, hierarchically organized loops. We describe a new phenotypic trait of reticulate vascular networks based on the topology of the nested loops. This phenotypic trait encodes information orthogonal to widely used geometric phenotypic traits, and thus constitutes a new dimension in the leaf venation phenotypic space. We apply our metric to a database of 186 leaves and leaflets representing 137 species, predominantly from the Burseraceae family, revealing diverse topological network traits even within this single family. We show that topological information significantly improves identification of leaves from fragments by calculating a "leaf venation fingerprint" from topology and geometry. Further, we present a phenomenological model suggesting that the topological traits can be explained by noise effects unique to specimen during development of each leaf which leave their imprint on the final network. This work opens the path to new quantitative identification techniques for leaves which go beyond simple geometric traits such as vein density and is directly applicable to other planar or sub-planar networks such as blood vessels in the brain.

Structural self-assembly and avalanchelike dynamics in locally adaptive networks.

Transport networks play a key role across four realms of eukaryotic life: slime molds, fungi, plants, and animals. In addition to the developmental algorithms that build them, many also employ adaptive strategies to respond to stimuli, damage, and other environmental changes. We model these adapting network architectures using a generic dynamical system on weighted graphs and find in simulation that these networks ultimately develop a hierarchical organization of the final weighted architecture accompanied by the formation of a system-spanning backbone. In addition, we find that the long term equilibration dynamics exhibit behavior reminiscent of glassy systems characterized by long periods of slow changes punctuated by bursts of reorganization events.