Biological neuronal networks excel over artificial ones in many ways, but the origin of this is still unknown. Our symbolic dynamics-based tool of excess entropies suggests that neuronal cultures naturally implement data structures of a higher level than what we expect from artificial neural networks, or from close-to-biology neural networks. This points to a new pathway for improving artificial neural networks towards a level demonstrated by biology.

Multi-component systems often exhibit dynamics of a high degree of complexity, rendering it difficult to assess whether a proposed model’s description is adequate. For the multitude of systems that allow for a symbolic encoding, we provide a symbolic-dynamics based entropy measure that quantifies the degree of deviation obtained by a systems’s internal dynamics from random dynamics using identical average symbol probabilities. We apply this measure to several well-studied theoretical models and show its ability to characterize differences in internal dynamics, thus providing a means to accurately compare model and experiment. Data from neuronal cultures on a multi-electrode array chip validate the usefulness of our approach, revealing inadequacies of existing models and providing guidelines for their improvement. We propose our measure to be systematically used to develop future models and simulations.

The main discovery of the paper by Imran Alsous et al is that spatial arrangement of cells in the forming fruit fly egg is to a large extent governed by simple geometrical constraints. The purpose of my comment is much more narrow, I want to attract attention to the connection of their results with some seemingly unrelated problems in soft matter and polymer physics. In brief, the picture considered by Imran Alsous et al is as follows. In the initial stages of egg production, a tree-like structure of 16 cells is formed from a single stem cell by a series of (incomplete) divisions. Of these 16, one will eventually become the egg, while 15 others play some supporting role (unlike the usual biological terminology, incomprehensible for a non-professional, these supporting cells are called nurse cells). The divisions are incomplete in the sense that each daughter cell remains connected to its mother cell by a tiny tube, called ring canal, through which presumably nurse cells can supply nutrients to the future egg. Through the series of these incomplete divisions, a structure arises with a tree-like topology. Moreover, this tree reflects the history of cell divisions and thus represents the cell lineage tree (which is why it is called CLT). For a better description of this process interested reader is referred to either the original article by Imran Alsous et al, or an accompanying N&V [1]. Imran Alsous et al pointed out that this tree-like structure, consisting of cells with a significant (excluded) volume, must be embedded in 3D and placed in a limited space on the internal surface of a convex shaped domain. They discovered that topologically different trees can be packed in different numbers of ways, interpreted this as difference in packing entropy, and argued that this entropic effect is significant for the real system in question. As stated, my goal is to point out parallels to some other problems. To begin with, consider a dendrimer – a regular branched polymer with one three-functional root, 3 threefunctional branched points of generation one, 3 · 2 three-functional points of generation two, . . . , 3 · 2`−1 three-functional points of generation `, and 3 · 2`−1 of single-functional ends, or “leaves” (see, e.g., [2]). The well known (and obvious) fact is that large generation dendrimers cannot fit in space: assuming bond lengths are fixed, the maximal size of an

Microfluidic experiments and simulations reveal an anomalous percolation transition for the flow of a yield stress fluid through a random porous medium. The evolution of the fluidized network results in a highly nonlinear flow conductivity and reveals a novel dispersion mechanism.

Mechanical and phononic metamaterials exhibiting negative elastic moduli, gapped vibrational spectra, or topologically protected modes enable precise control of structural and acoustic functionalities. While much progress has been made in their experimental and theoretical characterization, the inverse design of mechanical metamaterials with arbitrarily programmable spectral properties and mode localization remains an unsolved problem. Here, we present a flexible computational inverse-design framework that allows the efficient tuning of one or more gaps at nearly arbitrary positions in the spectrum of discrete phononic metamaterial structures. The underlying algorithm optimizes the linear response of elastic networks directly, is applicable to ordered and disordered structures, scales efficiently in 2D and 3D, and can be combined with a wide range of numerical optimization schemes. We illustrate the broad practical potential of this approach by designing mechanical bandgap switches that open and close pre-programmed spectral gaps in response to an externally applied stimulus such as shear or compression. We further show that the designed structures can host topologically protected edge modes, and validate the numerical predictions through explicit 3D finite element simulations of continuum elastica with experimentally relevant material parameters. Generally, this network-based inverse design paradigm offers a direct pathway towards manufacturing phononic metamaterials, DNA origami structures and topolectric circuits that can realize a wide range of static and dynamic target functionalities.

We introduce spectral Morse-Smale analysis as a robust method to identify topological phase transitions in disordered continuous media. Combining microfluidic experiments with large-scale, pore-resolved simulations of porous media flow, we demonstrate that invariants of Morse-Smale graphs of flow speed provide a well-defined measure of the effects of spatial disorder on fluid transport. By systematically perturbing a microfluidic lattice, the fluid flow topology undergoes a phase transition from periodic to filamentous flow structure, which corresponds to a change in the spectral density of the Morse-Smale graphs and carries important implications for advective transport and front dispersion. Due to its generic formulation, spectral Morse-Smale analysis can be applied to detect and characterize topological transformations in a wide range of complex physical, chemical or biological fluid systems.

An open question in biological neural networks is whether changes in firing modalities are mainly an individual network property or whether networks follow a joint pathway. For the early developmental period, our study focusing on a simple network class of excitatory and inhibitory neurons suggests the following answer: Networks with considerable variation of topology and dynamical parameters follow a universal firing paradigm that evolves as the overall connectivity strength and firing level increase, as seen in the process of network maturation. A simple macroscopic model reproduces the main features of the paradigm as a result of the competition between the fundamental dynamical system notions of synchronization vs chaos and explains why in simulations the paradigm is robust regarding differences in network topology and largely independent from the neuron model used. The presented findings reflect the first dozen days of dissociated neuronal in vitro cultures (upon following the developmental period bears similarly universal features but is characterized by the processes of neuronal facilitation and depression that do not require to be considered for the first developmental period).

In the version of this Letter originally published, the citations to equation (1) in Fig. 3 caption and the main text were incorrect; they should have been to equation (2). This has now been corrected.

Topological defects shape the material and transport properties of physical systems. Examples range from vortex lines in quantum superfluids, defect-mediated buckling of graphene, and grain boundaries in ferromagnets and colloidal crystals, to domain structures formed in the early universe. The Kibble-Zurek (KZ) mechanism describes the topological defect formation in continuous non-equilibrium phase transitions with a constant finite quench rate. Universal KZ scaling laws have been verified experimentally and numerically for second-order transitions in planar Euclidean geometries, but their validity for non-thermal transitions in curved and topologically nontrivial systems still poses open questions. Here, we use recent experimentally confirmed theory to investigate topological defect formation in curved elastic surface crystals formed by stress-quenching a bilayer material. For both spherical and toroidal crystals, we find that the defect densities follow KZ-type power laws. Moreover, the nucleation sequences agree with recent experimental observations for spherical colloidal crystals. Our results suggest that curved elastic bilayers provide an experimentally accessible macroscopic system to study universal properties of non-thermal phase transitions in non-planar geometries.

Optimal packings1,2 of unconnected objects have been studied for centuries3–6, but the packing principles of linked objects, such as topologically complex polymers7,8 or cell lineages9,10, are yet to be fully explored. Here, we identify and investigate a generic class of geometrically frustrated tree packing problems, arising during the initial stages of animal development when interconnected cells assemble within a convex enclosure10. Using a combination of 3D imaging, computational image analysis and mathematical modelling, we study the tree packing problem in Drosophila egg chambers, where 16 germline cells are linked by cytoplasmic bridges to form a branched tree. Our imaging data reveal non-uniformly distributed tree packings, in agreement with predictions from energy-based computations. This departure from uniformity is entropic and affects cell organization during the first stages of the animal’s development. Considering mathematical models of increasing complexity, we investigate spherically confined tree packing problems on convex polyhedra11 that generalize Platonic and Archimedean solids. Our experimental and theoretical results provide a basis for understanding the principles that govern positional ordering in linked multicellular structures, with implications for tissue organization and dynamics12,13. The organization of small clusters of connected cells confined to an egg chamber during early development can be mapped onto a tree packing problem. Entropically preferred packing configurations are shown to arise more readily in experiment.

Fracture fundamentally limits the structural stability of macroscopic and microscopic matter, from beams and bones to microtubules and nanotubes. Despite substantial recent experimental and theoretical progress, fracture control continues to present profound practical and theoretical challenges. While bending-induced fracture of elongated rod-like objects has been intensely studied, the effects of twist and quench dynamics have yet to be explored systematically. Here, we show how twist and quench protocols may be used to control such fracture processes, by revisiting Feynman's observation that dry spaghetti typically breaks into three or more pieces when exposed to large pure bending stresses. Combining theory and experiment, we demonstrate controlled binary fracture of brittle elastic rods for two distinct protocols based on twisting and nonadiabatic quenching. Our experimental data for twist-controlled fracture agree quantitatively with a theoretically predicted phase diagram, and we establish asymptotic scaling relations for quenched fracture. Due to their general character, these results are expected to apply to torsional and kinetic fracture processes in a wide range of systems.

Mechanical and phononic metamaterials with exotic characteristics, including negative elastic moduli, negative effective mass density, and gapped vibrational spectra, enable precise control of mechanical functionality. A promising approach to designing these and other desirable material properties uses network-based techniques, which are also well-suited to describing many naturally occurring and engineered systems such as actin--myosin networks in biological cells or stiff pin--jointed frames used in structural engineering. Here we show that discrete networks allow efficient algorithmic tuning of gaps at almost arbitrary positions in the spectrum, as well as multiple gaps, isolated modes in the center of a gap, and extensions to periodic 2D and 3D crystals. An additional practical advantage of network-based architectures is their ability to accommodate arbitrary topologies and connectivities and the relative ease of fabrication. The proposed spectral network design framework thus offers a direct path towards manufacturing flexible phononic metamaterials.

Induced by proteins within the cell membrane or by differential growth, heating, or swelling, spontaneous curvatures can drastically affect the morphology of thin bodies and induce mechanical instabilities. Yet, the interaction of spontaneous curvature and geometric frustration in curved shells remains poorly understood. Via a combination of precision experiments on elastomeric spherical shells, simulations, and theory, we show how a spontaneous curvature induces a rotational symmetry-breaking buckling as well as a snapping instability reminiscent of the Venus fly trap closure mechanism. The instabilities, and their dependence on geometry, are rationalized by reducing the spontaneous curvature to an effective mechanical load. This formulation reveals a combined pressurelike term in the bulk and a torquelike term in the boundary, allowing scaling predictions for the instabilities that are in excellent agreement with experiments and simulations. Moreover, the effective pressure analogy suggests a curvature-induced subcritical buckling in closed shells. We determine the critical buckling curvature via a linear stability analysis that accounts for the combination of residual membrane and bending stresses. The prominent role of geometry in our findings suggests the applicability of the results over a wide range of scales.

Fundamental biological and biomimetic processes, from tissue morphogenesis to soft robotics, rely on the propagation of chemical and mechanical surface waves to signal and coordinate active force generation. The complex interplay between surface geometry and contraction wave dynamics remains poorly understood, but it will be essential for the future design of chemically driven soft robots and active materials. Here, we couple prototypical chemical wave and reaction-diffusion models to non-Euclidean shell mechanics to identify and characterize generic features of chemomechanical wave propagation on active deformable surfaces. Our theoretical framework is validated against recent data from contractile wave measurements on ascidian and starfish oocytes, producing good quantitative agreement in both cases. The theory is then applied to illustrate how geometry and preexisting discrete symmetries can be utilized to focus active elastic surface waves. We highlight the practical potential of chemomechanical coupling by demonstrating spontaneous wave-induced locomotion of elastic shells of various geometries. Altogether, our results show how geometry, elasticity, and chemical signaling can be harnessed to construct dynamically adaptable, autonomously moving mechanical surface waveguides.

Topological defects shape the material and transport properties of physical systems. Examples range from vortex lines in quantum superfluids, defect-mediated buckling of graphene, and grain boundaries in ferromagnets and colloidal crystals, to domain structures formed in the early universe. The Kibble-Zurek (KZ) mechanism describes the topological defect formation in continuous non-equilibrium phase transitions with a constant finite quench rate. Universal KZ scaling laws have been verified experimentally and numerically for second-order transitions in planar Euclidean geometries, but their validity for discontinuous first-order transitions in curved and topologically nontrivial systems still poses an open question. Here, we use recent experimentally confirmed theory to investigate topological defect formation in curved elastic surface crystals formed by stress-quenching a bilayer material. Studying both spherical and toroidal crystals, we find that the defect densities follow KZ-type power laws independent of surface geometry and topology. Moreover, the nucleation sequences agree with recent experimental observations for spherical colloidal crystals. These results suggest that KZ scaling laws hold for a much broader class of dynamical phase transitions than previously thought, including non-thermal first-order transitions in non-planar geometries.

We investigate how thin structures change their shape in response to non-mechanical stimuli that can be interpreted as variations in the structure’s natural curvature. Starting from the theory of non-Euclidean plates and shells, we derive an effective model that reduces a three-dimensional stimulus to the natural fundamental forms of the mid-surface of the structure, incorporating expansion, or growth, in the thickness. Then, we apply the model to a variety of thin bodies, from flat plates to spherical shells, obtaining excellent agreement between theory and numerics. We show how cylinders and cones can either bend more or unroll, and eventually snap and rotate. We also study the nearly isometric deformations of a spherical shell and describe how this shape change is ruled by the geometry of a spindle. As the derived results stem from a purely geometrical model, they are general and scalable.