arXiv subjects

Alain Goriely

Publications and source records attributed to Alain Goriely.

At least 19 recordsLinked to original sources

arXiv2026

Tensional wrinkling of thin elastic sheets with two circular holes

A paradigm for the study of wrinkling in elastic sheet is the Lam\'{e} configuration, in which azimuthal wrinkles form in an annular sheet subjected to tensile loads at both edges. Since wrinkles are spatially extended, this instability provides a mechanism for stress transmission over long distances. A natural extension of this problem is wrinkling in sheets with multiple holes or broken symmetry. Here, we investigate tension-induced wrinkling in thin elastic sheets containing two identical circular holes by combining analytical modeling and experiments. The pre-buckled state is solved analytically using bipolar coordinates within the framework of linear elasticity, enabling the identification of the wrinkling threshold as a function of the distance between the two holes. Near-threshold wrinkling and interactions between wrinkles are analyzed, and we validate our theoretical predictions against experimental observations obtained through video imaging of spin-coated polystyrene sheets floating on liquid surfaces with controlled surface tension. Our results demonstrate that geometric symmetry breaking, such as the presence of a second hole, strongly influences wrinkle nucleation, orientation, and spatial extent. Beyond mechanics, these findings further support the idea that mechanical cues can be amplified by instabilities, providing a mechanical analogue for long-range cellular mechanosensing.

cond-mat.soft↗

arXiv2026

Drift-Diffusion Matching: Embedding dynamics in latent manifolds of asymmetric neural networks

Recurrent neural networks (RNNs) provide a theoretical framework for understanding computation in biological neural circuits, yet classical results, such as Hopfield's model of associative memory, rely on symmetric connectivity that restricts network dynamics to gradient-like flows. In contrast, biological networks support rich time-dependent behaviour facilitated by their asymmetry. Here we introduce a general framework, which we term drift-diffusion matching, for training continuous-time RNNs to represent arbitrary, nonlinear stochastic differential equations (SDEs), with given drift and diffusion coefficients, within a low-dimensional latent subspace. Allowing asymmetric connectivity, we show that RNNs can faithfully embed the drift and diffusion of a given SDE, including nonlinear and nonequilibrium dynamics such as chaotic attractors. As an application, we construct RNN realisations of stochastic systems that transiently explore various attractors through both input-driven switching and autonomous transitions driven by nonequilibrium currents, which we interpret as models of associative and sequential (episodic) memory. To elucidate how these dynamics are encoded in the network, we introduce decompositions of the RNN based on its asymmetric connectivity and its time-irreversibility. Our results extend attractor neural network theory beyond equilibrium, showing that asymmetric neural populations can implement a broad class of dynamical computations within low-dimensional manifolds, unifying ideas from associative memory, nonequilibrium statistical mechanics, and neural computation.

cond-mat.dis-nn↗

arXiv2026

A whole-brain model of amyloid beta accumulation and cerebral hypoperfusion in Alzheimer's disease

Accumulation of amyloid beta proteins is a defining feature of Alzheimer's disease, and is usually accompanied by cerebrovascular pathology. Evidence suggests that amyloid beta and cerebrovascular pathology are mutually reinforcing; in particular, amyloid beta suppresses perfusion by constricting capillaries, and hypoperfusion promotes the production of amyloid beta. Here, we propose a whole-brain model coupling amyloid beta and blood vessel through a hybrid model consisting of a reaction-diffusion system for the protein dynamics and porous-medium model of blood flow within and between vascular networks: arterial, capillary and venous. We discretize the resulting parabolic--elliptic system of PDEs by means of a high-order discontinuous Galerkin method in space and an implicit Euler scheme in time. Simulations in realistic brain geometries demonstrate the emergence of multistability, implying that a sufficiently large pathogenic protein seeds is necessary to trigger disease outbreak. Motivated by the "two-hit vascular hypothesis" of Alzheimer's disease that hypoperfusive vascular damage triggers amyloid beta pathology, we also demonstrate that localized hypoperfusion, in response to injury, can destabilize the healthy steady state and trigger brain-wide disease outbreak.

math.NA↗

arXiv2025

A universal phase-plane model for in vivo protein aggregation

Neurodegenerative diseases are driven by the accumulation of protein aggregates in the brain of affected individuals. The aggregation behaviour in vitro is well understood and driven by the equilibration of a super-saturated protein solution to its aggregated equilibrium state. However, the situation is altered fundamentally in living systems where active processes consume energy to remove aggregates. It remains unclear how and why cells transition from a state with predominantly monomeric protein, which is stable over decades, to one dominated by aggregates. Here, we develop a simple but universal theoretical framework to describe cellular systems that include both aggregate formation and removal. Using a two-dimensional phase-plane representation, we show that the interplay of aggregate formation and removal generates cell-level bistability, with a bifurcation structure that explains both the emergence of disease and the effects of therapeutic interventions. We explore a wide range of aggregate formation and removal mechanisms and show that phenomena such as seeding arise robustly when a minimal set of requirements on the mechanism are satisfied. By connecting in vitro aggregation mechanisms to changes in cell state, our framework provides a general conceptual link between molecular-level therapeutic interventions and their impact on disease progression.

q-bio.BM↗

arXiv2025

Coarse-graining nonequilibrium diffusions with Markov chains

We investigate nonequilibrium steady-state dynamics in both continuous- and discrete-state stochastic processes. Our analysis focuses on planar diffusion dynamics and their coarse-grained approximations by discrete-state Markov chains. Using finite-volume approximations, we derive an approximate master equation directly from the underlying diffusion and show that this discretisation preserves key features of the nonequilibrium steady-state. In particular, we show that the entropy production rate of the approximation converges as the number of discrete states goes to the limit. These results are illustrated with analytically solvable diffusions and numerical experiments on nonlinear processes, demonstrating how this approach can be used to explore the dependence of the entropy production rate on model parameters. Finally, we address the problem of inferring discrete-state Markov models from continuous stochastic trajectories. We show that discrete-state models significantly underestimate the true entropy production rate. However, we also show that they can provide tests to determine if a stationary planar diffusion is out of equilibrium. This property is illustrated with both simulated data and empirical trajectories from schooling fish.

cond-mat.stat-mech↗

arXiv2025

The illusion of illusions: There are no optical corrections in the Parthenon

One of the oldest and most enduring myths in human history is the belief that the Parthenon was cleverly designed with various curved structures and sizes in order to correct optical illusions, and therefore appear straight and regular. The myth has its origin in the writings of Vitruvius more than 2,000 years ago and was renewed in the nineteenth century when curved profiles were carefully measured. At least twelve different \textit{optical corrections} have been proposed, all following the same basic principle. The myth is still widely acknowledged as an obvious truth despite a complete absence of historical or scientific evidence. This paper analyzes these corrections scientifically and demonstrates that the illusions they are supposed to correct are either non-existent or so small as to be imperceptible.

math.HO↗

arXiv2025

A multiscale theory for network advection-reaction-diffusion

Mathematical network models are extremely useful to capture complex propagation processes between different regions (nodes), for example the spread of an infectious agent between different countries, or the transport and replication of toxic proteins across different brain regions in neurodegenerative diseases. In these models, transport is modeled at the macroscale through an operator, the so-called graph Laplacian, based on the edge properties and topology, capturing the fluxes between different nodes of the network. However, this phenomenological approach fails to take into account the physical processes taking place at the microscale within the edge. A fundamental problem is then to obtain a transport operator from mechanistic principles based on the underlying transport process. Using advection-reaction-diffusion as a generic mechanism for inter-nodal exchanges, we derive a multiscale network transport model and obtain the corresponding linear transport operator at the macroscale from first principles. This effective graph Laplacian is fully determined by the transport mechanisms along the edges at the microscale. We show that this operator correctly captures the transport, and we study its scaling properties with respect to edge length.

physics.bio-ph↗

arXiv2025

Generalized invariants meet constitutive neural networks: A novel framework for hyperelastic materials

The major challenge in determining a hyperelastic model for a given material is the choice of invariants and the selection how the strain energy function depends functionally on these invariants. Here we introduce a new data-driven framework that simultaneously discovers appropriate invariants and constitutive models for isotropic incompressible hyperelastic materials. Our approach identifies both the most suitable invariants in a class of generalized invariants and the corresponding strain energy function directly from experimental observations. Unlike previous methods that rely on fixed invariant choices or sequential fitting procedures, our method integrates the discovery process into a single neural network architecture. By looking at a continuous family of possible invariants, the model can flexibly adapt to different material behaviors. We demonstrate the effectiveness of this approach using popular benchmark datasets for rubber and brain tissue. For rubber, the method recovers a stretch-dominated formulation consistent with classical models. For brain tissue, it identifies a formulation sensitive to small stretches, capturing the nonlinear shear response characteristic of soft biological matter. Compared to traditional and neural-network-based models, our framework provides improved predictive accuracy and interpretability across a wide range of deformation states. This unified strategy offers a robust tool for automated and physically meaningful model discovery in hyperelasticity.

cond-mat.soft↗

arXiv2025

The Darboux Classification of Curl Forces

We study particle dynamics under curl forces. These forces are a class of non-conservative, non-dissipative, position-dependent forces that cannot be expressed as gradient of a potential function. We show that the fundamental quantity of particle dynamics under curl forces is a work $1$-form. By using the Darboux classification of differential $1$-forms on $\mathbb{R}^2$ and $\mathbb{R}^3$, we establish that any curl force in two dimensions has at most two generalized potentials, while in three dimensions, it has at most three. These potentials generalize the single potential of conservative systems. For any curl force field, we introduce a corresponding conservative force field -- the conservative auxiliary force. The Hamiltonian of this conservative force is a conserved quantity of motion for the dynamics of a particle under the curl force, although it is not the physical energy.

math-ph↗

arXiv2025

Post-buckling of fiber-reinforced soft tissues

Fiber-reinforcement is a universal feature of many biological tissues. It involves the interplay between fiber stiffness, fiber orientation, and the elastic properties of the matrix, influencing pattern formation and evolution in layered tissues. Here, we investigate the deformation of a compressed film bonded to a half-space, where either the film or the substrate exhibits anisotropy. Within the framework of finite elasticity, we formulate nonlinear incremental equations, enabling linear and weakly nonlinear analyses. These analyses yield exact bifurcation conditions and an amplitude equation for surface wrinkling. In particular, for a simple fiber-reinforced model, we show that the bifurcation can be supercritical or subcritical depending on the ratio between the substrate and the film moduli. These findings underscore the pivotal role of fiber-reinforcement in shaping pattern formation in anisotropic tissues and provide insights into the morphological evolution of biological tissues.

cond-mat.soft↗

arXiv2025

Nonequilibrium physics of brain dynamics

Information processing in the brain is coordinated by the dynamic activity of neurons and neural populations at a range of spatiotemporal scales. These dynamics, captured in the form of electrophysiological recordings and neuroimaging, show evidence of time-irreversibility and broken detailed balance suggesting that the brain operates in a nonequilibrium stationary state. Furthermore, the level of nonequilibrium, measured by entropy production or irreversibility appears to be a crucial signature of cognitive complexity and consciousness. The subsequent study of neural dynamics from the perspective of nonequilibrium statistical physics is an emergent field that challenges the assumptions of symmetry and maximum-entropy that are common in traditional models. In this review, we discuss the plethora of exciting results emerging at the interface of nonequilibrium dynamics and neuroscience. We begin with an introduction to the mathematical paradigms necessary to understand nonequilibrium dynamics in both continuous and discrete state-spaces. Next, we review both model-free and model-based approaches to analysing nonequilibrium dynamics in both continuous-state recordings and neural spike-trains, as well as the results of such analyses. We briefly consider the topic of nonequilibrium computation in neural systems, before concluding with a discussion and outlook on the field.

q-bio.NC↗

arXiv2025

From reductionism to realism: Holistic mathematical modelling for complex biological systems

At its core, the physics paradigm adopts a reductionist approach, aiming to understand fundamental phenomena by decomposing them into simpler, elementary processes. While this strategy has been tremendously successful in physics, it has often fallen short in addressing fundamental questions in the biological sciences. This arises from the inherent complexity of biological systems, characterised by heterogeneity, polyfunctionality and interactions across spatiotemporal scales. Nevertheless, the traditional framework of complex systems modelling falls short, as its emphasis on broad theoretical principles has often failed to produce predictive, empirically-grounded insights. To advance towards actionable mathematical models in biology, we argue, using neuroscience as a case study, that it is necessary to move beyond reductionist approaches and instead embrace the complexity of biological systems - leveraging the growing availability of high-resolution data and advances in high-performance computing. We advocate for a holistic mathematical modelling paradigm that harnesses rich representational structures such as annotated and multilayer networks, employs agent-based models and simulation-based approaches, and focuses on the inverse problem of inferring system dynamics from observations. We emphasise that this approach is fully compatible with the search for fundamental biophysical principles, and highlight the potential it holds to drive progress in mathematical biology over the next two decades.

physics.bio-ph↗

arXiv2025

The multiscale mechanics of axon durotaxis

During neurodevelopment, neuronal axons navigate through the extracellular environment, guided by various cues to establish connections with distant target cells. Among other factors, axon trajectories are influenced by heterogeneities in environmental stiffness, a process known as durotaxis, the guidance by substrate stiffness gradients. Here, we develop a three-scale model for axonal durotaxis. At the molecular scale, we characterise the mechanical interaction between the axonal growth cone cytoskeleton, based on molecular-clutch-type interactions dependent on substrate stiffness. At the growth cone scale, we spatially integrate this relationship to obtain a model for the traction generated by the entire growth cone. Finally, at the cell scale, we model the axon as a morphoelastic filament growing on an adhesive substrate, and subject to durotactic growth cone traction. Firstly, the model predicts that, depending on the local substrate stiffness, axons may exhibit positive or negative durotaxis, and we show that this key property entails the existence of attractive zones of preferential stiffness in the substrate domain. Second, we show that axons will exhibit reflective and refractive behaviour across interface between regions of different stiffness, a basic process which may serve in the deflection of axons. Lastly, we test our model in a biological scenario wherein durotaxis was previously identified as a possible guidance mechanism in vivo. Overall, this work provides a general mechanistic theory for exploring complex effects in axonal mechanotaxis and guidance in general.

physics.bio-ph↗

arXiv2024

Nonlinear Cauchy Elasticity

Most theories and applications of elasticity rely on an energy function that depends on the strains from which the stresses can be derived. This is the traditional setting of Green elasticity, also known as hyper-elasticity. However, in its original form the theory of elasticity does not assume the existence of a strain-energy function. In this case, called Cauchy elasticity, stresses are directly related to the strains. Since the emergence of modern elasticity in the 1940s, research on Cauchy elasticity has been relatively limited. One possible reason is that for Cauchy materials, the net work performed by stress along a closed path in the strain space may be nonzero. Therefore, such materials may require access to both energy sources and sinks. This characteristic has led some mechanicians to question the viability of Cauchy elasticity as a physically plausible theory of elasticity. In this paper, motivated by its relevance to recent applications, such as the modeling of active solids, we revisit Cauchy elasticity in a modern form.

physics.class-ph↗

arXiv2024

Soft cells, Kelvin's foam and the minimal surfaces of Schwarz

Recently, we introduced a new class of shapes, called soft cells which fill space as soft tilings without gaps and overlaps while minimizing the number of sharp corners. We introduced the edge bending algorithm that deforms a polyhedral tiling into a soft tiling and we proved that an infinite class of polyhedral tilings can be smoothly deformed into standard soft tilings. Here, we demonstrate that certain triply periodic minimal surfaces naturally give rise to non-standard soft tilings. By extending the edge-bending algorithm, we further establish that the soft tilings derived from the Schwarz P and Schwarz D surfaces can be continuously transformed into one another through a one-parameter family of intermediate non-standard soft tilings. Notably, by carrying its combinatorial structure, both resulting tilings belong to the first order equivalence class of the Dirichlet-Voronoi tiling on the body-centered cubic bcc lattice, highlighting a deep geometric connection underlying these minimal surface configurations. By requiring identical end-tangents for edges in a first order class, we also define second order equivalence classes among tilings and prove that there exist exactly two such classes among soft tilings which share the full symmetry group of the DV-bcc tiling. Additionally, we construct a one-parameter family of tilings bridging standard and non-standard soft tilings, explicitly including the classic Kelvin foam structure as an intermediate configuration. This construction highlights that both the soft cells themselves and the geometric methods employed in their generation provide valuable insights into the structural principles underlying natural forms. We also present the soft tiling induced by the gyroid structure.

cs.CG↗

arXiv2024

A Mechanical Model for the Failure of Reconstructive Breast Implant Surgery Due to Capsular Contracture

Capsular contracture is a pathological response to implant-based reconstructive breast surgery, where the ``capsule'' (tissue surrounding an implant) painfully thickens, contracts and deforms. It is known to affect breast-cancer survivors at higher rates than healthy women opting for cosmetic cosmetic breast augmentation with implants. We model the early stages of capsular contracture based on stress-dependent recruitment of contractile and mechanosensitive cells to the implant site. We derive a one-dimensional continuum spatial model for the spatio-temporal evolution of cells and collagen densities away from the implant surface. Various mechanistic assumptions are investigated for linear versus saturating mechanical cell responses and cell traction forces. Our results point to specific risk factors for capsular contracture, and indicate how physiological parameters, as well as initial states (such as inflammation after surgery) contribute to patient susceptibility.

q-bio.CB↗

arXiv2024

Elephant trunk wrinkles: A mathematical model of function and form

A notable feature of the elephant trunk is the pronounced wrinkling that enables its great flexibility. Here, we devise a general mathematical model that accounts for the characteristic skin wrinkles formed during morphogenesis in the elephant trunk. Using physically realistic parameters and operating within the theoretical framework of nonlinear morphoelasticity, we elucidate analytically and numerically the effect of skin thickness, relative stiffness, and differential growth on the physiological pattern of transverse wrinkles distributed along the trunk. We conclude that since the skin and muscle components have similar material properties, geometric parameters, such as curvature, play an important role. In particular, our model predicts that, in the proximal region close to the skull, where the curvature is lower, fewer wrinkles form and will form sooner than in the distal narrower region, where more wrinkles develop. Similarly, less wrinkling is found on the ventral side, which is flatter, compared to the dorsal side. In summary, the mechanical compatibility between the skin and the muscle enables them to grow seamlessly, while the wrinkled skin acts as a protective barrier that is both thicker and more flexible than the unwrinkled skin.

physics.bio-ph↗

arXiv2024

Decomposing force fields as flows on graphs reconstructed from stochastic trajectories

Disentangling irreversible and reversible forces from random fluctuations is a challenging problem in the analysis of stochastic trajectories measured from real-world dynamical systems. We present an approach to approximate the dynamics of a stationary Langevin process as a discrete-state Markov process evolving over a graph-representation of phase-space, reconstructed from stochastic trajectories. Next, we utilise the analogy of the Helmholtz-Hodge decomposition of an edge-flow on a contractible simplicial complex with the associated decomposition of a stochastic process into its irreversible and reversible parts. This allows us to decompose our reconstructed flow and to differentiate between the irreversible currents and reversible gradient flows underlying the stochastic trajectories. We validate our approach on a range of solvable and nonlinear systems and apply it to derive insight into the dynamics of flickering red-blood cells and healthy and arrhythmic heartbeats. In particular, we capture the difference in irreversible circulating currents between healthy and passive cells and healthy and arrhythmic heartbeats. Our method breaks new ground at the interface of data-driven approaches to stochastic dynamics and graph signal processing, with the potential for further applications in the analysis of biological experiments and physiological recordings. Finally, it prompts future analysis of the convergence of the Helmholtz-Hodge decomposition in discrete and continuous spaces.

cond-mat.stat-mech↗