A numerical simulation of neural fields on curved geometries

Despite the highly convoluted nature of the human brain, neural field models typically treat the cortex as a planar two-dimensional sheet of ne;urons. Here, we present an approach for solving neural field equations on surfaces more akin to the cortical geometries typically obtained from neuroimaging data. Our approach involves solving the integral form of the partial integro-differential equation directly using collocation techniques alongside efficient numerical procedures for determining geodesic distances between neural units. To illustrate our methods, we study localised activity patterns in a two-dimensional neural field equation posed on a periodic square domain, the curved surface of a torus, and the cortical surface of a rat brain, the latter of which is constructed using neuroimaging data. Our results are twofold: Firstly, we find that collocation techniques are able to replicate solutions obtained using more standard Fourier based methods on a flat, periodic domain, independent of the underlying mesh. This result is particularly significant given the highly irregular nature of the type of meshes derived from modern neuroimaging data. And secondly, by deploying efficient numerical schemes to compute geodesics, our approach is not only capable of modelling macroscopic pattern formation on realistic cortical geometries, but can also be extended to include cortical architectures of more physiological relevance. Importantly, such an approach provides a means by which to investigate the influence of cortical geometry upon the nucleation and propagation of spatially localised neural activity and beyond. It thus promises to provide model-based insights into disorders like epilepsy, or spreading depression, as well as healthy cognitive processes like working memory or attention.     

Muscle wrapping on arbitrary meshes with the heat method

Muscle paths play an important role in musculoskeletal simulations by determining a muscle’s length and how its force is distributed to joints. Most previous approaches estimate the way in which muscles ‘wrap’ around bones and other structures with smooth analytical wrapping surfaces. In this paper, we employ Newton’s method with discrete differential geometry to permit muscle wrapping over arbitrary polygonal mesh surfaces that represent underlying bones and structures. Precomputing distance fields allows us to speed up computations for the common situation where many paths cross the same wrapping surfaces. We found positive results for the accuracy, robustness, and efficiency of the method. However the method did not exhibit continuous changes in path length for dynamic simulations. Nonetheless this approach provides a valuable step toward fast muscle wrapping on arbitrary meshes.     

Effects of Hydraulic Soil Properties on Vegetation Pattern Formation in Sloping Landscapes

Current models of vegetation pattern formation rely on a system of weakly nonlinear reaction–diffusion equations that are coupled by their source terms. While these equations, which are used to describe a spatiotemporal planar evolution of biomass and soil water, qualitatively capture the emergence of various types of vegetation patterns in arid environments, they are phenomenological and have a limited predictive power. We ameliorate these limitations by deriving the vertically averaged Richards’ equation to describe flow (as opposed to “diffusion”) of water in partially saturated soils. This establishes conditions under which this nonlinear equation reduces to its weakly nonlinear reaction–diffusion counterpart used in the previous models, thus relating their unphysical parameters (e.g., diffusion coefficient) to the measurable soil properties (e.g., hydraulic conductivity) used to parameterize the Richards equation. Our model is valid for both flat and sloping landscapes and can handle arbitrary topography and boundary conditions. The result is a model that relates the environmental conditions (e.g., precipitation rate, runoff and soil properties) to formation of multiple patterns observed in nature (such as stripes, labyrinth and spots).     

Pattern formation in morphogenesis

We first treat the Gierer-Meinhardt equations by linear stability analysis to determine the critical parameter, at which the homogeneous distributions of activator and inhibitor concentrations become unstable. We find two types of instabilities: one leading to spatial pattern formation and another one leading to temporal oscillations. We consider the case where two instabilities are present. Using the method of generalized Ginzburg-Landau equations introduced earlier we then analyze the nonlinear equations. As we are mainly interested in spatial pattern formation on a sphere we consider the problem under an appropriate constraint. Combining the two occurring solutions we find patterns well-known in biology, such as a gradient system and temporal oscillations.     

Labyrinthine versus straight-striped patterns generated by two-dimensional Turing systems

Striped patterns are often observed on fish skin. Such patterns have been accounted for by reaction-diffusion (RD) Turing-type models, in which two substances can spontaneously form a spatially heterogeneous pattern in a homogeneous field. Among the striped patterns generated by Turing-type models, some are "straight-striped patterns," with many stripes running in parallel, while others are "labyrinthine patterns," in which the stripes often change direction, merge with each other, and frequently branch out. RD models differ in terms of their tendency to generate either labyrinthine or straight-striped patterns. Here, we studied the conditions under which either a labyrinthine or straight-striped pattern would emerge. First, we defined an index for stripe clearness, Sh. Straight-striped patterns (large Sh) are formed if only a narrow range of spatial periods corresponds to an unstable mode. Labyrinthine patterns (small Sh) are formed when a wide range of spatial periods is unstable. More specifically, labyrinthine patterns are formed when the maximum spatial period of unstable modes is more than twice that of the minimum spatial period of unstable modes; otherwise, straight-striped patterns are formed. We then examined RD models with nonlinear reaction terms, including both activator-inhibitor and substrate-depletion models, and we demonstrated that the same conclusions hold with respect to the conditions required for labyrinthine versus straight-striped patterns.     

Interactions Between Pattern Formation and Domain Growth

In this paper we develop a theoretical framework for investigating pattern formation in biological systems for which the tissue on which the spatial pattern resides is growing at a rate which is itself regulated by the diffusible chemicals that establish the spatial pattern. We present numerical simulations for two cases of interest, namely exponential domain growth and chemically controlled growth. Our analysis reveals that for domains undergoing rapid exponential growth dilution effects associated with domain growth influence both the spatial patterns that emerge and the concentration of chemicals present in the domain. In the latter case, there is complex interplay between the effects of the chemicals on the domain size and the influence of the domain size on the formation of patterns. The nature of these interactions is revealed by a weakly nonlinear analysis of the full system. This yields a pair of nonlinear equations for the amplitude of the spatial pattern and the domain size. The domain is found to grow (or shrink) at a rate that depends quadratically on the pattern amplitude, the particular functional forms used to model the local tissue growth rate and the kinetics of the two diffusible species dictating the resulting behaviour.     

An Integrative Approach for Modeling and Simulation of Heterocyst Pattern Formation in Cyanobacteria Filaments

Heterocyst differentiation in cyanobacteria filaments is one of the simplest examples of cellular differentiation and pattern formation in multicellular organisms. Despite of the many experimental studies addressing the evolution and sustainment of heterocyst patterns and the knowledge of the genetic circuit underlying the behavior of single cyanobacterium under nitrogen deprivation, there is still a theoretical gap connecting these two macroscopic and microscopic processes. As an attempt to shed light on this issue, here we explore heterocyst differentiation under the paradigm of systems biology. This framework allows us to formulate the essential dynamical ingredients of the genetic circuit of a single cyanobacterium into a set of differential equations describing the time evolution of the concentrations of the relevant molecular products. As a result, we are able to study the behavior of a single cyanobacterium under different external conditions, emulating nitrogen deprivation, and simulate the dynamics of cyanobacteria filaments by coupling their respective genetic circuits via molecular diffusion. These two ingredients allow us to understand the principles by which heterocyst patterns can be generated and sustained. In particular, our results point out that, by including both diffusion and noisy external conditions in the computational model, it is possible to reproduce the main features of the formation and sustainment of heterocyst patterns in cyanobacteria filaments as observed experimentally. Finally, we discuss the validity and possible improvements of the model.     

A new multicompartmental reaction-diffusion modeling method links transient membrane attachment of <Emphasis Type="Italic">E. coli</Emphasis> MinE to E-ring formation

Many important cellular processes are regulated by reaction-diffusion (RD) of molecules that takes place both in the cytoplasm and on the membrane. To model and analyze such multicompartmental processes, we developed a lattice-based Monte Carlo method, Spatiocyte that supports RD in volume and surface compartments at single molecule resolution. Stochasticity in RD and the excluded volume effect brought by intracellular molecular crowding, both of which can significantly affect RD and thus, cellular processes, are also supported. We verified the method by comparing simulation results of diffusion, irreversible and reversible reactions with the predicted analytical and best available numerical solutions. Moreover, to directly compare the localization patterns of molecules in fluorescence microscopy images with simulation, we devised a visualization method that mimics the microphotography process by showing the trajectory of simulated molecules averaged according to the camera exposure time. In the rod-shaped bacterium Escherichia coli, the division site is suppressed at the cell poles by periodic pole-to-pole oscillations of the Min proteins (MinC, MinD and MinE) arising from carefully orchestrated RD in both cytoplasm and membrane compartments. Using Spatiocyte we could model and reproduce the in vivo MinDE localization dynamics by accounting for the previously reported properties of MinE. Our results suggest that the MinE ring, which is essential in preventing polar septation, is largely composed of MinE that is transiently attached to the membrane independently after recruited by MinD. Overall, Spatiocyte allows simulation and visualization of complex spatial and reaction-diffusion mediated cellular processes in volumes and surfaces. As we showed, it can potentially provide mechanistic insights otherwise difficult to obtain experimentally.     

Spatio-temporal patterns in a mechanical model for mesenchymal morphogenesis

We present an in-depth study of spatio-temporal patterns in a simplified version of a mechanical model for pattern formation in mesenchymal morphogenesis. We briefly motivate the derivation of the model and show how to choose realistic boundary conditions to make the system well-posed. We firstly consider one-dimensional patterns and carry out a nonlinear perturbation analysis for the case where the uniform steady state is linearly unstable to a single mode. In two-dimensions, we show that if the displacement field in the model is represented as a sum of orthogonal parts, then the model can be decomposed into two sub-models, only one of which is capable of generating pattern. We thus focus on this particular sub-model. We present a nonlinear analysis of spatio-temporal patterns exhibited by the sub-model on a square domain and discuss mode interaction. Our analysis shows that when a two-dimensional mode number admits two or more degenerate mode pairs, the solution of the full nonlinear system of partial differential equations is a mixed mode solution in which all the degenerate mode pairs are represented in a frequency locked oscillation.     

Cell patterning using magnetite nanoparticles and magnetic force

Technologies for fabricating functional tissue architectures by patterning cells precisely are highly desirable for tissue engineering. Although several cell patterning methods such as microcontact printing and lithography have been developed, these methods require specialized surfaces to be used as substrates, the fabrication of which is time consuming. In the present study, we demonstrated a simple and rapid cell patterning technique, using magnetite nanoparticles and magnetic force, which enables us to allocate cells on arbitrary surfaces. Magnetite cationic liposomes (MCLs) developed in our previous study were used to magnetically label the target cells. When steel plates placed on a magnet were positioned under a cell culture surface, the magnetically labeled cells lined on the surface where the steel plate was positioned. Patterned lines of single cells were achieved by adjusting the number of cells seeded, and complex cell patterns (curved, parallel, or crossing patterns) were successfully fabricated. Since cell patterning using magnetic force may not limit the property of culture surfaces, human umbilical vein endothelial cells (HUVECs) were patterned on Matrigel, thereby forming patterned capillaries. These results suggest that the novel cell patterning methodology, which uses MCLs, is a promising approach for tissue engineering and studying cell-cell interactions in vitro.     

Patient-specific computational haemodynamics: generation of structured and conformal hexahedral meshes from triangulated surfaces of vascular bifurcations

Measuring the blood flow is still limited by current imaging technologies and is generally overcome using computational fluid dynamics (CFD) which, because of the complex geometry of blood vessels, has widely relied on tetrahedral meshes. Hexahedral meshes offer more accurate results with lower-density meshes and faster computation as compared to tetrahedral meshes, but their use is limited by the far more complex mesh generation. We present a robust methodology for conformal and structured hexahedral mesh generation - applicable to complex arterial geometries as bifurcating vessels - starting from triangulated surfaces. Cutting planes are used to slice the lumen surface and to construct longitudinal Bezier splines. Afterwards, an isoparametric transformation is used to map a parametrically defined quadrilateral surface mesh into the vessel volume, resulting in stacks of sections which can then be used for sweeping. Being robust and open source based, this methodology may improve the current standard in patient-specific mesh generation and enhance the reliability of CFD to patient-specific haemodynamics.     

Patient-specific computational haemodynamics: generation of structured and conformal hexahedral meshes from triangulated surfaces of vascular bifurcations

Measuring the blood flow is still limited by current imaging technologies and is generally overcome using computational fluid dynamics (CFD) which, because of the complex geometry of blood vessels, has widely relied on tetrahedral meshes. Hexahedral meshes offer more accurate results with lower-density meshes and faster computation as compared to tetrahedral meshes, but their use is limited by the far more complex mesh generation. We present a robust methodology for conformal and structured hexahedral mesh generation – applicable to complex arterial geometries as bifurcating vessels – starting from triangulated surfaces. Cutting planes are used to slice the lumen surface and to construct longitudinal Bezier splines. Afterwards, an isoparametric transformation is used to map a parametrically defined quadrilateral surface mesh into the vessel volume, resulting in stacks of sections which can then be used for sweeping. Being robust and open source based, this methodology may improve the current standard in patient-specific mesh generation and enhance the reliability of CFD to patient-specific haemodynamics.     

Templating membrane assembly, structure, and dynamics using engineered interfaces

The physical and chemical properties of biological membranes are intimately linked to their bounding aqueous interfaces. Supported phospholipid bilayers, obtained by surface-assisted rupture, fusion, and spreading of vesicular microphases, offer a unique opportunity, because engineering the substrate allows manipulation of one of the two bilayer interfaces as well. Here, we review a collection of recent efforts, which illustrates deliberate substrate-membrane coupling using structured surfaces exhibiting chemical and topographic patterns. Vesicle fusion on chemically patterned substrates results in co-existing lipid phases, which reflect the underlying pattern of surface energy and wettability. These co-existing bilayer/monolayer morphologies are useful both for fundamental biophysical studies (e.g., studies of membrane asymmetry) as well as for applied work, such as synthesizing large-scale arrays of bilayers or living cells. The use of patterned, static surfaces provides new models to design complex membrane topographies and curvatures. Dynamic switchable-topography surfaces and sacrificial trehalose based-substrates reveal abilities to dynamically introduce membrane curvature and change the nature of the membrane-substrate interface. Taken together, these studies illustrate the importance of controlling interfaces in devising model membrane platforms for fundamental biophysical studies and bioanalytical devices.     

Zebrafish leopard gene as a component of the putative reaction-diffusion system

It has been suggested, on a theoretical basis, that a reaction-diffusion (RD) mechanism underlies pigment pattern formation in animals, but as yet, there is no molecular evidence for the putative mechanism. Mutations in the zebrafish gene, leopard, change the pattern from stripes to spots. Interestingly each allele gives a characteristic pattern, which varies in spot size, density and connectivity. That mutations in a single gene can generate such a variety of patterns can be understood using a RD model. All the pattern variations of leopard mutants can be generated in a simulation by changing a parameter value that corresponds to the reaction kinetics in a putative RD system. Substituting an intermediate value of the parameter makes the patterns similar to the heterozygous fish. These results suggest that the leopard gene product is a component of the putative RD mechanism.     

Subject specific finite element mesh generation of the pelvis from biplanar x-ray images: application to 120 clinical cases

Several Finite Element (FE) models of the pelvis have been developed to comprehensively assess the onset of pathologies and for clinical and industrial applications. However, because of the difficulties associated with the creation of subject-specific FE mesh from CT scan and MR images, most of the existing models rely on the data of one given individual. Moreover, although several fast and robust methods have been developed for automatically generating tetrahedral meshes of arbitrary geometries, hexahedral meshes are still preferred today because of their distinct advantages but their generation remains an open challenge. Recently, approaches have been proposed for fast 3D reconstruction of bones based on X-ray imaging. In this study, we adapted such an approach for the fast and automatic generation of all-hexahedral subject-specific FE models of the pelvis based on the elastic registration of a generic mesh to the subject-specific target in conjunction with element regularity and quality correction. The technique was successfully tested on a database of 120 3D reconstructions of pelvises from biplanar X-ray images. For each patient, a full hexahedral subject-specific FE mesh was generated with an accurate surface representation.     

The interplay between phenotypic and ontogenetic plasticities can be assessed using reaction-diffusion models

Every morphological, behavioral, or even developmental character expression of living beings is coded in its genotype and is expressed in its phenotype. Nevertheless, the interplay between phenotypic and ontogenetic plasticities, that is, the capability to manifest trait variations, is a current field of research that needs morphometric, numerical, or even mathematical modeling investigations. In the present work, we are searching for a phenotypic index able to identify the underlying correlation among phenotypic, ontogenetic, and geographic distribution of the evolutionary development of species of the same genus. By studying the case of Pseudoplatystoma fishes, we use their skin patterns as an auxiliary trait that can be reproduced by means of a reaction diffusion (RD) model. From this model, we infer the phenotypic index in terms of one of the parameters appearing in the mathematical equations. To achieve this objective, we perform extensive numerical simulations and analysis of the model equations and link the parameter variations with different environmental and physicochemical conditions in which the individuals develop, and which may be regulated by the ontogenetic plasticity of the species. Our numerical study indicates that the patterns predicted by a set of reaction diffusion equations are not uniquely determined by the value of the parameters of the equation, but also depend on how the process is initiated and on the spatial distribution of values of these parameters. These factors are therefore significant, since they show that an individual’s growth dynamics and apparent secondary transport processes, like advection, can be determinant for the alignment of motifs in a skin pattern. Our results allow us to discern the correlation between phenotypic, ontogenetic, and geographic distribution of the different species of Pseudoplatystoma fishes, thus indicating that RD models represent a useful taxonomic tool able to quantify evolutionary indexes.     

Analytical treatment of pattern formation in the Gierer-Meinhardt model of morphogenesis

Summary We first perform a linear stability analysis of the Gierer-Meinhardt model to determine the critical parameters where the homogeneous distribution of activator and inhibitor concentrations becomes unstable. There are two kinds of instabilities, namely, one leading to spatial patterns and another one leading to temporal oscillations. Focussing our attention on spatial pattern formation we solve the corresponding nonlinear equations by means of our previously introduced method of generalized Ginzburg-Landau equations. We explicitly consider the two-dimensional case and find both rolls and hexagon-like structures. The impact of different boundary conditions on the resulting patterns is also discussed. The occurrence of the new patterns has all the features of nonequilibrium phase transitions.     

Surface topology assisted alignment of Min protein waves

Self-organization of proteins into large-scale structures is of pivotal importance for the organization of cells. The Min protein system of the bacterium Escherichia coli is a prime example of how pattern formation occurs via reaction–diffusion. We have previously demonstrated how Min protein patterns are influenced by compartment geometry. Here we probe the influence of membrane surface topology, as an additional regulatory element. Using microstructured membrane-clad soft polymer substrates, Min protein patterns can be aligned. We demonstrate that Min pattern alignment starts early during pattern formation and show that macroscopic millimeter-sized areas of protein patterns of well-defined orientation can be generated.