Hamiltonian Computational Chemistry: Geometrical Structures in Chemical Dynamics and Kinetics

The common geometrical (symplectic) structures of classical mechanics, quantum mechanics, and classical thermodynamics are unveiled with three pictures. These cardinal theories, mainly at the non-relativistic approximation, are the cornerstones for studying chemical dynamics and chemical kinetics. Working in extended phase spaces, we show that the physical states of integrable dynamical systems are depicted by Lagrangian submanifolds embedded in phase space. Observable quantities are calculated by properly transforming the extended phase space onto a reduced space, and trajectories are integrated by solving Hamilton's equations of motion. After defining a Riemannian metric, we can also estimate the length between two states. Local constants of motion are investigated by integrating Jacobi fields and solving the variational linear equations. Diagonalizing the symplectic fundamental matrix, eigenvalues equal to one reveal the number of constants of motion. For conservative systems, geometrical quantum mechanics has proved that solving the Schrödinger equation in extended Hilbert space, which incorporates the quantum phase, is equivalent to solving Hamilton's equations in the projective Hilbert space. In classical thermodynamics, we take entropy and energy as canonical variables to construct the extended phase space and to represent the Lagrangian submanifold. Hamilton's and variational equations are written and solved in the same fashion as in classical mechanics. Solvers based on high-order finite differences for numerically solving Hamilton's, variational, and Schrödinger equations are described. Employing the Hénon-Heiles two-dimensional nonlinear model, representative results for time-dependent, quantum, and dissipative macroscopic systems are shown to illustrate concepts and methods. High-order finite-difference algorithms, despite their accuracy in low-dimensional systems, require substantial computer resources when they are applied to systems with many degrees of freedom, such as polyatomic molecules. We discuss recent research progress in employing Hamiltonian neural networks for solving Hamilton's equations. It turns out that Hamiltonian geometry, shared with all physical theories, yields the necessary and sufficient conditions for the mutual assistance of humans and machines in deep-learning processes.

Extracting the gradient component of the gamma index using the Lie derivative method

Objective. The gamma index (γ) has been extensively investigated in the medical physics and applied in clinical practice. However,γhas a significant limitation when used to evaluate the dose-gradient region, leading to inconveniences, particularly in stereotactic radiotherapy (SRT). This study proposes a novel evaluation method combined withγto extract clinically problematic dose-gradient regions caused by irradiation including certain errors.Approach. A flow-vector field in the dose distribution is obtained when the dose is considered a scalar potential. Using the Lie derivative from differential geometry, we definedL,S, andUto evaluate the intensity, vorticity, and flow amount of deviation between two dose distributions, respectively. These metrics multiplied byγ(γL,γS,γU), along with the threshold valueσ, were verified in the ideal SRT case and in a clinical case of irradiation near the brainstem region using radiochromic films. Moreover, Moran's gradient index (MGI), Bakai's χ factor, and the structural similarity index (SSIM) were investigated for comparisons.Main results. A highL-metric value mainly extracted high-dose-gradient induced deviations, which was supported by highSandUmetrics observed as a robust deviation and an influence of the dose-gradient, respectively. TheS-metric also denotes the measured similarity between the compared dose distributions. In theγdistribution,γLsensitively detected the dose-gradient region in the film measurement, despite the presence of noise. The thresholdσsuccessfully extracted the gradient-error region whereγ> 1 analysis underestimated, andσ= 0.1 (plan) andσ= 0.001 (film measurement) were obtained according to the compared resolutions. However, the MGI, χ, and SSIM failed to detect the clinically interested region.Significance. Although further studies are required to clarify the error details, this study demonstrated that the Lie derivative method provided a novel perspective for the identifying gradient-induced error regions and enabled enhanced and clinically significant evaluations ofγ.

A novel geometry-based analysis of hippocampal morphometry in mesial temporal lobe epilepsy

Hippocampal volumetry is an essential tool in researching and diagnosing mesial temporal lobe epilepsy (mTLE). However, it has a limited ability to detect subtle alterations in hippocampal morphometry. Here, we establish and apply a novel geometry-based tool that enables point-wise morphometric analysis based on an intrinsic coordinate system of the hippocampus. We hypothesized that this point-wise analysis uncovers structural alterations not measurable by volumetry, but associated with histological underpinnings and the neuropsychological profile of mTLE. We conducted a retrospective study in 204 individuals with mTLE and 57 age- and gender-matched healthy subjects. FreeSurfer-based segmentations of hippocampal subfields in 3T-MRI were subjected to a geometry-based analysis that resulted in a coordinate system of the hippocampal mid-surface and allowed for point-wise measurements of hippocampal thickness and other features. Using point-wise analysis, we found significantly lower thickness and higher FLAIR signal intensity in the entire affected hippocampus of individuals with hippocampal sclerosis (HS-mTLE). In the contralateral hippocampus of HS-mTLE and the affected hippocampus of MRI-negative mTLE, we observed significantly lower thickness in the presubiculum. Impaired verbal memory was associated with lower thickness in the left presubiculum. In HS-mTLE histological subtype 3, we observed higher curvature than in subtypes 1 and 2 (all p < .05). These findings could not be observed using conventional volumetry (Bonferroni-corrected p < .05). We show that point-wise measures of hippocampal morphometry can uncover structural alterations not measurable by volumetry while also reflecting histological underpinnings and verbal memory. This substantiates the prospect of their clinical application.

A Differential-Geometric Approach to Quantum Ignorance Consistent with Entropic Properties of Statistical Mechanics

In this paper, we construct the metric tensor and volume for the manifold of purifications associated with an arbitrary reduced density operator ρS. We also define a quantum coarse-graining (CG) to study the volume where macrostates are the manifolds of purifications, which we call surfaces of ignorance (SOI), and microstates are the purifications of ρS. In this context, the volume functions as a multiplicity of the macrostates that quantifies the amount of information missing from ρS. Using examples where the SOI are generated using representations of SU(2), SO(3), and SO(N), we show two features of the CG: (1) A system beginning in an atypical macrostate of smaller volume evolves to macrostates of greater volume until it reaches the equilibrium macrostate in a process in which the system and environment become strictly more entangled, and (2) the equilibrium macrostate takes up the vast majority of the coarse-grained space especially as the dimension of the total system becomes large. Here, the equilibrium macrostate corresponds to a maximum entanglement between the system and the environment. To demonstrate feature (1) for the examples considered, we show that the volume behaves like the von Neumann entropy in that it is zero for pure states, maximal for maximally mixed states, and is a concave function with respect to the purity of ρS. These two features are essential to typicality arguments regarding thermalization and Boltzmann's original CG.

Effective optimization strategy for large optimization volume object, remaining volume at risk (RVR):α-value selection and usage from generalized equivalent uniform dose (gEUD) curve deviation perspective

Objective.A large optimization volume for intensity-modulated radiation therapy (IMRT), such as the remaining volume at risk (RVR), is traditionally unsuitable for dose-volume constraint control and requires planner-specific empirical considerations owing to the patient-specific shape. To enable less empirical optimization, the generalized equivalent uniform dose (gEUD) optimization is effective; however, the utilization of parametera-values remains elusive. Our study clarifies thea-value characteristics for optimization and to enable effectivea-value use.Approach.The gEUD can be obtained as a function of itsa-value, which is the weighted generalized mean; its curve has a continuous, differentiable, and sigmoid shape, deforming in its optimization state with retained curve characteristics. Using differential geometry, the gEUD curve changes in optimization is considered a geodesic deviation intervened by the forces between deforming and retaining the curve. The curvature and gradient of the curve are radically related to optimization. The vertex point (a=a_k) was set and thea-value roles were classified into the following three parts of the curve with respect to thea-value: (i) high gradient and middle curvature, (ii) middle gradient and high curvature, and (iii) low gradient and low curvature. Then, a strategy for multiplea-values was then identified using RVR optimization.Main results.Eleven head and neck patients who underwent static seven-field IMRT were used to verify thea-value characteristics and curvature effect for optimization. The lowera-value (i) (a= 1-3) optimization was effective for the whole dose-volume range; in contrast, the effect of highera-value (iii) (a= 12-20) optimization addressed strongly the high-dose range of the dose volume. The middlea-value (ii) (arounda=a_k) showed intermediate but effective high-to-low dose reduction. Thesea-value characteristics were observed as superimpositions in the optimization. Thus, multiple gEUD-based optimization was significantly superior to the exponential constraints normally applied to the RVR that surrounds the PTV, normal tissue objective (NTO), resulting in up to 25.9% and 8.1% improvement in dose-volume indices D2% and V10Gy, respectively.Significance.This study revealed an appropriatea-value for gEUD optimization, leading to favorable dose-volume optimization for the RVR region using fixed multiplea-value conditions, despite the very large and patient-specific shape of the region.

Information geometry for multiparameter models: new perspectives on the origin of simplicity

Complex models in physics, biology, economics, and engineering are oftensloppy, meaning that the model parameters are not well determined by the model predictions for collective behavior. Many parameter combinations can vary over decades without significant changes in the predictions. This review uses information geometry to explore sloppiness and its deep relation to emergent theories. We introduce themodel manifoldof predictions, whose coordinates are the model parameters. Itshyperribbonstructure explains why only a few parameter combinations matter for the behavior. We review recent rigorous results that connect the hierarchy of hyperribbon widths to approximation theory, and to the smoothness of model predictions under changes of the control variables. We discuss recent geodesic methods to find simpler models on nearby boundaries of the model manifold-emergent theories with fewer parameters that explain the behavior equally well. We discuss a Bayesian prior which optimizes the mutual information between model parameters and experimental data, naturally favoring points on the emergent boundary theories and thus simpler models. We introduce a 'projected maximum likelihood' prior that efficiently approximates this optimal prior, and contrast both to the poor behavior of the traditional Jeffreys prior. We discuss the way the renormalization group coarse-graining in statistical mechanics introduces a flow of the model manifold, and connect stiff and sloppy directions along the model manifold with relevant and irrelevant eigendirections of the renormalization group. Finally, we discuss recently developed 'intensive' embedding methods, allowing one to visualize the predictions of arbitrary probabilistic models as low-dimensional projections of an isometric embedding, and illustrate our method by generating the model manifold of the Ising model.

A Probabilistic-Geometric Approach for UAV Detection and Avoidance Systems

This paper proposes a collision avoidance algorithm for the detection and avoidance capabilities of Unmanned Aerial Vehicles (UAVs). The proposed algorithm aims to ensure minimum separation between UAVs and geofencing with multiple no-fly zones, considering the sensor uncertainties. The main idea is to compute the collision probability and to initiate collision avoidance manoeuvres determined by the differential geometry concept. The proposed algorithm is validated by both theoretical and numerical analysis. The results indicate that the proposed algorithm ensures minimum separation, efficiency, and scalability compared with other benchmark algorithms.

Transversality Conditions for Geodesics on the Statistical Manifold of Multivariate Gaussian Distributions

We consider the problem of finding the closest multivariate Gaussian distribution on a constraint surface of all Gaussian distributions to a given distribution. Previous research regarding geodesics on the multivariate Gaussian manifold has focused on finding closed-form, shortest-path distances between two fixed distributions on the manifold, often restricting the parameters to obtain the desired solution. We demonstrate how to employ the techniques of the calculus of variations with a variable endpoint to search for the closest distribution from a family of distributions generated via a constraint set on the parameter manifold. Furthermore, we examine the intermediate distributions along the learned geodesics which provide insight into uncertainty evolution along the paths. Empirical results elucidate our formulations, with visual illustrations concretely exhibiting dynamics of 1D and 2D Gaussian distributions.

A Hierarchy of Probability, Fluid and Generalized Densities for the Eulerian Velocivolumetric Description of Fluid Flow, for New Families of Conservation Laws

The Reynolds transport theorem occupies a central place in continuum mechanics, providing a generalized integral conservation equation for the transport of any conserved quantity within a fluid or material volume, which can be connected to its corresponding differential equation. Recently, a more generalized framework was presented for this theorem, enabling parametric transformations between positions on a manifold or in any generalized coordinate space, exploiting the underlying continuous multivariate (Lie) symmetries of a vector or tensor field associated with a conserved quantity. We explore the implications of this framework for fluid flow systems, based on an Eulerian velocivolumetric (position-velocity) description of fluid flow. The analysis invokes a hierarchy of five probability density functions, which by convolution are used to define five fluid densities and generalized densities relevant to this description. We derive 11 formulations of the generalized Reynolds transport theorem for different choices of the coordinate space, parameter space and density, only the first of which is commonly known. These are used to generate a table of integral and differential conservation laws applicable to each formulation, for eight important conserved quantities (fluid mass, species mass, linear momentum, angular momentum, energy, charge, entropy and probability). The findings substantially expand the set of conservation laws for the analysis of fluid flow and dynamical systems.

Characterization of surface motion patterns in highly deformable soft tissue organs from dynamic MRI: An application to assess 4D bladder motion

BACKGROUND AND OBJECTIVES

Dynamic Magnetic Resonance Imaging (MRI) may capture temporal anatomical changes in soft tissue organs with high-contrast but the obtained sequences usually suffer from limited volume coverage which makes the high-resolution reconstruction of organ shape trajectories a major challenge in temporal studies. Because of the variability of abdominal organ shapes across time and subjects, the objective of the present study is to go towards 3D dense velocity measurements to fully cover the entire surface and to extract meaningful features characterizing the observed organ deformations and enabling clinical action or decision.

METHODS

We present a pipeline for characterization of bladder surface dynamics during deep respiratory movements. For a compact shape representation, the reconstructed temporal volumes were first used to establish subject-specific dynamical 4D mesh sequences using the large deformation diffeomorphic metric mapping (LDDMM) framework. Then, we performed a statistical characterization of organ dynamics from mechanical parameters such as mesh elongations and distortions. Since we refer to organs as non-flat surfaces, we have also used the mean curvature change as metric to quantify surface evolution. However, the numerical computation of curvature is strongly dependant on the surface parameterization (i.e. the mesh resolution). To cope with this dependency, we employed a non-parametric method for surface deformation analysis. Independent of parameterization and minimizing the length of the geodesic curves, it stretches smoothly the surface curves towards a sphere by minimizing a Dirichlet energy. An Eulerian PDE approach is used to derive a shape descriptor from the curve-shortening flow. Intercorrelations between individuals' motion patterns are computed using the Laplace-Beltrami Operator (LBO) eigenfunctions for spherical mapping.

RESULTS

Application to extracting characterization correlation curves for locally-controlled simulated shape trajectories demonstrates the stability of the proposed shape descriptor. Its usability was shown on MRI acquired for seven healthy participants for which the bladder was highly deformed by maximum of inspiration. As expected, the study showed that deformations occured essentially on the top lateral regions.

CONCLUSION

Promising results were obtained, showing the organ in its 3D complexity during deformation due to strain conditions. Smooth genus-0 manifold reconstruction from sparse dynamic MRI data is employed to perform a statistical shape analysis for the determination of bladder deformation.

Multi-Frequency Image Completion via a Biologically-Inspired Sub-Riemannian Model with Frequency and Phase

We present a novel cortically-inspired image completion algorithm. It uses five-dimensional sub-Riemannian cortical geometry, modeling the orientation, spatial frequency and phase-selective behavior of the cells in the visual cortex. The algorithm extracts the orientation, frequency and phase information existing in a given two-dimensional corrupted input image via a Gabor transform and represents those values in terms of cortical cell output responses in the model geometry. Then, it performs completion via a diffusion concentrated in a neighborhood along the neural connections within the model geometry. The diffusion models the activity propagation integrating orientation, frequency and phase features along the neural connections. Finally, the algorithm transforms the diffused and completed output responses back to the two-dimensional image plane.

Energy contour exploration with potentiostatic kinematics

We introduce a method of exploring potential energy contours (PECs) in complex dynamical systems based on potentiostatic kinematics wherein the systems are evolved with minimal changes to their potential energy. We construct a simple iterative algorithm for performing potentiostatic kinematics, which uses an estimate curvature to predict new configuration-space coordinates on the PEC and a potentiostat term component to correct for errors in prediction. Our methods are then applied to atomic structure models using an interatomic potential for energy and force evaluations as would commonly be invoked in a molecular dynamics simulation. Using several model systems, we assess the stability and accuracy of the method on different hyperparameters in the implementation of the potentiostatic kinematics. Our implementation is open source and available within the atomic simulation environment package.

Computing the Riemannian curvature of image patch and single-cell RNA sequencing data manifolds using extrinsic differential geometry

Most high-dimensional datasets are thought to be inherently low-dimensional-that is, data points are constrained to lie on a low-dimensional manifold embedded in a high-dimensional ambient space. Here, we study the viability of two approaches from differential geometry to estimate the Riemannian curvature of these low-dimensional manifolds. The intrinsic approach relates curvature to the Laplace-Beltrami operator using the heat-trace expansion and is agnostic to how a manifold is embedded in a high-dimensional space. The extrinsic approach relates the ambient coordinates of a manifold's embedding to its curvature using the Second Fundamental Form and the Gauss-Codazzi equation. We found that the intrinsic approach fails to accurately estimate the curvature of even a two-dimensional constant-curvature manifold, whereas the extrinsic approach was able to handle more complex toy models, even when confounded by practical constraints like small sample sizes and measurement noise. To test the applicability of the extrinsic approach to real-world data, we computed the curvature of a well-studied manifold of image patches and recapitulated its topological classification as a Klein bottle. Lastly, we applied the extrinsic approach to study single-cell transcriptomic sequencing (scRNAseq) datasets of blood, gastrulation, and brain cells to quantify the Riemannian curvature of scRNAseq manifolds.

An Elementary Introduction to Information Geometry

In this survey, we describe the fundamental differential-geometric structures of information manifolds, state the fundamental theorem of information geometry, and illustrate some use cases of these information manifolds in information sciences. The exposition is self-contained by concisely introducing the necessary concepts of differential geometry. Proofs are omitted for brevity.

A new method for protein characterization and classification using geometrical features for 3D face analysis: An example of tubulin structures

This article reports on the results of research aimed to translate biometric 3D face recognition concepts and algorithms into the field of protein biophysics in order to precisely and rapidly classify morphological features of protein surfaces. Both human faces and protein surfaces are free-forms and some descriptors used in differential geometry can be used to describe them applying the principles of feature extraction developed for computer vision and pattern recognition. The first part of this study focused on building the protein dataset using a simulation tool and performing feature extraction using novel geometrical descriptors. The second part tested the method on two examples, first involved a classification of tubulin isotypes and the second compared tubulin with the FtsZ protein, which is its bacterial analog. An additional test involved several unrelated proteins. Different classification methodologies have been used: a classic approach with a support vector machine (SVM) classifier and an unsupervised learning with a k-means approach. The best result was obtained with SVM and the radial basis function kernel. The results are significant and competitive with the state-of-the-art protein classification methods. This leads to a new methodological direction in protein structure analysis.

Geometrical Aspects in the Analysis of Microcanonical Phase-Transitions

In the present work, we discuss how the functional form of thermodynamic observables can be deduced from the geometric properties of subsets of phase space. The geometric quantities taken into account are mainly extrinsic curvatures of the energy level sets of the Hamiltonian of a system under investigation. In particular, it turns out that peculiar behaviours of thermodynamic observables at a phase transition point are rooted in more fundamental changes of the geometry of the energy level sets in phase space. More specifically, we discuss how microcanonical and geometrical descriptions of phase-transitions are shaped in the special case of ϕ 4 models with either nearest-neighbours and mean-field interactions.

A review of mathematical representations of biomolecular data

Recently, machine learning (ML) has established itself in various worldwide benchmarking competitions in computational biology, including Critical Assessment of Structure Prediction (CASP) and Drug Design Data Resource (D3R) Grand Challenges. However, the intricate structural complexity and high ML dimensionality of biomolecular datasets obstruct the efficient application of ML algorithms in the field. In addition to data and algorithm, an efficient ML machinery for biomolecular predictions must include structural representation as an indispensable component. Mathematical representations that simplify the biomolecular structural complexity and reduce ML dimensionality have emerged as a prime winner in D3R Grand Challenges. This review is devoted to the recent advances in developing low-dimensional and scalable mathematical representations of biomolecules in our laboratory. We discuss three classes of mathematical approaches, including algebraic topology, differential geometry, and graph theory. We elucidate how the physical and biological challenges have guided the evolution and development of these mathematical apparatuses for massive and diverse biomolecular data. We focus the performance analysis on protein-ligand binding predictions in this review although these methods have had tremendous success in many other applications, such as protein classification, virtual screening, and the predictions of solubility, solvation free energies, toxicity, partition coefficients, protein folding stability changes upon mutation, etc.

A novel speed model for safety evaluation of freeway alignment in Euclidean 3D space

Objective: Operating speed is a critical indicator to evaluate consistency of road alignment and safety. Although extensive studies have been conducted on developing operating speed models, few researchers have considered the interactive influence of horizontal and vertical alignment in 3D space. The purpose of this study is to develop a speed model based on 3D alignment in Euclidean space rather than traditional horizontal and vertical alignment. Methods: According to the curve theory of differential geometry, a novel method to estimate operating speed is proposed in our study using 3D space curvature instead of traditional horizontal or vertical parameters to describe the spatial geometric properties for a freeway alignment. Speeds of 54 different alignment segments are observed to develop the speed model. Several observing sites of each segment are selected beforehand, and the speeds of more than 300 vehicles in each site are observed. Space curvature is used as an important index to estimate operating speed. Results: The findings of this study indicated that both horizontal alignment and vertical alignment contribute to space curvature. Space curvature mainly affects direction control operating performance. However, vehicles overcome the effects of gravity along the vertical alignment in the z direction. Results indicate that operating speed exponentially declines with space curvature and that quadratic parabola decline with vertical grade. Conclusions: It can be concluded that there is a clear correlation between velocity and spatial curvature, which is proved by variance analysis. The estimation results of the speed models are reliable as tested using a real engineering example. The study would provide a scientific basis for safety evaluation of freeway alignment.

Canonical Divergence for Measuring Classical and Quantum Complexity

A new canonical divergence is put forward for generalizing an information-geometric measure of complexity for both classical and quantum systems. On the simplex of probability measures, it is proved that the new divergence coincides with the Kullback-Leibler divergence, which is used to quantify how much a probability measure deviates from the non-interacting states that are modeled by exponential families of probabilities. On the space of positive density operators, we prove that the same divergence reduces to the quantum relative entropy, which quantifies many-party correlations of a quantum state from a Gibbs family.

A superposition free method for protein conformational ensemble analyses and local clustering based on a differential geometry representation of backbone

Here a differential geometry (DG) representation of protein backbone is explored on the analyses of protein conformational ensembles. The protein backbone is described by curvature, κ, and torsion, τ, values per residue and we propose 1) a new dissimilarity and protein flexibility measurement and 2) a local conformational clustering method. The methods were applied to Ubiquitin and c-Myb-KIX protein conformational ensembles and results show that κ\τ metric space allows to properly judge protein flexibility by avoiding the superposition problem. The d_max measurement presents equally good or superior results when compared to RMSF, especially for the intrinsically unstructured protein. The clustering method is unique as it relates protein global to local dynamics by providing a global clustering solutions per residue. The methods proposed can be especially useful to the analyses of highly flexible proteins. The software written for the analyses presented here is available at https://github.com/AMarinhoSN/FleXgeo for academic usage only.

A Geometrically Exact Model for Soft Continuum Robots: The Finite Element Deformation Space Formulation

Mathematical modeling of soft robots is complicated by the description of the continuously deformable three-dimensional shape that they assume when subjected to external loads. In this article we present the deformation space formulation for soft robots dynamics, developed using a finite element approach. Starting from the Cosserat rod theory formulated on a Lie group, we derive a discrete model using a helicoidal shape function for the spatial discretization and a geometric scheme for the time integration of the robot shape configuration. The main motivation behind this work is the derivation of accurate and computational efficient models for soft robots. The model takes into account bending, torsion, shear, and axial deformations due to general external loading conditions. It is validated through analytic and experimental benchmark. The results demonstrate that the model matches experimental positions with errors <1% of the robot length. The computer implementation of the model results in SimSOFT, a dynamic simulation environment for design, analysis, and control of soft robots.

Representation and Characterization of Nonstationary Processes by Dilation Operators and Induced Shape Space Manifolds

We proposed in this work the introduction of a new vision of stochastic processes through geometry induced by dilation. The dilation matrices of a given process are obtained by a composition of rotation matrices built in with respect to partial correlation coefficients. Particularly interesting is the fact that the obtention of dilation matrices is regardless of the stationarity of the underlying process. When the process is stationary, only one dilation matrix is obtained and it corresponds therefore to Naimark dilation. When the process is nonstationary, a set of dilation matrices is obtained. They correspond to Kolmogorov decomposition. In this work, the nonstationary class of periodically correlated processes was of interest. The underlying periodicity of correlation coefficients is then transmitted to the set of dilation matrices. Because this set lives on the Lie group of rotation matrices, we can see them as points of a closed curve on the Lie group. Geometrical aspects can then be investigated through the shape of the obtained curves, and to give a complete insight into the space of curves, a metric and the derived geodesic equations are provided. The general results are adapted to the more specific case where the base manifold is the Lie group of rotation matrices, and because the metric in the space of curve naturally extends to the space of shapes; this enables a comparison between curves’ shapes and allows then the classification of random processes’ measures.

Perfect Detection of Spikes in the Linear Sub-threshold Dynamics of Point Neurons

Spiking neuronal networks are usually simulated with one of three main schemes: the classical time-driven and event-driven schemes, and the more recent hybrid scheme. All three schemes evolve the state of a neuron through a series of checkpoints: equally spaced in the first scheme and determined neuron-wise by spike events in the latter two. The time-driven and the hybrid scheme determine whether the membrane potential of a neuron crosses a threshold at the end of the time interval between consecutive checkpoints. Threshold crossing can, however, occur within the interval even if this test is negative. Spikes can therefore be missed. The present work offers an alternative geometric point of view on neuronal dynamics, and derives, implements, and benchmarks a method for perfect retrospective spike detection. This method can be applied to neuron models with affine or linear subthreshold dynamics. The idea behind the method is to propagate the threshold with a time-inverted dynamics, testing whether the threshold crosses the neuron state to be evolved, rather than vice versa. Algebraically this translates into a set of inequalities necessary and sufficient for threshold crossing. This test is slower than the imperfect one, but can be optimized in several ways. Comparison confirms earlier results that the imperfect tests rarely miss spikes (less than a fraction 1/10⁸ of missed spikes) in biologically relevant settings.

Free energy of the edge of an open lipid bilayer based on the interactions of its constituent molecules

Lipid-bilayers are the fundamental constituents of the walls of most living cells and lipid vesicles, giving them shape and compartment. The formation and growing of pores in a lipid bilayer have attracted considerable attention from an energetic point of view in recent years. Such pores permit targeted delivery of drugs and genes to the cell, and regulate the concentration of various molecules within the cell. The formation of such pores is caused by various reasons such as changes in cell environment, mechanical stress or thermal fluctuations. Understanding the energy and elastic behaviour of a lipid-bilayer edge is crucial for controlling the formation and growth of such pores. In the present work, the interactions in the molecular level are used to obtain the free energy of the edge of an open lipid bilayer. The resulted free-energy density includes terms associated with flexural and torsional energies of the edge, in addition to a line-tension contribution. The line tension, elastic moduli, and spontaneous normal and geodesic curvatures of the edge are obtained as functions of molecular distribution, molecular dimensions, cutoff distance, and the interaction strength. These parameters are further analyzed by implementing a soft-core interaction potential in the microphysical model. The dependence of the elastic free-energy of the edge to the size of the pore is reinvestigated through an illustrative example, and the results are found to be in agreement with the previous observations.

Quantitative phenotyping of leaf margins in three dimensions, demonstrated on KNOTTED and TCP trangenics in Arabidopsis

The geometry of leaf margins is an important shape characteristic that distinguishes among different leaf phenotypes. Current definitions of leaf shape are qualitative and do not allow quantification of differences in shape between phenotypes. This is especially true for leaves with some non-trivial three-dimensional (3D) configurations. Here we present a novel geometrical method novel geometrical methods to define, measure, and quantify waviness and lobiness of leaves. The method is based on obtaining the curve of the leaf rim from a 3D surface measurement and decomposing its local curvature vector into the normal and geodesic components. We suggest that leaf waviness is associated with oscillating normal curvature along the margins, while lobiness is associated with oscillating geodesic curvature. We provide a way to integrate these local measures into global waviness and lobiness quantities. Using these novel definitions, we analysed the changes in leaf shape of two Arabidopsis genotypes, either as a function of gene mis-expression induction level or as a function of time. These definitions and experimental methods open the way for a more quantitative study of the shape of leaves and other growing slender organs.

Exact Relation between Potential of Mean Force and Free-Energy Profile

We apply concepts of covariant and contravariant vector space in differential geometry and general relativity to derive new, general, exact relations between potential of mean force and free-energy profile. These relations are immensely practical in free-energy simulations because a full Jacobian transformation (which is usually unknown) is not required; rather, only knowledge of the (constraint) coordinate of interest is needed. We reveal that in addition to the Jacobian determinant, the Jacobian scale factor and Leibnizian contributions must also be considered, as well a Fixman term with correct mass dependence. Our newly derived relations are verified with new non-trivial benchmark numerical examples for which exact results can be computed, and compared with relations available in the literature that turn out to exhibit significant deviations from the exact values.

MANIFOLD LEARNING FOR ANALYSIS OF LOW-ORDER NONLINEAR DYNAMICS IN HIGH-DIMENSIONAL ELECTROCARDIOGRAPHIC SIGNALS

The dynamical structure of electrical recordings from the heart or torso surface is a valuable source of information about cardiac physiological behavior. In this paper, we use an existing data-driven technique for manifold identification to reveal electrophysiologically significant changes in the underlying dynamical structure of these signals. Our results suggest that this analysis tool characterizes and differentiates important parameters of cardiac bioelectric activity through their dynamic behavior, suggesting the potential to serve as an effective dynamic constraint in the context of inverse solutions.

Representation Theoretic Patterns in Three-Dimensional Cryo-Electron Microscopy II-The Class Averaging Problem

In this paper we study the formal algebraic structure underlying the intrinsic classification algorithm, recently introduced in Singer et al. (SIAM J. Imaging Sci. 2011, accepted), for classifying noisy projection images of similar viewing directions in three-dimensional cryo-electron microscopy (cryo-EM). This preliminary classification is of fundamental importance in determining the three-dimensional structure of macromolecules from cryo-EM images. Inspecting this algebraic structure we obtain a conceptual explanation for the admissibility (correctness) of the algorithm and a proof of its numerical stability. The proof relies on studying the spectral properties of an integral operator of geometric origin on the two-dimensional sphere, called the localized parallel transport operator. Along the way, we continue to develop the representation theoretic set-up for three-dimensional cryo-EM that was initiated in Hadani and Singer (Ann. Math. 2010, accepted).

A differential geometric approach to automated segmentation of human airway tree

Airway diseases are frequently associated with morphological changes that may affect the physiology of the lungs. Accurate characterization of airways may be useful for quantitatively assessing prognosis and for monitoring therapeutic efficacy. The information gained may also provide insight into the underlying mechanisms of various lung diseases. We developed a computerized scheme to automatically segment the 3-D human airway tree depicted on computed tomography (CT) images. The method takes advantage of both principal curvatures and principal directions in differentiating airways from other tissues in geometric space. A "puzzle game" procedure is used to identify false negative regions and reduce false positive regions that do not meet the shape analysis criteria. The negative impact of partial volume effects on small airway detection is partially alleviated by repeating the developed differential geometric analysis on lung anatomical structures modeled at multiple iso-values (thresholds). In addition to having advantages, such as full automation, easy implementation and relative insensitivity to image noise and/or artifacts, this scheme has virtually no leakage issues and can be easily extended to the extraction or the segmentation of other tubular type structures (e.g., vascular tree). The performance of this scheme was assessed quantitatively using 75 chest CT examinations acquired on 45 subjects with different slice thicknesses and using 20 publicly available test cases that were originally designed for evaluating the performance of different airway tree segmentation algorithms.

A Geometric Approach to Joint 2D Region-Based Segmentation and 3D Pose Estimation Using a 3D Shape Prior

In this work, we present an approach to jointly segment a rigid object in a two-dimensional (2D) image and estimate its three-dimensional (3D) pose, using the knowledge of a 3D model. We naturally couple the two processes together into a shape optimization problem and minimize a unique energy functional through a variational approach. Our methodology differs from the standard monocular 3D pose estimation algorithms since it does not rely on local image features. Instead, we use global image statistics to drive the pose estimation process. This confers a satisfying level of robustness to noise and initialization for our algorithm and bypasses the need to establish correspondences between image and object features. Moreover, our methodology possesses the typical qualities of region-based active contour techniques with shape priors, such as robustness to occlusions or missing information, without the need to evolve an infinite dimensional curve. Another novelty of the proposed contribution is to use a unique 3D model surface of the object, instead of learning a large collection of 2D shapes to accommodate the diverse aspects that a 3D object can take when imaged by a camera. Experimental results on both synthetic and real images are provided, which highlight the robust performance of the technique in challenging tracking and segmentation applications.