Willy Govaerts

MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs

MATCONT is a graphical MATLAB software package for the interactive numerical study of dynamical systems. It allows one to compute curves of equilibria, limit points, Hopf points, limit cycles, period doubling bifurcation points of limit cycles, and fold bifurcation points of limit cycles. All curves are computed by the same function that implements a prediction-correction continuation algorithm based on the Moore-Penrose matrix pseudo-inverse. The continuation of bifurcation points of equilibria and limit cycles is based on bordering methods and minimally extended systems. Hence no additional unknowns such as singular vectors and eigenvectors are used and no artificial sparsity in the systems is created. The sparsity of the discretized systems for the computation of limit cycles and their bifurcation points is exploited by using the standard Matlab sparse matrix methods. The MATLAB environment makes the standard MATLAB Ordinary Differential Equations (ODE) Suite interactively available and provides computational and visualization tools; it also eliminates the compilation stage and so makes installation straightforward. Compared to other packages such as AUTO and CONTENT, adding a new type of curves is easy in the MATLAB environment. We illustrate this by a detailed description of the limit point curve type.

Numerical methods for bifurcations of dynamical equilibria

Preface Notation Introduction 1. Examples and Motivation 2. Manifolds and Numerical Continuation 3. Bordered Matrices 4. Generic Equilibrium Bifurcations in One-Parameter Problems 5. Bifurcations Determined by the Jordan Form of the Jacobian 6. Singularity Theory 7. Singularity Theory with a Distinguished Bifurcation Parameter 8. Symmetry-Breaking Bifurcations 9. Bifurcations with Degeneracies in the Nonlinear Terms 10. An Introduction to Large Dynamical Systems Bibliography Index.

Atypical E2F activity restrains APC/CCCS52A2 function obligatory for endocycle onset

The endocycle represents an alternative cell cycle that is activated in various developmental processes, including placental formation, Drosophila oogenesis, and leaf development. In endocycling cells, mitotic cell cycle exit is followed by successive doublings of the DNA content, resulting in polyploidy. The timing of endocycle onset is crucial for correct development, because polyploidization is linked with cessation of cell division and initiation of terminal differentiation. The anaphase-promoting complex/cyclosome (APC/C) activator genes CDH1, FZR, and CCS52 are known to promote endocycle onset in human, Drosophila, and Medicago species cells, respectively; however, the genetic pathways governing development-dependent APC/CCDH1/FZR/CCS52 activity remain unknown. We report that the atypical E2F transcription factor E2Fe/DEL1 controls the expression of the CDH1/FZR orthologous CCS52A2 gene from Arabidopsis thaliana. E2Fe/DEL1 misregulation resulted in untimely CCS52A2 transcription, affecting the timing of endocycle onset. Correspondingly, ectopic CCS52A2 expression drove cells into the endocycle prematurely. Dynamic simulation illustrated that E2Fe/DEL1 accounted for the onset of the endocycle by regulating the temporal expression of CCS52A2 during the cell cycle in a development-dependent manner. Analogously, the atypical mammalian E2F7 protein was associated with the promoter of the APC/C-activating CDH1 gene, indicating that the transcriptional control of APC/C activator genes by atypical E2Fs might be evolutionarily conserved.

Recycling, clustering, and endocytosis jointly maintain PIN auxin carrier polarity at the plasma membrane.

Cell polarity reflected by asymmetric distribution of proteins at the plasma membrane is a fundamental feature of unicellular and multicellular organisms. It remains conceptually unclear how cell polarity is kept in cell wall‐encapsulated plant cells. We have used super‐resolution and semi‐quantitative live‐cell imaging in combination with pharmacological, genetic, and computational approaches to reveal insights into the mechanism of cell polarity maintenance in Arabidopsis thaliana. We show that polar‐competent PIN transporters for the phytohormone auxin are delivered to the center of polar domains by super‐polar recycling. Within the plasma membrane, PINs are recruited into non‐mobile membrane clusters and their lateral diffusion is dramatically reduced, which ensures longer polar retention. At the circumventing edges of the polar domain, spatially defined internalization of escaped cargos occurs by clathrin‐dependent endocytosis. Computer simulations confirm that the combination of these processes provides a robust mechanism for polarity maintenance in plant cells. Moreover, our study suggests that the regulation of lateral diffusion and spatially defined endocytosis, but not super‐polar exocytosis have primary importance for PIN polarity maintenance.

Chapter 4 – Numerical Continuation, and Computation of Normal Forms

This chapter describes numerical continuation methods for analyzing the solution behavior of the dynamical system. Time-integration of a dynamical system gives much insight into its solution behavior. However, once a solution type has been computed—for example, a stationary solution (equilibrium) or a periodic solution (cycle)—then continuation methods become very effective in determining the dependence of this solution on the parameter α. Once a co-dimension-1 bifurcation has been located, it can be followed in two parameters—that is, with α e ℝ 2 . However, in many cases, detection of higher co-dimension bifurcations requires computation of certain normal forms for equations restricted to center manifolds at the critical parameter values. Pseudo-arclength continuation method allows the continuation of any regular solution, including folds. Geometrically, it is the most natural continuation method. The periodic solution continuation method is very suitable for numerical computations, and it is not difficult to establish the Poincare continuation with the help of it.

Emergence of tissue polarization from synergy of intracellular and extracellular auxin signaling

Plant development is exceptionally flexible as manifested by its potential for organogenesis and regeneration, which are processes involving rearrangements of tissue polarities. Fundamental questions concern how individual cells can polarize in a coordinated manner to integrate into the multicellular context. In canalization models, the signaling molecule auxin acts as a polarizing cue, and feedback on the intercellular auxin flow is key for synchronized polarity rearrangements. We provide a novel mechanistic framework for canalization, based on up‐to‐date experimental data and minimal, biologically plausible assumptions. Our model combines the intracellular auxin signaling for expression of PINFORMED (PIN) auxin transporters and the theoretical postulation of extracellular auxin signaling for modulation of PIN subcellular dynamics. Computer simulations faithfully and robustly recapitulated the experimentally observed patterns of tissue polarity and asymmetric auxin distribution during formation and regeneration of vascular systems and during the competitive regulation of shoot branching by apical dominance. Additionally, our model generated new predictions that could be experimentally validated, highlighting a mechanistically conceivable explanation for the PIN polarization and canalization of the auxin flow in plants.

New features of the software MatCont for bifurcation analysis of dynamical systems

Bifurcation software is an essential tool in the study of dynamical systems. From the beginning (the first packages were written in the 1970s) it was also used in the modelling process, in particular to determine the values of critical parameters. More recently, it is used in a systematic way in the design of dynamical models and to determine which parameters are relevant. MatCont and Cl_MatCont are freely available matlab numerical continuation packages for the interactive study of dynamical systems and bifurcations. MatCont is the GUI-version, Cl_MatCont is the command-line version. The work started in 2000 and the first publications appeared in 2003. Since that time many new functionalities were added. Some of these are fairly simple but were never before implemented in continuation codes, e.g. Poincaré maps. Others were only available as toolboxes that can be used by experts, e.g. continuation of homoclinic orbits. Several others were never implemented at all, such as periodic normal forms for codimension 1 bifurcations of limit cycles, normal forms for codimension 2 bifurcations of equilibria, detection of codimension 2 bifurcations of limit cycles, automatic computation of phase response curves and their derivatives, continuation of branch points of equilibria and limit cycles. New numerical algorithms for these computations have been published or will appear elsewhere; here we restrict to their software implementation. We further discuss software issues that are in practice important for many users, e.g. how to define a new system starting from an existing one, how to import and export data, system descriptions, and computed results.

Stable solvers and block elimination for bordered systems

Linear systems with a fairly well conditioned matrix M of the form \[ \begin{gathered} \begin{pmatrix} A & b \\ c & d \end{pmatrix} \begin{matrix} n \\ 1 \end{matrix}, \\ \begin{matrix} n & 1 \end{matrix} \end{gathered} \] for which a “black-box” solver for A is available, are considered. To solve systems with M, a mixed block elimination algorithm, called BEM, is proposed. It has the following advantages: (1) It is easier to understand and to program than the widely accepted deflated block elimination (DBE) proposed by Chan, yet allows the same broad class of solvers and has comparable accuracy. (2) It requires one less solve with A. (3) It allows a rigorous error analysis that shows why it may fail in exceptional cases (all other black-box methods known to us also fail in these cases).BEM is also compared to iterative refinement of Crout block elimination (BEC) introduced by Pryce and Govaerts. BEC allows a more restricted class of solvers than BEM but is faster in cases where a solver is given not for ...

Computation of the Phase Response Curve: A Direct Numerical Approach

Neurons are often modeled by dynamical systems—parameterized systems of differential equations. A typical behavioral pattern of neurons is periodic spiking; this corresponds to the presence of stable limit cycles in the dynamical systems model. The phase resetting and phase response curves (PRCs) describe the reaction of the spiking neuron to an input pulse at each point of the cycle. We develop a new method for computing these curves as a by-product of the solution of the boundary value problem for the stable limit cycle. The method is mathematically equivalent to the adjoint method, but our implementation is computationally much faster and more robust than any existing method. In fact, it can compute PRCs even where the limit cycle can hardly be found by time integration, for example, because it is close to another stable limit cycle. In addition, we obtain the discretized phase response curve in a form that is ideally suited for most applications. We present several examples and provide the implementation in a freely available Matlab code.