updated on 27/Aug/14 [7]

HeKKSaGOn Summer School @ Göttingen

Inference on Pattern Formation:
Applications to Biology and Materials Science

September 11 -- September 20, 2014

Venue:    Sitzungszimmer, Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstraße 3-5, Göttingen

Abstracts of Lectures

Serial Lectures

Lectures A1--A3:    Yasumasa NISHIURA (Sendai)
Pattern formation and applications to materials science, I--III

Abstract: My talk consists of two main parts: one is an application of persistence homology to materials science and the other is on the dynamics of localized patterns in dissipative systems and its applications. If time allows, I will talk about more recent results on self-recovery dynamics for a class of network system.
  1. Topological approach in materials science
  Topological approach is a new non-invasive mathematical measurement potentially applicable to various fields including materials science. In this talk I will present two case studies in which computational homology and its variants play a key role to extract essential information on morphological dynamics and its ordering. One is the diblock copolymer problem in a three-dimensional space. It is known that the double gyroid and orthorhombic morphologies are obtained as energy minimizers. By investigating the geometric properties of these bicontinuous morphologies, we demonstrate the underlying mechanism affecting the triply periodic energy minimizers in terms of a balanced scaling law. Then we apply computational homology to characterize the topological changes during the morphology transition in three-dimensional space. It should be noted that even the “transient” morphologies can be detected during the time course of the transition, since integer-valued index (Betti numbers) can be observed for certain time before reaching the final state, for instance, transient perforated layers are found from layers to cylinders. The scaling law for Betti number in the phase ordering process is also presented. The other study is about a chracterization of glasses from topological point of view. The glass state is loosely speaking in between perfect crystal and liquid, namely it is not perfectly symmetric nor completely random. Using the method of persistent homology, we discuss about its characterization.
  2. Dynamics of localized patterns in heterogeneous media
  Localized waves are one of the main carriers of information and the effect of heterogeneity of the media in which it propagates is of great importance for the understanding of signaling processes in biological and chemical problems. A typical and simple heterogeneity is a spatially localized bump or dent in 1D or 2D, which in general creates associated defects in the media. One of the main issues is how the geometry of heterogeneity influences over the dynamics of waves. Here the geometry means slope, height, size, curvature and so on. Localized waves are sensitive to those factors and in fact present a variety of dynamics including rebound, pinning, splitting, and traveling motion around the defect. A reduction method to finite-dimensional system is presented, which clarifies the mathematical structure for those dynamics. In the reference below we mainly focus on a class of one-dimensional traveling pulses the associated parameters of which are close to drift and/or saddle-node bifurcations. The great advantage to study the dynamics in such a class is two-fold: firstly it gives us a perfect microcosm for the variety of outputs in general setting when pulses encounter heterogeneities. Secondly it allows us to reduce the original PDE dynamics to tractable finite dimensional system. Such pulses are sensitive when they run into the heterogeneities and show rich responses such as annihilation, pinning, splitting, rebound as well as penetration. The reduced ODEs explain all these dynamics and the underlying bifurcational structure controlling the transitions among different dynamic regimes. It turns out that there are hidden ordered patterns associated with the critical points of ODEs which play a pivotal role to understand the responses of the pulse. We mainly focus on a bump and periodic types of heterogeneity, however our approach is also applicable to general case. It should be noted that there appears spatio-temporal chaos for periodic type of heterogeneity when its period becomes comparable with the size of the pulse.
  3. Behaviors of amoeba (Physarum plasmodium) in heterogeneous environments
  We present a mathematical model that describes adaptive behaviors of an amoeboid organism, the Physarum plasmodium of true slime mold. The plasmodium migrating in a narrow lane stops moving for a period of time (several hours but the duration differs for each plasmodium) when it encounters the presence of a chemical repellent, quinine. After stopping period, the organism suddenly begins to move again in one of three different ways as the concentration of repellent increases:
going through the repulsive place (penetration), splitting into two fronts of going throught it and turning (splitting) and turning back (rebound). In relation to the physiological mechanism for tip migration in the plasmodium, we found that the frontal tip is capable of moving further although the tip is divided from a main body of organism. This means that a motive force of front locomotion is produced by a local process at the tip. Based on this finding, a mathematical model for front locomotion is considered in order to understand the dynamics for both the long period of stopping and three kinds of behavior. A model based on reaction-diffusion equations succeeds to reproduce the experimental observation. The origin of long-time stopping and three different outputs may be reduced to the hidden instabilities of internal dynamics of the pulse, which may be a skeleton structure extracted from much more complex dynamics imbedded in the Physarum plasmodium.

Lectures B1--B3:    Takashi SUZUKI (Osaka)
Space-time patterns in biological model: thermodynamics of multi-species, I--III

Abstract: I talk on the recent study on the space-time patterns in biological models concerning the interaction of multi-species. New methods in nonlinear analysis such as Lyapunov functions, relative entropy, gradient inequalities, monotonicity formula, scaling limits and so on are used to clarify the principle of spatial homoginization of isolated systems. Then the interaction of multi-spaccies is described in accordance with the directed graph.
  1. master equation, chemotaxis, cross diffusion
  2. Shigesada model - relative entropy
  3. interactions of multi-species
  4. Lotka-Volterra system - weak scaling limit

Lectures C1--C3:    Stephan HUCKEMANN (Göttingen)
Non-Euclidean Statistics with Applications in Biology, I--III

Abstract: This set of lectures is motivated by the question how to deal statistically with non-Euclidean data arising in many applications of biology, biometry, imaging and medicine. Here, due to restrictions, or when equivalence classes are considered, originally vector-valued data often become non-Euclidean.
   (a) Typical examples will be given for such data on circles, spheres, tori, real and complex projective spaces, orbit spaces (shape spaces), SPD matrices (diffusion tensors) and phylogenetic tree spaces.
For statistical analysis, we will address key challenges as the development of
   (b) appropriate data descriptors (generalized Frechet means),
   (c) their employment for dimension reduction methods inspired by PCA,
   (d) their asymptotics (consistency and central limit theorems) leading to
   (e) inferential non-Euclidean statistics.
The lectures will conclude with a discussion of
   (f) open issues (e.g. uniqueness of Frechet means) and
   (g) a few known results about the effects of topology and curvature on asymptotics (stickiness and smeariness).


Tutorials M1-M2:    Izumi TAKAGI and Sohei TASAKI (Sendai)
1. Structure-preserving discretization
2. Practical introduction to numerical methods for diffusion equations

Abstract: The objective of the first tutorial is to introduce a special type of discretization for solving systems of ordinary differential equations which have, for instance, a family of periodic solutions. A typical example is the prey-predator system by Lotka and Volterra, where all the solutions are periodic in time; however, the standard discretizations such as the forward Euler scheme, or the Runge-Kutta method, do not preserve this property. New discretization methods are needed.
The second tutorial is about the standard schemes for discretization of reaction-diffusion equations with emphasis on dos and don'ts in programming.

Tutorial W1:    Andrea KRAJINA (Göttingen)
Some aspects of extreme value theory


Tutorial W2:    Timo ASPELMEIERA (Göttingen)
Stochastic nano-scale imaging in biology

Abstract: This tutorial gives an introduction to current experimental methods for superresolution microscopy, showing that stochasticity plays a crucial role for achieving high resolution.


M1 (14 September, 13:30--14:20):    Tsuyoshi KATO (Kyoto)
Spectral coincidence of transition operators, automata groups and BBS in tropical geometry

Abstract: I will talk on our discovery on spectral coincidence of the Markov operators between the lamplighter group and BBS. The former appears in group theory and the latter is the automaton of KdV equation in soliton theory, passing through tropical geometry.
This is a joint work with S.Tsujimoto and A.Zuk.

W1 (17 September, 13:30--14:20):     Sohei TASAKI (Sendai)    Prediction of motile bacterial colonies

Abstract: Bacteria form a great variety of colonies. The spatio-temporal population structures have a major impact on many scientific fields, and so it becomes more desirable to predict the development of colonies; nevertheless the prediction still remains difficult because the growth kinetics and its dependence on the environmental conditions have not yet been fully understood. Here we show morphological changes of colonies grown on nutrient agar media due to pH and nutrient-richness (peptone concentration) variations for pre-cultured, motile cells of bacterial species Bacillus subtilis. Then we perform a reproduction analysis consisting of a series of culture and measurement experiments, modelling and simulations on the basis of theoretical considerations. The resulting model enables us to predict the development of colonies of motile bacteria for diverse environmental settings. Furthermore, our findings provide insights into bacterial survival strategies and the role of each bacterial activity.
This is a joint work with Madoka Nakayama (Sendai National College of Technology) and Wataru Shoji (Tohoku University).

Th1 (18 September, 13:30--14:20):     Ryokichi TANAKA (Sendai)     Phase transitions of random infinite sums

Abstract: I present some results about distributions of several random infinite sums which appear in random walks on groups as harmonic measures, for example.

Th2 (18 September, 14:30--15:20):     Anderea KRAJINA (Göttingen)     An application of extreme value theory in non-Euclidean statistics


F1 (19 September, 13:30--14:20):     Timo ASPELMEIER (Göttingen)     Statistical methods for superresolution fluorescence microscopy

Abstract: In this talk I show some mathematical methods needed to deal with the strongly non-Gaussian and discrete distributions appearing in fluorescence microscopy.

F2 (19 September, 16:00-16:50):     Izumi TAKAGI (Sendai)     Control of patterns by spatial heterogeneity

Abstract: Patterns are spatial (or temporal) heterogeneity with a certain extent of orderliness. What Turing tried to say was "even if the environment is perfectly uniform, pattern can emergy as a result of diffusion-driven instability". No biological pattern formation takes place in the perfect spatial homogeneity, rather it happens in a fairly non-uniform environment. Sometimes spatial heterogeneity is very important for a group of cells to develop a form. If the heterogeneity is not strong enough, it takes too long to get started with forming a pattern. We hence face a difficulty of tautological explanation "patterns emergy because there is already a pattern". In this talk we give a very simple two-stage model to avoid this "chicken or egg?" argument. The first stage is to enhance the built-in spatial heterogeneity, and the second stage is to form a pattern by resorting to a reaction-diffusion mechanism.

Short Communications

Benjamin EITZNER (Göttingen)     Reliable extraction of filament data from microscopic live cell images
Abstract: A reliable extraction of filament data from microscopic images, especially of living cells, is of great interest in the analysis of the acto-myosin fiber structures. These serve as an early morphological marker in mechanically guided differentiation of human mesenchymal stem cells and the understanding of the underlying fiber originating processes. In this communication we give a brief overview over imaging methods and propose the novel filament sensor (FS), a fast and robust processing sequence which detects and records location, orientation, length and width for each single filament of an image. We sketch some of various promising future statistical analyses that are enabled by this tool.

Hiroko YAMAMOTO (Sendai)     Location of concentration points in solutions of a reaction-diffusion equation with variable coefficients
Abstract: We are interested in a point-condensation phenomenon in solutions of a spatially heterogeneous reaction-diffusion equation. This means that the distribution of a solution concentrates in a very narrow region around a finitely many points. Hence, it is most important to find the candidates for the concentration points. In this talk we consider the singularly perturbed Neumann problem for a semilinear elliptic equation with variable coefficients, and reveal the effects of the heterogeneity on the location of concentration points.

Poster Presentations

Zhijun GAO (Sendai)     Dynamics of traveling spots with oscillatory tails for the generalized three-component FitzHugh-Nagumo equations
Abstract: The research is concerned with the dynamics of traveling spots with oscillatory tails arising in a three-component reaction diusion systems called the generalized three-component FitzHugh-Nagumo equations. As is known by Yves Couder's oil droplet experiments on a liquid surface, particles (droplets) associated with waves (surface waves caused by bouncing of particles) show a macroscopic quantum-like behaviors similar to Young's slit experiments and tunnel's phenomena. The main aim is to show numerically that such traveling spots also display a similar wave-particle duality, and moreover by using center manifold reduction theory, we are able to show that the dynamics of single spot dynamics can be reduced to a 4D system of ODEs. Via the analysis of such ODEs, we can show at least partially the origin of such quantum-like behaviors that is never observed for monotone-tail spots. This is a joint work with Prof. Yasumasa Nishiura.

Steffen HAERTING (Heidelberg)    A receptor-based model with coexistence of Diffusion-Driven Instability and hysteresis
Abstract: The aim of this poster is to contribute to the understanding of stability of steady states of systems of ordinary differential equations coupled to reaction-diffusion equations. Such systems arise naturally from the modeling of biological phenomena such as cell-receptor-ligand binding, so called receptor- based models. We present conditions for stability of irregular, discontinuous steady states of a system of one ordinary differential equation coupled to one reaction-diffusion equation. Moreover, we present a model exhibiting Diffusion-Driven Instability and stable discontinuous irregular steady states, i.e. de-novo formation of discontinuous, non-periodic stable steady states. The poster is based on joint research with I. Takagi (Tohoku University) and A. Marciniak-Czochra (University of Heidelberg).

Kurumi HIRUKO (Sendai)    A dynamical aspect of hybrid system describing intermittent androgen suppression therapy of prostate cancer
Abstract: Since a prostate tumor is influenced by androgen, continuous androgen suppression (CAS) therapy is the most famous therapy of prostate cancer in Japan. However the relapse of tumors often occurs in spite of under the CAS therapy. Recently, clinical studies suggested that intermittent androgen suppression (IAS) therapy may delay or prevent the relapse. In the IAS therapy, medication is stopped when the size of tumor decreases less than a lower threshold, and resumed when the size exceeds an upper threshold.
We deal with a hybrid system describing IAS therapy and prove mathematically that the size of tumor remains in some bounded interval for any time under the IAS therapy.

Dhisa MINERVA (Osaka)    Pathway network analysis of the MMP2 activation model in the early stage of cancer cell invasion
Abstract: Cancer is the leading causes of death in the world, particularly in developing countries. Metastasis is believed as a factor that would make the situation worse. Cancer cell has the ability to invade and metastasize to a distant part of body. Sato, et al [2] has revealed the mechanism of cancer cell invasion. The mechanism itself involved three molecules found attached on cancer cell membrane, MT1-MMP, and in the extracellular matrix cell, TIMP2 and MMP2, that the interaction between these molecules leads to the MMP2 activation. We believe the MMP2 activation is the initial process causes the cancer cell metastasis.
We study the MMP2 activation model based on mass conservation and action laws. We propose a method based on conservation law and attachment and detachment rules of basic molecules to analyze and solve the ordinary differential equation system of pathway network associated with the MMP2 activation. The idea is to simplify the original model without affecting its dynamic. The method leads us to derive the explicit solution of such system with some analysis about its equilibrium state. The method is expected to be extended into general case for any system of the pathway network.
References [1] Hoshino, D., et al. 2012. Establishment and validation of computational model for MT1-MMP dependent ECM degradation and intervention strategies. PLoS Computational Bio, vol8:e1002479.
[2] Sato, H., et al. 1994. A matrix metalloproteinase expressed on the surface of invasive tumor cells. Nature, 370: 61-65.

Max SOMMERFELD (Göttingen)    Mathematical analysis and modelling of stress fiber skeletons
Abstract: The seminal study of Engler et al. (2006) demonstrated that the mechanical properties (Young痴 modulus E) of the micro-environment alone can direct stem cell differentiation. Here we are concerned with the statistical analysis of these effects based on flourescence imaging of the acto-myosin filament cell skeleton structures of human mesenchymal stem cells under various matrix elasticities. In order to reliably extract filament structures we have designed the Filament Sensor. For an analysis of cumulative histograms (total filament length over orientations) we have generalized the SiZer of Chaudhuri and Marron (1999, 2000) for the detection of shape parameters of densities on the real line to the case of circular data giving an exhaustive circular scale space theory not available previously. Future challenges of this research lie in including more parameters (beyond cumulative histograms) of filament processes, analyses of time series of filament processes and in a 3D analysis of the filament structures.

Jan-Eric STECHER (Heidelberg)    Mass concentration in a nonlocal model of clonal selection
Abstract: Self-renewal is a constitutive property of stem cells. Testing the cancer stem cell hypothesis requires investigation of the impact of self-renewal on cancer expansion. To understand better this impact, we propose a mathematical model describing the dynamics of a continuum of cell clones structured by the self-renewal potential. This model consists of a system of integro-differential equations, which is coupled by nonlinear, nonlocal term. The integral term acts as a feedback loop in cell differentiation and proliferation process. The solution of this model exhibits a mass concentration, which is located in the maximum of the self-renewal potential. In a second step we incorporate the idea that stem cells have the ability to mutate, which is modeled via a diffusive term in the first equation. We give an existence and uniqueness result for this system of partial integro-differential equations, as well as numerical simulations to illustrate the long-term behaviour of the solutions.

Fabian TELSCHOW (Göttingen)    Removing unwanted experimental side-effects in gait analysis for discrimination
Abstract: Recording paths from the motion of the human knee joint in practice involves marker placements on volunteers' thighs and shanks. Although done by experts, realistic markers have varying loci resulting in varying coordinate frames for the femur and tibia, respectively. In order to discriminate gait of volunteers more effectively, we remove these unwanted experimental side-effect. This allows, moreover, to detect possible changes in gait patterns after biomechanical interventions.

Tomoyuki TERADA (Sendai)    Characterization of the nonautonomous system and autonomous system using the topological invariants
Abstract: In this poster, we will introduce the classical characterization of nonautonomous system and autonomous system using the topologial invariants for linear differential operator on one dimensional spaces. In addition, we will present several topics via dynamical system.

Hoang Duy THAI (Göttingen)    Fingerprint image segmentation via total variation
Abstract: Segmentation is an important step separating foreground from background in images because the wrong segmentation will have impacted on the performance of the recognition system. For fingerprints, foreground regions contain structure-texture information. Based on ideas from Aujol et al. [1, 3], we have developed a technique for fingerprint segmentation via total variation to decompose the original image into the parts of cartoon, texture, and noise; then, the region of interest can be extracted by texture information via image processing steps.

Haruki UMAKOSHI (Osaka)    Global solution to quadratic systems of reaction-diffusion
Abstract: We study the global-in-time behavior of quadratic reaction-diffusion systems. The nonlinearity of this systems satisfies ”entropy inequality” which plays an important role to prove the existence of global weak solution. Furthermore, we can prove the existence of the global solution provided that the space dimension is less than or equal to 2.

Kazuyoshi WATANABE (Kyoto)    Continuous closed 1-form and the related numbers
Abstract: We can define a continuous closed 1-form on topological space. Similariy as smooth closed 1-form, this 1-form represent a singular cohomology class,and any cohomology class may be realized by a continuous 1-form. From any cohomology class, the novikov betti number and the category number with respect to the class are defined, and category number is related to a homoclinic cycle and a number of zero-point. I show the examples of any relation of their numbers.

Time Table

Monday, September 15, 2014

10:00--10:50   Lecture A1: Y. Nishiura Pattern formation and applications to materials science, I
11:05--11:55   Lecture A2: Y. Nishiura Pattern formation and applications to materials science, II
12:00--13:30   lunch
13:30--14:20   Lecture M1 (T. Kato) Spectral coincidence of transition operators, automata groups and BBS in tropical geometry
14:20--14:50   coffee
14:50--15:40   Tutorial M1 (I. Takagi, S. Tasaki) Structure-preserving discretization
15:50--16:40   Tutorial M2 (I. Takagi, S. Tasaki) Practical introduction to numerical methods for diffusion equations
18:00--20:00   Welcome Party

Tuesday, September 16, 2014

10:00--10:50   Lecture A3: Y. Nishiura Pattern formation and applications to materials science, III
11:00--11:50   Special Lecture (TBA)
12:00--13:30   lunch
13:30--14:20   Short communications by poster presenters
14:30--15:20   Short communications by poster presenters
15:20--15:50   coffee
15:50--17:00   Poster Session

Wednesday, September 17, 2014

10:00--10:50   Lecture B1: T. Suzuki Space-time patterns in biological model: thermodynamics of multi-species, I
11:00--11:50   Lecture B2: T. Suzuki Space-time patterns in biological model: thermodynamics of multi-species, II
12:00--13:30   lunch
13:30--14:20   Lecture W1: S. Tasaki Prediction of motile bacterial colonies
14:20--14:50   coffee
14:50--15:40   Tutorial W1 (A. Krajina) Some aspects of extreme value theory
15:50--16:40   Tutorial W2 (T. Aspelmeier) Stochastic nano-scale imaging in biology
16:45--17:15   Round Table

Thursday, September 18, 2014

10:00--10:50   Lecture B3: T. Suzuki Space-time patterns in biological model: thermodynamics of multi-species, III
11:00--11:50   Lecture C1: S. Huckemann Non-Euclidean Statistics with Applications in Biology, I
12:00--13:30   lunch
13:30--14:20   Lecture Th1: R. Tanaka Phase transitions of random infinite sums
14:30--15:20   Lecture Th2: A. Krajina An application of extreme value theory in non-Euclidean statistics
15:20--15:50   coffee
16:00--17:30   Göttingen observatory tour

Friday, September 19, 2014

10:00--10:50   Lecture C2: S. Huckemann Non-Euclidean Statistics with Applications in Biology, II
11:00--11:50   Lecture C3: S. Huckemann Non-Euclidean Statistics with Applications in Biology, III
12:00--13:30   lunch
13:30--14:50   Lecture F1: T. Aspelmeier Statistical methods for superresolution fluorescence microscopy
14:40--15:40   Short communications
B. Eitzner   Reliable extraction of filament data from microscopic live cell images
H. Yamamoto   Location of concentration points in solutions of a reaction-diffusion equation with variable coefficients
15:40--16:00   coffee
16:00--16:50   Lecture F2: I. Takagi Control of patterns by spatial heterogeneity
18:00--20:00   Farewell Party

created on 23/jul/14