Research



(Hover over to see inter-species
collaboration in the Martens lab ... )


Scientific Interests

My main research interest lies in the study of complex nonlinear (dynamical) systems. I apply mathematical methods from nonlinear dynamics, bifurcation theory, numerical methods and simulations. To study complex behavior, I investigate problems in mathematical biology (evolutionary and ecological models, cancer development, coupled oscillators) and (bio-)physical systems (coupled oscillator systems and networks, fluid dynamics and pattern formation). I am also working on applying concepts from complex systems to applications of physiological/biomedical interest, such as the behavior of vascular networks.


Overview


Network dynamics in physiological systems

The vascular system as a complex adaptive network


Synchronization patterns in complex networks

Chimera states in oscillator networks with nonlocal coupling

The synchronization of coupled oscillators is a striking manifestation of self-organization that nature employs to orchestrate essential processes of life, such as the beating of the heart. While it was long thought that synchrony or disorder were mutually exclusive steady states for a network of identical oscillators, numerous theoretical studies over the last 10 years revealed the intriguing possibility of `chimera states', in which the symmetry of the oscillator population is broken into a synchronous and an asynchronous part. Particularly, numerous analytical studies, involving different network topologies, and various sources of random perturbations establish chimeras as a robust theoretical concept and suggest that they exist in complex systems in nature. Yet, a striking lack of empirical evidence raises the question of whether chimeras are indeed characteristic to natural systems. This calls for a palpable realization of chimera states without any fine-tuning, from which physical mechanisms underlying their emergence can be uncovered. Here, we devise a simple experiment with mechanical oscillators coupled in a hierarchical network to show that chimeras emerge naturally from a competition between two antagonistic synchronization patterns. We identify a wide spectrum of complex states, encompassing and extending the set of previously described chimeras. Our mathematical model shows that the self-organization observed in our experiments is controlled by elementary dynamical equations from mechanics that are ubiquitous in many natural and technological systems. The symmetry breaking mechanism revealed by our experiments may thus be prevalent in systems exhibiting collective behaviour, such as power grids, opto-mechanical crystals or cells communicating via quorum sensing in microbial populations.




Strong spring coupling: In-phase synchronization on two swings.



Intermediate spring coupling: Chimera state on two swings.



Weak spring coupling: Anti-phase synchronization on two swings.

Collaboration with S. Thutupalli, Princeton University; D. Abrams, Northwestern University; O. Hallatschek, UCL Berkeley; A. Fourrière, MPI Dynamics & Self-Organization, Göttingen, Carlo Laing, Massey University

Publications



Applying and controlling chimera states in technological and biological settings

Coupled phase oscillators model a variety of dynamical phenomena in nature and technological applications. Chimera state are characterized by localized phase synchrony while the remaining oscillators move incoherently. Here, we apply the idea of control to chimera states: through a new dynamic control scheme that exploits drift, a chimera will attain and maintain any desired target position. Our control approach extends beyond chimera states as it may also be used to optimize more general objective functions. If we are able to control the position of a chimera state, the localized nature of chimeras becomes accessible for novel applications: for instance, local synchrony could encode information.

With C. Bick, Rice University, USA; D. Abrams, Northwestern University; Mark Panaggio, Hillsdale College, USA; A. Torcin, Istituto dei Sistemi Complessi (CNR), Florence.

Publications



Population dynanmics in ecology: Seasonal forcing of trophic chains.

I aim at understanding how seasonality and its varying strength along different latitudes affects marine ecosystem structure: how does seasonal dynamics affect predator-prey interactions in the ecosystem? How do seasonal variations influence overwintering and reproductive strategies of marine species? To answer these questions, I study mathematicalmodels of the trophic chain in marine systems. Rather than employing the concept of species or functional groups, I use a trait-based approach, considering individuals with mechanistically based traits that are described by few parameters. By disposing of the species concept, the trait-based approach arrives at a succinct description with few basic parameters, and sidesteps the complexity trap of species-centric modeling approaches. Further my research advances trait-based methods by extending current models to include seasonal forcing and multiple trophic levels, and in particular to consider all the links ranging from plankton to fish. To obtain an integral understanding of the dynamics of marine ecosystems, it is crucial to include all trophic levels. The trait parameters along the trophic chain follow a power law which gives rise to interesting slow-fast dynamics on multiple length scales.

Collaboration with Ken H. Andersen and Thomas Kiørboe, VKR Centre for Ocean Life.

Publications

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Size, sense and allometry of marine life and animals.

Why don’t bacteria have eyes? Why do fish not echolocate? How far can they see? The answer to these questions is closely linked to an age old obsession of ours - size! Our analysis shows that size matters very much for how marine organisms sense their environment. We analyzed the underwater physics of various sensory systems - smelling, mechanosensing, vision, hearing, and echolocation - to find the size limits where these senses can and cannot function. Our theoretical predictions match surprisingly well with observations of minimum and maximum size limits of these senses.

Collaboration with the Centre for Ocean Life, Technical University of Denmark.

Publications



Early warning signs of critical transitions in high dimensional systems

A large variety of complex systems in ecology, climate science, biomedicine, engineering, and financial markets have been observed to exhibit tipping points, where the dynamical state of a system abruptly changes. For instance, such critical transitions may result in the sudden change of ecological environments and climate conditions. Data and models suggest that detectable warning signs may precede some of these drastic events. While direct detection of critical points is difficult, work in a wide class of systems suggests the existence of generic early-warning signals (e.g., increasing variance or auto-correlation in time-series data) that indicate an approaching critical point (Scheffer et al., 2009). This view is also corroborated by mathematical theory for generic bifurcations in stochastic multi-scale systems, exhibiting scaling laws that may be used as warning signs. In a first study, we have affirmatively answered the question: ``Can stochastic warning signs known from other areas be detected in large-scale social media data?'' I am interested in identifying warning signs of tipping points in complex biological systems, in particular in settings of biomedical interest: for instance, warning signs have been found for the on-/offset of epileptic seizures in electroencephalographic time series.

Collaboration with C. Kuehn (TU Munich) and D. Romero (University of Michigan).

Publications



Population Dynamics in Evolution of Species in Spatially Extended Habitats

How fast do species adapt and improve their fitness in new environments? This is one of the most fundamental and perhaps less well understood aspects of evolution and population genetics. Many theoretical models are restricted to the case of well-mixed (non-spatial) populations. I currently work on models of spatially extended populations, and derive experimentally testable laws for the adaptation rate, applicable to simple organisms such as microbes. My research also includes biologically relevant mechanisms such as recombination and long-range dispersal. Thereby I use methods from theoretical physics, applied mathematics and computer models. My recent findings on power laws describing the speed of evolution in spatially extended populations are summarized here:

Interfering Waves of Adaptation Promote Spatial Mixing, Genetics, 189:1045 (2011).

Once a mutation occurs somewhere in the habitat (shown as stars in the figure below), its fate is not yet decided. Mutations are subject to stochastic number fluctuations (genetic drift), and may get extinct; their probability of establishment is a function of their selective advantage. When mutations survive number fluctuations and get established, they may spread throughout the entire population (fixation). Each time a mutation is fixated, the fitness of the entire population is increased by the selective fitness effect of the mutation. The fitness is indicated by changing colors in the figure.

Two important paradigms are observed in evolutionary models of asexual species: The scenario where mutations spread strictly consequently is called periodic selection (A). In this case, the time to fixation (tfix) is always shorter than the waiting time until the next mutations occurs (tmut). In the converse case, when a second mutation occurs before the first one has reached fixation, multiple clones may compete to reach fixation (B). This scenario is known as clonal interference, because clonal waves interfere with one another. Because of this interference, actually beneficial mutations may lose the competition, and as a consequence, evolution is slowed down. In other words, the speed of adaptation is decreased, in comparison to the case where all mutations could be absorbed.

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In well-mixed (non-structured) populations, mutations are subject to logistic growth; in contrast, beneficial mutations spread in spatially structured populations via waves. This concept of wave-like spread of mutations goes back to a seminal study conducted by R. A. Fisher in 1937, "The wave of advance of advantageous genes". Because the wave-like spread of mutations is much slower than the logistic growth in well-mixed populations, and fixation times thus are increased, mutations are more likely to compete with other mutations that occur meanwhile.

One solution that nature has found out of this dilemma is recombination: In sexual species, the genotypes of individuals can be combined (or simply: merged). It was first noted by Fisher (1930) and Muller (1932) that sex relaxes clonal competition and speeds up the process of adaptation as it allows beneficial mutations to be combined in a single genome even if they first appeared in different lineages. Today, this Fisher-Muller argument is one of the most important explanations for why most organisms engage in some form of genetic exchange. In our study, we show that in spatially structured populations, not only recombination is a a viable way to relax clonal interference, but that even small long-range migration is a powerful mechanism to relax the detrimental effects of clonal interference.

Questions

Collaboration with O. Hallatschek, MPI for Dynamics and Self-Organization, UC Berkeley.

Publications



Cancer progression in spatially extended epithelia

Progression to cancer is a process of somatic evolution within the body. The idea that cancer can be understood as an evolutionary process was already formulated by P.C. Nowell in 1976. Cancer results from a sequence of genetic and epigenetic changes which lead to a variety of abnormal phenotypes including increased proliferation and survival of somatic cells, and thus, to a selective advantage of pre-cancerous cells. Many efforts have been made to better understand and predict the progression to cancer using mathematical models; these mostly consider the evolution of a well-mixed cell population. However, some pre-cancerous often evolve in epithelial tissues with distinct spatial structure. In a recent study, we investigate a novel model of cancer progression that considers a spatially structured cell population where clones expand via adaptive waves, i.e. I apply the concept of Fisher waves (explained above) to model clonal expansions involved in the progression of epithelial cancers:


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Collaboration with O. Hallatschek, MPI for Dynamics and Self-Organization, UC Berkeley; R. Kostadinov Penn State; C. Maley UCSF.

Publications


Synchronization Theory in Coupled Networks and Nonlinear Dynamics

Globally coupled oscillators with bimodal frequency distribution (Kuramoto model)

A paradigmatic model explaining the synchronization in coupled oscillator networks was introduced by Y. Kuramoto. This model describes the time evolution of an infinite number of oscillators connected in a network of oscillators with equal coupling strength (global coupling). In this study I generalize the original model by introducing a bimodal frequency distribution of the oscillators. This generic problem has been posed over 30 years ago. In my work I provide an exact solution describing the macroscopic dynamics (mean field order parameter dynamics).

Symmetry breaking: nonlocal coupling and chimera states

Studies In networks of coupled oscillators often assume that every unit is coupled globally (with equal strength), or then, only locally (only via their nearest neighbors). But what happens in the intermediate, nonlocal case? This question can be addressed by introducing a coupling strength that attenuates depending on the distance of the oscillators in a network, e.g.,

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We consider phase oscillator systems of the Kuramoto type, i.e.,

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Symbols are here defined as follows:

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If we have a finite set of populations corresponding to spatial locations on a ring, we find so-called chimera states:



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The x-axis represents the spatial coordinate, and the y-axis corresponds the phase of individual oscillators. Space is subdivided into subpopulations, represented by the 'slots' delineated by red vertical lines. Each subpopulation is inhabited by the same number of oscillators. The figure above shows the time-dependence of individual oscillator phases as a movie (loop). The phases of individual oscillators are shown in the co-moving frame defined by the phase of the locked oscillators. It is seen that the oscillators form two groups, where one is phase-locked (left and right), and the other one is oscillating in an incoherent fashion (center).

This state of coexisting synchronized and desynchronized oscillators is stable in time and was first discovered by Kuramoto et al. in 2002. Due to its incongruous nature, the state was later dubbed chimera, alluding to the same named monster in Greek mythology ("... a thing of immortal make, not human, lion-fronted and snake behind, a goat in the middle, ... ", Homer's Iliad, book 6). The chimera state is remarkable because the (circular) all-in-phase symmetry is broken even though all oscillators are identical.

The chimera is born a saddle-node bifurcation for a certain class of networks. Chimera states may also undergo a subcritical Hopf bifurcation. States that undergo this change of stability are referred to as breathing chimeras, because the unlocked oscillators (seen in the middle above) periodically transition through phases of smaller and larger synchronicity. The Hopf and saddle-node bifurcation curves intersect then in a Bogdanov-Takens point.

Spiral wave chimeras

Chimera states may also appear in the shape of spiral waves. Spiral waves may be observed in a planar lattice, where each site is occupied with an oscillator. An example of such a spiral wave is shown in the figure above, where the colors indicate phases of individual oscillators at different sites. When the coupling has nonlocal character, a chimera state may be observed: the topological defect in the spiral center is replaced with a circular 'hole' of desynchronized oscillators, as seen in the figure.


Spiral wave chimera in its corotating frame (video by C. Laing).

Questions

Publications



Symmetry breaking and pattern formation in open surface flows

The Polygonal Hydraulic Jump

A stationary hydraulic jump forms when a liquid jet with circular cross section impinges on a flat surface. From the point where the jet impacts the surface, the fluid spreads radially in a thin layer to all directions, and a steady open surface flow is established. This flow is very fast and therefore liquid jet impingement has applications in the cooling of technical devices, such as is required for high-efficiency generators, microelectronics, and laser systems; however, liquid jet impingiment is also of fundamental physical interest.

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Jet of fluid impinging on flat surface. A flow separation occurs after the jump, and is indicated by the smaller of the two circles; the larger one indicates the roller vortex structure as it may appear for Type II jumps.

At a critical radius away from the impact point, the fluid height h rises suddenly and a circular jump is formed. As a consequence, the flow changes from being supercritical to being subcritical (i.e., the flow velocity u becomes larger than the surface wave velocity  g h , i.e. the Froude number is Fr = u/√ g h  > 1). Further away from this critical radius, the flow separates and a vortex is formed. This flow configuration is shown as "Type I" in the figure below. The circular hydraulic jump has been found to be mathematically analogous to so-called white holes, which are the time-reversed analog of a black hole. This analogy has recently been demonstrated experimentally by a French research group (Jannes et al.) (see also Wired on how to create a white hole in your kitchen sink).


jump transition
Transition from so-called Type I jump with separation vortex to Type II jump with roller structure at the surface.



Under ideal lab conditions, it is observed that the jump may undergo another transition from this circular structure (Type I) to more complex structures (Type II). In this state, an additional vortex is formed (see figure above (right)), which seems to lie on top of the previous Type I flow structure, similar to a stationary breaking wave (it is often referred to as roller vortex). As flow parameters are changed, the symmetry may break spontaneously and the previously circular shape of the jump takes on the shape of a polygon with varying number of corners (see Ellegaard et al., Nature 392, 767 (1998)). Under special conditions, even more adventurous shapes may be seen, known as bowties and clovers, discovered by Bush et al., Journ. Fluid Mech. (2006) (see also further below).


Polygonal hydraulic jump shapes

The forces acting on the roller include viscosity, hydrostatic pressure and surface tension. Based on these forces, it is possible to develop a nonlinear ODE model, which reproduces both circular as well as polygonal structures. In certain parameter regions of the model one may also find coexisting solutions for constant flow parameters. This indicates the possibility of hysteretical effects which are also observed in experiments.

Below, three examples of polygonal jumps are displayed and compared with solutions of shapes as obtained from the theoretical model that I have been working on (Master's thesis). Pictures from polygonal jumps, as seen in experiments (C. Ellegaard et al.), are photographed through an impact plate which is made of glass (shown on the left). For a visualization of the peculiar flow structure of a polygonal jump see also here.

Connection to toroidal vortices (under construction)

Experimentally observed jumps
(Ellegaard et al., Nonlinearity (1999), 1-12)


Shapes produced by mathematical model (E. Martens)

2-cornered polygon (lens shaped) 2-cornered polygon (lens shaped)
Jump with two corners

3-cornered polygon 3-cornered polygon
Three-sided jump

5-cornered polygon 5-cornered polygon
5-cornered polygon



hexagon bowtie
Hexagon in water. It is hard to stabilize noncircular structures in low viscosity fluids, but this hexagonal structure was observed in water in a pretty stable mode (E.Martens, TSL M&AE Cornell University). Bowties and clover structures were first observed in John Bush's group at MIT (Viscous Hydraulic Jumps (Physics of Fluids (2004), 16 - 9). This picture of a bowtie was taken at my own lab (TSL, Mech. & Aer. Eng., Cornell University) using green anti freeze.

Model of polygonal jumps and their instabilities

Collaboration with T. Bohr, TU Denmark; S. Watanabe, Ibaraki University;

My colleagues and I discuss in a recent paper a phenomenological model that describes the polygon shapes and the instability mechanism leading to them (see publication below). In there, we discuss how the instability leading to the symmetry breaking of the circular jump may be explained in parallel to the so-called Rayleigh-Plateau instability (Savart-Rayleigh-Plateau), which is known to cause the breaking up of a falling fluid jet such as the jet falling from a water tap (see for instance simulation by S. Popinet). For further information, see also our summary text.

Publications

Open Questions


2010 Erik (technique by Sven)