Transient dynamics in complex systems

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For decades, the main theoretical focus of dynamical systems research was on the long-time limits of dynamical processes, where the system remained in a certain state unless it was kicked out by external perturbations. During transient dynamics, however, the system may look stable for a while, but is not truly stable. In ecological systems, it may remain quasi-stable for hundreds of generations before a sudden transition to another dynamical regime occurs. Professor Hildegard Meyer-Ortmanns from Jacobs University in Bremen, Germany, investigates the use of heteroclinic dynamics to describe applications including ecological systems, but also brain dynamics.

What is behind phenomena such as the sudden extinction of species in population dynamics? Which dynamics are involved in transient features such as the temporary coexistence of species in ecological systems before some become extinct, or in the cognitive processes in the brain, transient but reproducible? What generates the spiral patterns that appear in density profiles or travelling waves? What is responsible for different gaits in animals and humans? Dr Hildegard Meyer-Ortmanns, Professor of Physics at Jacobs University Bremen, explains how such processes can be understood using heteroclinic dynamics.

Heteroclinic dynamics

Professor Meyer-Ortmanns studies heteroclinic networks in dynamical systems. Heteroclinic networks are networks in a phase space. A phase space is the space in which all possible states of a system are represented. The time evolution of the system is traced along phase trajectories in phase space. A heteroclinic network represents possible trajectories along which the dynamical system may move and comprises nodes, corresponding to saddles and heteroclinic connections. In the simplest case, saddles are stationary points, here called saddle equilibria (or saddle fixed points), that is, points in phase space with stable and unstable directions. Like saddles in a mountainous landscape, it depends on the direction from which a saddle is approached whether the approaching object is repelled from the saddle or attracted by it. More generally, saddles in heteroclinic dynamics may be more complicated objects such as entire heteroclinic cycles.

In particular, different saddle points may be connected by heteroclinic paths so that the repulsive direction of one saddle becomes the attractive direction of another. A particle moving along such a heteroclinic connection is then attracted to a saddle and remains in its vicinity for a while before a sudden switch occurs that repels it from the first saddle and attracts it to the next saddle. Such heteroclinic connections may form sequences, cycles, or entire networks. On top of these constructions, heteroclinic networks can be coupled together on spatial grids.

An ecological application

Say we have a population of three species (such as side-blotched lizards), here termed A, B, C. Observations of the time evolution of these species may reveal that species A is dominant for a long time (quasi-static), while B and C coexist at low population levels with A. Suddenly the density of A decreases drastically, while B gets dominant, and C remains at a low level. This is followed by the temporary dominance of C and low level of A and B, until the cycle repeats. The cycle realises a game that is termed rock-paper-scissors. The corresponding trajectory of the three species approaches a heteroclinic cycle.

The game rock-paper-scissors is played on many scales, ranging from the macroscopic level, where it is played by children, to the genetic level. A dynamical realisation of the game as transient dominance of one species followed by a sudden switch to another dominant one, as described above, can be formulated as a heteroclinic cycle between the temporary winners. It is an example of winnerless competition, as winners are only transient.

Cognitive processes in the brain

Now, instead of species of ecological systems, one may think of cognitive items in dynamical processes of the brain. Our thoughts are intrinsically transient. We do not want to get stuck in the same thought forever. Thoughts and sequences of thoughts, however, are reproducible, and so are heteroclinic sequences if they represent cognitive items. For a long time, it was an open question as to which mathematical framework could best capture the intrinsically transient, but reproducible and temporarily ordered sequences of such events. Heteroclinic dynamics provides such a framework.

“For a long time, it was an open question as to which mathematical framework could best capture intrinsically transient dynamics.”

Inspired by neural activities

Professor Meyer-Ortmanns is working on systems where entire heteroclinic networks are networks of heteroclinic connections in phase space. These networks are constructed so that they show hierarchies in time scales. The motivation comes from experimental observations of neural activities, in which the amplitudes of fast oscillations of neural activities are modulated by slow oscillations, possibly iteratively with several time scales in the modulated oscillations. A particular manifestation of inherent hierarchical organisation is chunking, in which long sequences of symbols, such as numbers, are split into shorter sequences for better memorising the long ones. Professor Meyer-Ortmanns’ research team has constructed hierarchical heteroclinic dynamics that reproduce these features.

A heteroclinic cycle of heteroclinic cycles

The solution is a heteroclinic cycle of heteroclinic cycles; this means that the first heteroclinic cycle does not connect simple saddle points, but objects which are heteroclinic cycles themselves. Again, based on experimental evidence, an important conjecture regarding the overall organisation of brain dynamics is that brain dynamics are organised hierarchically and run in parallel. Accordingly, the researchers considered a whole set of heteroclinic networks, individually designed to capture the feature of hierarchical organisation but coupled with each other and assigned to a spatial grid to allow the dynamics to run in parallel. This revealed that for a whole range of parameters, individual hierarchical heteroclinic trajectories synchronise between different sites. Thus, the entire set of many heteroclinic networks behaves in a simple way as if it were a single heteroclinic network, demonstrating that synchronisation is achievable in this mathematical framework. Partial synchronisation plays a pivotal role in neural dynamics.

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Nested dynamics

Apart from possible applications to the brain, nested hierarchical heteroclinic networks on a spatial grid have a different interpretation in terms of ecological applications. This is apparent when the predation rules between the competing species in the winnerless competition game are chosen to reproduce a rock-paper-scissors game, played on several spatial and time scales simultaneously. The researchers selected two hierarchy levels of nine species that have arranged themselves into three subpopulations, each comprising three types of species. Rock-paper-scissors is then played within the subpopulations on the scale of individuals, and between the subpopulations on the scale of the domain inhabited by the subpopulation. This kind of nested dynamics is self-similar and as such certainly special. However, it illustrates how in a general setting a simple set of predation rules is able to control possible cooperation or competition that emerge as self-organised when the game between the species is on.

Actually, nested dynamics is quite common. It means that dynamical processes are simultaneously ongoing on different scales within a large system, ranging from microscopic to macroscopic ones. Not common is the feature of self-similarity on different scales, neither the fact that the nested dynamics is generated by a simple set of rules as here from the construction of a heteroclinic network.

Brain dynamics and criticality

Returning to brain dynamics, criticality in relation to brain dynamics refers to what extent the brain works at the borderline between order and chaos, or more generally between order and disorder. Criticality summarises the characteristic features at critical points in phase transitions or at certain bifurcation points; in both cases the dynamical regime of the system radically changes at these points. Typical features at critical points include a high sensitivity to perturbations, long correlations in space, critical slowing down of the dynamics in time and a rich dynamical repertoire. These features may support the information processing in a constructive way, allowing information storage on the ‘ordered’ side and facilitating information transmission on the ‘chaotic’ side of the critical threshold.

Experimental evidence from analysing resting state fluctuations in the brain

Experimental data of resting state fluctuations in the brain are best fitted in a neural mass model when this network with the connectivity structure between brain areas are described in terms of coupled Stuart-Landau oscillators in the immediate vicinity of their Hopf bifurcation points. Stuart-Landau oscillators are prototypes of nonlinear oscillating systems with Hopf bifurcations where the system’s stability switches between a stable equilibrium and a stable periodic solution or vice versa. Thus, Hopf bifurcations in this theoretical description appear as suitable working points to match the data from resting state fluctuations.

“Upon synchronisation, the entire set of heteroclinic networks can simplify as if it were a single network.”

Zooming into bifurcation points of heteroclinic dynamics

The researchers ask a similar question in heteroclinic dynamics as a distinguished framework for cognitive processes and other brain dynamics: do bifurcation points in heteroclinic dynamics exist which are also candidates for working points in brain dynamics with emerging features of criticality? Do they provide a large repertoire of dynamical processes, depending on the initial conditions, and slow time scales in the critical regime?

The answer is positive, though it is completely open as to whether brain dynamics actually makes use of such working points and tunes itself to these parameters.

From a more general perspective, Professor Meyer-Ortmanns explains that the results provide a further example for ‘the importance of being borderline’. A new type of dynamics is just emerging at and in the immediate vicinity of singular points, here between certain bifurcation points that separate different dynamical regimes.

Schematic view of a heteroclinic network with selected trajectories. https://doi.org/10.1103/PhysRevResearch.1.023008

Work on heteroclinic dynamics opens a wide field of possible realisations by varying the kind of spatial coupling, the time scales in hierarchies, delay in the interactions, etc. From the information processing perspective, heteroclinic dynamics can be exploited for computation in artificial systems such as robots, in particular exploring the specific dynamics in the vicinity of bifurcation points.

Precision has its price

The research team has also been examining the versatile role of stochastic fluctuations (or noise) on dynamical behaviour. It has a wide range of applications such as applied ones in power grids, synchronisation phenomena in oscillatory systems, and ageing. Ageing is understood in the sense of error accumulation in processes that require high precision including manufacturing production processes and information transfer in biological systems. Usually, these processes are inherently error prone and require an investment of resources for the maintenance of their accuracy.

When ageing may be an unavoidable fate of dynamical systems

The researchers explore the final fate of systems with such processes under the assumption that perfect precision costs an infinite amount of resources. This assumption is based on results on different trade-off relations, demonstrating that high precision generally has its price, and that there are fundamental limits to the achievable precision.

Resources are taken from a reservoir that is in principle infinite, but only a finite amount is available in a finite period of time. Under these constraints – error-prone production and finite accessible resources – the researchers have analysed whether accumulating errors can be kept below a certain threshold that is considered as tolerable. They established that the result depends on the ratio of the costs spent on the production process (or the information transfer) in relation to the costs spent on the maintenance (or repair, itself assumed to be error prone).

Analogous to the consequence where the tolerable threshold is exceeded is the need to use all resources for maintenance so that the production process or information transfer would stop. The resources can represent energy, (disk-) space or time (finite performance speed). Indefinite error accumulation to the extent where the processes lose part of their function is a characteristic feature of ageing. Thus, the answer to the question as to whether ageing can be avoided is: it depends. It depends on the functional dependence of the costs of achieving a certain degree of precision in relation to the maintenance costs whether the errors accumulate indefinitely or converge to a finite tolerable ratio. Professor Meyer-Ortmanns remarks that to change the cost-precision functions on a fundamental biological level such as to avoid an indefinite increase of errors may violate fundamental bounds from physics (thermodynamics and kinetics); thus, ageing may be an unavoidable fate of a dynamical, in particular a biological system. Analysing the price for precision and a related sensible allocation of resources in processes on the nano- and microscale remains a hot topic in current research.

What inspired you to conduct this research?
I think that heteroclinic dynamics is a promising mathematical framework that allows the physical realisation of extremely versatile, transient and reproducible processes underlying in particular cognition. Beyond that, it illustrates how complex behaviour with multiple time scales and nested dynamics can be generated out of sets of simple rules. About the second topic, the role of intrinsic uncertainties, my main interest is in pursuing the impact of fundamental physical limits from thermodynamics and kinetics on biological processes on the micro-, meso- and macroscale. I like to understand fundamental trade-offs between costs, precision and speed when they play a fundamental role comparable to Heisenberg’s uncertainty relation.

Why did you switch between rather different fields in physics?
I had posed myself a number of basic questions from different realms of physics during the first years of high school, though during high school I learned five foreign languages rather than focusing on natural science. While studying physics I realised that this is the right place where my earlier questions are addressed and my interest in those questions holds on until today. They ranged indeed from gravitation to particle physics to cognitive processes in the brain.

 

References

  • Voit, M and Meyer-Ortmanns, H, (2018). A hierarchical heteroclinic network. The European Physical Journal Special Topics, 227, 1101–1115. Available at: doi.org/10.1140/epjst/e2018-800040-x
  • Voit, M and Meyer-Ortmanns, H, (2019). Predicting the separation of time scales in a heteroclinic network. Applied Mathematics and Nonlinear Sciences, 4(2), 273–282. Available at: dx.doi.org/10.2478/AMNS.2019.1.00024
  • Voit, M and Meyer-Ortmanns, H, (2019). Dynamics of nested, self-similar winnerless competition in time and space. Physical Review Research, 1, 023008. Available at: doi.org/10.1103/PhysRevResearch.1.023008
  • Voit, M and Meyer-Ortmanns, H, (2020). Emerging criticality at bifurcation points in heteroclinic dynamics. Physical Review Research, 2, 043097. Available at: doi.org/10.1103/PhysRevResearch.2.043097
  • Ritmeester, T and Meyer-Ortmanns, H, (2021). Minority games played by arbitrageurs on the energy market. Physica A: Statistical Mechanics and its Applications, 573, 125927. Available at: doi.org/10.1016/j.physa.2021.125927
  • Ionita, F, Labavić, D, Zaks, M and Meyer-Ortmanns, H, (2013). Order-by-disorder in classical oscillator systems. Eur Phys J B, 86(12), 511–524. Available at: doi.org/10.1140/epjb/e2013-40998-8
  • Voit, M and Meyer-Ortmanns, H, (2019). How aging may be an unavoidable fate of dynamical systems. New J Phys, 21, 043045. Available at: iopscience.iop.org/article/10.1088/1367-2630/ab1043
  • Rabinovich, MI, Varona, P, Tristan, I and Afraimovich, VS, (2014). Chunking dynamics: heteroclinics in mind. Front. Comput. Neurosci. 8 (22), 1. Available at: doi.org/10.3389/fncom.2014.00022
  • Grossberg, S, (2000). The complementary brain: Unifying brain dynamics and modularity. Trends Cognit Sci, 4, 233. Available at: doi.org/10.1016/S1364-6613(00)01464-9
  • Deco, G, Kringelbach, ML, Jirsa, VK and Ritter, P, (2017). The dynamics of resting fluctuations in the brain: Metastability and its dynamical cortical core. Sci. Rep., 7, 3095. Available at: doi.org/10.1038/s41598-017-03073-5
  • Chibbaro, S, Rondoni, L, Vulpiani, A, (2014). Reductionism, Emergence and Levels of Reality: The Importance of Being Borderline. Springer, Heidelberg.
DOI
10.26904/RF-138-1782561499

Research Objectives

Professor Meyer-Ortmanns studies complex systems with methods from nonlinear dynamics and statistical physics. One current research topic is heteroclinic dynamics, another one is the impact of stochastic fluctuations.

Funding

  • German Research Foundation (DFG, Grant No. ME-1332/28-1)
  • Bundesministerium für Bildung und Forschung (BMBF, Grant No. 03EK3055D)

Collaborators

  • Darka Labavić
  • Maximilian Voit
  • Tim Ritmeester

Bio

After a diploma in general relativity, a PhD in particle physics and several postdoc positions, Hildegard Meyer-Ortmanns changed her field of research around 2001 to network science with methods from statistical physics and nonlinear dynamics. Since 2002, she has been Professor of Physics at Jacobs University Bremen. Her current research is in the field of complex systems.

Hildegard Meyer-Ortmanns

Contact
Department of Physics and Earth Sciences
Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany

E: h.ortmanns@jacobs-university.de
T: +49 421 200 3221
W: https://www.jacobs-university.de/directory/meyer-ortmanns

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