Coupled heteroclinic networks

• Applicant: Professorin Dr. Hildegard Meyer-Ortmanns
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
• Term: from 2017 to 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 363137745

• Final Report Year: 2024

Final Report Abstract

Heteroclinic dynamics provides a framework for describing transient processes with sudden changes between metastable states and possibly long dwell times in the vicinity of metastable states. We considered heteroclinic dynamics with applications to evolutionary game theory and metastable features of brain dynamics. We demonstrated the possibility to design alliances in games of competition between species at wish. This is achieved by a suitable choice of the adjacency matrix in phase space that determines the strength of mutual competition in the framework of generalized Lotka-Volterra equations. In particular, this construction supports a nested attractor space with a hierarchy in time scales. In applications to brain dynamics this is reflected in a kind of chunking dynamics that is also experimentally observed. When our heteroclinic networks are assigned to a spatial grid, we realize a conjecture by S. Grossberg according to which the brain is organized by parallel processing of individually hierarchical dynamics. Our choice of diffusive coupling can lead to partially or fully synchronized systems, strongly reduced in the dimension of phase space. Information, for example on sensory input, which is said to be encoded in the temporal sequence of excited neural subpopulations (i.e., visited saddles of the heteroclinic network at a certain location) can be transferred over a spatial grid if only a small set of pacemakers entrains heteroclinic units in a resting state. No finetuning of the parameters of the resting units is needed. The dynamics in the very vicinity of certain bifurcation points of hierarchical heteroclinic dynamics displays all features of criticality so that these points may serve as working points for information transfer. To further assess heteroclinic dynamics as effective description of metastable brain dynamics, a fast adaptation to new external input is required. In view of this, we analyzed the relaxation times after a quench into the regime in which heteroclinic oscillations should be arrested. These relaxation times turn out to be nonnegligible and may play a role when heteroclinic dynamics is used as effective description of gaits, in particular when gaits are affected in neurodegenerative diseases with a delayed response in motion to external input. Moreover, we performed a detailed bifurcation analysis of coupled heteroclinic networks from a more abstract point of view: In general, they display a rich dynamical repertoire already for very small sets, ranging from limit cycles to transient chaos and chaos. Rather than entire heteroclinic networks in a deterministic description, we also considered stochastic systems and individual heteroclinic trajectories as connecting different saddles, when they provide an escape where no escape would be possible in the deterministic limit. We analyzed the statistics of rare events in the May-Leonard model of three cyclically competing species, in which all three species go extinct by strong but rare fluctuations. The mean time to extinction is very sensitive to the distance from the bifurcation point, even in the very same dynamical regime. We compared results based on a WKB approach for classical stochastic systems with Gillespie simulations in regions where both methods apply. By the same type of approaches we determined the mean time to extinction of a population of bacteria, composed of so-called normals and persisters, exposed to both demographic noise and an oscillating environment. Again, a rare extinction event may be mediated by heteroclinic trajectories. Here, the survival probability of the bacteria depends non-monotonically on the frequency of application of adverse environmental conditions. A minimal survival time of the bacteria is achieved by choosing the frequency out of a small interval of intermediate values, whereas the dose should be chosen as large as possible while still tolerable for the host of the bacteria.

Publications

• A hierarchical heteroclinic network. The European Physical Journal Special Topics, 227(10-11), 1101-1115.
  Voit, Maximilian & Meyer-Ortmanns, Hildegard
  
• Dynamical Inference of Simple Heteroclinic Networks. Frontiers in Applied Mathematics and Statistics, 5.
  Voit, Maximilian & Meyer-Ortmanns, Hildegard
  
• Dynamics of nested, self-similar winnerless competition in time and space. Physical Review Research, 1(2).
  Voit, Maximilian & Meyer-Ortmanns, Hildegard
  
• Predicting the separation of time scales in a heteroclinic network. Applied Mathematics and Nonlinear Sciences, 4(1), 279-288.
  Voit, Maximilian & Meyer-Ortmanns, Hildegard
  
• Coupled heteroclinic networks in disguise. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(8).
  Voit, Maximilian; Veneziale, Sara & Meyer-Ortmanns, Hildegard
  
• Emerging criticality at bifurcation points in heteroclinic dynamics. Physical Review Research, 2(4).
  Voit, Maximilian & Meyer-Ortmanns, Hildegard
  
• Rare extinction events in cyclic predator–prey games. Journal of Physics A: Mathematical and Theoretical, 54(23), 235001.
  Serrao, Shannon R.; Ritmeester, Tim & Meyer-Ortmanns, Hildegard
  
• Heteroclinic units acting as pacemakers: entrained dynamics for cognitive processes. Journal of Physics: Complexity, 3(3), 035003.
  Thakur, Bhumika & Meyer-Ortmanns, Hildegard
  
• Controlling the Mean Time to Extinction in Populations of Bacteria. Entropy, 25(5), 755.
  Thakur, Bhumika & Meyer-Ortmanns, Hildegard
  
• On relaxation times of heteroclinic dynamics. Chaos: An Interdisciplinary Journal of Nonlinear Science, 33(10).
  Aravind, Manaoj & Meyer-Ortmanns, Hildegard