Many real-world systems can be seen as networks of dynamical nodes. Examples range from power grid networks to synchronously firing neural oscillators in the brain. The dynamics – and therefore the function – of such networks are determined by the individual dynamics of each node as well as the interaction between nodes.
Crucially, in many cases the interactions between two nodes are not static: They may be influenced by the state of other nodes or change over time, e.g., via external forcing, intrinsic adaptation such plastic connections. In this case the network is adaptive or co-evolutionary. That adaptive effects are crucial for the network dynamics has been demonstrated, for example, in ecological networks and networks of neuronal oscillators. Despite this, there are only realtively limited mathematical insights into the dynamics of networks with state-dependent and adaptive interactions: Many commonly used models either assume static connections, or they assume static node dynamics. Even if there is co-evolutionary dynamics on and of the network, direct numerical simulation or formal moment approximations are some of the predominant approaches. Hence, there is still a very serious challenge to understand the mathematical fundamentals of adaptive networks. One key goal of the Focus Group "Network Dynamics" is to push the boundaries on the mathematical understanding on network dynamical systems with state-dependent and adaptive interactions.
Furthermore, robust function of networks may be thought of as an emergent dynamic property of the network resulting from the intrinsic dynamics within nodes, the structure and dynamics of connections between nodes, and the interplay of the network with the external environment. Consequently important transient and excitable behaviour is widespread in such networks, e.g., in biomedical applications. Given perturbations different transient and emergent dynamics arise in networks. The significance of such transient effects has been demonstrated in avalanches of gene activation in gene regulatory pathways to drive cell differentiation, development and cancer as well as cell fate in biofilm formation, just to name a few. However, our mathematical understanding of transient network dynamics is in its infancy in spite of some recents efforts. Accordingly, the Focus Group "Network Dynamics"– consisting of Prof. Dr. Krasimira Tsaneva-Atanasova (Hans Fischer Senior Fellow), Dr. Christian Bick (Hans Fischer Fellow) and Prof. Dr. Christian Kühn (TUM-Professor for Multiscale and Stochastic Dynamics) – aims at developing the mathematical understanding on transient emergent network dynamics.
TUM-IAS funded doctoral candidates:
Tobias Böhle
Maria Elena Gonzalez
Selected Publications by the Focus Group
Salman, M., Bick, C., & Krischer, K. (2020). Collective oscillations of globally coupled bistable, nonresonant components. Physical Review Research, 2(4), 043125. doi:10.1103/PhysRevResearch.2.043125
Ashwin, Peter; Creaser, Jennifer; Tsaneva-Atanasova, Krasimira: Quasipotentials for coupled escape problems and the gate-height bifurcation. Physical Review E 107 (1), 2023 mehr…
Bick, Christian; Böhle, Tobias; Kuehn, Christian: Higher-Order Network Interactions through Phase Reduction for Oscillators with Phase-Dependent Amplitude. , 2023 mehr…
Bick, Christian; Böhle, Tobias; Omel'chenko, Oleh E.: Hopf Bifurcations of Twisted States in Phase Oscillators Rings with Nonpairwise Higher-Order Interactions. , 2023 mehr…
Bick, Christian; Gross, Elizabeth; Harrington, Heather A.; Schaub, Michael T.: What Are Higher-Order Networks? SIAM Review 65 (3), 2023, 686-731 mehr…
Böhle, Tobias; Thalhammer, Mechthild; Kuehn, Christian: Community integration algorithms (CIAs) for dynamical systems on networks. Journal of Computational Physics 469, 2022, 111524 mehr…
Böhle, Tobias; Thalhammer, Mechthild; Kuehn, Christian: Community Integration Algorithms (CIAs) for Dynamical Systems on Networks. arXiv, 2022 mehr…
Kuehn, Christian; Berglund, Nils; Bick, Christian; Engel, Maximilian; Hurth, Tobias; Iuorio, Annalisa; Soresina, Cinzia: A general view on double limits in differential equations. Physica D: Nonlinear Phenomena 431, 2022, 133105 mehr…
Mulas, Raffaella; Kuehn, Christian; Böhle, Tobias; Jost, Jürgen: Random walks and Laplacians on hypergraphs: When do they match? Discrete Applied Mathematics 317, 2022, 26-41 mehr…
Mulas, Raffaella; Kuehn, Christian; Böhle, Tobias; Jost, Jürgen: Random walks and Laplacians on hypergraphs: When do they match? Discrete Applied Mathematics 317, 2022, 26-41 mehr…
Wang, Ai; Sieber, Jan; Young, William R.; Tsaneva-Atanasova, Krasimira: Time series analysis and modelling of the freezing of gait phenomenon. , 2022 mehr…
2021
Bick, Christian; Sclosa, Davide: Dynamical Systems on Graph Limits and Their Symmetries. , 2021 mehr…
Böhle, T.; Kuehn, C.; Thalhammer, M.: On the reliable and efficient numerical integration of the Kuramoto model and related dynamical systems on graphs. International Journal of Computer Mathematics 99 (1), 2021, 31-57 mehr…
Böhle, Tobias; Kuehn, Christian; Thalhammer, Mechthild: On the reliable and efficient numerical integration of the Kuramoto model and related dynamical systems on graphs. 2021 mehr…
Gonzalez Herrero, Maria Elena; Kuehn, Christian: A qualitative mathematical model of the immune response under the effect of stress. Chaos: An Interdisciplinary Journal of Nonlinear Science 31 (6), 2021, 061104 mehr…
Gonzalez Herrero, Maria Elena; Kuehn, Christian; Tsaneva-Atanasova, Krasimira: Reduced Models of Cardiomyocytes Excitability: Comparing Karma and FitzHugh–Nagumo. Bulletin of Mathematical Biology 83 (8), 2021 mehr…
Kuehn, Christian; Bick, Christian: A universal route to explosive phenomena. Science Advances 7 (16), 2021, eabe3824 mehr…
Mac Aogáin, Micheál; Narayana, Jayanth Kumar; Tiew, Pei Yee; Ali, Nur A’tikah Binte Mohamed; Yong, Valerie Fei Lee; Jaggi, Tavleen Kaur; Lim, Albert Yick Hou; Keir, Holly R.; Dicker, Alison J.; Thng, Kai Xian; Tsang, Akina; Ivan, Fransiskus Xaverius; Poh, Mau Ern; Oriano, Martina; Aliberti, Stefano; Blasi, Francesco; Low, Teck Boon; Ong, Thun How; Oliver, Brian; Giam, Yan Hui; Tee, Augustine; Koh, Mariko Siyue; Abisheganaden, John Arputhan; Tsaneva-Atanasova, Krasimira; Chalmers, James D.; Chotirmall, Sanjay H.: Integrative microbiomics in bronchiectasis exacerbations. Nature Medicine 27 (4), 2021, 688-699 mehr…
Narayana, Jayanth Kumar; Mac Aogáin, Micheál; Goh, Wilson Wen Bin; Xia, Kelin; Tsaneva-Atanasova, Krasimira; Chotirmall, Sanjay H.: Mathematical-based microbiome analytics for clinical translation. Computational and Structural Biotechnology Journal 19, 2021, 6272-6281 mehr…
Prince, Luke Y.; Bacon, Travis; Humphries, Rachel; Tsaneva-Atanasova, Krasimira; Clopath, Claudia; Mellor, Jack R.: Separable actions of acetylcholine and noradrenaline on neuronal ensemble formation in hippocampal CA3 circuits. PLOS Computational Biology 17 (10), 2021, e1009435 mehr…
Salman, Munir; Bick, Christian; Krischer, Katharina: Bifurcations of clusters and collective oscillations in networks of bistable units. Chaos: An Interdisciplinary Journal of Nonlinear Science 31 (11), 2021, 113140 mehr…
Wedgwood, Kyle C. A.; Słowiński, Piotr; Manson, James; Tsaneva-Atanasova, Krasimira; Krauskopf, Bernd: Robust spike timing in an excitable cell with delayed feedback. Journal of The Royal Society Interface 18 (177), 2021 mehr…
2020
Baldemir, Harun; Avitabile, Daniele; Tsaneva-Atanasova, Krasimira: Pseudo-plateau bursting and mixed-mode oscillations in a model of developing inner hair cells. Communications in Nonlinear Science and Numerical Simulation 80, 2020, 104979 mehr…
Herrero, Maria Elena Gonzalez; Kuehn, Christian; Tsaneva-Atanasova, Krasimira: Reduced models of cardiomyocytes excitability: comparing Karma and FitzHugh-Nagumo. 2020 mehr…
Horstmeyer, Leonhard; Kuehn, Christian: Adaptive voter model on simplicial complexes. Physical Review E 101 (2), 2020 mehr…
Horstmeyer, Leonhard; Kuehn, Christian; Thurner, Stefan: Balancing quarantine and self-distancing measures in adaptive epidemic networks. 2020 mehr…
Jardón-Kojakhmetov, Hildeberto; Kuehn, Christian: On Fast–Slow Consensus Networks with a Dynamic Weight. Journal of Nonlinear Science 30 (6), 2020, 2737-2786 mehr…
Jardón-Kojakhmetov, Hildeberto; Kuehn, Christian; Pugliese, Andrea; Sensi, Mattia: A geometric analysis of the SIRS epidemiological model on a homogeneous network. 2020 mehr…
Kuehn, Christian: Network dynamics on graphops. New Journal of Physics 22 (5), 2020, 053030 mehr…
Mulas, Raffaella; Kuehn, Christian; Jost, Jürgen: Coupled dynamics on hypergraphs: Master stability of steady states and synchronization. Physical Review E 101 (6), 2020 mehr…
