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Partially unstable attractors in networks of forced integrate-and-fire oscillators

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Abstract

The asymptotic attractors of a nonlinear dynamical system play a key role in the long-term physically observable behaviors of the system. The study of attractors and the search for distinct types of attractor have been a central task in nonlinear dynamics. In smooth dynamical systems, an attractor is often enclosed completely in its basin of attraction with a finite distance from the basin boundary. Recent works have uncovered that, in neuronal networks, unstable attractors with a remote basin can arise, where almost every point on the attractor is locally transversely repelling. Herewith we report our discovery of a class of attractors: partially unstable attractors, in pulse-coupled integrate-and-fire networks subject to a periodic forcing. The defining feature of such an attractor is that it can simultaneously possess locally stable and unstable sets, both of positive measure. Exploiting the structure of the key dynamical events in the network, we develop a symbolic analysis that can fully explain the emergence of the partially unstable attractors. To our knowledge, such exotic attractors have not been reported previously, and we expect them to arise commonly in biological networks whose dynamics are governed by pulse (or spike) generation.

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Acknowledgements

This research is supported by the National Natural Science Foundation of China (11502200, 91648101), “The Fundamental Research Funds for the Central Universities” (No. 3102014JCQ01036), and SRF for ROCS, SEM. This research is also supported by the Aihara Project, the FIRST program from JSPS, initiated by CSTP, and CREST, JST. YCL is supported by ARO under Grant No. W911NF-14-1-0504.

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Authors and Affiliations

  1. School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xian, 710072, China

    Hai-Lin Zou, Zi-Chen Deng & Wei-Peng Hu

  2. Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo, 153-8505, Japan

    Kazuyuki Aihara

  3. School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ, 85287, USA

    Ying-Cheng Lai

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  1. Hai-Lin Zou
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Correspondence to Hai-Lin Zou.

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Zou, HL., Deng, ZC., Hu, WP. et al. Partially unstable attractors in networks of forced integrate-and-fire oscillators. Nonlinear Dyn 89, 887–900 (2017). https://doi.org/10.1007/s11071-017-3490-5

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