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Role of graph architecture in controlling dynamical networks with applications to neural systems

Nature Physics volume 14, pages 91–98 (2018)Cite this article

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Abstract

Networked systems display complex patterns of interactions between components. In physical networks, these interactions often occur along structural connections that link components in a hard-wired connection topology, supporting a variety of system-wide dynamical behaviours such as synchronization. Although descriptions of these behaviours are important, they are only a first step towards understanding and harnessing the relationship between network topology and system behaviour. Here, we use linear network control theory to derive accurate closed-form expressions that relate the connectivity of a subset of structural connections (those linking driver nodes to non-driver nodes) to the minimum energy required to control networked systems. To illustrate the utility of the mathematics, we apply this approach to high-resolution connectomes recently reconstructed from Drosophila, mouse, and human brains. We use these principles to suggest an advantage of the human brain in supporting diverse network dynamics with small energetic costs while remaining robust to perturbations, and to perform clinically accessible targeted manipulation of the brain’s control performance by removing single edges in the network. Generally, our results ground the expectation of a control system’s behaviour in its network architecture, and directly inspire new directions in network analysis and design via distributed control.

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Figure 1: Network control of the drosophila, mouse, and human connectomes.
Figure 2: The simplified network representation offers a reasonable prediction for the full network’s control energy.
Figure 3: Geometric interpretation of simplified, first-order networks with corresponding control energies and trajectories.
Figure 4: Topological characteristics and energetic performance of networks with energetically favourable and unfavourable topologies.
Figure 5: Energetically favourable organization of topological features in networks.
Figure 6: Modifying the Drosophila, mouse and human connectomes to decrease the minimum energy required for control.

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Acknowledgements

J.Z.K. acknowledges support from National Institutes of Health T32-EB020087, PD: F. W. Wehrli, and the National Science Foundation Graduate Research Fellowship No. DGE-1321851. J.M.S. and D.S.B. acknowledge support from the John D. and Catherine T. MacArthur Foundation, the Alfred P. Sloan Foundation, the US Army Research Laboratory and the US Army Research Office through contract numbers W911NF-10-2-0022 and W911NF-14-1-0679, the National Institute of Health (2-R01-DC-009209-11, 1R01HD086888-01, R01-MH107235, R01-MH107703, R01MH109520, 1R01NS099348 R21-M MH-106799, and T32-EB020087), the Office of Naval Research, and the National Science Foundation (BCS-1441502, CAREER PHY-1554488, BCS-1631550, and CNS-1626008). A.E.K. and J.M.V. acknowledge support from the US Army Research Laboratory contract number W911NF-10-2-0022. F.P. acknowledges support from the National Science Foundation (BCS-1430280 and BCS 1631112). The content is solely the responsibility of the authors and does not necessarily represent the official views of any of the funding agencies.

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Authors and Affiliations

  1. Department of Bioengineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

    Jason Z. Kim, Jonathan M. Soffer, Jean M. Vettel & Danielle S. Bassett

  2. Department of Neuroscience, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

    Ari E. Kahn

  3. US Army Research Laboratory, Aberdeen, Maryland 21001, USA

    Ari E. Kahn

  4. Human Research & Engineering Directorate, US Army Research Laboratory, Aberdeen, Maryland 21001, USA

    Jean M. Vettel

  5. Department of Psychological and Brain Sciences, University of California, Santa Barbara, California 93106, USA

    Jean M. Vettel

  6. Department of Mechanical Engineering, University of California, Riverside, California 92521, USA

    Fabio Pasqualetti

  7. Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

    Danielle S. Bassett

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Contributions

J.Z.K., D.S.B. and F.P. wrote and revised the bulk of the manuscript. J.Z.K. developed the mathematical framework and analysed the data with feedback from F.P. and D.S.B. J.M.S. collected the human diffusion data, and A.E.K. processed the data to produce structural connectivity matrices with support from J.M.V.

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Correspondence to Danielle S. Bassett.

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Kim, J., Soffer, J., Kahn, A. et al. Role of graph architecture in controlling dynamical networks with applications to neural systems. Nature Phys 14, 91–98 (2018). https://doi.org/10.1038/nphys4268

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