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

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.

References

  1. 1

    Newman, M. E. J. Networks: An Introduction (Oxford Univ. Press, 2010).

    Google Scholar 

  2. 2

    Newman, M. E. J. The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. 3

    Watts, D. J. & Strogatz, S. H. Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998).

    ADS  Article  Google Scholar 

  4. 4

    Simon, H. The architecture of complexity. Proc. Am. Phil. Soc. 10, 467–482 (1962).

    Google Scholar 

  5. 5

    Bassett, D. S. & Sporns, O. Network neuroscience. Nat. Neurosci. 20, 353–364 (2017).

    Article  Google Scholar 

  6. 6

    Bettencourt, L. M., Stephens, G. J., Ham, M. I. & Gross, G. W. Functional structure of cortical neuronal networks grown in vitro. Phys. Rev. E 75, 021915 (2007).

    Article  ADS  MathSciNet  Google Scholar 

  7. 7

    Bassett, D. S. & Bullmore, E. T. Small-world brain networks revisited. Neuroscientist https://doi.org/10.1177/1073858416667720 (2016).

  8. 8

    Sporns, O. & Betzel, R. F. Modular brain networks. Annu. Rev. Psychol. 67, 613–640 (2016).

    Article  Google Scholar 

  9. 9

    Barabasi, A. L., Gulbahce, N. & Loscalzo, J. Network medicine: a network-based approach to human disease. Nat. Rev. Genet. 12, 56–68 (2011).

    Article  Google Scholar 

  10. 10

    Ching, S., Brown, E. N. & Kramer, M. A. Distributed control in a mean-field cortical network model: implications for seizure suppression. Phys. Rev. E 86, 021920 (2012).

    Article  ADS  Google Scholar 

  11. 11

    Khambhati, A. N., Davis, K. A., Lucas, T. H., Litt, B. & Bassett, D. S. Virtual cortical resection reveals push-pull network control preceding seizure evolution. Neuron 91, 1170–1182 (2016).

    Article  Google Scholar 

  12. 12

    Gonen, T. et al. Intra-operative multi-site stimulation: expanding methodology for cortical brain mapping of language functions. PLoS ONE 12, e0180740 (2017).

    Article  Google Scholar 

  13. 13

    Mohanty, S. K. & Lakshminarayananan, V. Optical techniques in optogenetics. J. Mod. Opt. 62, 949–970 (2015).

    Article  ADS  Google Scholar 

  14. 14

    Sprott, J. C. & Xiong, A. Classifying and quantifying basins of attraction. Chaos 25, 083101 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. 15

    Cornelius, S. P., Kath, W. L. & Motter, A. E. Realistic control of network dynamics. Nat. Commun. 4, 1942 (2013).

    Article  ADS  Google Scholar 

  16. 16

    Shine, J. M., Koyejo, O. & Poldrack, R. A. Temporal metastates are associated with differential patterns of time-resolved connectivity, network topology, and attention. Proc. Natl Acad. Sci. USA 113, 9888–9891 (2016).

    Article  Google Scholar 

  17. 17

    Mantzaris, A. V. et al. Dynamic network centrality summarizes learning in the human brain. J. Complex Netw. 1, 83–92 (2013).

    Article  Google Scholar 

  18. 18

    Bassett, D. S., Wymbs, N. F., Porter, M. A., Mucha, P. J. & Grafton, S. T. Cross-linked structure of network evolution. Chaos 24, 013112 (2014).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. 19

    Kalman, R. E. Mathematical description of linear dynamical systems. J. SIAM Control Ser. A 1, 152–192 (1963).

    MathSciNet  MATH  Google Scholar 

  20. 20

    Lin, C. T. Structural controllability. IEEE Trans. Autom. Control 19, 201–208 (1974).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. 21

    Liu, Y. Y., Slotine, J. J. & Barabasi, A. L. Controllability of complex networks. Nature 473, 167–173 (2011).

    Article  ADS  Google Scholar 

  22. 22

    Ruths, J. & Ruths, D. Control profiles of complex networks. Science 343, 1373–1376 (2014).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. 23

    Gu, S. et al. Controllability of structural brain networks. Nat. Commun. 6, 8414 (2015).

    Article  ADS  Google Scholar 

  24. 24

    Gu, S. et al. Optimal trajectories of brain state transitions. Neuroimage 148, 305–317 (2017).

    Article  Google Scholar 

  25. 25

    Betzel, R. F., Gu, S., Medaglia, J. D., Pasqualetti, F. & Bassett, D. S. Optimally controlling the human connectome: the role of network topology. Sci. Rep. 6, 30770 (2016).

    Article  ADS  Google Scholar 

  26. 26

    Muldoon, S. F. et al. Stimulation-based control of dynamic brain networks. PLoS Comput. Biol. 12, e1005076 (2016).

    Article  Google Scholar 

  27. 27

    Oh, S. W. et al. A mesoscale connectome of the mouse brain. Nature 508, 207–214 (2014).

    Article  ADS  Google Scholar 

  28. 28

    Rubinov, M., Ypma, R. J., Watson, C. & Bullmore, E. T. Wiring cost and topological participation of the mouse brain connectome. Proc. Natl Acad. Sci. USA 112, 10032–10037 (2015).

    Article  ADS  Google Scholar 

  29. 29

    Shih, C. T. et al. Connectomics-based analysis of information flow in the Drosophila brain. Curr. Biol. 25, 1249–1258 (2015).

    Article  Google Scholar 

  30. 30

    Kailath, T. Linear Systems (Prentice-Hall, 1980).

    Google Scholar 

  31. 31

    Fernandez, G. R. On how network architecture determines the dominant patterns of spontaneous neural activity. PLoS ONE 3, e2148 (2008).

    Article  Google Scholar 

  32. 32

    Honey, C. J. et al. Predicting human resting-state functional connectivity from structural connectivity. Proc. Natl Acad. Sci. USA 106, 2035–2040 (2009).

    Article  ADS  Google Scholar 

  33. 33

    Alstott, J., Breakspear, M., Hagmann, P., Cammoun, L. & Sporns, O. Modeling the impact of lesions in the human brain. PLoS Comput. Biol. 5, e1000408 (2009).

    Article  ADS  Google Scholar 

  34. 34

    Aristidou, A. A., San, K.-Y. & Bennett, G. N. Modification of central metabolic pathway in Escherichia coli to reduce acetate accumulation by heterologous expression of the Bacillus subtilis acetolactate synthase gene. Biotechnol. Bioeng. 44, 944–951 (1994).

    Article  Google Scholar 

  35. 35

    Sander, J. D. & Joung, J. K. CRISPR-Cas systems for editing, regulating and targeting genomes. Nat. Biotechnol. 32, 347–355 (2014).

    Article  Google Scholar 

  36. 36

    Motter, A. E. Networkcontrology. Chaos 25, 097621 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  37. 37

    Pasqualetti, F., Zampieri, S. & Bullo, F. Controllability metrics, limitations and algorithms for complex networks. IEEE Trans. Control Netw. Syst. 1, 40–52 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  38. 38

    Lanteaume, L. et al. Emotion induction after direct intracerebral stimulations of human amygdala. Cereb. Cortex 17, 1307–1313 (2007).

    Article  Google Scholar 

  39. 39

    Park, H., Niida, A., Miyano, S. & Imoto, S. Sparse overlapping group lasso for integrative multi-omics analysis. J. Comput. Biol. 22, 73–84 (2015).

    Article  MathSciNet  Google Scholar 

  40. 40

    Liu, Z., Lin, S., Deng, N., McGovern, D. P. & Piantadosi, S. Sparse inverse covariance estimation with L0 penalty for network construction with omics data. J. Comput. Biol. 23, 192–202 (2016).

    Article  MathSciNet  Google Scholar 

  41. 41

    Clauset, A., Moore, C. & Newman, M. E. Hierarchical structure and the prediction of missing links in networks. Nature 453, 98–101 (2008).

    Article  ADS  Google Scholar 

  42. 42

    Zhu, B. & Xia, Y. An information-theoretic model for link prediction in complex networks. Sci. Rep. 5, 13707 (2015).

    Article  ADS  Google Scholar 

  43. 43

    Chen, H. I. et al. Harnessing plasticity for the treatment of neurosurgical disorders: an overview. World Neurosurg. 82, 648–659 (2014).

    Article  Google Scholar 

  44. 44

    Chrysikou, E. G. & Hamilton, R. H. Noninvasive brain stimulation in the treatment of aphasia: exploring interhemispheric relationships and their implications for neurorehabilitation. Restor. Neurol. Neurosci. 29, 375–394 (2011).

    Google Scholar 

  45. 45

    Bassett, D. S., Khambhati, A. N. & Grafton, S. T. Emerging frontiers of neuroengineering: a network science of brain connectivity. Annu. Rev. Biomed. Eng. 19, 327–352 (2017).

    Article  Google Scholar 

  46. 46

    Johnson, M. D. et al. Neuromodulation for brain disorders: challenges and opportunities. IEEE Trans. Biomed. Eng. 60, 610–624 (2013).

    Article  Google Scholar 

  47. 47

    Rubinov, M., Sporns, O., Thivierge, J. P. & Breakspear, M. Neurobiologically realistic determinants of self-organized criticality in networks of spiking neurons. PLoS Comput. Biol. 7, e1002038 (2011).

    Article  MathSciNet  Google Scholar 

  48. 48

    Shew, W. L. et al. Adaptation to sensory input tunes visual cortex to criticality. Nat. Phys. 11, 659–663 (2015).

    Article  Google Scholar 

  49. 49

    Caeyenberghs, K., Verhelst, H., Clemente, A. & Wilson, P. H. Mapping the functional connectome in traumatic brain injury: what can graph metrics tell us? Neuroimage S1053–8119, 30694–30692 (2016).

    Google Scholar 

  50. 50

    van der Horn, H. J. et al. Altered wiring of the human structural connectome in adults with mild traumatic brain injury. J. Neurotrauma 34, 1035–1044 (2017).

    Article  Google Scholar 

  51. 51

    Bollobas, B. Random Graphs (Academic, 1985).

    Google Scholar 

  52. 52

    Klimm, F., Bassett, D. S., Carlson, J. M. & Mucha, P. J. Resolving structural variability in network models and the brain. PLoS Comput. Biol. 10, e1003491 (2014).

    Article  ADS  Google Scholar 

  53. 53

    Sizemore, A., Giusti, C. & Bassett, D. S. Classification of weighted networks through mesoscale homological features. J. Complex Netw. 5, 245–273 (2017).

    MathSciNet  Google Scholar 

  54. 54

    Pfeil, T. et al. Six networks on a universal neuromorphic computing substrate. Front Neurosci. 7, 11 (2013).

    Article  Google Scholar 

  55. 55

    Giusti, C., Papadopoulos, L., Owens, E. T., Daniels, K. E. & Bassett, D. S. Topological and geometric measurements of force-chain structure. Phys. Rev. E 94, 032909 (2016).

    Article  ADS  Google Scholar 

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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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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.

Corresponding author

Correspondence to Danielle S. Bassett.

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The authors declare no competing financial interests.

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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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