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Synchronization and Connectivity of Discrete Complex Systems

Synchronization and Connectivity of Discrete Complex Systems. Michael Holroyd. The neural mechanisms of breathing in mammals. Christopher A. Del Negro, Ph.D. John A. Hayes, M.S. Ryland W. Pace, B.S. Dept. of Applied Science The College of William and Mary

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Synchronization and Connectivity of Discrete Complex Systems

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  1. Synchronization and Connectivity ofDiscrete Complex Systems Michael Holroyd

  2. The neural mechanisms of breathing in mammals Christopher A. Del Negro, Ph.D. John A. Hayes, M.S. Ryland W. Pace, B.S. Dept. of Applied Science The College of William and Mary Del Negro, Morgado-Valle, Mackay, Pace, Crowder, and Feldman. The Journal of Neuroscience 25, 446-453, 2005. Feldman and Del Negro. Nature Reviews Neuroscience, In press, 2006.

  3. Networks Cells Molecules Neural basis for behavior Behavior Networks Networks Cells Molecules Genes

  4. In vitro breathing Neonatal rodent 500 µm Smith et al. J.Neurophysiol. 1990

  5. PreBötzinger Complex In vitro breathing

  6. Experimental Preparation

  7. Questions • What does the PreBötzinger Complex network look like? • What type of networks are best at synchronizing?

  8. Laplacian Matrix • Laplacian = Degree – Adjacency matrix • Positive semi-definite matrix • All eigenvalues are real numbers greater than or equal to 0.

  9. Algebraic Connectivity • λ1 = 0 is always an eigenvalue of a Laplacian matrix • λ2 is called the algebraic connectivity, and is a good measure of synchronizability. Despite having the same degree sequence, the graph on the left seems weakly connected. On the left λ2 = 0.238 and on the right λ2 = 0.925

  10. Geometric graphs Construction: Place nodes at random locations inside the unit circle, and connect any nodes within a radius r of each other.

  11. λ2 of Poisson random graphs

  12. λ2 of preferential attachment graphs

  13. λ2 of geometric graphs

  14. Degree preserving rewiring A C A C B D B D This allows us to sample from the set of graphs with the same degree sequence.

  15. Scale-free metric -- s(G) • First defined by Li et. al. in Towards a Theory of Scale-free Graphs • Graphs with low s(G) are scale-free, while graphs with high s(G) are scale-rich.

  16. λ2 vs. s(G)

  17. λ2 vs. clustering coefficient

  18. Back to the PreBötzinger Complex • Using a simulation of the PreBötzinger Complex, we can simulate networks with different λ2 values.

  19. Synchronizability • Neuron output from PreBötzinger complex simulation. Synchronization when λ2=0.024913 (left) is relatively poor compared to λ2=0.97452 (right).

  20. Correlation analysis • Closer values of λ2 can be difficult to distinguish from a raster plot.

  21. Autocorrelation analysis Autocorrelation analysis confirms that the higher λ2 network displays better synchronization.

  22. Further work • Find a physical network characteristic associated with high algebraic connectivity. • Maximal shortest path looks like a good candidate:

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