# Synchronization and Connectivity of Discrete Complex Systems Michael Holroyd.

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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 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.

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

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

In vitro breathing PreBötzinger Complex

Experimental Preparation

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

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

Algebraic Connectivity λ 1 = 0 is always an eigenvalue of a Laplacian matrix λ 2 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 λ 2 = 0.238 and on the right λ 2 = 0.925

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

λ 2 of Poisson random graphs

λ 2 of preferential attachment graphs

λ 2 of geometric graphs

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

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.

λ 2 vs. s(G)

λ 2 vs. clustering coefficient

Back to the PreBötzinger Complex Using a simulation of the PreBötzinger Complex, we can simulate networks with different λ 2 values.

Synchronizability Neuron output from PreBötzinger complex simulation. Synchronization when λ 2 =0.024913 (left) is relatively poor compared to λ 2 =0.97452 (right).

Correlation analysis Closer values of λ 2 can be difficult to distinguish from a raster plot.

Autocorrelation analysis Autocorrelation analysis confirms that the higher λ 2 network displays better synchronization.

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