Presentation on theme: "Synchronization and Connectivity of Discrete Complex Systems Michael Holroyd."— Presentation transcript:
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.
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.