Synchronization and Connectivity of Discrete Complex Systems同步与离散复杂系统的连接.pptVIP

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Synchronization and Connectivity of Discrete Complex Systems同步与离散复杂系统的连接.ppt

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Synchronization and Connectivity of Discrete Complex Systems同步与离散复杂系统的连接

Synchronization and Connectivity ofDiscrete 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 In vitro breathing In vitro breathing Experimental Preparation Questions What does the PreB?tzinger 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 λ1 = 0 is always an eigenvalue of a Laplacian matrix λ2 is called the algebraic connectivity, and is a good measure of synchronizability. Geometric graphs λ2 of Poisson random graphs λ2 of preferential attachment graphs λ2 of geometric graphs Degree preserving rewiring Scale-free metric -- s(G) λ2 vs. s(G) λ2 vs. clustering coefficient Back to the PreB?tzinger Complex Using a simulation of the PreB?tzinger Complex, we can simulate networks with different λ2 values. Synchronizability Correlation analysis Autocorrelation analysis Further work Find a physical network characteristic associated with high algebraic connectivity. Maximal shortest path looks like a good candidate: Now with that out of the way, let me introduce a type of graph which you might not be familiar with. [Describe construction] This model is appropriate for networks that take place in physical space, where connections are expensive or difficult at long distances, and so might be appropriate for modeling neuron networks. So let me show you three quick plots for different types of graphs to give you a feel for how this measure reacts. You’ll see in all of these that increased average degree results in