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 ScienceThe 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 breathingNeonatal

rodent

Smith et al. J.Neurophysiol. 1990

500 µm

In vitro breathing

PreBötzingerComplex

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.

nk

k

k

}1,0{

}1,0{

2

1

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

Geometric graphs

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

λ2 of Poisson random graphs

λ2 of preferential attachment graphs

λ2 of geometric graphs

Degree preserving rewiring

A

B D

C A

B D

C

This allows us to sample from the set of graphs with the same degree sequence.

Scale-free metric -- s(G)

Eji

ji kkGs),(

)()(

•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.

λ2 vs. s(G)

λ2 vs. clustering coefficient

Back to the PreBötzinger Complex

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

Synchronizability

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

Correlation analysis

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

Autocorrelation analysis

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

Further work

• Find a physical network characteristic associated with high algebraic connectivity.

• Maximal shortest path looks like a good candidate: