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
In vitro breathing
Neonatal
rodent
500 µm
Smith et al. J.Neurophysiol. 1990
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
{0,1}
 k1


k
2







kn 
{0,1}
• 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.
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
C
A
C
B
D
B
D
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
s (G ) 
 (k  k
( i , j )E
i
j
)
•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: