Synchronization and Connectivity of Discrete Complex Systems

1
Synchronization and Connectivity ofDiscrete
Complex Systems

• Michael Holroyd

2
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.

3
Neural basis for behavior
Behavior
Networks
Networks
Cells
Molecules
Genes
4
In vitro breathing
Neonatal rodent
500 µm
Smith et al. J.Neurophysiol. 1990
5
In vitro breathing
6
Experimental Preparation
7
Questions

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

8
Laplacian Matrix

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

9
Algebraic Connectivity

• ?1 0 is always an eigenvalue of a Laplacian
  matrix
• ?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 ?2 0.238 and on the right ?2 0.925
10
Geometric graphs
Construction Place nodes at random locations
inside the unit circle, and connect any nodes
within a radius r of each other.
11
?2 of Poisson random graphs
12
?2 of preferential attachment graphs
13
?2 of geometric graphs
14
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.
15
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.

16
?2 vs. s(G)
17
?2 vs. clustering coefficient
18
Back to the PreBötzinger Complex

• Using a simulation of the PreBötzinger Complex,
  we can simulate networks with different ?2 values.

19
Synchronizability

• Neuron output from PreBötzinger complex
  simulation. Synchronization when ?20.024913
  (left) is relatively poor compared to ?20.97452
  (right).

20
Correlation analysis

• Closer values of ?2 can be difficult to
  distinguish from a raster plot.

21
Autocorrelation analysis
Autocorrelation analysis confirms that the higher
?2 network displays better synchronization.
22
Further work

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