1. PAGERANKING THE BRAIN
Cortical Parcellation using Clustering
Niels van Galen Last

2. Brain Basics
Defined at micro, mesa and
macro level.
Functional regions at macro
level.
Figure 12: Functional regions of the left hemisphere of the cerebral cortex.
(Essentials of Anatomy & Physiology, Seeley et al. p. 210.)

3. PageRanking the Brain
Idea from Kaustubh Supekar.
Problem: PageRanking and undirected graphs.

4. Granger Causality Analysis
Granger Causality Analysis 232 A. Roebroeck et al. / NeuroImage 25 (2005) 230–242
on fMRI data.
Signals as zero-mean vector
time series x[n] = (x1[n], . . . , xM[n])T
modeled as a vector
autoregressive (VAR)p
process: -x[n] = A[i]x[n − i] + u[n]
Fig. 1. A schematic illustration of the procedure to generate simulated time series (in the leftmost column), examples of the generated series at various stages (in
the middle column), and of resulting distributions of computed influence values for 5000 simulations (in the rightmost column). The top row depicts the
A. Roebroeck et al. / Neuro Image 25 (2005) 230-242
generation of simulated local field potential (LFP) signals of X and Y at high temporal resolution. The simulation model implements a temporally directed
influence from X to Y. The middle row represents the filtering of the LFP signals through a canonical hemodynamic response model to obtain simulated blood
i=1 oxygenation level dependent (BOLD) signals. The bottom row shows how a temporal down-sampling of the BOLD signals then gives the simulated fMRI
signal. Influence measures F x Yy, F y Yx , and F x d y can be computed from the generated time series at all three stages. If the simulation is repeated many times
(e.g., 5000), distributions of the influence measures can be obtained. These are shown in the rightmost column, where the distributions of F x Yy values is shown
in blue, F y Yx distributions are shown in green and F x d y distributions are shown in red. The set of distributions for the simulated fMRI signal (in the red box) is
of most interest in these investigations.
The signals x[n] and y[n] of two interacting neuronal The time-step of the simulation was taken to be 10 ms. In
populations X and Y were generated as a realization of a bi- every simulation, the model was simulated for 10,000 time-steps
dimensional first-order VAR process with: (100 s), where additionally an initial 2000 + D time-steps were
simulated and later discarded to allow the system to enter a steady
À 0:9 0 1 0 state, to introduce the delay D and to avoid boundary effects in
A½1Š ¼ ;Æ ¼
I À 0:9 0 1 subsequent filtering. After simulation and introduction of addi-
tional delay, the channels were individually filtered by convolu-

5. Granger Causality Analysis
p
x[n] = − A[i]x[n − i] + u[n]
i=1
A[i] are the autoregression coefficients, regressing x[n] into its
past. u[n] is (multivariate) white noise.
Linear Prediction model!
So xi [n] is predicted based on combination of past values and
components.
Uses temporal precedence to identify the direction.

6. Granger Causality Analysis
p
x[n] = − A[i]x[n − i] + u[n]
i=1
fMRI is voxel-by-time-by-person.
Given x[n] and y[n] we try to identify the influence.
Fx,y = Fx→y + Fy→x + Fx.y
Fx.y denotes the improvement
Problem: Spurious connectivity!

7. Ranking the Brains
Directed graph from the Neuroscience Institute, SD.
G(10000, 957853)
Eigenspectra analysis

8. Ranking the Brains (2)
Strange eigenvalues!
To be continued....

9. Clustering the Brain
Functional regions at macro
level.
Problem: what are the
locations of the regions?
Exploring clustering Figure 12: Functional regions of the left hemisphere of the cerebral cortex.
(Essentials of Anatomy Physiology, Seeley et al. p. 210.)

10. Clustering the Brain (2)
Only five papers on this topic
identifying 2 to 7 regions.
Explored:
kmeans
normalized cut group
clustering (2)
Ideas:
kmeans++
Figure 12: Functional regions of the left hemisphere of the cerebral cortex.
Max-weight Subgraph (Essentials of Anatomy Physiology, Seeley et al. p. 210.)
Heat Kernel PageRank

11. Right now...
Investigating the PageRank
values.
Transforming fMRI data
into a graph.
Reading up on clustering
algorithms. Figure 12: Functional regions of the left hemisphere of the cerebral cortex.
(Essentials of Anatomy Physiology, Seeley et al. p. 210.)

12. THE END