Modeling functional networks of
the primate cortex
With:
Maria Ercsey-Ravasz
Department of Physics &
Interdisciplinary Center for Network Science and Applications (iCeNSA)
University of Notre Dame
Henry Kennedy, Kenneth Knoblauch,
Nikola Markov, Marie-Alice Gariel, etc.
Stem Cell and Brain Research Institute, INSERM U846 LYON
Markov N, Misery P, Falchier A, Lamy C, Vezoli J, Quilodran R, Giroud P, Gariel MA, Ercsey-Ravasz M, Pilaz LJ, Huissoud C, Barone P, Dehay C,
Toroczkai Z, Van Essen DC, Kennedy H, Knoblauch K. Area-specific visual cortical connectivity profiles revealed by quantitative brain-wide
analysis. Cereb Cortex, in press. (2010)
Markov N, Ercsey-Ravasz MM, Gariel MA, Falchier A, Quilodran R, Vezoli J, Lamy C, Misery P, Giroud P, Clavagnie S, Sallet J, Barone P, Dehay C,
Toroczkai Z, Kennedy H, Knoblauch K. Heterogeneity of connection strengths and a distance rule specify cortical networks. subm. (2010)
How does the brain work?
Understanding the brain would greatly benefit:  Medicine  Computing, science, industry
Artificial Intelligence (AI) started out with huge optimism:
 1965: H.A. Simon: “Machines will be capable, within twenty years, of doing any work a man can do.”
 1970: M. Minsky: “In from three to eight years we will have a machine with the general intelligence of an
average human being”
 1980’s: Progress stalled
 2005: Jeff Hawkins: “Neural network research ignores essential properties of the human cortex”
AI tries to get around real brain research, led by the belief that computing paradigms are not so
many, and that they could be “guessed”.
Nature already figured
Well, the brain is not this:
out efficient computing:
- 0.09g (green lizard)…
1400g (human)…
7800g (sperm whale).
 Get clues from brain
experiments?!
So what’s taking so long?
It is this “goo”:
Networks of the real world
Networks are physical systems that can be represented as graphs with processes on them.
Where are networks?
• Infrastructures: transportation nw-s (airports, highways, roads, rail,
water) energy transport nw-s (electric power, petroleum, natural gas)
• Communications: telephone, microwave backbone, internet, email,
www, etc.
• Biology: brain, gene regulatory network, protein-protein interactions,
metabolism, cell-signaling, the food web, etc.
• Social systems: acquaintance (friendship) nw-s, collaboration networks,
epidemic networks
• Geology: river networks
Real-world networks perform functions and are
characterized by heterogeneous structures
This typically, already shows up at the level of connectivity (degree) distributions:
Binomial Random graph
(Erdős-Rényi graph)
Poisson
Capacity achieving degree distribution of
Tornado code. The decay exponent -2.02.
[ M. Luby, et.al. ,Proc. 29th ACM Symp.
Theor. Comp. pg. 150 (1997). ]
networks in biology
the metabolic pathway
Chemicals
Bio-Chemical reactions
Cellular Networks: The Bio-Map
GENOME
Protein-gene
interactions
PROTEOME
Protein-Protein
interactions
METABOLISM
Bio-chemical
reactions
Citrate Cycle
social networks
person
relationship
R. S. Burt (2000) “Network structure of social capital”
Networks can be dynamic and evolving.
Visualization by Aaron Koblin, http://www.aaronkoblin.com/info.html
Human built networks are purposely functional, and they seem to be heterogeneous, as it was seen in the
case of the Internet and WWW. More examples:
Electronic circuits and devices:
Software (subroutine is a node, a call is a link):
N=104 element logic
circuits (iscas89, itc99).
R. F. I Cancho, C. Janssen and R.V. Sole. Phys. Rev. E
64, 046119 (2001).
S. Valverde and R.V. Sole. Europhys. Lett. 72, 858 (2005).
Biological networks are nature-built, via evolutionary
processes, and they are clearly functional.
Archae
a
Bacteria
Bacteria
Eukaryote
s
H. Jeong et.al. Nature 407, 651 (2000).
Eukaryote
s
The cerebral cortex
The neuron
(brainmaps.org)
An adult primate brain contains about 100 billion (1011)
neurons, with branches that connect at more than 100
trillion (1014) points.
The network of ~1011 neurons and ~1014 synapses is too large to be genetically encoded synapse
by synapse.
 Has randomness, modified during learning: Information encoding emerges at larger scales.
 Statistical description should provide reproducible measures.
 The best method is functional decomposition: a TOP-DOWN approach
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Functional areas of the cortex
Segmentation based on histological criteria. Cytoarchitectonic details.
Macaque cortex: 83 areas
11
Experiments: Retrograde Tracing
Kennedy team (Lyon):
(70 man-years work!)
Retrograde tracing uncovers inter-areal
connectivity.
Tracers:
Fast Blue (FsB), Diamino Yellow (Dy)
A neuron is a directed element
“Target”
NO secondary labeling (of pre-synaptic neurons)!
Measurements:
- number of labeled neurons in source areas
- projection lengths through the white matter
An injection reveals the FULL set of INCOMING
projections.
Macaque cortex: 26 areas
injected:
V1, V2, V4, TEO, DP; PBr, STPr, STPi,
STPc, 7A, 7B, 2, 5, F1, F2, F5, F7,
24c, ProM, 46d, 9/46d, 9/46V, 8A,
8B, 45B
?
?
“Sources”
“Target”
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Injections sites in the macaque cortex
Low-power fluorescence photomicrographs
obj X10 filter
Injection in V1, horiz section plane
Injection in V2, coronal section plane
Injection in V4, coronal section plane
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40 μm histological
sections
14
Fraction of Labeled Neurons (FLN)
Assume that an injection was performed in area X.
Define:
FLN(X  Y ) 
NY
Ne
- the “probability” that an extrinsic labeled neuron is projecting into
X from Y.

The FLN values are highly characteristic across animals:
FLN
• Varies over 6 orders of
magnitude!
• This heterogeneity is a
signature of functional
organization.
Distribution by reach
~79% within 2mm (local)
~16% to neighboring areas
~4% long-range
~1% sub-cortical
Inter-areal
network
The FLN distribution
Close to a lognormal
(f=FLN)
if
    2
with
Strong heterogeneity in weight distribution.
log10(FLN)
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Why Lognormal?
I. Projection
lengths:
Exponential Distance Rule (EDR):
p(d)   exp(d)
  0.088mm1
1 11.36mm
 This can be interpreted as the probability of a single neuronal projection having length d.
 this is an average physical property across the whole cortex:
We assume that
FLN( j i)  f ij  p(dij )   exp( dij )
ln f ij  dij  ln 

Thus, on average, distances are in linear relationship with the
corresponding ln(FLN) values.
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II. Distribution of inter-areal distances
(Geometry)
This is the fraction of area pairs i,j separated by a given distance d.
Expectation: unimodal distribution.
Fitted by a (truncated) Gaussian:
Substituting
( f ) 


d  1 ln f  1 ln 
from the exponential distance rule:
 1
 1
2 
1

2 
exp  2 1 ln f  1 ln    
exp  2 ln f    
 2
  2
 2
 2

  ln   
  
lognormal
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Others have also found lognormals in the
brain:
Distance rule locally:
N(d)  N0 exp(sd)
s: “Song” decay rate
(mm-1)
Kennedy measurements, intrinsic
labeling, distance distribution.
22% loss at d1 = 50μm = 0.05mm  78% are not lost at
this distance 
N(d1)/N0  0.78  exp(sd
1)
s  4.96mm 1
  4.35mm1
 d2 = 100μm = 0.1mm  56% are not lost at
44% loss at
this distance 

s  5.79mm 1

Network properties
• directed
• weighted
• spatially embedded
?
- 26 injected nodes
- all 83 nodes projecting into the set of 26
- found total of E=1232 directed edges/projections
- there are M=430 edges between the 26 nodes
The graph amongst the 26 nodes forms
which is an edge-completed sub-graph of
Density of
G26x26 :
 2626
?
G26x26
G26x26
G83x83
G83x83
430
430


 0.66..
26  25 650
In-degree distrib. of the 26 nodes within
G83x83
It is very dense! 66%
At such high densities,
the density of the graph is the most

determining factor for its “binary” properties:
Necessarily it will have:
-short average path length ( = 1.34 for G26x26)
-small diameter ( = 2)
-high clustering coefficients (= 0.87).
SOUNDS LIKE SMALL-WORLD…
But it really isn’t… too dense.
Neither is
Worst case:
1232/83(83-1) = 0.181 (18%). Extrapolating the same mean: 57%
G83x83
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Two Random Graph models:
Constant distance rule (CDR): p(d) = constant.
Exponential distance rule (EDR): p(d) = λexp(λd).
Using the set of distances D measured between the areas:
- choose a connection length d from D according to the distance rule p(d)
- pick uniformly at random an area pair (i,j) whose separation is d (same bin)
- insert a connection in the graph directed from j to i
- multiple connections from j to i are allowed, thus generating its weight
- the process stops once all the 430 binary connections are reached
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Degree and 3-motif distributions
EDR clearly outperforms CDR
A non-local, binary property:
Average (binary) shortest path length as function of
density, while sequentially removing the weakest links.
A non-local weighted property:
Communication capacity:
ln pk  i j  ln f k  i  ln f i j .
j
k
pk  i j  f k  i f i j ,
 
 positive, additive
along paths
i
Link “resistance”:wij  ln f ij  ln 1/ f ij  0
rij = lowest resistance path weight (sum of the w-s along paths).

Global “conductance”:

Eg 
1
1

N(N 1) i j rij
V Latora, M Marchiori, Europhys J B (2003)

Local “conductance”:
El 
1
1

N i ki (ki 1)

j k{i}
1
rjk / i
I Vragovic, E Louis, A Diaz-Guilera, PRE (2005)
High bandwidth backbone:
(Kamada-Kawai algorithm used for layout)
Unweighted network
Weighted network
visual
prefrontal
parietal
temporal
sensori-motor
80 strongest connections
just before un-reachability
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Conclusions – so far
• Massive involvement in local circuitry.
• Link strengths are consistent across individuals
• Range of strengths varies over 6 orders of magnitude
• The interareal cortical network has high density.
• Binary features are not very informative.
• Weights, distances and geometry are important
• Exponential distance rule: strong economy of lengths.
• An EDR based simple model captures binary properties
• Fluctuations around the EDR encodes bio-specificity:
e.g., new projections of PERI onto amygdala support recent theories of
inference based on memory prediction (Hawkins et.al. 2009).
• There is direct access between almost any two areas,
likely ensuring efficient high-level integration
• Massive processing of information happens locally.
• To block out cacophony, information is filtered/encoded
using hierarchies of weights (6 orders)
• Weight/geometry - based hierarchical layering in cortical
architecture and processing.
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Plasticity of the cortex:
Brainport: http://www.pbs.org/kcet/wiredscience/video/286-mixed_feelings.html
Rewiring ferret’s brain at birth: L. Melchner, S.L. Pallas & M.Sur. Nature, 404, 871 (2000)
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