Transcript The C. elegans neural network

Pattern Retrieval Performance and Role of
Wiring Cost
in the evolution of
C. elegans neural network
Yong-Yeol Ahn, Beom Jun Kim, Hawoong Jeong
Caenorhabditis elegans
• It’s a transparent nematode.
• All C. elegans have same neurons and
synapses.
• We know all of them!
Putting a “Neural Network Model” on a
“Neural Network”
The ability to recognize patterns may be
crucial for surviving and mating
• Let’s try Hopfield model on C. elegans
neural network. (Beom Jun Kim, 2004)
Hopfield Model
Neurons can have two states. At each time step, each neuron’s
state is determined by other neurons which have links to it.
+1
2
-1
-3
+1
1
Ex)
 (t )  2  (1)  3  (1)  1 (1)  1
Performance of Hopfield model
(Pattern recognition performance)
1. Generate several random patterns.
2. Teach the neural network with these patterns.
3. Make test patterns which are slightly different
from the one of the learned patterns.
4. Give the test patterns to the neural network
and calculate the overlap between the test
pattern and the learned pattern.
The pattern retrieval
performance of C. elegans
Performance m: overlap
fraction between original
pattern and retrieved pattern.
• C. elegans shows more
poor performance than BA
model and WS(p=1.0) model.
• Clustering coefficient
determines the performance
of network (under degree
conserving rewiring).
(clustering coefficient)
Beom Jun Kim 2004
What’s the problem?
• If we assume that the Hopfield model is an
appropriate model for measuring the neural
network’s performance,
• Then some other constraint limits the
performance of C. elegans neural network.
• Possible constraint is ‘wiring cost’.
– Assume that the wiring cost of a connection between
two neurons is equal to the Euclidean distance between
them.
– We can find a neuron’s geometric position at
http://wormatlas.org .
The C. elegans neural network
Side view
Front view
(Drawn by pajek)
Neurons I
Neurons II
Neurons III
General informations
•
•
•
•
•
•
# of nodes : 279
# of links(undirected) : 1963
# of links(directed) : 2170
Clustering coeff. : 0.28
Closeness : 0.422
Maximum degree(undirected) : 74
Degree distribution
-3
Clustering distribution
-0.5
knn
Two methods
Node replacement
(conserving topology)
Degree
conserving
rewiring
Distribution of distance(cost)
between two neurons
• There exist large
number of very
long-range
wirings
Distribution of distance after
node position optimized
Distance distributions
Randomly shuffled network
• Node replacement
– Cost: 588.9
• Degree conserving rewiring
– Cost: 556.1
– Clustering coeff. : 0.12
The wiring cost is playing important role in
the C. elegans neural network?
We shuffled the network randomly using above node
replacement method & (degree conserving) link
rewiring method
C. Elegans network’s cost
Randomly shuffled network’s cost ~
0.66
 Cost does matter.
Then, are C. elegans neurons
“optimized” by wiring cost?
• Using node replacement optimization method, we minimize
C.elegans neural network’s cost.
Node replacement
optimization
(conserving topology)
 Original C. Elegans network’s cost: 367.1
 Position optimized network’s cost: 199.7
• Is C. elegans neural network optimized by cost?
 Not really
Then, are C. elegans neurons
“optimized” by wiring cost?
Original: 367.1
• Using degree conserving rewiring,
– Optimized cost: 65.4
• Using node replacement,
– Optimized cost: 199.7
High performance network vs.
Poor performance network
• Using degree conserving rewiring, we can make highly
clustered network and poorly clustered network from original
C. elegans network
Degree
conserving
rewiring
High cc
C. elegans
Low cc
clustering
0.70
0.28
0.00
Cost(npo)
163.6
199.7(367.1)
291.5
performance
0.69
0.79
0.83
(npo: node position optimized)
Best performance
High performance network vs.
Poor performance network
High cc
C. elegans
Low cc
clustering
0.70
0.28
0.00
Cost(npo)
163.6
199.7(367.1)
291.5
performance
0.69
0.79
0.83
Clustering
Clustering
 performance
 performance
 low cost limit
 low cost limit
Another candidates?
 Ganglia structure (module)
• Neurons aggregate and make ‘ganglia’
• Let’s assume that connection
between ganglia can’t be modified
and the neurons in one ganglion are
optimized to show high performance,
to reduce cost.
In Ganglia
Ganglia
Cost
(node-position)
optimized cost
Anterior
0.79
0.53
Lateral-ventral
6.41
3.41
Retro-vesicular
0.61
0.36
Pre-Anal
0.15
0.08
Lumbar
0.11
0.07
 Neurons in a ganglia do not show
the evidence of cost optimization. : (
Euclidean distance vs. network
distance
• Mathias & Gopal, ‘Small world: how
and why’ : ‘neural network는 이 두 가
지의 절충일 것이다.’
• pD + (1-p)d 로 fitness를 놓고 p를 0에
서 1까지 변화시키면서 optimize 해봤
으나 celegans는 optimize되어있지 않
다.
Distance vs. distance
ce
ce shuffled
Optimized by
only diameter
Optimized by
only cost
Diameter
2.63
2.61
2.06
3.15
cost
367.1
387.1
558.1
51.3
p를 적당히 정하고 optimize 하면 언제나 fitness가 ce
보다 좋은 네트워크를 만들 수 있다.
Too many long range connections
• There are too many long range
connections
• Cut edges which are longer than
some threshold, then try optimizing
node position
– 특별한 결과 없음.
Too many long range connections
• Cost = D^{alpha} ?
• Cost  p D^{alpha} + (1p)d^{beta} ?
Better performance measure?
Conclusion
• We constructu the C. elegans neural network with
geometrical information
• Cherniak’s remark which state that ganglia position are
optimized for low cost isn’t true anymore in neuronal scale.
• Under the C. elegans neuron position topology, the higher
clustering coefficient, the smaller the cost.
• But, C. elegans neural network is not optimized to have
minimal cost.
• C. elegans neural network is small, and specific. We show
that cost, performance(Hopfield model) are not the central
organizing principle of C. elegans neural network.
• What is the design principle of C. elegans neural network?
 Still open question.