Yong-Yeol Ahn, Beom Jun Kim , Hawoong Jeong

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”

• Let’s try Hopfield model on C. elegans neural network. (Beom Jun Kim, 2004)

The ability to recognize patterns may be crucial for surviving and mating

Hopfield Model

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Neurons can have two states. At each time step, each neuron’s state is determined by other neurons which have links to it.

Ex) 1)1(1)1(3)1(2)( t

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).

Beom Jun Kim 2004

(clustering coefficient)

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

(Drawn by pajek)

Side view Front view

Distribution of distance(cost) between two neurons

• There exist large number of very long-range wirings (power-law like decay)

Is C. elegans neural network optimized by wiring cost?

• Using node replacement optimization method, we minimize C.elegans neural network’s cost.

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!! : (

Node replacement optimization (conserving topology)

Distance distribution of cost optimized network

Original network Node position optimized network

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

High cc C. elegans Low cc

clustering 0.70 0.28 0.00Cost(npo) 163.6 199.7(367.1) 291.5performance

0.69 0.79 0.83

(npo: node position optimized)

Degree conserving rewiring

Best performance

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 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

Lumber 0.11 0.07

Neurons in a ganglia do not show the evidence of cost optimization! : (

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.