Wiring Cost in the Organization of a Biological Network Yong-Yeol Ahn*, Beom Jun Kim, Hawoong Jeong KAIST, Republic of Korea * email@example.com 2 Putting a “Neural Network Model” on a “Neural Network” The ability to recognize patterns must be crucial for surviving and mating • 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. -> The state vector falls into some attractors. +1 2 -1 -3 1 +1 Ex) (+1) x 2 + (-1) x (-3) + … Performance Measure • By the Hebb’s rule, the Hopfield network can learn several patterns. • After learning process, we try with the testing pattern which has some error in it. • Does this testing pattern end up to the learned pattern? How much is the overlap? 5 6 So, Why? • If we assume that the measure is good, Possible other constraint is ‘wiring cost’. • Energy consumption acts a crucial role in the operation of the brain. (A brain consumes more energy than heart, about 20% of all energy spent) • Ganglia order in C. elegans is the global minima. (Cherniak1994) The C. elegans neuronal map Assumptions: • Head neurons are placed in the virtual cylinder which wraps the pharynx. • Body neurons are placed just below the cuticle layer (because of the body cavity) Is C. elegans neural network optimized by wiring cost? Two variational methods Node Swapping (conserving topology) Edge Excange (conserving positions and degrees of all neurons) Optimized Network vs. Random Network The C. elegans network is far from random nework If we neglect the body-spanning links, It approaches toward the optimal network. Cumulative Distribution of distance(cost) between two neurons • Exponential Decay • Three length scales • Body spanning Links (like spinal cord) Is Clustering Coefficient Playing an Important Role? • We equilibrate the network by rewiring randomly around given cost. • C. elegans has much bigger c.c. than this randomly rewired networks. 12 Conclusion • We constructu the C. elegans neural network with geometrical information. • The distribution of distances between neurons in C. elegans neural network follows piecewise exponential decaying funtion. • Although the neural network is not optimal, it is far from random. • The performance measure (Hopfield model) does not play a crucial role in the organization of C. elegans neural network.
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