Generally, components of a system can deviate from optimality at different rates. To visualize this, think of a two component system, with x1 and x2. Imagine that x1 has a higher probability of being in a non-optimal state, or in other words, has a more slowly decreasing objective function:

on the left the region of high prob is wider for x1 because the objective function decreases more slowly, on the right are contour plots, so the lines have equal value; doi: 10.1073/pnas.0905336106

Perez-Escudero et al (’09) were interested in the deviations from the minimum wiring configuration in the current connectome of C. elegans. Their assumption for optimality is that neurons should be in positions that minimize the cost of the “edge” between them. This is their objective function.

First they calculate the deviation of each neuron’s position from its position in the theoretical minimum wiring config. Then they show that neurons with fewer wires or “connections” to other neurons tend to have smaller deviations. This makes sense because the cost of their deviation from optimality is lower.

A = neuron positions on the line indicate no deviation from optimality, B = blue line is an inverse quadratic fit, indicating that deviations from optimality have an parabolic cost w/r/t number of connections, C = random redistribution of the deviations of neuron positions from optimum, note only 0.033% of permutations have a lower cost ; doi: 10.1073/pnas.0905336106

They say that ~ 15% of C. elegans neurons have significant deviations from optimality. Additional analysis reveals that some of the neurons deviate from optimality due to local minima in the cost of wiring, which is a common tendency in evolved systems. This analysis is very interesting, and one of the reasons it is able to be done is because the connectome of C. elegans has been partially solved.

**Reference**

Perez-Escudero A, et al. 2009 Structure of deviations from optimality in biological systems, PNAS, doi: 10.1073/pnas.0905336106.

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