Chapter 17 - Neural wiring optimization

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Abstract

Combinatorial network optimization theory concerns minimization of connection costs among interconnected components in systems such as electronic circuits. As an organization principle, similar wiring minimization can be observed at various levels of nervous systems, invertebrate and vertebrate, including primate, from placement of the entire brain in the body down to the subcellular level of neuron arbor geometry. In some cases, the minimization appears either perfect, or as good as can be detected with current methods. One question such best-of-all-possible-brains results raise is, what is the map of such optimization, does it have a distinct neural domain?

Introduction

Neuroconnectivity architecture sometimes shows virtually perfect network optimization, rather than just network satisficing. Long-range connections are a critically constrained resource in the brain, hence, there may be great selective pressure to optimize finely their deployment. The formalism of scarcity of interconnections is network optimization theory, which characterizes efficient use of limited connection resources. The field matured decades ago for microcircuit design, typically to minimize the total length of wire needed to make a given set of connections among components. For layout of neural components, such connection minimization has been reported for the nematode nervous system (Cherniak, 1994a), rat amygdala and olfactory cortex (Cherniak and Rodriguez-Esteban, 2010), cat sensory cortex, and macaque visual cortex (Cherniak et al., 2004). Corresponding arbor optimization also applies for some types of dendrites and axons (Cherniak et al., 1999). Results for more primitive nervous systems help fill in some of the evolutionary trajectory of neural optimization phenomena.

Such optimality contrasts with the familiar picture for biological design, of only moderately good engineering: for example, the first chapter of Descent of Man (Darwin, 1871) enumerated many instances of rudimentary structures in humans that are no longer in use (although the neural examples are in fact functional). Instead, it is almost as if neural connections had an unbounded cost. When this simple “save wire” idea is treated as a generative principle for nervous system organization, it turns out to have some applicability: To an extent, across evolutionary levels, wire-minimization yields brain structure.

A caveat is that, in general, network optimization problems are easy to state, but vastly computationally costly to solve exactly. These connection cost-minimization problems are a major hurdle of microcircuit design and are known to be NP-complete (nondeterministic polynomial time complete), that is, de facto intractable (Garey and Johnson, 1979). Computation costs of solving problems of comparatively small size typically grow exponentially, to cosmic scale: exactly solving some could consume more space and/or time than exists in the known Universe. The archetypal example of an NP-complete problem is Traveling Salesman: For a given set of points on a map, simply find the shortest roundtrip tour.

Section snippets

Neuron arbor optimization

The basic concept of an optimal tree is as follows: Given a set of loci in 3D space, find the minimum-cost tree that interconnects them, for instance, the set of interconnections of least total volume. If branches are allowed to join at points other than the given terminal loci (the “leaves” and “root”), the minimum tree is of the most economical type, a Steiner tree. If the synapse sites and origin of a dendrite or axon are treated in this way, optimization of the dendrite or axon can be

Component placement optimization

Another key problem in microcircuit design is component placement optimization (also characterized as a quadratic assignment problem). Given a set of interconnected components, find the location of the components on a 2D surface that minimizes total cost of connections (e.g., wirelength). A familiar example is siting of computer chips on a motherboard. Again, this concept seems to account for aspects of neuroanatomy at multiple hierarchical levels.

Why the brain is in the head is a one-component

Related optimization results

Some other recently reported instances of biological network optimization provide perspective on the above neural optimization cases. For example, an amoeboid organism, the plasmodium of the slime mold Physarum polycephalum, is capable of solving a maze, that is, not just finding some path across a labyrinth, but a shortest path through it to food sources (Nakagaki et al., 2000). Generating such a minimum-length solution is a network optimization feat for any simple creature. However, it should

Mapping neural optimization

Mechanisms of neural optimization are best understood against the background that the key problems of network optimization theory are NP-complete; hence, exact solutions in general are computationally intractable. For example, blind trial and error exhaustive search for the minimum-wiring layout of a 50-component system (such as all areas of a mammalian cerebral cortex hemisphere), even at a physically unrealistic rate of one layout per picosecond, would still require more than the age of the

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