Part of Advances in Neural Information Processing Systems 12 (NIPS 1999)

*Dmitri Chklovskii, Charles Stevens*

The complexity of cortical circuits may be characterized by the number of synapses per neuron. We study the dependence of complexity on the fraction of the cortical volume that is made up of "wire" (that is, ofaxons and dendrites), and find that complexity is maximized when wire takes up about 60% of the cortical volume. This prediction is in good agree(cid:173) ment with experimental observations. A consequence of our arguments is that any rearrangement of neurons that takes more wire would sacrifice computational power.

Wiring a brain presents formidable problems because of the extremely large number of con(cid:173) nections: a microliter of cortex contains approximately 105 neurons, 109 synapses, and 4 km ofaxons, with 60% of the cortical volume being taken up with "wire", half of this by axons and the other half by dendrites. [ 1] Each cortical neighborhood must have exactly the right balance of components; if too many cell bodies were present in a particular mm cube, for example, insufficient space would remain for the axons, dendrites and synapses. Here we ask "What fraction of the cortical volume should be wires (axons + dendrites)?" We ar(cid:173) gue that physiological properties ofaxons and dendrites dictate an optimal wire fraction of 0.6, just what is actually observed.

To calculate the optimal wire fraction, we start with a real cortical region containing a fixed number of neurons, a mm cube, for example, and imagine perturbing it by adding or sub(cid:173) tracting synapses and the axons and dendrites needed to support them. The rules for per(cid:173) turbing the cortical cube require that the existing circuit connections and function remain intact (except for what may have been removed in the perturbation), that no holes are cre(cid:173) ated, and that all added (or subtracted) synapses are typical of those present; as wire volume is added, the volume of the cube of course increases. The ratio of the number of synapses per neuron in the perturbed cortex to that in the real cortex is denoted by 8, a parameter we call the relative complexity. We require that the volume of non-wire components (cell bodies, blood vessels, glia, etc) is unchanged by our perturbation and use 4> to denote the volume fraction of the perturbed cortical region that is made up of wires (axons + dendrites; 4> can vary between zero and one), with the fraction for the real brain being 4>0. The relation between relative complexity 8 and wire volume fraction 4> is given by the equation (derived in Methods)

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