Have you ever wondered what determines the sizes of dendritic and axonal arbors? Why is it that sometimes dendritic arbors are wider than axonal (e.g. projection from bipolar to ganglion cells in the retina) and sometimes vice versa (e.g. neocortical pyramidal cells and parallel fiber to Purkinje cell projection in the cerebellum)?

Illustration from Neuroscience by Purves et al.

We suggest that the sizes of axonal and dendritic arbors are chosen to minimize wiring volume. Here is a simple example to illustrate this idea. Suppose you need to wire up a topographic projection between two layers of neurons - circles and squares in (A) below. Two arrangements that implement this wiring diagram with realistic neurons are shown below (B, C). These two arrangements have the same electrical properties (barring non-linear interactions in dendrites) but differ in the placement of synapses. Arrangement (B) is preferred because it has shorter total length of wires (i.e. axons + dendrites) than arrangement (C).Therefore, topographic projections from many neurons to few should have wide dendritic and narrow axonal arbors. In projections from few neurons to many the opposite should be true: wide axonal arbors should synapse onto narrow dendritic arbors.

Predictions of this theory agree with the data on retinal, cerebellar, olfactory, and cortical neurons, for which connectivity and morphology are known. We suggest to use this theory to infer neuronal connectivity from their morphology. You can find more details about this theory here: Optimal sizes of dendritic and axonal arbors in a topographic projection by Dmitri B. Chklovskii. Journal of Neurophysiology 83: (4) 2113-2119 (2000).

Ocular dominance patterns are found in many mammals with significant binocular overlap. Here is an example:

Why should ocular dominance patterns have the appearance they do? Wiring economy principle can help answer this question. Here is a two-part PowerPoint presentation on the subject. Below is a brief summary.

We find that ocular dominance patterns minimize wire length provided each neuron makes more connections with the same eye neurons than opposite eye neurons. The local structure of the patterns depends on the parameters of neuronal circuitry. In particular, we predict a transition from stripy to patchy pattern when the fraction of minority neurons is 40%. This prediction is consistent with the data from macaque and Cebus monkeys.

You can find out more details about this theory here: A wire length minimization approach to ocular dominance patterns in mammalian visual cortex by Dmitri B. Chklovskii and Alexei A. Koulakov. Physica A 284: (1-4) 318-334 (2000).

In the stripy ocular dominance pattern, what determines stripe orientation? I propose that the stripe orientation is such that, when the stripes are mapped back onto the retina, stripes are aligned with the locally dominant binocular disparity direction. This prediction is consistent with the data from macaque and Cebus monkeys.

You can find more details about this theory here: Binocular disparity can explain the orientation of ocular dominance stripes in primate primary visual by Dmitri B. Chklovskii. Vision Research 40: (13) 1765-1773 (2000).

Another map present in the visual cortex is the orientation preference pattern. This pattern often contains singularities called pinwheels and fractures.

Why do these singularities exist? Why are there regions (left side of the map) with no singularities? We suggest that the map appearance reflects the statistics of intra-cortical connectivity according to the wiring economy principle. Here is a brief PowerPoint presentation on the subject.

You can find more details about this theory here: Orientation preference maps in mammalian visual cortex: A wire length minimization approach by Alexei A. Koulakov and Dmitri B. Chklovskii. Neuron 29: (2) 519-527 (2001).