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I have heard and read about Peter's rule informally in the past, but never saw a formal definition or description.
Informally I have learned to understand Peter's rule as the assumed correlation between the amount of overlap between axons and dendrites and the post synaptic density in the synapse. Here is a sample of the search results in Google.
Is my interpretation correct? Is there a more rigorous, perhaps quantitative, definition of Peter's rule?
I think that your interpretation is correct, although Principles of Neural Science (Kandel et al.) doesn't mention Peter's rule (which is where I would have hoped to find a simple explanation).
According to Binzegger et al. (2004), the idea behind Peter's rule dates back to Peters and Feldman (1976) and Peters and Payne (1993), though the Rule per se may have been coined by Braitenberg and Schultz (1991; at which point it was still called Peters' rule). Binzegger et al. describe it as:
They assumed that axons simply connect in direct proportion to the occurrence of all synaptic targets in the neuropil. The rule can be extended to provide more precise estimates by taking into account the detailed morphometrics of the dendritic and axonal trees of the different neuronal types.
To paraphrase Binzegger et al., in the simplest form of Peter's rule, the number of synapses in a cortical layer are evenly distributed across the potential target neurons. So there would be a direct proportionality between synapses and targets.
The second sentence of the quote above is the meat of the Binzegger paper. They also present a mathematical model of a more complex and realistic variation of Peter's rule which takes into account different types of postsynaptic neurons and the fact that not all cells receive synapses.
Braitenberg V, Schüz A (1991) Peters' Rule and White's Exceptions. In: Anatomy of the Cortex, pp 109–112. Berlin: Springer.
Peters A, Feldman ML (1976) The projection of the lateral geniculate nucleus to area 17 of the rat cerebral cortex. I. General description. J Neurocytol 5:63–84.
Peters A, Payne BR (1993) Numerical relationships between geniculocortical afferents and pyramidal cell modules in cat primary visual cortex. Cereb Cortex 3:69–78.
Here is a definition taken from a Scholarpedia article on computational neuroanatomy:
The critical underlying question remains: how does the number of synapses between two neurons depend on the overlap between their dendritic and axonal trees? According to "Peters rule", synaptic contacts occur where dendrites and axons happen to be in apposition. These "potential synapses" are required but not sufficient for an actual synapse formation; and the expected number of connections between two neurons is proportional to the product of their dendritic and axonal trees densities (e.g. Peters et al., 91).
This seems to be saying that there are really two pieces to Peter's rule-- first that "synaptic contacts" (potential synapses) actually occur where axons and dendrites share space, and secondly that the proportion of these contacts that are real synapses is proportional to tree density.
