  Qualitatively, some real networks in the brain could be characterized as
'small worlds', in the sense that the structure of their connections is
intermediate between the extremes of an orderly geometric arrangement and of a
geometry-independent random mesh. Small worlds can be defined more precisely in
terms of their mean path length and clustering coefficient; but is such a
precise description useful to better understand how the type of connectivity
affects memory retrieval?
  We have simulated an autoassociative memory network of integrate-and-fire
units, positioned on a ring, with the network connectivity varied
parametrically between ordered and random. We find that the network retrieves
when the connectivity is close to random, and displays the characteristic
behavior of ordered nets (localized 'bumps' of activity) when the connectivity
is close to ordered. Recent analytical work shows that these two behaviours can
coexist in a network of simple threshold-linear units, leading to localized
retrieval states. We find that they tend to be mutually exclusive behaviours,
however, with our integrate-and-fire units. Moreover, the transition between
the two occurs for values of the connectivity parameter which are not simply
related to the notion of small worlds.
