We will demonstrate, through the use of the Ising model, that a phase transition can correspond to the propagation of transcategorial changes. Though we have limited our inquiry to the emergence of functional vocabulary in a population of users, we assume that transcategorial changes are reflected in the dynamics of synaptic structures as well.
Percolation is of particular interest because it is a process in which stable structures can emerge in a low connectivity environment. Consider an array whose sites can either be occupied or not. This array may be one or several dimensions. Sites may be occupied at random or according to some rules. One of which can be that there are neighbouring occupied sites. Percolation theory deals with the properties of clusters of occupied sites that form in this lattice. The word percolation reflects the possibility of having a cluster spreading over the whole lattice. Percolation usually demonstrates second-order phase transition in state as defined by the connectivity between sites. This state change is usually the result of a critical number of connections between clusters.
This is a relevant concept in optimizing resources.
For Calvin, most neural structure is the result of spatio-temporal cues which provide many stimuli for a high number of neurons to be entrained and connected in a specific synaptic pattern. We assume that this requires much synaptic space. But the question is, can neurons be entrained into patterns without the benefit of spatio-temporal cues, that is, in times in which connectivity is low and the possibility of entrainment seems unlikely?
A percolation problem is well illustrated by the 60's game Kerplunk!. In this game, marbles are suspended in a tube by many toothpicks running across the diameter of the tube. As the toothpicks are removed successively, the marbles may shift slightly. Eventually only one crucial point of support will remain. Once the crucial last toothpick is removed, marbles will start falling until the tube is empty. Marbles will follow, one after the other, a path traced from the architectural constraints imposed by the removal of the crucial toothpick.
Percolation is a second order phase transition. The point of our Kerplunk! example is to show how percolation can entrain a system in a low connectivity environment. Imagine that the toothpicks are attractors of sorts that are too weak individually in a high connectivity environment - many toothpicks - to influence the rate of firing of neurons around them (to direct the orientation and path of the marbles). In a low connectivity environment (few toothpicks), these attractors will be strong enough to entrain neurons around them, and those neurons will entrain the ones around them and so on. Soon enough a new structure will emerge. Now, because of the low connectivity, we assume that these structures may be stripped of many features that direct perceptual cues would usually provide. We suggest that some schematization is born from low connectivity. These are exploited in a strategy to maximize neural resources. This may apply to many kinds of abstractions especially those that do not seem to have any ties to spatio-temporal cues such as the grammatical features of language.