The emergence of connectivity in neuronal networks: from bootstrap percolation to auto-associative memory

We consider a random synaptic pruning in an initially highly interconnected network. It is proved that a random network can maintain a self-sustained activity level for some parameters. For such a set of parameters a pruning is constructed so that in the resulting network each neuron/node has almost equal numbers of in- and out-connections. It is also shown that the set of parameters which admits a self-sustained activity level is rather small within the whole space of possible parameters. It is pointed out here that the threshold of connectivity for an auto-associative memory in a Hopfield model on a random graph coincides with the threshold for the bootstrap percolation on the same random graph. It is argued that this coincidence reflects the relations between the auto-associative memory mechanism and the properties of the underlying random network structure. This article is part of a Special Issue entitled "Neural Coding".