  A binary state on a graph means an assignment of binary values to its
vertices. For example, if one encodes a network of spiking neurons as a
directed graph, then the spikes produced by the neurons at an instant of time
is a binary state on the encoding graph. Allowing time to vary and recording
the spiking patterns of the neurons in the network produces an example of
binary dynamics on the encoding graph, namely a one-parameter family of binary
states on it. The central object of study in this article is the closed
neighbourhood of a vertex $v$ in a graph $\mathcal{G}$, namely the subgraph of
$\mathcal{G}$ that is induced by $v$ and all its neighbours in $\mathcal{G}$.
We present a topological/graph theoretic method for extracting information out
of binary dynamics on a graph, based on a selection of a relatively small
number of vertices and their neighbourhoods. As a test case we demonstrate an
application of the method to binary dynamics that arises from sample activity
on the Blue Brain Project reconstruction of cortical tissue of a rat.
