An application of neighbourhoods in digraphs to the classification of binary dynamics

Abstract

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 oneparameter 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 G, namely the subgraph of G that is induced by v and all its neighbours in 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.A binary state on a graph means an assignment of binary values to its vertices. A motivating example in this article appears in the context of neuroscience. If one encodes a network of 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 a binary dynamics on the encoding graph, namely a one-parameter family of binary states on it. A network of neurons that receives external signals and responds to those signals thus generates a binary dynamics. Binary dynamics appear in other contexts as well, but in this paper we use networks of spiking neurons as a primary example.The task of correctly pairing a signal injected into a neuronal network with the response of the network, or in other words, identifying the incoming signal from the response, is generally very challenging. This paper proposes a methodology by which this task can be approached.Considering raw binary states on a large graph is generally quite problematic for a number of reasons. First, the sheer number of theoretically possible states makes analysing a collection of them a daunting task. Moreover, natural systems such as neuronal networks tend to be very noisy, in the sense that the emerging dynamics from the same stimulus may take a rather large variety of forms. Finally, it is a general working hypothesis in studying network dynamics that the network structure affects its function (see for instance [20]), and therefore instead of considering individual vertices in the network, it makes sense to examine ensembles of vertices and the way that they behave as dynamical sub-units.In previous studies we considered cliques in a directed graph, with various orientations of the connections between nodes, as basic units from which one could extract information about binary dynamics [18,8]. However, the results in these papers fell short of suggesting an efficient classifier of b...