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Geometric Constraints on the Dynamics of Networks with Refractory Node States Gabriel a Silva

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Geometric Constraints on the Dynamics of Networks with Refractory Node States Gabriel a Silva Silva RESEARCH Geometric constraints on the dynamics of networks with refractory node states Gabriel A Silva Correspondence: [email protected] Department of Bioengineering, Abstract Department of Neurosciences, and Center for Engineered Natural Understanding how local interactions among connected nodes in a network result Intelligence, University of in global network dynamics and behaviors remains a critical open problem in California, San Diego, 9500 Gilman Drive, La Jolla, California network theory. This is important both for understanding complex networks, 92037-0412, United States including the brain, and for the controlled design of networks intended to achieve Full list of author information is a specific function. Here, we describe the construction and theoretical analysis of available at the end of the article a framework derived from canonical neurophysiological principles that models the competing dynamics of incident signals into nodes along directed edges in a network. The framework describes the dynamics between the offset in the latencies of propagating signals, which reflect the geometry of the edges and conduction velocities, and the internal refractory dynamics and processing times of the downstream node. One of the main theoretical results is the definition of a ratio between the speed of signaling or information flow, which is bounded by the spatial geometry, and the internal time it takes for individual nodes to process incoming signals. We show that an optimal ratio is one where the speed of information propagation between connected nodes does not exceed the internal dynamic time scale of the nodes. A mismatch of this ratio leads to sub-optimal signaling and information flows in a network, and even a breakdown in signaling all together. Keywords: Geometric networks; Spatial networks; Temporal networks; Neural signaling; Neural dynamics; Arfificial neural networks 1 Introduction There is significant value in understanding how local interactions among connected nodes in a network result in global network dynamics and behaviors. This is impor- tant both for understanding complex networks, including the brain, and for the con- arXiv:1510.08729v4 [q-bio.MN] 12 May 2017 trolled design of networks intended to achieve a specific function or computational task. Here, we describe the construction and theoretical analysis of a framework that models the competing dynamics of incident signals into nodes along directed edges in a network. As a model this work draws on and extends classical approaches dating back to the original work of McCulloch and Pitts' abstraction of a neuron, Rosenblat's notion of a perceptron, and much of the work by others that has fol- lowed. From a neurobiological perspective we are abstractly modeling the interplay of postsynaptic spatial and temporal summation. The framework is constructed in order to explicitly analyze the dynamics between the offset in the latencies of prop- agating signals, which are the result of the geometry (morphology) of the edges and conduction velocities, and the internal refractory dynamics and processing times of the downstream node. Silva Page 2 of 30 Our main interest in this work was the construction of a framework that supports a formal theoretical analysis of this dynamics in order to allow us to make predic- tions and arrive at a deep understanding. This is in contrast to the construction of models that primarily only support computational experimental investigation through numerical simulations. Despite the significant literature on the types of models related to the framework we developed here, a theoretical analysis of this kind has not been previously investigated. This analysis produced a number of interesting results, most significantly the definition of a ratio between the speed of signaling or information flow, which is bounded by the spatial geometry, and the internal time it takes for individual nodes to process incoming signals. We show that an optimal ratio is one where the speed of information propagation between connected nodes does not exceed the internal dynamic time scale of the nodes. In other words, it represents a balance between how fast signals propagate through the network relative to the time needed for each node to process incoming signals. An optimal ratio serves to maximize the amount of information individual edges between node pairs can support. A mismatch of this ratio leads to sub-optimal signaling and information flows in a network, and even a breakdown in signaling all together. 2 Theoretical development We define a network as a physically constructible object consisting of nodes with unique internal dynamics and a physical connectivity in space between specific pairs of nodes. We assume there exists a finite signaling speed associated with the propagation of information between nodes and as a result a finite temporal delay or latency. Formally, we consider graph models of such physical networks. In general we will explicitly refer to the graph model of a network, and therefore refer to the vertices and edges of the graphs, but it should be understood that they model the nodes and connections of physical networks of interest. We consider the flow of information, or signaling, along the path lengths of di- rected edges that have a physical representation in space (or the plane) that connect two vertices of the graph in space (or the plane; Fig.1). In this sense edges are rep- resented by smooth and continuous Jordan arcs. The term 'geometric graph' is often used in a somewhat ambiguous way depending on the author or context. Topolog- ical graph theory typically refers to the embedding and study of graphs in some topological space, including three dimensional Euclidean spaces and surfaces [2,3]. However, other authors for example have distinguished geometric graphs from topo- logical graphs as graphs in the plane with distinct points connected by straight line edges in the former case versus graphs in the plane with distinct points connected by Jordan arcs in the latter case, i.e. both in R2 [18]. In this paper we only need to consider graphs with a physical representation in the Euclidean plane R2 or three dimensional space R3, and not in any other topological space. As such, we will use the term 'geometric graph' to refer to any graph in R2 or R3. Some authors use the term spatial graph instead. We formalize these notions below. 2.1 Preliminaries and definitions Definition 2.1 A graph is an ordered pair of finite disjoint sets G = (V; E) such that V is the set of vertices in the graph and E is the set of edges joining the vertices. Silva Page 3 of 30 Definition 2.2 A geometric graph G = (V;E¯ ) is a graph who's vertices have a physical position in R2 or R3 in an Euclidean coordinate system. Physical coordi- nates v¯i :=x ¯ for an ordered triplet x¯ = (x1; x2; x3) are assigned to each vertex vi for all vertices i = 1 :::N in the graph, with the set of all vertices given by V¯ = fvig [6]. Definition 2.3 A complete geometric graph G = (V;¯ E¯) is a geometric graph with directed edges that also have a physical representation in space: An edge eij 2 E is defined as a directed connection from vertex i to vertex j, i.e. from vi to vj. Let 3 an edge between vi and vj be defined geometrically for some function f(·) in R , restricted to simple non-intersecting Jordan arcs. The physical distance of the edge between any two vertices is determined by the path integral taken by the edge in space where for some scalar field f : U ⊆ Rn ! R. Following standard notation, the path integral dij from v¯i to v¯j formed by a piecewise smooth curve C ⊂ U is defined as Z Z v¯j 0 jeijj = dij := f ds = f (¯r(t)) jr¯ (t)j dt v¯i C where r : [¯vi; v¯j] ! C is an arbitrary bijective parametrization of the curve C such that r(¯vi) and r(¯vj) give the endpoints of C for v¯i < v¯j. As a technical comment, it should be noted that our use of the term `complete geometric graph' here differs from the use of the term 'complete graph' referring to a simple undirected graph in which every pair of distinct vertices is connected by a unique edge, or that of a complete digraph which is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges. In addition, we do not notationally distinguish between 'geometric graph' and 'complete geometric graph', i.e. between definitions 2.2 and 2.3 in the rest of this paper in order to avoid introducing additional (unnecessary) notation. This distinction is not necessary because we strictly deal with complete geometric graphs in the work that follows, and the notation G = (V;¯ E¯) should be assumed to refer accordingly. Definition 2.4 The set of all geometric edges in the geometric graph G = (V;¯ E¯) is given by E¯ = feijg. Definition 2.5 We distinguish a non-minimal complete geometric graph, denoted by G, from the 'minimal' complete geometric graph representation Gmin, in the sense that the edges of Gmin are all (geodesic) straight line edges. For E¯min with [eij]min < eij for all edges eij 2 E¯ in G. We define dij as the length of the edge eij: dij = jeijj. The graphs Gmin and G share the same vertex set V¯ and structural connectivity adjacency matrix, but the edge set of Gmin given by E¯min are straight lines while the edge set of of G given by E¯ follow longer convoluted path integrals in space determined by the actual physical nature of the network under consideration, c.f. Silva Page 4 of 30 definition 2.3 above.
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