Two’s company three or more is a simplex

Authors: Chad Giusti Robert Ghrist Danielle S Bassett

Publish Date: 2016/06/11

Volume: 41, Issue: 1, Pages: 1-14

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Abstract

The language of graph theory or network science has proven to be an exceptional tool for addressing myriad problems in neuroscience Yet the use of networks is predicated on a critical simplifying assumption that the quintessential unit of interest in a brain is a dyad – two nodes neurons or brain regions connected by an edge While rarely mentioned this fundamental assumption inherently limits the types of neural structure and function that graphs can be used to model Here we describe a generalization of graphs that overcomes these limitations thereby offering a broad range of new possibilities in terms of modeling and measuring neural phenomena Specifically we explore the use of simplicial complexes a structure developed in the field of mathematics known as algebraic topology of increasing applicability to real data due to a rapidly growing computational toolset We review the underlying mathematical formalism as well as the budding literature applying simplicial complexes to neural data from electrophysiological recordings in animal models to hemodynamic fluctuations in humans Based on the exceptional flexibility of the tools and recent groundbreaking insights into neural function we posit that this framework has the potential to eclipse graph theory in unraveling the fundamental mysteries of cognitionExtensions of network models provide insights into neural data a Network models are increasingly common for the study of wholebrain activity b Neuronlevel networks have been a driving force in the adoption of network techniques in neuroscience c Two potential activity traces for a trio of neural units top Activity for a “pacemaker”like circuit whose elements are pairwise active in all combinations but never as a triple bottom Activity for units driven by a common strong stimulus thus are simultaneously coactive d A network representation of the coactivity patterns for either population in c Networks are capable of encoding only dyadic relationships so do not capture the difference between these two populations e A simplicial complex model is capable of encoding higher order interactions thus distinguishing between the top and bottom panels in c f A similarity measure for elements in a large neural population is encoded as a matrix thought of as the adjacency matrix for a complete weighted network and binarized using some threshold to simplify quantitative analysis of the system In the absence of complete understanding of a system it is difficult or impossible to make a principled choice of threshold value g A filtration of networks is obtained by thresholding at every possible entry and arranging the resulting family of networks along an axis at their threshold values This structure discards no information from the original weighted network g Graphs of the number of connected components as a function of threshold value for two networks reveals differences in their structure top homogeneous network versus bottom a modular network dotted lines Thresholding near these values would suggest inaccurately that these two networks have similar structureAll graphbased models consist of a choice of vertices which represent the objects of study and a collection of edges which encode the existence of a relationship between pairs of objects Figs 1a–b 4a However in many real systems such dyadic relationships fail to accurately capture the rich nature of the system’s organization indeed even when the underlying structure of a system is known to be dyadic its function is often understood to be polyadic In largescale neuroimaging for example cognitive functions appear to be performed by a distributed set of brain regions Gazzaniga 2009 and their interactions Medaglia et al 2015 At a smaller scale the spatiotemporal patterns of interactions between a few neurons is thought to underlie basic information coding Szatmary and Izhikevich 2010 and explain alterations in neural architecture that accompany development Feldt et al 2011Drawing on techniques from the field of algebraic topology we describe a mathematically wellstudied generalization of graphs called simplicial complexes as an alternative often preferred method for encoding nondyadic relationships Fig 4 Different types of complexes can be used to encode cofiring of neurons Curto and Itskov 2008 coactivation of brain areas Crossley et al 2013 and structural and functional connections between neurons or brain regions Bullmore and Sporns 2009 Fig 5 After choosing the complex of interest quantitative and theoretical tools can be used to describe compare and explain the statistical properties of their structure in a manner analogous to graph statistics or network diagnosticsWe then turn our attention to a method of using additional data such as temporal processes or frequency of observations to decompose a simplicial complex into constituent pieces called a filtration of the complex Fig 1f–h Filtrations reveal more detailed structure in the complex and provide tools for understanding how that structure arises Fig 7 They can also be used as an alternative to thresholding a weighted complex providing a principled approach to binarizing which retains all of the data in the original weighted complexIn what follows we avoid introducing technical details beyond those absolutely necessary as they can be found elsewhere Ghrist 2014 Nanda and Sazdanović 2014 Kozlov 2007 though we include boxed mathematical definitions of the basic terms to provide context for the interested reader These ideas are also actively being applied in the theory of neural coding and for details we highly recommend the recent survey Curto 2016 Finally although the field is progressing rapidly we provide a brief discussion of the current state of computational methods in the AAppendix

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