Analysis of Structure and Dynamics in Three-Neuron Motifs.

Patrick Krauss,Claus Metzner,Alexandra Zankl,Holger Schulze,Achim Schilling

doi:10.3389/fncom.2019.00005

Abstract

Recurrent neural networks can produce ongoing state-to-state transitions without any driving inputs, and the dynamical properties of these transitions are determined by the neuronal connection strengths. Due to non-linearity, it is not clear how strongly the system dynamics is affected by discrete local changes in the connection structure, such as the removal, addition, or sign-switching of individual connections. Moreover, there are no suitable metrics to quantify structural and dynamical differences between two given networks with arbitrarily indexed neurons. In this work, we present such permutation-invariant metrics and apply them to motifs of three binary neurons with discrete ternary connection strengths, an important class of building blocks in biological networks. Using multidimensional scaling, we then study the similarity relations between all 3,411 topologically distinct motifs with regard to structure and dynamics, revealing a strong clustering and various symmetries. As expected, the structural and dynamical distance between pairs of motifs show a significant positive correlation. Strikingly, however, the key parameter controlling motif dynamics turns out to be the ratio of excitatory to inhibitory connections.

Highlights

A number of projects seek to map the human connectome, aiming to connect its structure to function and behavior (Markram, 2012; Van Essen et al, 2013; Glasser et al, 2016)
The problem is complicated by the fact that very similar dynamics of a neural network at a macroscopic level might be realized by very different structures at the microscopic level (Newman, 2003)
Human observers may not be able to deduce the function of a system from its structure alone: even if we know all neural connection strengths in some sub-network of the animal brain, as well as all its input and output signals, the specific purpose of this sub-network within the whole of the organism may remain elusive (Hobert, 2003; Gray et al, 2005; Jonas and Kording, 2017)

Summary

Introduction

A number of projects seek to map the human connectome, aiming to connect its structure to function and behavior (Markram, 2012; Van Essen et al, 2013; Glasser et al, 2016). Even if the connectome would be known completely, it remains an unresolved problem how to translate this detailed structural data into meaningful information processing functions and algorithms (Jonas and Kording, 2017). The connectome of C. elegans has been known for decades, and involves only 302 neurons. Even this relatively small system is not yet understood in terms of its dynamics, let alone at a functional level (Hobert, 2003; Gray et al, 2005). An important step toward extracting function from structure is a tool to quantitatively compare different structures and dynamics

Methods

Results

Conclusion