eMB: Multi-state bootstrap percolation on digraphs as a framework for neuronal network analysis

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

To study the spread of disease or opinions, it has proven useful to consider a collection of interacting individuals as a mathematical structure called a graph. A graph consists of a set of nodes and a set of edges, each of which links a pair of nodes. PI will extend this framework to the study of neuronal activity. In the neuronal case, each node is a neuron, and edges between them represent the one-way or directed synaptic connections through which neurons interact. Moreover, to capture the different levels of activity that neurons exhibit, each node will be in one of three states, with state changes determined by interactions along edges. PI will develop novel mathematical methods to analyze the patterns of activity that emerge in such multi-state, directed graphs. These methods will allow the project team to study how activity spreads in networks of neurons, how this spread depends on the connection patterns in the network, and what properties characterize sets of nodes that are especially effective at spreading activity. The research results will generate novel predictions about the properties and function of networks of neurons in the brain, with a focus on the key networks that drive breathing and other essential, rhythmic behaviors. The project will train students, will generate openly available computer code, and will include public outreach through online videos and a summer math program for girls. Functional outputs from brain circuits driving certain critical, rhythmic behaviors require the widespread emergence of an elevated, bursting state of neuronal activity. The main goal of this project is to advance knowledge about how the localized initiation of activity can rapidly evolve into widespread bursting in synaptically coupled neuronal networks, exemplified by the respiratory brainstem circuit. The project team will achieve this goal by representing such networks as digraphs in which each node can assume one of three possible activity states – inactive, weakly active, and fully active or bursting -- with updates based on in-neighbors’ states. Little theory exists to characterize this multi-state bootstrap percolation framework, and we will develop new analytical approaches involving mean field and master equations, asymptotic and probabilistic estimates, and graph design based on combinatorial principles. The analysis will address differences between dynamics in this framework and that in bootstrap percolation with only two possible states per node, the impact of global graph properties on activity spread, and the characteristics of local initiation sites that result in especially effective activity propagation. Overall, the project will support a new interdisciplinary collaboration and the completed work, involving trainee mentorship and open sharing of code, will provide important insights and predictions about neuronal dynamics and interactions as well as a range of mathematical advances. This project is jointly funded by the Mathematical Biology Program in the Division of Mathematical Sciences and the Research Resources Cluster in the Division of Biological Infrastructure in the Directorate for Biological Sciences. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.