ESTIMATING AND ANALYZING NEURAL FLOW USING SIGNAL PROCESSING ON GRAPHS Felix Schwock 1* , Les Atlas 1 , Shima Abadi 1 , Julien Bloch 2 , Azadeh Yazdan-Shahmorad 1,2 1 Department of Electrical and Computer Engineering University of Washington, Seattle 2 Department of Bioengineering, University of Washington, Seattle ABSTRACT Neural communication is fundamentally linked to the brain’s over- all state and health status. We demonstrate how communication in the brain can be estimated from recorded neural activity using con- cepts from graph signal processing. The communication is modeled as a flow signals on the edges of a graph and naturally arises from a graph diffusion process. We apply the diffusion model to local field potential (LFP) measurements of brain activity of two non-human primates to estimate the communication flow during a stimulation experiment. Comparisons with a baseline model demonstrate that adding the neural flow can improve LFP predictions. Finally, we demonstrate how the neural flow can be decomposed into a gradi- ent and rotational component and show that the gradient component depends on the location of stimulation. Index Terms— network neuroscience, neural communication, graph diffusion, edge flows 1. INTRODUCTION Analyzing communication and flow processes in the brain has re- ceived much attention in past years [1]. Understanding and mod- eling such processes is thought to be crucial for developing better treatments for neurological diseases [2]. Especially using graphs to study neural processes is a promising direction as they can naturally model structural and functional brain connectivity [3, 4]. While most research has focused on analyzing the graph that is associated with a certain pattern of neural activity using concepts from graph the- ory, more recently, graph signal processing has been proposed as a way to study neural signals that are observed at the nodes of an un- derlying brain network [5, 6, 7]. The central idea of graph signal processing is that, rather than focusing on the graph itself, we are interested in processing signals indexed by the nodes of the graph, for example, by finding a graph spectral representation of the signal, or performing filtering in the graph domain [8]. Graph signal processing in neuroscience has, to the best of our knowledge, exclusively focused on signals defined on the nodes of an underlying brain graph. Our goal is to shift the focus from signals at the nodes to flow signals that live on the edges of a graph and that characterize the communication flow in the brain. Specifically, we will address the following questions: (1) How can we leverage methods from graph signal processing to estimate the neural flow in the brain? (2) How can theory of signal processing on higher order networks [9, 10] be used to analyze neural flow signals? This new perspective allows for modeling neural flow on a finer temporal scale than other techniques used in neuroscience such as Granger causality [11, 12, 13]. To test our theoretical findings, we * corresponding author: fschwock@uw.edu will apply our model to estimate neural flow from electrophysiolog- ical recordings obtained from monkeys during a stimulation experi- ment. The remainder of the paper is outlined as follows: In Sec. 2 we describe the model that turns neural recordings into a flow signal. In Sec. 3 the model is used to estimate the neural flow in two non- human primates during a stimulation experiment. Sec. 4 describes ways to analyze the neural flow signal by decomposing it into differ- ent components. Finally, Sec. 5 summarizes the main findings and points towards future research directions. 2. ESTIMATING NEURAL FLOWS FROM TIME SERIES OF NEURAL ACTIVITY In this section, we describe how a graph diffusion model can be used to estimate neural flow signals from vector time series of neural ac- tivity. Diffusion models have been used in the graph signal process- ing community to describe the dynamic behavior of network data [8, 14, 15], as well as in the neuroscience community to model func- tional connectivity [16]. Instead of focusing on functional connec- tivity, here we demonstrate how neural activity on the nodes of a graph can be transformed into an edge flow signal using a diffusion process. We start by assuming a graph with N nodes and E edges. Fur- thermore, time series of neural activity are measured at the nodes of the graph. We will use s[t] ∈ R N×1 to denote the neural activ- ity at time t across all nodes. An example of this is shown on the left of Fig. 1, with the graph topology (nodes and edges) in black and the node time series si [t] in blue. Our goal is to use the given graph topology and observed node time series to estimate the time dependent information flow along the edges of the graph. This flow is illustrated on the right in Fig. 1. 2.1. 1 st Order Diffusion Model To estimate the flow, we use a parameterized diffusion model that naturally gives rise to an edge flow and whose parameters can be estimated from the observed node time series. In a nutshell, for each node the model expresses the node signal at the current time step as its weighted own past (memory) plus the inflow minus the outflow from all neighboring nodes. To describe this mathematically, we first encode the graph topology using the node-to-incidence matrix B ∈ R N×E , where each column represents an edge e =(i, j ) with tail node i and head note j . B is defined such that Bie = −Bje = −1 and B ke =0 otherwise. For every edge, it is thereby arbitrary which of the incident nodes is the tail and head node. For more details and examples for B, we refer the reader to [9]. Using B, our diffusion arXiv:2205.13719v2 [eess.SP] 11 Jun 2022