Dynamic Graph Learning based on Graph Laplacian

The increased and ubiquitous emergence of graphs is becoming an excellent tool for quantifying interaction between different elements in great variety of network systems. Analyzing and discovering the underlying structure of a graph for a given data set has become central to a variety of different signal processing research problems which may be interpreted as graph structure recovery. For example, in social media [1], such as Facebook and LinkedIn, the basic interaction/relation between two individuals being represented by a link, yields the notion of a graph known as the social network, which is used for inferring further characteristics among all involved individuals [2]. Similarly, in physics [3] and chemistry [4], graphs are widely used to capture the interactions among atoms/molecules to study the behavior of different materials. The rather recent connectome paradigm[5] in neuros-science, is based on the hypothesis that the brain is a massively connected network and its behavior variation and connectivity, particularly in response to controlled external, can be used to investigate brain’s structures and ideally its functionality [6, 7].
Existing analysis approaches of connectivity of neuron signals and associated problems can be categorized into two main groups, $(i)$ Noise correlation analysis[8, 9], which is often applied by neuro- scientists to uncover the connectivity among neurons, $(ii)$ Static graph learning [10, 11], which, by way of an optimization procedure, tries to attain a fixed graph over time. Noise correlation is commonly used in neuro-science to establish connectivity between every pair of neurons if their noise correlation over a short time is significant. This method, however, requires many observations, making it hard to obtain an acceptable connectivity estimation over that interval. Additionally the acquired connectivity does not lend to simple rationalization following an experiment with a specific stimilus. In the second track, research on estimating graph structures for a given data set has been active and includes graph learning. Research on the latter has primarily been based on graphs’ topology and signals’ smoothness, and the application of the graph Laplacian has been predominant. Other recent work includes deep neural network modeling, and the training/testing was performed on graph datasets to ultimately generate a graph to represent patterns under given signals. These studies have primarily focussed on a static graph, with signals non-sparse and the assumption of consistency of the graph over time. These models require sufficiently adequate samples for training and testing, once again making difficult to use on neuronal signals with a typically low sample size, in order to glean the desired variations over small time intervals. Graphs’ dynamics with clear potential impact on temporal data characterization, have also been of interest by many researchers [12, 13]. In this theme, the models are used to predict the links given previous graphs and associated signals. All these approaches require much prior information on known structures and plenty of data for training and predicting future graphs.
Building on the wealth of prior research work in neuro-science and graph learning [11], we propose a new model, with a dynamic structure goal to track neurons’ dynamic connectivity over time. To that end, our proposed graph will include vertices/nodes for nodes, and their connectivity is reflected by the graph edges whose attribute is determined by the probability/intensity of connection between every pair of neurons.
To proceed, the outline of our paper follows in order our contributions in the sequel. Firstly, exploiting the insight from prior research on graph learning with graph Laplacian [11, 14], we propose an optimization formulation, which can yield a graph over each short time interval which in turn reflects the evolving transformation of the connectivity. Secondly, we modify our model to fit sparse signals so that we can verify our optimized solution on a neuronal signal dataset. Thirdly, we apply three alternative methods to simplify the optimization procedures, to simplify the solution procedures of the optimization problems, and help reach the optimal points. We finally proceed to test our proposed model on a neuronal dataset, to help improve our understanding on the neuronal interaction and their process of transferring signals.

For notation clarity and throughout the paper, we will adopt an upper and lower case bold letter to respectively denote a matrix and a vector, and the superscripts $T,-1$ to respectively denote its tranpose and inverse. The operator $tr(\cdot)$ denotes a matrix trace. The identity, zero and ”1” matrices are respectively denoted by $\mathbf{I}$, $\mathbf{0}$ and $\mathbf{1}$, while $x_{ij}$ represents the $i$-th row, $j$-th column element of $\mathbf{X}$.
Our neuronal-activity dataset will consist of $\mathnormal{N}$ neuron/nodes, and will be characterized by a connectivity graph $\mathnormal{G}=\{V,E,W\}$, where $\mathnormal{V}$ denotes the vertex set $V=\{v_{1},v_{2},\dots,v_{N}\}$, $\mathnormal{E}$ is the edge set with attributes reflecting the connectivity between each pair of vertices quantified by $0\leq w\leq 1,\forall w\in W$ as a connectivity strength. A time series $y_{n}(t)$ of observations with $t=1,2,\ldots,T$, is associated with each node $v_{n}$. For simplicity in our derivations, we will aggregate the nodes’ finite length time series into a $N\times T$ matrix $\mathbf{Y}$, where $(\mathbf{Y})_{n,t}=y_{n}(t)$. Our problem formulation will seek for each observed $\mathbf{Y}$, either a static graph $G$ or a time dependent graph series of graphs $G_{1},G_{2},\ldots$.
The well known graph Laplacian of an undirected graph can describe its topology, and can serve as the second derivative operator of the underlying graph. The corresponding Laplacian matrix $\mathcal{L}$ is commonly defined as [15], with $\mathcal{L}(i,j)=-w_{i}j$, for $i$,$j$ adjacent nodes, and $\mathcal{L}(i,j)=0$ otherwise, and $\mathcal{L}(i,i)=d_{i}$ where $d_{i}=\sum_{j}w_{ij}$ denotes the degree of node $i$ . Its simple matrix expression is $\mathcal{L}=D-W$, where $\mathnormal{D}$ is a diagnal matrix of degrees.
The Laplacian matrix may also usefully adopt, in some context, a second derivative interpretation of graphs: Given an assignment $x=(x_{1},x_{2},\ldots,x_{n})$ of real number to the nodes, the Laplacian matrix may be found as the second derivative of $x$ as $\mathcal{L}(\mathbf{W},x)=\sum_{i}\sum_{j>i}w_{ij}\mathbf{a}_{ij}\mathbf{a}^{T}_{ij}x$, where $\mathbf{a}_{ij}$ denotes a N-dimensional vector whose elements are all 0s, except the $i$-th element being respectively 1 and $j$-th element -1. As may be seen, $a_{ij}$ represents the first derivative of the edge between the $i$-th and $j$-th node. The notion of a Laplacian will be exploited in this sequel as a structural regularizer when an optimal graph is sought for a given data set.

The conventional use of Lagrangian duality for solving the above optimization model is costly in time and memory, and thus begs for an alternative.

The data in these experiments were measured in S. L. Smith’s Laboratory at UCSB[8]. They use a two-photon electro-microscope [17] to collect fluorescent signals. The whole experiment consists of 3 specific scenarios with a 20 trial measurement, and the stimuli in each trial are the same. The stimuli for each of the scenarios are shown in Figure 1, and consist of ”gray” movie, artificial movie, natural movie 1 and natural movie 2. The dataset includes 590 neurons in V1 area and 301 neurons in AL area, and the sample rate is approximately 13.3 Hz. To select the most consistent 50 neurons in V1 area , We calculate the correlation between signals in every 2 trials for each neuron, and we choose the 50 neurons with the highest mean correlation.

This paper introduces an optimization model for learning dynamics of sparse graphs with sparse signals based on the graph Laplacian and its smoothness assumption without prior knowledge on the signals. Through applying three alternative solution methods, this model learns a single graph in a short time interval and a set of graphs over the whole signals, and it can capture the small change of graphs in a brief time interval. In the experiment, we solve the difficulty of the low sample rate for detecting graphs, and discover the functional connectivity on specific stimuli instead of revealing the physical connections of neurons. More future research should focus on discovering brains with more datasets and measuring methods, which will support future discovery on understanding how neurons collaborate with each other and how brains work. A future plan is to optimize the model to learn the transformation of graphs.