Network Analysis of Time Series: Novel Approaches to Network Neuroscience

Abstract

In the last two decades, there has been an explosion of interest in modeling the brain as a network, where nodes correspond variously to brain regions or neurons, and edges correspond to structural or statistical dependencies between them. This kind of network construction, which preserves spatial, or structural, information while collapsing across time, has become broadly known as "network neuroscience." In this work, we provide an alternative application of network science to neural data: network-based analysis of non-linear time series and review applications of these methods to neural data. Instead of preserving spatial information and collapsing across time, network analysis of time series does the reverse: it collapses spatial information, instead preserving temporally extended dynamics, typically corresponding to evolution through some kind of phase/state-space. This allows researchers to infer a, possibly low-dimensional, "intrinsic manifold" from empirical brain data. We will discuss three methods of constructing networks from nonlinear time series, and how to interpret them in the context of neural data: recurrence networks, visibility networks, and ordinal partition networks. By capturing typically continuous, non-linear dynamics in the form of discrete networks, we show how techniques from network science, non-linear dynamics, and information theory can extract meaningful information distinct from what is normally accessible in standard network neuroscience approaches.

Keywords: complex system; information theory; manifold learning; network science; ordinal partition network; recurrence analysis; time series analysis; visibility graph.

Figure 1

Visualization of how G_R changes when the value of ϵ is varied. The networks are constructed from ECoG data from the Neurotycho database and the distance function is the cosine distance (Nagasaka et al., 2011). (A) The weighted adjacency matrix and the associated RN thresholded at 10% of the maximal distance in the point cloud. It is clear that this is too low of a threshold, since every point is only similar to it's immediate past and future, creating a path graph. (B) The same network, this time thresholded at 20% of the maximum distance. Note that clear cyclic structures, indicating recurrences have started to appear, suggesting that the system is returning to particular regions of phase-space at distinct points in time. (C) The same network, this time thresholded at 30% of the maximum distance. This one captures even more meaningful recurrences, although at the cost of a much denser network.

Figure 2

Community detection in recurrence networks. The effects of ϵ on the higher-order community structure of the RNs. Communities were determined using a greedy modularity maximization function in NetworkX (Hagberg et al., 2008). (A) The community structure when ϵ is 20% of the maximum distance. Here, there are four distinct communities corresponding to temporally distinct “regions” of the phase space that the system visits in sequence. (B) The community structure when ϵ is 30% of the maximum distance, which only returns two communities that are not as restricted in time. Depending on the threshold, two moments can be lumped into the same, or different macro-states.

Figure 3

Complete and horizonal visibility networks. A comparison of the VN and HVN networks for the same Human Connectome Project BOLD data (Van Essen et al., 2013). A single time series was selected from a subject at random for demonstration purposes. (A) A cartoon of the visibility graph algorithm. For each origin point, an edge is drawn between it and all the points that it can “see”, illustrated by the multiple blue lines. (B) A cartoon of the horizontal visibility graph, where an edge is only drawn between an origin point at the next point at the same height. (C,D) The binary adjacency matrices for the above univariate time series. (E,F) The networks for the associated binary adjacency matrices. The colors correspond to the flow of time, as in Figure 1.

Figure 4

Constructing an Ordinal Partition Network from a single time series. An example of an OPN constructed from a single ECoG time series taken from the Neurotycho database (Nagasaka et al., 2011). (A) The time series itself: an electrophysiological time series recorded using invasive intra-cortical arrays. (B) The transition probability matrix for the OPN. (C) The OPN itself, nodes colored by community assignment determined using the Informap algorithm (Rosvall and Bergstrom, ; Rosvall et al., 2009). Note the mixture of long, path-like cycles corresponding to rare excursions through state-space, as well as denser regions with a high degree of interconnectivity.

Figure 5

Exploring the space of possible ordinal partition networks. Possible OPNs for a single time series with lags ranging from 12 to 24 (rows, in increments of 2) and dimensions ranging from 3 to 8 (columns). Nodes are colored according to community assignment determined using the Infomap algorithm (Rosvall and Bergstrom, ; Rosvall et al., 2009). Note that, as the embedding dimension gets large, the networks become increasingly path-like, as every embedded vector gets it's own unique ordinal partition vector, creating an illusion of determinism. The data is the same Neurotycho time series as what was used above in Figure 4.