Modeling and interpreting mesoscale network dynamics

Abstract

Recent advances in brain imaging techniques, measurement approaches, and storage capacities have provided an unprecedented supply of high temporal resolution neural data. These data present a remarkable opportunity to gain a mechanistic understanding not just of circuit structure, but also of circuit dynamics, and its role in cognition and disease. Such understanding necessitates a description of the raw observations, and a delineation of computational models and mathematical theories that accurately capture fundamental principles behind the observations. Here we review recent advances in a range of modeling approaches that embrace the temporally-evolving interconnected structure of the brain and summarize that structure in a dynamic graph. We describe recent efforts to model dynamic patterns of connectivity, dynamic patterns of activity, and patterns of activity atop connectivity. In the context of these models, we review important considerations in statistical testing, including parametric and non-parametric approaches. Finally, we offer thoughts on careful and accurate interpretation of dynamic graph architecture, and outline important future directions for method development.

FIG. 1. Mesoscale network methods can address activity, connectivity, or the two together

In the human brain, the structural connectome supports a diverse repertoire of functional brain dynamics, ranging from the patterns of activity across individual brain regions to the dynamic patterns of connectivity between brain regions. Current methods to study the brain as a networked system usually address connectivity alone (either static or dynamic) or activity alone. Methods developed to address the relations between connectivity and activity are few in number, and further efforts connecting them will be an important area for future growth in the field. In particular, the development of methods in which activity and connectivity can be weighted differently – such as is possible in annotated graphs, which we review later in this article – could provide much-needed insight into their complimentary roles in neural processing.

FIG. 2. Dynamic network modules and subgraphs

(Top) Network science enables investigators to study dynamic architecture of complex brain networks in terms of the collective organization of nodes and of edges. Clusters of strongly interconnected nodes are known as modules, and clusters of edges whose strengths, or edge weights, vary together in time are known as subgraphs. Nodes and edges of the same module or subgraph are shaded by color. Each module represents a collection of nodes that are highly interconnected to one another and sparsely connected to nodes of other modules, and each node may only be a member of a single module. Each subgraph is a recurring pattern of edges that link information between nodes at the same points in time, and each edge can belong to multiple subgraphs. (Bottom Left) Dynamic community detection assigns nodes to time-varying modules. Nodes may shift their participation between modules over time based on the demands of the system. (Bottom Right) Non-negative matrix factorization pursues a parts-based decomposition of the dynamic network into subgraphs and time-varying coefficients, which quantify the level of expression of each subgraph over time.

FIG. 3. State space of brain activity patterns

(A) The time-by-time graph captures similarities in neural activation patterns between different points in time. In practice, one can compute the average brain activity for individual brain regions within discrete time windows and compare the resulting pattern of activation between two time windows using a similarity function, such as the Pearson correlation. (B) The resulting time-by-time graph has an adjacency matrix representation in which each time window is a node and the neural activation similarity between a pair of time windows is an edge. (C) Clustering tools based on graph theory or machine learning can identify groups of time windows – or states – with similar patterns of neural activation. By parametrically varying the number of clusters, or their size, one can examine the dynamic states over multiple time scales (Adapted from [136]). (D) Multidimensional scaling (MDS) [137] can be used to trace the trajectory of a dynamical system through state space by projecting a high-dimensional, time-by-time graph onto a two dimensional subspace. In this subspace, each point is a neural activation pattern in a time window and the spatial proximity between two points represents the similarity of the neural activation pattern between time windows. The example shown here is an MDS projection of a time-by-time graph derived from ECoG in an epilepsy patient during a 2 s, resting period onto a two-dimensional space. Each point represents a 2 ms time window and is shaded based on its occurrence in the 2 s interval. The depicted state space demonstrates an interleaved trajectory in which the system revisits and crosses through paths visited at earlier time points. These tools can be readily adapted to characterize the evolution of a neural system in conjunction with changes in behavior.

FIG. 4. Mapping temporal structure with algebraic topology

(A) Schematic of Reeb graph construction. Given a topological space X, here a torus, the Reeb graph is constructed by examining the evolution of the level sets from h(X). (B) Illustration of the Mapper algorithm. Beginning with a point cloud and parameter space Z, points are binned and clustered. Resulting clusters are collapsed to nodes in the final Mapper graph, and edges between nodes exist if the two corresponding clusters share points from the original point cloud. (C) Example use of topological data analysis for dynamic networks. The initial point cloud here is the collection of brain states across time and parameter space the distance from the initial state of the system. Following the path of time (right, red curve) on the Mapper graph may yield insights to system evolution.

FIG. 5. Brain activity on brain graphs

(A) Annotated graphs enable the investigator to model scalar or categorical values associated with each node. These graphs are represented by both an adjacency matrix A of dimension N × N, and a vector x of dimension N × 1 (Adapted from [151]). (B) Graph signal processing allows one to interpret and manipulate signals atop nodes in a mathematical space defined by their underlying graphical structure. A graph signal is defined on each vertex in a graph. For example, the signal could represent the level of BOLD activity at brain regions interconnected by a network of fiber tracts. Graph filters can be constructed using the eigenvectors of A that are most and least aligned with its structure. Applying these filters to a graph signal decomposes it into aligned/misaligned components. Elements of the aligned component will tend to have the same sign if they are joined by a connection. The elements of the misaligned or “liberal” component, on the other hand, may change sign frequently, even if joined by a direct structural connection.

FIG. 6. Pharmacologic modulation of network dynamics

(A) By blocking or enhancing neurotransmitter release through pharmacologic manipulation, investigators can perturb the dynamics of brain activity. For example, an NMDA receptor agonist might hyper-excite brain activity [–187], while a NMDA receptor antagonist might reduce levels of brain activity [188, 189]. (B) Hypothetically speaking, by exogenously modulating levels of a neurotransmitter, one might be able to titrate the dynamics of brain activity and the accompanying functional connectivity to avoid potentially damaging brain states.