A novel mathematical construction for identifying attractors from task-driven fMRI data

1946 

Abstract Submission 

Hassan Abdallah¹, John Kopchick², Andrew Salch¹, Vaibhav Diwadkar¹

¹Wayne State University, Detroit, MI, ²Wayne State University, Department of Psychiatry, Detroit, MI

Hassan Abdallah  
Wayne State University
Detroit, MI

John Kopchick  
Wayne State University, Department of Psychiatry
Detroit, MI

Andrew Salch  
Wayne State University
Detroit, MI

Vaibhav Diwadkar  
Wayne State University
Detroit, MI

Functional brain network data provides a complex and complicated window into dynamics shaping brain function and dysfunction. Conventional connectivity matrices (based on bivariate correlations) can be interpreted as a "connectivity terrain" of the brain's functional state. In principle, we might be able to recover structure in this terrain by traversing it node by node and moving towards its nearest functional neighbor (i.e., its maximally correlated node). If there is meaningful functional structure in this terrain, any such traversal would settle on "attractor nodes." Those nodes that flow into a shared attractor would form an "attractor basin". Extant methods handle global summaries, such as degree distribution and characteristic path length [2], while being unable to effectively identify local structures, such as attractor nodes and basins. Here, we solve this problem by motivating the construction of a new relation from the connectivity terrain (bivariate correlation matrix), which we call transitive maximal correlation (TMC): thus, Node A is transitively maximally correlated to node B if and only if B is an attractor into which A flows. Remarkably, this method identifies attractors whose nearest functional neighbor is their interhemispheric homologue and whose attractor basins are significantly different across conditions during a learning task.

Data (3T) were acquired during learning [3]. In each subject (n=39), zero-lag undirected functional connectivity was estimated from time series (246 regions)([1]) in each condition (Encoding, Post-Encoding Consolidation, Retrieval, Post-Retrieval Consolidation) over eight repetitions (32 matrices per subject).
The resulting connectivity terrains are analyzed using transitive maximal correlation (TMC). Region B1 is "transitively maximally correlated" with Bm if there exists a sequence of regions B2, B3, … , Bm-1 such that B1 is maximally correlated with B2, and B2 is maximally correlated with B3, etc., and Bm-1 is maximally correlated with Bm. We interpret this relation as B1 "flowing" into Bm in the connectivity terrain.
We introduce the TMC matrix, a new construction: it is the square matrix whose entry in row x and column y is one if region x is transitively maximally correlated with y and with all regions transitively maximally correlated with y, and zero otherwise. Intuitively, the presence of ones in a column y indicates that region y is an attractor for information flow. The collection of regions that flow into region y form an attractor basin.
We develop a test statistic using the Frobenius norm to compare sets of TMC matrices from different conditions in the learning task. A paired permutation test is utilized to determine the p-value for the observed test statistics when comparing conditions pairwise [4].

We apply our statistical test to make pairwise comparisons of TMC matrices between encoding, post-encoding consolidation, retrieval, and post-retrieval consolidation using 10k permutations (see Figure 1). We identified the five percent most influential regions driving these significant results. These eight regions primarily reside in Brodmann areas 1, 2, 3, 6, 28, 34, 35, and 36. We capture the average composition of their attractor basins in Figure 2.

Our study employs a novel construction, TMC, to evaluate the structure of the brain's connectivity terrain. TMC enables the identification of attractor nodes and attractor basins, unveiling a nuanced window into information flow within the connectivity terrain. We implement a statistical test on TMC matrices that indicates significant differences across all pairwise comparisons of conditions during a learning task. Our findings highlight attractor nodes with interhemispheric homologues as nearest functional neighbors whose attractor basins exhibit condition-specific variation. These results underscore the efficacy of TMC to capture striking functional properties of the fMRI signal.

Learning and Memory Other

Activation (eg. BOLD task-fMRI)

fMRI Connectivity and Network Modeling ²

Methods Development ¹

BOLD fMRI

Data analysis

FUNCTIONAL MRI