Functional connectivity mapping using the ferromagnetic Potts spin model

Abstract

An unsupervised stochastic clustering method based on the ferromagnetic Potts spin model is introduced as a powerful tool to determine functionally connected regions. The method provides an intuitively simple approach to clustering and makes no assumptions of the number of clusters in the data or their underlying distribution. The performance of the method and its dependence on the intrinsic parameters (size of the neighborhood, form of the interaction term, etc.) is investigated on the simulated data and real fMRI data acquired during a conventional periodic finger tapping task. The merits of incorporating Euclidean information into the connectivity analysis are discussed. The ability of the Potts model clustering to uncover the hidden structure in the complex data is demonstrated through its application to the resting-state data to determine functional connectivity networks of the anterior and posterior cingulate cortices for the group of nine healthy male subjects.

The Conditional-Potts Clustering Model

“…These observations indicate that there is a vast region in the bandwidth-temperature space for which good clustering results may be found. This has already been observed by Stanberry, Murua, and Cordes (2008) on fMRI data, and has been suggested by Blatt, Domany, and Wiseman (1996) as a result of what is known about the Potts model in ferromagnetism. Note from Table 3 that the optimal temperature for the SPMC is sometimes very low.…”

“…Stanberry, Murua, and Cordes (2008) observed that these regions are also stable regions for the number of clusters and may correspond to phase-free regions of the random cluster model (i.e., no giant component appears in these regions; see Section 4.1.1). Very often in our experiments, the informative data-driven prior maximizer (σ p , T p ) lies in the high-density region of the posterior, and the clustering evidence is similar for both choices of the parameters: the MAP estimates and the informative prior maximizer.…”

“…There is a large literature on the subject in the physics community (Agrawal and Domany 2003;Ott et al 2004;Reichardt and Bornholdt 2004). Its impact has reached the medical (Stanberry, Murua, and Cordes 2008), bioinformatics (Getz et al 2000;Einav et al 2005), and the computer science and machine learning communities as well (Domany et al 1999;Quiroga, Nadasdy, and Ben-Shaul 2004). It also has been mentioned in the statistical literature, but as Potts model clustering (Murua, Stanberry, and Stuetzle 2008), where its link with other kernel-based methods and nonparametric density estimation was presented.…”

“…One way to accomplish this is by using the K-nearest-neighbor graph, that is, a graph where the only edges associated with each x i are those associated with the Knearest-neighbor points of x i . In this case, the work of Blatt, Domany, and Wiseman (1997) recommends using K = 10 for high-dimensional data, whereas the work of Stanberry, Murua, and Cordes (2008) recommends using a moderate value of K in the range 5 < K ≤ 30. In many applications, such as in bioinformatics (e.g., microarray data), the similarities correspond to the correlation between two signals (e.g., signals coming from two different probe sets or genes that measure the expression levels under several experimental conditions or over certain elapsed time).…”

On Potts Model Clustering, KernelK-Means and Density Estimation

“…Reichardt and Bornholdt (2004) introduced a spin glass Hamiltonian with a global diversity constraint to identify probable community assignments in complex networks. Stanberry, Murua, and Cordes (2007) applied the method to study functional connectivity patterns in fMRI data and examined the dependence of the method on neighborhood structure, signal-to-noise ratio, and spatial dependence in the data. Potts model clustering has been applied to different fields such as computer vision (Domany et al 1999), gene expression data (Getz et al 2000;Einav et al 2005), high-dimensional chemical data (Ott et al 2004(Ott et al , 2005 and neuronal spike detection (Quiroga, Nadasdy, and Ben-Shaul 2004).…”