Functional Brain Networks Formed Using Cross-Sample Entropy Are Scale Free

Abstract

Over the previous decade, there has been an explosion of interest in network science, in general, and its application to the human brain, in particular. Most brain network investigations to date have used linear correlations (LinCorr) between brain areas to construct and then interpret brain networks. In this study, we applied an entropy-based method to establish functional connectivity between brain areas. This method is sensitive to both nonlinear and linear associations. The LinCorr-based and entropy-based techniques were applied to resting-state functional magnetic resonance imaging data from 10 subjects, and the resulting networks were compared. The networks derived from the entropy-based method exhibited power-law degree distributions. Moreover, the entropybased networks had a higher clustering coefficient and a shorter path length compared with that of the LinCorr-based networks. While the LinCorr-based networks were assortative, with nodes with similar degrees preferentially connected, the entropy-based networks were disassortative, with high-degree hubs directly connected to low-degree nodes. It is likely that the differences in clustering and assortativity are due to ''megahubs'' in the entropy-based networks. These mega-hubs connect to a large majority of the nodes in the network. This is the first work clearly demonstrating differences between functional brain networks using linear and nonlinear techniques. The key finding is that the nonlinear technique produced networks with scale-free degree distributions. There remains debate among the neuroscience community as to whether human brains are scale free. These data support the argument that at least some aspects of the human brain are perhaps scale free.

BOLD fMRI complexity predicts changes in brain processes, interactions and patterns, in health and disease

“…Other available methods are Hurst exponent [22], Lempel-Ziv complexity [7] and causation entropy [28], just to mention a few. The entropy method of estimating the complexity of a time series has been shown to be efficient with fMRI data [17,24,33].…”

“…Bivariate or cross entropy estimates the complexity between the time series of two voxels, which characterises the complexity of "the interaction" between the two voxel time series. This has been shown to be significant in people with familial Alzheimer's disease [10] and in healthy subjects [17]. "Patterns of complexity" are estimated when the interactions among many voxel time series are characterised at different thresholds by using multivariate or multiscale entropy.…”

“…These connectomics and network analysis approaches which probe the temporal associations or coherence between brain nodes are based on linear correlations. Pearson's correlation coefficient is one of the methods widely used to create a linear correlation between brain nodes due to its computational ease [17]. Pearson correlation coefficient is only able to discriminate a linear association between two variables [17].…”

“…Pearson's correlation coefficient is one of the methods widely used to create a linear correlation between brain nodes due to its computational ease [17]. Pearson correlation coefficient is only able to discriminate a linear association between two variables [17]. Consequently, it may miss any nonlinear association among brain nodes [17].…”

“…Pearson correlation coefficient is only able to discriminate a linear association between two variables [17]. Consequently, it may miss any nonlinear association among brain nodes [17].…”