The geometry of correlation fields with an application to functional connectivity of the brain

We introduce two new types of random field. The cross correlation field R(s, t) is the usual sample correlation coefficient for a set of pairs of Gaussian random fields, one sampled at point s ∈ <M , the other sampled at point t ∈ <N . The homologous correlation field is defined as R(t) = R(t, t), that is, the ‘diagonal’ of the cross correlation field restricted to the same location s = t. Although the correlation coefficient can be transformed pointwise to a t-statistic, neither of the two correlation fields defined above can be transformed to a t-field, defined as a standard Gaussian field divided by the roit mean square of i.i.d. standard Gaussian fields. For this reason, new results are derived for the geometry of the excursion set of these correlation fields that extend those of Adler (1981). The results are used to detect functional connectivity (regions of high correlation) in 3D Positron Emission Tomography (PET) images of human brain activity.

[1]  D. Slepian,et al.  Large Excursions of Gaussian Processes , 1959 .

[2]  G. Lindgren Local maxima of Gaussian fields , 1972 .

[3]  A. M. Hasofer Upcrossings of Random Fields , 1978 .

[4]  R. Adler,et al.  The Geometry of Random Fields , 1982 .

[5]  Robert J. Adler,et al.  Extrema and level crossings of χ2 processes , 1986, Advances in Applied Probability.

[6]  D. Aldous Probability Approximations via the Poisson Clumping Heuristic , 1988 .

[7]  R. Adler,et al.  Sample path behaviour of Χ 2 surfaces at extrema , 1988, Advances in Applied Probability.

[8]  D. Naiman,et al.  INCLUSION-EXCLUSION-BONFERRONI IDENTITIES AND INEQUALITIES FOR DISCRETE TUBE-LIKE PROBLEMS VIA EULER CHARACTERISTICS , 1992 .

[9]  Karl J. Friston,et al.  Functional Connectivity: The Principal-Component Analysis of Large (PET) Data Sets , 1993, Journal of cerebral blood flow and metabolism : official journal of the International Society of Cerebral Blood Flow and Metabolism.

[10]  Karl J. Friston,et al.  Assessing the significance of focal activations using their spatial extent , 1994, Human brain mapping.

[11]  Sergio N. Torres Topological Analysis of COBE-DMR Cosmic Microwave Background Maps , 1994 .

[12]  Changbom Park,et al.  Topological analysis of the CfA redshift survey , 1994 .

[13]  K. Worsley,et al.  Local Maxima and the Expected Euler Characteristic of Excursion Sets of χ 2, F and t Fields , 1994, Advances in Applied Probability.

[14]  F. Gonzalez-Lima,et al.  Structural equation modeling and its application to network analysis in functional brain imaging , 1994 .

[15]  D. Siegmund,et al.  Testing for a Signal with Unknown Location and Scale in a Stationary Gaussian Random Field , 1995 .

[16]  R. Turner,et al.  Characterizing Dynamic Brain Responses with fMRI: A Multivariate Approach , 1995, NeuroImage.

[17]  R. Woods,et al.  Principal Component Analysis and the Scaled Subprofile Model Compared to Intersubject Averaging and Statistical Parametric Mapping: I. “Functional Connectivity” of the Human Motor System Studied with [15O]Water PET , 1995, Journal of cerebral blood flow and metabolism : official journal of the International Society of Cerebral Blood Flow and Metabolism.

[18]  K. Worsley,et al.  Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics , 1995, Advances in Applied Probability.

[19]  K. Worsley Estimating the number of peaks in a random field using the Hadwiger characteristic of excursion sets, with applications to medical images , 1995 .

[20]  E. Bullmore,et al.  Functional Magnetic Resonance Image Analysis of a Large-Scale Neurocognitive Network , 1996, NeuroImage.

[21]  J. Haxby,et al.  Brain functional connectivity changes as task difficulty is altered , 1996, NeuroImage.

[22]  Karl J. Friston,et al.  Psychophysiological and Modulatory Interactions in Neuroimaging , 1997, NeuroImage.

[23]  Alan C. Evans,et al.  Time-Related Changes in Neural Systems Underlying Attention and Arousal During the Performance of an Auditory Vigilance Task , 1997, Journal of Cognitive Neuroscience.

[24]  J. Cao The size of the connected components of excursion sets of χ2, t and F fields , 1999, Advances in Applied Probability.

[25]  R. Adler On excursion sets, tube formulas and maxima of random fields , 2000 .