Testing Nonlinearity In Topological Organization Of Functional Brain Network Based On Surrogate Data Method

For the study of brain functional networks,a complex brain functional image is constructed by describing and estimating the statistical correlation of signals between different brain region of interest(ROI)or nodes.From a linear perspective,many explorations of functional and effectual connections has been existed,while brain signal has been shown to be nonlinear.However,the researches concerning the nonlinear characteristics of brain functional signals mainly focus on the nature of the signal itself,rather than directly evaluating the characteristics of brain networks.Here we propose a surrogate data method for nonlinearity of brain network research..Surrogate data method is a typical indirect method in nonlinear theory.Based on the hypothesis test,a new set of surrogate data is built and the feature parameters of the data are calculated.If there is significant difference between the original data and the surrogate data,the original hypothesis is rejected and the signals are considered to have nonlinear characteristics.The brain functional images,which are obtained by positron emission tomography(PET)and functional magnetic resonance imaging(fMRI).We collected the old group(mean age 56.3 years,110)and the young group(mean age 36.5 years,113)for PET images,while for fMRI,we collected the normal control group(mean age 21.13 years,198).After preprocessing,including correction,smoothing and segmentation,we set filter range and brain node so that each group of brain functional signal can be gained.Using surrogate data method,we can acquire the surrogate data set by disrupting the original signals,then each group of original network and surrogate network can be established by Pearson correlation.The small-world parameters(clustering coefficient Cp,the shortest path Lp,local efficiency Eloc,global efficiency Eg)are calculated for each population’s original network and surrogate network respectively.The results show that the surrogate network has the characteristics of both small-world network and random network in both PET and fMRI.Aiming at the sparsity of 0.23,the small-world parameters of the surrogate network agrees with normal distribution,the difference between parameters of original networks and surrogate networks is large while the standard deviation is very small in all groups.Therefore,we reject the null hypothesis and assume that the brain function network can present nonlinear characteristics.Furthermore,we adjust the number of subjects in the three groups,and make a significance test on the small-world characteristic parameters of the original network and surrogate network in different situations.It can be discovered clearly that the brain functional network is nonlinear invariably.In this paper,we presents a direct evaluation method for brain networks.Compared with direct methods in nonlinear theory,including correlation dimension,complexity and Lyapunov index,the surrogate data method can be used for brain functional images with short scan time and not depend on the length of time series,and is not susceptible to noise interference in signals,which will provide theoretical guidance for medical research in brain network.