Zhenhai Zhang^{1}, Kaiming Li^{2}, and Xiaoping Hu^{2}

^{1}Department of Electrical and Computer Engineering, University of California, Riverside, Riverside, CA, United States, ^{2}Department of Bioengineering, University of California, Riverside, Riverside, CA, United States

The work aims to investigate the nonlinear dynamics in CEN/DMN networks. To this end, we applied phase-space embedding and multivariate autoregressive modeling of the phase-space trajectory. Furthermore, the AR coefficients were analyzed with a linear discriminant analysis to identify principle features that distinguish between patients and controls. The method was able to reveal differences in nonlinear dynamics in CEN and DMN networks respectively and jointly.

The first dataset is from the UCLA Consortium for Neuropsychiatric Phenomics LA5c. We selected 50 schizophrenia (SZ) patients and 50 age-matched healthy control (HC) subjects

We constructed the phase space of the normalized ICA percentile time course $$$x_n$$$ for each subject, n is the length of the time course. We formed the matrix $$$Y$$$ related to the phase portrait as shown in eq. (1).

$$Y=(\mathbf{x}_t,\mathbf{x}_{t+\tau},\cdots,\mathbf{x}_{t+(m-1)\tau}), \qquad\qquad\qquad(1)$$

The optimal embedding dimension $$$m$$$ was determined by false nearest neighbor algorithm

After construction of the phase space, we used an MVAR model to compute the autoregression coefficients of the phase space

where $$$Y[j]$$$ is the $$$j^{th}$$$ row of $$$Y$$$, the model coefficients $$$A[i]$$$ are matrices of size $$$m×m$$$, $$$\mathbf{e}[j]$$$ is the error vector of the model, and $$$P$$$ is the order of the autoregression model. Thus, for each subject, we obtained $$$P×m×m$$$ coefficients. To reduce the number of features, a feature vector $$$\mathbf{x}$$$ corresponding to the largest eigenvalue was extracted with dimension $$$d_x$$$ less than $$$P×m×m$$$ using LDA. The autoregression order $$$P$$$ was set to 1. The entire approach is summarized in the flow diagram in Fig.1. The input is a normalized ICA percentile time course and the output is the feature vector $$$\mathbf{x}$$$.

The joint phase space between the two networks, $$$Y_M$$$, was constructed as the following four-dimensional (we set $$$m$$$ equal to $$$1$$$ and $$$\tau$$$ equal to $$$4$$$) phase space matrix.

$$Y_M (t)=(\mathbf{c}_t,\mathbf{c}_{t+τ},\mathbf{d}_t,\mathbf{d}_{t+τ}), \qquad\qquad\qquad (3)$$

where $$$\mathbf{c}$$$ and $$$\mathbf{d}$$$ (length equals $$$n-(m-1)\tau$$$ ) represent the CEN and DMN percentile time series respectively. The model coefficients $$$A[i]$$$ had a size of size $$$4×4$$$ in this case. A similar process was performed to obtain the features.

The comparison with the principal feature vector between SZ and HC for the two datasets is shown in Table I and II, respectively. ANOVA with post-hoc analysis using Fisher's Least Significant Difference (LSD) showed a significant difference between SZ and HC for both datasets ($$$p<0.05$$$) in CEN, DMN and their interaction.

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Fig.1. Processing of a single network time course.

Fig.2. (a) The optimal time delay for UCLA dataset; (b) The optimal time delay for COBRE dataset; (c) The optimal embedding dimension for UCLA dataset; (d) The optimal embedding dimension for COBRE dataset.

TABLE I: Principal component
vector of autoregression matrices comparison between two groups of UCLA
subjects

TABLE II: Principal component vector of autoregression
matrices comparison between two groups of COBRE subjects