Joint Analysis of Band-Specific Functional Connectivity and Signal Complexity in Autism

Abstract

Examination of resting state brain activity using electrophysiological measures like complexity as well as functional connectivity is of growing interest in the study of autism spectrum disorders (ASD). The present paper jointly examined complexity and connectivity to obtain a more detailed characterization of resting state brain activity in ASD. Multi-scale entropy was computed to quantify the signal complexity, and synchronization likelihood was used to evaluate functional connectivity (FC), with node strength values providing a sensor-level measure of connectivity to facilitate comparisons with complexity. Sensor level analysis of complexity and connectivity was performed at different frequency bands computed from resting state MEG from 26 children with ASD and 22 typically developing controls (TD). Analyses revealed band-specific group differences in each measure that agreed with other functional studies in fMRI and EEG: higher complexity in TD than ASD, in frontal regions in the delta band and occipital-parietal regions in the alpha band, and lower complexity in TD than in ASD in delta (parietal regions), theta (central and temporal regions) and gamma (frontal-central boundary regions); increased short-range connectivity in ASD in the frontal lobe in the delta band and long-range connectivity in the temporal, parietal and occipital lobes in the alpha band. Finally, and perhaps most strikingly, group differences between ASD and TD in complexity and FC appear spatially complementary, such that where FC was elevated in ASD, complexity was reduced (and vice versa). The correlation of regional average complexity and connectivity node strength with symptom severity scores of ASD subjects supported the overall complementarity (with opposing sign) of connectivity and complexity measures, pointing to either diminished connectivity leading to elevated entropy due to poor inhibitory regulation or chaotic signals prohibiting effective measure of connectivity.

Appendices

Appendix 1: Computation of Multi-Scale Entropy

Let ${\mathbf{x}}_{N} (t_{i} ) = \left[ {x(t_{i} ),x(t_{i} + T),\ldots ,x(t_{i} + (N - 1)T)} \right]$ represent a vector of length N starting at time point t _i of a signal (or a time series) recorded by the sampling period T. SE is defined as the logarithmic conditional probability that two similar sequences of length m, i.e. x _m (t _p ) and x _m (t _q ), remain similar when added one sample to their length, i.e. x _m+1 (t _p ) and x _m+1 (t _q ). This can be quantified as follows

$$ SE = \log \frac{{\sum\limits_{i = 1}^{N - m} {n_{i}^{m} (r)} }}{{\sum\limits_{i = 1}^{N - m} {n_{i}^{m + 1} (r)} }} $$

where $n_{i}^{m} (r)$ is the number of m-length vectors x _m (t _j ) which are within a distance of r to x _m (t _i ) when self matches are not counted (Richman and Moorman 2000; Richman et al. 2004; Costa et al. 2005). The distance based on which the similarity between the vectors x _m (t _i ) and x _m (t _j ) are calculated is defined as the maximum absolute difference between their elements (Costa et al. 2005). Figure 10 illustrates how this SE is calculated.

Fig. 10

A simulated time series x(t) of 50 samples illustrating the SE calculation when m = 2 and r = 0.2σ _x . The green, pink, and red dashed lines around the first three datapoints x(0), x(0.02), and x(0.04) represent the area of ±r around these points, respectively. Here, a data point is determined matched with the first data point, if it is within the range of x(0) ± r. All those similar to the first data point are shown by green circles, and accordingly, pink and red circles are used for those matched with the second and third data points, respectively. In the segment of time series shown here, there are two two-component templates of green-pink matched with [x(0), x(0.02)] while there is only one three-component templates of subsequent green-pink-red points similar to [x(0), x(0.02), x(0.04)]. This procedure is repeated to calculate the ration of all two- and three-component template matches, from which the SE is calculated (Color figure online)

Multi-scale entropy (MSE) is then defined as the sample entropies measured at consecutive coarse grained time series y ^τ corresponding to the scale τ. The coarse grained procedure is performed by

$$ y^{\tau } (t_{j} ) = \frac{1}{\tau }\sum\limits_{{i = \left( {j - 1} \right)\tau + 1}}^{j\tau } {x(t_{j} )} ,\quad \quad \quad 1 \le j \le \frac{N}{\tau } $$

in which at each scale τ the data points are first divided into nonoverlapping segments of length τ, and then the data points inside each segment are averaged to calculate the corresponding $y_{j}^{\tau }$. Figure 11 illustrates the coarse grained time series at the scale 3.

Fig. 11

An illustration of the coarse-grained time series at the scale 3

By computing SE at various scales, MSE curves can be generated and used to compare the relative complexity of the power-normalized time series. The minimum, maximum, or average of the MSE measures along the scales can be used as different scalar measures of complexity. The two parameters m and r in MSE need to be set for proper analysis. It has been shown (Costa et al. 2005; Takahashi et al. 2010; Bosl et al. 2011; Catarino et al. 2011) that a good statistical validity of MSE is achieved when m = 1 or 2 and r = 0.1σ – 0.25σ where σ represents the SD of the time series at scale τ = 1 (i.e. SD of the original time series). In this work, we empirically chose m = 2 and r = 0.2σ after investigating different values. This procedure is performed for each sensor and each frequency band. A higher MSE value implies lower repeatability and accordingly higher signal complexity.

Appendix 2: Synchronization Likelihood

The time–frequency SL technique assumes that two signals are synchronized if a pattern of one signal repeats itself at certain time instants for a number of times within a certain period and another pattern in the other signal repeats itself at those same time instants (Montez et al. 2006).

For a given signal at channel k, i.e. x _k (t), the signal pattern at any given time instant t _i can be represented by an embedding vector $\mathbf{x}_{{k,t_{i} }} = \left[ {x_{k} (t_{i} ),x_{k} (t_{i +l} ), \ldots ,x_{k} (t_{i + (m - 1)l} )} \right]$ where l is the lag and m is the length of the embedding vector. l and m are typically set to $l = \frac{{f_{s} }}{{3h_{f} }}$ and $m = \frac{{3h_{f} }}{{l_{f} }} + 1$ where f _s is the sampling frequency, and h _f and l _f are the high and low frequency contents of the signal, respectively. At each time instant t _i , the Euclidean distance is then measured between the reference embedding vector ${\mathbf{x}}_{{k,t_{i} }}$ and the set of all other embedding vectors at times t _j , i.e. ${\mathbf{x}}_{{k,t_{j} }}$, where t _j lies in the range $t_{i} - \frac{{t_{{w_{2} }} }}{2} < t_{j} < t_{i} - \frac{{t_{{w_{1} }} }}{2}$ or $t_{i} + \frac{{t_{{w_{1} }} }}{2} < t_{j} < t_{i} + \frac{{t_{{w_{2} }} }}{2}$ where $t_{{w_{1} }} = \frac{{2l\left( {m - 1} \right)}}{{f_{s} }}$ and $t_{{w_{1} }} < t_{{w_{2} }}$. Then, n _ref nearest embedding vectors ${\mathbf{x}}_{{k,t_{j} }}$ are retained. This procedure is conducted for each channel k and each time instant t _i . The SL between channel k ₁ and channel k ₂ at time instant t _i is the number of simultaneous embedding vector recurrences in the two channels divided by the total number of recurrences, i.e. $SL_{{t_{i} }} = \frac{{n_{{k_{1} k_{2} }} }}{{n_{ref} }}$. Figure 12 illustrates how SL is calculated.

Fig. 12

An illustration of synchronization likelihood between two signals from two channels plotted by different colors. A reference pattern was selected at the time 0.074 s (thick rectangle). The recurrences in both signals are shown by thin rectangles for n _ref  = 10. Vertical arrows show simultaneous recurrences in both channels. SL at time 0.074 is therefore equal to the ratio between number of simultaneous recurrences and the number of n _ref , i.e. SL = 5/10 = 0.5

In this work, the parameters adjustments were performed for each frequency band according to the aforementioned equations, which are shown in Table 2.

Table 2 The parameters in measuring the synchronization likelihood for connectivity networks

Recording from 274 sensors, we obtained a three dimensional SL matrix of 274 × 274 × T per recording per frequency, where T is the number of time instants at which $SL_{{t_{i} }}$ is calculated. The functional connectivity matrix is then defined as the average of all the T matrices along the time instants, yielding a symmetric 274 × 274 SL matrix (per band) whose elements range between 0 and 1.

The SL connectivity matrix of each subject at each band represents a weighted graph in which each sensor represents the graph node and each SL connection between two sensors represents the graph edge weighted by their SL value. The node strength in a graph is defined as the summation of the edges between a node and all other graph nodes, i.e. $NS_{i} = \sum\nolimits_{j = 1,j \ne i}^{274} {SL\left( {i,j} \right)}$ where NS _i is the node strength of sensor i and SL(i,j) is the SL value between sensors i and j. Node strength of the connectivity graph can be interpreted as the relative strength of the corresponding node region to connect with other regions.