Symbolic time series analysis of fNIRS signals in brain development assessment

Abstract

Objective:

Assessing infant’s brain development remains a challenge for neuroscientists and pediatricians despite great technological advances. As a non-invasive neuroimaging tool, functional near-infrared spectroscopy (fNIRS) has great advantages in monitoring infant’s brain activity. To explore the dynamic features of hemodynamic changes in infants, in-pattern exponent (IPE), anti-pattern exponent (APE), as well as permutation cross-mutual information (PCMI) based on symbolic dynamics are proposed to measure the phase differences and coupling strength in oxyhemoglobin (HbO) and deoxyhemoglobin (Hb) signals from fNIRS.

Approach:

First, simulated sinusoidal oscillation signals and four coupled nonlinear systems were employed for performance assessments. Hilbert transform based measurements of hemoglobin phase oxygenation and deoxygenation (hPod) and phase-locking index of hPod (hPodL) were calculated for comparison. Then, the IPE, APE and PCMI indices from resting state fNIRS data of preterm, term infants and adults were calculated to estimate the phase difference and coupling of HbO and Hb. All indices’ performance was assessed by the degree of monotonicity (DoM). The box plots and coefficients of variation (CV) were employed to assess the measurements and robustness in the results.

Main Results:

In the simulation analysis, IPE and APE can distinguish the phase difference of two sinusoidal oscillation signals. Both hPodL and PCMI can track the strength of two coupled nonlinear systems. Compared to hPodL, the PCMI had higher DoM indices in measuring the coupling of two nonlinear systems. In the fNIRS data analysis, similar to hPod, the IPE and APE can distinguish preterm, term infants, and adults in 0.01–0.05 Hz, 0.05–0.1 Hz, and 0.01–0.1 Hz frequency bands, respectively. PCMI more effectively distinguished the term and preterm infants than hPodL in the 0.05–0.1 Hz frequency band. As symbolic time series measures, the IPE and APE were able to detect the brain developmental changes in subjects of different ages. PCMI can assess the resting-state HbO and Hb coupling changes across different developmental ages, which may reflect the metabolic and neurovascular development.

Significance:

The symbolic-based methodologies are promising measures for fNIRS in estimating the brain development, especially in assessing newborns’ brain developmental status.

1. Introduction

Evaluating brain development is always a challenge in neuroscience and a clinical concern for the pediatricians. Existing studies indicate that earlier exposure to the extra-uterine environment (Watanabe et al., 2017), postmenstrual age (Arimitsu et al., 2018), birth weight (Streimish et al., 2012), and the intrauterine/neonatal insults (Mwaniki et al., 2012; Thomason et al., 2017) may influence preterm infants’ brain developmental trajectories. Preterm infants have a higher rate of neurodevelopmental diseases than term infants (Volpe, 2009;Watanabe et al., 2017). However, present studies still lack effective methods to access neonates’ brain development status.

Functional near-infrared spectroscopy (fNIRS) is an optical brain imaging tool used in neuroscience. With the portable and non-invasive characteristics, as well as the merit of being tolerated by infants and children, the fNIRS is more suitable for assessing neonates’ brain development (Issard and Gervain, 2018;Quaresima et al., 2012). fNIRS also allows bedside monitoring for long periods. All these features give unique advantages of fNIRS over functional Magnetic Resonance Imaging (fMRI) in the studies of newborns (Benavides-Varela et al., 2017; Issard and Gervain, 2017).

In developmental neuroscience, fNIRS studies of infants have shown that the hemodynamic response has canonical and non-canonical responses and changes with age in different cognitive tasks and brain regions (Issard and Gervain, 2018; Lloyd-Fox et al., 2017). In term and preterm infants, Watanabe et al. found that the time-averaged phase differences between oxyhemoglobin (HbO) and deoxyhemoglobin (Hb) in spontaneous low-frequency oscillation (<0.1 Hz) were highly correlated with the chronological age (CA) (Watanabe et al., 2017). The phase of oxygenation and deoxygenation hemoglobin (hPod) changes from in-phase to anti-phase as the CA increases. More fluctuations in hPod values were found in early preterm infants than in late preterm and term infants before 40 weeks postmenstrual age. However, early preterm infants had a slower development at later CAs (i.e., after 8–13 weeks). Furthermore, Taga et al. employed the hPod and the phase-locking index of hPod (hPod_L) to investigate the spatial variation in infants’ developing brain cortex (Taga et al., 2018). They found that hPod exhibited spatial dependency in different brain development stages. Significant hPod_L increases occurred in 3-month-old infants compared to the neonate group. All these studies suggest that the dynamic phase difference between HbO and Hb have potential value in assessing the development of underlying neurovascular functions, as well as hemodynamic and metabolic changes, in infants.

In these studies, the instantaneous phase estimated in hPod and hPod_L methodologies are based on the Hilbert transform, which is based on the assumption that the signal has narrow frequency band and stationary (Cohen et al., 1999; Oliveira and Barroso, 1999). However, brain development is a complex process that may be influenced by nonlinear circulatory and neurovascular developmental changes in different states, especially in infants (Norman and O’Kusky, 1986; Franceschini et al., 2007a; Roche-Labarbe et al., 2010). Many studies have shown that brain activities’ dynamic responses measured by fNIRS signals are very likely to be nonlinear (Khoa et al., 2008; Pouliot et al., 2012). Nonlinearity from neural activity, blood flow, and metabolism makes the evaluations based on linear analyzing methods inadequate (Fantini, 2014; Bießmann et al., 2011; Sassaroli et al., 2016). Moreover, oscillations are a prevalent feature of neuronal and hemodynamic recordings. Some methods used to analyze these physiological oscillations assume that they are largely sinusoidal (Cole and Voytek, 2017). However, physiological oscillations are usually pseudo-periodic and contain multiple physiological fluctuations (Cole and Voytek, 2017). Phase synchronization measurement is an important methodology used to interpret the mechanisms of electrophysiological and hemodynamic variations in cognitive neuroscience and mental diseases. Unfortunately, most phase synchronization analytical methods based on the Fourier transform and Hilbert transform cannot optimally represent a non-stationary signal (Li et al., 2011).

The biological system’s nonlinear characteristics have motivated researchers to investigate its underlying mechanism via nonlinear methodologies. Various measurements have been proposed for fNIRS signal analysis, such as wavelet phase coherence (Tan et al., 2016), graph theory (Homae et al., 2010), entropy (Gu et al., 2017; Perpetuini et al., 2018), and mutual information (Dalmis and Akin, 2015; Yin et al., 2015). In all these nonlinear measures, the symbolic dynamic analysis is an important nonlinear measurement, which has been widely applied to physiological system analysis (Edwards et al., 2001; Ray, 2004; Daw et al., 2003; Bießmann et al., 2011). A central procedure of the symbolic dynamic analysis is discretizing unprocessed time-series records into a corresponding sequence of symbols by comparing neighboring time-series (Bandt and Pompe, 2002). Amigo et al. stated that the signal’s ordinal patterns are not only ad hoc symbols but also contain quantitative information regarding the underlying data’s temporal scale (Amigo et al., 2015). Due to the advantages described above, various symbolic dynamic-based measurements, such as permutation entropy, symbolic transfer entropy, and permutation min-entropy, have been widely used in neurological disease diagnosis and brain state monitoring (Zunino et al., 2015; Martini et al., 2011; Ferlazzo et al., 2014), especially in fNIRS recording analysis (Gu et al., 2017). A multi-scale symbolic information-theory approach has also been proposed to discriminate delayed synchronization and anticipated synchronization (Montani et al., 2015). In this study, we propose two new exponents based on the patterns of symbolic dynamics, namely in-pattern exponent (IPE) and anti-pattern exponent (APE), to measure the phase relationship between HbO and Hb.

Although many synchronization measurements have been proposed, detecting the weak couplings in physiological recordings remains challenging considering the complex, nonlinear, and nonstationary characteristics of a physiological system (Palus and Vejmelka, 2007; Bahraminasab et al., 2008). Symbolic dynamic-based permutation analysis and conditional mutual information were proposed to estimate the weak coupling in two cardiorespiratory series (Bahraminasab et al., 2008). The similar approaches have been used to estimate neuronal population coupling during seizures (Li and Ouyang, 2010) and EEG oscillations in patients under anesthesia (Liang et al., 2015). It was suggested that the permutation cross mutual information (PCMI) approach can track time-dependent coupling strength changes, which is associated with a better anti-noise effect. This encouraged us to utilize the PCMI in measuring HbO and Hb coupling in fNIRS and apply it to distinguishing brain development at different stages.

In this study, we used the IPE and APE, as well as PCMI, which are derived from symbolic dynamics and information theory to assess the brain development changes at different stages. These methods were applied to the same data as the hPod and hPod_L methods proposed in the previous literature (Watanabe et al., 2017; Taga et al., 2018). The whole process is described below: Section 2 presents IPE, APE, PCMI, hPod, hPod_L and statistical analysis methods used in detail. In Section 3, comparisons are made based on simulated sinusoidal oscillations, coupled nonlinear models are described in detail. An analysis of fNIRS recordings in preterm and term infants and adults based on these measures is detailed in section 4. Finally, the discussion and conclusions are presented in Section 5.

2. Methods

2.1. In-pattern exponent and anti-pattern exponent

Symbolic dynamic measurements used to encode nonlinear systems based on time-series analysis have been systematically described in previous studies(Daw et al., 2003; Ray, 2004). In this study, we explored the temporal interrelationship of specific patterns in two time-series (i.e., HbO and Hb signals). The resulting exponent was used to evaluate the degree of in-phase or anti-phase in these two time-series. The percentage of in-phase and anti-phase measurements was calculated as an exponent to measure synchronization in these two time-series. We termed these two indices in-pattern exponent (IPE) and anti-pattern exponent (APE), to separate from the phase measures of the Hilbert transform. An algorithm of the diagram was shown in Figure 1. When the embedding dimension m = 3, there are 3!= 6 patterns (as shown in Figure 1A). We termed these 6 patterns as M #1 to M #6. For these 6 patterns, the anti-pattern pairs and in-pattern pairs were shown in Figure 1B. The calculation of IPE and APE is straightforward. For two time series, such as HbO and Hb, the symbolic patterns can be constructed at each time point using a moving window. Then, the number of in-pattern pairs N_i and anti-pattern pairs N_a can be achieved. If the data length is L, the maximum number of in-pattern pair or anti-pattern pairs would be L – m + 1. Finally, the IPE and APE can be calculated as shown below:

$$\{\begin{array}{c}IPE=\frac{N_i}{L-m+1}\times 100\%\\ APE=\frac{N_a}{L-m-1}\times 100\%\end{array}$$ (1)

When m ≥ 4 the symbolic patterns can describe more sophisticated signals, however, the in-pattern pairs and anti-pattern pairs will become much more complicated than when m = 3. Since the hemodynamic changes are relatively slow, the choice of m = 3 is sufficient. A more detailed description of the parameter selection was shown in Appendix B.

Figure 1.

The diagram of anti-pattern and in-pattern percent measurements. (A) The motifs of the order 3 (3!). (B) The lists of the anti-pattern and in-pattern pairs. (C) Two original hemodynamic parameters (HbO and Hb) epochs of 30 s segments. In the time point near 5 s, the patterns of HbO and Hb are anti-pattern with each other (M #1 vs M#6). While the patterns of HbO and Hb are in-pattern with each other at the time near 20 s (M# 6 vs M#6). (D) The in-pattern and anti-pattern percent of the signal in (C).

2.2. Permutation cross mutual information

Mutual information is a measurement of synchronization based on information theory, which calculates the amount of shared information between two time-series (Paluš, 1996). The PCMI proposed by Li et al. (Li and Ouyang, 2010) has been applied to estimate the synchronization between two EEG signals (Liang et al., 2016b; Liang et al., 2015). The details of PCMI are described as follows:

• (i)A phase space reconstruction method was used to construct the vectors $X_t\left[x_t,x_{t+\tau },\dots ,x_{t+m\tau }\right]$ and $Y_t\left[y_t,y_{t+\tau },\dots ,y_{t+m\tau }\right]$ based on two time series of x(t) and y(t), t =1,2,⁖n where m and τ are the embedding dimension and time lag, respectively.
• (ii)X_t and Y_t were ranked in increasing order, $\left[x_{t+(j_1-1)\tau }\le x_{t+(j_2-1)\tau }\le \cdots \le x_{t+(j_m-1)\tau }\right]$ and$\left[y_{t+(j_1-1)\tau }\le y_{t+(j_2-1)\tau }\le \cdots \le y_{t+(j_m-1)\tau }\right]$ , respectively.
• (iii)Probability distribution functions of the time series x(t) and y(t) were calculate based on the emerged probability of ordinal patterns and termed p(x) and p(y).
  The entropy of X_t and Y_t is defined as follows:

  $$H(X)=-{\displaystyle \sum _{j=1}^J{p}_j(x)\log p_j(x)}$$ (2)

  and

  $$H(Y)=-{\displaystyle \sum _{j=1}^J{p}_j(y)\log p_j(y)}$$ (3)
• (iv)
  The joint probability function of X_t and Y_t is termed as p(x, y). The joint entropy of H(X, Y) is defined as

  $$H(X,Y)=-{\displaystyle \sum _{x\in X}{\displaystyle \sum _{y\in Y}p(x,y)\log p(x,y)}}$$ (4)
  (v) The PCMI of time series X_t and Y_t is described as

  $$PCMI(X;Y)=H(X)+H(Y)-H(X,Y)$$ (5)

  The embedding dimension m is crucial in PCMI calculation, same as the lag τ that is the number of sample points spanned by each section of the motif (Li and Ouyang, 2010). A detailed description of the parameters’ selections is shown in Appendix C

2.3. hPod and hPod_L

hPod and hPod_L are the two measurements of phase synchronization. The hPod algorithms described in (Watanabe et al., 2017) are similar to the spatial analytic phase difference index presented in (Pockett et al., 2009). And the hPod_L measurement is a classical phase-locking value proposed by Lachaux et al. (Lachaux et al., 1999) and has been widely used in the neurophysiological signal analysis (Wang et al., 2014; Li et al., 2011). Considering two time-series of x(t) and y(t) , the calculations of hPod and hPod _L are described as below.

(1) The analytic signal representation of x(t) and y(t) was calculate based on the Hilbert transform.

$$\{\begin{array}{c}{x}_a(t)=x(t)+jHT[x(t)]\\ y_a(t)=y(t)+jHT[y(t)]\end{array}$$ (6)

Where the HT[·] is the Hilbert transform.

(2) The signal’s instantaneous phase (IP) was estimated:

$${\phi }_x(t)={\tan }^{-1}\frac{\mathrm{Im}(x_a(t))}{\mathrm{Re}(x_a(t))}$$

and

$${\phi }_y(t)={\tan }^{-1}\frac{\mathrm{Im}(y_a(t))}{\mathrm{Re}(y_a(t))}.$$

(3) hPod was calculated based on the IP. Because the IP is wrapped around [−π, π], we needed to unwrap the IP on the real axis. Firstly, we estimated the absolute difference between the IP of φ_x and φ_y, termed Δφ_xy. Then, Δφ_xy was projected into the interval [0,2π]. Finally, to achieve a consistent phase range with the measure in (Watanabe et al., 2017), the phase differences in the [0,π] range were projected into the [π,2π] range.

(4) hPod_L was calculated. The measurement of hPod _L is defined as:

$$hPo{d}_L=\frac{1}{N}\left|{\displaystyle {\sum }_{t=1}^N{e}^{j\Delta {\phi }_{xy}(t)}}\right|$$ (7)

Where the N is the length of the time series x(t) and y(t). The hPod_L is bounded between 0 and 1.

2.4. Statistical analysis

The aim of this study was to (1) evaluate the performance of symbolic measurements (i.e., IPE, APE, and PCMI) in assessing the brain developmental stages, and (2) compare them with calculated hPod and hPod_L. Given the indices of hPod are angle values (i.e., different from numerical values), a circular statistics toolbox was used to perform statistical analysis (Berens, 2009). The Watson-Williams test was used to evaluate whether the mean phase of two or more groups is identical, and the parametric Watson-Williams multi-sample test was used to determine the significant differences between groups. For other indices, the Liliefors test (lillietest.m) was performed to determine whether the data had a normal distribution. Kruskal-Wallis test (kruskalwallis.m) and Multiple comparison tests (multcomare.m) were used to determine the significant differences between the indices in different age groups. The Bonferroni correction was used to prevent multiple comparison problems with p<0.05/(number of channels). The coefficient of variation (CV), calculated from the ratio of the standard deviation (SD) over the mean, was employed to assess the index stability in brain development measurements (Li et al., 2008).

3. Stimulation and results

3.1. The simulated signals

To evaluate IPE’s and APE’s performance in phase difference estimation, we Simulated two time-series by combining two sinusoid waves with different frequencies across three situations (i.e., in-phase, anti-phase, and orthogonal-phase). Furthermore, two random time-series were used to assess the index’s performance in evaluating the noise signals’ impact. All the formulas are described as follows: (1) In phase sinusoid equations

$$\{\begin{array}{c}x(t)=20\sin (0.4\pi t)+15\sin \left(0.2\pi t\right)\\ y(t)=0.3x(t)\end{array}$$ (8)

(2) Anti-phase sinusoid equations

$$\{\begin{array}{c}x(t)=20\sin (0.4\pi t)+15\sin (0.2\pi t)\\ y(t)=20\sin (0.4\pi t-\pi )+15\sin (0.2\pi t-\pi )\end{array}$$ (9)

(3) Orthogonal-phase sinusoid equations

$$\{\begin{array}{c}x(t)=20\sin (0.4\pi t)+15\sin (0.2\pi t)\\ y(t)=20\sin (0.4\pi t+0.5\pi )+15\sin (0.2\pi t+0.5\pi )\end{array}$$ (10)

Random signals were generated by the MATLAB function randn.m. The amplitude of the signal y(t) is 3 times larger than x(t). The data length of all the simulation signals is 500 s.

3.2. The coupling models

In this study, we employed four models to evaluate the coupling performance of PCMI and hPod_L. The coupling strength of bivariate dynamics was controlled by one continuous parameter. The first is the Rossler-Lorenz model (Andrzejak et al., 2003):

$$\{\begin{array}{l}{\dot{x}}_1=-6(x_2+x_3)\hfill \\ {\dot{x}}_2=6(x_1+0.2x_2)\hfill \\ {\dot{x}}_3=6[(x_1-5.7)x_3+0.2]\hfill \\ {\dot{y}}_1=10(-y_1+y_2)\hfill \\ {\dot{y}}_2=28y_1-y_2-y_1{y}_3+\epsilon x_2^2\hfill \\ {\dot{y}}_3=y_1{y}_2-\frac{8}{3}{y}_3\hfill \end{array}$$ (11)

The coupling strength parameter ranged from ɛ = 0 to ɛ = 5 in a step of 0.2. The discrete time series X_n and Y_n from x₁ and y₁ were resampled at the sampling rate at 40 Hz (Liang et al., 2015).

The second model system consisted of two coupled Henon maps proposed in the literature (Schiff et al., 1996). The third and fourth models were made by two coupled Rossler (Palus and Stefanovska, 2003) and Lorenz systems (Kreuz et al., 2007), respectively. In this study, the coupling strength parameter in Henon was ranged from 0 to 0.8 with a step of 0.01. The coupling strength parameters in Rossler and Lorenz systems were set from 0 to 2 in step of 0.025 (Liang et al., 2016a).

3.3. Evaluation of the model

We hypothesized that an increased coupling strength is linearly correlated with increased of synchronization. The degree of monotonicity (DoM) was used to evaluate the dependence of hPod_L and PCMI measures on the coupling strength (Kreuz et al., 2007).

The formula of DoM is as follows:

$$DoM=\frac{2}{r(r-1)}{\displaystyle \sum _{i=1}^{r-1}{\displaystyle \sum _{j=i+1}^{r}sign(s_j-s_i)}}$$ (12)

Where s_i and s_j are the coupling measure indices (i.e., hPod_L and PCMI) at monotonously-increased coupling strengths. i, j = 1,2,…,r is the number of the discretized coupling strengths. If indices of s monotonically increased with the enhancing coupling strength ɛ, then s_i < s_j , i ≤ j . DoM = 1 when the sequence of $s_1,s_2,\dots ,s_r$ is strictly monotonous increases with the enhanced coupling strengths, while DoM = –1 when the indices monotonically decrease with the increasing coupling strengths.

3.4. Results

To compare IPE’s and APE’s performance with hPod via the phase relationship between two time-series, we employed simulated and random noise signals. hPod_L and PCMI were also compared in estimating coupling strength in different coupled models. We studied the anti-phases, in-phases, and orthogonal-phases of two sinusoidal signals, which were shown in Figures 2A, S1A, and S2A, respectively. Two random noise signals, which had random phase and no coupling between them, were used for assessment (see Figure S3A). The hPod index was able to accurately reflect anti-phases, in-phases, and orthogonal-phases (see Figures 2B, S1B, and S2B). Like hPod, the IPE and APE can precisely measure the same phase difference between these three situations. The IPEs were equal to 1 and 0 in in-phase and anti-phase patterns respectively (see Figures 2D and S1D). For the orthogonal phase relationship, IPE and APE were distributed around 0.42, which means the percent of anti-patterns and in-patterns in this situation is close to 0.5 (see Figure S2D). All these results illustrate that the IPE and APE have similar performance in measuring phase differences.

Figure 2.

The comparison of the hPod and symbolic dynamic based measurements. (A) Two in-phase time series. (B) The hPod indices of the two time series in (A). (C) The hPod length of the time series in (A). (D) The anti-pattern percent and in-pattern percent values of the two time series. (D) The PCMI indices of the two time series.

For the coupling strength measurement, the hPod_L indices remained 1 at in-phase, anti-phase, and orthogonal-phase scenarios (see Figures 2C, S1C, and S2C). The analysis of parameter selection presented in Appendix C shows that PCMI with an embedding dimension of m = 3 , combined with lag τ = 11 is the optimal selection. The PCMI indices in in-phase, anti-phase, and orthogonal-phase patterns were about 1.68, 1.68, and 0.64, respectively. Unlike hPod_L , the PCMI measurement regards the orthogonal-phase pattern to be a different coupling pattern from the in-phase and anti-phase patterns. We calculated the cross-correlation index of these two orthogonal-phase signals and we found that the correlation coefficient was equal to 0 when the lag=0 s. The different results for PCMI, hPod_L , and cross-correlation may be derived from the different calculation principles and mechanisms while estimating synchronization (Liang et al., 2016b).

Furthermore, the random noise simulation showed that the hPod indices were randomly distributed around the circle (see in Figure S3B). The hPod_L indices calculated from this time course are ranged from 0 to 0.2, which means that the two time-series are not correlated with each other. Hence, in this situation, the hPod values are invalid. The IPE and APE of these two random noise signals ranged in 0.1–0.3 and 0.3–0.5, respectively. The PCMI indices ranged from 0.02 to 0.1.

Additionally, to analyze the indices’ range in all these measures at random time series, we calculated 500 pairs random noise segments. The distributions of hPod_L , as well as the sums of IPE, APE (SPE), and PCMI were presented in Figures 3A, 3B, and 3C, respectively. Based on these distributions, we proposed a significant threshold for each measure (p<0.01) to avoid spurious results. The significant thresholds of hPod_L , SPE and PCMI are 0.20, 0.64 and 0.19, respectively.

Figure 3.

The histogram distributions of hPod_L (A), SPE (B) and PCMI (C) derived from the 500 random time series epochs.

hPod_L ‘s and PCMI’s performance in tracking the coupling strength was assessed further in four coupled nonlinear systems. Figures 4A and 4B showed the two simulated datasets in a coupled Rossler-Lorenz system, where coupling strength varied from 0 to 5 with a step of 0.2. The time series at the coupling strength of ɛ = 0 and 5 were presented in Figures 4C and 4D. Figure 4E showed the hPod_L , and PCMI of a coupled Rossler-Lorenz system at different coupling strengths. All three indices showed a rising trend with an increase in coupling strength. However, we also observed an abrupt increase in the coupling strength value around 2.2 that gradually decreased when the coupling strength value was greater than 3.5 in the hPod_L curve (FIGURE 4E).

Figure 4.

The simulation data generated based on the coupled Rossler-lorenz model. (A) and (B) are the time series from the variables of v₁ and w₁ in Rossler-lorenz model with dt = 0.025. (C)–(D) Time series of X_n and Y_n at two specific coupling strength ɛ = 0 and 5. (E) The hPod_L and PCMI indices versus different coupling strength of Rossler-Lorenz system.

DoM was used to evaluate hPod_L’s and PCMI’s dependence on the coupling strength. DoM values of hPod_L and PCMI for the coupled Hénon, Rössler, Lorenz, and Rossler-Lorenz (R-L) systems were shown in Figure 5. All the DoM values were shown in Table 1. Notably, the DoM values of PCMI were higher than those of hPod_L in every model. The results illustrate that the PCMI more accurately assessed coupling strength than hPod_L.

Figure 5.

The degree of monotonicity of hPod_L and PCMI indices for the coupled Henon, Rossler, Lorenz and Rossler-Lorenz systems.

Table 1.

The DoM values of hPod_L and PCMI in measuring the model of coupled Henon, Rossler, Lorenz and R-L.

	hPodL	PCMI
Henon	0.70	0.96
Rossler	0.85	0.94
Lorenz	0.73	0.77
R-L	0.40	0.77

4. Applications in fNIRS

4.1. fNIRS data recordings

We analyzed data from 10 term infants (6 boys and 4 girls), and 10 preterm infants (5 boys and 5 girls) during sleep. A detailed description of the participants was presented in Table 2. Ethical approval was obtained from the ethical committee of Keio University Hospital (No. 20090189), and written informed consent was acquired from the participants’ parent(s) prior to beginning the test. All the neonates were tested in a dimly-lit room when they were in active sleeping status as judged by their frequent motor activity and rapid eye movements.

Table 2.

Demographic characteristics of the neonates.

characteristics	Preterm	term
total(boy/girl)	10(6/4)	10(5/5)
GA(mean(min-max))(wk)	32.5(30.1–33.4)	38.3(37–39.9)
PNA(day)	23.7(17–35)	4.2(3–6)
Birth weight (mean(min-max)) (g)	1644.3(1302–1887)	2874.8(2476–3936)

GA= Gestational age;

PNA = Post natal age;

wk= week;

A NIRS system (ETG-4000, Hitachi Medical Corporation) with 46 channels was used to measure the relative fluctuations of HbO and Hb concentrations (millimolar millimeter (mM*mm)). Two sets of 3*3 array probes were mounted on the left and right temporal regions. And one 3*5 holder was mounted on the frontal region. The distance between the source and detector was about 2 cm. The sampling rate of the fNIRS recording was 10 Hz and the wavelengths of NIR lights were 695 and 830 nm. fNIRS coverage included frontal and temporal areas.

To compare developing and mature brains, we enrolled 8 adult volunteers (aged 22–40 years, 7 male, 1 female) for resting state testing. A fNIRS neuroimaging system (NIRx, NIRx Medical Technologies, LLC) was used for the experiment. All the volunteers were asked to close their eyes and lay flat on the bed in a dimly-lit room. Volunteers signed the written informed consent approved by the Purdue University ethics committee. The duration of the resting-state NIRS measurement was 15 min and a total of 20 channels were used to cover the prefrontal brain area.

4.2. Data preprocessing

Motion artifact is a major source of noises in fNIRS, especially in clinical tests. In this study, we used a kurtosis-based wavelet filtering for motion artifact correction (Chiarelli et al., 2015). Figure 6 showed the denoising process of HbO and Hb from one channel. Figure 6A presented the HbO and Hb time series before and after denoising. In one example, the transient noise caused by motion was detected at around 250 s. Figure 6B showed an example of motion correction for HbO and Hb (between 210s–270s). To analyze hemodynamic changes in preterm and term infants and adults, we used a zero-phase digital filter (Matlab function of filtfilt.m, the order of butterworth=3) for band-pass filtering. All the data were motion corrected and resampled to 2 Hz for the further calculation (Pinti et al., 2015).

Figure 6.

The motion denoising and bandpass filtering of HbO and Hb for one channel. (A) The motion artifact detection and denoising based on the Kurtosis-based wavelet algorithm. The red and blue curves are the original HbO and Hb, respectively. The cyan and black curves are the signals after removal of motion noise. The signals in the time range from 210 s to 270 s (pink rectangle) were contaminated by head movement. (B) The enlarged signals from the time range from 210s to 270s. (C) The low frequency band signals (0.01–0.1 Hz) of HbO (red) and Hb (blue), respectively.

In a previous study (Watanabe et al., 2017), the authors analyzed the hPod of three frequency bands: 0.05–0.1 Hz (main frequency), 0.01–0.05 Hz (low frequency), and 0.1–5 Hz (high frequency, which is dominated by pulsation/respiration and not the focus of our study). To be consistent with the previous study, we analyzed the phase difference and coupling of HbO and Hb in the 0.01–0.05 Hz, 0.05–0.1 Hz, 0.01–0.1 Hz frequency bands. Considering the low cut-off frequency of 0.01 Hz, the data length of IPE, APE, PCMI, hPod, and hPod_L used in the calculation was set as 120 s in 0.01–0.05 Hz and 0.01–0.1 Hz frequency bands. However, in the frequency band of 0.05–0.1 Hz, we have found that the minimum length of the data (i.e., experiment) can be as short as 30 s, which is preferable in the studies of infants. Detailed evaluations can be found in Appendix C and D (we only showed PCMI, which is the most complicated parameter).

4.3. Results

Figure 7 showed the distributions of APE versus IPE for preterm and term infants and adults in the 0.01–0.05 Hz (A), 0.05–0.1 Hz (B) and 0.01–0.1 Hz (C) frequency bands. We averaged the indices for all the channels and each circle represented the scatter distribution of one subject. The distributions of the adults’ data were clearly separated from the other two groups (i.e., term and preterm infants) in all three frequency bands. Because IPE and APE are complementary measures, the sum of the in-phase pattern and anti-phase proportions is close to 1 under the slow oscillations of HbO and Hb. For this reason, the IPE and APE indices were mostly scattered around the backslash line, which is a feature of the method we used and does not reflect physiological information. The clear separation of the clustering of IPE and APE along the line indicates the power of the method rather than the line itself. Figure 8 presented the hpod and hPod_L of preterm and term infants and adult subjects in the 0.01–0.05 Hz, 0.05–0.1 Hz, and 0.01–0.1 Hz frequency bands. It was shown that hPod indices of adults are near phase π in all these frequency bands, while the indices of the term and preterm infants are close to 3π/2 , especially in the 0.05–0.1 Hz frequency band. et al., 2017) and (Taga et al., 2018).

Figure 7.

The scatter plot of in-pattern percent versus anti-pattern percent of preterm, term, and adult in 0.01–0.05 Hz (A), 0.05–0.1 Hz (B) and 0.01–0.1 Hz (C), respectively.

Figure 8.

Vector representation of hPod and hPod_L (length of the vector) for preterm (A), term (B) and adults (C) in 0.01–0.05 Hz, 0.05–0.1 Hz, and 0.01–0.1 Hz, respectively.

To further investigate the ability of different measures to assess brain development, especially in the term and preterm infants, the indices of each measurement in each channel were considered samples for statistical analysis. A corresponding box plot was presented in Figures 9 and 10 for each measurement in each frequency band. In this study, the number of fNIRS recording channels in neonates (preterm and term) and adults are 46 and 20, respectively. To identify significant differences between them, we set the p-value at p 0.05/ 46 = 0.001. The indices were not normally distributed, so the statistics of the different groups were expressed as medians (min-max), as shown in Tables 3–5. The box plots in Figure 9 showed that the hPod, IPE and APE of the term and preterm infants and adults differed significantly (p<0.0002). Notably, hPod, IPE, and APE more precisely demarcated different brain development states. The overlaps of these three types of measures (i.e., hPod, IPE, and APE) between the adults and neonates (i.e., both term and preterm) were smaller than those between the term and preterm infants. Compared to the phase difference measures, the coupling strength measures (i.e., hPod_L and PCMI) were less precise in distinguishing the preterm and term (see Figure 10) neonates except for PCMI in the 0.05–0.1 Hz frequency band. The significant p-values of all indices were shown in Table 6. PCMI with epoch lengths of the 30 s, 60 s, and 90 s can distinguish preterm and term stages in the 0.05–0.1 Hz frequency band, which is better than the parallel measure of hPod_L (see Figures 10 and S6).

Figure 9.

The statistics of the phase relationship measures indices. (A)–(C) The box plots of the hPod indices for preterm, term, and adults in 0.01–0.05 Hz, 0.05–0.1 Hz and 0.01–0.1 Hz, respectively. (D)–(F) The box plots of IPE indices for preterm, term and adults in the similar frequency bands with (A)–(C). (G)–(I) The box plots of IPE indices for preterm, term, and adult in the similar frequency bands with (A)–(C). The symbol of “***” in each legend means that the p-value p < 0.001/46=0.00002.

Figure 10.

The statistics of the coupling strength measurements indices. (A)–(C) The box plots of hPod_L indices for preterm, term, and adults in 0.01–0.05 Hz, 0.05–0.1 Hz and 0.01–0.1 Hz, respectively. (D)–(F) The box plots of PCMI indices for preterm, term, and adults in the similar frequency bands with (A)–(C). The symbol of “***” in each legend means that the p-value p < 0.001/46=0.00002.

Table 3:

The statistics of different measurements with preterm, term, and adults in 0.01–0.05 Hz.

	Preterm	Term	Adults
	Median(min-max)	Median(min-max)	Median(min-max)
hPod	1.23(1.00–1.99)	1.37(1.00–1.99)	1.06(1.00–1.61)
IPE	0.37(0.09–0.79)	0.43(0.06–0.95)	0.18(0–0.52)
APE	0.56(0.13–0.85)	0.51(0.01–0.85)	0.75(0.42–0.98)
hPodL	0.51(0.02–0.97)	0.50(0.04–0.99)	0.73(0.08–0.99)
PCMI	0.43(0.16–0.94)	0.41(0.13–1.04)	0.59(0.23–1.46)

hPod = hemoglobin phase of oxygenation and deoxygenation;

IPE= in-pattern exponent;

APE = anti-pattern exponent;

hPod_L = phase-locking index of hPod;

PCMI= permutation cross mutual information with m = 3 ,τ = 11;

Table 5:

The statistics of different measurements with preterm, term, and adults in 0.01–0.1 Hz.

	Preterm	Term	Adults
	Median(min-max)	Median(min-max)	Median(min-max)
hPod	1.29(1.00–1.99)	1.41(1.00–1.99)	1.05(1.00–1.56)
IPE	0.40(0.13–0.78)	0.45(0.12–0.81)	0.16(0–0.49)
APE	0.49(0.11–0.79)	0.45(0.13–0.75)	0.75(0.42–0.99)
hPodL	0.48(0.03–0.90)	0.50(0.01–0.99)	0.73(0.07–0.99)
PCMI	0.41(0.17–0.97)	0.42(0.19–1.00)	0.63(0.21–1.68)

Table 6.

The p-values of different measurements with preterm, term, and adults in 0.01–0.1 Hz, 0.05–0.1 Hz and 0.01–0.1 Hz.

	Preterm-term	Preterm-adults	Term-adults
hPod	***/**/***	***/***/***	***/***/***
IPE	***/***/***	***/***/***	***/***/***
APE	***/***/***	***/***/***	***/***/***
hPodL	o/o/o	***/***/***	***/***/***
PCMI	o/ ***/ o	***/***/***	***/***/***

The symbols, “o”,”*”, “**” and “***” indicate the p-values p>0.05/46=0.001, p < 0.05/46=0.001, p < 0.01/46=0.0002 and p < 0.001/46=0.00002, respectively. The combination symbols “***/**/***” represent the p-values in 0.01–0.05 Hz, 0.05–0.1 Hz and 0.01–0.1 Hz, respectively. The Kruskal-Wallis test and Multiple comparison were applied.

Furthermore, the index assessments’ robustness in different states is important in assessing brain development, especially in neonates. We calculated the CV of all measurements within all subjects in different frequency bands. The CV values were shown in Tables 7–9. Because the CV value is vulnerable to the mean and standard deviation, to compare the CV of IPE, APE, and hPod consistently, we projected the hPod indices into the range of 0 to 1 (Liang et al., 2016b). All the CV values of IPE and APE were smaller than those of hPod in all frequency bands and the PCMI values in the 0.05–0.1 Hz frequency band, while the 30 s epoch length had the lowest CVs among all the measurements (0.07, 0.08, and 0.09). These results illustrated that IPE, APE, and PCMI had a higher tolerance for noise than hPod and hPod_L.

Table 7:

The CV indices of different measurements with preterm, term, and adults in 0.01–0.05 Hz.

	Preterm	Term	Adults
hPod	0.80	0.65	1.08
IPE	0.36	0.32	0.48
APE	0.24	0.29	0.12
hPodL	0.44	0.41	0.25
PCMI	0.34	0.37	0.30

Table 9:

The CV indices of different measurements with preterm, term, and adults in 0.01–0.1 Hz.

	Preterm	Term	Adults
hPod	0.67	0.57	1.04
IPE	0.25	0.22	0.51
APE	0.22	0.23	0.11
hPod L	0.40	0.37	0.25
PCMI	0.32	0.32	0..34

Spatial differences are also an important issue in brain development evaluation. We analyzed all the measurements in the prefrontal and left and right temporal areas of the infants. However, in this study, the fNIRS data were collected only from the prefrontal area of adults. Hence, we can only compare the prefrontal area measurements of infants and adults. The arrangement of fNIRS channels and brain partition in infants are shown in Figure S7, and a detailed analysis of all measurements was presented in Appendix E. There were significant differences in the prefrontal area among preterm term neonates and adults based on the phase difference measurements (i.e., APE, IPE, and hPod) (p<0.00002). However, these indices were not consistent with each other in distinguishing preterm and term infants in the left and right temporal regions. The coupling strength measurements (hPod_L and PCMI) can also differentiate between infant and adult brains in the prefrontal area, although there was no significant difference between preterm and term. Only the PCMI in the 0.05–0.1 Hz frequency band more precisely distinguished between preterm and term infants in the left and right temporal areas (see Figure S12).

5. Discussion and conclusions

Although brain development in neonates has been studied for a long time, and many fNIRS/fMRI algorithms have been proposed, no widely accepted theory has been established (Taga et al., 2018; Watanabe et al., 2017; Gu et al., 2017). The fNIRS and fMRI literature indicated that the hemodynamic response curve changes with age (Issard and Gervain, 2018) and the relative phase between the HbO and Hb can reflect complex interactions in neurovascular and metabolic development (Obrig and Villringer, 2003; Fantini, 2002; Boas et al., 2008). There are many different ways to measure age-dependent phase changes in neonates, especially the phase synchronization methodology based on the Hilbert transform (Taga et al., 2000; Imai et al., 2014). The goal of these studies was to investigate more effective and robust methodologies to evaluate brain development. Motivated by this goal, in this study, we attempted to evaluate several novel methods (i.e., IPE, APE, and PCMI) based on symbolic dynamics and information theory in assessing brain development. IPE’s, APE’s and PCMI’s performance were compared to those of hPod and hPod_L indices in both simulated and real fNIRS data. Results indicated that IPE and APE can effectively demarcate preterm and term infants and adults, and had smaller CV indices in all three different frequency bands than hPod. Furthermore, PCMI was superior in distinguishing neonates and adults than hPod_L , especially in the 0.05–0.1 Hz frequency band. A comparison and statistical analysis based on coupled nonlinear models also showed that symbolic dynamics-based measurements made more accurate predictions in tracking coupling strength.

One possible explanation is that IPE, APE, and PCMI are calculated based on the time series’ order patterns, which combines mutual information, symbolic dynamics, and nonlinear system theories. hPod and hPod_L are based on the FFT and Hilbert transform, which implicitly assumes that the physiological records are sums of periodical stationary signals. However, IPE, APE, and PCMI are based on the patterns’ sequential information , so this symbolic transform focuses on the temporal relationship between neighboring points instead of the oscillation magnitude (Bandt and Pompe, 2002). Hence, the phase relationship (APE and IPE) and coupling strength (PCMI) obtained from HbO and Hb signals at different stages will be presented in symbolic synchronization and mutual information. Moreover, compared to FFT and Hilbert transform based algorithms (hPod and hPod_L), symbolic dynamics and mutual information methodologies are nonparametric and required limited computing power (Bahraminasab et al., 2008). Furthermore, it does not need any underlying assumption that the time series or oscillations are stationary or sinusoidal (Talebinejad et al., 2011). More importantly, mutual information and symbolic transform analysis offer a unique perspective to estimate complex hemodynamic and neurovascular changes during brain development (e.g., the coupling strength of the HbO and Hb become stronger as the cerebral vascular system and neurovascular and metabolic functions develop). Lastly, as described in our previous studies, the symbolic dynamic and mutual information measures were insensitive to the time series’ amplitude, making them more tolerant to noise (Abásolo et al., 2006; Ferenets et al., 2006; Liang et al., 2015). Therefore, all these merits indicated the great potential of IPE, APE, and PCMI to be outstanding nonlinear measurements in evaluating brain development changes using fNIRS signals.

Although fNIRS can not measure neuronal activation directly, it can measure HbO and Hb concentration changes, which can be caused by changes in blood flow/volume, or neuronal activation through the neurovascular coupling. In this study, we systematically measured phase changes between Hb and HbO among infants during sleep (no stimulation). We will interpret the results from the perspective of the cerebrovascular development. From a previous study, it is known that brain development will cause the following vascular changes. (1) increased blood volume due to growing capillary density; (2) increased capillary and venous blood flow; and (3) increased mean arterial pressure (limited by cerebral auto-regulation (Greisen, 2005)). In addition to these vascular changes, Franceschini, et al. found that there were regionally specific increases in oxygen consumption in healthy infants during their first year (Franceschini et al., 2007b). Interestingly, based on the model (Fantini, 2014), increased cerebral blood volume has an in-phase contribution to the phase difference of HbO and Hb, whereas increases in the partial pressure of oxygen, oxygen utilization rate, and speed of blood flow have an anti-phase effect (Watanabe et al., 2017). In this study, we found that infants with a higher post-natal age (preterm infants (mean: 23.7 d, range: 17–35 d) have a higher anti-phase trend (compared to term infants with mean: 4.2 d, range: 3–6 d). Thus, our study implies that growth of the partial pressure of oxygen, oxygen utilization rate and speed of blood flow is the dominant effect in infants’ development, which outweigh that of cerebral blood volume increase. This growth likely stabilizes after 6 months, resulting in a constant phase difference between HbO and Hb that are maintained into adulthood (Watanabe et al., 2017).

We reached the conclusions that the hPod, IPE, and APE indices have the similar group mean values in 0.01–0.05 Hz and 0.05–0.1 Hz frequency bands, which are consistent with a previous study (Watanabe et al., 2017). From the perspective of coupling measurement, the coupling measure of PCMI indices shows a significant difference between term and preterm infants only in the 0.05–0.1 Hz frequency band (see Figures 10 and S6) (p<0.0002). However, we do not know what underlying physiological changes made the 0.05–0.1 Hz signal more sensitive to brain development for PCMI. Frequency dependence is an open question in fNIRS and fMRI resting-state studies. We believe that multimodal studies using EEG and fNIRS/fMRI (Vasung et al., 2018) , or animal studies could be exploited to deepen our understanding of this issue.

In this study, the most important finding is the identification of parameters that can distinguish term and preterm infants. For these two groups of infants, channel locations, measurement devices, and recording durations were exactly the same. The fNIRS measurements of adults only covered the prefrontal area, which was not ideal. However, we do not think it would change the results for the following reasons. First, many studies of healthy adults have observed the same anti-phase relationship of HbO and Hb oscillations in many regions of the brain in both resting and task experiments (Watanabe et al., 2017; Issard and Gervain, 2017; Lloyd-Fox et al., 2017). The measurements we performed should only be served as an example. Second, in this study, we found that the prefrontal region gave the most robust results for many parameters (e.g., IPE, APE, and hPod). This region was shared by infants and adults. Third, although different measurement devices were used for infants and adults, all the indices we analyzed were based on the phase or symbolic pattern of the signal, which is less sensitive to device selection. Finally, in both infants and adults, the measurement times extended beyond 2 minutes, which has been proven sufficient to obtain a robust index.

However, there are some limitations that should be addressed. First, in this study, the number of datasets is relatively small and the preterm data set only contains early preterm neonates. In future studies, more subjects from a wider age range should be recruited. Second, the adults’ experimental conditions were different from those of the infants, which may make direct comparisons difficult. More consistent experimental conditions will be sought in future studies.

In conclusion, the symbolic dynamic-based measures, IPE, APE, and PCMI, can measure brain development changes based on fNIRS signals. PCMI reflects the coupling strength of hemodynamic signals and, to some extent, reveals the underlying mechanisms of brain development. The potential of IPE, APE, and PCMI in estimating brain development has been demonstrated in this study. Furthermore, our new methods (IPE, APE, and PCMI) can be applied to fMRI data. However, instead of measuring HbO and Hb phase differences within one channel, these parameters can be used to assess coupling strength or phase differences between channels (voxels/regions of interest). We believe it will approach interesting brain interactions from a new perspective.

Table 4:

The statistics of different measurements with preterm, term, and adults in 0.05–0.1 Hz.

	Preterm	Term	Adults
	Median(min-max)	Median(min-max)	Median(min-max)
hPod	1.48(1.00–1.99)	1.53(1.00–1.99)	1.05(1.00–1.56)
IPE	0.43(0.16–0.82)	0.48(0.16–0.85)	0.12(0–0.39)
APE	0.43(0.06–0.72)	0.38(0.05–0.73)	0.75(0.49–0.99)
hPodL	0.57(0.03–0.95)	0.58(0.01–0.97)	0.75(0.14–0.99)
PCMI	0.95(0.74–1.17)	1.00(0.74–1.26)	1.07(0.91–1.60)

Table 8:

The CV indices of different measurements with preterm, term, and adults in 0.05–0.1 Hz.

	Preterm	Term	Adults
hPod	0.43	0.37	1.04
IPE	0.26	0.21	0.57
APE	0.25	0.26	0.12
hPodL	0.40	0.36	0.24
PCMI	0.07	0.08	0.09

Highlights:

►Through simulated sinusoidal oscillatory signals and nonlinear coupled model analysis, the in-pattern exponent (IPE), anti-pattern exponent (APE), and permutation cross-mutual information (PCMI) based on symbolic dynamics precisely measure phase difference variation and coupling strength change.

►IPE and APE can distinguish the hemoglobin phase of oxygenation and deoxygenation differences in preterm, term infants and adults.

►The proposed IPE and APE, as well as PCMI, have been proven to be suitable measures in brain development assessment.