# Detecting Chronotaxic Systems from Single-Variable Time Series with Separable Amplitude and Phase

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Chronotaxic Systems

**p**∈ R

^{n},

**x**∈ R

^{m}, f : R

^{n}→ R

^{n}, g : R

^{m}× R

^{n}→ R

^{m}, in which n and m can be any positive integers. Importantly, the solution

**x**(t, t

_{0}, x

_{0}) of Equation (1) depends on the actual time t as well as on the initial conditions (t

_{0},

**x**

_{0}), whereas the solution p(t − t

_{0},

**p**

_{0}) depends only on initial condition p

_{0}and on the time of evolution t − t

_{0}. The subsystem

**x**is nonautonomous in the sense that it can be described by an equation which depends on time explicitly, e.g., $\dot{\mathbf{x}}=\mathrm{g}\left(\mathbf{x},\mathrm{p}\left(t\right)\right)$. A chronotaxic system is described by

**x**which is assumed to be observable, and

**p**which may be inaccessible for observation, as often occurs when studying real systems. Rather than assuming or approximating the dynamics of p, we focus on the dynamics of

**x**and use only the following simple assumption: system

**p**is assumed to be such that it creates a time-dependent steady state in the dynamics of

**x**, which is schematically shown in Figure 1a.

**x**is divided into two parts. The first part is given by

**p**which is only that part which makes the system

**x**chronotaxic (defined below), i.e., an unperturbed chronotaxic system. The second part contains the rest of the environment, and is therefore considered as external perturbations.

**x**

^{A}(t).

^{m}of

**x**is a basin of attraction, i.e., that for any initial condition

**x**

_{0}at time t

_{0}the solution of the system asymptotically approaches the time-dependent steady state

**x**

^{A}, a condition of forward attraction for

**x**

^{A}is the following,

**x**

^{A}(t), this condition is not satisfactory in terms of defining the time-dependent point attractor. Any solution $\tilde{\mathbf{x}}\left(t,{t}_{0},{\mathbf{x}}_{0}\right)$ which satisfies Equation (2) with given

**x**

^{A}(t) can also be considered as a time-dependent point attractor. Moreover, when dealing with living systems, it is crucially important to describe stability at the current time t and not in the infinite future. This problem is resolved by employing a condition of pullback attraction, which should also be satisfied by

**x**

^{A}(t) in a chronotaxic system,

**x**

^{A}is a solution of the system Equation (1),

**x**(t, t

_{0},

**x**

_{0}) to deviate from

**x**

^{A}during a certain finite time interval. Thus, during this time-interval the ability to resist continuous external perturbations will be absent. Therefore, in order to characterize the ability of living systems to sustain their time-dependent dynamics at finite time intervals, a chronotaxic system should satisfy the condition of contraction, or equivalently the attraction at all times. This means that in the phase space R

^{m},

**x**∈ R

^{m}, there should be a contraction region C(t) such that for any two trajectories

**x**

_{1},

**x**

_{2}of a system inside the contraction region

**x**

_{i}(t, t

_{0},

**x**

_{0}

_{i}) ∈ C(t), i = 1, 2, the distance between them can only decrease,

^{′}, A

^{′}⊂ C, such that solutions of the system starting in A

^{′}never leave it, ∀t

_{0}< t, ∀x

_{0}∈ A

^{′}(t

_{0}), x(t, t

_{0}, x

_{0}) ∈ A

^{′}(t).

^{A}is located inside the area A

^{′}inside the contraction region C.

**x**

^{A}(t) can be viewed as a linearly attracting uniformly hyperbolic trajectory [13], so that the distance between a neighboring trajectory and

**x**

^{A}(t) can only decrease in an unperturbed chronotaxic system. For more details and for relations between chronotaxic and other dynamical systems see Reference [12]. A simple example of an unperturbed chronotaxic system is given by unidirectionally coupled phase oscillators with unwrapped phase φ

_{x}∈ (−∞, ∞) driven by a phase φ

_{p}∈ (−∞, ∞):

_{0}(t)−ω(t)|.

_{0}(t) = 0, the equation can be integrated, and the limit t

_{0}→ −∞ can be calculated, leading to the explicit expression for a time-dependent point attractor of an unperturbed chronotaxic system, ${\phi}_{\mathbf{x}}^{A}\left(t\right)={\phi}_{\mathrm{p}}\left(t\right)-\pi /2+2\pi k$, and k is any integer.

**p**can be complex, stochastic, or chaotic, provided that the above conditions are met. Nevertheless, the dynamics of

**x**will be determined by the dynamics of p, and therefore it will be deterministic, at least in unperturbed chronotaxic systems. When considering perturbed chronotaxic systems, for simplicity it is sufficient to consider perturbations only to the

**x**component, as any perturbations to

**p**can be included in its dynamics assuming that

**x**does not influence p. In perturbed chronotaxic systems, which model real life systems, the general external perturbations will create complex dynamics of

**x**with a stochastic component. Such dynamics may look very complex: perturbations can push trajectories away from the contraction region, therefore they can temporarily deviate before they converge. Despite this, due to the existence of the contraction region, the system will resist continuous external perturbations. The time-dependent dynamics of a perturbed chronotaxic system will be very close to the dynamics of the unperturbed chronotaxic system provided the perturbations are weak enough. In the case of very strong continuous perturbations, such perturbations may override the driving system p, and become effectively a new driver, causing the initial point attractor of the chronotaxic system to disappear. However, it may be restored once the perturbations again become sufficiently weak.

**x**and its driver p, [10,14]. For such complex systems as living systems, it has the potential to extract properties of the system which were previously neglected.

## 3. Inverse Approach for Chronotaxic Systems

#### 3.1. Inverse Approaches to Nonautonomous Dynamical Systems

#### 3.2. Detecting Chronotaxicity

_{0}(t) is the time-dependent natural frequency and the observed phase ϕ

_{x}is perturbed by noise fluctuations η(t). Integrating we find,

_{0}(t) > 0 and η(t) is an uncorrelated Gaussian process, this means that the dynamics of φ

_{x}will consist of a monotonically increasing phase perturbed by a random walk noise (Brownian motion). However, the situation is different for a chronotaxic phase oscillator, e.g.,

_{p}is an external phase and |ε| > 1. In this case the stability provided by the point attractor causes each noise perturbation to decay over time, preventing η(t) from being integrated over to the same extent. The perturbations still do not decay instantly, as the system takes time to return to the point attractor, meaning that some integration of the noise still takes place. However, the size of the observed perturbations over longer time scales is greatly reduced, causing a change in the overall distribution from that expected for Brownian motion.

#### 3.3. Extracting the Perturbed and Unperturbed Phases

_{0}is a parameter known as central frequency which defines the time/frequency resolution [27].

_{T}(s, t) using either a ridge-extraction method [29,30] or the synchrosqueezed wavelet transform (SWT) [18]. These extraction methods can be used to estimate the instantaneous frequencies of the oscillatory components in a time series, allowing identification of harmonics, which can be used to determine the intra-cycle dynamics. The phase φ

_{x}of the observed system is then arg(W

_{T}(s, t)), where s and t denote the positions of the oscillation in the s − t plane.

#### 3.4. Dynamical Bayesian Inference

_{i}is the natural frequency of the oscillation, f

_{i}(φ

_{i}) are the self-dynamics of the phase, g

_{i}(φ

_{i}, φ

_{j}) are the cross couplings, and ξ(t) is a two-dimensional white Gaussian noise 〈ξ

_{i}(t)ξ

_{j}(τ)〉 = δ(t − τ)E

_{ij}. Based on the periodic nature of the system, the basis functions are modeled using the Fourier bases

_{i,k}(φ

_{i}, φ

_{j}). The corresponding parameters from ${\tilde{a}}_{i}$ and ${\tilde{b}}_{i}$ then form the parameter vector ${c}_{k}^{(i)}$. The inference of these parameters utilises Bayes’ theorem,

^{−1}, the stationary point of S can be calculated recursively from

_{1}in the direction 2 → 1, and the even parameters correspond to the coupling terms inferred for φ

_{2}in the direction 1 → 2. See [32] for further details and an in depth tutorial on dynamical Bayesian inference and its implementation.

_{x}.

#### 3.5. Phase Fluctuation Analysis

_{x}and φ

_{p}are almost synchronized. Given the estimates of φ

_{x}and ${\phi}_{\mathbf{x}}^{A}$, the next step is to analyse $\mathrm{\Delta}{\phi}_{\mathbf{x}}={\phi}_{\mathbf{x}}^{\ast}-{\phi}_{\mathbf{x}}^{A\ast}$ to find the distribution of fluctuations in the system relative to the unperturbed trajectory.

^{α}.

_{x}is integrated in time and divided into sections of length n. For each section the local trend is removed by subtracting a fitted polynomial—usually a first order linear fit [6,33]. The root mean square fluctuation for the scale equal to n is then given by

_{x}. This would cause perturbations over large time scales (i.e., greater than one cycle) to not decay even if the system was chronotaxic. In these cases another approach should be used instead [14].

_{x}to change by 2π, but these are part of a continuous probability distribution, in contrast to the chronotaxic case. Phase slips can be detected by calculating the distribution of the difference between the phase fluctuations Δφ

_{x}(t) and these fluctuations delayed by a time scale τ. $d\mathrm{\Delta}{\phi}_{\mathbf{x}}^{\tau}\left(t\right)=\mathrm{\Delta}{\phi}_{\mathbf{x}}\left(t+\tau \right)-\mathrm{\Delta}{\phi}_{\mathbf{x}}\left(t\right)$ therefore gives information about the perturbations of the system over that time scale. When phase slips are present, the distribution of $|d\mathrm{\Delta}{\phi}_{\mathbf{x}}^{\tau}|$ changes with respect to τ [14]. An example of this difference is shown in Figure 2g,h, and can also be seen in real biological systems, as previously demonstrated in the heart rate variability [14].

## 4. Application of Inverse Approach Methods

#### 4.1. Numerical Simulations

_{p}and φ

_{x}are the instantaneous phases of the driving and the driven oscillators, respectively, ω

_{p}> 0 and ω

_{x}> 0 are the natural frequencies of the oscillators, ε > 0 is the strength of the coupling and η is white Gaussian noise with standard deviation $\sigma =\sqrt{2E}$ where 〈η(t)〉 = 0, 〈η(t)η(τ)〉 = δ(t − τ)E. Note that when ε = 0 the system is reduced to ${\dot{\phi}}_{\mathbf{x}}={\omega}_{\mathbf{x}}+\eta \left(t\right)$ and becomes non-chronotaxic; when η = 0 and ε > |ω

_{x}− ω

_{p}| the system becomes chronotaxic with ${\phi}_{\mathbf{x}}^{A}(t)={\phi}_{\mathrm{p}}(t)-\text{arcsin}(({\omega}_{\mathrm{p}}-{\omega}_{\mathbf{x}})/\epsilon )$. The system was integrated using the Heun scheme [15], with an integration step of 0.001 and noise strength σ = 0.3. Δφ

_{x}, shown in Figure 2, was obtained by subtracting the unperturbed phase $({\phi}_{\mathbf{x}}^{A}\left(t\right)$ and ω

_{x}t in the chronotaxic and non-chronotaxic cases, respectively) from the perturbed phase φ

_{x}, as obtained numerically. DFA was then performed on Δφ

_{x}, with exponents shown in Figure 2. The values of the exponents demonstrate the differences in the noise distributions between chronotaxic and non-chronotaxic systems. In the chronotaxic case, the noise is closer to white, whereas in the non-chronotaxic case it is closer to a random walk. It is this difference which is exploited in the PFA method.

_{p}and f

_{x}are the average frequencies of oscillation in Hz of the chronotaxic and non-chronotaxic case, respectively, and f

_{m}is the frequency of variation. Frequencies of oscillation were chosen to vary around 1 and 0.25 Hz in the non-chronotaxic and chronotaxic cases, respectively, with f

_{m}= 0.003. Both systems were perturbed with white Gaussian noise with strength σ = 0.5. The logarithmic frequency scale of the wavelet transform is very useful for identifying and separating the presence of oscillatory modes, which may otherwise appear as merged in other time–frequency representations, such as the windowed Fourier transform. Figure 3 shows the results of PFA on the signal. It correctly identifies mode A (around 0.25 Hz) as chronotaxic, and mode B (around 1 Hz) as non-chronotaxic.

_{3}was allowed to vary in time in Equation (23), whilst ε

_{1}= ε

_{2}= 0.1 and ε

_{4}= 0, resulting in intermittent chronotaxicity of the oscillator ${\phi}_{\mathbf{x}2}$. ${\phi}_{\mathbf{x}2}^{A\ast}$ and ${\phi}_{\mathbf{x}2}^{\ast}$ were extracted from the synchrosqueezed wavelet transform of $\mathrm{sin}\phantom{\rule{0.2em}{0ex}}({\phi}_{\mathbf{x}2})$. Results of the application of dynamical Bayesian inference are shown in Figure 5. This method is able to track the intermittent changes in chronotaxicity, through changes in synchronization and direction of coupling, demonstrating its usefulness for the detection of chronotaxicity in systems where the interactions between oscillators are time-varying.

#### 4.2. Practical Considerations

_{min}and n

_{max}, the lower and upper values for the range of the first order polynomial fits performed in order to calculate $\mathrm{\Delta}{\phi}_{\mathbf{x}}$. The lower value, n

_{min}, is set to be 2 cycles of the slowest oscillation, to ensure observation of the dynamics over a longer range than one cycle. The smallest n

_{max}required to still obtain a reliable DFA exponent was observed to be n

_{max}= 3 cycles of oscillation (see Figure 7), provided that the time series is sufficiently long. The second test seeks to identify the required length of the whole time series when using these values of n

_{min}and n

_{max}in DFA. The DFA exponent was calculated from varying lengths of the same noise signals, from 3 to 10 times n

_{max}, to identify the point where the result is no longer reliable. It was found that the time series should be at least 8 times n

_{max}in order to obtain a reliable result, therefore at least 24 cycles of the slowest oscillation are required to test for chronotaxicity. However, if possible, the time series should be at least 10 times n

_{max}[38], to reduce noise by providing more data windows. Overlapping within DFA is also possible, and will go some way toward reducing noise, and improve reliability. The results shown in Figure 7 were obtained with an overlap of 0.8.

#### 4.3. Application to Experimental Data

## 5. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**

**(a)–(c)**5 s time series of sin(φ

_{x}) (red line) in 3 cases: chronotaxic, non-chronotaxic, and chronotaxic with phase slips, from Equation (21). The grey line shows φ

_{p}(chronotaxic), and ω

_{x}t (non-chronotaxic).

**(d)–(f)**Δφ

_{x}for the whole time series, detrended with a moving average of 200 s. In all cases ω

_{x,p}= 2π, h = 0.001, L = 1000 s and σ = 0.3. ε = 5 and 0 in the chronotaxic and non-chronotaxic cases, respectively. detrended fluctuation analysis (DFA) exponents, α, are shown. The DFA exponent of

**(f)**incorrectly suggests the system is non-chronotaxic. To distinguish between a non-chronotaxic system and a chronotaxic system with phase slips, the delayed distributions were calculated (see Section 3.5) in the non-chronotaxic

**(g)**and chronotaxic

**(h)**case.

**Figure 3.**Identifying chronotaxicity in signals with more than one oscillatory mode.

**(a)**The first 250 s of a time series of a simulated signal containing two distinct oscillations, with coupling strengths ε = 2 for mode A (chronotaxic) and ε = 0 for mode B (non-chronotaxic).

**(b)**The continuous wavelet transform of the signal in

**(a)**.

**(c)**The instantaneous frequency (light grey) of both components is extracted from the wavelet transform, with central frequency f

_{0}= 0.5, and smoothed (red), using a polynomial fit. The smoothed frequency is then integrated in time to obtain an estimate of the unperturbed phase, ${\phi}_{\mathbf{x}}^{A\ast}$, which is then subtracted from the perturbed phase ${\phi}_{\mathbf{x}}^{\ast}$ as extracted directly from the wavelet transform.

**(d)**and

**(e)**show $\mathrm{\Delta}{\phi}_{\mathbf{x}}={\phi}_{\mathbf{x}}^{\ast}-{\phi}_{\mathbf{x}}^{A\ast}$ for each mode. (f) and (g) show the results of DFA on Δφ

_{x}, with DFA exponents α correctly identifying mode A as chronotaxic and mode B as not chronotaxic.

**Figure 4.**Identifying chronotaxicity using phase fluctuation analysis in a system of bidirectionally coupled oscillators. The system presented in Equation (23) was simulated in two different states of chronotaxicity. (

**a**) Phase trajectories for the system when ε

_{1}=0.1, ε

_{2}= 20, ε

_{3}= 0.1 and ε

_{4}= 10. (

**b**) Phase trajectories of the system with ε

_{1}=0.5,ε

_{2}=0.1, ε

_{3}= 0.1 and ε

_{4}= 15. (

**c**) Five seconds of the time series of both drivers and oscillators for parameters shown in (

**a**). (

**d**) A 5 s time series for parameters shown in (

**b**). (

**e**) and (

**g**) phase fluctuations from PFA on $\mathrm{sin}\left({\phi}_{\mathbf{x}1}\right)$ and $\mathrm{sin}\left({\phi}_{\mathbf{x}2}\right),$ respectively. (

**f**) and (

**h**) phase fluctuations extracted with PFA on $\mathrm{sin}\left({\phi}_{\mathbf{x}1}\right)$ and $\mathrm{sin}\left({\phi}_{\mathbf{x}2}\right),$ respectively.

**Figure 5.**Identifying intermittent chronotaxicity using dynamical Bayesian inference. Bayesian inference was performed on ${\phi}_{\mathbf{x}2}^{\ast}$ and ${\phi}_{\mathbf{x}2}^{A\ast}$ extracted from $\mathrm{sin}({\phi}_{\mathbf{x}2})$ (see Equation (23)) with ε

_{3}varying as shown in (

**d**). (

**a**) the CWT of $\mathrm{sin}({\phi}_{\mathbf{x}2})$. (

**b**) Instantaneous frequencies extracted from the wavelet transform. ${\phi}_{\mathbf{x}2}^{A\ast}$ was extracted with ${f}_{\mathrm{o}}=2$ and smoothed using a polynomial fit (red line), whilst ${\phi}_{\mathbf{x}2}^{\ast}$ was extracted from the wavelet transform with ${f}_{\mathrm{o}}=0.5$ (grey line). Bayesian inference was applied, using a time window of 90 s. The inferred direction of coupling can be seen in (

**c**). Positive values show coupling from the driver to the oscillator only. (

**d**) I

_{sync}was calculated and shows excellent agreement with changes in ε

_{3}. I

_{chrono}was also calculated, and was slightly less accurate due to the direction of coupling becoming negative very briefly, because of reduced information flow between systems to accurately infer parameters during synchronization.

**Figure 6.**An example of the application of phase fluctuation analysis to an electroencephalogram (EEG) signal obtained from the forehead of an anaesthetised patient, shown in (

**a**). (

**b**) The continuous wavelet transform of the electroencephalogram (EEG) signal in (

**a**). (

**c**) Using NMD, a significant oscillatory mode in the alpha frequency band was identified and extracted (dark grey line). (

**d**) The instantaneous frequency extracted using NMD (grey line), and smoothed using a moving average of 4 s (red line). (

**e**) The extracted phases of the mode from NMD (grey), smoothed NMD (red), and from the CWT (black) with ${f}_{\mathrm{o}}=1.5$. (

**f**) $\mathrm{\Delta}{\phi}_{\mathbf{x}}$ was calculated as ${\phi}_{\mathbf{x}}^{\ast}-{\phi}_{\mathbf{x}}^{A\ast}$. The DFA exponent was calculated and was 1.57, suggesting that the system is not chronotaxic. Checking for phase slips in (

**g**) shows no change in distribution.

**Figure 7.**In order to test the reliability of the DFA exponent when reducing n

_{max}, the maximum number of cycles of oscillation used in its calculation was varied. (

**a**) Chronotaxic oscillation of 1 Hz. (

**b**) Chronotaxic oscillation of 0.1 Hz. (

**c**) Non-chronotaxic oscillation of 1 Hz. (

**d**) Non-chronotaxic oscillation of 0.1 Hz. The same noise signals were then tested with n

_{max}= 3 for different lengths of the time series from 10 to 3 times n

_{max}. Based on these results, the time series should be at least 8 times n

_{max}, thus, there should be at least 24 oscillations in the time series. However, to ensure universal applicability, the length of the time series should be at least 10 times n

_{max}, the generally accepted value in DFA [38], resulting in the requirement of 30 cycles.

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lancaster, G.; Clemson, P.T.; Suprunenko, Y.F.; Stankovski, T.; Stefanovska, A.
Detecting Chronotaxic Systems from Single-Variable Time Series with Separable Amplitude and Phase. *Entropy* **2015**, *17*, 4413-4438.
https://doi.org/10.3390/e17064413

**AMA Style**

Lancaster G, Clemson PT, Suprunenko YF, Stankovski T, Stefanovska A.
Detecting Chronotaxic Systems from Single-Variable Time Series with Separable Amplitude and Phase. *Entropy*. 2015; 17(6):4413-4438.
https://doi.org/10.3390/e17064413

**Chicago/Turabian Style**

Lancaster, Gemma, Philip T. Clemson, Yevhen F. Suprunenko, Tomislav Stankovski, and Aneta Stefanovska.
2015. "Detecting Chronotaxic Systems from Single-Variable Time Series with Separable Amplitude and Phase" *Entropy* 17, no. 6: 4413-4438.
https://doi.org/10.3390/e17064413