# Oscillating systems with cointegrated phase processes

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## Abstract

We present cointegration analysis as a method to infer the network structure of a linearly phase coupled oscillating system. By defining a class of oscillating systems with interacting phases, we derive a data generating process where we can specify the coupling structure of a network that resembles biological processes. In particular we study a network of Winfree oscillators, for which we present a statistical analysis of various simulated networks, where we conclude on the coupling structure: the direction of feedback in the phase processes and proportional coupling strength between individual components of the system. We show that we can correctly classify the network structure for such a system by cointegration analysis, for various types of coupling, including uni-/bi-directional and all-to-all coupling. Finally, we analyze a set of EEG recordings and discuss the current applicability of cointegration analysis in the field of neuroscience.

## Keywords

Coupled oscillators Synchronization Cointegration Phase process Interacting dynamical system Winfree oscillator EEG signals## Mathematics Subject Classification

37N25 62M10 92B25 62F03## 1 Introduction

Since the first scientific discovery of two pendulums synchronizing by Christiaan Huygens in the seventeenth century, this naturally occurring phenomenon has now been observed in diverse areas such as fireflies synchronizing their flashing behavior, a theatre audience applauding after a show and also in chemical and biological systems, such as the brain and the heart beats of a mother and her fetus, where coupled oscillators appear, see also Pikovsky et al. (2001). Due to it’s pervasive presence, understanding synchronization is of key interest for researchers to understand biological networks, such as the connectivity of the nervous system, circadian rhythms or the cardiovascular system. To a statistician this presents a fascinating challenge of modelling complex behavior in large scale systems and how to infer the data-generating mechanisms. To this day, synchronization is not fully understood, but has been the centre of research for decades as evident in Ermentrout (1985), Kuramoto (1984), Strogatz (1987, 2000), Taylor and Holmes (1998), Winfree (1967), even the phenomenon of synchronizing pendulums as observed by Huygens, still attracts attention today, see Martens et al. (2013), Oliveira and Melo (2015). Many innovative ideas have been presented since Winfree (1967) began a mathematical treatment of the subject. When Kuramoto (1984) first presented his model of coupled oscillators, this made a huge impact in the field and spawned a new generation of research on synchronization. Kuramotos model is still considered among one of the most significant advancements in the study of synchronization in oscillating systems as acknowledged by Strogatz (2000), and the study of coupled oscillators still attracts a fair interest from researchers Ashwin et al. (2016), Burton et al. (2012), Fernandez and Tsimring (2014), Ly (2014), Ly and Ermentrout (2011).

A long standing problem in neuroscience is to recover the network structure in a coupled system. This could for example be to infer the functional connectivity between units in a network of neurons from multiple extracellularly recorded spike trains, or how traces of EEG signals from different locations on the scalp affect each other, which we will treat in this paper. To the authors knowledge, this challenge is still lacking a sound statistical framework to model and test for interaction in a system, as well as impose statistical hypotheses on the network structure. For this task, cointegration analysis offers a refined statistical toolbox, where detailed information on the connections can be inferred, such as the direction and proportional strength of the coupling. The theory of cointegration was originally conceived by Granger (1981), and has since then also been the subject of intense research, most notably within the field of econometrics. In the monograph by Johansen (1996), the full likelihood theory for linear cointegration models with Gaussian i.i.d. errors is derived, and a framework for estimation and inference on parameters using the quotient test is presented. This well acknowledged framework is popularly termed the Johansen procedure. Even though cointegration analysis has developed from within the field of econometrics, it may potentially be used for different models outside economics, such as biological models in continuous time as we explore here. It has also been applied in climate analysis, see Schmith et al. (2012).

In this paper, we demonstrate how to apply cointegration analysis to a system of linearly phase coupled oscillating processes. To display the applicability of the method, we present a simulation experiment, where we present a statistical analysis of phase coupled systems with varying network structures, including uni-/ bi-directional and all-to-all couplings. We show that we can identify the proportional coupling strengths and directions given by the estimated *cointegration matrix* parameter. Our work is inspired by Dahlhaus and Neddermeyer (2012), which also introduces cointegration analysis as a statistical toolbox to neuroscientists and new challenges for researchers in cointegration theory. However, in contrast to Dahlhaus and Neddermeyer (2012), we incorporate the fact that we are dealing with continuous systems and also ensure that the cointegration property of the system is well posed as a linear structure. This approach assures that the conclusion on the interaction in the data is accurate in terms of cointegration.

The paper is composed as follows. In Sect. 2 we define a class of phase coupled oscillators, in Sect. 3 we highlight some cointegration theory for the analysis including an extension to discretely observed, continuous time models. In Sect. 4 we present a statistical analysis of linearly phase coupled oscillating systems and in Sect. 5 we analyze EEG recordings from an epileptic subject experiencing a seizure, previously analyzed by Shoeb (2009). We discuss the model and findings, conclude on the research and give an outlook of the future direction of the research in Sect. 6. Technical details are presented in the appendix.

Throughout we use the following notation and conventions: unless explicitly stated otherwise, time \(t\in [0,\infty )\) is assumed continuous, and the process \((x_t,y_t)'\) is assumed observed with corresponding polar coordinates \((\phi _t,\gamma _t)'\). Here \('\) denotes transposition. For a \(p\times r\) matrix *M*, with \(r\le p\), we denote the orthogonal complement \(M_\perp \), a \(p \times (p-r)\) matrix such that \(M'_\perp M=0\) (zero matrix). Also denote by \(\text {sp}(A)\) the subspace spanned by the columns of a matrix *A*, and let \({{\mathrm{rank}}}(A)\) denote the rank of the matrix, i.e., the dimension of \(\text {sp}(A)\).

## 2 Oscillating systems

*oscillator*as a continuous time bi-variate process \(z_t = (x_t,y_t)'\in {\mathbb {R}}^2\), \(t\in [0,\infty )\), such that \(z_t\) revolve around some arbitrary center. Such a process can be derived from an equivalent process in polar coordinates \((\phi _t,\gamma _t)'\), where \(\phi _t\in {\mathbb {R}}\) is the

*phase*process and \(\gamma _t\in {\mathbb {R}}\) is the

*amplitude*process, such that

### 2.1 Defining a class of coupled oscillators

*coupled stochastic oscillators*, where we observe

*p*oscillators that interact, i.e., \(z_t\in {\mathbb {R}}^{2p}\). Define a class of oscillators with phase (\(\phi _t\in {\mathbb {R}}^p\)) and amplitude (\(\gamma _t\in {\mathbb {R}}^p\)) processes given by the multivariate stochastic differential equations (SDE)

*angular velocities*are positive. Note that we have defined the phase-trend as positive, corresponding to counter-clockwise rotation in accordance with the standard interpretation of the phase. However, for a negative trending process, one can either look at \(-\phi _t\) or simply interpret rotations as clockwise.

*xy*-plane, we derive a DGP from (2)–(3), see “Appendix 1”. Assuming that \(\varSigma _\phi = {{\mathrm{diag}}}(\sigma _1^\phi ,\ldots ,\sigma _p^\phi )\) and \(\varSigma _\gamma = {{\mathrm{diag}}}(\sigma _1^\gamma ,\ldots ,\sigma _p^\gamma )\) we find that

*f*and

*g*define the properties of the system, such as interaction. This broad definition of oscillating systems covers among others the Kuramoto model, see example (Sect. 2.5) below and other standard oscillators such as the FitzHugh–Nagumo and the Duffing oscillator. In this paper we will analyze

*phase coupled*oscillators, and therefore we assume that \(g_k(\phi _t,\gamma _t)=g_k(\gamma _{kt})\), such that there is no feedback from the phase process \(\phi _t\) into the amplitude and the

*k*’th amplitude is not dependent on the rest. Hence, interaction in the system is solely through \(f(\phi _t,\gamma _t)\), such that the phase processes are attracted by some interdependent relation.

### 2.2 Linear coupling

The arbitrary function *f* enables us to choose any transformation of the variables to obtain a coupled system, including unidirectional coupling between phases or periodic forcing of the system if we extend *f* to depend on *t* as well, intermittent synchronization dependent on a threshold in process differences, etc.

*f*is composed of a linear mapping of \(\phi _t\) and a function of \(\gamma _t\), with components,

*k*’th oscillator is only dependent on the intrinsic amplitude \(\gamma _{kt}\) through \(h(\gamma _{kt})\). We will refer to such a system as

*linearly phase coupled*.

Although we impose the linear restriction \(\varPi \) on the interaction between phases, we can still model a broad set of coupling structures as we show with examples below. Since the interaction is given by \(\varPi \phi _t\), we note that the *coupling strength* in the system is given as the absolute values of the entries of \(\varPi \) and that row *k* of \(\varPi \) define how oscillator *k* depends on the rest. Note also that \(\omega \) defines the attracting state for the phase relations, see example (Sect. 2.3) below. Normally \(h(\gamma _{kt})\) is restricted to a constant, but in Sect. 4 we will relax this and investigate systems where \(h(\gamma _{kt})\) is only approximately linear and has a sufficiently low variance. This implies a misspecified model, but as we will show, we can still identify the coupling structure, although inference on \(h(\gamma _{kt})\) itself is less meaningful.

### 2.3 Example: Linearly phase coupled system with a degenerate \(\gamma _t\) process

*f*be defined as in (7) and assume that \(\gamma _t\) is a constant (positive) process such that \(h(\gamma _{kt})=\mu _k>0\). Then

*f*is of the form

*f*admits a linearly phase coupled system with intrinsic rotating frequencies \(\mu \). Note that if \(\varPi =0\) then there is no interaction in the system, and the individual oscillators will rotate according to their own \(\mu _k>0\), and we refer to the system as

*independent*.

*in-phase*, whereas \(\omega ^*=\pi \) would imply that the system is attracted towards being in

*anti-phase*. Considering that neither \(\alpha _1,\alpha _2\) or \(\omega ^*\) depend on time, the system cannot switch to a different attracting regime.

To illustrate possible coupling structures, consider again the system of two oscillators and assume that \(\omega =0\). Then with \(\alpha _2=0\) and \(\alpha _1\ne 0\) the coupling between \(\phi _{1t},\phi _{2t}\) is *uni-directional* \(\phi _{2t}\rightarrow \phi _{1t}\) where the arrow \(\rightarrow \) denote the direction of interaction. Likewise, if \(\alpha _1=0\) and \(\alpha _2\ne 0\) then \(\phi _{1t}\rightarrow \phi _{2t}\). However, if both \(\alpha _1,\alpha _2\ne 0\) then \(\phi _{2t}\leftrightarrow \phi _{1t}\) and the coupling is *bi-directional*. In general, if \(\phi _{kt}\) appears in the expression \(f_l(\phi _t)\) for oscillator \(l\ne k\), then \(\phi _{kt}\rightarrow \phi _{lt}\). If the opposite is true, then \(\phi _{lt}\rightarrow \phi _{kt}\) and if both directions exist, then \(\phi _{lt}\leftrightarrow \phi _{kt}\). For \(f_k(\phi _t)=0\) oscillator *k* is (oneway) independent from the rest, but it can still possibly influence others.

When \(\gamma _t\) is a constant vector process the properties of the system are fully identified by (2). Furthermore, if the noise level of the phases \(\varSigma _\phi \) is sufficiently small, we can use the Hilbert transform^{1} to derive the phase process \(\phi _t\) from observations of either \(x_t\) or \(y_t\). This is a commonly used technique in signal processing and has been applied to oscillating systems as well, see Dahlhaus and Neddermeyer (2012), Pikovsky et al. (2001). For systems where \(\phi _t\) is very noisy, this method is less applicable.

### 2.4 Example: Winfree oscillator

*Winfree oscillator*. Note that the formulation of \(d\gamma _{kt}\) implies that the amplitude fluctuates around \(\kappa _k\). Due to this, we can for sufficiently small noise \(\varSigma _\gamma \) insist that \(\gamma _{kt} \approx \kappa _k\) for \(k=1,\ldots ,p\) and therefore analyze the Winfree oscillator using the cointegration toolbox, assuming a constant \(\gamma _{t}\) in \(d\phi _t\). In Sect. 4 we analyze the range of noise, \(\varSigma _\gamma \), where the cointegration analysis still performs well.

### 2.5 Example: Kuramoto model

*k*’th and

*j*’th oscillators. In the classic version, \(K_{kj}=K\) \(\forall k,j\), such that for a certain threshold \(K_c\), then with \(K > K_c\) the oscillators exhibit synchronization. For an arbitrary \(\gamma _t\) process we cannot simplify (6), but with a degenerate \(\gamma _t\) we obtain the same expression as in (9) with \(f_k(\phi _t)\) as in (11).

*f*is a nonlinear function, hence it is not directly applicable to a standard cointegration analysis where

*f*is assumed linear. To emphasize this fact, consider the special case \(p=2\), where the Kuramoto model is particularly simple and (11) can be written explicitly as,

## 3 Cointegration

Cointegration theory was originally developed for discrete time processes, however the ubiquitous use of continuous time models has inspired development of continuous time cointegration theory, see Kessler and Rahbek (2004, 2001). In order to present cointegration analysis as a framework for phase-processes, we therefore review some background on *discrete* time processes before entering into continuous time cointegrated models. The first part of this section is based on Johansen (1996) and Ltkepohl (2005).

### 3.1 Integrated process

*characteristic polynomial*for (15) is the determinant of \(I_p-A\zeta \) for \(\zeta \in {\mathbb {C}}\), where \(I_p\) is the

*p*-dimensional identity matrix. If the roots of the characteristic polynomial are all outside the unit circle, then the initial values of \(\phi _n\) can be given a distribution such that \(\phi _n\) is stationary, see Johansen (1996).

If the characteristic polynomial of (15) contains one or more roots at \(\zeta =1\), then there is no stationary solution of \(\phi _n\), and we say that the process is *integrated*. In particular, see Johansen (1996), \(P=A-I_p\) will have reduced rank \(r<p\) and can be written as \(P= ab'\) with \(a,b(p\times r)\) matrices of rank *r*. Moreover, the process \(\phi _n\) is integrated of order one, *I*(1) with *r* cointegrating relations \(b'\phi _n\) under regularity conditions presented in Sect. 3.2. Note that the order of integration is a stochastic property and hence including deterministic terms in a model does not change the order of integration.

In this paper we will only deal with *I*(1) processes, so when we refer to \(\phi _n\) as integrated, we implicitly mean that \(\phi _n\) is integrated of order 1.

### 3.2 Cointegrated process

*I*(0) process). In particular,

*P*has full rank

*p*and all linear combinations of \(\phi _n\) are stationary. If the \((p\times p)\)-dimensional matrix

*P*has reduced rank \(r<p\) then \(P =ab'\) with \(a,b,p\times r\) dimensional matrices of rank

*r*. Moreover, the process \(\phi _n\) is integrated of order one,

*I*(1) with

*r*cointegrating stationary relations \(b'\phi _n\) provided \(\rho (I_{r}+b'a) <1\) with \(\rho \left( \cdot \right) \) denoting the spectral radius. This we refer to as the

*I*(1) conditions in the following.

*I*(1) with no cointegration, while if \(r=p\) (and \(\rho (A) <1\)) then \(\phi _n\,\)is

*I*(0), or

*p*stationary linear combinations exist. Under the reduced rank

*r*, the system is written as,

*b*containing the

*r*cointegration vectors and

*a*the

*loadings*or

*adjustment coefficients*. Note that the entries of

*a*and

*b*are not uniquely identified, since we can use any non-singular transformation to obtain similar results. Rather we identify the subspaces \(\text {sp}(a),\text {sp}(b)\in {\mathbb {R}}^r\), that is, the subspaces spanned by the columns of

*a*,

*b*, where we use the normalization

*b*in order to identify parameters uniquely. Furthermore, let \(m_\perp \) denote the matrix such that \(\text {sp}(m_\perp )\) is orthogonal to \(\text {sp}(m)\), then a necessary condition for an

*I*(1) process is that \(|a_\perp 'b_\perp |\ne 0\). For more on estimation and inference in cointegration models, see “Appendix 2”.

### 3.3 Continuous time cointegrated models

*f*as in (8) and for simplicity \(\omega =0\). This is a

*p*-dimensional Ornstein–Uhlenbeck process. The exact solution is

*not*of full rank, then \(\phi _t\) is necessarily not stationary.

*P*is of reduced rank, and can be decomposed \(P=ab'\) with \(a,b\in {\mathbb {R}}^{p\times r}\) of full rank \(r\le p\). However, it also holds that

*a*and

*b*.

In the numerical part, we will refer to estimates of \(\alpha \) and \(\beta \), implicitly referring to the *discrete time* estimates. In terms of subspaces, there is no difference between the discrete and continuous time, but in order to interpret the *continuous time* \(\varPi \) matrix, one must translate the discrete estimate to a continuous estimate using (19).

It is important to note that when working with continuous time models, one must be careful with regard to the relation (19) between discrete and continuous time and the sampling timestep \(\delta \). Kessler and Rahbek (2004) refer to this issue as the *embedding problem*, and to ensure that the continuous time model is appropriate, one must check for \(\exp (\delta \varPi )\) in (18) that it is non-singular, i.e., \(|\exp (\delta \varPi )|\ne 0\), and that it has no negative eigenvalues. If this is the case and the underlying process is in fact cointegrated, the results above hold.

### 3.4 Likelihood ratio test for \({{\mathrm{rank}}}(\varPi )=r\)

*bootstrap*simulations as presented by Cavaliere et al. (2012), in order to determine critical values. Specifically, given the data \(\{\phi _{t_n}\}_{n=1}^N\) bootstrap sequences \(\{\phi _{t_n}^{*(m)}\}_{n=1}^N\) for \(m=1,\ldots ,M\) are simulated and for each sequence the LRT statistic \(\text {LRT}^{*(m)}\) is re-computed. The empirical quantiles of \(\{\text {LRT}^{*(m)}\}_{m=1}^M\) are then used for testing. With

*r*determined, \(\hat{\beta }\) is given by the

*r*eigenvectors corresponding to \(\hat{\lambda }_i,i=1,\ldots ,r\) and the parameter estimates \(\hat{\alpha },\hat{\mu },\hat{\varSigma }\) follow by ordinary least squares estimation as outlined in “Appendix 2”.

### 3.5 Inference for \(\alpha \) and \(\beta \)

*r*eigenvectors corresponding to the

*r*largest eigenvalues. This ensures that

*A*and

*B*represent the linear hypotheses and \(\psi \) and \(\xi \) are parameters to be estimated. It is also possible to combine the hypotheses for \(\alpha \) and \(\beta \) and we denote this \(H_{\alpha ,\beta }\).

*l*:

*n*coupling, can be specified by appropriately designing the matrices

*A*and

*B*. Thus, a broad variety of linear hypotheses on the parameter \(\varPi =\alpha \beta '\) can be investigated, notably inference on the coupling directions and the effect of system disequilibrium on individual oscillators.

*A*and/or

*B*obtain eigenvalues \(\lambda _i^*\) for the restricted model. The LRT statistic is then given by

*m*and

*s*are the column dimensions of the matrices

*A*and

*B*, respectively. This shows that once \({{\mathrm{rank}}}(\varPi )\) is determined, statistical inference for \(\alpha \) and \(\beta \) can be carried out, relatively straightforward. As for the rank determination, an alternative to the \(\chi ^2\) approximation for inference on \(\alpha \) and \(\beta \) is to perform bootstrapping for the test (26), see Boswijk et al. (2016).

## 4 Numerical simulations

### 4.1 General setup

For each experiment we simulate 1.000.000 iterations of the oscillator (10) using the Euler–Maruyama scheme with timestep \(\widetilde{\varDelta }t = 0.0002\) and then subsample for \(\varDelta t=0.1\), thus obtaining \(N=2000\) (equidistant and discrete) observations of \(z_{t}\) for \(t\in [0,200)\). Subsampling every 5000th values diminishes the discretization error of the simulation scheme.

Note that the \(\kappa \) parameter for \(\phi _{2t}\) is set equal to \(\phi _{3t}\) to obtain similar simulated outcomes for some experiments to investigate whether we can distinguish between interaction and independence between these two. We set the cointegration parameters for each experiment individually to impose different coupling structures, and will refer to the relevant model by it’s \(\varPi _k,k=0,1,2,3\) matrix, where *k* defines the model structure (see Fig. 1).

*mean phase coherence measure*, see Mormann et al. (2000), bilaterally to the

*wrapped*phases (i.e., \(\phi _{it}\in [0,2\pi )\) for \(i=1,2,3\))

*R*measures, we bootstrapped critical values for the hypothesis \(R=0\). Hence, these values are the same for all experiments and presented along with the measured

*R*values. We compare the resulting value of

*R*to the conclusion of the cointegration analysis.

*dW*. First we run a simulation with uncoupled oscillators as a benchmark, and then continue with coupled systems as presented in Fig. 1. Figure 2 display the

*x*-coordinates for \(t\in [100,150]\) from a simulation of these four systems.

### 4.2 Independent oscillators

*R*for the phases indicates that this is not the case.

Rank tests for models \(\varPi _i, i=0,1,2,3\) with the selected models indicated in bold

Model | \(H_r\) | Test values | |
---|---|---|---|

\(\varPi _0\) | \(\mathbf {r=0}\) | | |

\(r\le 1\) | 3.94 | 0.753 | |

\(r\le 2\) | 0.05 | 0.812 | |

\(\varPi _1\) | \(r=0\) | 118.39 | 0.000 |

\(\mathbf {r\le 1}\) | | | |

\(r\le 2\) | 0.00 | 0.958 | |

\(\varPi _2\) | \(r=0\) | 104.48 | 0.000 |

\(\mathbf {r\le 1}\) | | | |

\(r\le 2\) | 0.03 | 0.843 | |

\(\varPi _3\) | \(r=0\) | 157.81 | 0.000 |

\(r\le 1\) | 63.82 | 0.000 | |

\(\mathbf {r\le 2}\) | | |

The test does not reject the hypothesis \(H_r: r=0\), thus suggesting that there is no cointegration present in the system. This in turn implies that the oscillators are independent in terms of synchronization, in accordance with the DGP for \(\varPi _0\), and with the mean phase coherence measure.

### 4.3 Uni-directional coupling

The unwrapped phases for the simulation of model \(\varPi _1\) are seen in the top-right of Fig. 3. The dashed lines indicate the independent phases from the top-left of Fig. 3, and we see that phases \(\phi _{2t},\phi _{3t}\) are equal to their independent versions, whereas we now clearly find that \(\phi _{1t}\) is attracted towards \(\phi _{2t}\) due to the coupling structure in the system.

Results from the rank test are in the second part of Table 1. Here we see that \(r={{\mathrm{rank}}}(\varPi _1)=0\) is clearly rejected, whereas \(r=1\) cannot be rejected with a *p* value of 0.568. This indicates the presence of a single cointegration relation, in accordance with the construction of the model.

Fitted model \(\varPi _1\)

Parameter | True value | Unrestricted estimates | Restricted \(\alpha \) and \(\beta \) | ||||
---|---|---|---|---|---|---|---|

Estimate | SE | | Estimate | SE | | ||

\(\alpha _1\) | \(-\)0.5 | \(-\)0.527 | 0.049 | <0.001 | \(-\)0.514 | 0.048 | <0.001 |

\(\alpha _2\) | 0 | \(-\)0.050 | 0.049 | 0.307 | 0 | ||

\(\alpha _3\) | 0 | 0.059 | 0.048 | 0.223 | 0 | ||

\(\beta _1\) | 1 | 1 | 1 | ||||

\(\beta _2\) | \(-\)1 | \(-\)0.981 | \(-\)1 | ||||

\(\beta _3\) | 0 | \(-\)0.016 | 0 | ||||

\(\kappa _1\) | 0.75 | 0.765 | 0.076 | <0.001 | 0.638 | 0.081 | <0.001 |

\(\kappa _2\) | 1 | 1.035 | 0.075 | <0.001 | 1.063 | 0.080 | <0.001 |

\(\kappa _3\) | 1 | 1.119 | 0.074 | <0.001 | 1.086 | 0.080 | <0.001 |

*A*fix \(\alpha _2=\alpha _3=0\) and

*B*restricts to a 1:1 coupling. This yields the test statistic 3.617 which is \(\chi ^2\) distributed with 4 degrees of freedom and hence implies a

*p*value of 0.460. Thus, we recover the true uni-directional coupling structure of the simulated phases. The fitted model is presented in the right of Table 2.

The conclusion is that we have successfully identified the coupling structure of uni-directional coupled phases in a three dimensional system, with two independent phases, and one dependent. Since \(\phi _{3t}\) is completely independent of \(\phi _{1t}\) and \(\phi _{2t}\) and \(r=1\) we can discard \(\phi _{3t}\) when interpreting the cointegration in the system. Then we can interpret the cointegration parameter \(\alpha \) as the coupling strength and \(\beta \) as the coupling scheme, here 1:1. If we had analyzed different data, with a \(\beta \) estimate close to \(\hat{\beta }=(1,-n,0)'\), we could then identify a *n*:1 coupling between \(\phi _{1t}\) and \(\phi _{2t}\). This can be seen from the fact that in this case \(\alpha _k(\phi _{1t}-n\phi _{2t})\) would be a stationary relation, and thus \(\phi _{2t}\) would rotate \(\approx \) *n* times slower than \(\phi _{1t}\).

### 4.4 A bi-directional coupling with one independent oscillator

*p*value of 0.707. Hence, we recover the correct dimension of the column space of \(\beta \), and fitting a model with \(r=1\) yields the parameters in the left of Table 3.

Fitted model \(\varPi _2\)

Parameter | True value | Unrestricted estimates | Restricted \(\alpha \) and \(\beta \) | Restricted \(\alpha \) and \(\beta \) with \(A^*\) | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Estimate | SE | | Estimate | SE | | Estimate | SE | | ||

\(\alpha _1\) | \(-\)0.5 | \(-\)0.530 | 0.071 | <0.001 | \(-\)0.506 | 0.069 | <0.001 | \(-\)0.475 | 0.069 | <0.001 |

\(\alpha _2\) | 0.5 | 0.450 | 0.069 | <0.001 | 0.443 | 0.067 | <0.001 | 0.475 | 0.067 | <0.001 |

\(\alpha _3\) | 0 | 0.087 | 0.070 | 0.214 | 0 | 0 | ||||

\(\beta _1\) | 1 | 1 | 1 | 1 | ||||||

\(\beta _2\) | \(-\)1 | \(-\)0.970 | \(-\)1 | \(-\)1 | ||||||

\(\beta _3\) | 0 | \(-\)0.022 | 0 | 0 | ||||||

\(\kappa _1\) | 0.75 | 0.754 | 0.072 | <0.001 | 0.646 | 0.076 | <0.001 | 0.660 | 0.076 | <0.001 |

\(\kappa _2\) | 1 | 0.943 | 0.070 | <0.001 | 1.040 | 0.074 | <0.001 | 1.053 | 0.074 | <0.001 |

\(\kappa _3\) | 1 | 1.103 | 0.071 | <0.001 | 1.086 | 0.075 | <0.001 | 1.086 | 0.075 | <0.001 |

*p*value of 0.342. The fitted model is given in the middle of Table 3. If we instead of \(H_{\alpha ,\beta }\) specify

*p*value of 0.423. Thus, we can also restrict the model to one where the coupling strengths are equal in magnitude. The fitted model is presented in the right part of Table 3.

Summing up, in a system of bi-directional coupled oscillators plus one independent, we can identify the correct coupling between them, including identifying the proportionally equal coupling strength between the coupled phases. Again we identify \(r=1\), and hence we can interpret the cointegration parameters as before, hence \(\alpha \) is the coupling strength, and \(\beta \) the interaction, again 1:1 coupling.

### 4.5 Fully coupled system

Fitted model \(\varPi _3\)

Parameter | True value | Unrestricted estimates | Restricted \(\alpha \) and \(\beta \) | ||||
---|---|---|---|---|---|---|---|

Estimate | SE | | Estimate | SE | | ||

\(\alpha _{11}\) | \(-\)0.50 | \(-\)0.584 | 0.075 | <0.001 | \(-\)0.569 | 0.075 | <0.001 |

\(\alpha _{21}\) | 0.25 | 0.232 | 0.073 | <0.001 | 0.241 | 0.072 | <0.001 |

\(\alpha _{31}\) | 0.25 | 0.326 | 0.072 | <0.001 | 0.328 | 0.072 | <0.001 |

\(\alpha _{12}\) | 0.25 | 0.223 | 0.067 | <0.001 | 0.224 | 0.067 | <0.001 |

\(\alpha _{22}\) | \(-\)0.50 | \(-\)0.423 | 0.064 | <0.001 | \(-\)0.423 | 0.064 | <0.001 |

\(\alpha _{32}\) | 0.25 | 0.201 | 0.064 | <0.001 | 0.199 | 0.064 | <0.001 |

\(\beta _{11}\) | 1 | 1 | 1 | ||||

\(\beta _{21}\) | 0 | 0 | 0 | ||||

\(\beta _{31}\) | \(-\)1 | \(-\)0.997 | \(-\)1 | ||||

\(\beta _{12}\) | 0 | 0 | 0 | ||||

\(\beta _{22}\) | 1 | 1 | 1 | ||||

\(\beta _{32}\) | \(-\)1 | \(-\)0.999 | \(-\)1 | ||||

\(\kappa _1\) | 0.75 | 0.712 | 0.076 | <0.001 | 0.607 | 0.083 | <0.001 |

\(\kappa _2\) | 1 | 1.054 | 0.074 | <0.001 | 1.061 | 0.080 | <0.001 |

\(\kappa _2\) | 1 | 1.023 | 0.073 | <0.001 | 1.130 | 0.080 | <0.001 |

*p*value of 0.785. Hence, we can reduce the model to one with restrictions that generates the true structure of \(\varPi \). The estimated model parameters are presented in Table 4, and the corresponding \(\hat{\varPi }\) is

### 4.6 Strength of coupling and identification of interaction

In this section, we compare the mean phase coherence measure to the cointegration analysis with respect to interactions in the system. More specifically, we look at how strong the coupling constants in \(\varPi \) must be in order for the two methods to conclude correctly on interaction in the system. We reuse the parameter settings (34) from the fully coupled experiment, but use a scaled \(\varPi \) matrix \(\varPi \rightarrow \varepsilon \varPi \), for \(\varepsilon \in [0,1]\), where \(\varepsilon \) controls the coupling strength. The higher \(\varepsilon \) is, the stronger the coupling, and hence the attraction between phases. Note that \(\varepsilon =0\) corresponds to the model \(\varPi _0\) and \(\varepsilon =1\) corresponds to \(\varPi _3\). The *p* values are calculated using bootstrapping as presented by Cavaliere et al. (2012) to obtain an estimate of the asymptotic distribution of the trace test statistics.

The experimental setup is 100 repetitions for each value of \(\varepsilon \), and in each repetition perform 500 bootstrap samples to estimate the *p* value for the hypotheses \(H_r: r=0,1,2\). Figure 4 presents the median *p* values for the rank test and median mean phase coherence measures against \(\varepsilon \). The top row of the figure shows the *p* values for \(H_r: r=0,1,2\) respectively, and the bottom row shows the mean phase coherence (*R*) measures for pairs of \(\phi _{1t},\phi _{2t}\) and \(\phi _{3t}\). The dotted lines indicate the \(p=0.05\) value, under which we reject the hypothesis. For the mean phase coherence measure, the 95% significance level for the hypothesis \(R=0\) has been determined numerically using bootstrapping and is indicated by the dotted lines. If the *R*-measure falls below this line, independence cannot be rejected.

Seen in the top row of Fig. 4, at least half the simulations reject \(H_r: r=0\) for \(\varepsilon > 0.12\), and at least half the simulations reject \(H_r: r=1\) for \(\varepsilon >0.11\). The test does not reject \(H_r: r=2\) for around 88% of the simulated samples for any values of \(\varepsilon \). Thus, for \(\varepsilon >0.11\), we can conclude that there is interaction present in the system, and in most of the simulations we also recognize the true \({{\mathrm{rank}}}(\varPi )=2\).

If we turn to the bottom row of Fig. 4, where the mean phase coherence measures are shown, we find that half the simulations does not reject the hypothesis \(R=0\) for \(\varepsilon <0.34,0.36\) and 0.35, respectively, for \(R(\phi _{1t},\phi _{2t})\),\(R(\phi _{1t},\phi _{3t})\) and \(R(\phi _{2t},\phi _{3t})\), thus clearly indicating an inferior detection of interaction for small values of \(\varepsilon \) equivalent to weak couplings.

Concluding on this experiment, we find that the rank test detects interaction in the system already at relatively weak coupling strengths. In contrast to this, the coupling must be significantly stronger for a sound conclusion on interaction in the system when using mean phase coherence as a measure of interaction. Furthermore, when detecting interaction in the system, the rank test is also very capable of identifying the true rank of the system, despite a misspecified model. Higher sample sizes will of course improve the inference results.

### 4.7 Consistency of the rank estimation

Percentage of conclusions on \({{\mathrm{rank}}}(\varPi )\), at a 5% significance level for a sample size of 2000

Model | \(r=0\) | \(r\le 1\) | \(r\le 2\) | \(r\le 3\) |
---|---|---|---|---|

Independent (%) | | 2.2 | 1.3 | 0.3 |

Uni-directional (%) | 1.7 | | 19.0 | 2.5 |

Bi-directional (%) | 2.4 | | 24.7 | 3.1 |

Fully coupled (%) | 0.0 | 1.3 | | 13.2 |

Percentage of conclusions on interaction indicated by the rank test and the mean phase coherence measures, at a 5% significance level for a sample size of 2000

Model | Rank test | \(R(\phi _{1t},\phi _{2t})\) | \(R(\phi _{1t},\phi _{3t})\) | \(R(\phi _{2t},\phi _{3t})\) |
---|---|---|---|---|

Independent (%) | 3.8 | 4.7 | 4.4 | 5.7 |

Uni-directional (%) | 98.3 | 99.8 | 5.6 | 4.4 |

Bi-directional (%) | 97.6 | 100.0 | 7.2 | 7.0 |

Fully coupled (%) | 100 | 100.0 | 100 | 100.0 |

These results show that identification of interaction in a system of coupled oscillators is quite precise, and the rank is underestimated in \(\le \)2.5% of the simulations for any model. In the case of independent or full interaction, the method is very good, whereas for systems with directed interaction, or interaction among some oscillators the frequency of overestimating the rank is \(\approx \)20–25%. This discrepancy seems intuitively correct, since for the latter systems the true model is a subset of the model of higher rank. As before higher sample sizes will of course improve the inference results.

In Table 6 we compare, in percentages, the conclusions on interaction in the systems, for each model. The values for the rank test presented here, are the summed values from Table 5 for \(r\ne 0\). We find that both methods are very adept in identifying interaction in these systems. The results, however, should be held against the previous section, where the rank test outperformed the mean phase coherence measure for weak coupling strength. Also noting the fact, that the mean phase coherence measure cannot account for uni-directional coupling, our overall conclusion is that in terms of identifying interaction in the system, the methods seem to perform equally well for stronger coupling, whereas in explaining the system architecture, a cointegration analysis leaves us with more information on how the network is constructed.

## 5 Analysis of EEG data

*prior to seizure*and

*during seizure*. With a sample frequency of 256 measurements each second there are a total of 10,240 measurements for each of the four signals during the 40 s intervals. For more details on the data, see Shoeb (2009). The objective is to compare two fitted cointegration models with interaction as in Eq. (8) for each period:

Mean phase coherence measures for EEG phases prior to and during the seizure

Prior to seizure | During seizure | |
---|---|---|

\(R_{(\text {FP1-F3; FP2-F4)}}\) | 0.480 | 0.542 |

\(R_{(\text {FP1-F3; FP1-F7)}}\) | 0.535 | 0.644 |

\(R_{(\text {FP1-F3; FP2-F8)}}\) | 0.295 | 0.184 |

\(R_{(\text {FP2-F4; FP1-F7)}}\) | 0.321 | 0.350 |

\(R_{(\text {FP2-F4; FP2-F8)}}\) | 0.486 | 0.342 |

\(R_{(\text {FP1-F7; FP2-F8)}}\) | 0.525 | 0.379 |

Average | 0.440 | 0.407 |

Rank tests for EEG phases in the bottom of Fig. 6

\(H_r\) | Prior to seizure | During seizure | ||
---|---|---|---|---|

Test values | | Test values | | |

\(r=0\) | 105.87 | 0.000 | 1132.64 | 0.000 |

\(r\le 1\) | 42.82 | 0.000 | 41.68 | 0.008 |

\(\mathbf {r\le 2}\) | | | | |

\(r\le 3\) | 0.46 | 0.439 | 0.72 | 0.786 |

In accordance with the indications from the mean phase coherence measure, the conclusion is a clear presence of cointegration during both periods. Prior to the seizure the rank test of \(r\le 2\) is close to the usual 5% significance level, hence the *p* value here is determined using 5000 bootstrap samples, in contrast to the 2000 bootstrap samples used in the other interval, as the conclusion here is quite clear with a *p* value \(\approx \)0.62. In both cases we choose the rank \(r=2\) for the system.

Fitted model for EEG phases F7-T7, T7-P7 and FP1-F7

Parameter | Prior to seizure | During seizure | ||||
---|---|---|---|---|---|---|

Estimate | SE | | Estimate | SE | | |

\(\alpha _{\text {FP1-F3},1}\) | \(-\)0.100 | 0.018 | <0.001 | \(-\)0.462 | 0.028 | <0.001 |

\(\alpha _{\text {FP1-F7},1}\) | \(-\)0.002 | 0.019 | 0.930 | \(-\)0.308 | 0.032 | <0.001 |

\(\alpha _{\text {FP2-F4},1}\) | \(-\)0.035 | 0.017 | 0.044 | \(-\)0.722 | 0.035 | <0.001 |

\(\alpha _{\text {FP2-F8},1}\) | \(-\)0.115 | 0.030 | <0.001 | \(-\)0.648 | 0.042 | <0.001 |

\(\alpha _{\text {FP1-F3},2}\) | \(-\)0.117 | 0.016 | <0.001 | 0.041 | 0.033 | 0.212 |

\(\alpha _{\text {FP1-F7},2}\) | \(-\)0.024 | 0.016 | 0.147 | 0.071 | 0.037 | 0.057 |

\(\alpha _{\text {FP2-F4},2}\) | \(-\)0.026 | 0.015 | 0.084 | 0.173 | 0.041 | <0.001 |

\(\alpha _{\text {FP2-F8},2}\) | \(-\)0.049 | 0.026 | 0.063 | 0.468 | 0.049 | <0.001 |

\(\beta _{\text {FP1-F3},1}\) | 1 | 1 | ||||

\(\beta _{\text {FP1-F7},1}\) | 0 | 0 | ||||

\(\beta _{\text {FP2-F4},1}\) | \(-\)3.424 | \(-\)0.036 | ||||

\(\beta _{\text {FP2-F8},1}\) | 2.610 | \(-\)0.573 | ||||

\(\beta _{\text {FP1-F3},2}\) | 0 | 0 | ||||

\(\beta _{\text {FP1-F7},2}\) | 1 | 1 | ||||

\(\beta _{\text {FP2-F4},2}\) | 2.486 | \(-\)0.840 | ||||

\(\beta _{\text {FP2-F8},2}\) | \(-\)3.631 | 0.188 | ||||

\(\mu _\text {FP1-F3}\) | 25.210 | 2.162 | <0.001 | 39.647 | 1.307 | <0.001 |

\(\mu _\text {FP1-F7}\) | 30.648 | 2.252 | <0.001 | 36.499 | 1.473 | <0.001 |

\(\mu _\text {FP2-F4}\) | 39.058 | 2.107 | <0.001 | 58.268 | 1.608 | <0.001 |

\(\mu _\text {FP2-F8}\) | 48.853 | 3.615 | <0.001 | 54.765 | 1.947 | <0.001 |

Fitted \(\hat{\varPi }\) matrices for the two periods

\(\hat{\varPi }\) prior to seizure | \(\hat{\varPi }\) during seizure | |||||||
---|---|---|---|---|---|---|---|---|

FP1-F3 | FP1-F7 | FP2-F4 | FP2-F8 | FP1-F3 | FP1-F7 | FP2-F4 | FP2-F8 | |

FP1-F3 | 4.388 | 1.572 | \(-\)11.120 | 5.743 | \(-\)5.305 | \(-\)11.021 | 9.447 | 0.971 |

FP1-F7 | 1.519 | 0.892 | \(-\)2.985 | 0.725 | \(-\)4.335 | \(-\)7.285 | 6.275 | 1.116 |

FP2-F4 | 0.540 | \(-\)0.050 | \(-\)1.971 | 1.589 | \(-\)10.265 | \(-\)17.047 | 14.686 | 2.681 |

FP2-F8 | \(-\)0.733 | \(-\)1.658 | \(-\)1.613 | 4.108 | \(-\)14.907 | \(-\)14.729 | 12.909 | 5.776 |

Here we can determine an all-to-all coupling during both periods and the estimated cointegration matrices show a clear difference for the two intervals. Prior to the seizure the right side signals FP2-F4 and FP2-F8 are much less influenced by the feedback in the system, whereas during the seizure both experience a much larger feedback from the left side signals FP1-F3 and FP1-F7 respectively. Surprisingly, the FP2-F8 signal does not seem to impose a large influence in the system in either interval. It is also interesting to note the changing signs in the two matrices. The two left side signals exhibit a positive feedback on themselves prior to the seizure, whereas during the seizure they impose a negative feedback both on themselves and the right side signals. This could possibly be part of an explanation of the slight kink seen in the phases around 3015–3020 s halfway through the seizure.

Concluding on this analysis we find, not surprisingly, a fully coupled 4 dimensional system with a clear change in the trends prior to and during the seizure. We find that during the seizure the interaction in the system is much stronger, suggesting the more distinctive phases shown in this interval. Including this temporal effect into a single cointegration model covering the full period by utilizing regime switching cointegration models, would be an interesting pursuit for future work.

## 6 Discussion

In this paper we have investigated the use of cointegration analysis to determine coupling structures in linearly phase coupled systems. Using these techniques we can with a good precision identify the coupling structure as a subspace for this type of model. A standard measure to identify synchronization in the literature is the mean phase coherence measure. Contrary to this standard measure, we can detect uni-directional coupling, and we can construct and test hypotheses on the model in form of linear restrictions in the estimated subspace. Furthermore, comparing the mean phase coherence measure with the cointegration analysis in Sect. 4.6, we found that cointegration detects interaction in a system more robustly and for weaker coupling strength than does the mean phase coherence measure. Combined with the fact that cointegration does not just provide a level of synchronization, but rather the structure of the synchronization mechanism, this technique can be used to infer system structures in a much more detailed manner. Of course this higher level of information comes at a cost, since the mean phase coherence measure is easily implemented for any system, whereas the cointegration analysis is more involved and time consuming.

Due to the linear nature of the cointegration theory used, we are not able to cover more complex models, such as the Kuramoto model. Thus, an important extension for future work would be to allow for nonlinear coupling functions. However, the linear structure appears naturally when considering a linearization around some phase-locked state, such as for systems showing synchrony or asynchrony. Another interesting pursuit is to extend the model framework to include nonlinear deterministic trends, such that also models like the FitzHugh–Nagumo or the van der Pol oscillator would be covered. The model considered in this paper was constructed from the starting point of the phase process in the spirit of the Kuramoto model, and noise was added on this level. Another approach would be to start from a biological model or a reduction thereof and introduce the noise on the DGP. This would also lead to non-linearities both in drift and diffusion of the phase process. Finally, high dimensional systems are a major challenge in the area of coupled oscillators, hence it would only be natural to investigate cointegration properties of high dimensional systems. A system of more than two synchronizing oscillators that are nonlinearly phase coupled, facilitate chaotic behavior since phases can then bilaterally attract and repel each other. When the number of oscillators increase, one quickly ends up with intuitive shortcomings. The number of parameters rapidly increase with the dimension of the system, possibly leading to a desirable reduction to a sparse interaction structure. This is a key issue with the cointegration framework, which take into account all individual oscillators, as opposed to a mean-field approach that does not run into the same curse of dimensionality. The quality of the estimators will rapidly decrease with increasing dimension of the parameter space or numerical problems may arise. This problem might be alleviated by imposing a sparse interaction structure through a LASSO \(L_1\) penalization.

Cointegration to identify coupling of oscillators has been attempted before in a neuroscience context by Dahlhaus and Neddermeyer (2012). There, the Kuramoto model is approximated for strongly phase coupled oscillators by setting \(\sin (\phi _j - \phi _i) \approx \phi _j-\phi _i\), since the phase differences are assumed to be small. We have used the idea from Dahlhaus and Neddermeyer (2012) of analyzing the unwrapped multivariate phase process. Contrary to Dahlhaus and Neddermeyer (2012), however, we have not linearized the sine function to replicate Kuramoto, since this will cause a discrepancy when the phase difference of two oscillators is closer to \(\pi \) than 0 (or \(2\pi \)). To mitigate this problem, we have instead taken the approach of designing a DGP with the properties we are interested in, and which allows for any phase differences. Furthermore, this DGP enables us to specify a cointegration model that comply with data from this DGP. Although it may not fully comply with a biological model, it can point to where necessary flexibility is needed in order to develop more realistic cointegration models for biological processes. A first attempt to analyze EEG signals with cointegration analysis with linear coupling structures has been presented. The results are promising, and reveal a finer dependence structure characterizing states of seizure and non-seizure in epileptic patients, which in this example was not possible from the simple Mean Phase Coherence measure. To fully explore the potential of the cointegration analysis for EEG signals, it would be useful to extend the model and analysis tools to allow for non-linearities and simultaneous treatment of many traces, as well as time varying coupling strengths.

Summing up, by applying cointegration as a technique to the field of coupled oscillators in biology, we open up for a whole new area of applications for this statistical theory. On the other hand, using cointegration methods, biologists can gain new insights into network structures, being able to fit models and carry out statistical hypothesis testing. If the cointegration framework presented in this paper can be extended to include the standard models currently used in the field, cointegration would prove a powerful analysis tool for researchers.

## Footnotes

- 1.
The Hilbert transform of a signal \(x_t\) is defined as \(H(x_t) = \pi ^{-1}\text {p.v.}\int _{-\infty }^\infty \frac{x_\tau }{t-\tau }d\tau = -\pi ^{-1}\lim _{\varepsilon \rightarrow 0}\int _\varepsilon ^\infty \frac{x_{t+\tau }-x_{t-\tau }}{\tau }d\tau \), where \(\text {p.v.}\int _{-\infty }^\infty \) denotes the principal value integral.

## Notes

### Acknowledgements

The work is part of the Dynamical Systems Interdisciplinary Network, University of Copenhagen.

## References

- Ashwin P, Coombes S, Nicks R (2016) Mathematical frameworks for oscillatory network dynamics in neuroscience. J Math Neurosci 6(1):1–92MathSciNetCrossRefzbMATHGoogle Scholar
- Bec F, Rahbek A (2004) Vector equilibrium correction models with non-linear discontinuous adjustments. Econom J 7(2):628–651MathSciNetCrossRefzbMATHGoogle Scholar
- Boswijk HP, Cavaliere G, Rahbek A, Taylor AMR (2016) Inference on co-integration parameters in heteroskedastic vector autoregressions. J Econom 192(1):64–85MathSciNetCrossRefzbMATHGoogle Scholar
- Burton SD, Ermentrout GB, Urban NN (2012) Intrinsic heterogeneity in oscillatory dynamics limits correlation-induced neural synchronization. J Neurophysiol 108(8):2115–2133CrossRefGoogle Scholar
- Cavaliere G, Rahbek A, Taylor AMR (2012) Bootstrap determination of the co-integration rank in vector autoregressive models. Econometrica 80(4):1721–1740MathSciNetCrossRefzbMATHGoogle Scholar
- Dahlhaus R, Neddermeyer J (2012) On the relationship between the theory of cointegration and the theory of phase synchronization. Working paper. arXiv:1201.0651
- Ermentrout GB (1985) Synchronization in a pool of mutually coupled oscillators with random frequencies. J Math Biol 22(1):1–9MathSciNetCrossRefzbMATHGoogle Scholar
- Fernandez B, Tsimring LS (2014) Typical trajectories of coupled degrade-and-fire oscillators: from dispersed populations to massive clustering. J Math Biol 68(7):1627–1652MathSciNetCrossRefzbMATHGoogle Scholar
- Granger C (1981) Some properties of time series data and their use in econometric model specification. J Econom 16(1):121–130MathSciNetCrossRefGoogle Scholar
- Johansen S (1996) Likelihood-based inference in cointegrated vector autoregressive models. Oxford University Press, OxfordzbMATHGoogle Scholar
- Kessler M, Rahbek A (2001) Asymptotic likelihood based inference for co-integrated homogenous Gaussian diffusions. Scand J Stat 28(3):455–470MathSciNetCrossRefzbMATHGoogle Scholar
- Kessler M, Rahbek A (2004) Identification and inference for multivariate cointegrated and ergodic Gaussian diffusions. Stat Infer Stoch Process 7(2):137–151MathSciNetCrossRefzbMATHGoogle Scholar
- Kristensen D, Rahbek A (2013) Testing and inference in nonlinear cointegrating vector error correction models. Econom Theory 29(6):1238–1288MathSciNetCrossRefzbMATHGoogle Scholar
- Kuramoto Y (1984) Chemical oscillations, waves and turbulence. Springer, BerlinCrossRefzbMATHGoogle Scholar
- Ltkepohl H (2005) New introduction to multivariate time series analysis. Springer, BerlinCrossRefGoogle Scholar
- Ly C (2014) Dynamics of coupled noisy neural oscillators with heterogeneous phase resetting curves. SIAM J Appl Dyn Syst 13(4):1733–1755MathSciNetCrossRefzbMATHGoogle Scholar
- Ly C, Ermentrout GB (2011) Analytic approximations of statistical quantities and response of noisy oscillators. Phys D Nonlinear Phenom 240(8):719–731CrossRefzbMATHGoogle Scholar
- Martens EA, Thutupalli S, Fourrire A, Hallatschek O (2013) Chimera states in mechanical oscillator networks. Proc Natl Acad Sci 110(26):10563–10567CrossRefGoogle Scholar
- Mormann F, Lehnertz K, David P, Elger CE (2000) Mean phase coherence as a measure for phase synchronization and its application to the EEG of epilepsy patients. Phys D: Nonlinear Phenom 144(3–4):358–369CrossRefzbMATHGoogle Scholar
- Oliveira HM, Melo LV (2015) Huygens synchronization of two clocks. Sci Rep 5:11548 EPCrossRefGoogle Scholar
- Pikovsky A, Rosenblum M, Kurths J (2001) Synchronization: a universal concept in nonlinear sciences. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
- R Core Team (2015) R: a language and environment for statistical computing. R Foundation for Statistical Computing, ViennaGoogle Scholar
- Schmith T, Johansen S, Thejll P (2012) Statistical analysis of global surface temperature and sea level using cointegration methods. J Clim 25(22):7822–7833CrossRefGoogle Scholar
- Shoeb AH (2009) Application of machine learning to epileptic seizure onset detection and treatment. Dissertation, MIT Health Sciences and Technology DivisionGoogle Scholar
- Strogatz SH (1987) Human sleep and circadian rhythms: a simple model based on two coupled oscillators. J Math Biol 25(3):327–347MathSciNetCrossRefzbMATHGoogle Scholar
- Strogatz SH (2000) From Kuramoto to Crawford exploring the onset of synchronization in populations of coupled oscillators. Phys D Nonlinear Phenom 143(1–4):1–20MathSciNetCrossRefzbMATHGoogle Scholar
- Taylor D, Holmes P (1998) Simple models for excitable and oscillatory neural networks. J Math Biol 37(5):419–446MathSciNetCrossRefzbMATHGoogle Scholar
- Winfree AT (1967) Biological rhythms and the behavior of populations of coupled oscillators. J Theor Biol 16(1):15–42CrossRefGoogle Scholar
- Winfree AT (2001) The geometry of biological time. Springer, BerlinCrossRefzbMATHGoogle Scholar

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