Original article
Bifurcation analysis of the Poincaré map function of intracranial EEG signals in temporal lobe epilepsy patients

https://doi.org/10.1016/j.matcom.2011.03.012Get rights and content

Abstract

In this paper, the Poincaré map function as a one-dimensional first-return map is obtained by approximating the scatter plots of inter-peak interval (IPI) during preictal and postictal periods from invasive EEG recordings of nine patients suffering from medically intractable focal epilepsy. Evolutionary Algorithm (EA) is utilized for parameter estimation of the Poincaré map. Bifurcation analyses of the iterated map reveal that as the neuronal activity progresses from preictal state toward the ictal event, the parameter values of the Poincaré map move toward the bifurcation points. However, following the seizure occurrence and in the postictal period, these parameter values move away from the bifurcation points. Both flip and fold bifurcations are analyzed and it is demonstrated that in some cases the flip bifurcation and in other cases the fold bifurcation are the dynamical regime underlying epileptiform events. This information can offer insights into the dynamical nature and variability of the brain signals and consequently could help to predict and control seizure events.

Introduction

The brain is a complex network of interacting subsystems and it is now well documented that synchronization plays an important role in normal and abnormal brain functioning. A well-known case for pathophysiologic neuronal synchronization is epilepsy [2], [20]. Epilepsy is a common neurological disorder, second only to stroke, that affects more than 50 million people worldwide. It is characterized by recurrent seizures or ictal events and impairs normal functions of the brain. These seizures are signs of excessive synchronous neuronal activity in the brain [1], [11]. Two-thirds of the patients achieve sufficient seizure control from anticonvulsive drugs and about 8–10% could benefit from resective surgery. For the remaining of patients, no satisfactory treatment is currently available [24].

The neurophysiologic signals such as electroencephalograph (EEG) that reflect the macroscopic spatio-temporal dynamics of the brain activity have received worthwhile attention in recent years [5]. Since EEG signal can be considered as the output of a highly nonlinear and multidimensional dynamic system, the framework of nonlinear dynamical systems provides new concepts and powerful tools in order to extract relevant information from EEG signals [37]. Several studies indicate that nonlinear methods can extract valuable information from neuronal dynamics [5]. Currently, epilepsy is probably the most important application for nonlinear EEG analysis [34]. Babloyantz and Destexhe were among the first to apply nonlinear dynamics to analyze absence seizure (3 Hz spike and wave discharges) [6]. The correlation dimension of this type of seizure was lower than the dimension of normal waking EEG. This suggested that epileptic seizures might be due to a pathological ‘loss of complexity’. In the same way, Iasemidis and his colleagues showed the decrease of the largest Lyapunov exponent for patients with temporal lobe epilepsy. They found that the EEG activity progressively becomes less chaotic as the seizure approaches [14], [15]. Since these pioneering studies, nonlinear dynamical system theory has been employed to quantify the changes in the brain dynamics in preictal, ictal and postictal periods. Other studies based on nonlinear associations in multivariate signals have reported that long distance functional connectivity is considerably changed during seizures [13] or indicated that the topology of networks alters as ictal activity grows [7], [30].

The fact that the interictal EEG is high dimensional and seizure activity is low dimensional, raises the question as tohow the transition between these two states can occur. This transition has two aspects: changes in the local dynamics of attractors and changes in the coupling between different brain areas [21], [34]. With respect to the second aspect, seizures are generally characterized by an increase in coupling among different brain areas. However, in some types of seizures, there is a decrease in the level of coupling [9], [25]. Regarding the first aspect, Lopes da Silva and his collaborators reviewed the dynamics of seizure generation. They proposed three different routes to epileptic seizures: (1) an abrupt transition, of the bifurcation type; this would be characteristic of absence seizure; (2) reflex epilepsy: deformation of the attractor is caused by an external stimulus and (3) a deformation of the attractor leading to a gradual evolution onto the ictal state (temporal lobe epilepsy) [21], [34]. Therefore, the theory of nonlinear dynamical systems suggests that the state transition is probably due to arising of one or more bifurcations when some critical parameters of a neuronal network change.

When studying a highly complex system, a conventional approach is to reduce the system's multidimensional continuous trajectory in the state space to a discrete low dimensional projection which is known as Poincaré map [33]. The question of how to determine the specific dynamical regimes of brain activity has caught considerable attention among neurophysiology and engineering communities. In this respect, Velazquez and his colleagues constructed a return map and qualitatively analyzed the dynamical regimes underlying epileptiform events [28], [29]. The main concept of their studies was to use the time interval between spikes in EEG recordings as a variable to construct inter-peak interval (IPI) plots. In this way, neuronal population activity during the transition to seizure as well as during seizures could be studied. For approximating the local system dynamics, a first-return one dimensional mapping function was obtained by a Levenberg–Marquradt (LM) fitting of the IPI plot with an inverted polynomial. It should be pointed out that this mapping function is not an accurate model of epileptic activity and cannot reveal the rich variety of brain dynamics. However, it can characterize the state of the system by capturing essential phenomena of the collective dynamics of brain network and provides a relatively good approximation to the dynamics observed [18], [26].

Drawing on these concepts, in this paper a quantitative analysis of some important dynamical mechanisms that may take place during epileptic neuronal activity is presented. Similar to the procedure that was used by Velazquez and colleagues, a first-return one dimensional map (Poincaré map) is obtained by approximating the scatter plots of IPI during preictal, ictal and postictal periods. For doing so, invasive EEG recordings of nine patients suffering from medically intractable focal epilepsy are used. In this way, a discrete representation of the original time series is provided. Evolutionary Algorithm (EA) is utilized for parameter estimation of the Poincaré map. Mathematical analyses reveal that the flip and fold bifurcations can occur during the transition to seizure.

The outline of the paper is as follows: in Section 2, some relevant definitions and theorems will be introduced and then the intracranial EEG data recorded at the Epilepsy Center of the University Hospital of Freiburg, Germany are explained. The construction process of the first-return IPI scatter plots and its approximation by an inverted quadratic polynomial are also covered in this section. In Section 3, based on the bifurcation theory, some explicit equations are derived which relate the bifurcation types to the Poincaré map parameter values. The results of some simulation are discussed in Section 4 and finally, Section 5 concludes the paper.

Section snippets

Preliminary remarks

Dynamical systems may be continuous or discrete, depending on whether they are described by differential or difference equations. The difference equation for a general time-invariant discrete dynamical system can be written as:Xk+1=f(Xk)k=0,1,where f :  n   n, X   n can be a linear or nonlinear function of Xk [3]. The following definitions and theorems are of interest in (1) [4], [19]:

Definition 1

A point x¯ is an equilibrium point for the dynamical system (1), or a fixed point for map f, if f(x¯)=x¯.

Definition 2.a

A

Bifurcation analysis

Bifurcation is commonly used in the study of nonlinear dynamics to describe qualitative changes of the behavior of the system as one or more control parameters are changed. In this section, we analyze the dynamical behavior of the Poincaré map function using nonlinear dynamical system theory. Consider the first order dynamical system in the form of:g(yk)=yk+1=d+1a1yk2+b1yk+c1a1,b1,c1,dchanging the variable y  d = x; we can writexk+1=1a1(xk+d)2+b1(xk+d)+c1After some calculation, it is

Results and discussion

In this section, it is shown that the suggested Poincaré map from the IPI plots is able to capture some features of the epileptic EEG data and provide insights into the underlying dynamics of the transition from preictal to ictal and then to postictal periods. Fig. 9 shows changes in the behavior of the first-return IPI plots before, during, and after the seizure for 100 s in patient number 2 with temporal lobe epilepsy [23]. The first-return IPI plot obtained for 40 s of preictal activity

Conclusion

Knowledge about the dynamics of epileptiform activity is an important scientific question with practical considerations in clinical control of seizure activity. Since the neural systems have strong nonlinear characteristics and are usually able to display different dynamics according to system parameters or external inputs, application of linear methods to signals generated by these nonlinear systems may result in spurious conclusions. In general, Fourier decomposition and similar methods are

Acknowledgements

The authors would like to thank the anonymous and esteemed reviewers for their valuable comments on earlier versions of this paper. M. Amiri would like to thank Prof. Olivier David for his insightful suggestions and appreciates his assistance.

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