Hidden Markov Chain modeling for epileptic network

Hidden Markov Chain modeling for epileptic network identification Steven Le Cam1 , Val´erie Louis-Dorr1 and Louis Maillard2

Abstract— Partial epileptic seizure genesis is rooted in a single structure area due to a wrong balance between inhibitory and excitatory interneurons connections. These abnormal balance translates in loss of functional connectivity between remote brain structures, while functional connectivity within the epileptogenic zone is enforced. The identification of the resulting epileptic network underlying these hyper-synchronies is likely to bring a decisive insight on the origin and the development of the seizure. In this objective, various thresholding strategies have been proposed, based on synchrony measurements computed from the brain electrophysiologic activity. Such methods are reported to be prone to errors and false alarms. A hidden Markov chain modeling of the synchrony states is proposed with the aim to develop a reliable machine learning methods for epileptic network inference. The method is applied on a real SEEG data set, demonstrating consistent results with regard to clinical evaluation and to current knowledge on partial epilepsies. Index Terms— Partial epilepsy, stereo-EEG, network inference, bayesian approach, hidden Markov chain

I. I NTRODUCTION Partial epilepsies are occuring due to the hyperexcitabe nature of a single structure area. This set of structures is known as the epileptogenic zone, commonly assumed to be organized in an epileptic network responsible for the genesis of paroxistic events. The topology of this network takes successive configurations all along the time course of the seizure [1], and their identification would be helpful in understanding the role of each of the incriminated structures in the epileptic process. The strength of connectivity of two structures are usually evaluated from the electrophysiologic activities given by EEG or Stereo-EEG (SEEG) measurements setup1 . Various quantification methods have been proposed [2], from those the network topology has to be infered [3]. In this paper, a bivariate cross-correlation quantification using MODWT (Maximum Overlap Discrete Wavelet Transform) has been chosen to evaluate time-delayed synchrony between SEEG channels. The efficiency of such frequency-dependent quantification has been demonstrated in the case of epilepsy [4], [5]. 1 Steven Le Cam and Val´ erie Louis-Dorr are with the Universit´e de Lorraine, CRAN, UMR 7039, 54500 Vandoeuvre les Nancy, France and with the CNRS, CRAN, UMR 7039, France steven.le-cam at

univ-lorraine.fr 2 Louis Maillard is with the Universit´ e de Lorraine, CRAN, UMR 7039, 54500 Vandoeuvre les Nancy, France, with the CNRS, CRAN, UMR 7039, France and with the CHU Nancy, Neurology Service, 54000 Nancy, France

l.maillard at chu-nancy.fr 1 The SEEG modality, observed on bipolar montage, provides local measurements of targeted brain structures activity by placing nailed electrodes very near from the brain generators suspected to be involved in the epileptic process.

A simple and often used method for graph inference is based on thresholding strategies [6], [3]. The choice of the threshold has a high impact on the deduced graph and is very sensible to concurrent activities and background noise correlation effect. In particular, such approach does not take into account of the synchrony persistence along the seizure time course. In this work, a probabilistic single-event method is proposed to derive the connectivity graph from a bayesian segmentation using Hidden Markov Chain (HMC). HMC is a well known model for robust signal segmentation and learning of data sequences, and are introduced here for the purpose of identifying stabilized synchrony states all along the epileptic event. The Bayesian approach combined with optimization algorithm is able to learn the model parameters, including synchrony delays and thresholds. The method is here illustrated on a Temporal Lobe Epilepsy (TLE) SEEG data set recorded at the Neurology Unit of Nancy Hospital (France).

II. C ONNECTIVITY

MEASUREMENT AND LIKELIHOOD

A. The epileptic context A wide range of measurements have been proposed in the recent past years [2], from linear to non-linear analysis of bivariate or multivariate EEG/SEEG time series, with the objective to characterize functionnal brain connectivity. Depending on the phenomena that are to be emphasized, the quantification method has to be chosen carefully, each having its specifities in capturing particular synchronies within the electrophysiologic data. In the case of TLE, linear correlation-frequency dependent methods can be advantageously chosen [4]. Indeed, these events are characterized by well identified oscillating activities in particular frequency bands, from beta rythm at seizure initiation to very fast oscillations, and usually show increased frequency-dependent synchronies as the seizure progress [3]. Such behavior is illustrated in figure 1, where the harmonic nature of the signal is well observed. On this figure, a phase switching occurs between the two signals, a feature that has its importance as it is an indicator of a reconfiguration within the EZ network. Such delays in synchrony are to be taken in account in the measurement method. The synchrony quantification is carried out using the maximal overlap discrete wavelet transform (MODWT) [5]. j Let wm and wnj be the j th level MODWT coefficients of two stochastic processes with zero-mean, called sm and sn . The MODWT estimator of the cross-correlation at scale j

60

considering a delay value of τ writes: − τ ]) ρτ (t) = arg max q j j τ var(wm [t])var(wn [t − τ ])

(1)

where cov(.) and var(.) refer to the empiric estimations of covariance and variance respectively, [t] denotes the considered time window, and the delay τ is allowed between fixed bounds (−τb ≤ τ ≤ τb ). Let τmax be the vector containing the delays for which the cross-correlation is maximized for each time window. The associated maximum cross-correlation values will be stored in the vector ρτmax . The delay τ is commonly assumed to bear an information on the timed causality of an activity over the second one [5]. However, there is few evidence to validate this statement, one has to be careful in analysing this delay parameter.

Hippocampe Entorhinal Cortex (EC)

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Fig. 1. A jitter in synchronization appearing during the paroxystic ictal event between hippocampus and entorhinal cortex (EC). Two phases in synchrony can be observed: a first on where the hippocampus is leading the EC with τ = 23ms, and a second phase where the EC is leading the hippocampus with τ = 22ms.

B. Connectivity likelihood The Beta distribution provide well adapted bell shapes to fit the histograms of observed correlation values. Such modeling has already been proposed in the field of bioinformatic to model the correlations of gene-expression [7], and is here applied on the MODWC cross-correlation of SEEG recordings (see fig. 2(a)&(b)). The beta distribution is given by: g(y) =

(1 − y)α−1 y β−1 B(α, β)

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Fig. 2. Cross-correlations likelihoods: (a) shows the histogram of crosscorrelation values in absence of synchrony (λ0 ), while (b) shows the histogram of the values related to the synchrony states λi, i>0 , with fitted beta distributions.

III. N ETWORK I NFERENCE Even though epilepsy is a chaotic event of hyper excitability and hyper synchronizations between distant brain structures, it can be reasonably assumed that some stable pathways of communications are followed by the flow of information from a structure to another. A finite number of stable synchrony configurations (including delays) between two brain regions will be considered during an epileptic occurrence, and a hidden Markov chain is used to model the time-evolution of these states.

τ=−22ms

τ=23ms

10

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j cov(wm [t], wnj [t

(2)

Γ(α+β) . Crossing-values of the resulting where B(α, β) = Γ(α)+Γ(β) distributions can be seen as an estimation of the threshold ρthr between synchrony and dyssynchrony cross-correlation values. A thresholding decision strategy is however not adequate, since it would be leading to false alarms and omissions [6], [3]. The electrophysologic data are produced by a biological systems, the uncertainty of the measurements being inherently high and the robustness of network inference methods has thus to be improved. In this work, we introduce a prior constraint of synchrony persistence in time modeled by a hidden Markov chain.

A. Hidden Markov Chain Let X = (Xt )1≤t≤T be the stochastic process modeling the state of synchrony between a couple of recorded channels. X is said to be hidden and takes its values in a finite set Λ = {λ0 , ..., λK }, λ0 being associated with the state of dyssynchrony, and each λk1≤k≤K being associated to a state of synchrony with a delay τk . Y stands for the observation process, with realization y = {(yt )}1≤t≤T , here given by the synchrony measurements ρτ (eq. 1), so that we have yt = {yt0 , yt1 , ..., ytK } = {ρτmax (t), ρτ1 (t), ..., ρτK (t)}. X is supposed to be Markovian and we wish to retrieve these hidden states from the observation Y. The priors on X are given by the initial probability πk (x1 ) = p(x1 = λk ) and by the (K + 1) × (K + 1) square transition matrix aij = p(xt = λj |xt−1 = λi ). X is supposed to be homogeneous, and the i.i.d assumption is made on the process Y conditionnally on X. The posterior probabilities at each time site P (xt |y) can then be computed [8], and the decision on X is finally obtained through the Maximal Posterior Mode criterion: xˆt = arg max P (xt |y) xt

B. Parameter Estimation The parameters of the model are estimated using the ICE procedure [9], which can handle stochastic reestimations. The set of parameters to estimate can be drawn as: θ = {πk , aij , αk , βk , τk }.

[q]

1) Initialization: • The number of synchrony classes K and the related [0] are initialized empiricaly from the signal values τi τmax . If a time delay brings sufficiently high correlation quantification over a sufficient length of time, then it is set a priori as an elligible synchrony state: a delay is supposed to be a priori elligible if its standard deviation is less than 5ms over a time-interval of at least 1s. An observation vector yt is then computed from the equation 1 for each selected synchrony class λi . • Prior parameters on X are initialized empirically as, [0] [0] 1 9 1 e.g., π [0] (i) = K+1 , aii = 10 and aij = 10(K) for i 6= j. • K + 1 beta distribution set of parameters {αk , βk }k∈{0,..,K} are to be initialized: an initial thresh[0] old value ρthr is computed, being the mean plus twice the standard deviation of the maximum cross-correlation observations ρτmax . Using this rough thresholding, the full set of observations y is splitted in two arrays {yl }l∈{0,1} : [0] [0] yti ∈ y0 if yti < ρthr and yti ∈ y1 if yti > ρthr . Empirical [0] [0] mean {µ }l∈{0,1} and variance {σ }l∈{0,1} of these set are computed. We then get preliminary estimated versions of the Beta distribution parameters [10]: [0]

αl

[0]

βl

[0]

[0]

[0]

µl (1 − µl )

=

µl (

=

(1 − µl )(

[0]

σl [0]

− 1)

[0]

(3)

[0]

µl (1 − µl ) [0]

σl

− 1)

(4)

The dyssynchrony beta distribution is then initialized [0] [0] with the parameters {α0 , β0 } = {α0 , β0 }, while the K synchrony beta distributions are initialized with the set of [0] [0] parameters {αk , βk }k>0 = {α1 , β1 }. 2) ICE estimation: For each q in N∗ , q ≤ Q (Q is the number of iteration set by the user), we compute the following ICE estimation for the parameters update: • Computation of the posterior probabilities: [q] [q] ξt (i) = P (xt = λi |Y ) and Ψt (i, j) = P (xt = λi , xt+1 = λj |Y) using the forward-backward probabilities. ˆ = {ˆ Computation of a realization X xt }1≤t≤T using these posterior parameters [9].

[q]

- Stochastic estimation of {α[q] , β [q] }: {µk , σk }k∈{0,K} ˆ are estimated using the stochastic realization X: PT k,[q] xt =k) yt [q] t=1 1(ˆ µk = (6) PT xt =k) t=1 1(ˆ PT k,[q] [q] − µk )2 xt =k) (yt [q] t=1 1(ˆ σk = (7) PT xt =k) t=1 1(ˆ We then get the estimated versions of the Beta distribution parameters at iteration q from equations (3) and (4) - Similarly, it is possible to compute a stochastic readjust[q] ˆ ment of the delays τk from the realization X: P k τmax [n] [q] τk = n∈J (8) Mk ˆ xˆn = k}, and Nk = card(J k ) the where J k = {k|ˆ xn ∈ X, k cardinal of the set J . IV. R ESULTS AND

DISCUSSION

The method has been applied on a SEEG data set of a patient suffering from partial TLE, sampled at Fs=512Hz. The intracranial recordings considered are related to the electrodes implanted in the incriminated regions of the temporal lobe: the hippocampus (B1-2), the entorhinal cortex (TB12), the agmydala (A1-2) and the temporal pole (P1-2). The observed activities lie mostly in the frequency band [8, 16]Hz (wavelet level j = 5), thus being the chosen frequency band for the identification task. A length of 3s has been chosen for the MODWT cross-correlation sliding window analysis, with an overlapping of 0.25s. The segmentation has been performed on a time window of length [0 − 200]s, containing approximatively 75s of ictal activity in the time segment [52 − 128]s. The results are given on figure 3 for the 4 channels considered, zoomed on the most informative part between 40s and 140s, few significative synchronies being found outside these bounds. Corresponding values of delays are given in table I. By selecting a synchronization window simultaneously on each of these 6 segmentation maps, a connectivity graph is derived, reflecting the configuration of the epileptic network for this specific time window. Three time-dependent graphs associated to three well identified phases of the seizure are given on figure 4. TABLE I D ELAYS ( IN MS ) ESTIMATED FOR EACH SYNCHRONY STATES VS COUPLE OF CHANNELS .

• Equation for the maximization step: - Estimation of the prior parameters: [q]

π (i) =

[q] ξ1 (i),

[q] aij

=

[q] t=1 Ψt (i, j) PT −1 [q] t=1 ξt (i)

PT −1

(5)

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λ2

λ3

P1-2 - A1-2

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Fig. 3.

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Results of segmentation for the 6 couples of channel.

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Fig. 4. Connectivity graphs derived from the segmentation results. (a) preictal spikings ([49 − 53]s). (b) rapid discharge ([64 − 72]s). (c) paroxistic event onset ([72 − 76]s).

No significative synchronizations are observed before or near after the ictal event. This is relevant with the common observation that the activity existing in the lobe temporal area are severely attenuated at the proximity of a seizure. Around time instant 50s, high amplitudes pre-ictal spikings are observed in the signal, resulting in synchrony states between hippocampus, entorhinal cortex and temporal lobe, while agmydala seems to have no role in this event (see fig. 4(a)). At the beginning of the following rapid discharge activity ([55 − 60]s) no synchronies are observed, as this activity does not lie in the studied frequency band. While the ictal activity is progressively deriving to its paroxism, with higher amplitudes while decreasing in frequency, synchronies are first mainly observed between hippocampus and EC ([64 − 72]s, see fig. 3 (B1-2 - TB1-2) and fig. 4(b)), then generalizing to the whole set of structures. Fully connected graph are thus obtained at the onset of the paroxistic event ([72−76]s, see fig. 4(c)). Such synchronies tend to disappear fastly as the epileptic event reaches its term. Another interesting feature, however having to be interpreted with caution, are the estimated delays. The hippocampus and the EC seem to have a major role in the development of the event, as these structures show high connectivity with leading delays toward agmydala and temporal pole. These observations are concordant with the electro-clinical evaluation of this data set, confirming that these four regions are indeed involved in the epileptic process, associated with an ictal behaviour typical of medial temporal lobe seizure (early elementary gestual and oro-alimentary automatisms [11]. Visual analysis of the signals however do not allow for identification of the relation between structures, thus of the underlying network. While being consistent

by many aspects with previous studies on partial epilepsy networks [1], the reliabiliy of the method has to be further adressed. A first step in this direction will be to rely on the high reproductibility of the TLE [1], by evaluating the capacity of the method to reproduce a similar succession of network patterns from an epileptic event to another within a given patient. However, since the current knowledge on brain information pathways are still partial, it is very difficult to establish if connections identified from the data are physiologically effective. This validation issue may be further addressed by exploiting in vitro/in vivo studies along with tractographic data, bringing quantitative information on the connections between distant cortical regions. Some studies already confirmed the relation between enthorinal cortex and the hippocampus region [12]. These information could be advantageously introduced as priors in the bayesian framework described in this paper. If the MODWT frequency-dependent measurement is relevant in the (partial) epileptic case [4], [3], it has to be emphasized that the bayesian approach presented in this paper is not dependant of the synchrony quantification. Any (e.g. non-linear) quantification of connectivity can be derived in this framework, offering a general and systematic method for network inference wathever the pathologic/functional relations are to be put in light. R EFERENCES [1] F. Wendling, P. Chauvel, A. Biraben, and F. Bartolomei, “From intracerebral eeg signals to brain connectivity: identification of epileptogenic networks in partial epilepsy,” Frontiers in systems neuroscience, vol. 4, 2010. [2] E. Pereda, R. Quiroga, and J. Bhattacharya, “Nonlinear multivariate analysis of neurophysiological signals,” Progress in neurobiology, vol. 77, no. 1-2, pp. 1–37, 2005. [3] M. Kramer, E. Kolaczyk, and H. Kirsch, “Emergent network topology at seizure onset in humans,” Epilepsy research, vol. 79, no. 2, pp. 173–186, 2008. [4] K. Ansari-Asl, J. Bellanger, F. Bartolomei, F. Wendling, and L. Senhadji, “Time-frequency characterization of interdependencies in nonstationary signals: application to epileptic eeg,” Biomedical Engineering, IEEE Transactions on, vol. 52, no. 7, pp. 1218–1226, 2005. [5] L. Amini, S. Achard, C. Jutten, H. Soltanian-Zadeh, G. Hossein-Zadeh, O. David, L. Vercueil, et al., “Sparse differential connectivity graph of scalp eeg for epileptic patients,” 2009. [6] S. Ponten, F. Bartolomei, and C. Stam, “Small-world networks and epilepsy: graph theoretical analysis of intracerebrally recorded mesial temporal lobe seizures,” Clinical neurophysiology, vol. 118, no. 4, pp. 918–927, 2007. [7] Y. Ji, C. Wu, P. Liu, J. Wang, and K. Coombes, “Applications of betamixture models in bioinformatics,” Bioinformatics, vol. 21, no. 9, pp. 2118–2122, 2005. [8] L. Baum, T. Petrie, G. Soules, and N. Weiss, “A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains,” The Annals of Mathematical Statistics, pp. 164– 171, 1970. [9] W. Pieczynski, “Champs de Markov cach´es et estimation conditionnelle it´erative,” Traitement du Signal, vol. 11, no. 2, pp. 141–153, 1994. [10] M. Evans, N. Hastings, and B. Peacock, Statistical distributions. Wiley-Interscience, 2000, ch. 5: ”Beta distribution”. [11] L. Maillard, J. Vignal, M. Gavaret, M. Guye, A. Biraben, A. McGonigal, P. Chauvel, and F. Bartolomei, “Semiologic and electrophysiologic correlations in temporal lobe seizure subtypes,” Epilepsia, vol. 45, no. 12, pp. 1590–1599, 2004. [12] M. Witter and F. Wouterlood, “The parahippocampal region, organization and role in cognitive function,” Recherche, vol. 67, p. 02, 2002.