Study of Phase Relationships in ECoG Signals Using Hilbert-Huang Transforms

Study of Phase Relationships in ECoG Signals Using Hilbert-Huang Transforms Gahangir Hossain, Mark H. Myers, and Robert Kozma Center for Large-Scale Integrated Optimization and Networks (CLION) The University of Memphis, Memphis, TN 38152, USA {ghossain,rkozma,mhmyers}@memphis.edu Abstract. This study investigates phase relationships between electrocorticogram (ECoG) signals through Hilbert-Huang Transform (HHT), combined with Empirical Mode Decomposition (EMD). We perform spatial and temporal filtering of the raw signals, followed by tuning the EMD parameters. It can be seen that carefully tuning of EMD filter, it is possible to capture distinct features of non-stationary data. This makes EMD, combined with HHT a valuable tool of complex brain signal analysis and modeling. Keywords: Electrocorticogram (ECoG), Hilbert Huang Transform (HHT), Empirical Mode Decomposition (EMD); Phase cone. 1 Introduction Hilbert-Huang transform (HHT) is a recent method which generates amplitude and frequency vs. time spectra using a powerful data analysis tool called empirical mode decomposition (EMD) [1, 2]. HHT is suitable to analyze non-stationary and nonlinear data. Global basis states must be replaced with adaptive, locally determined ones, a process the first stage of the HHT does perform. The resulting basis states are, in general, not strictly orthogonal. The goal of this study is to analysis different phase relationships in phase cone discovery from different types of EMD filtered datasets. The main idea behind EMD approach is to first compute the local median of a signal via a sifting procedure and then subtract the local median of a signal before applying the amplitude spectrum method to define instantaneous frequency. Therefore, in performance comparison of EMD filtering the very recent discovery of Hou and Shi, EMD performance depends on the sensitivity on the number of sifting and the stopping criteria [3, 4], is adopted. The variance of EMD filtering is performed by carefully tuning some dependent parameters in intrinsic mode function (IMF) that decomposes the signal into modes that are intrinsic to the function using an iterative or sifting process considering only local extrema. In ECoG analysis, the spatially ordered phase relationship between cortical signals is named as phase cone [5]. Instantaneous identification of phase cones therefore serves as markers by which to locate emergent AM patterns at varying latencies over sequential trials. To identify better phase cones, ECoG, data must be preprocessed

without changing its inherent properties. The present work aims at studying phase relationships in ECoG data to improve the identification of salient properties of spatio-temporal brain dynamics. This work starts with a brief introduction to the applied methodology. This is followed by describing the analyzed data obtained from intracranial experiments with chronically implanted rabbits. We study the performance of HHT processing algorithms and optimize parameters of EMD algorithm. Finally, we summarize the obtained results and conclude direction for future studies. 2 Background Study ECoG represents complex irregular brain signals by recording of tiny electrical potentials that underlie neural activities related to perception and action. This section provides some background materials, mostly fast Fourier transform and Hilbert- Huang transformation that are used to uncover phase cones from ECoG signals. The implemented signal processing approach is explained on the block diagram as shown in Figure 1. After a very brief review of the HHT, and Phase cone identification, the comparative behavior of these two transforms on various filtered data set is explored. Along this direction the IMF parameter number-of-sifting in iteration is varied. Fig. 1. Block Diagram of Experimental Steps 2.1 Hilbert Transformations For an arbitrary signal v, the analytic signal V(t) is a complex function of time defined as: 1 v(t') + v'(t) = PV π (t t') dt' (1)

where PV corresponds to the Cauchy Principal Value. At each digitizing step, the time series yielded a point in the complex plane [vj (t)]. Each signal denoted by v(t) was transformed to a vector, V(t) having a real part, v(t), and an imaginary part, v'(t). As seen from Eq. (1), the Hilbert transform v (t) of v(t) can be considered as the convolution of the function v(t) with 1/πt. Each ECoG signal denoted v(t) was transformed to a vector, V(t), having a real part, v(t), and an imaginary part, v'(t) (Freeman, Rogers, 2002): V(t) = v(t) + i v'(t) = AA(t) exp [iap(t)] (2) where the length of the vector gave the analytic amplitude, AA(t) = [ v 2 (t) + v' 2 (t) ].5 (3) and arc tangent of the vector gave the analytic phase, AP(t) = atan [ v'(t) / v(t) ]. (4) The instantaneous amplitude AA(t) and the instantaneous phase AP(t) of the signal v(t) are thus uniquely defined by Eqs. (3,4). The real data corresponds to the raw incoming data, while the Hilbert transform (HT) provides the imaginary frequency that is changing in time. The imaginary part is a version of the original real sequence with a 90 phase shift. Sin functions are therefore transformed to cosines and vice versa [5]. The Hilbert transformed series has the same amplitude and frequency content as the original real data and includes phase information that depends on the phase of the original data. The Hilbert transform is useful in calculating instantaneous attributes of a time series, especially the amplitude and frequency. The instantaneous amplitude is the amplitude of the complex Hilbert transform; the instantaneous frequency is the time rate of change of the instantaneous phase angle [5]. For a pure sinusoid, the instantaneous amplitude and frequency are constant. 2.2 The Hilbert-Huang Transform The Hilbert-Huang transform is the combination of empirical mode decomposition (EMD) and Hilbert transform (HT). EMD process deconstructs the signal into a set of intrinsic mode functions (IMF) and HT extracts frequency vs. time information from each of the IMF s. The EMD is a method of signal decomposition introduced for analysis of nonlinear and non-stationary signals. It is to identify proper time scales that reveal physical characteristics of the signals, and then decomposed the signal into modes that are intrinsic to the function, referred as Intrinsic Mode Functions (IMFs). IMFs interpret signals as the zero mean oscillations at each scale and the local mean of the signal respectively. IMFs are signals satisfying two conditions: (a) in the whole dataset, the number of extrema and the number of zero-crossings must either be equal or differ at

most by one, and (b) at any point, the mean value of the envelope defined by local maxima and the envelope defined by the local minima is zero. Condition one is similar to the traditional narrow band requirements for a stationary Gaussian process. Whereas, the second condition is necessary in order to avoid unwanted fluctuations induced by asymmetric waveforms in the instantaneous frequencies will not have. An IMF is not limited as a sinusoid in the classical sense (such as in Fourier Transforms), it can be an amplitude and frequency modulated signal and, can even be a non-stationary signal. This method enables us to eliminate the drawback of a traditional time-domain to frequency-domain transformation (like Fourier transform) where frequency contents are observed by sacrificing time resolution. Instead, IMFs provide amplitude and frequency information of a signal at any given time. Practically, EMD is implemented as an iterative or sifting process considering only local extrema. The EMD algorithm for amplitude and frequency extraction from a given discrete IMF is shown in Table 1. Table 1. EMD Algorithm (sifting algorithm) Given a discretely sampled signal y(t), Step-1: Find the locations of all the extrema of y(t) first IMF signal. Step-2: Interpolate between all the minima (respectively maxima) to obtain the lower signal envelope, ymin(t) (respectively ymax(t)). Step-3: Compute the local mean m(t) = [ymin(t) + ymax(t)]/2. Step-4: Subtract the mean from the signal to obtain the oscillatory mode d(t)= y(t) m(t) // removing the trend Step-5: If d(t) meets stopping criteria, then define c i (t)=d(t) and i = i +1 // increment i r(t)= y(t) d(t) // extract the residual If d(t) does not meets stopping criteria farther sifting is required. set y(t)=d(t) and repeat from step 1. Step-6: Repeat steps 1 through 5 until the residual no longer contains any useful frequency information. Amplitudes and frequencies are extracted from these IMF s in the second stage of the HHT process. The instantaneous amplitude and angular frequency associated with each IMF depend on the amplitude and phase of a complex number that the IMF and its Hilbert transform (HT) define. The real part of the complex number is the IMF; the imaginary part of the number is the IMF s HT. The instantaneous amplitude is the amplitude of this complex number. The instantaneous angular frequency associated with that IMF is the derivative of the unwrapped phase. The entire process is repeated for each IMF to extract the complete frequency versus time information from the original ECoG data set. The computation of the HT is essentially a convolution of an IMF, x(t), with 1=t and effective to emphasize the local properties of x(t). This locality preserves the time structure of the signal s amplitude and frequency. Generally, ECoG signals are represented equal to the sum of its parts. We have N

IMFs and a final residual rn (t), (5) The second stage of the HHT process extracts the amplitude and frequency information from each IMF (HT algorithm in Table 2). Table 2. Algorithm: HT Given a discretely sampled signal y(t), Step-1: Compute the IMF s discrete Fourier transform (DFT) using the series expression (1) for the transform. Step-2: Compute the HT. Use the real and imaginary parts of step 1 s DFT as coefficients (M = N/2): Step-3: Form the complex number z j = x j +iy j, extract the phase j = tan -1 (y j /x j ). Step-4: Unwrap the phase so that it becomes a monotonically increasing function. Step-5: Determine the frequency. Take the derivative of the phase Step-6: Determine the amplitude. 2.3 Phase Cone Detection Phase cones describe the spatially ordered phase relationship between cortical signals. Phase cones reveals the property: a state transition is not everywhere instantaneous but begins at a site of nucleation and spreads concentrically, like the formation of a snowflake around a grain of dust. Fig. 2. A 3D plot of a special distribution of analytic phase across 8 x 8 ECoG electrode array at the time frame t = 0.256 s. The phase lags confirm as like phase cone.

The apex that marks the site in the cortex is a random variable both in sign (lead or lag) and location. The cones can appear with positive and negative phase lags, corresponding to explosive and implosive transitions in the cortical spatiotemporal dynamics and there can be several phase cones simultaneously present in a measurement window [6,7]. Phase Cones depicting an "implosion" due to a phase lag of the ECoG time series. The power spectral analysis usually reveals a dominant peak at a central frequency in short segments and near 1/f power spectra in time segment > 1sec. This dominant component of each burst is constituted by the power at the point where the central frequency rises and it drops towards the end in temporal amplitude modulation (AM) on all channels [8,9]. These AM pattern have been found to be accompanied by pattern of phase modulation and forms like a cone, that is named as phase cone. More details on identification of propagating phase gradients in ECoG signals using Dynamic Logic (DL) approach is experimented [10]. Figure 2 shows 3D view of sample phase cones at timestamp t128. The figure explains special distribution of analytical phase across the 8 x 8 ECoG electrode array at that time frame. 3 Data and Methods 3.1 Data To demonstrate the comparative study 64-channel ECoG recordings of rabbit data is used. The data is captured in Walter Freeman's UC Berkeley lab [11,12]. ECoG was recorded monopolarly with respect to that cranial reference electrode nearest the array and amplified by fixed-gain (10K) WPI ISO 4/8 differential amplifiers. Each channel was filtered with single pole, first order analog filters set at 100 Hz and 0.1 Hz. Records of sixty-four 12-bit samples multiplexed at 10 µs were recorded at a 2 ms digitizing interval (500 Hz) for 6 seconds and stored as signed-16-bit integers. More specifically, a sample data is a 64 x 3000 matrix, which means 64 ECoG channels measured for 6s, at 2 ms sampling time, so in total 3000 points. The incremental time delay caused by multiplexing of the ECoG was corrected off-line. Bad channels associated with movement artifact or EMG were identified by visual editing and replaced off-line by averaging the signals of two adjacent channels. Using the EMD technique described in previous section and changing the value of number-of-sifting in IMF iteration, fix different types of filtered data sets are created. 3.2 Methods The comparative experimental procedure works as the block diagram shown in Figure 1. The filtering based system works for three steps. In first step, input ECoG signals are decomposed using EMD with couple of iterations for all signals band to be stabilized in terms of IMFs. After successful IMF iterations EMD phase ended. In step two, HT is applied on the filtered signal. That works as HHT on the applied signal. Tuning the EMD parameter finally tuned the HHT for different dataset. In farther step the reflection of amplitude and phase change is analyzed for meaningful pattern discovery. As ECoG time series are large in size, a significant sample

selection is a crucial part of the comparative study. To select a good and representative sample window average phase around 60 Hz for all 64 channels (8 X 8) are analyzed for different filtered dataset. An average phase spectrum of all 64 channels for different filtered dataset is plotted. By visual inspection on the figure matrix any filtered data is compared with the top row. 4 Results In this section three important properties of Hilbert transform namely analytic amplitude, analytic phase and unwrapped phase are plotted from all EMD filtered ECoG signal. Fig. 3. Hilbert analytic amplitude and phase comparison between original data set and EMD filtered datasets.

The original rabbit data is marked as not filtered (NF) whereas EMD filtered data sets are varied based on number of IMF sifting. These versions are marked as IMFs10, IMFs20, IMFs50, IMFs100, IMFs500, IMFs600, IMFs700, IMFs800, IMFs900 and IMFs1000.Hilbert analytic phase and amplitude comparison are shown in Figure 3. Fig.4. Hilbert unwrap phase comparison between original data set and EMD filtered datasets. Smoothing of the time series occurs in the analytic amplitude due to an increased EMD order. A change in the phase/frequency range from higher order EMD filtering seems to increase the number of phases over time. In Figure 4, the unwrapped phase changes dramatically between higher order EMD filtering. The challenge of filter selection may cause bifurcations of the unwrapped phase to occur due to noise and opposed to too much filtering which may cause important attributes of the signal to become smoothed over. An EMD order of less than 100 seems to enable the ECoG signal to retain its lower frequency attributes while discarding noisy artifacts. 5. Conclusion This study revealed that combination of EMD and HT is better in lower-order filtering in ECoG analysis mostly tuning the filtering parameters related to sifting in IMF iteration less than hundred times. Hence, EMD may be a useful and effective tool for filtering ECoG data before phase cone detection. Analysis of other EMD parameters (e.g. fidelity or noise related issues) for filter tuning can be experimented in the next phase. Standard EMD is limited to the analysis of single data channels, whereas modern applications require its multichannel extensions. The complex ECoG signal needs to

be processed using multivariate algorithms to get complex valued IMFs. In that case,, a bi-variate or tri-variate EMD algorithm may be useful. On the other hand, results from the EMD with HT (HHT) method showed relatively deprived phase behavior that need more inspection in further study. Additionally, EMD parameters that capture temporal information changes need a continuous, automatic assessment in phase gradient identification process. Therefore, a dynamic process of optimal approximation and robust identification of phase cones from filtered or noisy data can be experimented. Future work includes better phase cone detection mechanism, automation of the detection process, and performance improvement and analysis of the impact of filtering on phase relationship. Acknowledgments: This work has been supported in part by the University of Memphis Foundation, through a grant of the FedEx Institute of Technology. References 1. B en dat J. S.: The Hilbert Transform and Applications to Correlation Measurements, Bruel & Kjiaer, Denmark, 1985 2. Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, H. H., Zheng, Q., Yen, N. C., Tung, C. C. and Liu, H. H., The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis, Proc. Roy. Soc. Lond. A, 1998, pp. 903-1005. 3. Norden E. Huang et. al., "The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis," Proc. R. Soc. Lond. A., vol. 454, pp. 903-995, 1998. 4. Battista, B., Knapp, C., McGee, T. and Goebel. V.,Application of the Empirical Mode Decomposition and Hilbert-Huang Transform to Seismic Reflection Data. Geophysics, Vol.72, No.2, pages H29-H37., 2007. N. 5. Freeman WJ, Rogers LJ.,"Fine temporal resolution of analytic phase reveals episodic synchronization by state transitions in gamma EEGs",J Neurophysiol. 2002 Feb;87(2):937-45. 6. Barrie, J.M.,Freeman,W.J.& Lenhart,M.D. (1996), Spatiotemporal Analysis of Prepyriform, Visual, Auditory, and Somesthetic Surface EEGs in Trained Rabbits, J. Neurophysiology, Vol. 76, No. 1, pp.520-539. 7. Freeman W. J.,and Barrie, J. M., Analysis of Spatial Patterns of Phase in Neocortical Gamma EEGs in Rabbit, Journal of Neurophysiology 84: 1266-1278, 2000. 8. Freeman, W. J., Origin, structure, and role of background EEG activity, Part II, Analytic Phase, 2004, Clinical Neurophysiology, 9-47. 9. Freeman, W. J., Origin, structure, and role of background EEG activity, Part I, Analytic Amplitude, Clinical Neurophysiology (2005) 116 (5): 1118-1129. 10. Kozma, R., W.J. Freeman (2001) Analysis of Visual Theta Rhythm Experimental and Theoretical Evidence of Visual Sniffing, IEEE/INNS Int. Joint Conf. Neural Networks, Washington D.C., July 14-19, 2001, pp. 1118-1123. 11. Freeman, W.J., Mass Action in the Nervous System. Academic Press, New York, 1975. 12. Kozma, R.; Perlovsky, L.; Ankishetty, J.S., "Detection of propagating phase gradients in EEG signals using Model Field Theory of non-gaussian mixtures," Neural Networks, 2008. IJCNN 2008. (IEEE World Congress on Computational Intelligence). IEEE International Joint Conference on, vol., no., pp.3524-3529.