Sensitivity Analysis of Hilbert Transform with Band- Pass FIR Filters for Robust Brain Computer Interface

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1 Sensitivity Analysis of Hilbert Transform with Band- Pass FIR Filters for Robust Brain Computer Interface Jeffery Jonathan (Joshua) Davis (1,2) 1 Center for Large-Scale Integrated Optimization Networks University of Memphis, Memphis, TN 38152, USA 2 Embassy of Peace, Whitianga New Zealand Joshua_888@yahoo.com Robert Kozma Center for Large-Scale Integrated Optimization Networks Dept. Mathematical Sciences, University of Memphis Memphis TN USA rkozma@memphis.edu Abstract Transient cortical oscillations in the form of rapid synchronization-desynchronization transitions are key candidates of neural correlates of higher cognitive activity monitored by scalp EEG and intracranial ECoG arrays. The transition period is in the order of ms, and standard signal processing methodologies such as Fourier analysis are inadequate for proper characterization of the phenomenon. Hilbert transform-based (HT) analysis has shown great promise in detecting rapid changes in the synchronization properties of the cortex measured by highdensity EEG arrays. Therefore, HT is a primary candidate of operational principles of brain computer interfaces (BCI). Hilbert transform over narrow frequency bands has been applied successfully to develop robust BCI methods, but optimal filtering is a primary concern. Here we systematically evaluate the performance of FIR filters over various narrow frequency bands before applying Hilbert transforms. The conclusions are illustrated using rabbit ECoG data. The results are applicable for the analysis of scalp EEG data for advanced BCI devices. Keywords - Electrocorticogram, Hilbert Transform; Synchronization; Instantaneous Frequency; Analytic Amplitude; Analytic Phase; Cognition. I. INTRODUCTION Brains are the most complex substances in the known Universe. Neurophysiological processes underlying higher cognition and consciousness are intensively studied worldwide with many spectacular successes using fmri, ECoG, MEG, and EEG [1-5]. There is a need for more robust analysis of the filter properties on the outcome, in particular in BCI applications. The present study aims at better understanding of cortical processes and increasing the credibility and future dynamical development of this promising new research field. This work is a continuation of studies based on rabbit electrocorticogram (ECoG) experiments due to W. J. Freeman [1]. The present study helps to interpret the operation of brains without getting trapped into representational cognitivism as described by Dreyfus [6-7]. Instead, we work towards an embodied cognition model as a promising approach to deal with the complexities of cognition and consciousness [8-10]. This material is based upon work supported by the Science Foundation Program "Collaborative Research in Computational Neuroscience (CRCNS)" under Grant Number DMS Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. In the present approach brains are viewed as nonlinear dynamical systems, which lead to successful characterization of learning in neural systems and to the interpretation of cognitive functions [11-17]. Dynamic brain models can provide the basis for adaptation and give an account of how the brain of an intentional animal can obtain meaningful information on the world [6, 18]. We are exploring paradigms that can shed light to the understanding of creativity, health, and peace in the context of behavioral responses and universal values for the progress of the individual and the society. In the dynamic brain approach, learning establishes an attractor landscape for each cortical area, in which the basins of attractions are shaped by experience. Each attractor in the landscape corresponds to a class of stimulus that the animals have learned to discriminate and each attractor is accessed by the arrival of a learned stimulus of that class. The mathematical and computational formulation is expressed in Freeman K (Katchalsky) sets [19-20]. In previous work our findings are based on experimental ECoG studies using intracranial electrode arrays implanted in rabbits [22-25]. Recently we presented results based on a comprehensive analysis of ECoG signals starting form 2Hz until 40Hz, covering delta, theta, alpha, beta, and gamma bands. This allowed us to move in the direction to overcome the limitations of previous works covering only specific frequency bands, such as gamma or theta band. In this work, we analyze the impact of changing filter order on the Analytic Amplitude (AA), the Analytic Phase (AP), and the Instantaneous Frequency (IF). We explore how much information is lost after applying FIR filters, and how filters of low order have an influence on phase displacements. This work extends on previous studies band-pass filter designs generating beating patterns and associated null spikes in the analytic signal domain [26]. The goal is to build a reliable foundation concerning the mathematical procedures we are applying as part of a methodology for further BCI research. We start with describing the rabbit ECoG experiments, followed by introducing the Hilbert transform-based signal processing approach. Next we describe the obtained results and discuss their relevance to the reliability of the mathematical procedures concerning FIR band pass filters. We

2 study the accuracy of the experimental parameters (both AA and IF) with the expected theoretical values derived from the center frequencies when applying the FIR band pass filters on both Rabbit and simulated data, to obtain a signal for each narrow band in order to verify that the filter is working properly in capturing the signal component of each particular band of interest. Our results point to the role of every band in the cognitive process. We point to some limitations in our analysis and outlines avenues to overcome them in the future research. II. DESCRIPTION OF ECOG DATA Local areas of sensory cortex of 1 cm in diameter generate broad-spectrum aperiodic waves of dendritic activity that have the same waveform. This spatial coherence is shown by the similarity of the waveforms of ECoGs that are recorded simultaneously from 8 8 epidural electrode arrays giving windows 6 6 mm 2 in width onto the olfactory, visual, auditory and somatomotory cortices [21]; see Fig. 1. In this work, only the experiments with visual cortex are presented. The spatial amplitude modulation (AM) of this coherent waveform in brief time segments gives a spatial pattern that is determined by the synaptic connectivity within each cortex. The connections and patterns change with the training of animals to identify significant stimuli. Evidence for the dynamic systems theory of chaotic self-organization in cortices comes from the results of classification of the spatial AM patterns. The ECoG segments coming from a sensory area give clusters of points, each of which corresponds to a response to a sample of the class of stimulus that the animal has learned to identify. The animals under classical conditioning learn discrimination, in which one stimulus is reinforced (CS+) and the other is not (CS ). III. ANALYTIC AMPLITUDE, PHASE ANALYSIS AND INSTANTANEOUS FREQUENCY OF EXPERIMENTAL DATA Here we expand on our findings regarding various frequency bands in multiple runs. As presented in other papers [28, 32], first we analyzed the 64-channel ECoG signals, performed preprocessing and filtering over them, and evaluated various statistical measures as specified below. Next, we evaluated the same statistical measures for the analytic signals calculated after Hilbert-transforming the preprocessed and filtered ECoG signals. The applied Hilbert transform methodology and the analytic signal construction followed the approach described in [26-27]. Informationtheoretic measures have been evaluated previously for the theta band using the same rabbit ECoG data [28]. In the present work we extend earlier studies toward a comprehensive study of a range of frequency bands. In all evaluations, we kept the frequency bandwidth constant at 2Hz, while shifting the center frequency of the filter stepwise at 2Hz increments. The considered frequency windows are summarized in Table I. In this case this was done for the 39 trials. TABLE I. FREQUENCY WINDOWS ANALYZED Frequency Band Windows (Hz) f Low f High Delta 2-4 Theta 4-6, 6-8 Alpha 8-10, Beta 12-14, 14-16, 16-18, 18-20, 20-22, 22-24, Low Gamma 26-28, 28-30, 30-32, 32-34, 34-36, 36-38, For this work we will summarize and present our results based on Narrow Band analysis within the broadband of Theta (4-8 Hz), Alpha (8-12 Hz), Beta (12-26 Hz), Gamma (26-40 Hz). The following quantities have been evaluated both for the band-passed ECoG and their analytic counterparts for the 39 trials: Signal amplitude (SA or AA): This could be either the amplitude (SA) of the ECoG signal or the analytic amplitude (AA) of the analytic signal. Standard Deviation in Space (STDx): The standard deviation calculated across the 64 channels for SA or AA, respectively. Instantaneous Frequency (IFx): Evaluated using the temporal derivative of the analytic phase (AP) across all 64 channels. This analysis requires a massive amount of data processing and visual exploration of each trial for each band. Here we present the results for the gamma band for 2 trials (first and last) as shown in Figure 1 (a, b) and Figure 2 (c, d) describing SA and AA characteristics respectively, to give you an initial impression of the work at hand. The instantaneous frequency has been calculated as the spatial average over all 64 individual channels. This means we have an average of the IF time series instead of 64 time series for each band. Note that in all figures the first half of the time series corresponds to the resting period prior the stimulus, which is administered in the form of a light flash at moment t=3s. The second part of the experiments after 3s represents the poststimulus response. The first and last 0.5s are omitted as those periods are used for the proper windowing. We have the following major observations based on Figs. 1-2: 1. SA and AA: There are oscillations during the whole time period, even during the resting period, but the oscillations show significant increase between 3-4s, i.e., immediately after the stimulus. In the case of SA, the increased amplitude is more visible at higher frequencies. For the AA, on the other hand, the increase is very prominent over all frequencies. 2. In the case of AA: We can observe a fine structure of the amplitude measure. Namely, there is an increase of AA between followed by a drop between seconds. In this last period we can observe a small increase between seconds, followed by a drop between seconds.

3 (a) accompanied by a drop in analytic frequency in the narrow band signals within Low Gamma, Beta, Alpha and Theta, something that can be observed in Figures 1 and 2 for Low Gamma. The onset and termination of these events is clearly observable in the trials, which vary within the window (3-4 s). At the same time, the two peaks in some cases are very close to each other, they may be almost merged; sometimes there are three peaks and occasionally four. However, based on the average, as well as other indicators like standard deviation, analytic phase and analytic frequency, we conjecture that the immediate post stimuli window (3-4 s) is reflecting a process for knowledge creation, consolidation and intentional behavior and that the process taking place in this window can be interpreted as a nonlinear, far from stability one, with four stages in it. These stages we call: Awe, Chaotic Exploration, Aha moment and Chaotic Integration and finally after the 4 th second a return to usual linear behavior. To make sure that our cognitive hypothesis rest on a sound mathematical foundation we feel compelled to test the reliability of the FIR band pass filtering procedure band by band, as well as, the limitation imposed by the Filter itself for the beginning and end of the filtered data. For these purpose we will present three initial tests: A test on a simple sinusoidal wave A test on a more complex sinusoidal wave A test on a theoretical complex sum of sinusoidal waves representing each frequency band in our analysis. (b) Figure 1. ECoG signals filtered over the Low Gamma (26Hz-40Hz). Temporal frequency band at constant 2Hz bandwidth segments; each plot displays the signal amplitude (SA), standard deviation of the signal, and the average frequency over the given narrow band in linear (omit reading log) coordinates; (a) trial #1, (b) trial # 39. Multiple curves on a given figure indicate the signals obtained over 7 bands of width 2Hz each within the Low Gamma. 3. STDx for SA and AA: Here the most important feature is that the overall STDx increases significantly between 3-4 seconds. This effect is clear in both SA and AA analysis. 4. IFx: We consider the absolute value of IF in linear and log10 coordinates. We observe a drop between seconds followed by an increase between seconds (with more dispersion). 5. AA and IFx relationship: As a general trend, we observe that wherever there is a drop in IFx there is an increase in AA and vice versa; this is in line with previous research [26-27]. Now we can confirm this as a general behavior over a broad frequency band. These graphs, together with the other 37 trials showed the presence of synchronization-desynchronization periods. We also observe that in the window just after stimuli (between 3-4 s) two periods of increased amplitude (peaks) appear, IV. HILBERT ANALYSIS OVER SEVERAL BANDS (Analytic Amplitude and Phase Relationship of ECoG Signals) We calculate the analytic signals V(t) after Hilberttransforming the 64-channel ECoG array data. The applied Hilbert transform methodology follows the approach described in [27]. The ECoG of each channel v j (t) (j=1,,64) is transformed to a time series of complex numbers, V j (t), with a real part, v j (t), and an imaginary part, u j (t), V j (t) = v j (t) + i u j (t), j = 1, 64, (1) Here the real part is the ECoG signal, while the imaginary part is the Hilbert transform of v j (t). We use MATLAB Hilbert function to produce u j (t). Sequences of steps give a trajectory of the complex vector V(t) composed of 64 complex values evolving in time. The vector length at each digitizing step, t, is the analytic amplitude:!!!!! =!!(!) +!!(!), (2) while the analytic phase is defined as the arctangent of the angle of the vector, also composed of 64 values evolving in time:

4 !! = 12 cos 2! 5! + 6 cos 2! 12!!! = 12 cos 2! 5! + 6 cos 2! 12! + 3 cos 2! 23! (c) Figure 3. Illustration shows the effect of the Band-Pass filter when applied to a simulated signal!! = 12 cos (2! 5!) with amplitude, A=12 and frequency, f=5, same as the center frequency of the FIR filter with parameters: FstopL = 3 Hz, FpassL = 4 Hz, FpassH = 6 Hz, and FstopH = 7 Hz and with the order of the filter N=281. (d) Figure 2. ECoG signals filtered over the Low Gamma (26Hz-40Hz). Temporal frequency band at constant 2Hz bandwidth segments; each plot displays the analytic amplitude (AA), standard deviation of the signal, and the average frequency over the given narrow band in linear (omit reading log) coordinates; (c) trial #1, (d) trial # 39. Multiple curves on a given figure indicate the signals obtained over 7 bands of width 2Hz each within the Low Gamma.!!! = tan!!!! (!)!! (!), (3) The Instantaneous Frequency vector, IF(t) general formula based on the analytic phase is as follows:!"!! = V.!!!!!!! =!!!!!!!!!!!!!, (4) INITIAL TESTS ON SINUSOIDAL WAVES USING THE FIR BAND PASS FILTER In this section we analyzed a set of generated sinusoidal signals, to test the FIR filters with data we know theoretically. Following we describe the different generated signals according to the general formula!! =!!!!!! cos (2!!!!):!! = 12 cos (2! 5!) Here, we show detailed results obtained for a band between 4-6 Hz. This helps to illustrate the most important features when applying FIR band pass filters before Hilbert analysis. In the next section we introduce results over a broad range of Narrow Band generated frequencies based both on the sinusoidal theoretical formula and the parameters of amplitude and frequency obtained with real Rabbit data. Band-pass filter is applied to the simulated signals on the theta band with parameters: FstopL = 3 Hz, FpassL = 4 Hz, FpassH = 6 Hz, and FstopH = 7 Hz. These filters are applied together with two different values for the order of the filter, N=281 and N=600, to show the effects of this parameter on the filtered data. Figure 3 shows a filtered simulated signal of the form,!! = 12 cos (2! 5!) where the order of the filter is N=281 and where we can observe two important features of this process: (1) the filter data FIR(xt) preserves the amplitude but is shifted or out of phase (almost antiphase), (2) the Analytic Amplitude is very distorted and unstable for the first and last half of a second respectively. We observe a drastic departure from A=12 to almost A=0.5 at the beginning of the first second. It is only after around 0.5s that it return to values around A=12 with slight oscillations. The last 0.5s displays a similar situation. These are clearly some of the expected effects of the FIR filter. Following we show how when we increase substantially the order of the FIR filter to N=600, we solve the issue of having the FIR(xt) out of phase with the original data (xt). However, this comes with a cost in a slight variation in amplitude (different than the oscillations observed in analytic amplitude) and longer periods of initial and final distortions and instabilities.

5 (a) (b) Figure 4. Illustration shows the effect of the Band-Pass filter when applied to the simulated signal,!! = 12 cos (2! 5!) + 6 cos (2! 12!) with compound amplitudes of, A1=12 and A2=6 and frequencies, f1=5 and f2=12, with FIR filter parameters: FstopL = 3 Hz, FpassL = 4 Hz, FpassH = 6 Hz, and FstopH = 7 Hz. (a) shows the results for N=281, (b) displays the results for N=600. Figure 4 shows a similar situation, however with a more complex signal containing the original component plus a new cosine wave,!! = 12!"# 2! 5! + 6 cos 2! 12! with two different orders for the filter N=281 and N=600. Figure 5. Illustration shows the effect of the Band-Pass filter when applied to,!! = 12 cos 2! 5! + 6 cos 2! 12! + 3 cos 2! 23!, the simulated signal with compound amplitudes of, A1=12, A2=6 and A3=3 and frequencies, f1=5, f2=12 and f3=23, with FIR filter parameters: FstopL = 3 Hz, FpassL = 4 Hz, FpassH = 6 Hz, and FstopH = 7 Hz and with the order of the filter N=600. We can also observe very marked oscillations in analytic amplitude values, between and around 6 and 18 due to the compound nature of the signal together with the effects of the selected parameters of the filter. These values are in the range of the sum of both amplitudes A1=12 and A2=6 of the different cosine components of the signal and the minimum which corresponds to A2=6. This behavior is preserved with the different orders of the FIR filter applied to the signal, both N=281 and N=600. However as we have stated before, it is clear that the initial conditions have a longer period of instability for N=600. The third signal we have analyzed for these section is an even more complex signal,!! = 12 cos 2! 5! + 6 cos 2! 12! + 3 cos 2! 23! with three cosine components. It is important to note that we just added again a new component with a lower amplitude and a higher frequency and consistently we applied the same filter with N=600 to show, in figure 5, that regardless of: (1) the added complexity, (2) some initial instabilities and (3) a slight variation in amplitude values of the original signal (xt), the FIR filter captures well the behavior of the frequency band of interest. In the next set of graphs we will show the analytic phase (wrapped and unwrapped), the instantaneous frequency (IF) and the histogram for the IF for each of the signals described above.

6 Figure 6. These Illustration shows four types of graphs: Wrapped Analytic Phase (top left), Unwrapped Analytic Phase (top right), Analytic or Instantaneous Frequency, IF (bottom left) and the Histogram of IF (bottom right). All of the above were calculated after Band-Pass filter has been applied to the simulated signal!! = 12 cos (2! 5!) with amplitude, A=12 and frequency, f=5, same as the center frequency of the FIR filter with parameters: FstopL = 3 Hz, FpassL = 4 Hz, FpassH = 6 Hz, and FstopH = 7 Hz and with the order of the filter N=281. This is to show with more clarity the reliability of the filter as well as its limitations. It is important to mention at this stage that a significant loss of precision in amplitude or a significant out of phase result when filtering data could alter dramatically our observations and conclusions when analyzing real data. Because for example if the onset of a stimuli is supposed to show at time t, the out of phase data would show it at time t+ε. Also if the analytic amplitude is our order parameter, then we should be very careful to preserve its values as close as possible after filtering. The two important things to mention in figure 6 are the instabilities observed at the beginning and at the end periods for the computed instantaneous frequency (IF) also reflected in the histogram, with an observed average frequency very close to five (5) as expected. Figure 7 (a) shows a similar behavior as Figure 6, however, the average IF is very close to five (5) as expected. We observe some small oscillatory departures from the expected frequency. These we conjecture is due to some of the limitations and sensibilities of the FIR filter, something we need to consider carefully when analyzing real data. In figure 7 (b), it is clear that the instabilities are much larger in magnitude for the initial period. The average computed IF is still near the expected value of five (5) and it is showing similar yet more attenuated oscillatory behavior than the one in figure 7 (a). This is showing us the trade offs when increasing N from N=281 to N=600. Figure 8 shows a similar behavior as already observed in the rest of the above analysis. Basically, the FIR band pass filter captures relatively well the behavior of the instantaneous frequency with some trade offs in terms of: initial and end instabilities and amplitude values when changing the order of the filter. (a) (b) Figure 7. These Illustration shows four types of graphs: Wrapped Analytic Phase (top left), Unwrapped Analytic Phase (top right), Analytic or Instantaneous Frequency, IF (bottom left) and the Histogram of IF (bottom right). All of the above were calculated after Band-Pass filter has been applied to the simulated signal,!! = 12 cos (2! 5!) + 6 cos (2! 12!), with compound amplitudes of, A1=12 and A2=6 and frequencies, f1=5 and f2=12, with FIR filter parameters: FstopL = 3 Hz, FpassL = 4 Hz, FpassH = 6 Hz, and FstopH = 7 Hz. (a) shows the results for N=281, (b) displays the results for N=600. However, the refinement of the filtering process can be improved as shown in the following graphs in Figure 9, where we have plotted both IF and Analytic Amplitude (AA) for the most complex signal,!! = 12 cos 2! 5! + 6 cos 2! 12! + 3 cos 2! 23!, under three different scenarios of filter parameters. The general results obtained are shown in the following table. TABLE II. PASS BAND SCENARIOS Scenario Pass Band Wide Middle Narrow FstopL FpassL FpassH FstopH Mean IF Mean AA

7 achieve a very close precision in relation to both the expected frequency and amplitude of f=5 and A=12. Finally, it is important to mention that to properly observe this oscillations we had to remove half a second from the beginning and the end of the process, and that it would had been even more accurate to remove a whole second in the beginning of the process which is about the time were we start to observe stable values for both IF and AA. Figure 8. Illustration of four types of graphs: Wrapped Analytic Phase (top left), Unwrapped Analytic Phase (top right), Analytic or Instantaneous Frequency, IF (bottom left) and the Histogram of IF (bottom right). All of the above were calculated after Band-Pass filter has been applied to,!! = 12 cos 2! 5! + 6 cos 2! 12! + 3 cos 2! 23!, the simulated signal with compound amplitudes of, A 1=12, A 2=6 and A 3=3 and frequencies, f 1=5, f 2=12 and f 3=23, with FIR filter parameters: FstopL = 3 Hz, FpassL = 4 Hz, FpassH = 6 Hz, and FstopH = 7 Hz and with the order of the filter N=600. Though both the means of IF and AA are very similar for the three different scenarios, they show very different results when we take in consideration the oscillatory patterns imposed by the filtering process together around the mean center frequency of f 1 =5 and its associated amplitude A 1 =12 for that band, as shown in figure 5. Figure 9. These Illustration shows two types of graphs: Instantaneous Frequency, IF (top), Analytic Amplitude, AA (bottom). All of the above were calculated after Band-Pass filter was applied to,!! = 12 cos 2! 5! + 6 cos 2! 12! + 3 cos 2! 23!, for N=600 and with FIR filter parameters: (a) Wide (dashed line), (b) Narrow (dotted line), (c) Middle (solid line) bandwidth as shown in table II. These three scenarios show that as we narrow the Band- Pass filter s band and we keep the centered frequency f=5 the oscillations diminish. Our graphs also show that we can VI. DISCUSSION AND CONCLUSIONS This work introduced the complexities associated with applying FIR filters of different orders with different bandwidth parameters on both real and simulated data as a means to give a warning about the sensitivities observed both in AA, AP and IF after Hilbert transforming the data. When we increase the order of the filter we observe a more accurate behavior of both the AA and the IF in relation to the expected theoretical values imposed on simulated data. However, this comes with the cost of loosing reliable data at the beginning and the end periods of both the computed AA and IF and the amount of data lost depends on the order of the filter. We also observe some oscillatory behaviors on the IF that can be solved by narrowing the Low and High, Stop and Pass parameters. All of this has to be taken in consideration when choosing a filter for real data and some calibration might be needed together with a proper understanding of the nature of the data and the objectives of the analysis. Our methodology is based on the Hilbert transform after FIR band pass filter for a broad-range of frequencies, including gamma, beta, alpha, and theta bands. A set of indexes and ratios using the AA and AP are employed to detect the onset of synchronization of neural activity across large cortical areas during high-level cognitive functions. In order to properly filter data before applying Hilbert transform, detailed analysis is required to fine tune to a greater degree the accuracy of the filters in terms of both order and bandwidth, together with a clear measurement of the trade-offs in terms of precision and loss of data. The results introduced in this work give a suitable approximation satisfying the needs of interpreting ECoG data. The understanding we derived here, give us insight into how to approach future analysis of scalp EEG data with potential for BCI applications. ACKNOWLEDGMENTS Experimental data have been recorded by Dr. John Barrie at the Freeman Neurophysiology Laboratory, Division of Neuroscience, Department of Molecular & Cell Biology, the University of California at Berkeley. These data are available upon request. This work has been supported in part by NSF CRCNS Grant Number DMS REFERENCES [1] Freeman, W. J., Consciousness, Intentionality and Causality, In: Reclaiming Cognition: The Primacy of Action, Intention and Emotion. Núñez, R., Freeman, W. J. (eds.) Bowling Green, OH. Imprint Academic

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