


ARTICLE 

Year : 2012  Volume
: 2
 Issue : 2  Page : 8286 

Wavelet Entropy Measure to Quantify Information Transmission in Human Cerebral Cortex
Ranganadh Narayanam, Yuu Ono, Hilmi Dajani
School of Information Technology and Engineering University of Ottawa, Ottawa, Ontorio, K1N 6N5, Canada
Date of Web Publication  4Aug2012 
Correspondence Address: Ranganadh Narayanam School of Information Technology and Engineering University of Ottawa, Ottawa, Ontorio, K1N 6N5 Canada
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/09768580.99293
Abstract   
The electrical brain signal analysis is mostly qualitative, the development of new quantitative methods is crucial for restricting the subjectivity in the study of brain signals. These methods are particularly fruitful when they are strongly correlated with intuitive physical concepts that allow a better understanding of brain dynamics. These current waveforms were decomposed into different approximation and details using the wavelet analysis. The wavelet entropy of such decompositions is analyzed, reaching a successful methodology for information transmission. The suggested approach is tested using different eventrelated potential conditions, and different types of cognitive disorders have proven to be successful in the identification of the transmission of information. Keywords: Electroencephalogram, mutual information transmission, wavelet entropy
How to cite this article: Narayanam R, Ono Y, Dajani H. Wavelet Entropy Measure to Quantify Information Transmission in Human Cerebral Cortex. J Eng Technol 2012;2:826 
How to cite this URL: Narayanam R, Ono Y, Dajani H. Wavelet Entropy Measure to Quantify Information Transmission in Human Cerebral Cortex. J Eng Technol [serial online] 2012 [cited 2019 Aug 21];2:826. Available from: http://www.onlinejet.net/text.asp?2012/2/2/82/99293 
1. Introduction   
Although it is unclear how the immense numbers of neurons of the human brain interact to produce different functions, oscillatory activity is increasingly discussed as one of the possible mechanisms [1]. Owing to the anatomical and histological complexities of the cerebral cortex, all these studies present a difficulty in directly interpreting the results of the physiological states in terms of these physical parameters. A major interest is to investigate how brain electric oscillations get information transmission during pathological or physiological brain states (e.g., epileptic seizures, sleepwake stages, etc.), or by external and internal stimulation. This issue can be addressed by applying the methods of system analysis to the electroencephalogram (EEG) signals, because changes in EEG activity occur in temporal relation to triggering events, and correspond to the movement of mutual information transition in the neocortex.
2. Wavelet Transform   
Wavelet analysis gives us a powerful tool to confront very diverse problems in applied sciences or pure mathematics [2]. The wavelet is a smooth and quickly vanishing oscillating function, with good localization both in frequency and time. It can be interpreted as single signals, or atoms, of short time, with oscillating structures.
A wavelet family, ψ_{a ,b,} is a set of elemental functions generated by dilations and translations of a unique admissible mother wavelet ψ (t),
where a, b ε R, and a ≠ 0 are the scale and translation parameters, respectively, and t is the time. As a increases, the wavelet becomes more narrow. Thus, we have a unique analystic pattern and its replicas, at different scales and with variable localizations in time.
Given a finite energy signal S(t), the different correlations ‹S ψ_{a ,b} › indicate how precisely the wavelet function locally fits the signal at every scale a. This correlation operation defines the transformation that synthesizes the numerical information obtained in this manner [3]. From a different viewpoint, the wavelets of a family play the roles of elemental functions, representing the functions as a superposition of wavelets correlated with the function for different scales (different's). This makes it possible to organize the information in some particular structure, to distinguish, for example, trends, or the shape associated with long scales of the local details from the corresponding short scales [4].
The continuous wavelet transform of a signal S(t)εL^{2} (R) is defined as the correlation between the function S(t) with the family wavelet ψ_{a ,b,} for each a,
the asterisk denotes a complex conjugation.
For special selections of the function, ψ and a discrete net of parameters a _{j} _{=} 2 ^{j} and b _{j}_{,k =} 2 ^{j} k, with j,k ε Z and the scale 2 ^{j}, give us the shift parameter. The subfamily
constitutes an orthonormal basis of the Hilbert space L^{2} (R). In this manner, we can obtain discrete transformations, and it is possible to expand the signal in a series of wavelets [5]. Following this, we can join the advantages of the wavelet transform with the atomic decomposition of S(t).
The discrete wavelet transform associated with ψ is simply seen as a restriction of the continuous wavelet transform at the parameter set {aj, bj, k.}. In this case, as is well known, the information given by the discrete wavelet transform can be organized according to a hierarchical scheme of nested subspaces called the multiresolution analysis in L^{2} (R).
In the present analysis we used a multiresolution scheme based on cubic orthogonal spline functions as the mother wavelet, with a discretized version of the integral wavelet transform given by Eq.(2). We selected this wavelet due to the fact that it forms a base in L^{2} (R), with a very convenient characteristic of symmetry and simplicity [6]. Moreover, the smoothness of its derivatives is very suitable for representing the natural phenomena.
In the following we will assume that the EEG signal is given by sampled values {s0(n)}, n = 1, …, M, which correspond to a uniform time grid, with a sampling time of Δt . Without loss of generality, we can suppose that the sampling rate is Δt = 1. We define the representation of the signal by interpolating the sampled data in the form.
For any resolution level N<0, we can write the decomposition of the signal as:
where Cj(k) are the wavelet coefficients, and the sequence {s _{N}(k)} represents the coarser signal data at resolution level N, where N = Ln_{2} (M), the second term is the wavelet expansion. The wavelet coefficients Cj(k) can be interpreted as the local residual errors between successive signal approximations at scales j and j+1, and
is the detail signal at scale j. It contains the information of the signal S(t) corresponding to the frequencies 2 ^{j}π≤/ω/≤2 ^{j+1} ?. If the decomposition is carried out over all resolution levels, then the wavelet expansion will be
π. If the decomposition is carried out over all resolution levels, then the wavelet expansion will be
As the family {ψ_{j ,k} (t)}is an orthonormal basis for L^{2} (R), the concept of energy is linked with the usual notions derived from the Fourier theory. Then the wavelet coefficients are given by Cj(k) = ‹S, ψ_{j ,k} ›, and the total energy by //S// ^{2} = ∑_{j<</i>0} /C _{j}(k)/ ^{2} .
In the wavelet multiresolution framework (described earlier) it is possible to evaluate the energy corresponding to each level, and they can be used for detection of the characteristic epileptic events [7]. As we are using dyadic decomposition of the range of frequencies, from a signal of M samples, we have M/2 ^{j} coefficients at level j. In order to obtain an accurate event detection, we distribute the 'atoms' of energy in each level j uniformly along 2 ^{j} points. The energy in each resolution level j = 1, …. N, will be E _{j} _{=} ∑ _{k} /C _{j}(k)/ ^{2} , and the energy at each sampled time k will be
Natural time series are usually a combination of stochastic (noisy) and chaotic behaviors. Many applications, like nonlinear dynamics, require the separation of signal and noise. Otherwise, when nonlinear invariants are evaluated (i.e., characteristic dimensions or Lyapunov exponents) [8], the contaminating noise can give spurious results, and the obtained values will underestimate (if the noise has strong periodic or almost periodic components) or overestimate (if the noise is representative of a deterministic high dimensional or stochastic process) the real complexity of the system under study. When noise is present only in specific frequency bands, a filtering process can be implemented [9]. However, from the point of view of nonlinear dynamics, filtering could be suitable only in some cases, depending on the system under study. It is important to emphasize that poorly designed filtering of the signal can give spurious results (i.e., filtering frequencies that determine the real dynamics of the signal).
3. Wavelet Entropy   
Once the mean coefficients C _{i,j} are known, the energy for each time I and level j can be calculated as:
As the number of coefficients for each resolution level is different, we redefine the energy by calculating, for each level, its means value in successive time windows (Δt = 128 ms, this window being the minimum one, including at least one coefficient in every level) denoted by the index K, which will now give the time evolution.
Then, the mean energy will be
where k _{0} is the starting value if the time window (k _{0} = 1, 1+Δt, 1+2Δt,….) and N is the number of wavelet coefficients in the time window for each resolution level. For every time window k, the total mean energy can be evaluated as:
And the probability distribution for each level can be defined as:
Clearly, for each time window K,λ = 1, and then, following the definition of entropy given by Shannon, we define the timevarying wavelet entropy as
The multichannel human EEG is measured and the phase space from the EEG time series data by the mutual information theory is constructed. Suppose a dynamical system X (for instance, a time series of an EEG channel) has an invariant probability distribution μ, a measurement of X induces a partition of the phase space of X and locates X in an element of the partition. The probability that an isolated measurement will find the system in the i ^{th} element of the partition is the integral of μ over that element, which is p(i). If two systems X _{1} and X _{2} are measured simultaneously, then the relevant probability distributions are p(i _{1} ), p(i _{2} ), and the joint distribution p(i _{1} , i _{2} ). These distributions yield a dynamic invariant.
This is the mutual information, that is, the amount of information gained about one system, from the measurement of the other. The information transmission among various parts of the human cerebral cortex is computed by the timedelayed mutual information I(A _{t,} B _{t+τ} ) between two sites, with the changes of time delay τ.
A 10 channel EEG signal, with a sampling rate of 150 Hz, is collected from a healthy subject with eyes opened, eyes closed, and eye transition from open to closed states. The time delay τ varies from 1 to 768 ms. Therefore, we have 10 × 10 = 100 transmission channels and their information transmission time series. For example, I(A _{t,} B _{t+τ} ) means the information of A at time t transmitted to B at time t+r. On the other hand, I(B _{t,} A _{t+τ}) means the information of B at time t transmitted to A at time t+r. In view of these curves, it is necessary to have some coarse graining methods to show an overall picture of brain information transmission activities. As the functional state of the conscious human brain may change very rapidly, EEGs are not suitable for measuring the chaotic dynamics (e.g., correlation dimension). This has motivated the introduction of complexity measures.
4. Results and Discussion   
The information transmission time series from lead 4 to 5, of a normal subject, under three different functional states, and an epileptic case is measured. The three functional states are: Awake with his eyes closed, awake with transition from closed to open state, awake with his eyes open. These are displayed in [Figure 1], [Figure 2] and [Figure 3]. [Figure 4] was constructed from the data of an epileptic patient during the onset of a seizure.  Figure 2: Mutualinformation obtained in the case of Eyes Transition state
Click here to view 
 Figure 3: Mutual information obtained in the case of Eyes Opened condition
Click here to view 
4.1 Mutual information obtained from different tasks
 During Eyes Closed condition
 During Transition of Eyes from Closed to Open state
 During Eyes Opened state
 During Epileptic states
4.2 The mutual information transmitted between different channels in a relaxed state
An eightchannel EEG signal, with a sampling of 100 Hz, is collected from a healthy subject during a relaxed state. The time delay τ is varied from 1 to 230 ms. Therefore, we have 8 × 8 = 64 transmission channels and their information transmission time series. For illustration purposes the information transmission among 3, 4, 5, and 6 leads is shown in [Figure 5]. The row indicates the information transmitted from the i ^{th} position to other leads (including it). The column denotes the information received by the i ^{th} lead from the others.  Figure 5: The mutual information obtained from a relaxed healthy subject with closed eyes
Click here to view 
4.3 The mutual information transmitted between different channels in an epileptic patient
A 10channel EEG signal, with a sampling of 150 Hz, is collected from a subject undergoing an epileptic seizure. The time delay τ is varied from 1 to 230 ms. Therefore, we have 10 × 10 = 100 transmission channels and their information transmission time series. For illustration purposes, the information transmission among 3, 4, 5, and 6 leads is shown in [Figure 6]. The row indicates the information transmitted from the i ^{th} position to other leads (including it). The column denotes the information received by the i ^{th} lead from the others.
4.4 The mutual information transmitted between different channels in the MED1 segment
A 64channel EEG signal, with a sampling of 1000 Hz, is collected from a subject undergoing meditation. The time delay is varied from 1 to 500 ms. Therefore, we have 64 × 64 = 4096 transmission channels and their information transmission time series. For illustration purposes, the information transmission among 45, 46, 47, 48, and 49 (P _{7} , P _{5} , C _{1} , C _{2} , CP _{1} ) lead positions are shown in [Figure 7]. The row indicates the information transmitted from the i ^{th} position to other leads (including it). The column denotes the information received by the i ^{th} lead from the others. The fortyeighth (C _{2} ) lead receives more information from other leads  Figure 7: The mutual information transmitted between different channels in MED1
Click here to view 
5. Conclusion   
Finally, the information transmission between different parts of the human brain is extracted. This information transmission in various parts of the human cerebral cortex is obtained using the wavelet entropy for mutual information transmission (time delayed mutual information). Mutual information is nothing but the amount of information gained about one node using the measurement from others nodes. The mutual information under different conditions are obtained and analyzed. A strange periodicity is observed in the epileptic condition. On account of the functional state of the conscious human brain, it may change very rapidly EEG is not suitable for measuring the chaotic dynamics (e.g., correlation dimension). The degree of brightness gives the amount of complexity. The ITM's for the EEG record for eyes closed state, during transition of eyes from the closed to open state, and eyes open state, are evaluated and analyzed. This study of finding the complexity, direction, and information flow, provides potential information for analysis of the human brain, which can be applied to the study of various brain disorders, like epilepsy
References   
1.  L. F. LagoFernández, R. Huerta, F. Corbacho, and J. A. Sigüenza. "Fast response and temporal coherent oscillations in smallworld networks," Phys Rev Letter, vol. 84, pp. 275861, 2000. 
2.  M. Barahona, and L. M. Pecora. "Synchronization in smallworld systems." Phys Rev Letter, vol. 89, pp. 054101, 2002. 
3.  G. Tononi, G. M. Edelman, and O. Sporns. "Complexity and coherency: Integrating information in the brain," TICS, Vol. 2, pp. 47484, 1998. 
4.  Determining Lyapunov Exponents from a Time Series Alan Wolf, Jack B. Swift, Harry L. Swinney and John A. Vastano. 
5.  H. A. Alnashash, and J. S. Paul. "Wavelet entropy for subband segmentation of EEG during injury and recovery," Annals of Biomedical Engineering, vol. 31, pp. 653658, 2003. 
6.  R. Quian Quiroga, O. A. Rosso, E. Basar, and M. Schurmann, "Wavelet Entropy in Event  Related Potentials: A New Method Shows Ordering of EEG Oscillations," Biol. Cybern, vol. 84, pp. 291299, 2001. 
7.  O. A. Rosso, S. Blanco, and J. Yordanova, "Wavelet entropy: A new tool for analysis of short durationbrain electrical signals," Journal of Neuroscience Methods, vol. 105, pp. 6575, 2001. 
8.  Giorgos A. Giannakakis, Nikolaos N. Tsiaparas, MonikaFilitsa S. Xenikou, Charalabos C. Papageorgiou, and Konstantina S. Nikita. "Wavelet entropy differentiations of event related potentials in dyslexia," BIBE 2008:16. 
9.  S. El Safty, and A. ElZonkoly. "Applying Wavelet Entropy Principle in Fault Classification," Proceedings of World Academy of Science, Engineering and Technology, Vol. 40. pp. 133135, 2008. 
Authors   
Mr. Ranganadh Narayanam , is currently an assistant professor in the Department of ECE in the Marri Laxman Reddy Institute of Technology, Hyderabad. Mr. Narayanam had conducted research, as a research assistant in the University of Texas at San Antonio, University of California at Berkeley, and University of Ottawa under Dr. CJ Qian, Dr. Parimal Patel, Dr. Artyom Grigoryan, Dr. Sos Agaian, Dr. Ben Inglis, and Dr. Hilmi Dajani, in the areas of control systems, signal processing, digital systems, and neurological signal processing. This research was partially guided by Dr. Hilmi Dajani of the University of Ottawa and Dr. Yuu Ono of the Carleton University. Ranganadh Narayanam can be contacted at the Marri Laxman Reddy Institute of Technology, Department of Electronics and Communications Engineering, Hyderabad, AP, India.
[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7]
