CONCIOUSNESS NEUROMATH Limassol Cyprus 29 11 1 12

CONCIOUSNESS & NEUROMATH Limassol, Cyprus 29. 11. - 1. 12. 2009. ; , Detection of structural features in brain signals and causality tests Aleksandar Perovic***, Wlodzimierz Klonowski**, Wlodzisław Duch*, Zoran Djordjevic ***, Aleksandar Jovanovic*** *Department of Informatics, Nicolaus Copernicus University, Torun, Poland **Lab. Biosignal Analysis Fundamentals, Institute of Biocybernetics & Biomedical Engineering, Polish Academy of Sciences, Warsaw, Poland ***Group for Intelligent Systems, School of Mathematics, University of Belgrade, Serbia Correspondence: Aleksandar Jovanovic, School of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade, Serbia. email: aljosha. jovanovich@gmail. com cell +38164 1412527 phone +381 11 2027801, We are involved in the investigation and development of methods for automatic localization, extraction, analysis and comparison of features in spectra of biological signals. With diverse applications, different feature attributes turn out as significant and as characteristics of the investigated phenomena. Individual tunes and music representation in brain signals have attracted our attention since early nineties, including the possibility to use inner tunes in the brain for the Brain Computer Interfaces (BCI). These were some of the motives for the present study, but applicability of the developed algorithms is broader. Keywords: Spectral feature automatic detection; Inner Tones and Music; Brain Computer Interface; Introduction In the investigation of features in brain signals, the importance of connectivity between different loci engaged in the implementation of sensory motor, other mechanical controls and those involved in the pre-cognitive and cognitive processing is well established and exploited. A broad variety of network modeling applied in different fields found its use in the brain connectivity research. In BCI – brain computer interfaces, pattern identification, localization and transmition is usually related to the connectivity problem, especially when relatively very low or low energy features are subject of interest like in the case of inner tones and inner music in BCI applications, which is central in our interests. The focus of our attention in the connectivity problem is somewhat shifted from the usual approach to this problem. The amount of knowledge related to this topic and its variations is fastly growing, covering solutions in diverse fields. Within the brain connectivity problem there are different aspects, overlapping semantically and methodologically. Statistic measures of causality between two or more time series, introduced by Wiener – Granger, represent very important part of the method. For the two time series x and y, the intention of Wiener – Granger measures was to establish causal relationship between sequences x and y, if the knowledge of one supports prediction of future values of the other. The method was originally introduced by Granger and generalized and further developed by a number of researchers. We will discuss measures closer to those introduced by Geweke, presenting examples where this and other criteria fail, while the processes involved share a common information. Introducing PLD – partial linear dependence, we show it can work in limit cases, where the common information is practically negligible, or at the noise threshold. Figure 1. Examples of road network, with the flow measurement at the violet section level. The 3 highways traffic flow behaves randomly and is not causality related, though there are capillary interconnections. . The traffic accident at the red star point on the busy blue rd, down-bound direction at moment t causes traffic jamming in the middle high way down bound. Within a certain amount of time the capillary links would transfer viscosity increase to the nearby highways down bound, reducing the flow and stabilizing the reduced car pace, inducing the road flow shift to more deterministic behavior. Method Granger causality simple form is usually stated as follows. In the simplified AR - auto regressive form, one way causality: from x to y (1) y(t) = Figure 2. Top row: Brain signals with 1 KHz and 3 KHz stimulation, power spectra; lower row: (time) spectrograms, left; their dot product – left part of the right image; exhibiting linear independence in high frequency – HF, while increasing the LF part of spectra by 1 order of magnitude. If all coefficients B are zero, then x does not cause y, where e is the error function. The important development in causality characterizations, involving multivariate linear and non linear modeling established a number of causality criteria and measures. Among many variations, mention DTF – direct transfer function, PDC - partial directed coherence, RNAR – recursive non linear modeling, Involved by Kaminski, Ding, Sameshima, Doulamis-Ntalianis, et others, proved to be successful causality measures. Granger causality formulated in spectral form and elaborated in work of Geweke – “Measurement of linear dependence and feedback between multiple time series” involving Fourier spectral representations is of interest for our purposes and intentions. It provides estimation of feedback from x to y at frequency λ, formulated below. Write x and y in the autoregressive representation as follows. with the linear projection of Figure 3. Brain signals with inner tone c 2. Power spectra of these signals exhibit some artifacts at odd 50 Hz multiples, some other isolated HF, while the traces of the imagined – inner tone are close to the noise level. Multiple spectral dot products reduce overall spectral randomness – the coordinates with random fluctuations are mutually linearly independent, so their products converges to zero, while the coordinates with the linearly dependent values are enhanced relatively to the noise threshold and become locally prominent, even when their integral contribution to the spectral power is small or very small. Here, a major artifact spectral frequency at 250 Hz with value is 80 (energy**3), while the inner tone C 2 shows 15 units in the composite spectrogram – power spectrograms time*frequency*intensity matrix S is exponentiated by 3: S**3, corresponding to the 3 EEG channel power spectra dot product in time. & on their past values is then given by Let - the minimum square error. Then the measures of y to f and x to y are respectively defined as and the measure of instantaneous linear feedback by while suggested as a measure of linear dependence: Clearly, the last one equals the sum of previous three. By Gelfand – Yaglom decomposition of linear feedback, it follows where C(φ) is the coherence x of and y at frequency φ. Under certain restrictions, where , measure respectively linear feedback from x to y and y to x at frequency φ calculated via Fourier transform of and vectors and the corresponding spectral density. Figure 4. The major spectral frequency at 6. 1 Hz exhibits magnitude 8. 71 in the original signal power spectrum (shown left), while the inner tone C 2 corresponding frequency at 522. 78 Hz , marked line in the right picture, has the value of 0. 17 embedded in its spectral frequency neighborhood ~ “noise”, with the magnitude ratio over 50, consuming approximately one pro mile of spectral energy, elusive for all causality measures. If applied to a certain given frequency, a reduced size set of frequencies, or a known frequency band, this method can supply good answers with somewhat complex calculations. If the spectral parts contain frequencies with poor signal to noise ratio then similar approach exploiting multivariate linear dependence might work. Alternatively, if we start with independent sequences, feeding all of them with relatively small magnitude process, we should be able to establish the threshold level from the lower side, i. e. when the shared information becomes perceptible. By a MS – a modulation system we designate the usual meanings of signal modulation, i. e. coding or fusion of information process with some base function (carrier). Thus, a MS can be of any sort, e. g. AM, FM, PCM, BFH (base frequency/carrier-wave hopping), etc. , or some their meaningful combination. Thus, MS: (F, G) → H, where F is a subset of B - a system of base functions, while G = {g 1, g 2, …gn} a set of information contents, H the fusion output, all components of F, G, H are time functions- sequences. In simplest case F, G, H are singletons. Obviously, a brain connectivity path might accommodate lower frequency and high frequency information patterns. For two functions f 1 and f 2 we say that they are independent wrt. to causality measure μ, if μ(f 1, f 2) = 0. In practice, for experimental f 1, f 2, that would be μ(f 1, f 2) < ε for all ε > 0, down to the noise threshold. If we inject/modulate a sequence G of information sequences, into a couple of μ - independent f 1 and f 2, resulting in f 1’ and f 2’ and if ||G|| < ||f 1||, ||f 2|| for a suitable norm (where the norm of G could be ||G|| = sup {||g||: g ε G}, we can well have μ(f 1’, f 2’) ≈ 0, while f 1’ and f 2’ share a vector of information sequences G. The case becomes more complicated if f 1’ and f 2’ are modulated using different MS’s, or have different delays involved in modulation of a set G, or if we deal with spectral features with non constant frequencies, or if modulation processes involve some headers - protocols. With some rather simple examples we illustrate possible criteria instability. We introduce a pragmatic property – the local/partial linear dependence, as the linear dependence in a subset of all coordinates, in our treatment in the power spectra form. A set G of time functions/sequences is locally linearly dependent at frequency λ, if Π{F(g, λ): g ε G} > 0, where F(g, λ) is the λ-th coordinate of Fourier (power) spectrum of g. Similarly for Fourier spectrograms of G in time interval T, Π{S(g, λ, T): g ε G} > 0, where S(g, λ, T) is time integral of F(g, λ) in the epoch scrolling time interval T, i. e. the integral of the time trace of the spectral line at λ: S(g, λ, T) = ∑ {F(g, λ) : t ε T}. Figure 5. Left image: marked spectral frequency in the 3 -composite spectrogram at the 2. 08 Hz with the magnitude of 8510 in the cubed units; Right image, shows a composite part with 3 initial dominant lines – artifacts of 50 Hz at 250, 350 and 450 Hz, followed by arrow marked position of C 2 inner tone with the (cubed) value of 0. 0128. Relative ratio to the dominant line is 664843, energy**3 ratio of C 2 line to the spectral energy is ~10**7. The criteria express that all spectra of elements of G have a non zero λ coordinate. We define two simple quotient measures for power spectra, energy density indices for λ, ED(g, λ) = F(g, λ)/ ∑ {F(g, λ) : λ ε SR}, the energy at the frequency λ relative total spectral energy and the similar index for spectral neighborhood of λ, i. e. for Λ subset of SR - the set of all frequencies in the spectrum. Hence ED(g, Λ) = ∑ {ED(g, ν) : ν ε Λ}, where usually Λ is some (topological) neighborhood of λ. Clearly, the more prominent spectral line at λ, the higher the first index; the more prominent spectral line at λ locally, the higher the second index. Obviously we can have situations with the first index negligible while the second index is perceptible. Integrating those indices over a period of time T, we get similar measures for spectrograms. Clearly, the same definitions apply to composite spectra and spectrograms, for a set of g’s in G. In practice that means that we can search for a set G – a subset of electrode network measurements, which contain the same spectral component. After selectivity preprocessing of connectivity for certain application, we can use the linear dependent channels to enhance the periodic component present in all members of G, which might all be near or even below noise threshold, as illustrated in the spectra and spectrograms in figures. The advantage is that the noise behaves randomly and the frequencies of noise alone will be zero flushed by the above criteria, in both composite spectra and composite spectrograms. Even without knowing at which frequencies interesting periodic patterns might be expected, the above method provides a high resolution spectral and spectrogram scanning. If there artifacts which are characteristic for certain frequency bands, in case when the searched information is out of these bands, with sufficient frequency separation, we might be able to localize and extract even the features embedded in the noise. The criteria can be expressed for λ ε Λ, a subset of SR – spectral resolution. Thus in the composite spectra and spectrograms, the first index ED(g, λ) easily converges to the (practical) zero, while the second index ED(g, Λ), for certain spectral neighborhoods Λ of λ can locally amplify the hidden information. When searching for spectrogram features which have certain time frequency allocation lower and upper bounds, the spectrogram indices would apply. If G is modulated by certain MS’s, we can still separate the carriers if known, even within the same procedure as above. Discussion Figure 6. Towards elusive feature recognition. Left: the 3 composite power spectrogram matrix exhibits the dominant feature peak at 2. 03 Hz with relative magnitude of 41405. The locally linearly dependent – locally enhanced structure corresponding to inner C 2 tone at 522. 78 Hz with peak magnitude of 0. 035; their relative magnitude ratio **3 is 1183000. The total energy of C 2 time vector in the S**3 matrix is less than 10**7. Henceforth this feature is imperceptible by all global criteria, but extractable by local linear dependence. In various examples with brain signals and the automatic real time spectrogram feature recognition demands, we encounter the connectivity problem and its relation to the various causality criteria. Our problems are well related to the work of Geweke. We show examples which are very elusive for the otherwise well established criteria, and show to overcome some of these problems using PLD – partial linear dependence, even in cases when the target information is embedded below the noise level. Combined with the recalibrated spectroscopy this results give better examples, results and the stated problem resolution. The method is well applicable for other orthonormal representations. First 2 images, composite, extracted locally perceptible inner C 2, third: similar shown with S**2 spectrogram with the starting spectrograms green, showing C 2 original spectral neighborhood, forth and fifth single spectrograms with blood pressure variations in the important HF feature; whale song spectrogram with complex time*frequency structures. The first two images: Inner C 2, better responding channel selection: S**6 coordinate vise product of 3 D spectra of 6 channels, lateral and the top lateral electrodes, frequency interval 500 -545 Hz showing extracted Inner C 2 tone. .