{"title": "Independent Component Analysis of Intracellular Calcium Spike Data", "book": "Advances in Neural Information Processing Systems", "page_first": 931, "page_last": 937, "abstract": null, "full_text": "Independent Component Analysis of \n\nIntracellular Calcium Spike Data \n\nKlaus Prank, Julia Borger, Alexander von zur Miihlen, \n\nGeorg Brabant, Christof Schoil \nDepartment of Clinical Endocrinology \n\nMedical School Hannover \n\nD-30625 Hannover \n\nGermany \n\nAbstract \n\nCalcium (Ca2+)is an ubiquitous intracellular messenger which reg(cid:173)\nulates cellular processes, such as secretion, contraction, and cell \nproliferation. A number of different cell types respond to hormonal \nstimuli with periodic oscillations of the intracellular free calcium \nconcentration ([Ca2+]i). These Ca2+ signals are often organized \nin complex temporal and spatial patterns even under conditions \nof sustained stimulation. Here we study the spatio-temporal as(cid:173)\npects of intracellular calcium ([Ca 2+]i) oscillations in clonal J3-cells \n(hamster insulin secreting cells, HIT) under pharmacological stim(cid:173)\nulation (Schofi et al., 1996). We use a novel fast fixed-point al(cid:173)\ngorithm (Hyvarinen and Oja, 1997) for Independent Component \nAnalysis (ICA) to blind source separation of the spatio-temporal \ndynamics of [Ca2+]i in a HIT-cell. Using this approach we find two \nsignificant independent components out of five differently mixed in(cid:173)\nput signals: one [Ca2+]i signal with a mean oscillatory period of \n68s and a high frequency signal with a broadband power spectrum \nwith considerable spectral density. This results is in good agree(cid:173)\nment with a study on high-frequency [Ca2+]j oscillations (Palus \net al., 1998) Further theoretical and experimental studies have to \nbe performed to resolve the question on the functional impact of \nintracellular signaling of these independent [Ca2+]i signals. \n\n\f932 \n\nK. Prank et al. \n\n1 \n\nINTRODUCTION \n\nIndependent component analysis (ICA) (Comon, 1994; Jutten and Herault, 1991) \nhas recently received much attention as a signal processing method which has been \nsuccessfully applied to blind source separation and feature extraction. The goal of \nICA is to find independent sources in an unknown linear mixture of measured sen(cid:173)\nsory data. This goal is obtained by reducing 2nd-order and higher order statistical \ndependencies to make the signals as independent as possible. Mainly three different \napproaches for ICA exist. The first approach is based on batch computations min(cid:173)\nimizing or maximizing some relevant criterion functions (Cardoso, 1992; Comon, \n1994). The second category contains adaptive algorithms often based on stochastic \ngradient methods, which may have implementations in neural networks (Amari et \nal., 1996; Bell and Sejnowski, 1995; Delfosse and Loubaton, 1995; Hyvarinen and \nOja, 1996; Jutten and Herault, 1991; Moreau and Macchi, 1993; Oja and Karhunen, \n1995). The third class of algorithms is based on a fixed-point iteration scheme for \nfinding the local extrema of the kurtosis of a linear combination of the observed \nvariables which is equivalent to estimating the non-Gaussian independent compa(cid:173)\nnents (Hyvarinen and Oja 1997). Here we use the fast fixed-point algorithm for \nindependent component analysis proposed by Hyvarinen and Oja (1997) to analyze \nthe spatia-temporal dynamics of intracellular free calcium ([Ca2+]i) in a hamster \ninsulin secreting cell (HIT). \nOscillations of [Ca2+]i have been reported in a number of electrically excitable and \nnon-excitable cells and the hypotheses of frequency coding were proposed a decade \nago (Berridge and Galione, 1988). Recent experimental results clearly demonstrate \nthat [Ca2+]i oscillations and their frequency can be specific for gene activation con(cid:173)\ncerning the efficiency as well as the selectivity (Dolmetsch et al., 1998). Cells are \nhighly compartmentalized structures which can not be regarded as homogenous en(cid:173)\ntities. Thus, [Ca 2+]i oscillations do not occur uniformly throughout the cell but \nare initiated at specific sites which are distributed in a functional and nonunifortm \nmanner. These [Ca2+]i oscillations spread across individual cells in the form of \nCa2+ waves. [Ca2+]i gradients within cells have been proposed to initiate cell mi(cid:173)\ngration, exocytosis, lymphocyte, killer cell activity, acid secretion, transcellular ion \ntransport, neurotransmitter release, gap junction regulation, and numerous other \nfunctions (Tsien and Tsien, 1990). Due to this fact it is of major importance to \nstudy the spatia-temporal aspects of [Ca2+]i signaling in small sub compartments \nusing calcium-specific fluorescent reporter dyes and digital videomicroscopy rather \nthan studying the cell as a uniform entity. The aim of this study was to define the \nindependent components of the spatia-temporal [Ca2+]i signal. \n\n2 METHODS \n\n2.1 FAST FIXED-POINT ALGORITHM USING KURTOSIS FOR \n\nINDEPENDENT COMPONENT ANALYSIS \n\nIn Independent Component Analysis (ICA) the original independent sources are un(cid:173)\nknown. In this study we have recorded the [Ca2+]i signal in single HIT-cells under \npharmacological stimulation at different subcellular regions (m = 5) in parallel. \nThe [Ca2+]i signals (mixtures of sources) are denoted as Xl, X2, \u2022\u2022 \u2022 , X m . Each Xi \nis expressed as the weighted sum of n unknown statistically independent compa-\n\n\fIndependent Component Analysis of Intracellular Calcium Spike Data \n\n933 \n\nnents (ICs), denoted as SI, S2, \u2022.\u2022 , Sn. The components are assumed to be mutually \nstatistically independent and zero-mean. The measured signals Xi as well as the in(cid:173)\ndependent component variables can be arranged into vectors x = (Xl, X2, \u2022.. ,XIIl ) \nand 8 = (81,82, ... , 8 n ) respectively. The linear relationship is given by: \n\nX=A8 \n\n(I) \nHere A is a constant mixing matrix whose elements aij are the unknown coefficients \nof the mixtures. The basic problem of ICA is to estimate both the mixing matrix \nA and the realizations of the Si using only observations of the mixtures X j. In \norder to perform ICA, it is necessary to have at least as many mixtures as there \nare independent sources (m 2: n). The assumption of zero mean of the ICs is no \nrestriction, as this can always be accomplished by subtracting the mean from the \nrandom vector x. The ICs and the columns of A can only be estimated up to a \nmUltiplicative constant, because any constant multiplying an IC in eq. 1 could be \ncancelled by dividing the corresponding column of the mixing matrix A by the same \nconstant. For mathematical convenience, the ICs are defined to have unit variance \nmaking the (non-Gaussian) ICs unique, up to their signs (Comon, 1994). Here we \nuse a novel fixed-point algorithm for ICA estimation which is based on 'contrast' \nfunctions whose extrema are closely connected to the estimation of ICs (Hyvarinen \nand OJ a, 1997). This method denoted as fast fixed-point algorithm has a number \nof desirable properties. First, it is easy to use, since there are no user-defined \nparameters. Furthermore, the convergence is fast, conventionally in less than 15 \nsteps and for an appropriate contrast function, the fixed-point algorithm is much \nmore robust against outliers than most ICA algorithms. \n\nMost solutions to the ICA problem use the fourth-order cumulant or kurtosis of the \nsignals, defined for a zero-mean random variable x as: \n\n(2) \n\nwhere E{ x} denotes the mathematical expectation of x. The kurtosis is negative for \nsource signals whose amplitude has sub-Gaussian probability densitites (distribution \nflatter than Gaussian, positive for super Gaussian) sharper than Gaussian, and zero \nfor Gausssian densities. Kurtosis is a contrast function for ICA in the following \nsense. Consider a linear combination of the measured mixtures x, say wTx, where \nthe vector w is constrained so that E{(wT X}2} = 1. When w T x = \u00b1Si, for some i, \ni.e. when the linear combination equals, up to the sign, one of the ICs, the kurtosis \nof w T x is locally minimized or maximized. This property is widely used in ICA \nalgorithms and forms the basis of the fixed-point algorithm used in this study which \nfinds the relevant extrema of kurtosis also for non-whitened data. Based on this fact, \nHyvarinen and Oja (1997) introduced a very simple and highly efficient fixed-point \nalgorithm for computing ICA, calculated over sphered zero-mean vectors x, that is \nable to find the rows of the separation matrix (denoted as w) and so identify one \nindependent source at a time. The algorithm which computes a gradient descent \nover the kurtosis is defined as follows: \n\n1. Take a random initial vector Wo of unit norm. Let 1 = 1. \n\n2. Let WI = E{V(Wf-1V}3} - 3WI-l. The expectation can be estimated using \n\na large sample of Vk vectors. \n\n\f934 \n\nK. Prank et al. \n\n3. Divide WI by its norm (e.g. the Euclidean norm II W 11= .J~i wn\u00b7 \n4. If 1 WfWI-l 1 is not close enough to 1, let 1 = 1 + 1 and go back to step 2. \n\nOtherwise, output the vector WI. \n\nTo calculate more than one solution, the algorithm may be run as many times as \nrequired. It is nevertheless, necessary to remove the information contained in the \nsolutions already found, to estimate each time a different independent component. \nThis can be achieved, after the fourth step of the algorithm, by simply subtracting \nthe estimated solution 8 = w T v from the unsphered data x. \n\nIn the first step of analysis we determined the eigenvalues of the covariance matrix \nof the measured [Ca2+]i signals Si to reduce the dimensionality of the system. \nThen the fast fixed-point algorithm was run using the experimental [Ca2+]i data to \ndetermine the lOs. The resulting lOs were analyzed in respect to their frequency \ncontent by computing the Fourier power spectrum. \n\n2.2 MEASUREMENT OF INTRACELLULAR CALCIUM IN \n\nHIT-CELLS \n\nTo measure [Ca2+]i' HIT (hamster insulin secreting tumor)-cells were loaded with \nthe fluorescent indicator Fura-2/ AM and Fura-2 fluorescence was recorded at five \ndifferent subcellular regions in parallel using a dual excitation spectrofluorometer \nvideoimaging system. The emission wavelength was 510 nm and the excitation \nwavelengths were 340 nm and 380 nm respectively. The ration between the excita(cid:173)\ntion wavelength (F340nm/ F38onm) which correlates to [Ca2+]i was sampled at a rate \nof 1 Hz over 360 s. [Ca2+]i spikes in this cell were induced by the administration \nof 1 nM arginine vasopressin (AVP). \n\n3 RESULTS \n\nFrom the five experimental [Ca2+]i signals (Fig. 1) we determined two significant \neigenvalues of the covariance matrix. The fixed-point algorithm converged in less \nthan 15 steps and yielded two different lOs, one slowly oscillating component with \na mean period of 68 s and one component with fast irregular oscillations with a flat \nbroadband power spectrum (Fig. 2). The spectral density of the second component \nwas considerably larger than that for the high-frequency content of the first slowly \noscillating component. \n\n4 CONCLUSIONS \n\nOhanges in [Ca2+]i associated with Ca2+ oscillations generally do not occur uni(cid:173)\nformly throughout the cell but are initiated at specific sites and are able to spread \nacross individual cells in the form of intracellular Ca2+ waves. Furthermore, Ca2+ \nsignaling is not limited to single cells but occurs between adjacent cells in the form of \nintercellular Ca2+ waves. The reasons for these spatio-temporal patterns of [Ca2+]i \nare not yet fully understood. It has been suggested that information is encoded in \nthe frequency, rather than the amplitude, of Ca2+ oscillations, which has the ad(cid:173)\nvantage of avoiding prolonged exposures to high [Ca2+]i. Another advantage of \n\n\fIndependent Component Analysis of Intracellular Calcium Spike Data \n\n935 \n\n-4 \n\n200 \n\no \n\n50 \n\n300 \n\n100 \n\n150 \n\n250 \n\n=}0~~J~~~j \nj~~~~2i \n{k~~g \n-~ -4 \n=~~: : : : : ~j \n\n250 \n\nLL \n\n0 \n\no \n\no \n\n100 \n\n150 \n\n100 \n\n150 \n\n100 \n\n150 \n\n200 \n\n250 \n\n200 \n\n250 \n\n300 \n\n200 \n\n50 \n\n50 \n\n50 \n\n300 \n\n300 \n\nrim. (5) \n\nFigure 1: Experimental time series of [Ca2+]i in a ,B-cell (insulin secreting cell from \na hamster, HIT-cell) determined in five subcellular regions. The data are given as \nthe ratio between both exciation wavelengths of 340 nm and 380 nm respectively \nwhich correspond to [Ca2+k [Ca2+]i can be calculated from this ratio. The plotted \ntime series are whitened. \n\nfrequency modulated signaling is its high signal-to-noise ratio. In the spatial do(cid:173)\nmain, the spreading of a Ca2+ oscillation as a Ca2+ wave provides a mechanism \nby which the regulatory signal can be distributed throughout the cell. The exten(cid:173)\nsion of Ca2+ waves to adjacent cells by intercellular communication provides one \nmechanism by which multicellular systems can effect coordinated and cooperative \ncell responses to localized stimuli. In this study we demonstrated that the [Ca2+]i \nsignal in clonal ,B-cells (HIT cells) is composed of two independent components \nusing spatio-temporal [Ca2+]i data for analysis. One component can be described \nas large amplitude slow frequency oscillations whereas the other one is a high fre(cid:173)\nquency component which exhibits a broadband power spectrum. These results are \nin good agreement with a previous study where only the temporal dynamics of \n[Ca2+]i in HIT cells has been studied. Using coarse-grained entropy rates com(cid:173)\nputed from information-theoretic functionals we could demonstrate in that study \nthat a fast oscillatory component of the [Ca2+]i signal can be modulated phar(cid:173)\nmacologically suggesting deterministic structure in the temporal dynamics (Palu8 \net al., 1998). Since Ca2+ is central to the stimulation of insulin secretion from \npancreatic ,B-cells future experimental and theoretical studies should evaluate the \nimpact of the different oscillatory components of [Ca2+]i onto the secretory pro(cid:173)\ncess as well as gene transcription. One possibility to resolve that question is to \nuse a recently proposed mathematical model which allows for the on-line decoding \nof the [Ca2+]i into the cellular response represented by the activation (phospho-\n\n\f936 \n\nK. Prank et al. \n\n-0 '----- - -- - -----' \n\nsao \n\na \n\n100 \n\n200 \nrime (S) \n\n100 \n\n200 \nlimets) \n\n.00 \n\n'D. C \n\n'd. \n\n0 \n\n10 -\u00b70\u00b7'------:0,-, '-0::-\":,2-\n\n\"\"'0,':---:\"0,-:-0 --=-'0 .\u2022 \n\nfrequency (Hz) \n\n10- 6 L - -_\n\n_\n\n_ ___ - - - ' \n\na \n\n0.1 \n\n0 .2 \n\nO. S \n\n0 .4 \n\n0 .5 \n\nfrequency (Hz) \n\nFigure 2: Results from the independent component analysis by the fast fixed-point \nalgorithm. Two independent components of [Ca2+]i were found. A: slowlyoscillat(cid:173)\ning [Ca2+]i signal, B: fast oscillating [Ca2+]i signal. Fourier power spectra of the \nindependent components. C: the major [Ca2+]i oscillatory period is 68 s, D: flat \nbroadband power spectrum. \n\nrylation) of target proteins (Prank et al., 1998). Very recent experimental data \nclearly demonstrate that specificty is encoded in the frequency of [Ca2+]i oscil(cid:173)\nlations. Rapid oscillations of [Ca2+]j are able to stimulate a set of transcription \nfactors in T-Iymphocytes whereas slow oscillations activate only one transcription \nfactor (Dolmetsch et al., 1998). Frequency-dependent gene expression is likely to \nbe a widespread phenomenon and oscillations of [Ca2+]i can occur with periods \nof seconds to hours. The technique of independent component analyis should be \nable to extract the spatio-temporal features of the [Ca2+]i signal in a variety of \ncells and should help to understand the differential regulation of [Ca2+]i-dependent \nintracellular processes such as gene transcription or secretion. \n\nAcknowledgements \n\nThis study was supported by Deutsche Forschungsgemeinschaft under grants \nScho 466/1-3 and Br 915/4-4. \n\n\fIndependent Component Analysis of Intracellular Calcium Spike Data \n\n937 \n\nReferences \n\nAmari, S., Cichocki, A. & Yang, H. (1996) A new learning algorithm for blind source \nseparation. In Touretzky, D.S., Mozer, M. 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Cell BioI. 6:715-760. \n\n\f", "award": [], "sourceid": 1545, "authors": [{"given_name": "Klaus", "family_name": "Prank", "institution": null}, {"given_name": "Julia", "family_name": "B\u00f6rger", "institution": null}, {"given_name": "Alexander", "family_name": "von zur M\u00fchlen", "institution": null}, {"given_name": "Georg", "family_name": "Brabant", "institution": null}, {"given_name": "Christof", "family_name": "Sch\u00f6fl", "institution": null}]}