Tensor based Blind Source Separation for Current ...

8th International IEEE EMBS Conference on Neural Engineering Shanghai, China, May 25 - 28 , 2017

Tensor based Blind Source Separation for Current Source Density Analysis of Evoked Potentials from Somatosensory Cortex of Mice Jo˜ao Paulo C. L. da Costa∗†‡ , Ricardo Kehrle Miranda∗† , and Mateus da Rosa Zanatta §

∗ Department of Electrical Engineering, University of Brasilia, Brasilia, Brazil § Department of Mechanical Engineering, University of Brasilia, Brasilia, Brazil

† Institute for Information Technology, Ilmenau University of Technology, Ilmenau, Germany ‡ Fraunhofer Institute for Integrated Circuits IIS, Erlangen, Germany

Abstract—In order to understand brain mechanisms and functionalities, neural probes with electrode arrays are incorporated into mice and Local Field Potentials (LFP) are recorded indicating the activities of groups of neurons. Next, the brain activity can be analyzed in terms of Current Source Density (CSD), which are computed via the LFP. In this paper, we propose the analysis of the somatosensory cortex signals of a mouse applying Blind Source Separation (BSS) schemes. In contrast to the standard CSD, we show that signal separation using BSS schemes can be useful to identify groups of neurons of different layers of the somatosensory cortex that are associated. Another contribution of this work is to propose the use of the PARAFAC model on the analysis of somatosensory cortex signals, whose results are consistent with results obtained via Spatiotemporal Independent Component Analysis (stICA).

I. I NTRODUCTION The resulting knowledge arising from mice experiments can assist in the treatment of human brain diseases and to understand mechanisms and circuits associated with the processing of information in the brain. In this sense, the first studies using Local Field Potential (LFP) recordings date back to the 1940s [2], [3], [4]. LFPs are electric potentials recorded from the extracellular space of brain tissue. Since its introduction in 1952 [5], the Current Source Density (CSD) analysis has grown in popularity as a method for the analysis of LFPs. The one dimensional version of the CSD has become well known due to Nicholson and Freeman [6], [7]. According to Bellistri et al. [8], the CSD analysis is popularly used by the scientific community. Nevertheless, seldom, array signal processing schemes for Blind Source Separation (BSS) are applied to decompose the multichannel LFPs or the derived CSDs into their original informative sources. The scientific literature has shown the application of the following techniques for LFPs and CSDs: Principle Component Analysis (PCA) Barth et al. [9], Independent Component Analysis (ICA) Makarov et al. [10], and Spatiotemporal Independent Component Analysis (stICA) Leski et al. [11]. Recently, linear multielectrodes have been exploited by neuroscientists to investigate the behaviour of mouse somatosensory cortex. Current technology allows in vivo voltage recordings at tens or hundreds of contacts across cortex laminae [12], [13]. Such recordings can be used to verify which cortex layers are activated by different stimuli. In this sense, we show in this work that BSS techniques can assist the analysis of such signals. Moreover, BSS schemes can support the identification of specific activation groups for given stimuli. Therefore, we propose the use of multi-way analysis techniques to study one dimensional cortical recordings with linear multielectrodes. Multi-way analysis has many advantages over its two-way, i.e matrix, counterpart [14], [15], [16], [17]. For instance, multi-way analysis is unique under very mild conditions

978-1-5090-4603-4/17/$31.00 ©2017 IEEE

without the need of additional constraints [14], [17], and allows exploiting the diversity of multimodal datasets [16]. However, in order to use such techniques to this data, another dimension is necessary besides the spatial dimension of the electrodes and the time dimension of the time snapshots. Such artificial dimension can be created by decomposing the time dimension into frequency and time dimensions by applying Time- Frequency Analysis (TFA) [15], [18]. Nevertheless, the common approach used to analyze scalp electroencephalographic (EEG) signals, which is by taking only the magnitudes in the time-frequency representations and discarding the phases, is not sufficient in this case since the inverse TFA operation should be applied to the signals requiring both phase and amplitude information. In our experiments, the Parallel Factor Analysis (PARAFAC) has successfully managed to decompose the LFPs and recover the signal. This paper is organized as follows. In Section II, we present biological concepts necessary to understand the origin of the signals recorded by electrodes in the extracellular space of brain tissue. In Section III, we show how the data under consideration can be modeled as in the standard BSS problem. In Section IV, we describe the proposed approximation of the real-valued PARAFAC for complex-valued data. In Section V, we present the results obtained obtained using the stICA and our real-valued PARAFAC approximation, and in Section VI, the conclusions are drawn. II. C ONCEPTS ON S OMATOSENSORY C ORTEX M EASUREMENTS Generally in cortical Local Field Potential (LFP) recordings, linearly spaced electrodes are inserted perpendicularly to the cortical surface. The neocortex of the mammalian is composed of six well identifiable layers, each with specific neuronal population cell bodies. The cells that contribute the most to the signal recorded are pyramidal cells [12], which receive this name due to their triangular shaped cell bodies. These cells have elongated projections to receive information from other neurons called apical dendrites, as opposed to other projections with the same function that are not elongated, the basal dendrites. In this sense, the recorded signals represent a superposition of signals originated from individual populations of neurons. Such populations belongs to a similar location and extension. The Excitatory Postsynaptic Potentials (EPSP) and the Inhibitory Postsynaptic Potentials (IPSP) take place along the neurons [12][19], and the connection between two neurons is the synapse. In the EPSP, positive ions enter the cell through the cell membrane in the location of the excitatory synapse, while in IPSP negative ions

664

enter, or positive ions leave the cell through the cell membrane in the location of the inhibitory synapse [19]. Therefore, a current sink is created in the location of an excitatory synapse, which is compensated by a distributed current source along the soma dendritic membrane, and a current source is created at the location of an inhibitory synapse, which is compensated by a distributed current sink along the soma dendritic membrane. Since the conventional current flows out the extracellular space in the location of the current sink, which can be thought as positive charge leaving the extracellular space, the extracellular potential at this location becomes more negative. Since the conventional current flows in the extracellular space in the location of the current source, which can be thought as positive charge entering the extracellular space, the extracellular potential at this location becomes more positive. The superposition of similarly distributed currents in different neurons of the same population at the same time results in a higher power signal [20]. In the experiment under consideration, our goal is to analyze the evoked potentials (EPs). Since the EPs signals have a very small signal to noise ratio (SNR), several realizations of the same event with stimulus should be repeated and the EPs signals are averaged. For instance, suppose an experimental setting in which a rat received the same sensory stimulus every τ seconds in a single recording section. To isolate the evoked potential in one electrode, the continuously recorded data from that electrode has to be subdivided into τs epochs, i.e. pieces of signals of the same length in which a stimulus happened at the same instant after the beginning of the signal. Then, all these windows of the signals are averaged. Although some criticism can be made to this procedure, this is the method generally used to isolate evoked potentials [21]. A standard tool for the analysis of multielectrode Local Field Potentials has been the Current Source Density (CSD) analysis, which allows to estimate the net volume density of current entering or leaving the extracellular space at different locations [13]. Consider an arbitrarily small volume of the brain, located at point (x, y, z), where x, y and z are the cartesian coordinates. The net transmembrane current contributed by all cellular elements in this volume, or the Current Source Density (CSD), is related to the extracellular field potential φ(x, y, z) by the Poisson equation i(x, y, z) = −σ∇2 φ(x, y, z), where the conductivity σ is considered constant and the same in all directions and ∇2 is the laplacian operator [6] [12]. It is generally assumed that most variation of neural activity is in the vertical direction, due to the laminar structure of the cortex [22] [12]. Therefore, the CSD can be estimated by LFP recordings from linearly spaced electrodes inserted perpendicularly to the cortical surface as i(z) ≈ −σ

φ(z + ) − 2φ(z) + φ(z − ) , 2

(1)

where  is the spacing between vertical electrodes. III. DATA M ODEL From now on, current sources and sinks are referred as current sources, with a sink being a source with negative value. Following the description in [14], neglecting inductive effects and for a fixed time interval, the potential at sensor p at instant n is due to Q groups of synchronized point current sources, called aggregates, xp (n) =

Q X q=1

hpq sq (n) + vp (n),

(2)

where sq (n) is the time course common to the point current sources of the qth aggregate, hpq is a weight that depends on the magnitudes of the point current sources belonging to the qth aggregate and their location with respect to the pth sensor, and vp (n) is the noise at sensor p at instant n. Note that the synchronous point current sources of the qth aggregate do not need to be from the same population. According to [9], they can be from spatially distinct neuron populations of separate cortical lamina if their transmembrane currents substantially covary over time. With a slight change in notation, (2) can be rewritten as xpn =

Q X

hpq sqn + vpn ,

(3)

q=1

with p = 1, ..., P and n = 1, ..., N , which allows for the representation Q X X= hq sq + V, (4) q=1

where X is a P × N matrix with elements xpn , hq is a P × 1 column vector with elements hpq , sq is a 1 × N row vector with elements sqn , and V is a P × N matrix with elements vpn . This is equivalent to [17] X = HS + V, (5) where xp are the rows of the P × N matrix X, hq are the columns of the P × Q matrix H and sq are the rows of the Q × N matrix S. Our objective is to estimate vectors hq and sq so that we can plot the CSD figure for each aggregate q separately. In this work, whenever CSD analysis is performed before BSS, it will be considered as a preprocessing step, i.e. high pass filtering of the columns of X. IV. P ROPOSED APPROXIMATION OF THE REAL - VALUED PARAFAC FOR COMPLEX - VALUED DATA By applying a Time Frequency Analysis (TFA) approach and assuming the PARAFAC data model, the data in (2) is rewritten in the following tensor fashion, ypik =

Q X

˜ pq diq gkq + apik , h

(6)

q=1

where ypik is the tensor gotten after performing TFA on the signal X at tensor p evaluated at time i and frequency k, gkq is the qth component frequency domain representation, composed of complex numbers, whose amplitude is temporally modulated by the coefficients diq , and whose resulting time-frequency representation ˜ pq for sensor p. In addition, apik is the time-frequency is scaled by h representation of the noise at sensor p evaluated at time n and frequency k. In order to perform the complex valued PARAFAC model described by [23] to analyze each electrode signal, in this work, we adopt the Short Time Fourier Transform (STFT) to perform the TFA and to obtain a three way Y with elements ypik . Considering (6), each element ypik is a complex number composed of magnitude and phase, ypik = |ypik |ej∠ypik .

|ypik |ej∠ypik =

Q X

˜ pq ||diq ||gkq |ejθpikq + |apik |ej∠apik . |h

(7)

q=1

˜ pq |, |diq |, |gkq | and |apik | are all positive and θpikq is the where |h phase of the element of the qth component with indexes p, i and k.

665

˜ pq , Note that θpikq is the sum of the phases of the three variables h dpq and gkq . Empirically, for the data used in this paper, we have found that the following approximation holds for (7) |ypik | ≈

Q X

˜ pq ||diq ||gkq | + |apik |. |h

(8)

q=1

Given the approximation in (8), the real-valued PARAFAC decomposition can be first applied to the tensor |Y|, whose each element corresponds to |ypik |. Once the Q real-valued rank one tensors are obtained, their phases are obtained by simply using the phases of the elements of ej∠ypik . Finally, each complex-valued rank one tensor is then converted to the time domain by applying the overlap algorithm in [24]. V. R ESULTS In the experiment considered in this work, a mouse received the same sensory stimulus every τ = 7 seconds while Local Field Potentials were recorded from its somatosensory cortex with 32 electrodes spaced  = 50 micrometers from each other perpendicularly to the cortical surface. The signals were acquired using a sampling rate of 40 kHz and a referential montage. The proposed approach considered the voltage time courses from -0.005s to 0.095s, except for stICA, which was applied after CSD analysis. Third-party codes used were: the stICA from J.V. Stone and J. Porrill [25]; the runica function, part of the EEGLAB toolbox [26]; the N-WAY toolbox by Rasmus Bro [27], and the complex valued PARAFAC code from Nicholas D. Sidiropoulos and Rasmus Bro [28]. In Figure 1, four CSD analyses are depicted. The first one corresponds to the original signals, i.e. without applying any blind source separation approach. The second and third CSD analyses correspond to the first and second components with greatest power obtained via the stICA based on [11]. Finally, the fourth CSD analysis corresponds to the sum of the two components obtained via stICA. According to the first stICA component, the bottom layers are more associated to each other. According to the second stICA component, the top layers are more associated to each other. In [11] α ∈ [0, 1] quantifies how much weight we attribute to the temporal and spatial independence. We verified that stICA with α = 1, i.e. purely spatial decomposition, applied after CSD analysis returns good results. In order to apply proposed technique from Section IV to the measured data, each electrode signal is analyzed using STFT with a Kaiser window of duration N = 400 samples and short time analyses being performed every J = 10 samples. After obtaining a three way complex-valued data composed of the electrodes by time by frequency dimensions, the method proposed in Section IV. Figure 2 depicts the original signal, the two components with greatest power and the recovered signal by adding the two components. Note that the results obtained in Figure 2 are consistent with the ones obtained in Figure 1 using stICA.

Fig. 1. Four CSD analyses: first CSD analysis of the original measurements, second CSD analysis of the first component with greatest power obtained with stICA, third CSD analysis of the second component with greatest power obtained with stICA, and fourth CSD analysis with the sum of the two components obtained with stICA

Fig. 2. Four CSD analyses: first CSD analysis of the original measurements, second CSD analysis of the first component with greatest power obtained with real-valued PARAFAC approximation from Section IV, third CSD analysis of the second component with greatest power obtained with realvalued PARAFAC approximation from Section IV, and fourth CSD analysis with the sum of the two components obtained with real-valued PARAFAC approximation from Section IV

VI. C ONCLUSION In this work, we propose the analysis of the somatosensory cortex signals of a mouse by applying BSS schemes. We also propose the use of multi-way analysis techniques to study recordings from a one dimensional grid of electrodes. For that, besides electrodes and time, another dimension is necessary. This can be achieved by means of Time-Frequency Analysis techniques. We proposed

an approximation in order to apply the PARAFAC decomposition into the measured data. The obtained results are consistent with the results obtained using the stICA. Furthermore, as a future work, the proposed method could be applied to in vitro microelectrode array measurement analysis and more analyses should be performed considering different values of J and N .

666

ACKNOWLEDGMENT The authors thank Mr. Lucas Silva Lopes for supporting the research presented in this paper and also Prof. Patrik Krieger from Ruhr-University Bochum for providing them the measurements. This work has only been possible with the support of the RuhrUniversity Bochum via the Research Explorer Ruhr (RER) 2015 and the support of Brazilian research and innovation agencies FAPDF (Research Support Foundation of the Federal District), Coordenao de Aperfeioamento de Pessoal de Nvel Superior (CAPES) under the PVE grant number 88881.030392/2013-01, the productivity grant number 303905/2014-0, the postdoctoral scholarship abroad (PDE) number 207644/2015-2, the joint double degree scholarship number 88887.115692/2016-00, and the Program Science without BordersAerospace Technology supported by CNPq and CAPES. R EFERENCES [1] L. S. Lopes, “Blind source separation applied to the analysis of rat somatosensory evoked response,” Bachelors’s thesis, University of Brasilia, 2016. [2] T. H. Bullock, “Problems in the comparative study of brain waves,” Yale J Biol Med, vol. 17, no. 5, pp. 657–680.3, 1945. [3] R. Galambos, “Cochlear potentials from the bat,” Science, vol. 93, p. 215, 1941. [4] W. H. Marshall, C. N. Woolsey, and P. Bard, “Cortical representation of tactile sensibility as indicated by cortical potentials,” Science, vol. 85, pp. 388–390, 1937. [5] W. H. Pitts, “Investigations on synaptic transmission,” in Cybernetics, Trans. 9th Conf. Josiah Macy Foundation H. von Foerster, New York, 1952, pp. 159–166. [6] C. Nicholson and J. A. Freeman, “Theory of current source-density analysis and determination of conductivity tensor for anuran cerebellum,” Journal of Neurophysiology, vol. 38, no. 2, pp. 356–368, 1975. [7] ——, “Experimental optimization of current source-density technique for anuran cerebellum,” Journal of Neurophysiology, vol. 38, no. 2, pp. 369–382, 1975. [8] E. Bellistri, J. Aguilar, and L. M. d. J. R. Brotons-Mas, G. Foffani, “Basic properties of somatosensory-evoked responses in the dorsal hippocampus of the rat,” J Physiol., vol. 591, no. 10, pp. 2667–2686, 2013. [9] D. S. Barth, S. Di, and C. Baumgartner, “Laminar cortical interactions during epileptic spikes studied with principal component analysis and physiological modeling,” Brain Research, vol. 484, no. 1-2, pp. 13–35, 1989. [10] V. A. Makarov, J. Makarova, and O. Herreras, “Disentanglement of local field potential sources by independent component analysis,” Journal of Computational Neuroscience, vol. 29, no. 3, pp. 445–457, 2010. [11] S. Leski, E. Kublik, D. A. Swiejkowski, A. Wrobel, and D. Wojcik, “Extracting functional components of neural dynamics with independent component analysis and inverse current source density,” Journal of Computational Neuroscience, vol. 29, no. 3, pp. 459–473, 2012. [12] G. T. Einevoll, C. Kayser, N. K. Logothetis, and S. Panzeri, “Modeling and analysis of local field potentials for studying the function of cortical circuits,” Nature Reviews Neuroscience, vol. 14, no. 11, pp. 770–785, 2013. [13] H. T. Glabska, E. Norheim, A. Devor, A. M. Dale, G. T. Einevoll, and D. K. Wojcik, “Generalized laminar population analysis (gLPA) for interpretation of multielectrode data from cortex,” Frontiers in Neuroinformatics, vol. 10, pp. 1–15, 2016. [14] J. Mocks, “Decomposing event-related potentials: A new topographic components model,” Biological Psychology, vol. 26, no. 1-3, pp. 199– 215, June 1988. [15] M. Weis, “Multi-dimensional signal decomposition techniques for analysis of EEG data,” Ph.D. dissertation, Ilmenau University of Technology, 2015. [16] D. Lahat, T. Adali, and C. Jutten, “Multimodal data fusion: An overview of methods, challenges, and prospects,” Proceedings of the IEEE, vol. 103, no. 9, pp. 1449–1477, August 2015. [17] R. Bro, “Multiway analysis in the food industry,” Ph.D. dissertation, Universiteit van Amsterdam, 1998.

[18] F. Miwakeichi, E. Martnez-Montes, P. A. Valds-Sosa, N. Nobuaki, H. Mizuhara, and Y. Yamaguchi, “Decomposing EEG data into spacetime-frequency components using parallel factor analysis,” Neuroimage, vol. 22, no. 3, pp. 1035–1045, 2004. [19] F. Lopes da Silva, “EEG: Origin and measurement,” in EEG - fMRI: Physiological Basis, Technique, and Applications, C. Mulert and L. Lemieux, Eds. Springer Science and Business Media, 2009, pp. 19–38. [20] R. Nieuwenhuys, H. Donkelaar, and C. Nicholson, “The Central Nervous System of Vertebrates, Volume 1,” vol. 1, 1998. [21] K. Blinowska and P. J. Durka, “Electroencephalography (EEG),” in Wiley Encyclopedia of Biomedical Engineering, M. Akay, Ed. Willey, 2006. [22] D. K. Wojcik, “Current source density (CSD) analysis,” in Encyclopedia of Computational Neuroscience, D. Jaeger and R. Jung, Eds. New York: Springer, 2014. [23] R. Bro, N. Sidiropoulos, and G. Giannakis, “A fast least squares algorithm for separating trilinear mixtures,” Proc. First International Conference on Independent Component Analysis and Blind Source Separation ICA’99, no. 2, pp. 289–294, 1999. [24] J. B. Allen and L. R. Rabiner, “A unified approach to short-time Fourier analysis and synthesis,” Proceedings of the IEEE, vol. 65, no. 11, pp. 1558–1564, 1977. [25] J. Stone and J. Porrill, “Regularisation using spatiotemporal independence and predictability,” Computational Neuroscience Report, vol. 201, 1999. [26] A. Delorme and S. Makeig, “EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics,” Journal of Neuroscience Methods, vol. 134, pp. 9–21, 2004. [27] C. A. Andersson and R. Bro, “The n-way toolbox for matlab,” Chemometrics and Intelligent Laboratory Systems., vol. 52, no. 1, pp. 1–4, 2000. [28] R. Bro, N. Sidiropoulos, and G. Giannakis, “A fast least squares algorithm for separating trilinear mixtures,” in Proc. ICA99 - Int. Workshop on Independent Component Analysis and Blind Separation, Aussois, France, Jan. 1999, pp. 11–15.

667