ARTICLE IN PRESS

Neurocomputing 69 (2006) 1108–1111 www.elsevier.com/locate/neucom

Spatiotemporal clustering of synchronized bursting events in neuronal networks Uri Barkan, David Horn School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel Available online 13 March 2006

Abstract In vitro neuronal networks display synchronized bursting events (SBEs), with characteristic temporal width of 100–500 ms and frequency of once every few seconds. We analyze such data using preprocessing by SVD for dimensional reduction, after which we apply quantum clustering to sort out the SBEs into different groups. Each SBE is described within a spatiotemporal template, allowing us to distinguish between neuronal activities (firing rates) and relative temporal variations of the spiking of each neuron within the SBE. Clustering assignments according to the two different kinds of information are very different from one another. SBEs may thus serve as carriers of different information in these two different implementations of the neuronal code. r 2006 Elsevier B.V. All rights reserved. Keywords: Synchronized bursting events; Clustering; Spatiotemporal coding

1. Introduction In vitro neuronal networks display synchronized bursting events (SBEs), with characteristic temporal width of 100–500 ms and frequency of once every few seconds. These events can be registered over a period of many hours. We start our analysis of these experiments with preprocessing by singular value decomposition (SVD) or principal component analysis (PCA), applied to a matrix of neuronal activity vs SBE number, i.e. a matrix whose ith row specifies the number of spikes of the ith neuron for each burst. This has been used by Elhalal and Horn [1] who have demonstrated characteristic changes that take place over time scales of hours. They have applied simple clustering to the data in the reduced dimensions of the first few principal components. Here we extend this investigation in two directions. We distinguish between firing rate and temporal information and analyze these data separately, using the Quantum Clustering (QC) method [2] to reveal underlying structures.1 Corresponding author. Tel.: +972 3 6429305; fax: +972 3 6407932.

E-mail address: [email protected] (D. Horn). Available software for SVD and QC can be found at http:// adios.tau.ac.il/compact. 1

0925-2312/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2005.12.121

The data we analyze are from eight experiments carried out in the laboratory of Ben Jacob [3,4]. The experiments consist of registering the electrical activity of in vitro neuronal networks that are derived from cortical regions of rats, and are allowed to self-assemble into an active neuronal network for about a week, which is when the SBE activity is observed. We first process the data to select the SBEs. We define an SBE by setting a threshold to the total activity (namely the number of spikes in a certain time window) within a given temporal bin of 10 ms. Then, in order to define the SBE raster plot, we determine its beginning by the point where the total activity is 15 of the burst’s peak. The end of the SBE is defined by the time when the activity decreases to the same threshold. On the average we find 2000 SBEs in each experiment. All SBEs were fit into a spatiotemporal template determined by the SBE with the longest time span, such that all peaks are set at the same time and zeros are added to the prefix and suffix of each temporal sequence defining an SBE. Next we apply SVD processing to these data to achieve dimensional reduction. Let X denote an m
n matrix of real-valued data, e.g. firing rates of m neurons measured for n SBEs along the time-span of one experiment. The

ARTICLE IN PRESS U. Barkan, D. Horn / Neurocomputing 69 (2006) 1108–1111

singular value decomposition of X is

2. Clustering of firing rate data

X ¼ USV T

(1)

where S is diagonal, and U and V are orthonormal matrices. Consider a column vector in X . Clearly, it can be reconstructed from a linear combination of column vectors of U. Dimensional reduction means we limit ourselves to just a few of them, e.g. the first three columns of U. Afterward, we normalize the projections to the unit sphere, so that the squares of the three projections sum up to one. Within this three-dimensional representation we perform clustering using the quantum clustering (QC) method [2]. This method is based on a potential function 2 ðs2 =2Þr2 c d 1 X 2 V ðxÞ ¼ ¼ þ 2 ðx  xi Þ2 eðxxi Þ =2s , c 2 2s c i

(2) where cðxÞ ¼

X

eðxxi Þ

2

1109

=2s2

(3)

i

is the scale-space probability function (depending on a width-parameter s) associated with the data points located at xi in a d-dimensional space. Data points that fall within different valleys of the potential are assigned to different clusters. This is done by applying gradient-descent to the points, letting them fall to the nearest minimum of the potential. To assure good separability of the different clusters we have selected points within the bottom half of each valley, regarding all others as outliers. An example of the potential for one of the experiments (231000, this number indicating the date when the experiment was recorded) is shown in Fig. 1.

Firing rate data are obtained from our SBE representation by summing over time within each SBE. We reproduce the clusters of [1] in the three experiments analyzed there, and usually obtain three clusters in each of our eight experiments. However, whereas [1] have observed that different clusters occur at different times over the span of the experiment, we find that this is not a general characteristic. In four of the experiments that we have analyzed we observed a uniform temporal distribution, i.e. bursts belonging to a specific cluster may occur anytime within the many hours of the experiment. 3. Clustering of spatiotemporal data Now we turn to spatiotemporal data, applying our SVD procedure and clustering to a matrix whose columns carry the spatiotemporal information of the SBEs. The QC potential of one of the experiments is demonstrated in Fig. 1. The data in the three valleys of this potential are assigned to the three different clusters. The first important observation is that the assignment of SBEs to clusters according to the spatiotemporal structure is different, almost orthogonal, to the assignments derived from the firing-rate analysis. Hence the two different representations seems to carry different information. Next we wish to point out that all spatiotemporal clusters seem to have uniform distribution along the time scale of the experiment. Since the clusters were derived from the SVD reduced representation, we return now to the original data and ask whether the spatiotemporal clustering can be observed directly in it. We randomly select 100 SBEs from each one of the three clusters, and measure the Pearson correlations between their original raster plots. The results, shown in Fig. 2, demonstrate that correlations within the clusters are

231000 - Potential of Quantum Clustering 231000 - Pearson Correlation 1 0.4

V

0.8

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0.35 0.3

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di

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m

1

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0 dimension 3

-0.5

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-1

Fig. 1. The values (x) of the QC potential function for data points (dots) displayed in a plane spanned by the second and the third principal components of SVD.

0

300 50

100

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300

Fig. 2. Pearson correlations between the raster plots of SBEs assigned to the three clusters corresponding to the bottom halves of the three valleys in Fig. 1.

ARTICLE IN PRESS U. Barkan, D. Horn / Neurocomputing 69 (2006) 1108–1111

1110

270800 - cluster 2

231000 - Neuron #9

0.4 Activity level

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neuron # 4 neuron # 6

0.3 0.2 0.1

0 250 260 270 280 290 300 310 320 330 340 350 time [ms] 0.06 270800 - cluster 3 0.25 neuron # 4 neuron # 6

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Fig. 3. (a) Left: Activity characteristics of a particular neuron in the SBEs belonging to the three different clusters display clearly different profiles. (b) Right: Average profiles of two neurons in another experiment show different relative phases depending on the clusters.

significantly higher than correlations between SBEs that belong to different clusters, i.e. the clustering selection is indeed meaningful. The dimensional reduction, while very helpful in the clustering analysis, did not distort the important features in the data. To exemplify the differences between the spatiotemporal clusters we display in Fig. 3a the average activity of a specific neuron within the SBEs of three different clusters in one of the experiments. Clearly this profile is strongly cluster dependent. Moreover, we observe different interneuron relations in different clusters. As an example we display in Fig. 3b the profiles of two neurons in two different clusters, taken from another experiment, showing synchrony in one cluster and out-of-phase behavior in the other. The variation displayed in these two figures is specific to particular neurons or neuron pairs; other neurons may not show different behavior in different clusters. The extreme variation of neuron number 9 in Fig. 3a, displayed here for the average over all SBEs in each cluster, implies that similar differences are present also between characteristic individual SBEs belonging to different clusters.

this issue took place in 1995 between Softky [6], who claimed that temporal coding is much more efficient than rate coding, hence evolution was supposed to prefer it, and Shadlen and Newsome [5] who pointed out that there was no clear biological evidence found for the more complex temporal coding. In our case, we analyze experiments in which neurons self-assemble and act without any obvious external stimuli. Hence the information carried by their firing patterns cannot be specified because it is not clear what it should be associated with. We may regard the different clusters as representing candidates for neuronal cell assemblies that may be used as such if and when the neuronal system becomes part of a general information-carrying organism. Our contribution to the important neuronal code problem is that we observe independent clusters in the two different modes of firing rate and temporal variation within SBEs. In other words, information may be simultaneously and independently carried by these two different modes. Thus the two different codes may coexist within the same system.

Acknowledgments 4. Discussion The question may arise whether our results can contribute to understanding the nature of the neural code, i.e. is information carried only by neuronal firing rate or also by the timing of its spikes. A well-known debate on

We would like to thank Nadav Raichman and Itay Baruchi for sharing the data and for fruitful discussions. This research was supported by the Israel Science Foundation and by the Adams Super Center for Brain Studies at Tel Aviv University.

ARTICLE IN PRESS U. Barkan, D. Horn / Neurocomputing 69 (2006) 1108–1111

References [1] A. Elhalal, D. Horn, In vitro neuronal networks: evidence for synaptic plasticity, Neurocomputing 65–66 (2005) 31–34. [2] D. Horn, A. Gottlieb, Algorithm for data clustering in pattern recognition problems based on quantum mechanics, Phys. Rev. Lett. 88 (2002) 018702. [3] E. Hulata, R. Segev, Y. Shapira, M. Benveniste, E. Ben-Jacob, Detection and sorting of neural spikes using wavelet packet, Phys. Rev. Lett. 85 (2000) 4637–4640. [4] R. Segev, Self-organization of in-vitro neuronal networks, Ph.D. Thesis, Tel Aviv University, Israel, 2002. [5] M.N. Shadlen, W.T. Newsome, Is there a signal in the noise?, Current Opin. Neurobiol. 5 (1995) 248–250. [6] W.R. Softky, Simple codes versus efficient codes, Current Opin. Neurobiol. 5 (1995) 239–247. Uri Barkan received the B.Sc. (2003) and M.Sc (2005) degrees from Tel Aviv University, Tel Aviv, Israel. Since 2005 he works in the R&D department of Elisra Electronic Systems, Ltd. His research interests include computational physics, pattern recognition, classification methods, theory and applications of neural networks and quantum information theory.

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David Horn received his Ph.D. in Physics from the Hebrew University in Jerusalem in 1965. Since 1971 he has been Professor of Physics at Tel Aviv University. From 1993 to 2000 he has served as Director of the Adams Super Center for Brain Studies at Tel Aviv University. His current research interests are development and use of clustering methods, and motif and syntax extraction from texts and from biological sequences.