Spatiotemporal clustering of synchronized bursting events in neuronal networks Uri Barkan, David Horn, Tel-Aviv University Introduction
Email:
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We study recordings of spiking neuronal networks from the laboratory of Prof. Eshel Ben-Jacob1. The cell cultures consist of living cells from the cortex of one-day-old rats, composed of both neurons and glial cells. The method enables to extracellularly record the electrical activity of up to dozens of neurons. The network fires in Synchronized Bursting Events (SBEs). The sampling rate is 12 KHz.
This time, no clear correlation to the time in the experiment was found.
Results
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In this example, three clusters were found. A remarkable correlation was found2 between the time of occurrence of the SBEs that belong to the same cluster.
Spatial Picture
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Some network features: • Size: ~ 1 cm2. • Sampling duration: ~ hours. • Number of neurons in the network: ~ 106. • Number of neurons recorded: 12-60. • Duration of mean SBE: ~ 300 ms. • Mean duration between SBEs: ~ 10 sec.
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Spatiotempotal presentation– An instance is composed of a raster plot of a burst (Fig1, upper plot). The matrix of all bursts is constructed by reshaping each raster plot into a long vector. Rate presentation– An instance is a PST histogram of the raster plot, i.e. a count of the spikes of each neuron in each burst. (Fig. 1, bottom plot) 10
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Fig. 2. Applying SVD to the rate matrix - Here are the first three dimensions (normalized)
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Fig. 8. The times of the bursts in the experiment, according to the spatiotemporal cluster they belong to. Spatiotemporal clusters do not exhibit any relation to the temporal order of the bursts.
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Two presentation of the data were used:
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Fig. 5. The times of the bursts in the experiment, according to the cluster they belong to. Clearly the clusters in the rate presentation are related to temporal ordering of the bursts.
Pearson Correlation was computed for every two bursts, and the correlation matrix was arranged so that the bursts of each cluster would appear as a group (in a random order).
Spatiotemporal Picture
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Fig. 1: Spatiotemporal and rate presentations
Methods
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SVD – Singular Value Decomposition. Definition: Every (m x n; m>>n) matrix X can be represented as: X= USVt where S is diagonal and U and V are orthonormal (UtU = Im, VtV = In). Truncates S to r leading eigenvalues. Properties: Captures the ‘real’ characteristics of the data. Eliminates noise and serves as method for dimensional reduction.
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Fig. 3. Applying Quantum Clustering to the SVD picture
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Fig. 6. Applying SVD to the spatiotemporal matrix - again, here are the first three dimensions (normalized) Cluster 1 Cluster 2 Cluster 3 Outliers
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Fig. 9. Pearson Correlation matrix for all the bursts, organized by clusters. The fourth frame corresponds to the outliers
• SVD was found to be a useful method for dimensional reduction. • Quantum Clustering succeeded in finding meaningful clusters. • Rate and spatiotemporal presentations are not correlated, and divide the data into different groups. While the spatial picture is highly correlated to the time of the burst, the spatio-temporal clusters are distributed uniformly along the time axis.
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References
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Quantum Clustering3 – For each data point: • An estimator of the probability distribution (one free parameter) • A potential energy function (minima = cluster centers) • Find the minima by gradient descent (no need to predetermine the number of clusters)
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Conclusions
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Figure based on Wall et.al. (Practical Approach to Microarray Data Analysis, 03)
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Fig. 4. The mean PST histogram (i.e. rate presntation) of each Cluster.
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Fig. 7. Applying Quantum Clustering to the SVD picture
1. Ronen Segev, Self wiring of neural networks, School of Physics and Astronomy, The Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, PhD Thesis, 1998. 2. Anat Elhalal and David Horn, In-vitro neuronal networks: evidence for synaptic plasticity. To be published in proceedings of CNS04. 3. COMPACT - Comparative Package for Clustering Assessment http://adios.tau.ac.il/compact/
