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A Set Probability Technique for Detecting Relative Time Order Across Multiple Neurons

annesmith{at}ucdavis.edu Department of Anesthesiology and Pain Medicine, University of California at Davis, Davis, CA 95616, U.S.A.

Peter Smith

p.smith{at}maths.keele.ac.uk Department of Mathematics, University of Keele, Keele, Staffordshire, ST5 5BG, U.K.

With the development of multielectrode recording techniques, it is possible to measure the cell firing patterns of multiple neurons simultaneously, generating a large quantity of data. Identification of the firing patterns within these large groups of cells is an important and a challenging problem in data analysis. Here, we consider the problem of measuring the significance of a repeat in the cell firing sequence across arbitrary numbers of cells. In particular, we consider the question, given a ranked order of cells numbered 1 to N, what is the probability that another sequence of length n contains j consecutive increasing elements? Assuming each element of the sequence is drawn with replacement from the numbers 1 through N, we derive a recursive formula for the probability of the sequence of length j or more. For n < 2j, a closed-form solution is derived. For n 2 j, we obtain upper and lower bounds for these probabilities for various combinations of parameter values. These can be computed very quickly. For a typical case with small N (<10) and large n (<3000), sequences of 7 and 8 are statistically very unlikely. A potential application of this technique is in the detection of repeats in hippocampal place cell order during sleep. Unlike most previous articles on increasing runs in random lists, we use a probability approach based on sets of overlapping sequences.

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