Enumerating longest increasing subsequences and patience sorting
Information Processing Letters
Longest increasing subsequences in sliding windows
Theoretical Computer Science
Probability of repeating patterns in simultaneous neural data
Neural Computation
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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 j, a closed-form solution is derived. For n ≥ 2j, 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.