Statistical Definition of Bursting in Neural Spike Trains by Means of Novelty and Surprise Measure

We present a method for calculating the statistical significance ofbursts in a spike train relative to a 0-hypothesis of independent Poisson processes. Mathematically, the method is based on the calculation ofnovelty and surprise for bursts as defined by Palm (2012) [1,2], where a’burst’ signifies the event that a large number of spikes have occurred ina short interval of time.Let (X_k) k∈ Z be a sequence of positive random variables (inter-spike-intervals)and X_{i,k} = \sum_{j=0}^{i-1} X_{k−j} for 1 ≤ i. Obviously the distribution of X_{i,k} isthe same for each k and X_{1,k} = X_k . Let F_i(t) = P[X_{i,0} < t], we define the burst novelty at spike k as:N_k := − log_2 [min F_i (X_{i,k} )].Again the distribution of N_k is the same for each k with cumulative distribution function F , i.e. F (x) = P[N_0< x]. Now we can define the surprise function as:S: R^+ → R^+,S : x → − log_2(1 − F (x))The burst surprise at spike k is defined as the random variable S_k :=S(N_k ). It is the logarithm of the significance probability of the burstnovelty at spike k.In order to apply those measures in practice to experimentally observedspike-trains, the task is to precompute the functions F_i and S eitheranalytically or by sampling. To this end we assume as 0-hypothesis that thespike trains are sampled from a Poisson process. For exponentially distributed X_k the F_i are well-known, and we have computed or estimated S by sampling.This method can be easily generalized to more general inter-spike interval distributions. In addition to the characterization of burstiness themethod may also be used to determine response latency of single neuronsor small groups of neurons based on single trials. It is related to someprevious approaches by [3,4,5].Acknowledgements. Supported by EU Grant 785907(HBP) and EU Grant720270(HBP)References[1] Palm, G. (2012). Novelty, Information and Surprise. Springer Science &Business Media.[2] Palm, G. (1981). Evidence, Information, and Surprise. Biological Cyber-netics, 42(1), 57-68.[3] Baker, S. N., & Gerstein, G. L. (2001). Determination of Response Latencyand its Application to Normalization of Cross-Correlation Measures. NeuralComputation, 13(6), 1351-1377.[4] Legendy, C. R., & Salcman, M. (1985). Bursts and Recurrences of Bursts inthe Spike Trains of Spontaneously Active Striate Cortex Neurons. Journalof Neurophysiology, 53(4), 926-939.[5] Chen, L., Deng, Y., Luo, W., Wang, Z., & Zeng, S. (2009). Detection ofBursts in Neuronal Spike Trains by the Mean Inter-Spike Interval Method.Progress in Natural Science, 19(2), 229-235