Robust entropy rate estimation for nonstationary neuronal calcium spike trains based on empirical probabilities

Objective. Temporal patterns in neuronal spiking encode stimulus uncertainty, and convey information about high-level functions such as memory and cognition. Estimating the associated information content and understanding how that evolves with time assume significance in the investigation of neuronal coding mechanisms and abnormal signaling. However, existing estimators of the entropy rate, a measure of information content, either ignore the inherent nonstationarity, or employ dictionary-based Lempel-Ziv (LZ) methods that converge too slowly for one to study temporal variations in sufficient detail. Against this backdrop, we seek estimates that handle nonstationarity, are fast converging, and hence allow meaningful temporal investigations.Approach. We proposed a homogeneous Markov model approximation of spike trains within windows of suitably chosen length and an entropy rate estimator based on empirical probabilities that converges quickly.Main results. We constructed mathematical families of nonstationary Markov processes with certain bi/multi-level properties (inspired by neuronal responses) with known entropy rates, and validated the proposed estimator against those. Further statistical validations were presented on data collected from hippocampal (and primary visual cortex) neuron populations in terms of single neuron behavior as well as population heterogeneity. Our estimator appears to be statistically more accurate and converges faster than existing LZ estimators, and hence well suited for temporal studies.Significance. The entropy rate analysis revealed not only informational and process memory heterogeneity among neurons, but distinct statistical patterns in neuronal populations (from two different brain regions) under basal and post-stimulus conditions. Taking inspiration, we envision future large-scale studies of different brain regions enabled by the proposed tool (estimator), potentially contributing to improved functional modeling of the brain and identification of statistical signatures of neurodegenerative diseases.