  Quantification of information content and its temporal variation in
intracellular calcium spike trains in neurons helps one understand functions
such as memory, learning, and cognition. Such quantification could also reveal
pathological signaling perturbation that potentially leads to devastating
neurodegenerative conditions including Parkinson's, Alzheimer's, and
Huntington's diseases. Accordingly, estimation of entropy rate, an
information-theoretic measure of information content, assumes primary
significance. However, such estimation in the present context is challenging
because, while entropy rate is traditionally defined asymptotically for long
blocks under the assumption of stationarity, neurons are known to encode
information in short intervals and the associated spike trains often exhibit
nonstationarity. Against this backdrop, we propose an entropy rate estimator
based on empirical probabilities that operates within windows, short enough to
ensure approximate stationarity. Specifically, our estimator, parameterized by
the length of encoding contexts, attempts to model the underlying memory
structures in neuronal spike trains. In an example Markov process, we compared
the performance of the proposed method with that of versions of the Lempel-Ziv
algorithm as well as with that of a certain stationary distribution method and
found the former to exhibit higher accuracy levels and faster convergence.
Also, in experimentally recorded calcium responses of four hippocampal neurons,
the proposed method showed faster convergence. Significantly, our technique
detected structural heterogeneity in the underlying process memory in the
responses of the aforementioned neurons. We believe that the proposed method
facilitates large-scale studies of such heterogeneity, which could in turn
identify signatures of various diseases in terms of entropy rate estimates.
