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

Sathish Ande, Srinivas Avasarala, Sarpras Swain, Ajith Karunarathne, Lopamudra Giri, Soumya Jana
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Abstract

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.

基于经验概率的非稳态神经元钙离子尖峰列车稳健熵率估计
目的:神经元尖峰振荡的时间模式编码刺激的不确定性,并传递有关记忆和认知等高级功能的信息。估算相关的信息含量并了解其如何随时间演变,对于研究神经元编码机制和异常信号具有重要意义。然而,现有的熵率估算器(一种信息含量测量方法)要么忽略了固有的非平稳性,要么采用基于字典的 Lempel-Ziv (LZ) 方法,这种方法收敛速度太慢,无法对时间变化进行足够详细的研究。在此背景下,我们寻求能够处理非平稳性、快速收敛、从而进行有意义的时间研究的估算方法:我们提出了在适当长度的窗口内对尖峰列车进行同质马尔可夫模型近似的方法,以及基于经验概率的熵率估计器,该估计器收敛速度很快:我们构建了具有某些双/多级特性(受神经元反应的启发)的非平稳马尔可夫过程数学族,这些数学族具有已知的熵率,并针对这些数学族验证了所提出的估计器。从单个神经元行为和群体异质性的角度,对从海马(和初级视觉皮层)神经元群体收集的数据进行了进一步的统计验证。 我们的估计器在统计上似乎比现有的 LZ 估计器更准确,收敛速度更快,因此非常适合时间研究:熵率分析不仅揭示了神经元之间的信息和过程记忆异质性,还揭示了神经元群(来自两个不同脑区)在基础和刺激后条件下的不同统计模式。受此启发,我们设想未来将利用所提出的工具(估计器)对不同脑区进行大规模研究,从而为改进大脑功能建模和识别神经退行性疾病的统计特征做出潜在贡献。
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