In this paper, we develop methods for efficient and accurate information extraction from calcium-imaging-based neural signals. The particular form of information extraction we investigate involves predicting behavior variables linked to animals from which the calcium imaging signals are acquired. More specifically, we develop algorithms to systematically generate compact deep neural network (DNN) models for accurate and efficient calcium-imaging-based predictive modeling. We also develop a software tool, called NeuroGRS, to apply the proposed methods for compact DNN derivation with a high degree of automation. GRS stands for Greedy inter-layer order with Random Selection of intra-layer units, which describes the central algorithm developed in this work for deriving compact DNN structures. Through extensive experiments using NeuroGRS and calcium imaging data, we demonstrate that our methods enable highly streamlined information extraction from calcium images of the brain with minimal loss in accuracy compared to much more computationally expensive approaches.