State-of-the-art algorithms for visual place recognition, and related visual navigation systems, can be broadly split into two categories: computer-science-oriented models including deep learning or image retrieval based techniques with minimal biological plausibility, and neuroscience-oriented dynamical networks that model temporal properties found in neural cells underlying spatial navigation in the brain. In this paper, we propose a new compact and high-performing place recognition hybrid model that bridges this divide for the first time. Our approach comprises two key components that incorporate neural models of these two categories: (1) FlyNet, a compact, sparse twolayer neural network inspired by brain architectures of fruit flies, Drosophila melanogaster, and (2) a one-dimensional continuous attractor neural network (CANN). The resulting FlyNet+CANN network combines the compact pattern recognition capabilities of our FlyNet model with the powerful temporal filtering capabilities of an equally compact CANN, replicating entirely in a hybrid neural implementation the functionality that yields high performance in algorithmic localization approaches like SeqSLAM. We evaluate our approach, and compare it to three state-of-the-art place recognition methods, on two benchmark real-world datasets with small viewpoint variations and extreme environmental changes; including day/night cycles where it achieves an AUC performance of 87% compared to 60% for Multi-Process Fusion, 46% for LoST-X and 1% for SeqSLAM, while being 6.5, 310, and 1.5 times faster respectively.