This article proposes a robust brain-inspired audio feature extractor (RBA-FE) model for depression diagnosis, using an improved hierarchical network architecture. Most deep learning models achieve state-of-the-art performance for image-based diagnostic tasks, ignoring the counterpart audio features. In order to tailor the noise challenge, RBA-FE leverages six acoustic features extracted from the raw audio, capturing both spatial characteristics and temporal dependencies. This hybrid attribute helps alleviate the precision limitation in audio feature extraction within other learning models like deep residual shrinkage networks. To deal with the noise issues, our model incorporates an improved spiking neuron model, called adaptive rate smooth leaky integrate-and-fire (ARSLIF). The ARSLIF model emulates the mechanism of ``retuning of cellular signal selectivity" in the brain attention systems, which enhances the model robustness against environmental noises in audio data. Experimental results demonstrate that RBA-FE achieves state-of-the-art accuracy on the MODMA dataset, respectively with 0.8750, 0.8974, 0.8750 and 0.8750 in precision, accuracy, recall and F1 score. Extensive experiments on the AVEC2014 and DAIC-WOZ datasets both show enhancements in noise robustness. It is further indicated by comparison that the ARSLIF neuron model suggest the abnormal firing pattern within the feature extraction on depressive audio data, offering brain-inspired interpretability.