Despite deep learning's transformative impact on various domains, the reliability of Deep Neural Networks (DNNs) is still a pressing concern due to their complexity and data dependency. Traditional software fault localization techniques, such as Spectrum-based Fault Localization (SBFL), have been adapted to DNNs with limited success. Existing methods like DeepFault utilize SBFL measures but fail to account for fault propagation across neural pathways, leading to suboptimal fault detection. Addressing this gap, we propose the NP-SBFL method, leveraging Layer-wise Relevance Propagation (LRP) to identify and verify critical neural pathways. Our innovative multi-stage gradient ascent (MGA) technique, an extension of gradient ascent (GA), activates neurons sequentially, enhancing fault detection efficacy. We evaluated the effectiveness of our method, i.e. NP-SBFL-MGA, on two commonly used datasets, MNIST and CIFAR-10, two baselines DeepFault and NP- SBFL-GA, and three suspicious neuron measures, Tarantula, Ochiai, and Barinel. The empirical results showed that NP-SBFL-MGA is statistically more effective than the baselines at identifying suspicious paths and synthesizing adversarial inputs. Particularly, Tarantula on NP-SBFL-MGA had the highest fault detection rate at 96.75%, surpassing DeepFault on Ochiai (89.90%) and NP-SBFL-GA on Ochiai (60.61%). Our approach also yielded results comparable to those of the baselines in synthesizing naturalness inputs, and we found a positive correlation between the coverage of critical paths and the number of failed tests in DNN fault localization.

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