Cross-lingual chain-of-thought prompting techniques have proven effective for investigating diverse reasoning paths in Large Language Models (LLMs), especially for low-resource languages. Despite these empirical gains, the mechanisms underlying cross-lingual improvements remain perplexing. This study, therefore, addresses whether the benefits of cross-lingual prompting arise from language-specific reasoning structures intrinsic to each language, or are simply a consequence of improved comprehension through cross-linguistic exposure. We employ neuron intervention and perturbation techniques to analyze and deactivate language-specific reasoning neurons during cross-lingual prompting, leading to performance disparities across languages, up to 27.4%. Our findings disentangle that these neurons are essential for reasoning in their respective languages, but have minimal effect on reasoning in other languages, providing evidence for the existence of language-specific local reasoning structures and guiding the development of more interpretable and effective multilingual AI systems.