Small changes in word choice can lead to dramatically different interpretations of narratives. How does the brain accumulate and integrate such local changes to construct unique neural representations for different stories? In this study, we created two distinct narratives by changing only a few words in each sentence (e.g., "he" to "she" or "sobbing" to "laughing") while preserving the grammatical structure across stories. We then measured changes in neural responses between the two stories. We found that differences in neural responses between the two stories gradually increased along the hierarchy of processing timescales. For areas with short integration windows, such as early auditory cortex, the differences in neural responses between the two stories were relatively small. In contrast, in areas with the longest integration windows at the top of the hierarchy, such as the precuneus, temporal parietal junction, and medial frontal cortices, there were large differences in neural responses between stories. Furthermore, this gradual increase in neural differences between the stories was highly correlated with an area's ability to integrate information over time. Amplification of neural differences did not occur when changes in words did not alter the interpretation of the story (e.g., sobbing to "crying"). Our results demonstrate how subtle differences in words are gradually accumulated and amplified along the cortical hierarchy as the brain constructs a narrative over time.