Title: On Sparse Critical Paths of Neural Response. Abstract: Is critical input information encoded in specific sparse paths within the network? The pruning objective --- finding a subset of neurons for which the response remains unchanged --- has been used to discover such paths. However, we show that paths obtained from this objective do not necessarily encode the input features and also encompass (dead) neurons that were not originally contributing to the response. We investigate selecting paths based on neurons' contributions to the response to ensure that the paths envelop the critical segments of the encoded input information. We show that these paths have the property of being provably locally linear in an $\ell_{2}$-ball of the input, thus having stable gradients. This property is leveraged for proposing a feature attribution paradigm that is guided by neurons, therefore inherently taking interactions between input features into account. We evaluate the attribution methodology quantitatively in mainstream benchmarks.