Sarcasm detection has been a well-studied problem for the computational linguistic researchers. However, research related to different categories of sarcasm has still not gained much attention. Self-Deprecating Sarcasm (SDS) is a special category of sarcasm in which users apply sarcasm over themselves, and it is extensively used in social media platforms, mainly as an advertising tool for the brand endorsement, product campaign, and digital marketing with an aim to increase the sales volume. In this paper, we present a deep learning approach for detecting SDS on Twitter. We propose a novel Convolution and Attention with Bi-directional Gated Recurrent Unit (CAT-BiGRU) model, which consists of an input, embedding, convolutional, Bi-directional Gated Recurrent Unit (BiGRU), and two attention layers. The convolutional layer extracts SDS-based syntactic and semantic features from the embedding layer, BiGRU layer retrieves contextual information from the extracted features in both preceding and succeeding directions, and attention layers are used to retrieve SDS-based comprehensive context representation from the input texts. Finally, sigmoid function is employed to classify the input texts as a self-deprecating or non-self-deprecating sarcasm. Experiments are conducted over seven Twitter datasets to evaluate the proposed (CAT-BiGRU) model using standard evaluation metrics. The experimental results are impressive and significantly better than many neural network-based baselines and state-of-the-art methods. In this paper, we have highlighted biologically inspired and psychologically motivated basis of the proposed approach to examine its affective capabilities with respect to SenticNet. The efficacy of the proposed model is evaluated on two SenticNet-based sentic computing resources-Amazon word embedding and AffectiveSpace. Based on the experimental results, we conclude that deep learning-based approaches have potential to detect SDS in social media texts accurately.