This paper introduces a novel approach to Dynamic Artificial Neural Networks (D-ANNs) for multi-task demand forecasting called Neuroplastic Multi-Task Network (NMT-Net). Unlike conventional methods focusing on inference-time dynamics or computational efficiency, our proposed method enables structural adaptability of the computational graph during training, inspired by neuroplasticity as seen in biological systems. Each new task triggers a dynamic network adaptation, including similarity-based task identification and selective training of candidate ANN heads, which are then assessed and integrated into the model based on their performance. We evaluated our framework using three real-world multi-task demand forecasting datasets from Kaggle. We demonstrated its superior performance and consistency, achieving lower RMSE and standard deviation compared to traditional baselines and state-of-the-art multi-task learning methods. NMT-Net offers a scalable, adaptable solution for multi-task and continual learning in time series prediction. The complete code for NMT-Net is available from our GitHub repository.