Continual learning is the ability to acquire a new task or knowledge without losing any previously collected information. Achieving continual learning in artificial intelligence (AI) is currently prevented by catastrophic forgetting, where training of a new task deletes all previously learned tasks. Here, we present a new concept of a neural network capable of combining supervised convolutional learning with bio-inspired unsupervised learning. Brain-inspired concepts such as spike-timing-dependent plasticity (STDP) and neural redundancy are shown to enable continual learning and prevent catastrophic forgetting without compromising standard accuracy achievable with state-of-the-art neural networks. Unsupervised learning by STDP is demonstrated by hardware experiments with a one-layer perceptron adopting phase-change memory (PCM) synapses. Finally, we demonstrate full testing classification of Modified National Institute of Standards and Technology (MNIST) database with an accuracy of 98% and continual learning of up to 30% non-trained classes with 83% average accuracy.