The ability to learn continuously from an incoming data stream without catastrophic forgetting is critical to designing intelligent systems. Many approaches to continual learning rely on stochastic gradient descent and its variants that employ global error updates, and hence need to adopt strategies such as memory buffers or replay to circumvent its stability, greed, and short-term memory limitations. To address this limitation, we have developed a biologically inspired lightweight neural network architecture that incorporates synaptic plasticity mechanisms and neuromodulation and hence learns through local error signals to enable online continual learning without stochastic gradient descent. Our approach leads to superior online continual learning performance on Split-MNIST, Split-CIFAR-10, and Split-CIFAR-100 datasets compared to other memory-constrained learning approaches and matches that of the state-of-the-art memory-intensive replay-based approaches. We further demonstrate the effectiveness of our approach by integrating key design concepts into other backpropagation-based continual learning algorithms, significantly improving their accuracy. Our results provide compelling evidence for the importance of incorporating biological principles into machine learning models and offer insights into how we can leverage them to design more efficient and robust systems for online continual learning.