Neural networks have become the key technology of artificial intelligence and have contributed to breakthroughs in several machine learning tasks, primarily owing to advances in deep learning applied to Artificial Neural Networks (ANNs). Simultaneously, Spiking Neural Networks (SNNs) incorporating biologically-feasible spiking neurons have held great promise because of their rich temporal dynamics and high-power efficiency. However, the developments in SNNs were proceeding separately from those in ANNs, effectively limiting the adoption of deep learning research insights. Here we show an alternative perspective on the spiking neuron that casts it as a particular ANN construct called Spiking Neural Unit (SNU), and a soft SNU (sSNU) variant that generalizes its dynamics to a novel recurrent ANN unit. SNUs bridge the biologically-inspired SNNs with ANNs and provide a methodology for seamless inclusion of spiking neurons in deep learning architectures. Furthermore, SNU enables highly-efficient in-memory acceleration of SNNs trained with backpropagation through time, implemented with the hardware in-the-loop. We apply SNUs to tasks ranging from hand-written digit recognition, language modelling, to music prediction. We obtain accuracy comparable to, or better than, that of state-of-the-art ANNs, and we experimentally verify the efficacy of the in-memory-based SNN realization for the music-prediction task using 52,800 phase-change memory devices. The new generation of neural units introduced in this paper incorporate biologically-inspired neural dynamics in deep learning. In addition, they provide a systematic methodology for training neuromorphic computing hardware. Thus, they open a new avenue for a widespread adoption of SNNs in practical applications.