`Biologically inspired' activation functions, such as the logistic sigmoid, have been instrumental in the historical advancement of machine learning. However in the field of deep learning, they have been largely displaced by rectified linear units (ReLU) or similar functions, such as its exponential linear unit (ELU) variant, to mitigate the effects of vanishing gradients associated with error back-propagation. The logistic sigmoid however does not represent the true input-output relation in neuronal cells under physiological conditions. Here, bionodal root unit (BRU) activation functions are introduced, exhibiting input-output non-linearities that are substantially more biologically plausible since their functional form is based on known biophysical properties of neuronal cells. In order to evaluate the learning performance of BRU activations, deep networks are constructed with identical architectures except differing in their transfer functions (ReLU, ELU, and BRU). Multilayer perceptrons, stacked auto-encoders, and convolutional networks are used to test supervised and unsupervised learning based on the MNIST and CIFAR-10/100 datasets. Comparisons of learning performance, quantified using loss and error measurements, demonstrate that bionodal networks both train faster than their ReLU and ELU counterparts and result in the best generalised models even in the absence of formal regularisation. These results therefore suggest that revisiting the detailed properties of biological neurones and their circuitry might prove invaluable in the field of deep learning for the future.