We propose a plausible model for learning by back-error propagation in biological neurons. Forwards propagation occurs as action potentials propagate signals along branching axons and transmit those signals across axo-dendritic synapses, whereupon post-synaptic neurons sum their incoming signals. In our model, back-error propagation is proposed to occur via signals within intraneuronal cytoskeletal microtubules. These signals modify the effective strengths of synapses during learning. Differences between network output and desired (target) outputs are computed at synapses or by synaptic complexes. Biophysical mechanisms are suggested for the summing of errors and the propagation of errors backwards through microtubules within each neuron of the network. We discuss issues and assumptions of the model, alternative candidate mechanisms, and the degree of biological plausibility.