Transhumeral amputation has a considerable detrimental effect on the amputee's quality of life and independence. Previous work has already established the potential for exploiting proximal humerus myoelectric and kinematic signals for the effective control of a myoelectric prosthesis. That previous work used a Time-Delay Neural Network (TDNN) to perform the mapping of six electromyographic (EMG) and six kinematic proximal humerus signals to predict elbow flexion/extension. Since that earlier work alternative deep learning and recurrent neural network architectures, well-suited to the processing of high-dimensional time-series data, have come to the fore. The work reported here is a comparative evaluation using the metric of RMS error for the predicted elbow flexion/extension angles output by TDNN, Long Short Term Memory (LSTM) and Echo State Network (ESN) architectures. For the most effective comparison we successfully reproduced TDNN results that were comparable to previous work (with average RMSE of 11.8 degrees on unseen test data). Then using the same training and testing datasets, and networks of broadly similar complexity, we evaluated the effectiveness of the LSTM and ESN approaches. The LSTMs trained here delivered an average RMSE of 10.4 degrees on unseen test data. The ESNs delivered an average RMSE of 16.3 degrees on unseen test data. The current work was not intended to find the best possible LSTM or ESN solution for this problem. Instead the intention was to see if any particular aspects of the network architectures worked better with the particular challenges of transhumeral biomedical engineering data of this sort.