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A generalized asynchronous cellular automaton-based neuron model is a special kind of cellular automaton that is designed to mimic the nonlinear dynamics of neurons. The model can be implemented as an asynchronous sequential logic circuit and its control parameter is the pattern of wires among the circuit elements that is adjustable after implementation in a field-programmable gate array (FPGA) device. In this paper, a novel theoretical analysis method for the model is presented. Using this method, stabilities of neuron-like orbits and occurrence mechanisms of neuron-like bifurcations of the model are clarified theoretically. Also, a novel learning algorithm for the model is presented. An equivalent experiment shows that an FPGA-implemented learning algorithm enables an FPGA-implemented model to automatically reproduce typical nonlinear responses and occurrence mechanisms observed in biological and model neurons.