The low-voltage low-power sinh-domain (SD) implementations of integer- and fractional-order FitzHugh-Nagumo (FHN) neuron model have been introduced in this paper. Besides, the effect of fractional-orders on the external excitation current and dynamics of the neuron has been examined in this paper. The proposed SD designs of FHN neuron model have the benefits of: 1) low-voltage operation; 2) integrability, due to resistor-less design and the employment of only grounded components; 3) electronic tunability of performance parameters; and 4) low-power implementation due to the inherent properties of SD technique. HSPICE simulator tool and Taiwan Semiconductor Manufacturing Company, Hsinchu, Taiwan 130-nm CMOS process was used to evaluate and verify the correct functioning of the model. In addition, to experimentally verify the operation of the proposed fractional-order FHN neuron model, field-programmable analog array (FPAA) implementation of the model has been presented, and the proper functioning of the model has been verified. To the best of the authors' knowledge, this is the first paper that describes the electronic realization of the fractional-order FHN neuron model. In addition, it is the first time that the FPAA implementation of any fractional-order neuron model has been presented.