Spiking neural networks are able to control with high precision the rotation and force of single-joint robotic arms when shape memory alloy wires are used for actuation. Bio-inspired robotic arms such as anthropomorphic fingers include more junctions that are actuated simultaneously. Starting from the hypothesis that the motor cortex groups the control of multiple muscles into neural synergies, this work presents for the first time an SNN structure that is able to control a series of finger motions by activation of groups of neurons that drive the corresponding actuators in sequence. The initial motion starts when a command signal is received, while the subsequent ones are initiated based on the sensors’ output. In order to increase the biological plausibility of the control system, the finger is flexed and extended by four SMA wires connected to the phalanges as the main tendons. The results show that the artificial finger that is controlled by the SNN is able to smoothly perform several motions of the human index finger while the command signal is active. To evaluate the advantages of using SNN, we compared the finger behaviours when the SMA actuators are driven by SNN, and by a microcontroller, respectively. In addition, we designed an electronic circuit that models the sensor’s output in concordance with the SNN output.