Significant progress has been made in the research of all-optical neural networks in recent years. In this paper, we theoretically explore the properties of a neural system composed of semiconductor ring lasers (SRLs). Our study demonstrates that external optical signals generated by a tunable laser (TL) are injected into the first semiconductor ring laser photonic neuron (SRL1). Subsequently, the responses of SRL1 in the clockwise (CW) and counterclockwise (CCW) directions are unidirectionally injected into the CW and CCW directions of the second semiconductor ring laser photonic neuron (SRL2), respectively, which then exhibits similar spiking inhibition behaviors. Numerical simulations reveal that the spiking inhibition behavior of the SRL response can be precisely controlled by adjusting the perturbation time and intensity of the external injection signal, and this behavior is highly repeatable. Most importantly, we successfully achieve the stable transmission of these responses between the two SRL photonic neurons. These inhibition behaviors are analogous to those of biological neurons, but with a response speed reaching the sub-nanosecond level. Additionally, we indicate that SRL photonic neurons undergo a refractory-period-like phenomenon when subjected to two consecutive perturbations. These findings highlight the immense potential for the design and implementation of future all-optical neural networks, providing critical theoretical foundations and support for them.