In networked control systems (NCS), a certain class of attacks on the communication network is known to raise traffic flows causing delays and packet losses to increase. This paper presents a novel neural network (NN)-based attack detection and estimation scheme that captures the abnormal traffic flow due to a class of attacks on the communication links within the feedback loop of an NCS. By modeling the unknown network flow as a nonlinear function at the bottleneck node and using a NN observer, the network attack detection residual is defined and utilized to determine the onset of an attack in the communication network when the residual exceeds a predefined threshold. Upon detection, another NN is used to estimate the flow injected by the attack. For the physical system, we develop an attack detection scheme by using an adaptive dynamic programming-based optimal event-triggered NN controller in the presence of network delays and packet losses. Attacks on the network as well as on the sensors of the physical system can be detected and estimated with the proposed scheme. The simulation results confirm theoretical conclusions.