First, this research work establishes the theory of novel artificial neural architectures with higher-order nonlinear synaptic neural operation and with adaptable time delays in both the state feedback and neural inputs. The higher-order nonlinear aggregation function, especially together with time-delays as neural parameters of neural units, increases the computational capability of the static and dynamic neural units and thus simplifies the neural architecture and minimizes the number of neural parameters necessary for complex system approximation. The parallel between the novel neural architectures and a real biological neuron is drawn especially with focus on higher computational capability expected from single neurons. The applications to the approximation and control of systems are shown. Second, a novel methodology based on utilization of the proposed neural architectures for adaptive evaluation of variability in complex signals is established. The methodology is based on observation of neural parameters during the adaptation. Results are shown on sensitive sample-to-sample detection of changes in dynamics of highly chaotic time series and HRV (R-R diagrams).