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In this paper, a four-neuron delayed bidirectional associative memory (BAM) model with inertia is considered. Weak resonant double Hopf bifurcations are completely analyzed in the parameter space of the coupling weight and the coupling delay by the perturbation-incremental scheme (PIS). Numerical simulations are given for justifying the theoretical results. To the best of our knowledge, the paper is the first one to introduce inertia to a four-neuron delayed system and clarify the relationship between system parameters and dynamical behaviors.

Interaction between distant neuronal populations is essential for communication within the nervous system and can occur as a highly nonlinear process. To better understand the functional role of neural interactions, it is important to quantify the nonlinear connectivity in the nervous system. We introduce a general approach to measure nonlinear connectivity through phase coupling: the multi-spectral phase coherence (MSPC). Using simulated data, we compare MSPC with existing phase coupling measures, namely n : m synchronization index and bi-phase locking value. MSPC provides a system description, including (i) the order of the nonlinearity, (ii) the direction of interaction, (iii) the time delay in the system, and both (iv) harmonic and (v) intermodulation coupling beyond the second order; which are only partly revealed by other methods. We apply MSPC to analyze data from a motor control experiment, where subjects performed isotonic wrist flexions while receiving movement perturbations. MSPC between the perturbation, EEG and EMG was calculated. Our results reveal directional nonlinear connectivity in the afferent and efferent pathways, as well as the time delay $(43\pm 8\mathrm{\text{ms}})$ between the perturbation and the brain response. In conclusion, MSPC is a novel approach capable to assess high-order nonlinear interaction and timing in the nervous system.

Communication between neuronal populations is facilitated by synchronization of their oscillatory activity. Although nonlinearity has been observed in the sensorimotor system, its nonlinear connectivity has not been widely investigated yet. This study investigates nonlinear connectivity during the human stretch reflex based on neuronal synchronization. Healthy participants generated isotonic wrist flexion while receiving a periodic mechanical perturbation to the wrist. Using a novel cross-frequency phase coupling metric, we estimate directional nonlinear connectivity, including time delay, from the perturbation to brain and to muscle, as well as from brain to muscle. Nonlinear phase coupling is significantly stronger from the perturbation to the muscle than to the brain, with a shorter time delay. The time delay from the perturbation to the muscle is 33 ms, similar to the reported latency of the spinal stretch reflex at the wrist. Source localization of nonlinear phase coupling from the brain to the muscle suggests activity originating from the motor cortex, although its effect on the stretch reflex is weak. As such nonlinear phase coupling between the perturbation and muscle activity is dominated by the spinal reflex loop. This study provides new evidence of nonlinear neuronal synchronization in the stretch reflex at the wrist joint with respect to spinal and transcortical loops.

Graphic simulation based on virtual reality technology is applied to the tele-operation robot system with tele-presence to make full use of the immersion feeling and the forecast property of the 3-D graphic stereoscopic display, which not only provides the operator with friendly work windows but also can overcome the effect of the communication time delay in tele-operation. In order to investigate the ineluctability communication time delay in force tele-presence of the tele-operation robot system, a tele-presence system based on the virtual reality is setup in this paper. The system structure, geometrical noumenon modeling, and man-machine interface are introduced in detail. In the tele-operation system, graphical robot technology is used to solve the time delay, and the relating experiments have proved the validity of the reappearance of the movement of an actual robot by a graphical robot.