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The "infinitesimal phase response curve" (PRC) is a common tool used to analyze phase resetting in the natural sciences in general and neuroscience in particular. We make the observation that the PRC with respect to a coordinate v actually depends on the choice of other coordinates. As a consequence, a complete delay embedding reconstruction of the dynamics using v which would allow phase to be computed still does not allow the v PRC to be computed. We give a coordinate-free definition of the PRC making this observation obvious. This leads to an experimental protocol: first collect an appropriate ensemble of measurements by intermittently controlling neuron voltage. Then, for any suitable current carrier dynamic postulated, we show how the ensemble can be used to compute the voltage PRC with that current carrier. The approach extends to many oscillators measured and controlled through a subset of their coordinates. [ABSTRACT FROM AUTHOR]

Movement is one of the essential characteristics of living beings. Despite the diversity of animal species and the apparent differences, standard features exist between their movement systems that follow a particular pattern. The movements can mainly be divided into discrete and rhythmic categories controlled by the central nervous system. Scientists usually consider these two types of motion separately in the control system and use different methods and resources to produce and model them. Proposing a unified and comprehensive model for generating and controlling rhythmic and discrete movement with the same control system is more valuable, albeit challenging. This is essential because such a model would address a fundamental problem in the field of motor control, offering a holistic solution to understanding how living beings generate and control movement. A unified model could revolutionize various fields, including robotics, rehabilitation, and neuroscience, by providing a versatile framework applicable to various applications. In this study, we employed the Hodgkin-Huxley (HH) equations in our computational model; their suitability lies in their ability to capture the intricate dynamics of neural oscillations and the behavior of neural networks, making them an ideal choice for our investigation. Our comprehensive analysis of the model, factors influencing motion, and oscillation revealed crucial insights. We found that supraspinal input and motor neuron feedback, as the key motor control parameters, play pivotal roles in generating and controlling rhythmic and discrete movements. These findings contribute to our understanding of how the nervous system orchestrates both types of motion within a single framework. Developing a neuromechanical model capable of creating rhythmic and discrete movements holds promising implications. This research can potentially advance fields such as robotics, biomechanics, and rehabilitation by providing a unified framework for motor control. Moreover, understanding the switching mechanism between rhythmic and discrete states could lead to innovative strategies for designing more versatile and adaptive robotic systems and improving rehabilitation protocols for individuals with motor impairments. [ABSTRACT FROM AUTHOR]

The study on neuromorphic oscillation behaviors and their oscillators have been considered as one of the most straightforward approaches to mimic some biological neurons via the configured nonlinear memristive equivalent circuits. Also, neuromorphic oscillation is one classical oscillation phenomenon, both the frequency response and zero-pole analysis could be clearly regarded as a complete description of the sinusoidal oscillation behavior for a nonlinear circuit. As one of the most classical memristors, Chua corsage memristor (CCM) is so famous to exhibit chaotic oscillation and neuromorphic dynamics due to its locally active and edge of chaos. However, the parasitic phenomena in the practical circuits are unique and tiny but existed as one of the inherent characteristics, which could lead to some unexpected results. Even a pretty small perturbation may heavily affect the quality of the entire system. In this paper, one modified CCM with the parasitic parameter (named $g_{p}$ ) is proposed. Then, some unique and unusual phenomena are captured and analyzed. Moreover, the analysis on the distributions for the locally-active domains (LADs) and edge of chaos are presented. Furthermore, one small-signal equivalent circuit with a positive capacitance is introduced, as well as its impedance and admittance functions. Also, both types of neuromorphic oscillators are captured and observed via external inductor and capacitor, respectively. Finally, the applications in neural networks are explored, which herald the proposed model could be more suitable to transmit the mental, physical fatigue, memory load and closer to simulating the actual neuromorphic systems.

Neural oscillators, which are a mathematical model of a central pattern generator, have been used to investigate human gait and control mobile robots. A typical neural oscillator uses two neurons that mutually inhibit each other’s activity. The exact orbitally-stable conditions of the neural activity of neural oscillators without external signals have been reported. However, the behavior of neural oscillators with external signals is unclear because their neural activity depends on the external signals, which have many types. In this study, for simplicity, external signals were regarded as a sinusoidal wave with a period Tex and an amplitude Aex. The connectivity aij, ratio for time constants τz/τx, and fatigue coefficient b were changed for neurons i and j, while Tex and Aex were changed for external signals. The orbit-stability of the output signals from a neuron was decided based on the transient time (≦ 3 s) and the duration (≧ 30 s). The period Tout and the amplitude Aout of the output signals were evaluated. Tout had discrete values of Tex, 2Tex, or 0.5Tex or was non-orbitally-stable (value of 0). When Aex was greater than or equal to ui, the neural oscillator became synchronized. For a small Tex, some combinations of τz/τx and aij values led to instability. For a large Tex, some combinations of b and aij values led to instability. In contrast to Tout, the amplitude Aout of the output signal showed continuous changes depending on τz/τx, b, and aij. The amplitude Aout could be expressed as the sum of ui and Aex. Aout was not significantly affected by τz/τx but decreased with decreasing b.

We celebrate the 60th anniversary of Biological Cybernetics. It has also been 30 years since "Self-organized control of bipedal locomotion by neural oscillators in unpredictable environment" was published in Biological Cybernetics (Taga et al. in Biol Cybern 65(3):147–159, 1991). I would like to look back on the creation of this paper and discuss its subsequent development and future perspectives. [ABSTRACT FROM AUTHOR]

In this paper, construction of a newly designed snake robot is suggested along with the algorithm for generation of different rhythmic motions. The proposed robot system has modular structures with extendable length. It is subdivided into body, neck, head and tail modules. Each body module has two rotary motors to generate pitch and yaw motions of the snake robot. A linear actuator is also installed inside each body module in order to change the length of robot. The neck module in the robot is provided with two rotary motors to make the spherical motion of head module. Neural oscillator based central pattern generators (CPG) are used to produce rhythmic patterns for various snake robot movements, for example, serpentine, side-winding, two-step-concertina and four-step-concertina motions are generated in the snake robot using the proposed CPG algorithm. For serpentine motion, the body of robot is bent to form the planar sinusoidal waveform using whole body modules on the ground. To generate side-winding motion, the robot body is bent in such a way that it makes two dimensional sinusoidal waveform and only a few points of its body make contacts with the ground. By using the CPG algorithm, these contact points are propagated from tail to head, and the robot is ultimately moved along one side. In concertina motion, the body is sequentially pushed forward in steps from tail to head using the proposed algorithm. Finally several experiments are conducted on a laboratory floor in order to confirm the authenticity of robot design and CPG algorithm so that the comparison between different motions can be achieved. [ABSTRACT FROM AUTHOR]

The prediction of the joint angle position, especially during tremor bursts, can be useful for detecting, tracking, and forecasting tremors. Thus, this research proposes a new model for predicting the wrist joint position during rhythmic bursts and inter-burst intervals. Since a tremor is an approximately rhythmic and roughly sinusoidal movement, neural oscillators have been selected to underlie the proposed model. Two neural oscillators were adopted. Electromyogram (EMG) signals were recorded from the extensor carpi radialis and flexor carpi radialis muscles concurrent with the joint angle signals of a stroke subject in an arm constant-posture. The output frequency of each oscillator was equal to the frequency corresponding to the maximum value of power spectrum related to the rhythmic wrist joint angle signals which had been recorded during a postural tremor. The phase shift between the outputs of the two oscillators was equal to the phase shift between the muscle activation of the wrist flexor and extensor muscles. The difference between the two oscillators′ output signals was considered the main pattern. Along with a proportional compensator, an adaptive neural controller has adjusted the amplitude of the main pattern in such a way so as to minimize the wrist joint prediction error during a stroke patient′s tremor burst and a healthy subject′s generated artificial tremor. In regard to the range of wrist joint movement during the observed rhythmic motions, a calculated prediction error is deemed acceptable.

For the crawl stroke in swimming, it is important that the stroke made by the upper limbs and the flutter kick made by the lower limbs are well coordinated in order to enhance swimming performance. However, the training method to acquire the appropriate flutter kick timing has not been sufficiently established. In the present study, a biofeedback training system for swimmers to acquire appropriate kick timing was proposed. In this system, the most difficult and important part is the estimation of the appropriate kick timing. Therefore, the objective of this study was to develop a kick timing estimation algorithm in the crawl stroke for biofeedback training system using neural oscillators. First, a CPG network which outputs the kick estimation timing according to the input was constructed. In order to synchronize the roll angle in the CPG network with the actual one measured by a sensor, a special algorithm to change the cycle of the oscillation for the CPG network was introduced. Validation for the output of sinusoidal input accompanying sudden change in cycle was examined. It was found that the output signal for the roll tracked the input signal well, despite the sudden change in cycle. Validation for the actual input obtained in the experiment was next examined. It was found that the output from the CPG network was sufficiently consistent with the experimental values, suggesting sufficient performance of the proposed estimation algorithm.

The involvement of glial cells in the regulation of physiological functions is being increasingly recognized, yet their role in plasticity of neural oscillators has remained largely elusive. An excellent model system to address the latter function is the pacemaker nucleus of the weakly electric fish, Apteronotus leptorhynchus. This brainstem oscillator drives the fish's electric organ discharge in a one-to-one fashion, with median frequencies of 880 Hz in males and 740 Hz in females. Morphometric analysis of the pacemaker nucleus has shown that astrocytes outnumber mature neurons seven-fold, and oscillator neurons even 200-fold. A similar dominance of astrocytes occurs among the adult-born cells that differentiate into glia and neurons. The astrocytes form a dense meshwork of cells interconnected by gap junctions. The degree of association of astrocytic fibers with the neural oscillator cells, and the gap-junction coupling between individual astrocytes, exhibit a sexual dimorphism, which parallels the sexual dimorphisms in the output frequency of the pacemaker nucleus, and ultimately in the electric organ discharge of the fish. It is hypothesized that the dynamics in astroglial structure mediate differences in the capacity to buffer potassium, which increases during the generation of action potentials. These differences, in turn, affect the excitability of the neural oscillator cells, and thus the output frequency of the pacemaker nucleus. Comparison of the pacemaker nucleus with other brain oscillators suggests that modulation of the output activity is one of the chief functions of the interaction of glia with the neural oscillator cells.

The objective of this study was to propose a CPG network which can generate the swimming motion of the crawl stroke. First, the CPG network for legs performing a flutter kick was constructed by connecting the neural oscillators for the leg joints. The flutter kick motion was successfully generated by the proposed CPG network. The propulsion by the generated flutter kick motion was confirmed by the simulation of the swimming movement. Next, the CPG network for both the arms and legs were constructed, in which the neural oscillator for the arms initiated the trigger signal to start the prescribed stroke motion. By changing the intrinsic cycle of the neural oscillators for the legs, both six- and two-beat crawls could be realized. It was also found that a stable region with respect to the relationship between the intrinsic cycles of the neural oscillators for the arms and legs certainly existed for the six-beat crawl, although the intrinsic cycles of the arms were three times longer than those of the legs in this case. The propulsion by the generated swimming motion was confirmed by the simulation of the swimming movement both for the six- and two-beat crawls. Finally, the roll angle of the swimmer was fed back into the CPG network in order to restore the balance in the roll direction. Restoring the balance in the roll direction was successfully realized by the proposed feedback algorithm. The resultant motion showed a complicated behavior, such as skipping strokes.

This study proposes a neural oscillator-based brain–computer interface (BCI) that controls a bipedal neuromusculoskeletal (NMS) model by inputting electroencephalogram (EEG) signals. In this BCI system, while the bipedal NMS system realizes bipedal locomotion through internal entrainment among neural oscillators and a musculoskeletal system, the locomotion of the system is controlled via external entrainment of the neural oscillators to the external input of EEG signals. As the first step in developing the neural oscillator-based BCI controlling a bipedal NMS model, exploratory numerical simulations were conducted to investigate the behavior of the proposed BCI when sinusoidal waves and alpha waves were inputted. The following tendencies were observed: (a) inputting sinusoidal waves with small amplitudes and high frequencies did not affect the natural walking behavior of the bipedal NMS model that was generated by including only offset values in the external input, (b) inputting sinusoidal waves with small amplitudes and low frequencies disturbed and decelerated the walking behavior, (c) inputting sinusoidal waves with large amplitudes accelerated the walking behavior, (d) inputting sinusoidal waves with large amplitudes and a particular frequency changed walking behavior to running behavior, (e) changing the external input of alpha waves between an eyes-open condition and an eyes-closed condition successfully changed the walking behavior. The eyes-open condition led to faster walking compared with the eyes-closed condition.

Taxis behaviour in Drosophila larva is thought to consist of distinct control mechanisms triggering specific actions. Here, we support a simpler hypothesis: that taxis results from direct sensory modulation of continuous lateral oscillations of the anterior body, sparing the need for ‘action selection’. Our analysis of larvae motion reveals a rhythmic, continuous lateral oscillation of the anterior body, encompassing all head-sweeps, small or large, without breaking the oscillatory rhythm. Further, we show that an agent-model that embeds this hypothesis reproduces a surprising number of taxis signatures observed in larvae. Also, by coupling the sensory input to a neural oscillator in continuous time, we show that the mechanism is robust and biologically plausible. The mechanism provides a simple architecture for combining information across modalities, and explaining how learnt associations modulate taxis. We discuss the results in the light of larval neural circuitry and make testable predictions.