Analysis of a neural oscillator

Bifurcation analysis of a two-neuron central pattern generator model for both oscillatory and convergent neuronal activities

The neural oscillator model proposed by Matsuoka is a piecewise affine system that exhibits distinctive periodic solutions. Although such typical oscillation patterns have been widely studied, little is understood about the dynamics of convergence to certain fixed points and bifurcations between the periodic orbits and fixed points in this model. We performed fixed point analysis on a two-neuron version of the Matsuoka oscillator model, the result of which explains the mechanism of oscillation and the discontinuity-induced bifurcations such as subcritical/supercritical Hopf-like, homoclinic-like and grazing bifurcations. Furthermore, it provided theoretical predictions concerning a logarithmic oscillation-period scaling law and noise-induced oscillations observed around those bifurcations. These results are expected to underpin further investigations into oscillatory and transient neuronal activities concerning central pattern generators.

Central Pattern Generator with Defined Pulse Signals for Compliant-Resistant Control of Biped Robots

For biped robots, the ability to maintain balance under external forces is an essential requirement. Inspired by human beings’ behaviors to resist external forces, a compliant-resistant balance-control method is proposed to keep the biped robot balance subjected to an external force. A model-free trajectory generator is designed based on the central pattern generator (CPG) to generate compliant-resistant human-like behavior. The CPG pattern generator generates the desired pulse signal utilizing Matsuoka’s CPG. The signal modulator applies the defined signal to the robot’s center of mass (CoM) to generate the workspace trajectory when standing on double feet. Moreover, when standing on single foot, the output signal of the CPG will directly act on the hip joint of the robot to generate the joint space trajectory. Furthermore, the motion engine calculates the workspace trajectory into joint sequence values. The proposed control strategy can generate defined pulse signals to realize compliant-resistant balance control for biped robots. The control strategy proposed in this paper is verified in the NAO simulation environment.

Central pattern generator network model for the alternating hind limb gait of rats based on the modified Van der Pol equation

Herein, we employed a central pattern generator (CPG), a spinal cord neural network that regulates lower-limb gait during intra-spinal micro-stimulation (ISMS). Particularly, ISMS was used to determine the spatial distribution pattern of CPG sites in the spinal cord and the signal regulation pattern that induced the CPG network to produce coordinated actions. Based on the oscillation phenomenon of the single CPG neurons of Van der Pol (VDP) oscillators, a double-cell CPG neural network model was constructed to realise double lower limbs, six-joint modelling, the simulation of 12 neural circuits, the CPG loci characterising stimuli-inducing alternating movements and changes in polarity stimulation signals in rat hindlimbs, and leg-state change movements. The feasibility and effectiveness of the CPG neural network were verified by recording the electromyographic burst-release mode of the flexor and extensor muscles of the knee joints during CPG electrical stimulation. The results revealed that the output pattern of the CPG presented stable rhythm and coordination characteristics. The 12-neuron CPG model based on the improved VDP equation realised single-point control while significantly reducing the number of stimulation electrodes in the gait training of spinal cord injury patients. We believe that this study advances motor function recovery in rehabilitation medicine.

Generic Mechanism for Waveform Regulation and Synchronization of Oscillators: An Application for Robot Behavior Diversity Generation

While nonlinear oscillators have been widely used for central pattern generators to produce basic rhythmic signals for robot locomotion control, methods to shape and regulate the signal waveform without changing the characteristics of the oscillators have not been fully investigated, especially during the network synchronization process. To illustrate the principle and process of waveform regulation of nonlinear oscillators in detail and ensure that the influence can be controlled, we present a method for waveform regulation and synchronization and analyze the relationship of different factors (e.g., initial conditions, network parameters, phase, and waveform regulation factors) in synchronization deviation. Then, the method is indicated to be effective in other commonly used nonlinear oscillators and neural oscillators. As an example application, a three-layer behavioral control architecture for a legged robot is constructed based on the proposed method. Modules for the body behavior, leg coordination, and single-leg adjustment are established to realize diverse robot behaviors. The effectiveness of the method is validated by a series of experiments. The results prove that the method performs well in terms of signal control accuracy, behavior pattern diversity, and smooth motion transition.

A new biological central pattern generator model and its relationship with the motor units

The central pattern generator (CPG) is a key neural-circuit component of the locomotion control system. Recently, numerous molecular and genetic approaches have been proposed for investigating the CPG mechanisms. The rhythm in the CPG locomotor circuits comes from the activity in the ipsilateral excitatory neurons whose output is controlled by inter-neuron inhibitory connections. Conventional models for simulating the CPG mechanism are complex Hodgkin-Huxley-type models. Inspired by biological investigations and the continuous-time Matsuoka model, we propose new integral-order and fractional-order CPG models, which consider time delays and synaptic interfaces. The phase diagrams exhibit limit cycles and periodic solutions, in agreement with the CPG biological characteristics. As well, the fractional-order model shows state transitions with order variations. In addition, we investigate the relationship between the CPG and the motor units through the construction of integral-order and fractional-order coupling models. Simulations of these coupling models show that the states generated by the three motor units are in accordance with the experimentally-obtained states in previous studies. The proposed models reveal that the CPG can regulate limb locomotion patterns through connection weights and synaptic interfaces. Moreover, in comparison to the integral-order models, the fractional-order ones appear to be more effective, and hence more suitable for describing the dynamics of the CPG biological system.

Emergence of robust self-organized undulatory swimming based on local hydrodynamic force sensing

Undulatory swimming represents an ideal behavior to investigate locomotion control and the role of the underlying central and peripheral components in the spinal cord. Many vertebrate swimmers have central pattern generators and local pressure-sensitive receptors that provide information about the surrounding fluid. However, it remains difficult to study experimentally how these sensors influence motor commands in these animals. Here, using a specifically designed robot that captures the essential components of the animal neuromechanical system and using simulations, we tested the hypothesis that sensed hydrodynamic pressure forces can entrain body actuation through local feedback loops. We found evidence that this peripheral mechanism leads to self-organized undulatory swimming by providing intersegmental coordination and body oscillations. Swimming can be redundantly induced by central mechanisms, and we show that, therefore, a combination of both central and peripheral mechanisms offers a higher robustness against neural disruptions than any of them alone, which potentially explains how some vertebrates retain locomotor capabilities after spinal cord lesions. These results broaden our understanding of animal locomotion and expand our knowledge for the design of robust and modular robots that physically interact with the environment.

Memristor Circuits for Simulating Neuron Spiking and Burst Phenomena

Since the introduction of memristors, it has been widely recognized that they can be successfully employed as synapses in neuromorphic circuits. This paper focuses on showing that memristor circuits can be also used for mimicking some features of the dynamics exhibited by neurons in response to an external stimulus. The proposed approach relies on exploiting multistability of memristor circuits, i.e., the coexistence of infinitely many attractors, and employing a suitable pulse-programmed input for switching among the different attractors. Specifically, it is first shown that a circuit composed of a resistor, an inductor, a capacitor and an ideal charge-controlled memristor displays infinitely many stable equilibrium points and limit cycles, each one pertaining to a planar invariant manifold. Moreover, each limit cycle is approximated via a first-order periodic approximation analytically obtained via the Describing Function (DF) method, a well-known technique in the Harmonic Balance (HB) context. Then, it is shown that the memristor charge is capable to mimic some simplified models of the neuron response when an external independent pulse-programmed current source is introduced in the circuit. The memristor charge behavior is generated via the concatenation of convergent and oscillatory behaviors which are obtained by switching between equilibrium points and limit cycles via a properly designed pulse timing of the current source. The design procedure takes also into account some relationships between the pulse features and the circuit parameters which are derived exploiting the analytic approximation of the limit cycles obtained via the DF method.

Matsuoka's CPG With Desired Rhythmic Signals for Adaptive Walking of Humanoid Robots

The desired rhythmic signals for adaptive walking of humanoid robots should have proper frequencies, phases, and shapes. Matsuoka's central pattern generator (CPG) is able to generate rhythmic signals with reasonable frequencies and phases, and thus has been widely applied to control the movements of legged robots, such as walking of humanoid robots. However, it is difficult for this kind of CPG to generate rhythmic signals with desired shapes, which limits the adaptability of walking of humanoid robots in various environments. To address this issue, a new framework that can generate desired rhythmic signals for Matsuoka's CPG is presented. The proposed framework includes three main parts. First, feature processing is conducted to transform the Matsuoka's CPG outputs into a normalized limit cycle. Second, by combining the normalized limit cycle with robot feedback as the feature inputs and setting the required learning objective, the neural network (NN) learns to generate desired rhythmic signals. Finally, in order to ensure the continuity of the desired rhythmic signals, signal filtering is applied to the outputs of NN, with the aim of smoothing the discontinuous parts. Numerical experiments on the proposed framework suggest that it can not only generate a variety of rhythmic signals with desired shapes but also preserve the frequency and phase properties of Matsuoka's CPG. In addition, the proposed framework is embedded into a control system for adaptive omnidirectional walking of humanoid robot NAO. Extensive simulation and real experiments on this control system demonstrate that the proposed framework is able to generate desired rhythmic signals for adaptive walking of NAO on fixed and changing inclined surfaces. Furthermore, the comparison studies verify that the proposed framework can significantly improve the adaptability of NAO's walking compared with the other methods.

Comparative study of forced oscillators for the adaptive generation of rhythmic movements in robot controllers

The interest of central pattern generators in robot motor coordination is universally recognized so much so that a lot of possibilities on different scales of modeling are nowadays available. While each method obviously has its advantages and drawbacks, some could be more suitable for human-robot interactions. In this paper, we compare three oscillator models: Matsuoka, Hopf and Rowat-Selverston models. These models are integrated to a control architecture for a robotic arm and evaluated in simulation during a simplified handshaking interaction which involves constrained rhythmic movements. Furthermore, Hebbian plasticity mechanisms are integrated to the Hopf and Rowat-Selverston models which can incorporate such mechanisms, contrary to the Matsuoka. Results show that the Matsuoka oscillator is subpar in all aspects and for the two others, that plasticity improves synchronization and leads to a significant decrease in the power consumption.

Motion Simulation of Ionic Liquid Gel Soft Actuators Based on CPG Control

The ionic liquid gel (ILG), a new type of soft actuator material, is a mixture of 1-butyl-3-methylimidazolium tetrafluoroborate (BMIMBF₄), hydroxyethyl methacrylate (HEMA), diethoxyacetophenone (DEAP), and ZrO₂ polymerized into a gel state under ultraviolet (UV) light irradiation. The soft actuator structure consists of a layer of ionic liquid polymer gel sandwiched between two layers of activated carbon capped with gold foil. The volume of the cationic BMIM⁺ in the ionic liquid BMIMBF₄ is much larger than that of the anionic BF₄ ^-. When voltages are applied to both sides of the actuator, the anions and cations move toward the anode and cathode of the electrode, respectively, under the electric field. The volume of the ILG cathode side therefore expands, and the volume of the ILG anode side shrinks, hence bending the entire actuator toward the anode side. The Ogden model was selected as the hyperelastic constitutive model to study the mechanical properties of the ILG by nonlinear analysis. As the ILG is an ideal material for the preparation of a supercapacitor, the equivalent circuit of the ILG can be modeled by the supercapacitor theory to identify the transfer function of the soft actuator. The central pattern generator (CPG) control is widely used in the area of biology, and CPGs based on bioinspired control methods have attracted great attention from researchers worldwide. After the continuum soft actuator is discretized, the CPG-based bioinspired method can be used to control the soft robot drivers. According to the simulation analysis results, the soft actuator can be smooth enough to reach the specified location.

The self-organization of ball bouncing

The hybrid rhythmic ball-bouncing task considered in this study requires a participant to hit a ball in a virtual environment by moving a paddle in the real environment. It allows for investigation of the online visual control of action in humans. Changes in gravity acceleration in the virtual environment affect the ball dynamics and modify the ball-paddle system limit cycle. These changes are shown to be accurately reproduced through simulation by a model integrating continuous information-movement couplings between the ball trajectory and the paddle trajectory, giving rise to a resonance-tuning phenomenon. On the contrary, the tested models integrating only intermittent sensorimotor couplings were unable to replicate the observed human behavior. Results suggest that the visual control of action is achieved online, in a prospective way. Human rhythmic motor control would benefit from the timing and phase control emerging from the low-level continuous coupling between the central pattern generator and the visual perception of the ball trajectory. This control strategy, which precludes the need for internal clock and explicit environmental representation, is also able to explain the empirical result that the bounces tend to converge toward a passive stability regime during human ball bouncing.

Neuromechanical Simulation of Hand Pronation and Supination Task in Parkinson's disease

Parkinson's disease is a prevalent and debilitating neurological disorder, where the severity of motor symptoms are frequently monitored using clinical tests that include a hand pronation and supination task. Objective quantification of motor symptoms in persons with Parkinson's disease and detection of dopamine-induced dyskinesias during treatment is important for the management of the most common symptoms in persons with Parkinson's disease. Thus, the development of a neuromechanical model of rhythmic hand pronation and supination may further our understanding of the mechanisms underlying motor symptoms during rhythmic upper extremity tasks in persons with Parkinson's disease. The aim of this study was to create a model for a rhythmic hand pronation and supination task. This was done to create a simulation of a popular diagnostic task used in determining the severity of motor impairments in persons with Parkinson's disease. It is imperative to understand the neural dynamics as well as the physiological constraints placed on a system such as this in both the creation of a usable model as well as understanding the neuromechanical interactions occurring during this diagnostic task. This model of either normal or slowed, clinical behavior, can then serve as a springboard for the creation of models that characterize disordered motor movement and perhaps even the creation of models that could be incorporated into the diagnostic process.

Model of rhythmic ball bouncing using a visually controlled neural oscillator

The present paper investigates the sensory-driven modulations of central pattern generator dynamics that can be expected to reproduce human behavior during rhythmic hybrid tasks. We propose a theoretical model of human sensorimotor behavior able to account for the observed data from the ball-bouncing task. The novel control architecture is composed of a Matsuoka neural oscillator coupled with the environment through visual sensory feedback. The architecture's ability to reproduce human-like performance during the ball-bouncing task in the presence of perturbations is quantified by comparison of simulated and recorded trials. The results suggest that human visual control of the task is achieved online. The adaptive behavior is made possible by a parametric and state control of the limit cycle emerging from the interaction of the rhythmic pattern generator, the musculoskeletal system, and the environment.NEW & NOTEWORTHY The study demonstrates that a behavioral model based on a neural oscillator controlled by visual information is able to accurately reproduce human modulations in a motor action with respect to sensory information during the rhythmic ball-bouncing task. The model attractor dynamics emerging from the interaction between the neuromusculoskeletal system and the environment met task requirements, environmental constraints, and human behavioral choices without relying on movement planning and explicit internal models of the environment.

Coupling relationship between the central pattern generator and the cerebral cortex with time delay

Brain activity is a cooperative process among neurons and involves the coupling relationship, which is crucial to perform operational tasks in various specialized areas of the nervous system. A finite signal transmission speed along the axons results in a space-dependent time delay. The central pattern generator (CPG) can in principle produce basic locomotor rhythm in the absence of inputs from higher brain centers and peripheral sensory feedback. To study the dynamic performance of CPG with time delay and its coupling relationship with the cerebral cortex, a new CPG model with time delay and a model of the neural mass model (NMM) and the CPG are developed. The coupling model is based on biological experimental results. Bifurcation theories and maximal Lyapunov exponent are used to analyze the dynamic performance. From the results, some CPGs are suggested to be embedded in limbs and composed of the parameters space which corresponds to the one of the cerebral cortex. This embodiment of humans can reduce the burden of the brain and simplify the control of the locomotion. The results also show that the phase diagram of the CPG cannot keep the limit cycle, and that the state of the NMM becomes increasingly chaotic as time delay increases. This finding implies that a person with slow reaction can easily lose the stability of his or her locomotion.

Effects on hypothalamus when CPG is fed back to basal ganglia based on KIV model

The KIV model approximates the operation of the basic vertebrate forebrain together with the basal ganglia and motor systems. In KIV model, the hypothalamus and the basal ganglia which are two important parts in the midline forebrain are closely associated with the locomotion. The CPG model with time delay is established in this paper and the stability of this CPG model is discussed. The CPG output is treated as the proprioception and fed back to the basal ganglia. We focus on the effects on the hypothalamus and the basal ganglia when the time delay parameter a d , the CPG amplitude parameter e and the CPG frequency parameter T r are changed. Through analysis, we find that there exists optimum value of the parameters a d or T r which can make the synchronization of the hypothalamus optimum when the CPG is added into the basal ganglia. The results could have important implications for biological processes which are about interaction between the neural network and the CPG.

Simulation of Octopus Arm Based on Coupled CPGs

The octopus arm has attracted many researchers’ interests and became a research hot spot because of its amazing features. Several dynamic models inspired by an octopus arm are presented to realize the structure with a large number of degrees of freedom. The octopus arm is made of a soft material introducing high-dimensionality, nonlinearity, and elasticity, which makes the octopus arm difficult to control. In this paper, three coupled central pattern generators (CPGs) are built and a 2-dimensional dynamic model of the octopus arm is presented to explore possible strategies of the octopus movement control. And the CPGs’ signals treated as activation are added on the ventral, dorsal, and transversal sides, respectively. The effects of the octopus arm are discussed when the parameters of the CPGs are changed. Simulations show that the octopus arm movements are mainly determined by the shapes of three CPGs’ phase diagrams. Therefore, some locomotion modes are supposed to be embedded in the neuromuscular system of the octopus arm. And the octopus arm movements can be achieved by modulating the parameters of the CPGs. The results are beneficial for researchers to understand the octopus movement further.

Synchronization and stochastic resonance of the small-world neural network based on the CPG

According to biological knowledge, the central nervous system controls the central pattern generator (CPG) to drive the locomotion. The brain is a complex system consisting of different functions and different interconnections. The topological properties of the brain display features of small-world network. The synchronization and stochastic resonance have important roles in neural information transmission and processing. In order to study the synchronization and stochastic resonance of the brain based on the CPG, we establish the model which shows the relationship between the small-world neural network (SWNN) and the CPG. We analyze the synchronization of the SWNN when the amplitude and frequency of the CPG are changed and the effects on the CPG when the SWNN's parameters are changed. And we also study the stochastic resonance on the SWNN. The main findings include: (1) When the CPG is added into the SWNN, there exists parameters space of the CPG and the SWNN, which can make the synchronization of the SWNN optimum. (2) There exists an optimal noise level at which the resonance factor Q gets its peak value. And the correlation between the pacemaker frequency and the dynamical response of the network is resonantly dependent on the noise intensity. The results could have important implications for biological processes which are about interaction between the neural network and the CPG.

Coupling effect analysis between the central nervous system and the CPG network with proprioception

Human rhythmic movement is generated by central pattern generators (CPGs), and their application to robot control has attracted interest of many scientists. But the coupling relationship between the central nervous system and the CPG network with external inputs is still not unveiled. According to biological experiment results, the CPG network is controlled by the neural system; in other words, the interaction between the central nervous system and the CPG network can control human movement effectively. This paper offers a complex human locomotion model, which illustrates the coupling relationship between the central nervous system and the CPG network with proprioception. Based on Matsuoka's CPG model (K. Matsuoka, Biol. Cybern. 52(6), 367–376 (1985)), the stability and robustness of the CPG network are analyzed with external inputs. In order to simulate the coupling relationship, the Radial Basis Function (RBF) neural network is used to simulate the cerebral cortex, and the Credit-Assignment Cerebellar Model Articulation Controller algorithm is employed to realize the locomotion mode conversion. A seven-link biped robot is chosen to simulate the walking gait. The main discoveries include: (1) the output of a new CPG network, which is stable and robust, can be treated as proprioception. Proprioception provides the central nervous system with the information about all joint angles; (2) analysis on a new locomotion model reveals that the cerebral cortex can modulate CPG parameters, leading to adjustment in walking gait.