Models of central pattern generators for quadruped locomotion I. Primary gaits

In this paper we continue the analysis of a network of symmetrically coupled cells modeling central pattern generators for quadruped locomotion proposed by Golubitsky, Stewart, Buono, and Collins. By a cell we mean a system of ordinary differential equations and by a coupled cell system we mean a network of identical cells with coupling terms. We have three main results in this paper. First, we show that the proposed network is the simplest one modeling the common quadruped gaits of walk, trot, and pace. In doing so we prove a general theorem classifying spatio-temporal symmetries of periodic solutions to equivariant systems of differential equations. We also specialize this theorem to coupled cell systems. Second, this paper focuses on primary gaits; that is, gaits that are modeled by output signals from the central pattern generator where each cell emits the same waveform along with exact phase shifts between cells. Our previous work showed that the network is capable of producing six primary gaits. Here, we show that under mild assumptions on the cells and the coupling of the network, primary gaits can be produced from Hopf bifurcation by varying only coupling strengths of the network. Third, we discuss the stability of primary gaits and exhibit these solutions by performing numerical simulations using the dimensionless Morris-Lecar equations for the cell dynamics.     

The Adaptive Significance of Tetrapod Gait Selection

At the slow walk tetrapods avoid lateral couplets gaits to minimizesupport by ipsilateral bipods. Most of them use the lateralsequence because tripods then make larger triangles than forthe diagonal sequence. Of the running symmetrical gaits thesingle foots (in each sequence) permit the smoothest and fastesttravel without suspensions. The trot and pace allow two legsto thrust in unison, the former giving the most stability toanimals not placing the feet well under the body and the latteravoiding interference for long legged runners. The bound andhalf bound are most used by small agile mammals for bursts ofspeed and for maneuvering on rough terrain by a series of leaps.Such animals use the extended suspension. Large cursors on openterrain usually select the shorter more economical, gatheredsuspension. The fastest runners use both suspensions to gainlong strides. At moderate speed the transverse gallop has theadvantages over the rotary gallop that both bipods and tripodsare more stable and that interference may be avoided. At highspeed using both suspensions, none of these advantages pertains.The rotary gallop may then increase maneuverability.     

Analysis of the gait generation principle by a simulated quadruped model with a CPG incorporating vestibular modulation

This study aims to understand the principles of gait generation in a quadrupedal model. It is difficult to determine the essence of gait generation simply by observation of the movement of complicated animals composed of brains, nerves, muscles, etc. Therefore, we build a planar quadruped model with simplified nervous system and mechanisms, in order to observe its gaits under simulation. The model is equipped with a mathematical central pattern generator (CPG), consisting of four coupled neural oscillators, basically producing a trot pattern. The model also contains sensory feedback to the CPG, measuring the body tilt (vestibular modulation). This spontaneously gives rise to an unprogrammed lateral walk at low speeds, a transverse gallop while running, in addition to trotting at a medium speed. This is because the body oscillation exhibits a double peak per leg frequency at low speeds, no peak (little oscillation) at medium speeds, and a single peak while running. The body oscillation autonomously adjusts the phase differences between the neural oscillators via the feedback. We assume that the oscillations of the four legs produced by the CPG and the body oscillation varying according to the current speed are synchronized along with the varied phase differences to keep balance during locomotion through postural adaptation via the vestibular modulation, resulting in each gait. We succeeded in determining a single simple principle that accounts for gait transition from walking to trotting to galloping, even without brain control, complicated leg mechanisms, or a flexible trunk.     

Central pattern generators for bipedal locomotion

Golubitsky, Stewart, Buono and Collins proposed two models for the achitecture of central pattern generators (CPGs): one for bipeds (which we call leg) and one for quadrupeds (which we call quad). In this paper we use symmetry techniques to classify the possible spatiotemporal symmetries of periodic solutions that can exist in leg (there are 10 nontrivial types) and we explore the possibility that coordinated arm/leg rhythms can be understood, on the CPG level, by a small breaking of the symmetry in quad, which leads to a third CPG architecture arm. Rhythms produced by leg correspond to the bipedal gaits of walk, run, two-legged hop, two-legged jump, skip, gallop, asymmetric hop, and one-legged hop. We show that breaking the symmetry between fore and hind limbs in quad, which yields the CPG arm, leads to periodic solution types whose associated leg rhythms correspond to seven of the eight leg gaits found in leg; the missing biped gait is the asymmetric hop. However, when arm/leg coordination rhythms are considered, we find the correct rhythms only for the biped gaits of two-legged hop, run, and gallop. In particular, the biped gait walk, along with its arm rhythms, cannot be obtained by a small breaking of symmetry of any quadruped gait supported by quad.     

Hard-wired central pattern generators for quadrupedal locomotion

Animal locomotion is generated and controlled, in part, by a central pattern generator (CPG), which is an intraspinal network of neurons capable of producing rhythmic output. In the present work, it is demonstrated that a hard-wired CPG model, made up of four coupled nonlinear oscillators, can produce multiple phase-locked oscillation patterns that correspond to three common quadrupedal gaits — the walk, trot, and bound. Transitions between the different gaits are generated by varying the network's driving signal and/or by altering internal oscillator parameters. The above in numero results are obtained without changing the relative strengths or the polarities of the system's synaptic interconnections, i.e., the network maintains an invariant coupling architecture. It is also shown that the ability of the hard-wired CPG network to produce and switch between multiple gait patterns is a model-independent phenomenon, i.e., it does not depend upon the detailed dynamics of the component oscillators and/or the nature of the inter-oscillator coupling. Three different neuronal oscillator models — the Stein neuronal model, the Van der Pol oscillator, and the FitzHugh-Nagumo model -and two different coupling schemes are incorporated into the network without impeding its ability to produce the three quadrupedal gaits and the aforementioned gait transitions.     

Reflexes and preflexes: on the role of sensory feedback on rhythmic patterns in insect locomotion

Neuromuscular systems are stabilized and controlled by both feedforward and feedback signals. Feedforward pathways driven by central pattern generators (CPGs), in conjunction with preflexive mechanical reaction forces and nonlinear muscle properties, can produce stable stereotypical gaits. Feedback is nonetheless present in both slow and rapid running, and preflexive mechanisms can join with neural reflexes originating in proprioceptive sensors to yield robust behavior in uncertain environments. Here, we develop a single degree-of-freedom neuromechanical model representing a joint actuated by an agonist/antagonist muscle pair driven by motoneurons and a CPG in a periodic rhythm characteristic of locomotion. We consider two characteristic feedback modes: phasic and tonic. The former encodes states such as position in the timing of individual spikes, while the latter can transmit graded measures of force and other continuous variables as spike rates. We use results from phase reduction and averaging theory to predict phase relationships between CPG and motoneurons in the presence of feedback and compare them with simulations of the neuromechanical model, showing that both phasic and tonic feedback can shift motoneuronal timing and thereby affect joint motions. We find that phase changes in neural activation can cooperate with preflexive displacement and velocity effects on muscle force to compensate for externally applied forces, and that these effects qualitatively match experimental observations in the cockroach.     

Contribution to the analysis of gaits: practical elements to complement the Hildebrand method

This paper suggests three additions to the Hildebrand method for gait-pattern specification. The first allows an extension of this method by the use of the forelimb as reference. Thus, dataset could be gathered indifferently from fore or hindlimbs cycles to identify a symmetrical gait or compare gaits of diverse species. On the basis of Hildebrand's definitions, the second suggestion permits to check the proportion of symmetrical and asymmetrical gaits adopted by different individuals of one or different species. The third addition makes a graphical clear-cut between four basic modes of gallops: rotary, transverse, half-bound and bound. These additions will facilitate extensive data comparison either to evaluate the range of variation within a single species or to specify the preferred gaits in diverse species.     

Multiple transmitter neurons in Tritonia. III. Modulation of central pattern generator controlling feeding

Feeding behavior in the gastropod mollusc Tritonia diomedea is controlled by a central pattern generator (CPG) in the buccal ganglia. The medially located, large dorsal white cells (B11) have been shown to contain two small cardioactive peptides (SCPs). A smaller nearby neuron (B12) also appears to contain the SCPs. B11's have also been shown to contain acetylcholine (ACh), whereas B12's do not. We have shown earlier that intracellular stimulation of B11's drives contractions of the foregut. Here we show that intracellular electrical stimulation of B11's also elicits excitation of neurons B5 and stimulates the patterned motor output of the CPG. We showed earlier that B12's also stimulate contractions in the foregut, but they are in the opposite direction from those elicited by B11. We show here that electrical stimulation of B12's inhibits the output of the CPG. We showed earlier that superfusion of the isolated gut with SCPB enhances peristalsis, and here we report that superfusion of the buccal ganglion with SCPB elicits enhanced coordinated motor output from the CPG. The peptide has two effects on the bursting output of motor neurons. It produces an increase in (1) the rate of bursting and (2) the spike frequency during each burst. On the other hand, we reported earlier that ACh applied directly to isolated foregut inhibits ongoing peristalsis. Here we demonstrate that ACh superfusion of the buccal ganglion also inhibits the CPG output. Our evidence supports the view that in addition to stimulating foregut contractility, B11's modulate the output of the swallowing CPG by releasing a peptide from central terminals. We suggest roles for B11, B12, the SCPs, and ACh in controlling both central and peripheral aspects of feeding behavior.     

The metabolic and mechanical costs of step time asymmetry in walking

Animals use both pendular and elastic mechanisms to minimize energy expenditure during terrestrial locomotion. Elastic gaits can be either bilaterally symmetric (e.g. run and trot) or asymmetric (e.g. skip, canter and gallop), yet only symmetric pendular gaits (e.g. walk) are observed in nature. Does minimizing metabolic and mechanical power constrain pendular gaits to temporal symmetry? We measured rates of metabolic energy expenditure and calculated mechanical power production while healthy humans walked symmetrically and asymmetrically at a range of step and stride times. We found that walking with a 42 per cent step time asymmetry required 80 per cent (2.5 W kg⁻¹) more metabolic power than preferred symmetric gait. Positive mechanical power production increased by 64 per cent (approx. 0.24 W kg⁻¹), paralleling the increases we observed in metabolic power. We found that when walking asymmetrically, subjects absorbed more power during double support than during symmetric walking and compensated by increasing power production during single support. Overall, we identify inherent metabolic and mechanical costs to gait asymmetry and find that symmetry is optimal in healthy human walking.     

Hormone‐mediated modulation of the electromotor CPG in pulse‐type weakly electric fish. Commonalities and differences across species

Like stomatogastric activity in crustaceans, vocalization in teleosts and frogs, and locomotion in mammals, the electric organ discharge (EOD) of weakly electric fish is a rhythmic and stereotyped electromotor pattern. The EOD, which functions in both perception and communication, is controlled by a two‐layered central pattern generator (CPG), the electromotor CPG, which modifies its basal output in response to environmental and social challenges. Despite major anatomo‐functional commonalities in the electromotor CPG across electric fish species, we show that Gymnotus omarorum and Brachyhypopomus gauderio have evolved divergent neural processes to transiently modify the CPG outputs through descending fast neurotransmitter inputs to generate communication signals. We also present two examples of electric behavioral displays in which it is possible to separately analyze the effects of neuropeptides (mid‐term modulation) and gonadal steroid hormones (long‐term modulation) upon the CPG. First, the nonbreeding territorial aggression of G. omarorum has been an advantageous model to analyze the status‐dependent modulation of the excitability of CPG neuronal components by vasotocin. Second, the seasonal and sexually dimorphic courtship signals of B. gauderio have been useful to understand the effects of sex steroids on the responses to glutamatergic inputs in the CPG. Overall, the electromotor CPG functions in a regime that safeguards the EOD waveform. However, prepacemaker influences and hormonal modulation enable an enormous versatility and allows the EOD to adapt its functional state in a species‐, sex‐, and social context‐specific manners.     

Hexapodal gaits and coupled nonlinear oscillator models

The general, model-independent features of different networks of six symmetrically coupled nonlinear oscillators are investigated. These networks are considered as possible models for locomotor central pattern generators (CPGs) in insects. Numerical experiments with a specific oscillator network model are briefly described. It is shown that some generic phase-locked oscillation-patterns for various systems of six symmetrically coupled nonlinear oscillators correspond to the common forward-walking gaits adopted by insects. It is also demonstrated that transitions observed in insect gaits can be modelled as standard symmetry-breaking bifurcations occurring in such systems. The present analysis, which leads to a natural classification of hexapodal gaits by symmetry and to natural sequences of gait bifurcations, relates observed gaits to the overall organizational structure of the underlying CPG. The implications of the present results for the development of simplified control systems for hexapodal walking robots are discussed.     

Evolution of talking plants in a tritrophic context: conditions for uninfested plants to attract predators prior to herbivore attack

Herbivory induces plants to emit volatile chemicals that attract enemies of the herbivores (bodyguards of plants). In this way, the plant acquires protection and the bodyguards gain food. These plant signals cause neighboring plants, not under attack, to release signals as well. We hypothesize that such "secondary" signals help to reduce damage from future herbivore attacks by the protection received from the bodyguards. By modeling we explore the conditions for such secondary signals to evolve. Three kinds of strategies are considered: plants of the first strategy always emit a signal, those of the second strategy emit a signal only when infested, and those of the third strategy emit a signal not only when infested, but also when a certain number of neighbors are infested (i.e. secondary signaling). When signaling is much less (much more) costly than damage from herbivory, the first (second) strategy will be favored by selection, whereas for intermediate costs the third strategy, i.e. secondary signaling, will evolve. However, secondary signaling will not evolve when the primary signals lure the bodyguards too effectively. This is because the undamaged plant gains associational defense when the infested individual is defending very well; therefore, the need for secondary signaling decreases.     

The biomechanics of skipping gaits: a third locomotion paradigm?

Skipping, a gait children display when they are about four- to five-years-old, is revealed to be more than a behavioural peculiarity. By means of metabolic and biomechanical measurements at several speeds, the relevance of skipping is shown to extend from links between bipedal and quadrupedal locomotion (namely galloping) to understanding why it could be a gait of choice in low-gravity conditions, and to some aspects of locomotion evolution (ground reaction forces of skipping seem to originate from pushing the walking gait to unnaturally high speeds). When the time-courses of mechanical energy and the horizontal ground reaction force are considered, a different locomotion paradigm emerges, enabling us to separate, among the bouncing gaits, the trot from the gallop (quadrupeds) and running from skipping (bipeds). The simultaneous use of pendulum-like and elastic mechanisms in skipping gaits, as shown by the energy curve analysis, helps us to understand the low cost of transport of galloping quadrupeds.     

Trot Gait Design and CPG Method for a Quadruped Robot

Developing efficient walking gaits for quadruped robots has intrigued investigators for years. Trot gait, as a fast locomotion gait, has been widely used in robot control. This paper follows the idea of the six determinants of gait and designs a trot gait for a parallel-leg quadruped robot, Baby Elephant. The walking period and step length are set as constants to maintain a relatively fast speed while changing different foot trajectories to test walking quality. Experiments show that kicking leg back improves body stability. Then, a steady and smooth trot gait is designed. Furthermore, inspired by Central Pattern Generators （CPG）, a series CPG model is proposed to achieve robust and dynamic trot gait. It is generally believed that CPG is capable of producing rhythmic movements, such as swimming, walking, and flying, even when isolated from brain and sensory inputs. The proposed CPG model, inspired by the series concept, can automatically learn the previous well-designed trot gait and reproduce it, and has the ability to change its walking frequency online as well. Experiments are done in real world to verify this method.     

Coding conspecific identity and motion in the electric sense

Interactions among animals can result in complex sensory signals containing a variety of socially relevant information, including the number, identity, and relative motion of conspecifics. How the spatiotemporal properties of such evolving naturalistic signals are encoded is a key question in sensory neuroscience. Here, we present results from experiments and modeling that address this issue in the context of the electric sense, which combines the spatial aspects of vision and touch, with the temporal aspects of audition. Wave-type electric fish, such as the brown ghost knifefish, Apteronotus leptorhynchus, used in this study, are uniquely identified by the frequency of their electric organ discharge (EOD). Multiple beat frequencies arise from the superposition of the EODs of each fish. We record the natural electrical signals near the skin of a "receiving" fish that are produced by stationary and freely swimming conspecifics. Using spectral analysis, we find that the primary beats, and the secondary beats between them ("beats of beats"), can be greatly influenced by fish swimming; the resulting motion produces low-frequency envelopes that broaden all the beat peaks and reshape the "noise floor". We assess the consequences of this motion on sensory coding using a model electroreceptor. We show that the primary and secondary beats are encoded in the afferent spike train, but that motion acts to degrade this encoding. We also simulate the response of a realistic population of receptors, and find that it can encode the motion envelope well, primarily due to the receptors with lower firing rates. We discuss the implications of our results for the identification of conspecifics through specific beat frequencies and its possible hindrance by active swimming.     

Systems-level modeling of neuronal circuits for leech swimming

This paper describes a mathematical model of the neuronal central pattern generator (CPG) that controls the rhythmic body motion of the swimming leech. The systems approach is employed to capture the neuronal dynamics essential for generating coordinated oscillations of cell membrane potentials by a simple CPG architecture with a minimal number of parameters. Based on input/output data from physiological experiments, dynamical components (neurons and synaptic interactions) are first modeled individually and then integrated into a chain of nonlinear oscillators to form a CPG. We show through numerical simulations that the values of a few parameters can be estimated within physiologically reasonable ranges to achieve good fit of the data with respect to the phase, amplitude, and period. This parameter estimation leads to predictions regarding the synaptic coupling strength and intrinsic period gradient along the nerve cord, the latter of which agrees qualitatively with experimental observations.     

Pattern generating and reflex-like processes controlling aiming movements in the presence of inertia,damping and gravity

A model is proposed, in which goal-directed movements of the forearm are controlled by a central pattern generator (CPG) initiated for exactly one period, and by reflex-analogous processes. Movement width is proportional to the amplitude factor of the CPG's output, and to the square of the CPG's period length. The period duration can be freely selected, thus enabling the CPG to accommodate its time scale to the period of others CPG's. Parameters which influence movement accuracy can be adjusted by means of closed control loop, which are discrete with respect to time: The time unit corresponds to the period of the CPG. For instance, momentum adjustment balances the CPG in such a manner that the velocity of the arm becomes zero on termination of the period, while gain adjustment serves to attain a correct movement length in the presence of an inertial load. Friction, stiffness and gravitational force are neutralized by additional reflex-type processes, interpretable as positive feedback loops with adjustable gain factors, using position and velocity signals.     

Worldwide frequency distribution of the ‘Gait keeper’ mutation in the DMRT3 gene

For centuries, domestic horses have represented an important means of transport and served as working and companion animals. Although their role in transportation is less important today, many horse breeds are still subject to intense selection based on their pattern of locomotion. A striking example of such a selected trait is the ability of a horse to perform additional gaits other than the common walk, trot and gallop. Those could be four‐beat ambling gaits, which are particularly smooth and comfortable for the rider, or pace, used mainly in racing. Gaited horse breeds occur around the globe, suggesting that gaitedness is an old trait, selected for in many breeds. A recent study discovered that a nonsense mutation in DMRT3 has a major impact on gaitedness in horses and is present at a high frequency in gaited breeds and in horses bred for harness racing. Here, we report a study of the worldwide distribution of this mutation. We genotyped 4396 horses representing 141 horse breeds for the DMRT3 stop mutation. More than half (2749) of these horses also were genotyped for a SNP situated 32 kb upstream of the DMRT3 nonsense mutation because these two SNPs are in very strong linkage disequilibrium. We show that the DMRT3 mutation is present in 68 of the 141 genotyped horse breeds at a frequency ranging from 1% to 100%. We also show that the mutation is not limited to a geographical area, but is found worldwide. The breeds with a high frequency of the stop mutation (>50%) are either classified as gaited or bred for harness racing.     

The cortex as a central pattern generator

Vertebrate spinal cord and brainstem central pattern generator (CPG) circuits share profound similarities with neocortical circuits. CPGs can produce meaningful functional output in the absence of sensory inputs. Neocortical circuits could be considered analogous to CPGs as they have rich spontaneous dynamics that, similar to CPGs, are powerfully modulated or engaged by sensory inputs, but can also generate output in their absence. We find compelling evidence for this argument at the anatomical, biophysical, developmental, dynamic and pathological levels of analysis. Although it is possible that cortical circuits are particularly plastic types of CPG ('learning CPGs'), we argue that present knowledge about CPGs is likely to foretell the basic principles of the organization and dynamic function of cortical circuits.