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Abstract

Rhythmic motor activity requires coordination of different muscles or muscle groups so that they are all active with the same cycle duration and appropriate phase relationships. The neural mechanisms for such phase coupling in vertebrate locomotion are not known. Swimming in the lamprey is accomplished by the generation of a travelling wave of body curvature in which the phase coupling between segments is so controlled as to give approximately one full wavelength on the body at any swimming speed. This article reviews work that has combined mathematical analysis, biological experimentation and computer simulation to provide a conceptual framework within which intersegmental coordination can be investigated. Evidence is provided to suggest that in the lamprey, ascending coupling is dominant over descending coupling and controls the intersegmental phase lag during locomotion. The significance of long-range intersegmental coupling is also discussed.

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... Here, we focus on a chain of unidirectionally coupled limit cycle oscillators, a relevant model in many biological settings [6,21,[34][35][36][37]. For example, sustained volumetric esophageal distension triggers stretch receptors along the distended region. ...
... Coupled oscillators models capable of producing both direct and retrograde propagating waves have previously been studied to model biological systems [34,[40][41][42]. Studying crawling in Drosophila larvae via a bidirectionally coupled chain of oscillators, Gjorgjieva et al. [40] obtained a direct wave when applying a stimulating input to only the anterior (proximal) oscillator, and a retrograde wave when applying an input to only the posterior (distal) oscillator. ...
... Thus, switching the direction of wave propagation required switching a localized input from the proximal to the distal oscillator. Other studies suggested that transitions in the direction of wave propagation occur through changes in the local intrinsic frequency of oscillation, a mechanism that can be implemented through frequency gradients, external perturbations, or varying local inputs [34,[42][43][44][45][46]. Varying the strength and orientation of the couplings along the chain has also been shown to influence the direction of wave propagation [13,14,42]. ...
Article
Some biological systems exhibit both direct and retrograde propagating signal waves despite unidirectional coupling. To explain this phenomenon, we study a chain of unidirectionally coupled Wilson-Cowan oscillators. Surprisingly, we find that changes in the homogeneous global input to the chain suffice to reverse the wave propagation direction. To obtain insights, we analyze the frequencies and bifurcations of the limit cycle solutions of the chain as a function of the global input. Specifically, we determine that the directionality of wave propagation is controlled by differences in the intrinsic frequencies of oscillators caused by the differential proximity of the oscillators to a homoclinic bifurcation.
... By tuning the coordination between the component oscillators, a neural circuit can generate a variety of activity patterns. For example, the spinal cord circuit can produce different behavior such as walking, trotting, or galloping by changing the phase differences between the neural oscillators that control various types of muscles in different limbs [10][11][12]. ...
... Despite numerous studies on coupled oscillators, the dynamic tuning of their coordination remains poorly understood [13,14]. Changing either the strength of coupling between component oscillators or the frequencies of the oscillators relative to each other can affect their coordination [12,[15][16][17]. This may be achieved through neuromodulation, separately changing the inputs to each oscillator, or by varying the rhythm frequency [18][19][20]. ...
... In the field of neuroscience, the rhythmic motor control of locomotion in vertebrates requires the phase coordination of propagating activity in the spinal cord [1,12,28]. ...
Article
Full-text available
Coupled neuronal oscillators generate a wide range of dynamic activity patterns that drive various behaviors in animals. A single neural circuit can generate multiple activity patterns and switch between them by tuning the coordination between its component oscillators. However, the mechanisms underlying how neural circuits dynamically tune the phase relationship between oscillators are not fully understood. We studied the phase relationships between two unidirectionally coupled oscillators and found that the duration of synaptic input to the driven oscillator controls the phase difference between them. The phase difference can smoothly change over a wide range, from large phase advances to phase delays, with variations in the duration of synaptic input. The control of the phase difference derives from the entrainment properties shared by commonly used neural oscillators, particularly when they are close to their relaxation limit. We applied our findings on the control of phase difference to a chain of unidirectionally synaptically coupled identical oscillators. We demonstrated that the direction and speed of activity propagation in the chain can be controlled by the duration of synaptic input.
... In animals like lamprey, leech, and fish larvae the basic motion consists of an undulation wave traveling along the body. A striking experimental observation in the undulatory motion of various species is that the timing of the activity of the different segments of the animal depends only weakly on the frequency of the motion (Williams, 1992). More precisely, the delay in the oscillatory motion from one segment to the next is not given by a fixed time delay determined by synaptic time constants or axonal propagation speed but rather by a fraction of the period that is quite independent of the frequency of the wave. ...
... The steady-state phase relationship of spinal-cord models has been studied extensively and comprehensively in the context of coupled-oscillator theory, where each segment corresponds to an individual oscillator that is coupled to its nearest and possibly further neighbors (Kopell and Ermentrout, 1986;Kopell et al., 1991;Cohen et al., 1992). For the commonly considered case of weakly interacting oscillators the coupling arises through the corresponding phase differences. ...
... For the commonly considered case of weakly interacting oscillators the coupling arises through the corresponding phase differences. For bidirectionally coupled oscillators it has been shown that the phase difference can be frequency-independent if the coupling function does not depend on the oscillation period (Kopell and Ermentrout, 1986;Kopell et al., 1991;Cohen et al., 1992). This has been obtained explicitly in some models that were formulated in terms of firing rates (Williams, 1992;Varkonyi et al., 2008). ...
Preprint
Full-text available
A significant component of the repetitive dynamics during locomotion in vertebrates is generated within the spinal cord. The legged locomotion of mammals is most likely controled by a hierarchical, multi-layer spinal network structure, while the axial circuitry generating the undulatory swimming motion of animals like lamprey is thought to have only a single layer in each segment. Recent experiments have suggested a hybrid network structure in zebrafish larvae in which two types of excitatory interneurons (V2a-I and V2a-II) both make first-order connections to the brain and last-order connections to the motor pool. These neurons are connected by electrical and chemical synapses across segments. Through computational modeling and an asymptotic perturbation approach we show that this interleaved interaction between the two neuron populations allows the spinal network to quickly establish the correct activation sequence of the segments when starting from random initial conditions and to reduce the dependence of the intersegmental phase difference (ISPD) of the oscillations on the swimming frequency. The latter reduces the frequency dependence of the waveform of the swimming motion. In the model the reduced frequency dependence is largely due to the different impact of chemical and electrical synapses on the ISPD and to the significant spike-frequency adaptation that has been observed experimentally in V2a-II neurons, but not in V2a-I neurons. Our model makes experimentally testable predictions and points to a benefit of the hybrid structure for undulatory locomotion that may not be relevant for legged locomotion.
... Several types of models have proved useful. First, a mathematical model of phase-coupled oscillators (PCO) has been applied to the lamprey swimming (reviewed in Cohen et al. 1992) and crayfish swimmeret system (Skinner at al. 1997). Second, detailed cellular models have been constructed for these systems, and these have replicated some experimental findings (lamprey: Wadden et al. 1997;crayfish: Skinner and Mulloney 1998a). ...
... The amount of this phase shift of the nth oscillator, ⌬ n , is determined by a phase response curve (PRC). In our model, the interactions mediated by individual INs are characterized by sinusoidal PRCs (Fig. 2) (see Cohen et al. 1992;Pearce and Friesen 1988;Skinner et al. 1997) and have the generic form: ⌬ T ϭ A ϫ sin( T Ϫ x), when the coupling is active, i.e., when S is within an appropriate range of values; otherwise ⌬ T ϭ 0, where S is the phase of the oscillator that sends out the coupling signal and T is the phase of the target oscillator (we use the term n when describing a specific segmental oscillator, n); A is the maximum amplitude; and x is a phase parameter that is determined by the phase of target IN (see RESULTS) and is different for each channel (Table 1). Therefore the phase of any oscillator n at time t ϩ ⌬t is given by n ͑t ϩ ⌬t͒ ϭ n ͑t͒ ϩ ⌬t ϫ 360°/P n ϩ ⌬ n where P n is the intrinsic period of the nth oscillator, ⌺⌬ n is the total phase shift summed over all channels from all connected oscillators that are active (up to 6 oscillators away in either direction). ...
... Recent modeling studies have helped clarify the central mechanisms of intersegmental coordination (reviewed in Skinner and Mulloney 1998b). A mathematical model of phase-coupled oscillators (PCOs) has been applied to the lamprey swimming (reviewed in Cohen et al. 1992) and crayfish swimmeret system (Skinner et al. 1997). The prediction of the PCO models that intersegmental coordination is produced by asymmetric couplings has been confirmed in the crayfish swimmeret system. ...
Article
Sensory feedback as well as the coupling signals within the CNS are essential for leeches to produce intersegmental phase relationships in body movements appropriate for swimming behavior. To study the interactions between the central pattern generator (CPG) and peripheral feedback in controlling intersegmental coordination, we have constructed a computational model for the leech swimming system with physiologically realistic parameters. First, the leech swimming CPG is simulated by a chain of phase oscillators coupled by three channels of coordinating signals. The activity phase, the projection direction, and the phase response curve (PRC) of each channel are based on the identified intersegmental interneuron network. Output of this largely constrained model produces stable coordination in the simulated CPG with average phase lags of 8–10°/segment in the period range from 0.5 to 1.5 s, similar to those observed in isolated nerve cords. The model also replicates the experimental finding that shorter chains of leech nerve cords have larger phase lags per segment. Sensory inputs, represented by stretch receptors, were subsequently incorporated into the CPG model. Each stretch receptor with its associated PRC, which was defined to mimic the experimental results of phase-dependent phase shifts of the central oscillator by the ventral stretch receptor, can alter the phase of the local central oscillator. Finally, mechanical interactions between the muscles from neighboring segments were simulated by PRCs linking adjacent stretch receptors. This model shows that interactions between neighboring muscles could globally increase the phase lags to the larger value required for the one-wavelength body form observed in freely swimming leeches. The full model also replicates the experimental observation that leeches with severed nerve cords have larger intersegmental phase lags than intact animals. The similarities between physiological and simulation results demonstrate that we have established a realistic model for the central and peripheral control of intersegmental coordination of leech swimming.
... The steady-state phase relationship of spinal-cord models has been studied extensively and comprehensively in the context of coupled-oscillator theory, where each segment corresponds to an individual oscillator that is coupled to its nearest and possibly further neighbors (Kopell & Ermentrout, 1986;Kopell et al., 1991;Cohen et al., 1992). For the commonly considered case of weakly interacting oscillators the coupling arises through the corresponding phase differences. ...
... For the commonly considered case of weakly interacting oscillators the coupling arises through the corresponding phase differences. For bidirectionally coupled oscillators it has been shown that the phase difference can be frequencyindependent if the coupling function does not depend on the oscillation period (Kopell & Ermentrout, 1986;Kopell et al., 1991;Cohen et al., 1992). This has been obtained explicitly in some models that were formulated in terms of firing rates (Williams, 1992;Varkonyi et al., 2008). ...
Article
Full-text available
A significant component of the repetitive dynamics during locomotion in vertebrates is generated within the spinal cord. The legged locomotion of mammals is most likely controled by a hierarchical, multi-layer spinal network structure, while the axial circuitry generating the undulatory swimming motion of animals like lamprey is thought to have only a single layer in each segment. Recent experiments have suggested a hybrid network structure in zebrafish larvae in which two types of excitatory interneurons (V2a-I and V2a-II) both make first-order connections to the brain and last-order connections to the motor pool. These neurons are connected by electrical and chemical synapses across segments. Through computational modeling and an asymptotic perturbation approach we show that this interleaved interaction between the two neuron populations allows the spinal network to quickly establish the correct activation sequence of the segments when starting from random initial conditions, as needed for a swimming spurt, and to reduce the dependence of the intersegmental phase difference (ISPD) of the oscillations on the swimming frequency. The latter reduces the frequency dependence of the waveform of the swimming motion. In the model the reduced frequency dependence is largely due to the different impact of chemical and electrical synapses on the ISPD and to the significant spike-frequency adaptation that has been observed experimentally in V2a-II neurons, but not in V2a-I neurons. Our model makes experimentally testable predictions and points to a benefit of the hybrid structure for undulatory locomotion that may not be relevant for legged locomotion.
... Model organisms have been used to study the complex interactions between the nervous system, body mechanics, and environmental dynamics in generating and coordinating locomotion [24,33]. Some studies of locomotion in model organisms highlight feedforward control of locomotion, where the nervous system drives motor activity and sensory feedback plays only a modulatory role; these include swimming behavior in lamprey, crayfish, and leeches [8,36,45,46,47,50]. However, other organisms, such as cockroaches and stick insects, can only be understood as fully integrated neuromechanical systems because sensory feedback is essential to generate and coordinate movements [4,17,24,32,42]. ...
... This 1-D map is given by (C.4) α 2 = F (α 0 ) = −F 2 (F 1 (−F 2 (F 1 (α 0 )))), and is generated by intermediate maps along each solution branch α 1/2 = F 1 (α 0 ), (C.5) α 1 = F 2 (α 1/2 ), (C.6) α 3/2 = −F 1 (−α 1 ), (C.7) α 2 = −F 2 (−α 3/2 ), (C. 8) which we define in the following subsections. Note that depending on parameters, these maps may not be defined, which indicates that the system instead converges to a stable equilibrium along one of the solution branches, breaking the neuromechanical loop. ...
Thesis
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Understanding principles of neurolocomotion requires the synthesis of neural activity, sensory feedback, and biomechanics. The nematode C. elegans is an ideal model organism for studying lo- comotion in an integrated neuromechanical setting because its nervous system is well characterized and its forward swimming gait adapts to the surrounding fluid using sensory feedback. However, it is not understood how the gait emerges from mechanical forces, neuronal coupling, and sensory feedback mechanisms. Here, a modular neuromechanical model of C. elegans forward locomotion is developed and analyzed. The model captures the experimentally observed gait adaptation over a wide range of parameters, provided that the muscle response to input from the nervous system is faster than the body response to changes in internal and external forces. The model is analyzed as a system of coupled neuromechanical oscillator modules using the theory of weakly coupled oscillators to identify the relative roles of body mechanics, neural coupling, and proprioceptive coupling in coordinating the undulatory gait. The analysis shows that the wavelength of body undulations is set by the relative strengths of these three coupling forms and the experimentally observed decrease in wavelength in response to increasing fluid viscosity is the result of an increase in the relative strength of mechanical coupling, which promotes a short wavelength. Parameters of the neuromechanical modules were also explored to assess their effects on the existence, period, amplitude, and phase response properties of the oscillations; this analysis allows the coordination trend of the full neuromechanical model to be inferred from the properties of the individual neu- romechanical oscillator modules themselves. The neuromechanical module is also reduced to a form that is analytically piecewise solvable, which allows for the construction of a 1-D Poincar ́e map that captures the limit cycle dynamics. This builds a framework for future analysis of the biophysical mechanisms underlying the oscillator properties and thus the coordination of the full neuromechan- ical model. The systematic analysis of the neuromechanical model presented in this dissertation provides a deeper understanding of how the interactions between the neuromechanical components of the C. elegans forward locomotion system produce coordination and gait adaptation.
... Multistability [12][13][14], basin of attractions [15,16], and traveling waves [17] are some of the fundamental phenomena directly related with equilibriums in vari-ants of the Kuramoto model with both attractive and repulsive phase couplings. 1 These phenomena are also observed in real-world networks [18][19][20][21]. Such manifestations are mostly studied in the continuous thermodynamic limit and keep not yet well understood. ...
... initial configuration of phases, given by Eq.(18); b comparison between the theoretical prediction (obtained with Fourier decomposition and our equations for perturbations, blue curve) and the results obtained by direct numerical integration of 50 differential equations (red circles). (Colorfigure online) ...
Article
Full-text available
We study the Kuramoto–Sakaguchi model composed by N identical phase oscillators symmetrically coupled. Ranging from local (one-to-one, R=1) to global (all-to-all, R=N/2) couplings, we derive a general solution that describes the network dynamics close to an equilibrium. Therewith, we build stability diagrams according to N and R bringing to the light a rich scenery of attractors, repellers, saddles, and non-hyperbolic equilibriums. Our result also uncovers the obscure repulsive regime of the model through bifurcation analysis. Numerical simulations show great accordance with our analytical studies. The exact knowledge of the behavior close to equilibriums may be a fundamental step to investigate phenomena about synchronization in networks. As an example, in the end, we discuss the dynamics behind chimera states from our results.
... T raveling waves of mechanical actuation provide a versatile strategy for locomotion and transport in both natural [1][2][3] and engineered 4-8 systems across many scales. In vertebrates such as the aquatic lamprey 1,9 , these and other rhythmic motor patterns are orchestrated by networks of neurons called central pattern generators (CPGs) 10 , which are often idealized as systems of coupled oscillators 1,9 . The rhythmic output of these oscillators is relayed to actuators (e.g., muscles) to produce complex motions without the need for sensory feedback. ...
... T raveling waves of mechanical actuation provide a versatile strategy for locomotion and transport in both natural [1][2][3] and engineered 4-8 systems across many scales. In vertebrates such as the aquatic lamprey 1,9 , these and other rhythmic motor patterns are orchestrated by networks of neurons called central pattern generators (CPGs) 10 , which are often idealized as systems of coupled oscillators 1,9 . The rhythmic output of these oscillators is relayed to actuators (e.g., muscles) to produce complex motions without the need for sensory feedback. ...
Article
Full-text available
Traveling waves of mechanical actuation provide a versatile strategy for locomotion and transport in both natural and engineered systems across many scales. These rhythmic motor patterns are often orchestrated by systems of coupled oscillators such as beating cilia or firing neurons. Here, we show that similar motions can be realized within linear arrays of conductive particles that oscillate between biased electrodes through cycles of contact charging and electrostatic actuation. The repulsive interactions among the particles along with spatial gradients in their natural frequencies lead to phase-locked states characterized by gradients in the oscillation phase. The frequency and wavelength of these traveling waves can be specified independently by varying the applied voltage and the electrode separation. We demonstrate how traveling wave synchronization can enable the directed transport of material cargo. Our results suggest that simple energy inputs can coordinate complex motions with opportunities for soft robotics and colloidal machines.
... Neural circuits consisting of interconnected CPGs have been often modeled mathematically as chains of coupled phase oscillators, for example, Cohen et al. (1992), Williams et al. (1990), and Skinner et al. (1997). In phase models, the state of the kth oscillator is described completely by its phase θ k ∈ [0, 1). ...
... The functional implications of long-range connections have been considered in other systems. In a model of undulatory swimming in the lamprey, Cohen et al. (1992) showed that long-range coupling could alleviate the strict conditions necessary for 1% phase delays. Specifically, in nearestneighbor oscillator models of that system, the interaction functions must be finely tuned in order to generate such small and precise phase-differences. ...
Article
Full-text available
During forward swimming, crayfish and other long-tailed crustaceans rhythmically move four pairs of limbs called swimmerets to propel themselves through the water. This behavior is characterized by a particular stroke pattern in which the most posterior limb pair leads the rhythmic cycle and adjacent swimmerets paddle sequentially with a delay of roughly 25% of the period. The neural circuit underlying limb coordination consists of a chain of local modules, each of which controls a pair of limbs. All modules are directly coupled to one another, but the inter-module coupling strengths decrease with the distance of the connection. Prior modeling studies of the swimmeret neural circuit have included only the dominant nearest-neighbor coupling. Here, we investigate the potential modulatory role of long-range connections between modules. Numerical simulations and analytical arguments show that these connections cause decreases in the phase-differences between neighboring modules. Combined with previous results from a computational fluid dynamics model, we posit that this phenomenon might ensure that the resultant limb coordination lies within a range where propulsion is optimal. To further assess the effects of long-range coupling, we modify the model to reflect an experimental preparation where synaptic transmission from a middle module is blocked, and we generate predictions for the phase-locking properties in this system.
... Many neuronal circuits driving body or limb movements in segmented animals are composed of a chain of HCOs of different lengths (Mulloney and Smarandache 2010;Ijspeert 2008;Stein 2007;Grillner 2006;Marder and Calabrese 1996;Cohen and Kiemel 1993). A chain with four or five pairs of HCOs produces the metachronal swimmeret coordination in long-tailed crustaceans (Mulloney and Smarandache-Wellmann 2012), a chain of 3 HCOs helps produce different gaits in the legged locomotion of insects (Daun-Gruhn and Toth 2011;Proctor et al. 2010), and a chain of approximately 100 HCOs drives the undulatory body movement of lamprey (Cohen et al. 1992). All of these systems display phase-waves in chains of HCOs that are robust to variations in frequency. ...
... The coupled phase model is a commonly used mathematical framework for studying the dynamics of interconnected oscillators (Cohen et al. 1992;Williams et al. 1990; Kopell and Ermentrout 1988;Kuramoto 1984). In a phase model, the state of each oscillator is described completely by its phase θ k , in which k is the index of the oscillator. ...
Article
Full-text available
Many neuronal circuits driving coordinated locomotion are composed of chains of half-center oscillators (HCOs) of various lengths. The HCO is a common motif in central pattern generating circuits (CPGs); an HCO consists of two neurons, or two neuronal populations, connected by reciprocal inhibition. To maintain appropriate motor coordination for effective locomotion over a broad range of frequencies, chains of CPGs must produce approximately constant phase-differences in a robust manner. In this article, we study phase-locking in chains of nearest-neighbor coupled HCOs and examine how the circuit architecture can promote phase-constancy, i.e., inter-HCO phase-differences that are frequency-invariant. We use two models with different levels of abstraction: (1) a conductance-based model in which each neuron is modeled by the Morris–Lecar equations (the ML-HCO model); and (2) a coupled phase model in which the state of each HCO is captured by its phase (the phase-HCO model). We show that one of four phase-waves with inter-HCO phase-differences at approximately 0, 25, 50 or 75 % arises robustly as a result of the inter-HCO connection topology, and its robust existence is not affected by the number of HCOs in the chain, the difference in strength between the ascending and descending nearest-neighbor connections, or the number of nearest-neighbor connections. Our results show that the internal anti-phase structure of the HCO and an appropriate inter-HCO connection topology together can provide a mechanism for robust (i.e., frequency-independent) limb coordination in segmented animals, such as the 50 % interlimb phase-differences in the tripod gate of stick insects and cockroaches, and the 25 % interlimb phase-differences in crayfish and other long-tailed crustaceans during forward swimming.
... Following previous studies of CPGs for undulatory locomotion [44,45] to describe this system, we describe the oscillator dynamics by a limit cycle close to a Hopf bifurcation. In this case, phase reduction of the dynamics leads to [46] (2.2) Videomicroscopy from video 1 of the supplemental material of [18]. ...
Article
Full-text available
Dissipative environments are ubiquitous in nature, from microscopic swimmers in low-Reynolds-number fluids to macroscopic animals in frictional media. In this study, we consider a mathematical model of a slender elastic locomotor with an internal rhythmic neural pattern generator to examine various undulatory locomotion such as Caenorhabditis elegans swimming and crawling behaviours. By using local mechanical load as mechanosensory feedback, we have found that undulatory locomotion robustly emerges in different rheological media. This progressive behaviour is then characterized as a global attractor through dynamical systems analysis with a Poincaré section. Furthermore, by controlling the mechanosensation, we were able to design the dynamical systems to manoeuvre with progressive, reverse and turning motions as well as apparently random, complex behaviours, reminiscent of those experimentally observed in C. elegans. The mechanisms found in this study, together with our dynamical systems methodology, are useful for deciphering complex animal adaptive behaviours and designing robots capable of locomotion in a wide range of dissipative environments.
... These researchers concluded that since the dendrite length is fixed then so should be the wavelength. It can be shown, using a coupled oscillator model (Kopell and Ermentrout, 1988;Cohen et al. 1992; ...
Preprint
Undulatory locomotion is common to nematodes as well as to limbless vertebrates, but its control is not understood in spite of the identification of hundred of genes involved in Caenorhabditis elegans locomotion. To reveal the mechanisms of nematode undulatory locomotion, we quantitatively analyzed the movement of C. elegans with genetic perturbations to neurons, muscles, and skeleton (cuticle). We also compared locomotion of different Caenorhabditis species. We constructed a theoretical model that combines mechanics and biophysics, and that is constrained by the observations of propulsion and muscular velocities, as well as wavelength and amplitude of undulations. We find that normalized wavelength is a conserved quantity among wild-type C. elegans individuals, across mutants, and across different species. The velocity of forward propulsion scales linearly with the velocity of the muscular wave and the corresponding slope is also a conserved quantity and almost optimal; the exceptions are in some mutants affecting cuticle structure. In theoretical terms, the optimality of the slope is equivalent to the exact balance between muscular and visco-elastic body reaction bending moments. We find that the amplitude and frequency of undulations are inversely correlated and provide a theoretical explanation for this fact. These experimental results are valid both for young adults and for all larval stages of wild type C. elegans. In particular, during development, the amplitude scales linearly with the wavelength, consistent with our theory. We also investigated the influence of substrate firmness on motion parameters, and found that it does not affect the above invariants. In general, our biomechanical model can explain the observed robustness of the mechanisms controlling nematode undulatory locomotion.
... Some systems, such as lamprey (Cohen et al., 1992) and leech (Marder and Calabrese, 1996) swimming, require constant phase across frequencies. In either case, when such systems transition to new steady frequencies, they normally do so in a smooth, monotonic fashion. ...
Article
Full-text available
Motor systems operate over a range of frequencies and relative timing (phase). We studied the role of the hyperpolarization-activated inward current (I h ) in regulating these features in the pyloric rhythm of the stomatogastric ganglion (STG) of the crab, Cancer borealis, as temperature was altered from 11°C to 21°C. Under control conditions, rhythm frequency increased monotonically with temperature, while the phases of the pyloric dilator (PD), lateral pyloric (LP), and pyloric (PY) neurons remained constant. Blocking I h with cesium (Cs ⁺ ) phase advanced PD offset, LP onset, and LP offset at 11°C, and the latter two further advanced as temperature increased. In Cs ⁺ the frequency increase with temperature diminished and the Q 10 of the frequency dropped from ~1.75 to ~1.35. Unexpectedly in Cs ⁺ , the frequency dynamics became non-monotonic during temperature transitions; frequency initially dropped as temperature increased, then rose once temperature stabilized, creating a characteristic ‘jag’. Interestingly, these jags persisted during temperature transitions in Cs ⁺ when the pacemaker was isolated by picrotoxin, although the temperature-induced change in frequency recovered to control levels. Overall, these data suggest that I h plays an important role in maintaining smooth transitory responses and persistent frequency increases by different mechanisms in the pyloric circuitry during temperature fluctuations.
... The architecture of CPGs is often depicted as coupled oscillators with distributed sensing and actuation. For undulatory swimming animals like leeches and lampreys, for instance, the CPG is formed as a chain of segmental oscillators, each of which receives local sensory feedback and induces local muscle contraction [8,9]. Intersegmental communication was considered essential for coordinating the phase timing of the oscillators so that traveling body waves are generated. ...
... S2). Compared with similar types of models of swimming CPG circuits (49)(50)(51), the novelty here is the feedback mechanism and the study of its role in rhythm generation and coordination in both a sensorized robot interacting with real physics and in simulation. ...
... First, the tapered body plan and the bending rigidity of the elastic body were chosen to reflect those of lampreys (28,29). Second, the strength of the neural connections in the phase oscillator model of the CPG driving the motion was based upon experiments on lamprey (30,31). In addition, we use the work-dependent deactivation muscle model (32,33) with parameters estimated from lamprey experiments. ...
Article
Spinal injuries in many vertebrates can result in partial or complete loss of locomotor ability. While mammals often experience permanent loss, some nonmammals, such as lampreys, can regain swimming function, though the exact mechanism is not well understood. One hypothesis is that amplified proprioceptive (body-sensing) feedback can allow an injured lamprey to regain functional swimming even if the descending signal is lost. This study employs a multiscale, integrative, computational model of an anguilliform swimmer fully coupled to a viscous, incompressible fluid and examines the effects of amplified feedback on swimming behavior. This represents a model that analyzes spinal injury recovery by combining a closed-loop neuromechanical model with sensory feedback coupled to a full Navier-Stokes model. Our results show that in some cases, feedback amplification below a spinal lesion is sufficient to partially or entirely restore effective swimming behavior.
... In animals with a segmented CNS, with the possible exception of the small nervous system of C. elegans that does not have CPG interneurons [10], each CNS segment typically contains an independent CPG that generates its own rhythm. Phase shifts between segments are accomplished by differential coupling, i.e., gradient or asymmetrical excitatory projections between segmental CPGs [31][32][33][34][35][36]. The underlying circuit and synaptic basis for asymmetrical excitatory projections has been most extensively characterized in the crayfish swimmeret system [37][38][39]. ...
Article
Full-text available
Locomotion in mollusc Aplysia is implemented by a pedal rolling wave, a type of axial locomotion. Well-studied examples of axial locomotion (pedal waves in Drosophila larvae and body waves in leech, lamprey, and fish) are generated in a segmented nervous system via activation of multiple coupled central pattern generators (CPGs). Pedal waves in molluscs, however, are generated by a single pedal ganglion, and it is unknown whether there are single or multiple CPGs that generate rhythmic activity and phase shifts between different body parts. During locomotion in intact Aplysia , bursting activity in the parapedal commissural nerve (PPCN) was found to occur during tail contraction. A cluster of 20 to 30 P1 root neurons (P1Ns) on the ventral surface of the pedal ganglion, active during the pedal wave, were identified. Computational cluster analysis revealed that there are 2 phases to the motor program: phase I (centered around 168°) and phase II (centered around 357°). PPCN activity occurs during phase II. The majority of P1Ns are motoneurons. Coactive P1Ns tend to be electrically coupled. Two classes of pedal interneurons (PIs) were characterized. Class 1 (PI1 and PI2) is active during phase I. Their axons make a loop within the pedal ganglion and contribute to locomotor pattern generation. They are electrically coupled to P1Ns that fire during phase I. Class 2 (PI3) is active during phase II and innervates the contralateral pedal ganglion. PI3 may contribute to bilateral coordination. Overall, our findings support the idea that Aplysia pedal waves are generated by a single CPG.
... Spinal cord Ermentrout, 1986, 1988;Cohen et al., 1992;Grillner et al., 1995) It includes frequency shifts with sensory, cognitive or motor variables and frequency differences observed across structures. ...
Article
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Brain oscillations emerge during sensory and cognitive processes and have been classified into different frequency bands. Yet, even within the same frequency band and between nearby brain locations, the exact frequencies of brain oscillations can differ. These frequency differences (detuning) have been largely ignored and play little role in current functional theories of brain oscillations. This contrasts with the crucial role that detuning plays in synchronization theory, as originally derived in physical systems. Here, we propose that detuning is equally important to understand synchronization in biological systems. Detuning is a critical control parameter in synchronization, which is not only important in shaping phase-locking, but also in establishing preferred phase relations between oscillators. We review recent evidence that frequency differences between brain locations are ubiquitous and essential in shaping temporal neural coordination. With the rise of powerful experimental techniques to probe brain oscillations, the contributions of exact frequency and detuning across neural circuits will become increasingly clear and will play a key part in developing a new understanding of the role of oscillations in brain function.
... The undulation patterns of lampreys and leeches as they swim are mainly generated by internal coupling of CPGs (Cohen et al., 1992;Grillner et al., 1995;KristanJr. et al., 2005). ...
Article
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Multi-legged animals show several types of ipsilateral interlimb coordination. Millipedes use a direct-wave gait, in which the swing leg movements propagate from posterior to anterior. In contrast, centipedes use a retrograde-wave gait, in which the swing leg movements propagate from anterior to posterior. Interestingly, when millipedes walk in a specific way, both direct and retrograde waves of the swing leg movements appear with the waves' source, which we call the source-wave gait. However, the gait generation mechanism is still unclear because of the complex nature of the interaction between neural control and dynamic body systems. The present study used a simple model to understand the mechanism better, primarily how local sensory feedback affects multi-legged locomotion. The model comprises a multi-legged body and its locomotion control system using biologically inspired oscillators with local sensory feedback, phase resetting. Each oscillator controls each leg independently. Our simulation produced the above three types of animal gaits. These gaits are not predesigned but emerge through the interaction between the neural control and dynamic body systems through sensory feedback (embodied sensorimotor interaction) in a decentralized manner. The analytical description of these gaits' solution and stability clarifies the embodied sensorimotor interaction's functional roles in the interlimb coordination.
... S2). Compared with similar types of models of swimming CPG circuits (49)(50)(51), the novelty here is the feedback mechanism and the study of its role in rhythm generation and coordination in both a sensorized robot interacting with real physics and in simulation. ...
Article
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.
... Computational modeling studies based on experimental data in the lamprey then further explored how a distributed organization of unit burst generators along the spinal cord can endow the spinal networks with the flexibility to switch between coordination patterns (such as in forward and backward swimming) while simultaneously allowing the characteristic expression of constant phase differences between segments [20,[79][80][81]. Intersegmental coordination has been studied extensively for multiple model systems through computational modeling in several contexts and at various levels of complexity (e.g., [15,[80][81][82][83][84][85][86][87][88]). ...
Article
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Neuronal circuits in the spinal cord are essential for the control of locomotion. They integrate supraspinal commands and afferent feedback signals to produce coordinated rhythmic muscle activations necessary for stable locomotion. For several decades, computational modeling has complemented experimental studies by providing a mechanistic rationale for experimental observations and by deriving experimentally testable predictions. This symbiotic relationship between experimental and computational approaches has resulted in numerous fundamental insights. With recent advances in molecular and genetic methods, it has become possible to manipulate specific constituent elements of the spinal circuitry and relate them to locomotor behavior. This has led to computational modeling studies investigating mechanisms at the level of genetically defined neuronal populations and their interactions. We review literature on the spinal locomotor circuitry from a computational perspective. By reviewing examples leading up to and in the age of molecular genetics, we demonstrate the importance of computational modeling and its interactions with experiments. Moving forward, neuromechanical models with neuronal circuitry modeled at the level of genetically defined neuronal populations will be required to further unravel the mechanisms by which neuronal interactions lead to locomotor behavior.
... Model organisms have been used to study the complex interactions between the nervous system, body mechanics, and environmental dynamics in generating and coordinating locomotion [18,25]. Some studies of locomotion in model organisms highlight feedforward control of locomotion, where the nervous system drives motor activity and sensory feedback plays only a modulatory role; these include legged locomotion in cockroaches and swimming in lamprey, crayfish, and leeches [8,15,18,28,37,38,39,42,46]. However, other systems such as stick insect locomotion are better understood as integrated neuromechanical systems because sensory feedback is essential to coordinating movements [4,24,34]. ...
Article
Understanding principles of neurolocomotion requires the synthesis of neural activity, sensory feedback, and biomechanics. The nematode C. elegans is an ideal model organism for studying locomotion in an integrated neuromechanical setting because its neural circuit has a well-characterized modular structure and its undulatory forward swimming gait adapts to the surrounding fluid with a shorter wavelength in higher viscosity environments. This adaptive behavior emerges from the neural modules interacting through a combination of mechanical forces, neuronal coupling, and sensory feedback mechanisms. However, the relative contributions of these coupling modes to gait adaptation are not understood. Here, an integrated neuromechanical model of C. elegans forward locomotion is developed and analyzed. The model consists of repeated neuromechanical modules that are coupled through the mechanics of the body, short-range proprioception, and gap-junctions. The model captures the experimentally observed gait adaptation over a wide range of mechanical parameters, provided that the muscle response to input from the nervous system is faster than the body response to changes in internal and external forces. The modularity of the model allows the use of the theory of weakly coupled oscillators to identify the relative roles of body mechanics, gap-junctional coupling, and proprioceptive coupling in coordinating the undulatory gait. The analysis shows that the wavelength of body undulations is set by the relative strengths of these three coupling forms. In a low-viscosity fluid environment, the competition between gap-junctions and proprioception produces a long wavelength undulation, which is only achieved in the model with sufficiently strong gap-junctional coupling. The experimentally observed decrease in wavelength in response to increasing fluid viscosity is the result of an increase in the relative strength of mechanical coupling, which promotes a short wavelength. Read More: https://epubs.siam.org/doi/10.1137/20M1346122
... Model organisms have been used to study the complex interactions between the nervous system, body mechanics, and environmental dynamics in generating and coordinating locomotion [15,21]. Some studies of locomotion in model organisms highlight feedforward control of locomotion, where the nervous system drives motor activity and sensory feedback plays only a modulatory role; these include swimming behavior in lamprey, crayfish, and leeches [6,24,32,33,34,36,40]. However, other organisms, such as cockroaches and stick insects, can only be understood as fully integrated neuromechanical systems because sensory feedback is essential to generate and coordinate movements [3,12,15,20,29]. ...
Preprint
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Understanding principles of neurolocomotion requires the synthesis of neural activity, sensory feedback, and biomechanics. The nematode C. elegans\textit{C. elegans} is an ideal model organism for studying locomotion in an integrated neuromechanical setting because its nervous system is well characterized and its forward swimming gait adapts to the surrounding fluid using sensory feedback. However, it is not understood how the gait emerges from mechanical forces, neuronal coupling, and sensory feedback mechanisms. Here, an integrated neuromechanical model of C. elegans\textit{C. elegans} forward locomotion is developed and analyzed. The model captures the experimentally observed gait adaptation over a wide range of parameters, provided that the muscle response to input from the nervous system is faster than the body response to changes in internal and external forces. The model is analyzed using the theory of weakly coupled oscillators to identify the relative roles of body mechanics, neural coupling, and proprioceptive coupling in coordinating the undulatory gait. The analysis shows that the wavelength of body undulations is set by the relative strengths of these three coupling forms. The model suggests that the experimentally observed decrease in wavelength in response to increasing fluid viscosity is the result of an increase in the relative strength of mechanical coupling, which promotes a short wavelength.
... The phase convergence described between two coupled neurons can account for the fixed phase relation between CPGs cycles (Marder et al., 2005). In the lamprey, the intersegmental time delay represents 1% of the cycle and is constant, irrespective of the swimming frequency (Cohen et al., 1992). ...
... This circuit configuration would enable simultaneous activation of transverse muscles in one segment (via the delay circuit) and longitudinal muscles in the forward adjacent segment (by the wave front) regardless of the speed of wave propagation. The circuit configuration would also maintain the relative timing (phase) of the two groups of muscles: in many axial motions including larval crawling, phase of individual muscle movements is maintained independent of the axial speed, in order to generate functional motor outputs [1][2][3][4][5][6][33][34][35][36] . As shown in Fig. 8a, each segment receives excitatory inputs at two distinct time points: at the arrival of the wave front (time 0) and at a later phase via the intersegmental delay circuits (time 1). ...
Article
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Animal locomotion requires spatiotemporally coordinated contraction of muscles throughout the body. Here, we investigate how contractions of antagonistic groups of muscles are intersegmentally coordinated during bidirectional crawling of Drosophila larvae. We identify two pairs of higher-order premotor excitatory interneurons present in each abdominal neuromere that intersegmentally provide feedback to the adjacent neuromere during motor propagation. The two feedback neuron pairs are differentially active during either forward or backward locomotion but commonly target a group of premotor interneurons that together provide excitatory inputs to transverse muscles and inhibitory inputs to the antagonistic longitudinal muscles. Inhibition of either feedback neuron pair compromises contraction of transverse muscles in a direction-specific manner. Our results suggest that the intersegmental feedback neurons coordinate contraction of synergistic muscles by acting as delay circuits representing the phase lag between segments. The identified circuit architecture also shows how bidirectional motor networks could be economically embedded in the nervous system.
... CPGs are neural networks capable of producing rhythmic patterned outputs without sensory input. CPGs in animal locomotion have been studied such as swimming in salamander [3] or lamprey [4], and heartbeat system in leech [5,6]. ...
Article
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The design of biomimetic robot is one popular research. To achieve this goal, the reproduction of animal locomotion is mandatory. Animal locomotion is created by the activities of Central Pattern Generator (CPG). CPGs are neural networks capable of producing rhythmic patterned outputs without rhythmic sensory or central input. We propose a network of several biomimetic CPGs using biomimetic neuron model and synaptic plasticity. This network is implemented on a field programmable gate array. We designed one unsupervised snake robot using this network of CPG. It is composed of one head wagon followed by seven slave wagons. Infrared sensors are also embedded in the head wagon. This robot can reproduce the locomotion of one snake.
... In the future we will incorporate models in which the amplitude of the CPG output may be affected by feedback input, which would require shifting from a phase model to a model that incorporates neural properties (e.g. [33,54]). ...
Article
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Like other animals, lampreys have a central pattern generator (CPG) circuit that activates muscles for locomotion and also adjusts the activity to respond to sensory inputs from the environment. Such a feedback system is crucial for responding appropriately to unexpected perturbations, but it is also active during normal unperturbed steady swimming and influences the baseline swimming pattern. In this study, we investigate different functional forms of body curvature-based sensory feedback and evaluate their effects on steady swimming energetics and kinematics, since little is known experimentally about the functional form of curvature feedback. The distributed CPG is modeled as chains of coupled oscillators. Pairs of phase oscillators represent the left and right sides of segments along the lamprey body. These activate muscles that flex the body and move the lamprey through a fluid environment, which is simulated using a full Navier-Stokes model. The emergent curvature of the body then serves as an input to the CPG oscillators, closing the loop. We consider two forms of feedback, each consistent with experimental results on lamprey proprioceptive sensory receptors. The first, referred to as directional feedback, excites or inhibits the oscillators on the same side, depending on the sign of a chosen gain parameter, and has the opposite effect on oscillators on the opposite side. We find that directional feedback does not affect beat frequency, but does change the duration of muscle activity. The second feedback model, referred to as magnitude feedback, provides a symmetric excitatory or inhibitory effect to oscillators on both sides. This model tends to increase beat frequency and reduces the energetic cost to the lamprey when the gain is high and positive. With both types of feedback, the body curvature has a similar magnitude. Thus, these results indicate that the same magnitude of curvature-based feedback on the CPG with different functional forms can cause distinct differences in swimming performance.
... Les CPGs [HOO00] sont des réseaux de neurones produisant un pattern rythmique de manière autonome. Ces oscillateurs locomoteurs contrôlent par exemple la nage chez la salamandre [IJS07] et la lamproie [COH92] et aussi le battement cardiaque chez la sangsue [CYM02]. Suivant la complexité de la fonction à remplir, les CPGs vont varier en taille (variation dans le nombre de neurones et de synapses). ...
... Multistability [12,13], basin of attractions [14,15], and traveling waves [16] are some of fundamental phenomena directly related with equilibriums in variants of the Kuramoto model with both attractive and repulsive phase couplings 1 . These phenomena are also observed in real-world networks [17][18][19][20]. Such manifestations are mostly studied in the continuous thermodynamic limit and keep not yet well understood. ...
Preprint
We study the Kuramoto-Sakaguchi (KS) model composed by any N identical phase oscillators symmetrically coupled. Ranging from local (one-to-one, R = 1) to global (all-to-all, R = N/2) couplings, we derive the general solution that describes the network dynamics next to an equilibrium. Therewith we build stability diagrams according to N and R bringing to the light a rich scenery of attractors, repellers, saddles, and non-hyperbolic equilibriums. Our result also uncovers the obscure repulsive regime of the KS model through bifurcation analysis. Moreover, we present numerical evolutions of the network showing the great accordance with our analytical one. The exact knowledge of the behavior close to equilibriums is a fundamental step to investigate phenomena about synchronization in networks. As an example, at the end we discuss the dynamics behind chimera states from the point of view of our results.
... Prior to this, among limbed vertebrates, traveling waves of motoneuron activation had been reported in the newt (Delvolvé et al., 1997), and the rodent (Bonnot et al., 2002;Cazalets, 2005), and suggested in the cat (Yakovenko et al., 2002), and human (Ivanenko et al., 2006). In vertebrates without limbs, a traveling wave occurs during swimming in the lamprey (Cohen et al., 1992), and zebrafish (Wiggin et al., 2012). Among invertebrates with limbs, a traveling wave occurs in the crustacean swimmeret system (Davis, 1969;Heitler, 1980). ...
Article
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The central pattern generator (CPG) architecture for rhythm generation remains partly elusive. We compare cat and frog locomotion results, where the component unrelated to pattern formation appears as a temporal grid, and traveling wave respectively. Frog spinal cord microstimulation with N-methyl-D-Aspartate (NMDA), a CPG activator, produced a limited set of force directions, sometimes tonic, but more often alternating between directions similar to the tonic forces. The tonic forces were topographically organized, and sites evoking rhythms with different force subsets were located close to the constituent tonic force regions. Thus CPGs consist of topographically organized modules. Modularity was also identified as a limited set of muscle synergies whose combinations reconstructed the EMGs. The cat CPG was investigated using proprioceptive inputs during fictive locomotion. Critical points identified both as abrupt transitions in the effect of phasic perturbations, and burst shape transitions, had biomechanical correlates in intact locomotion. During tonic proprioceptive perturbations, discrete shifts between these critical points explained the burst durations changes, and amplitude changes occurred at one of these points. Besides confirming CPG modularity, these results suggest a fixed temporal grid of anchoring points, to shift modules onsets and offsets. Frog locomotion, reconstructed with the NMDA synergies, showed a partially overlapping synergy activation sequence. Using the early synergy output evoked by NMDA at different spinal sites, revealed a rostrocaudal topographic organization, where each synergy is preferentially evoked from a few, albeit overlapping, cord regions. Comparing the locomotor synergy sequence with this topography suggests that a rostrocaudal traveling wave would activate the synergies in the proper sequence for locomotion. This output was reproduced in a two-layer model using this topography and a traveling wave. Together our results suggest two CPG components: modules, i.e., synergies; and temporal patterning, seen as a temporal grid in the cat, and a traveling wave in the frog. Animal and limb navigation have similarities. Research relating grid cells to the theta rhythm and on segmentation during navigation may relate to our temporal grid and traveling wave results. Winfree’s mathematical work, combining critical phases and a traveling wave, also appears important. We conclude suggesting tracing, and imaging experiments to investigate our CPG model.
... The neural circuitry responsible for the rostrocaudal coordination is not identified yet. It was proposed that neurons with ascending axons play a significant role (Cohen et al. 1992;Guan et al. 2001), but their cellular properties are still unknown. The excitability gradient between segments was also found to be crucial (Matsushima and Grillner 1992;reviewed in Grillner and Wallén 2002). ...
Chapter
In both invertebrates and vertebrates the basic rhythmicity and pattern of activity of most rhythmic behaviors is generated by central pattern generators (CPGs), neural networks that can generate rhythmic, patterned activity in the absence of rhythmic central input or sensory feedback. The chapter presents coordination mechanisms used in several particularly well-studied invertebrate preparations. The stomatogastric nervous system (STNS) has the advantage of many invertebrate preparations that network neuron make-up and synaptic connectivity are the same across individuals. As in invertebrates, vertebrate rhythmic movements are generated by CPGs, and CPGs involved in different motor functions are located in different regions of the central nervous system. In vertebrates, the locomotor CPGs are located in the spinal cord and those for respiration, mastication, swallowing, and airway defensive behaviors are located in the brainstem. A common occurrence in the invertebrate examples, and a constant theme in the vertebrate examples, are multiple mechanisms working in concert to produce coordination.
... CPGs coordinate complex muscle activations to help the animal achieve proper timing to accomplish a given task. They have been found to be involved in a large variety of movement behaviors including the leech heartbeat (Arbas and Calabrese, 1987), human breathing and gasping (Tryba et al., 2006), lobster digestion (Meyrand et al., 1994), turtle scratching (Mortin and Stein, 1989), and locomotion in stick insects (Bässler and Büschges, 1998), lamprey (Cohen et al., 1992), cats (Brown, 1914), and mice (Hägglund et al., 2013). ...
Article
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Animals dynamically adapt to varying terrain and small perturbations with remarkable ease. These adaptations arise from complex interactions between the environment and biomechanical and neural components of the animal's body and nervous system. Research into mammalian locomotion has resulted in several neural and neuro-mechanical models, some of which have been tested in simulation, but few " synthetic nervous systems " have been implemented in physical hardware models of animal systems. One reason is that the implementation into a physical system is not straightforward. For example, it is difficult to make robotic actuators and sensors that model those in the animal. Therefore, even if the sensorimotor circuits were known in great detail, those parameters would not be applicable and new parameter values must be found for the network in the robotic model of the animal. This manuscript demonstrates an automatic method for setting parameter values in a synthetic nervous system composed of non-spiking leaky integrator neuron models. This method works by first using a model of the system to determine required motor neuron activations to produce stable walking. Parameters in the neural system are then tuned systematically such that it produces similar activations to the desired pattern determined using expected sensory feedback. We demonstrate that the developed method successfully produces adaptive locomotion in the rear legs of a dog-like robot actuated by artificial muscles. Furthermore, the results support the validity of current models of mammalian locomotion. This research will serve as a basis for testing more complex locomotion controllers and for testing specific sensory pathways and biomechanical designs. Additionally, the developed method can be used to automatically adapt the neural controller for different mechanical designs such that it could be used to control different robotic systems.
... This analysis quantifies the gait that is exhibited on a continuum (Hildebrand (1989)), which can be thought of as a gait space. Dynamical systems approaches have had success in describing and reducing general dynamics (Kelso et al. (1986); Scholz et al. (1987); Kepler et al. (1992)), as well as in the more specific context of locomotion (Diedrich and Warren (1995); Hogan and Sternad (2013); Aoi et al. (2013); Cohen et al. (1992)). These approaches typically utilize a description that considers the phase of oscillatory components of the system. ...
Article
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Legged animals utilize gait selection to move effectively and must re-cover from environmental perturbations. We show that on rough terrain domestic dogs, Canis lupus familiaris, spend more time in longitudinal quasi-statically stable patterns of movement. Here longitudinal refers to the rostro-caudal, axis. We used an existing model in the literature to quantify the longitudinal quasi-static stability of gaits neighbouring the walk, and found that trot-like gaits are more stable. We thus hypothesized that when perturbed, the rate of return to a stable gait would depend on the direction of perturbation, such that perturbations towards less quasi-statically stable patterns of movement would be more rapid than those towards more stable patterns of movement. The net result of this would be greater time spent in longitudinally quasi-statically stable patterns of movement. Limb movement patterns in which diagonal limbs were more synchronised (those more like a trot) have higher longitudinal quasi-static stability. We therefore predicted that as dogs explored possible limb configurations on rough terrain at walking speeds, the walk would shift towards trot. We gathered experimental data quantifying dog gait when perturbed by rough terrain and confirmed this prediction using GPS and inertial sensors (n=6, p<0.05). By formulating gaits as trajectories on the N -torus we are able to make tractable analysis of gait similarity. These methods can be applied in a comparative study of gait control which will inform the ultimate role of the constraints and costs impacting locomotion, and have applications in diagnostic procedures for gait abnormalities, and in the development of agile legged robots.
... In the absence of the higher brain center function the reciprocal walking motion on the treadmill is thought to be controlled at least in part by the spinal cord and is organized by the central pattern generators (CPGs), CPGs activated by lower brain centers, the brain stem and the basal ganglia, which activate muscles responsible for repetitive walking movements [12][13][14]. There is evidence of CPGs activity in human but with more complex activation process than that demonstrated in animals [15]. ...
Article
Neurological disorders affect millions globally and necessitate advanced treatments, especially with an aging population. Brain Machine Interfaces (BMIs) and neuroprostheses show promise in addressing disabilities by mimicking biological dynamics through biomimetic Spiking Neural Networks (SNNs). Central Pattern Generators (CPGs) are small neural networks that, emulated through biomimetic networks, can replicate specific locomotion patterns. Our proposal involves a real-time implementation of a biomimetic SNN on FPGA, utilizing biomimetic models for neurons, synaptic receptors and synaptic plasticity. The system, integrated into a snake-like mobile robot where the neuronal activity is responsible for its locomotion, offers a versatile platform to study spinal cord injuries. Lastly, we present a preliminary closed-loop experiment involving bidirectional interaction between the artificial neural network and biological neuronal cells, paving the way for bio-hybrid robots and insights into neural population functioning.
Preprint
Motor systems operate over a range of frequencies and relative timing (phase). We studied the contribution of the hyperpolarization-activated inward current (I h ) to frequency and phase in the pyloric rhythm of the stomatogastric ganglion (STG) of the crab, Cancer borealis as temperature was altered from 11°C to 21°C. Under control conditions, the frequency of the rhythm increased monotonically with temperature, while the phases of the pyloric dilator (PD), lateral pyloric (LP), and pyloric (PY) neurons remained constant. When we blocked I h with cesium (Cs + ) PD offset, LP onset, and LP offset were all phase advanced in Cs + at 11°C, and the latter two further advanced as temperature increased. In Cs + the steady state increase in pyloric frequency with temperature diminished and the Q 10 of the pyloric frequency dropped from ∼1.75 to ∼1.35. Unexpectedly in Cs + , the frequency displayed non-monotonic dynamics during temperature transitions; the frequency initially dropped as temperature increased, then rose once temperature stabilized, creating a characteristic “jag”. Interestingly, these jags were still present during temperature transitions in Cs + when the pacemaker was isolated by picrotoxin, although the temperature-induced change in frequency recovered to control levels. Overall, these data suggest that I h plays an important role in the ability of this circuit to produce smooth transitory responses and persistent frequency increases by different mechanisms during temperature fluctuations.
Preprint
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Motor systems operate over a range of frequencies and relative timing (phase). We studied the contribution of the hyperpolarization-activated inward current (Ih) to frequency and phase in the pyloric rhythm of the stomatogastric ganglion (STG) of the crab, Cancer borealis as temperature was altered from 11 degrees C to 21 degrees C. Under control conditions, the frequency of the rhythm increased monotonically with temperature, while the phases of the pyloric dilator (PD), lateral pyloric (LP), and pyloric (PY) neurons remained constant. When we blocked Ih> with cesium (Cs+) PD offset, LP onset, and LP offset were all phase advanced in Cs+ at 11 degrees C, and the latter two further advanced as temperature increased. In Cs+ the steady state increase in pyloric frequency with temperature diminished and the Q10 of the pyloric frequency dropped from ~1.75 to ~1.35. Unexpectedly in Cs+, the frequency displayed non-monotonic dynamics during temperature transitions; the frequency initially dropped as temperature increased, then rose once temperature stabilized, creating a characteristic jag. Interestingly, these jags were still present during temperature transitions in Cs+ when the pacemaker was isolated by picrotoxin, although the temperature-induced change in frequency recovered to control levels. Overall, these data suggest that Ih plays an important role in the ability of this circuit to produces smooth transitory responses and persistent frequency increases by different mechanisms during temperature fluctuations.
Article
Plane waves have commonly been observed in recordings of human brains. These waves take the form of spatial phase gradients in the oscillatory potentials picked up by implanted electrodes. We first show that long but finite chains of nearest-neighbor coupled phase oscillators can produce an almost constant phase gradient when the edge effects interact with small heterogeneities in the local frequency. Next, we introduce a continuum model with nonlocal coupling and use singular perturbation methods to show similar interactions between the boundaries and small frequency differences. Finally, we show that networks of Wilson–Cowan equations can generate plane waves with the same mechanism.
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Synopsis The central pattern generator (CPG) in anguilliform swimming has served as a model for examining the neural basis of locomotion. This system has been particularly valuable for the development of mathematical models. As our biological understanding of the neural basis of locomotion has expanded, so too have these models. Recently, there have been significant advancements in our understanding of the critical role that mechanosensory feedback plays in robust locomotion. This work has led to a push in the field of mathematical modeling to incorporate mechanosensory feedback into CPG models. In this perspective piece, we review advances in the development of these models and discuss how newer complex models can support biological investigation. We highlight lamprey spinal cord regeneration as an area that can both inform these models and benefit from them.
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Symmetries in the external world constrain the evolution of neuronal circuits that allow organisms to sense the environment and act within it. Many small “modular” circuits can be viewed as approximate discretizations of the relevant symmetries, relating their forms to the functions they perform. The recent development of a formal theory of dynamics and bifurcations of networks of coupled differential equations permits the analysis of some aspects of network behavior without invoking specific model equations or numerical simulations. We review basic features of this theory, compare it to equivariant dynamics, and examine the subtle effects of symmetry when combined with network structure. We illustrate the relation between form and function through examples drawn from neurobiology, including locomotion, peristalsis, visual perception, balance, hearing, location detection, decision-making, and the connectome of the nematode Caenorhabditis elegans.
Chapter
Central pattern generators (CPGs) are ubiquitous neural circuits that contribute to an eclectic collection of rhythmic behaviors across an equally diverse assortment of animal species. Due to their prominent role in many neuromechanical phenomena, numerous bioinspired robots have been designed to both investigate and exploit the operation of these neural oscillators. In order to serve as effective tools for these robotics applications, however, it is often necessary to be able to adjust the phase alignment of multiple CPGs during operation. To achieve this goal, we present the design of our phase difference control (PDC) network using a functional subnetwork approach (FSA) wherein subnetworks that perform basic mathematical operations are assembled such that they serve to control the relative phase lead/lag of target CPGs. Our PDC network operates by first estimating the phase difference between two CPGs, then comparing this phase difference to a reference signal that encodes the desired phase difference, and finally eliminating any error by emulating a proportional controller that adjusts the CPG oscillation frequencies. The architecture of our PDC network, as well as its various parameters, are all determined via analytical design rules that allow for direct interpretability of the network behavior. Simulation results for both the complete PDC network and a selection of its various functional subnetworks are provided to demonstrate the efficacy of our methodology.KeywordsMultistate central pattern generatorsFunctional subnetwork approachPhase difference control
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In oscillatory systems, neuronal activity phase is often independent of network frequency. Such phase maintenance requires adjustment of synaptic input with network frequency, a relationship that we explored using the crab, Cancer borealis, pyloric network. The burst phase of pyloric neurons is relatively constant despite a 2-fold variation in network frequency. We used noise input to characterize how input shape influences burst delay of a pyloric neuron, and then used dynamic clamp to examine how burst phase depends on the period, amplitude, duration, and shape of rhythmic synaptic input. Phase constancy across a range of periods required a proportional increase of synaptic duration with period. However, phase maintenance was also promoted by an increase of amplitude and peak phase of synaptic input with period. Mathematical analysis shows how short-term synaptic plasticity can coordinately change amplitude and peak phase to maximize the range of periods over which phase constancy is achieved.
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Locomotion is one of the most basic abilities in animals. Neurobiologists have established that locomotion results from the activity of half-center oscillators that provides alternation of bursts. Central Pattern Generators (CPGs) are neural networks capable of producing rhythmic patterned outputs without rhythmic sensory or central input. We propose a network of several biomimetic CPGs using biomimetic neuron model and synaptic plasticity. This network is implemented on a FPGA (Field Programmable Gate Array). The network implementation architecture operates on a single computation core and in real-time. The real-time implementation of this CPGs network is validated by comparing it with biological data of leech heartbeat neural network. From these biomimetic CPGs, we use them for robotic applications and also for biomedical research to restore lost synaptic connections.
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The spinal cord has been well established as the site of generation of the locomotor rhythm in vertebrates, but studies have suggested that the caudal hindbrain in larval fish and amphibians can also generate locomotor rhythms. Here, we investigated whether the caudal hindbrain of the adult lamprey (Petromyzon marinus and Ichthyomyzon unicuspis) has the ability to generate the swimming rhythm. The hindbrain-spinal cord transition zone of the lamprey contains a bilateral column of somatic motoneurons that project via the spino-occipital (S-O) nerves to several muscles of the head. In the brainstem-spinal cord-muscle preparation, these muscles were found to burst and contract rhythmically with a left-right alternation when swimming activity was evoked with a brief electrical stimulation of the spinal cord. In the absence of muscles, the isolated brainstem-spinal cord preparation also produced alternating left-right bursts in S-O nerves (i.e., fictive swimming), and the S-O nerve bursts preceded the bursts occurring in the first ipsilateral spinal ventral root. After physical isolation of the S-O region using transverse cuts of the nervous system, the S-O nerves still exhibited rhythmic bursting with left-right alternation when glutamate was added to the bathing solution. We conclude that the S-O region of the lamprey contains a swimming rhythm generator that produces the leading motor nerve bursts of each swimming cycle, which then propagate down the spinal cord to produce forward swimming. The S-O region of the hindbrain-spinal cord transition zone may play a role in regulating speed, turning, and head orientation during swimming in lamprey.
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We present a case study of how topology can affect synchronization. Specifically, we consider arrays of phase oscillators coupled in a ring or a chain topology. Each ring is perfectly matched to a chain with the same initial conditions and the same random natural frequencies. The only difference is their boundary conditions: periodic for a ring, and open for a chain. For both topologies, stable phase-locked states exist if and only if the spread or "width" of the natural frequencies is smaller than a critical value called the locking threshold (which depends on the boundary conditions and the particular realization of the frequencies). The central question is whether a ring synchronizes more readily than a chain. We show that it usually does, but not always. Rigorous bounds are derived for the ratio between the locking thresholds of a ring and its matched chain, for a variant of the Kuramoto model that also includes a wider family of models.
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Phase-locking in a system of oscillators that are weakly coupled can be predicted by examining a related system in which the coupling is averaged over the oscillator cycle. This fails if the coupling is large. It is shown that in the presence of large interactions, a pair or a chain of oscillators may develop a new stable equilibrium state that corresponds to the cessation of oscillation. This phenomenon is robust for neural type interactions and does not happen in systems that are weakly coupled.
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A very broad framework is given for the investigation of long chains of N weakly coupled oscillators. The framework allows nonmonotonic changes of natural frequency along the chain, differences in coupling strength, anisotropy in the two directions of coupling, and very general local coupling functions. It is shown that the phase-locked solutions of all the systems of oscillators within this framework converge for large N to solutions of a class of nonlinear singularly perturbed two-point boundary-value problems. Using the latter continuum equations, it is also shown that there are parameter regimes in which the solutions have qualitatively different behavior, with a phase-transition-like change in behavior across the boundary between parameter regimes in the limit as N goes to infinity.
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1. To elucidate the neural mechanisms responsible for coordinating undulatory locomotor movements, the intersegmental phase lag was analyzed from ventral roots along the spinal cord during fictive swimming. It was induced by bath application of N-methyl-D-aspartate (NMDA) in in vitro preparations of lamprey spinal cord, while the excitability of different segments were modified. The phase lag between consecutive segments during normal forward swimming is 1% of the cycle duration in a broad range of values. Rostral segments are activated before more caudal ones. 2. Under control conditions, whole preparations (12-24 segment long; n = 22) were perfused with NMDA solutions of the same concentration (100-150 microM). The intersegmental phase lag values varied in a continuous range with a single peak around a median value of forward +0.74% per segment (range: forward +2.23% to backward -0.97%). 3. To examine whether excitability differences along the spinal cord could modify the intersegmental phase lag, different levels of excitatory amino acids (NMDA) were applied to spinal cord preparations positioned in a partitioned chamber. Different portions of the cord could be perfused separately by NMDA solutions of different concentrations (50-150 microM). If rostral segments were perfused with the higher NMDA solution, the lag was inevitably in the forward direction. Conversely, if the caudal portion was perfused with the higher NMDA solution, caudally located ventral roots became activated before the rostral ventral roots in a caudorostral succession, thus reversing the direction of the fictive swimming wave to propagate as during backward swimming. If the middle portion was perfused by the highest NMDA solution, this portion instead became leading, and the activity propagated from this point in both the rostral and the caudal directions. The portion located in the pool with highest NMDA concentration always gave rise to a "leading" segment. 4. When a portion of the preparation was perfused with an NMDA solution of a high concentration (75-150 microM), the cycle duration was close to that recorded when the whole preparation was perfused with the same high NMDA solution. The ensemble cycle duration is, therefore, largely determined by the leading segment. 5. The phase lag changes were not restricted to the region around the barrier separating pools with different NMDA solutions.(ABSTRACT TRUNCATED AT 400 WORDS)
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The central pattern generator (CPG) for locomotion can be thought of as a chain of segmental oscillators coupled together that produces the basic locomotor pattern. The isolated spinal cord of the lamprey is an excellent preparation in which to formulate general principles for the operation of the CPG. The stability of the preparation and the ease with which surgical lesions can be made in the cord have allowed the study of the coordinating system with a convenience unobtainable in more complex vertebrates. Mathematical models have been developed to help analyze the CPG and other systems of coupled oscillators. The models have pointed to two important parameters for determining the relative timing of a system of coupled oscillators: the nature of the coupling and the difference in frequency among the oscillators. The latter is dealt with here. In lampreys, the frequency differences of the segmental oscillators along the cord can be quite large. This factor is shown to be related to changes in the intersegmental phase lags during serotonin modulation of fictive swimming. An understanding of some effects of the frequency difference is also shown to have been important in helping to formulate a protocol for the demonstration of functional regeneration in the isolated spinal cord preparation.
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Fictive swimming activity was induced in isolated spinal cords of adult lampreys Ichthyomyzon unicuspis and Petromyzon marinus by addition of D-glutamate or N-methyl-D,L-aspartate (NMA) to the bathing fluid. Propriospinal interneurons are defined as nerve cells within the spinal cord with projections longer than 1 segment. The hypothesis that propriospinal interneurons contribute to intersegmental coordination during fictive swimming was tested using electrical stimulation, extracellular recording, and separated compartments. Two hypothetical groups of propriospinal interneurons are proposed for the coordination of swimming activities in the isolated spinal cords of adult lampreys. 1) Crossed, ascending interneurons may be excited in phase with nearby motoneurons and may excite and entrain rostral pattern generators on the opposite side. 2) Short, commissural interneurons may be excited in phase with nearby motoneurons and may inhibit contralateral generators.
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A new class of excitatory premotor interneurons that are important in the generation of locomotion in the lamprey has now been described. In the isolated spinal cord, these neurons act simultaneously with their postsynaptic motoneurons during fictive swimming. They are small and numerous, and they monosynaptically excite both motoneurons and inhibitory premotor interneurons. The excitatory postsynaptic potentials are depressed by an antagonist of excitatory amino acids. These interneurons receive reticulospinal input from the brain stem and polysynaptic input form skin afferents. A model of the network underlying locomotion based on the synaptic interactions of these neurons can now be proposed for the lamprey.
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1. Application of D-glutamate to the isolated spinal cord of the lamprey produces phasic activity in ventral roots, which is similar to that of the muscles of the intact swimming animal (5,18). Therefore, the isolated spinal cord may be used as a convenient model for the investigation of the generation of locomotor rhythms in a vertebrate. 2. Almost all slow muscle fibers exhibited excitatory junctional potentials (EJPs) during swimming activity. The number of EJPs per cycle increased with the intensity of ventral root (VR) bursting. Few twitch fibers were active, and these fired action potentials only during high intensities of VR bursts. 3. As was found by Russell and Wallén (25), myotomal motoneurons had oscillating membrane potentials during fictive swimming which, on the average, reached a peak depolarization in the middle of the VR burst (phi = 0.21 +/- 0.05; phi = 0 is defined as the onset of the VR burst, and the duration of the cycle is set equal to 1). Membrane potential oscillations in fin motoneurons were antiphasic to those of nearby myotomal motoneurons (peak depolarization phi = 0.68 +/- 0.05). 4. Lateral interneurons had oscillating membrane potentials in synchrony with those of myotomal motoneurons (peak depolarization phi = 0.21 +/- 0.10). Interneurons with axons projecting contralaterally and caudally (CC interneurons) had oscillating membrane potentials that peaked significantly earlier in the cycle (peak depolarization phi = 0.06 +/- 0.12). 5. Edge cells were only weakly modulated during fictive swimming. Their peak depolarizations occurred near the end of the VR burst (phi = 0.33 +/- 0.10). Most giant interneurons were not phasically modulated during fictive swimming. 6. Repetitive intracellular stimulation of Müller cells during fictive swimming generally evoked an increased burst intensity in ipsilateral VRs and a decreased burst intensity in contralateral VRs. The cells M3, B1, and B2 also produced increases or decreases in the frequency of VR bursts. Repetitive intracellular stimulation of sensory dorsal cells could also change the intensities and timing of VR bursts. 7. This study is an initial survey of lamprey spinal interneurons that participate in swimming activity. Lateral interneurons and CC interneurons are active during fictive swimming and probably help coordinate the undulations of the body, but their roles in pattern generation are not known. The central pattern generator is subject to modification by descending and sensory inputs.
Article
An in vitro preparation of the lamprey spinal cord was developed to enable detailed studies of the neuronal organization of the central spinal network generating fish swimming movements, one basic type of vertebrate locomotion. 1. In the isolated lamprey spinal cord, stable bursting activity recorded in the ventral roots was initiated by adding, e.g., D-glutamate or L-DOPA to the bathing solution. Less stable rhythmic activity could also be induced by tonic electrical stimulation of the spinal cord. 2. The isolated spinal cord is capable of producing rhythmic activity with the same type of intra- and intersegmental coordination as in the swimming fish, i.e., with alternation between the two sides of the segment and and intersegmental phase coupling. Hence, the in vitro preparation of the lamprey spinal cord may be said to represent the neuronal correlate of the undulatory swimming movements of fish. 3. By performing spinal cord transections it was demonstrated that as few as four segments can produce rhythmic activity with maintained coordination. It was concluded that the capacity to produce coordinated burst activity is distributed throughout the lamprey spinal cord. 4. A longitudinal midline lesion as long as four segments did not prevent the ventral roots on each side from bursting with maintained coordination between adjacent hemisegments. Thus, one side of a segment can produce bursting without interaction with its opposite side, at least when connected to its rostral and caudal neighbors. 5. The rate of bursting was found to vary from one cycle to the next with the period length tending to oscillate about a mean value. Burst duration and intersegmental phase lag varied in the same manner.