Locomotion in mammals is controlled by a spinal central pattern generator (CPG) coupled to a biomechanical limb system, with afferent feedback to the spinal circuits and CPG closing the control loop. We have considered a simplified model of this system, in which the CPG establishes a rhythm when a supra-spinal activating drive is present and afferent signals from a single-joint limb feed back to affect CPG operation. Using dynamical systems methods, in a series of two papers, we analyze the mechanisms by which this model produces oscillations, and the characteristics of these oscillations, in the closed and open loop regimes. In this first paper, we analyze the phase transition mechanisms operating within the CPG and use the results to explain how afferent feedback allows oscillations to occur at a wider range of drive values to the CPG than the range over which oscillations occur in the CPG without feedback and to comment on why stronger feedback leads to faster oscillations. Linking these transitions to structure in the phase plane associated with the limb segment clarifies how increased weights of afferent feedback to the CPG can restore locomotion after removal of supra-spinal drive to simulate spinal cord injury.

Distinct rhythmic behaviors involving a common set of motoneurons and muscles can be generated by separate central nervous system (CNS) networks, a single network, or partly overlapping networks in invertebrates. Less is known for vertebrates. Simultaneous activation of two networks can reveal overlap or interactions between them. The turtle spinal cord contains networks that generate locomotion and three forms of scratching (rostral, pocket, and caudal), having different knee-hip synergies. Here, we report that in immobilized spinal turtles, simultaneous delivery of types of stimulation, which individually evoked forward swimming and one form of scratching, could 1) increase the rhythm frequency; 2) evoke switches, hybrids, and intermediate motor patterns; 3) recruit a swim motor pattern even when the swim stimulation was reduced to subthreshold intensity; and 4) disrupt rhythm generation entirely. The strength of swim stimulation could influence the result. Thus even pocket scratching and caudal scratching, which do not share a knee-hip synergy with forward swimming, can interact with swim stimulation to alter both rhythm and pattern generation. Model simulations were used to explore the compatibility of our experimental results with hypothetical network architectures for rhythm generation. Models could reproduce experimental observations only if they included interactions between neurons involved in swim and scratch rhythm generation, with maximal consistency between simulations and experiments attained using a model architecture in which certain neurons participated actively in both swim and scratch rhythmogenesis. Collectively, these findings suggest that the spinal cord networks that generate locomotion and scratching have important shared components or strong interactions between them. central pattern generation; rhythm generation; interneuron; swim; oscillator ANIMALS PERFORM a wide variety of behaviors with a limited number of neurons and muscles. Are different behaviors involving the same motoneurons and muscles generated by the same central nervous system (CNS) network or different networks? This question can conveniently be addressed for rhythmic behaviors, which are relatively simple and often generated by CNS networks. Individual neuron recordings have shown that CNS neurons can be rhythmically activated during multiple rhythmic behaviors involving the same motoneurons

We analyze a closed loop neuromechanical model of locomotor rhythm generation. The model is composed of a spinal central pattern generator (CPG) and a single-joint limb, with CPG outputs projecting via motoneurons to muscles that control the limb and afferent signals from the muscles feeding back to the CPG. In a preceding companion paper, we analyzed how the model generates oscillations in the presence or absence of feedback, identified curves in a phase plane associated with the limb that signify where feedback levels induce phase transitions within the CPG, and explained how increasing feedback strength restores oscillations in a model representation of spinal cord injury; from these steps, we derived insights about features of locomotor rhythms in several scenarios and made predictions about rhythm responses to various perturbations. In this paper, we exploit our analytical observations to construct a reduced model that retains important characteristics from the original system. We prove the existence of an oscillatory solution to the reduced model using a novel version of a Melnikov function, adapted for discontinuous systems and also comment on the uniqueness and stability of this solution. Our analysis yields a deeper understanding of how the model must be tuned to generate oscillations and how the details of the limb dynamics shape overall model behavior. In particular, we explain how, due to the feedback signals in the model, changes in the strength of a tonic supra-spinal drive to the CPG yield asymmetric alterations in the durations of different locomotor phases, despite symmetry within the CPG itself.

In this paper we discuss what is known about the classification of symmetry groups and patterns of phase-shift synchrony for periodic solutions of coupled cell networks. Specifically, we compare the lists of spatial and spatiotemporal symmetries of periodic solutions of admissible vector fields to those of equivariant vector fields in the three cases of R n (coupled equations), T n (coupled oscillators), and (R k) n where k ≥ 2 (coupled systems). To do this we use the H/K Theorem of Buono and Golubitsky applied to coupled equations and coupled systems and prove the H/K theorem in the case of coupled oscillators. Josić and Török [17] prove that the H/K lists for equivariant vector fields and admissible vector fields are the same for transitive coupled systems. We show that the corresponding theorem is false for coupled equations. We also prove that the pairs of subgroups H ⊃ K for coupled equations are contained in the pairs for coupled oscillators which are contained in the pairs for coupled systems. Finally, we prove that patterns of rigid phase-shift synchrony for coupled equations are contained in those of coupled oscillators and those of coupled systems.

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.