A dynamical systems analysis of afferent control in a neuromechanical model of locomotion: II. Phase asymmetry

Abstract

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.

A dynamical systems analysis of afferent control in a neuromechanical model of locomotion: I. Rhythm generation

“…It turns out that stronger feedback signals yield stronger motoneuron outputs, which speed up these stance/swing transitions and the associated epochs in the limb phase plane, enhancing the frequency increase expected from the CPG. The mechanisms responsible for these effects will be discussed in the next section and, in greater detail, in our subsequent paper [15]. The upshot of this analysis is that the CPG and limb responses to strengthened feedback provide complementary increases in the locomotor oscillation frequency in the model that we consider, as illustrated in Figure 7.…”

“…Since the definition of stance and swing phases is based on the direction of limb motion (and correspondingly the presence or absence of ground reaction force), these phases are shifted relative to the F and E phases. Thus, we can define eSwing, eStance, fSwing, and fStance subphases, which will be important in [15], where in each subphase name, the first letter denotes the active motoneuron (‘f’ for flexor, ‘e’ for extensor) and the subsequent string indicates whether the limb is in the swing or stance phase. One cycle through these phases defines a locomotor cycle in the model.…”

“…Provided that the feedback strength to the In is sufficiently large, the system will oscillate. In our companion paper, we will return to the issue of what can cause oscillations to be lost in the presence of feedback, for example as occurs as drive is reduced, using an analysis of limb dynamics subject to Mn outputs [15]. …”

“…In Section 4.2, we visualize the dynamics of the model in a completely new way, in limb phase space, and use this viewpoint to show how this feedback increase alters the qualitative form of the oscillation and the location of the transition curves defined in the previous section. This visualization will reveal a critical relationship that is necessary for oscillations to exist in the model, which we will investigate in more detail in [15]. …”

“…In this paper and a subsequent companion paper [15], our primary goal is to use dynamical systems analysis to explain the mechanisms underlying the performance of the computational model proposed by Markin et al [14], as a step towards establishing general conditions under which the observed simulation results can be expected to hold, clarifying which model features are essential to its behaviors, and enhancing the predictive capacity of the model. As a starting point in this endeavor, we first investigate how the model produces periodic oscillations.…”