Contribution of Afferent Feedback to Adaptive Hindlimb Walking in Cats: A Neuromusculoskeletal Modeling Study

Abstract

Mammalian locomotion is generated by central pattern generators (CPGs) in the spinal cord, which produce alternating flexor and extensor activities controlling the locomotor movements of each limb. Afferent feedback signals from the limbs are integrated by the CPGs to provide adaptive control of locomotion. Responses of CPG-generated neural activity to afferent feedback stimulation have been previously studied during fictive locomotion in immobilized cats. Yet, locomotion in awake, behaving animals involves dynamic interactions between central neuronal circuits, afferent feedback, musculoskeletal system, and environment. To study these complex interactions, we developed a model simulating interactions between a half-center CPG and the musculoskeletal system of a cat hindlimb. Then, we analyzed the role of afferent feedback in the locomotor adaptation from a dynamic viewpoint using the methods of dynamical systems theory and nullcline analysis. Our model reproduced limb movements during regular cat walking as well as adaptive changes of these movements when the foot steps into a hole. The model generates important insights into the mechanism for adaptive locomotion resulting from dynamic interactions between the CPG-based neural circuits, the musculoskeletal system, and the environment.

Keywords: afferent feedback; cat; central pattern generator; neuromusculoskeletal model; walking.

FIGURE 1

Neuromusculoskeletal model of a cat hindlimb composed of a neural network and a musculoskeletal model. The neural network model consists of a two-level central pattern generator (CPG) with rhythm generating (RG) and pattern formation (PF) circuits and motoneurons (Mns). This network generates motor commands through Mns and receives afferent feedback signals from the musculoskeletal model (orange arrows). The musculoskeletal model, which consists of three rigid links (black lines) and seven muscles (red and orange lines), walks on a treadmill.

FIGURE 2

Nullclines ${\widehat{N}}_i^{\mathrm{V}}$ and ${\widehat{N}}_i^{\mathrm{h}}$ for $i\in \left\{\mathrm{R}\mathrm{G}\right\}$ of RG neurons in the phase plane and fast and slow dynamics that the state (V _i , h _i ) for $i\in \left\{\mathrm{R}\mathrm{G}\right\}$ follows. (A) ${\widehat{N}}_i^{\mathrm{V}}$ has two different inflection points and intersection with ${\widehat{N}}_i^{\mathrm{h}}$ is saddle equilibrium point. (B) ${\widehat{N}}_i^{\mathrm{V}}$ changes monotonically and intersection with ${\widehat{N}}_i^{\mathrm{h}}$ is stable node equilibrium point. Bold and thin arrows represent fast and slow dynamics, respectively.

FIGURE 3

Change in membrane potential of PF-F neuron by stimulation: (A) No stimulation. (B) Stimulation to the flexor side during the active phase, which increases the duration of the current active phase and cycle (T′ > T). (C) Stimulation to the flexor side during the silent phase, which initiates the active phase and decreases the cycle duration (T′ < T). Red bars indicate the application of stimulation. Gray regions indicate active phases.

FIGURE 4

Phase-dependent response of CPG model to a stimulation. (A) Response of PF-F neuron in our model to stimulation of the flexor side compared with the response against flexor muscle stimulation during fictive locomotion in spinal cats [adapted from Schomburg et al. (1998)], where the dotted lines are approximate functions using a first order polynomial for each flexion and extension phase. (B) Response of PF-F neuron of our model to stimulation of the extensor side compared with the response against extensor muscle stimulation during fictive locomotion in decerebrate cats [adapted from Duysens (1977)], where the dotted line is an approximate function using an eighth order polynomial. R is the correlation coefficient between the active/silent phase of the simulation results and flexion/extension phase of the approximate function.

FIGURE 5

Simulation results for normal walking obtained by the optimization: (A) Membrane potentials of CPG neurons and afferent feedback from flexor and extensor muscles. (B) Joint angles and (C) muscle activations compared with measured data in cat [adapted from Prilutsky et al. (2016)]. R is the correlation coefficient and S is the cosine similarity. Liftoffs are represented by 0 and 100% in the gait cycle. Vertical lines indicate touchdowns.

FIGURE 6

Simulation results when on paw enters a hole compared with those during normal walking: (A) Joint angles and stick diagrams compared with measured data [adapted from Hiebert et al. (1994) for entering a hole and from Prilutsky et al. (2016) for normal walking]. These data are shown from the liftoff before entering a hole to the touchdown after entering the hole. R is the correlation coefficient of the results after entering the hole. The stick diagrams are shifted to the left by according to the distance traveled by the treadmill belt. (B) Activities of IP and GA muscles compared with measured data in spinal cat [adapted from Hiebert et al. (1994)]. Upper arrows indicate the activity duration, and lower ones indicate the intervals between the onsets of current and subsequent activities.

FIGURE 7

Comparison of the response of RG neurons to a foot entering a hole with steady oscillation during normal walking. (A) Time profiles of membrane potentials and afferent feedback from flexor and extensor muscles. “a” indicates the time at which the foot enters the hole (touchdown in normal walking). “b”–“e” indicate the lapsed times of 0.1, 0.29, 0.43, and 0.56 s, respectively, after “a”. (B) Trajectories of $\left(V_{\mathrm{R}\mathrm{G}‐\mathrm{F}},h_{\mathrm{R}\mathrm{G}‐\mathrm{F}}\right)$ and $\left(V_{\mathrm{R}\mathrm{G}‐\mathrm{E}},h_{\mathrm{R}\mathrm{G}‐\mathrm{E}}\right)$ in phase planes. (C) Changes of nullclines ${\widehat{N}}_{\mathrm{R}\mathrm{G}‐\mathrm{F}}^{\mathrm{V}}$ and ${\stackrel{‐}{N}}_{\mathrm{R}\mathrm{G}‐\mathrm{E}}^{\mathrm{V}}$ .