Silvia Daun-Gruhn

A mathematical modeling study of inter-segmental coordination during stick insect walking

The biomechanical conditions for walking in the stick insect require a modeling approach that is based on the control of pairs of antagonistic motoneuron (MN) pools for each leg joint by independent central pattern generators (CPGs). Each CPG controls a pair of antagonistic MN pools. Furthermore, specific sensory feedback signals play an important role in the control of single leg movement and in the generation of inter-leg coordination or the interplay between both tasks. Currently, however, no mathematical model exists that provides a theoretical approach to understanding the generation of coordinated locomotion in such a multi-legged locomotor system. In the present study, I created such a theoretical model for the stick insect walking system, which describes the MN activity of a single forward stepping middle leg and helps to explain the neuronal mechanisms underlying coordinating information transfer between ipsilateral legs. In this model, CPGs that belong to the same leg, as well as those belonging to different legs, are connected by specific sensory feedback pathways that convey information about movements and forces generated during locomotion. The model emphasizes the importance of sensory feedback, which is used by the central nervous system to enhance weak excitatory and inhibitory synaptic connections from front to rear between the three thorax-coxa-joint CPGs. Thereby the sensory feedback activates caudal pattern generation networks and helps to coordinate leg movements by generating in-phase and out-of-phase thoracic MN activity.

Quadrupedal gaits in hexapod animals - inter-leg coordination in free-walking adult stick insects

SUMMARY The analysis of inter-leg coordination in insect walking is generally a study of six-legged locomotion. For decades, the stick insect Carausius morosus has been instrumental for unravelling the rules and mechanisms that control leg coordination in hexapeds. We analysed inter-leg coordination in C. morosus that freely walked on straight paths on plane surfaces with different slopes. Consecutive 1.7 s sections were assigned inter-leg coordination patterns (which we call gaits) based on footfall patterns. Regular gaits, i.e. wave, tetrapod or tripod gaits, occurred in different proportions depending on surface slopes. Tetrapod gaits were observed most frequently, wave gaits only occurred on 90 deg inclining slopes and tripod gaits occurred most often on 15 deg declining slopes, i.e. in 40% of the sections. Depending on the slope, 36–66% of the sections were assigned irregular gaits. Irregular gaits were mostly due to multiple stepping by the front legs, which is perhaps probing behaviour, not phase coupled to the middle legs’ cycles. In irregular gaits, middle leg and hindleg coordination was regular, related to quadrupedal walk and wave gaits. Apparently, front legs uncouple from and couple to the walking system without compromising middle leg and hindleg coordination. In front leg amputees, the remaining legs were strictly coordinated. In hindleg and middle leg amputees, the front legs continued multiple stepping. The coordination of middle leg amputees was maladapted, with front legs and hindlegs performing multiple steps or ipsilateral legs being in simultaneous swing. Thus, afferent information from middle legs might be necessary for a regular hindleg stepping pattern.

An inter-segmental network model and its use in elucidating gait-switches in the stick insect

Animal locomotion requires highly coordinated working of the segmental neuronal networks that control the limb movements. Experiments have shown that sensory signals originating from the extremities play a pivotal role in controlling locomotion patterns by acting on central networks. Based on the results from stick insect locomotion, we constructed an inter-segmental model comprising local networks for all three legs, i.e. for the pro-, meso- and meta-thorax, their inter-connections and the main sensory inputs modifying their activities. In the model, the local networks are uniform, and each of them consists of a central pattern generator (CPG) providing the rhythmic oscillation for the protractor-retractor motor systems, the corresponding motoneurons (MNs), and local inhibitory interneurons (IINs) between the CPGs and the MNs. Between segments, the CPGs are connected cyclically by both excitatory and inhibitory pathways that are modulated by the aforementioned sensory inputs. Simulations done with our network model showed that it was capable of reproducing basic patterns of locomotion such as those occurring during tri- and tetrapod gaits. The model further revealed a number of elementary neuronal processes (e.g. synaptic inhibition, or changing the synaptic drive at specific neurons) that in the simulations were necessary, and in their entirety sufficient, to bring about a transition from one type of gait to another. The main result of this simulation study is that exactly the same mechanism underlies the transition between the two types of gait irrespective of the direction of the change. Moreover, the model suggests that the majority of these processes can be attributed to direct sensory influences, and changes are required only in centrally controlled synaptic drives to the CPGs.

A neuromechanical model explaining forward and backward stepping in the stick insect

The mechanism underlying the generation of stepping has been the object of intensive studies. Stepping involves the coordinated movement of different leg joints and is, in the case of insects, produced by antagonistic muscle pairs. In the stick insect, the coordinated actions of three such antagonistic muscle pairs produce leg movements and determine the stepping pattern of the limb. The activity of the muscles is controlled by the nervous system as a whole and more specifically by local neuronal networks for each muscle pair. While many basic properties of these control mechanisms have been uncovered, some important details of their interactions in various physiological conditions have so far remained unknown. In this study, we present a neuromechanical model of the coupled protractor-retractor and levator-depressor neuromuscular systems and use it to elucidate details of their coordinated actions during forward and backward walking. The switch from protraction to retraction is evoked at a critical angle of the femur during downward movement. This angle represents a sensory input that integrates load, motion, and ground contact. Using the model, we can make detailed suggestions as to how rhythmic stepping might be generated by the central pattern generators of the local neuronal networks, how this activity might be transmitted to the corresponding motoneurons, and how the latter might control the activity of the related muscles. The entirety of these processes yields the coordinated interaction between neuronal and mechanical parts of the system. Moreover, we put forward a mechanism by which motoneuron activity could be modified by a premotor network and suggest that this mechanism might serve as a basis for fast adaptive behavior, like switches between forward and backward stepping, which occur, for example, during curve walking, and especially sharp turning, of insects.

From neuron to behavior: dynamic equation-based prediction of biological processes in motor control

This article presents the use of continuous dynamic models in the form of differential equations to describe and predict temporal changes in biological processes and discusses several of its important advantages over discontinuous bistable ones, exemplified on the stick insect walking system. In this system, coordinated locomotion is produced by concerted joint dynamics and interactions on different dynamical scales, which is therefore difficult to understand. Modeling using differential equations possesses, in general, the potential for the inclusion of biological detail, the suitability for simulation, and most importantly, parameter manipulation to make predictions about the system behavior. We will show in this review article how, in case of the stick insect walking system, continuous dynamical system models can help to understand coordinated locomotion.

A neuromechanical model for the neuronal basis of curve walking in the stick insect.

The coordination of the movement of single and multiple limbs is essential for the generation of locomotion. Movement about single joints and the resulting stepping patterns are usually generated by the activity of antagonistic muscle pairs. In the stick insect, the three major muscle pairs of a leg are the protractor and retractor coxae, the levator and depressor trochanteris, and the flexor and extensor tibiae. The protractor and retractor move the coxa, and thereby the leg, forward and backward. The levator and depressor move the femur up and down. The flexor flexes, and the extensor extends the tibia about the femur-tibia joint. The underlying neuronal mechanisms for a forward stepping middle leg have been thoroughly investigated in experimental and theoretical studies. However, the details of the neuronal and mechanical mechanisms driving a stepping single leg in situations other than forward walking remain largely unknown. Here, we present a neuromechanical model of the coupled three joint control system of the stick insects middle leg. The model can generate forward, backward, or sideward stepping. Switching between them is achieved by changing only a few central signals controlling the neuromechanical model. In kinematic simulations, we are able to generate curve walking with two different mechanisms. In the first, the inner middle leg is switched from forward to sideward and in the second to backward stepping. Both are observed in the behaving animal, and in the model and animal alike, backward stepping of the inner middle leg produces tighter turns than sideward stepping.

Dominance of local sensory signals over inter-segmental effects in a motor system: experiments

Legged locomotion requires that information local to one leg, and inter-segmental signals coming from the other legs are processed appropriately to establish a coordinated walking pattern. However, very little is known about the relative importance of local and inter-segmental signals when they converge upon the central pattern generators (CPGs) of different leg joints. We investigated this question on the CPG of the middle leg coxa–trochanter (CTr)-joint of the stick insect which is responsible for lifting and lowering the leg. We used a semi-intact preparation with an intact front leg stepping on a treadmill, and simultaneously stimulated load sensors of the middle leg. We found that middle leg load signals induce bursts in the middle leg depressor motoneurons (MNs). The same local load signals could also elicit rhythmic activity in the CPG of the middle leg CTr-joint when the stimulation of middle leg load sensors coincided with front leg stepping. However, the influence of front leg stepping was generally weak such that front leg stepping alone was only rarely accompanied by switching between middle leg levator and depressor MN activity. We therefore conclude that the impact of the local sensory signals on the levator–depressor motor system is stronger than the inter-segmental influence through front leg stepping.

A neuro-mechanical model of a single leg joint highlighting the basic physiological role of fast and slow muscle fibres of an insect muscle system.

In legged animals, the muscle system has a dual function: to produce forces and torques necessary to move the limbs in a systematic way, and to maintain the body in a static position. These two functions are performed by the contribution of specialized motor units, i.e. motoneurons driving sets of specialized muscle fibres. With reference to their overall contraction and metabolic properties they are called fast and slow muscle fibres and can be found ubiquitously in skeletal muscles. Both fibre types are active during stepping, but only the slow ones maintain the posture of the body. From these findings, the general hypothesis on a functional segregation between both fibre types and their neuronal control has arisen. Earlier muscle models did not fully take this aspect into account. They either focused on certain aspects of muscular function or were developed to describe specific behaviours only. By contrast, our neuro-mechanical model is more general as it allows functionally to differentiate between static and dynamic aspects of movement control. It does so by including both muscle fibre types and separate motoneuron drives. Our model helps to gain a deeper insight into how the nervous system might combine neuronal control of locomotion and posture. It predicts that (1) positioning the leg at a specific retraction angle in steady state is most likely due to the extent of recruitment of slow muscle fibres and not to the force developed in the individual fibres of the antagonistic muscles; (2) the fast muscle fibres of antagonistic muscles contract alternately during stepping, while co-contraction of the slow muscle fibres takes place during steady state; (3) there are several possible ways of transition between movement and steady state of the leg achieved by varying the time course of recruitment of the fibres in the participating muscles.

Investigating inter-segmental connections between thoracic ganglia in the stick insect by means of experimental and simulated phase response curves

The neuronal networks that control the motion of the individual legs in insects, in particular in the stick insect, are located in the pro-, meso- and meta-thoracic ganglia. They ensure high flexibility of movement control. Thus, the legs can move in an apparently independent way, e.g., during search movements, but also in tight coordination during locomotion. The latter is evidently a very important behavioural mode. It has, therefore, inspired a large number of studies, all aiming at uncovering the nature of the inter-leg coordination. One of the basic questions has been as to how the individual control networks in the three thoracic ganglia are connected to each other. One way to study this problem is to use phase response curves. They can reveal properties of the coupling between oscillatory systems, such as the central pattern generators in the control networks in the thoracic ganglia. In this paper, we report results that we have achieved by means of a combined experimental and modelling approach. We have calculated phase response curves from data obtained in as yet unpublished experiments as well as from those in previously published ones. By using models of the connected pro- and meso-thoracic control networks of the protractor and retractor neuromuscular systems, we have also produced simulated phase response curves and compared them with the experimental ones. In this way, we could gain important information of the nature of the connections between the aforementioned control networks. Specifically, we have found that connections from both the protractor and the retractor “sides” of the pro-thoracic network to the meso-thoracic one are necessary for producing phase response curves that show close similarity to the experimental ones. Furthermore, the strength of the excitatory connections has been proven to be crucial, while the inhibitory connections have essentially been irrelevant. We, thus, suggest that this type of connection might also be present in the stick insect, and possibly in other insect species.

Dominance of local sensory signals over inter-segmental effects in a motor system: modeling studies

Recent experiments, reported in the accompanying paper, have supplied key data on the impact afferent excitation has on the activity of the levator–depressor motor system of an extremity in the stick insect. The main finding was that, stimulation of the campaniform sensillae of the partially amputated middle leg in an animal where all other but one front leg had been removed, had a dominating effect over that of the stepping ipsilateral front leg. In fact, the latter effect was minute compared to the former. In this article, we propose a local network that involves the neuronal part of the levator–depressor motor system and use it to elucidate the mechanisms that underlie the generation of neuronal activity in the experiments. In particular, we show that by appropriately modulating the activity in the neurons of the central pattern generator of the levator–depressor motor system, we obtain activity patterns of the motoneurons in the model that closely resemble those found in extracellular recordings in the stick insect. In addition, our model predicts specific properties of these records which depend on the stimuli applied to the stick insect leg. We also discuss our results on the segmental mechanisms in the context of inter-segmental coordination.