Extending to more segments

As more segments are added to the model, the significance of physical interactions be-tween them will increase. This is particularly significant for segments near the middle of the worm, as the body parts attached to either side must be moved in order for the segment to bend, adding significant mechanical load. Furthermore, motion of a given segment will

generally deform its neighbours, interfering with the sensory feedback loop. A simulation of the model with three segments (adding and additional “body” segment) should begin to reveal whether the control system is robust to these perturbations. As demonstrated in Figure 5.5, the three segment model continues to oscillate robustly. However, it is immediately apparent that the physical interactions are detrimental to the model’s ability to synchronize, leading to a breakdown of the correct phase relationship. The bending waveforms of the individual segments are also affected.

Figure 5.5: Oscillation of the model with two body segments (i = 1, 2) and one tail seg-ment (i = 3). While the model is still able to oscillate, the waveforms are distorted and the correct phase lags are not preserved.

If the number of segments is increased to 11 (as in the original neural model) similar behaviour is observed. Oscillations continue and forward progress is still made, but the waveforms of individual segments are again distorted and the correct phase relationship is not preserved. Specifically, the phase lags between adjacent segments are too large, resulting in a very short wavelength as shown in Figure 5.6. In fact, the direction of bending (dorsal or ventral) typically alternates every two to three segments.

Figure 5.6: Frames extracted from a movie of the 11 segment integrated model (See Supplementary movie C5 1). Numbers in the top right corners give the time in seconds at which each frame was taken. Notice that despite the lack of coordination, the model still moves forwards (to the left).

5.4 Discussion

The model presented here, while only partially successful, provides valuable insight into the importance of body physics. First, it demonstrates that a neural model developed with only the most rudimentary physical framework can continue to function only partially when a more realistic embodiment is added. Second, it shows that the inclusion of body physics can have a dramatic effect on the oscillation dynamics. When only a short section (two segments or less) is simulated, these effects are beneficial. Indeed, both the wave-form and frequency of oscillation are improved beyond what was possible for the isolated neural model [21]. However, when more segments are included the effect of the body becomes more detrimental, particularly to the inter-segment phase relationship.

To understand these different classes of effect it helps to consider the local and long range effects of the body separately. Locally the effect of the body and environment is primarily to filter the neural outputs, adding a delay to the sensory feedback loop with a similar effect to that used by Karbowski et al. [67]. In the original neural model, the frequency of oscillation depends only on the neural time-scale. Adding a physical model introduces an additional time-scale, as well as a nonlinearity in the mapping between neural activity and segment bending. When the neural states change, the body will begin bending at a rate that depends on the strength of the muscles relative to the resistance of the environment. The neural states depend critically on the stretch receptor feedback, which in turn is dependent on the body dynamics. Thus the neurons must “wait” for the body configuration to change. With the parameters used in this model, the frequency is dominated by the physical rather than the neural time-scale, so the oscillation slows down.

The observed change in waveform from roughly square to roughly sinusoidal is caused by the nonlinearity introduced by the body. As the segment bends the muscle force is increasingly resisted by the elasticity of the stretched or compressed cuticle, reducing the net force and therefore the rate of bending. It is this effect that smooths the waveform.

When a second segment is added, the effect remains largely unchanged. The middle rod stays roughly in place, while the oscillation of each segment causes the outer two rods (which are connected only to the middle rod) to move.

Let us now consider the long range effects. In the original model the angles of each segment were computed independently, and the resulting worm “shape” was visualized as a series of lines connected to each other at the specified angles. Adding a physical model means that bending of any of the inner segments applies forces to the adjacent segments. To accommodate this motion the nearby points must typically move laterally, but this is strongly resisted by the normal drag coefficient. If this load is too great, the

nearby points will not move and instead the body elements will have to stretch or com-press to accommodate this force, often leading to unrealistic, contorted body shapes. The result is that the segments are constantly perturbing each other physically, with these de-formations feeding back into the control system via the stretch receptors. The very short wavelength displayed by the 11 segment integrated model can also be explained by the fact that motion normal to the body surface is resisted more strongly that motion tangent to it. In the extreme case of zero phase lag (a standing wave), the net curvature of the body will be great and the ends of the worm would have to undergo large lateral displace-ment. If instead the phase lag between adjacent segments is 180^o, the net curvature of the body will be approximately zero, so all segments would experience only small lateral displacements, thereby reducing the load.

There are two main conclusions to be drawn from this work. First, the physical effects of a body can be both beneficial and detrimental, but must certainly be taken into account.

This should be incorporated at an early stage, as a control system developed in the ab-sence of long range physical interactions is unlikely to work once they are introduced.

Second, this investigation has shown that changing the physical drag is a viable method for modulating the frequency of oscillation.

Conductance based muscle model

6.1 Introduction

Models of C. elegans locomotion generally focus on the worm’s nervous system [20, 21, 118] or body mechanics [36, 89], but it is currently not clear which components of the locomotion subsystem are actively involved in generating and shaping locomotion. The candidates are the interneurons, head and ventral cord motor neurons, body wall muscles and the C. elegans body itself. While it was demonstrated in Chapters 4 and 5 that the body is an important part of the locomotion system, some other components of the system must generate the patterns of muscle activation required for locomotion. Thus the two alternatives are either that the patterned activity of the motor neurons activates the muscles which then act as actuators to deliver the mechanical contractions, or that in addition to neuronal activity, the muscles themselves are capable of generating oscillatory dynamics and/or of propagating such signals down the length of the worm. The former holds in most studied motor systems: the neural circuit generates a patterned output, and the muscles serve as actuators of that output. Interestingly, this division does not seem to be as clear in Ascaris lumbricoides[36, 131, 132] – a much larger but closely related nematode whose nervous system is structurally very similar to that of C. elegans. In Ascaris, the body wall muscles are electrically coupled by gap junctions and appear to form a functional syncytium which produces spontaneous myogenic activity: voltage spikes superimposed on slow depolarisations, which propagate independently of the nervous system [131,132].

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From this perspective, it may not be surprising if C. elegans muscles had a similar pattern generating (or pattern modulating) role in locomotion. In fact, a recent locomotion model due to Karbowski et al. [67] relies on a mechanism whereby oscillations generated in the head are imposed on the rest of the body in part via strong muscle gap junctions.

In the absence of a direct answer to this question, one may turn to behavioural evi-dence from locomotion-defective (or so called uncoordinated) mutant strains of the worm.

Particularly instructive are mutations that might disrupt electrical signal flow between muscles. There are two gap junction genes that have been implicated in C. elegans lo-comotion, namely unc-7 and unc-9 [110]. Mutations in both of these result in virtually identical phenotypes [92] where locomotion is severely impaired. Both are widely ex-pressed, but only unc-9 is expressed in muscles. Liu et al. have shown that worms treated with unc-9 RNA interference (RNAi) to suppress unc-9 gene expression exhibit substan-tially reduced locomotion velocities [76]. The authors suggest that this effect could be attributed specifically to the reduction in gap junction coupling between body wall mus-cles, based in part on the fact that C. elegans neurons are partially resistant to RNAi [109].

In this chapter, which was previously published as Ref. [16], I rely on electrophysio-logical data recorded from body wall muscles in acutely dissected preparations [64,65,76]

to construct a model of individual and coupled muscle cells. This model is then used to determine what possible active role may be attributed to individual C. elegans body wall muscles and, furthermore, to determine the consistency of such a model with the ob-served phenotype of gap junction defective worms. More specifically, I will attempt to address the following questions: Do the muscles typically fire action potentials? What is their contribution to the generation of rhythmic behaviour? And finally, how strong is the inter-muscular coupling, and to what extent does it affect locomotion?