A complete theory of locomotory behavior requires an integrative approach linking the nervous system in an organism to the body in which the nervous system lives and the environment that the body interacts with (Pearson et al., 2006; Tytell et al., 2011; Chiel et al., 2009). However, most studies focus on rhythmic gait and its maintenance in an organism driven by the presence of a central pattern generator (CPG) that drives coordinated motor activity (Marder and Bucher, 2001; Ijspeert, 2008). While the existence of a CPG has been validated in a variety of organisms (Marder and Bucher, 2001; Grillner, 2006) and exploited in artificial systems (Ijspeert, 2008; Boxerbaum et al., 2012), growing evidence suggests that sensory feedback plays an important role in maintaining robust and stable locomotion. Indeed recent studies on C. elegans (Wen et al., 2012; Boyle et al., 2012) focusing on local sensory feedback and proprioception show that these modalities suffice to modulate the locomotory pattern and explain gait transitions associated with undulatory swimming and crawling, without the need for a central pattern generator. This has led to recent attempts to include proprioceptive coupling and build an integrative theory of locomotion in examples such as anguilliform swimming in fish (Ekeberg, 1993; Ekeberg and Grillner, 1999), swimming in leech (Cang and Friesen, 2002), walking in insects (Kukillaya et al., 2009; Proctor et al., 2010; Proctor and Holmes, 2010; Holmes et al., 2006) and humanoids (Verdaasdonk et al., 2009). However, the complexity of the brain-body-environment coupling in these organismal systems has been a substantial impediment in the use of models to make testable predictions on the biological mechanisms regulating locomotion.

Here, we consider the rectilinear crawling behavior of a model organism, the larva of the fruit fly, D. melanogaster (Video 1), which has increasingly become the focus of molecular, cellular, genetic and behavioral studies using a variety of experimental probes (Suster and Bate, 2002; Fox et al., 2006; Hughes and Thomas, 2007; Crisp et al., 2008; Song et al., 2007; Lahiri et al., 2011; Inada et al., 2011; Berni et al., 2012; Heckscher et al., 2012; Crisp et al., 2011; Fushiki et al., 2013; Gjorgjieva et al., 2013; Vogelstein et al., 2014; Kohsaka et al., 2014; Itakura et al., 2015; Pulver et al., 2015). The larva is a soft bodied cylindrical organism about 4 mm in length and about 800 μm in diameter in the third instar stage. It moves in a manner similar to other long soft-bodied creatures such as earthworms and leeches by exploiting the peristaltic propagation of muscular relaxation and contraction waves along their bodies to induce forward locomotion, or crawling (Trueman, 1975); despite biomechanical differences between these different organisms, the crawling gait seems to be a convergent strategy across species. The dynamical process that triggers, coordinates and maintains the propagation of such waves has attracted the attention of researchers for a century (Garrey, 1915). Early experimental efforts tried to understand the macroscopic mechanics of soft bodied animal locomotion (Trueman, 1975) by focusing on one of the underlying subsystems: body mechanics, muscular force production and neural dynamics. More recently, there is a growing realization that the coupling between the nervous system, the body and the substrate in the presence of sensory feedback plays a major role in development and maintenance of crawling gaits, and perhaps even in evolution (Chiel et al., 2009). For example, the locomotory behavior of Manduca sexta larvae, an organism where proprioceptive sensing displays a wide range of behaviors (Simon and Trimmer, 2009), is known to be dependent on substrate stiffness that modulates the role of external stimuli and on body deformation rate (Lin and Trimmer, 2010). In the D. melanogaster larva, there is strong evidence that proprioception plays as important a role as the CPG in generating coordinated motion. Eliminating proprioception with genetic (Hughes and Thomas, 2007) and optogenetic (Inada et al., 2011) methods leads to qualitative changes in the crawling gait: peristaltic waves have a period that is ≈4 times longer or may even stop propagating (Song et al., 2007), and body segments contract ≈2 times more (Hughes and Thomas, 2007). Experiments that block sensory input in the embryo show that crawling behavior still develops in the larva, but with longer peristaltic wave periods (Suster and Bate, 2002; Fushiki et al., 2013), providing further evidence for the importance of proprioception. Together, the classical and modern studies on locomotory physiology and the modern studies on the molecular and cellular subsystems in the larva suggest that it is an excellent biophysical testbed for an integrative theory that spans multiple scales.

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In this paper, we present a mathematical model of crawling in D. melanogaster larvae that is guided by the anatomy and the kinematics of the gait of the organism. Our theory explicitly accounts for the mechanics of the passive deformable soft body, properties of the substrate on which the crawling occurs, active muscular forcing, neural dynamics and the interactions and feedbacks between these sub-systems. This allows us to reproduce the robust crawling gait that is consistent with experimental findings in first (Heckscher et al., 2012) and third (Hughes and Thomas, 2007) instar larvae. Furthermore, our model qualitatively and quantitatively captures the effects of a) optogenetic perturbations of neural activity (Inada et al., 2011; Kohsaka et al., 2014), and b) silencing proprioception with genetic (Hughes and Thomas, 2007) and optogenetic (Inada et al., 2011) methods.

Our integrated model also allows us to make specific experimentally testable predictions. In particular, by changing the strength of coupling between adjacent segments in the CPG, both in the absence and presence of proprioception, we show how proprioception increases the robustness of crawling. This leads to the prediction that there should be much more variability in crawling metrics among individuals with silenced or weakened proprioception. Furthermore, we predict that larvae could use the strength of CPG coupling as a means of controlling the speed of gait. Finally, we show that changing the frictional interactions of the organism with the substrate should yield observable effects on the efficiency of locomotion.

More broadly, our study also aims to provide a set of plausible scenarios for the biophysical mechanisms underlying crawling, by linking body mechanics, muscular forcing, neural dynamics, the properties of the substrate and their coupling, with natural implications for engineering applications.

We start with a quantitative description of our model results and their comparison to experiments before turning to make testable predictions.

Experimental validation of the model

The model produces sustained crawling with metrics matching experiments

Crawling is initiated with a short excitatory pulse to the excitatory neural population in the most posterior segment. We do not model the source of this initiation command; it could be, for instance, a descending signal from the brain initiating forward movement or sensory feedback from the tail initiating an escape. This yields a sustained crawling gait shown in Figure 2, where a kymograph of body segments of the larva (Figure 2A) as well as the corresponding muscular (Figure 2B) and neural activity (Figure 2C) are shown (also see Video 2). We see that head and tail segments move together in the visceral piston phase (Heckscher et al., 2012; Simon et al., 2010), followed by a peristaltic wave with neural activity leading in phase, followed by muscular and contraction activity (Figure 2), propagating from posterior to anterior segments. Crawling can be stopped by shutting down the activity of excitatory neurons in the most posterior segment by an inhibitory input (see also section 3.1.2).

Figure 2

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Video 2

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Our simulations show that the larva produces approximately 0.04 waves per unit time associated with the relaxation of the excitatory neurons ${\tau }_E$, where waves start when the tail moves off the ground, leading to a larval speed ≈$0.04L/{\tau }_E$. Hughes and Thomas (Hughes and Thomas, 2007) found that third instar larvae of typical length ${\displaystyle 10L}$≈4 mm (estimated from Figure 1 of (Hughes and Thomas, 2007)) produce ≈1.5 waves/s. Using this latter number, we can estimate the time scale: ${\tau }_E$≈25 ms, a reasonable time constant for activity of neural populations. Combining the time scale estimate and the length of the third instar larvae, our model predicts a speed of ≈0.5 mm/s, comparable with the observed speed of ≈1 mm/s (estimated from Figure 1 of Hughes and Thomas, [2007]). In another experiment, for first instar larvae with typical length ${\displaystyle 10L}$≈600 μm (estimated from Figure 1 of Heckscher et al. [2012]), Heckscher et. al. Heckscher et al. (2012) give a range of 0.5 − 1.5 waves produced per second, which suggests the neural time scale range ${\displaystyle {\tau }_E}$≈25 − 75 ms, again a reasonable time constant range. Combining the time scale estimate and the length of the first instar larvae, our model predicts a speed range of ≈30 − 90 μm/s, in agreement with the range reported in (Heckscher et al., 2012): ≈40 − 125 μm/s. Our simulations show that typically (median across time steps), three segments are off the ground, defined by the number of segments in which $f_i$ is above $\widehat{f}$ and thus friction vanishes, consistent with observations (Heckscher et al., 2012). Furthermore, we find that peak segmental contraction, averaged over segments and waves, is ≈30%, consistent with observations (Hughes and Thomas, 2007).

The model reproduces the effects of optogenetic VNC perturbations

Our model shows that normal segment-to-segment propagation of activity arises through two different but coordinated channels, the posterior-to-anterior neural coupling between excitatory populations in adjacent segments, and via proprioceptive coupling (Figure 1C and D). Experimental perturbations of these channels are known to change the crawling modalities.

For example, recent advances in optogenetics have allowed for targeted manipulations of specific neuron types in the VNC (Inada et al., 2011; Kohsaka et al., 2014; Itakura et al., 2015). When segment-to-segment propagation was perturbed with optogenetic inhibition of motor neurons in a VNC segment (Inada et al., 2011), crawling stopped when the wave reached that segment. Conversely, when the inhibition is removed after up to $10$ s, the larva resumed crawling from the same segment. To see whether this observation is reproduced in our model, we performed an acute shutdown of the excitatory population in a segment, modeling the effect of optogenetic inhibition on motor neurons. Figure 3A–C and Video 3 show the results of this simulation. Crawling is stopped at segment A6 ($u_8$) by setting $E_8=0$ when $t\in [65,95]$. We note that crawling continues in this time frame until it reaches segment A6, consistent with that observed experimentally (Figure 7 of (Inada et al., 2011)). Additionally, even after crawling stops, contraction of the body segment $A7$ keeps producing proprioceptive input to $E_8$ and $I_9$. This can be seen by the sustained activity of $I_9$ during stopping, while the activity of $E_9$ dies out. When the optogenetic inhibition is removed, allowing $E_8$ to evolve according to its normal dynamics (Equation (1)), proprioceptive input drives $E_8$ to firing and the larva resumes crawling from A6.

Figure 3

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Video 3

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In our model (Figure 3), during a stop, the memory of crawling phase is kept in the contraction of the body and the proprioceptive channel which drives the proprioceptive activity. In Inada et. al.’s experiments (Inada et al., 2011), it is possible to resume crawling even after the neighboring posterior segment to the one being inhibited completes a contraction and relaxes back to its equilibrium length (Figure 7 of (Inada et al., 2011)). Hence, Inada et. al. propose that the memory is held in the VNC. At first this might seem to contradict our result that the memory is kept in the body and the proprioceptive channel, but, although our model does not include such mechanism, it is possible for the proprioceptive neurons to exhibit self-sustained activity and be the location of memory. We also want to point that in (Inada et al., 2011) the memory can last as long as 10 s. Whether in the VNC or in the proprioceptive neurons, the mechanism that creates such long time intervals from neural time scales of a few tens of milliseconds is an open problem.

More recent experimental studies (Kohsaka et al., 2014; Itakura et al., 2015) identified pre-motor inhibitory neuron populations that play a role in locomotion. Specifically, during forward crawling, period-positive median segmental interneurons (PMSIs) were found to be activated slightly later than the motor neurons in the same segment (Kohsaka et al., 2014) and their optogenetic activation leads to inhibition of motor neuron activity locally, arresting peristaltic crawling. These observations match nicely with the dynamics of the inhibitory neurons in our model, which are activated slightly later than the excitatory neurons in the same segment (Figure 2C). Further, increasing their activity above what is normally seen during normal crawling led to the local arrest of crawling. In Figure 3D–F and Video 4, we show the results of a simulation where crawling is stopped at segment A6 ($u_8$) by setting $I_8=1$, the maximum inhibitory population activity in our model, over the time period $t\in [65,95]$. Increased inhibitory neuron activity prevents the excitatory neurons from becoming active and stops the peristaltic wave. This causes the behavior of the model to be similar to the previous case where $E_8$ was shut down, except that here $I_8$ is active in the time frame of perturbation. When $I_8$ is left to evolve according to to its normal dynamics (Equation (1)), proprioceptive input drives $E_8$ to fire and larva resumes crawling from A6.

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In the optogenetic inhibition experiments of Kohsaka et al. (2014), when the activation of PMSIs is removed, the peristaltic wave did not continue from the inhibited segment, but a new wave from the posterior end started. This is in contrast to our model, where crawling continues from the inhibited segment. Other observations on PMSIs (Kohsaka et al., 2014) that remain to be incorporated into future models are: when inhibited using genetic and optogenetic methods, speed of peristalsis greatly decreased, duration of motor neuron bursting and muscle contraction increased however degree of segmental contraction did not change.

The model reproduces effects of silencing proprioception

In another set of experiments, it was shown that silencing proprioceptive feedback to VNC using genetic (Hughes and Thomas, 2007; Suster and Bate, 2002) and optogenetic (Inada et al., 2011) methods leads to an increase in peak segmental contraction from ≈30% to ≈65% (Hughes and Thomas, 2007), and a reduction in rate to approximately one fourth (Inada et al., 2011) and to one tenth (Hughes and Thomas, 2007), and reduced speed (Hughes and Thomas, 2007). To understand this, we implemented a purely neural coupling in our model by setting proprioceptive couplings $w_{Ep}=0$ and $w_{Ip}=0$. With purely neural coupling, the model still produces sustained crawling but with a qualitative change in the crawling pattern (Figure 4 and Video 5). Only a single segment is off-the-ground at a time, while peak segmental contraction, averaged over all segments, increases to ≈65% of segmental length (≈70% for only A3-A4 segmental distance, Figure 4E) the larva produces a reduced rate of ≈0.01 waves per ${\tau }_E$ and moves with speed $\approx 0.01L/{\tau }_E$ (Figure 4A and D), in agreement with experiments. Furthermore, we see that inhibitory neural dynamics show a significant modulation, due to the removal of excitatory proprioceptive input to these populations, and peak activity of inhibitory population is reduced and delayed, as shown in Figure 4C. However, the phases of muscular and neural activity within a wave do not change, but the contraction phase is slightly earlier (Figure 4F).

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Video 5

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Predictions

Role of proprioceptive coupling

We tried the converse of silencing proprioception, and implemented a purely proprioceptive coupling with no neural coupling by setting the posterior-to-anterior neural coupling $w_{En}=0$. In this case, we find that our model produces sustained crawling almost identical to what is shown in Figure 2 (see also below) where both channels are intact. This shows that a spatially coordinated CPG is not necessary for crawling, as was also suggested and shown in the simpler model put forward in (Paoletti and Mahadevan, 2014). While this might not be biologically realistic in an adult, our model shows that this plausible arrangement might be relevant in engineering coordinated locomotion, and even of some interest in an evolutionary or developmental setting.

The relative insensitivity of our model to variations in the parameters that describe neural coupling and the qualitative changes that result from silencing proprioception led us to investigate the stabilizing effect of proprioception in more detail. To this end, we varied posterior-to-anterior neural coupling, $w_{En}$, and investigated its effect when proprioception was silenced or intact. In Figure 5, we quantify various metrics characterizing the dynamics of the segments. As a function of $w_{En}$, Figure 5A shows the frequency of peristaltic waves which increases with $w_{En}$: stronger neural coupling leads to faster propagation of peristaltic wave. While higher frequency of peristaltic waves leads to faster locomotion as shown in Figure 5B, the relationship between speed and wave frequency is not linear as the size of step taken per peristaltic wave also increases with $w_{En}$, see Figure 5C. Figure 5D shows that peak contraction falls as $w_{En}$ increases and Figure 5E shows that the number of simultaneously off-ground segments increases as $w_{En}$ increases: faster peristaltic waves leave less time for segments to contract.

Figure 5

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This manipulation suggests that proprioception increases robustness of locomotory behavior in two distinct ways. First, there is a wider range of neural coupling over which sustained crawling can be achieved. When proprioception is silenced, there is a minimum value of $w_{En}$ below which crawling is not possible, due to neural coupling being too weak to excite the neural population in the next segment (Figure 5). Proprioception provides the extra excitation that allows for sustained crawling till the neural coupling weight $w_{En}=0$. Second, in all metrics that we used to characterize crawling, the observed changes as a function of $w_{En}$ was smaller when proprioception was intact (Figure 5). Thus, proprioception has a stabilizing effect on crawling. Gjorgjieva et al. (2013) came to a similar conclusion with their purely neural model.

In Figure 5 all metrics of crawling with and without proprioception show a cross-over around $w_{En}\approx 0.65$, which also sets the threshold below which sustained crawling in the full model shows no dependence on $w_{En}$. To understand these further, we investigate the propagation delays through both the neural and proprioceptive channels. In Figure 5F, we plot three quantities: 1) the time it takes for excitatory neurons in neighboring segments to cross $\widehat{E}$ (threshold to activate muscular forces) in the presence of proprioception (blue line), which quantifies segment-to-segment signal propagation delay, 2) the time for supra-threshold activation of excitatory neurons without proprioception (dashed red line), which quantifies the propagation delay through the neural channel, and 3) the time it takes for the excitatory population in a segment to cross $\widehat{E}$ and the turning on of proprioceptive signal in that population (black line), which quantifies the propagation delay through the proprioceptive channel. We see that segment-to-segment signal propagation in the full model follows the faster channel, while in the proprioception silenced model it always follows the neural channel. Proprioceptive propagation delay becomes comparable to neural propagation delay around $w_{En}\approx 0.65$, the threshold when the metrics of crawling with and without proprioception cross-over. Below this threshold, proprioception is faster, explaining why the full model is insensitive to $w_{En}$. Above this threshold, neural propagation is faster, but proprioception still has a stabilizing effect in this regime because of its inhibitory effect on the neural activity in the moving segment, neutralizing strong excitation. Furthermore, the proprioceptive channel gets slower beyond the cross-over point: it takes longer for a segment to contract to the proprioceptive threshold point. This is consistent with decreased peak contraction and increased number of simultaneously moving segments.

Our results with varied proprioceptive coupling lead us to make the following predictions:

1. Crawling without any direct segment-to-segment coupling in the VNC is possible by segment-to-segment transmission of activity through the proprioceptive channel (Figure 5).

2. Variation in parameters can be associated with variations across individuals in a population. Then, there should be much more variability in crawling metrics among individuals with silenced or weakened proprioception, a testable prediction using existing genetic tools (Hughes and Thomas, 2007; Inada et al., 2011).

3. Our model predicts that in the presence of proprioception, increase in neural coupling leads to higher speeds, as was also observed in (Gjorgjieva et al., 2013) in a purely neural model. In an analogous scenario in stick insects, experiments suggest that slow, steady-walking is mainly coordinated by local proprioceptive signals but faster motion is driven by increased neural coupling between legs (Büschges, 2012). Then, our model suggests that the larva might modulate the strength of neural coupling to control speed. This suggestion brings accompanying predictions which can be read directly from Figure 5, e.g. faster speed comes with decreased peak contraction and a higher number of simultaneously off-ground segments.

Role of body-substrate frictional coupling

To quantify the effect of the environment on the locomotion, we varied the maximum scaled frictional force, $F_{\text{max}}/kL$ between body and the substrate. Our results are shown in Figure 6 and Video 6. Our main observation is that metrics of crawling are robust until friction falls below a threshold ($F_{max}/kL\sim 0.4$), when muscular forces become stronger than friction, leading to slippage. At this point the number of moving segments start differing from number of off-ground segments (Figure 6F). An example kymograph from the slipping regime ($F_{\text{max}}/kL=0.005$), and corresponding muscular and neural activities are shown in Figure 6A and Video 6. Below the threshold, number of peristaltic waves per unit time drops (Figure 6B), as is confirmed by increased segment-to-segment propagation delay in the VNC (Figure 6G). Thus we see an effect on neural propagation due to a change in the mechanical interaction with the substrate, clearly showing how we cannot ignore the triad of nervous system-body-substrate in continuous conversation with each other. The speed of the larva (Figure 6C) also follows an interesting trend; with decreasing friction, speed first increases due to larger step sizes (Figure 6D), even though the frequency of peristaltic waves decreases (Figure 6B), but eventually saturates. Peak contraction of segments increase (Figure 6E) with decreased friction. These effects on behavior are clear testable predictions of our model.

Figure 6

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Video 6

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Our theoretical model for the forward crawling of a D. melanogaster larva incorporates the coupled mechanics of the soft body, the neural dynamics of VNC and frictional interactions with the substrate, complementing earlier isolated studies of these sub-systems. It produces a robust, sustained crawling gait with metrics consistent with experimental findings (Heckscher et al., 2012; Hughes and Thomas, 2007), and furthermore, can reproduce qualitative and quantitative changes in crawling gait due to perturbations in proprioceptive (Hughes and Thomas, 2007; Inada et al., 2011) and neural (Inada et al., 2011; Kohsaka et al., 2014) channels of segment-to-segment wave propagation.

A surprising finding of our model is its ability to produce sustained crawling with purely proprioceptive coupling between segments, a scenario which was first suggested by us for a very general model of crawling (Paoletti and Mahadevan, 2014). While both in our model and in experiments (Hughes and Thomas, 2007; Heckscher et al., 2012), the larvae can crawl without any proprioception, albeit much slower, the demonstration of purely proprioceptive crawling challenges the central role of CPG in crawling locomotion. It will be interesting to see if this finding can be experimentally tested by disrupting intersegmental coupling in the VNC.

Our study suggests that proprioception increases the robustness of crawling by effectively stabilizing segment-to-segment neural coupling. Interpreting this variation in parameters to that due to variations across individuals in a population, we predict that there should be much more variability in crawling metrics among individuals with silenced or weakened proprioception, a scenario amenable to testing using existing genetic tools (Hughes and Thomas, 2007; Inada et al., 2011).

In the context of body-nervous system coupling, proprioception is necessary for adaptive behavior, but are both neural and proprioceptive intersegmental coupling needed, since crawling is possible with only one of them? Neural coupling may be used to control speed. Indeed, our model predicts that increase in neural coupling leads to higher speeds, even in the presence of proprioception. This suggestion is supported by experiments in stick insects, which show that slow, steady-walking is mainly coordinated by local proprioceptive signals but faster motion is driven by increased neural coupling between legs (Büschges, 2012).

Our model makes testable predictions on how crawling should change if body-substrate coupling is modified. In particular, decreasing friction below slippage threshold should lead to a higher speed with larger step sizes but a smaller number of waves per unit time, another testable prediction.

Our minimal model makes a number of assumptions and simplifications. Improvements, some of which are already pointed out, will be necessary to describe forward crawling as more experiments become available. For example, in a very recent experiment (Itakura et al., 2015), a new class of pre-motor inhibitory neurons, Glutamatergic Ventro-Lateral Interneurons (GVLIs), were identified (glutamate inhibits larva motor neurons (Rohrbough and Broadie, 2002)), which ceased locomotion in the same segment when optogenetically activated. During unperturbed peristaltic wave propagation, GVLIs’ activation lagged motor neurons by several segments, suggesting that GVLIs provide a contraction termination signal when the wave reaches anterior segments. In contrast, inhibitory neurons of our model get activated slightly later than the excitatory neurons in the same segment, and receive proprioceptive input only from the same segment. Therefore, our current model does not take into account the role of GLVIs in locomotion. Some other points of improvement could be moving beyond linear mechanics, taking into account different muscles, introducing dynamics for stretch receptors, incorporating observations on PMSI’s (Kohsaka et al., 2014), including further specialized neuron types and accompanying connectivity profiles in a segment of the VNC, introducing possible long range intersegmental neural and proprioceptive connections, and using more sophisticated parameter fitting procedures.

Our integrated approach suggests generalizations that can move us beyond prograde rectilinear locomotion. Backward crawling, for instance, could be achieved by a separate neural circuit running from anterior-to-posterior, perhaps similar to that in C. elegans (Haspel et al., 2010), or by introducing anterior to posterior neural coupling between individual segments as in the model of Gjorgjieva et al. (2013). Additional extensions of the body mechanics to account for bending by differential movement of hemisegments, and a bilateral VNC, similar to that in (Gjorgjieva et al., 2013), allowing for propagation of neural excitation in opposite directions in different hemisegments will allow us to account for turning, and thus the larger behavioral repertoire of Drosophila larvae (Vogelstein et al., 2014).

Finally, our model naturally suggests novel biomimetic designs for robotics crawlers. In fact, the interest in soft robots has significantly grown in the last few years thanks to these systems’ capability of moving in uncertain environments, a daunting task for traditional rigid robots. For example Boyle et al., (2013) presented a proprioceptive-driven articulated crawler based on C. elegans morphology and showed that it is able to navigate arenas with unknown obstacles without requiring complex sensory capability. On the other hand, Trimmer and colleagues are developing a soft platform to build artificial crawlers, see for example (Umedachi et al., 2013) and (Kim et al., 2013) for a review of current attempts. Our model can then be exploited to merge soft robotics and proprioception to create novel biomimetic crawlers with an electromechanical circuit implementing force production and proprioception in a soft gel.

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Thank you for submitting your work entitled "Integrative neuromechanics of crawling in D. melanogaster larvae" for consideration by eLife. Your article has been reviewed by three peer reviewers, and the evaluation has been overseen by Ronald L. Calabrese as the Reviewing Editor and K VijayRaghavan as the Senior Editor.

The reviewers have discussed the reviews with one another and the Reviewing editor has drafted this decision to help you prepare a revised submission.

Summary:

The neuro-mechanics underlying peristalsis in the Drosophila larva offers an excellent opportunity to understand the link between a body, its nervous system and the environment surrounding it. In recent years, the neural network underlying crawling in the larva has been investigated by numerous labs. Theorists have started integrating experimental data in quantitative models, as was done in pioneering work by Gjorgjieva et al. 2013. The present manuscript offers a detailed model of the dynamics produced by the neuro-mechanical coupling of larval segments. The model is minimally complex, which makes it elegant and powerful. Using this model, the authors explore the influence of proprioceptive inputs and body-substrate friction on quantitative aspects crawling behavior. Many behavioral observations are reproduced. Remarkably, the authors propose a set of experimentally testable predictions. The strength of the manuscript is the novelty and the elegance of the model, its ability to account for a number of experimental observed phenomena, and its clear experimental predictions.

Essential revisions:

The major weaknesses pointed out in the expert reviews and which require revision are:

1) At the moment, the model is presented for applied mathematicians but not biologists. Given that this paper targets an audience of biologists, an effort should be made to create a more intuitive understanding of the math behind the model. To help this issue, we suggest that the authors add a model overview section, preceding the current modeling section, that explains the physics and the biology of the model in a way that is accessible to a biologist, but without equations. Such a section would have to be written so that a reader who wants to struggle through the equations will have read the overview and therefore have a good intuitive understanding of what the model consists of/model assumptions etc., whereas a reader who chooses to skip the technical section can still fully follow the Results and Discussion of the paper. Particularly indicate how the parameters of the model pertain to specific interactions perhaps in reference to an expanded Figure 1. Given that the model contains a large number of parameters more effort must be made to explain how the different parameters were fitted to the model. It is important to discuss the limitations inherent in the assumptions of the model (i.e. obvious discrepancies with/simplifications of the biological process) and in the future how might these limitations be overcome?

2) In 2013, Gjorgjieva et al. proposed a model similar to that presented by the authors. Although the conclusions of both studies do not fully overlap, the authors should discuss more thoroughly how their findings compare to those of Gjorgjieva et al. One striking difference is related to the necessary involvement of a CPG to drive peristalsis. The authors demonstrate that peristalsis can emerge from a pure neuro-mechanical coupling between neighboring segments. This result should be highlighted in the context of the CPG-centric view prevailing in the field. Perhaps part of the discussion found in supplementary information should be moved to the main text? The authors should also emphasize the fundamental differences between a CPG-based and pure neuro-mechanical coupling model.

3) Currently, the model does not integrate the results of two recent studies from the lab of Aki Nose:

A) Kohsaka H, Takasu E, Morimoto T, Nose A. A group of segmental premotor interneurons regulates the speed of axial locomotion in Drosophila larvae. Curr Biol. 2014 Nov 17;24(22):2632-42. doi: 10.1016/j.cub.2014.09.026. Epub 2014 Oct 16. PubMed PMID: 25438948.

B) Itakura Y, Kohsaka H, Ohyama T, Zlatic M, Pulver SR, Nose A. Identification of Inhibitory Premotor Interneurons Activated at a Late Phase in a Motor Cycle during Drosophila Larval Locomotion. PLoS One. 2015 Sep 3;10(9):e0136660. doi: 10.1371/journal.pone.0136660. eCollection 2015. PubMed PMID: 26335437; PubMedCentral PMCID: PMC4559423.

These papers should be considered. See B. and D. below for more detail.

Below are excerpts from the expert reviews that expand on these needed revisions.

A) The model is simple and elegant. Necessarily, it includes a large number of assumptions and simplifications. These are in general clearly stated, but some of these lack justification (and references to the neurobiological evidence). For example, the proprioceptive coupling is only from the nearby segment (contraction of segment i+1 feeds into the neural activation of segment i). Furthermore, such coupling is subject to a highly nonlinear activation function (effectively a threshold), and is fast (the very strong nonlinearity is somewhat reminiscent of other models, e.g. in C. elegans, Bryden 2008, Boyle 2012) begging the question is such strong nonlinearity strictly necessary for the model to work and does this render the dynamics of each segment as effectively binary, up to the capacitive filtering?

B) The model assumes one excitatory and one inhibitory neuronal cell type. Recent literature has revealed more specialized cell types, e.g. excitatory PMSIs (Kohsaka et al., Curr Biol. 2014;24(22):2632-42.) and inhibitory GVLIs (Itakura et al. 2015, PLoS ONE 10(9):e0136660), with both classes implicated in peristaltic locomotion, and at least the inhibitory class also possibly consistent with the existing "mission accomplished" intuition. Is this more recent literature consistent with the model? Would it be possible to add a schematic with relevant cell types and lay out the reasoning of how these may be simplified/collapsed to the model presented?

C) The mathematical model builds on a 2014 paper by two of the authors (Paoletti and L Mahadevan, Proc. R. Soc. B, 2014). While this is stated, much more detail needs to be given to explain what elements of the previous model framework are built on, and what elements of the current model and results are novel (e.g. are the results in sections 3.2.1 new?). Specifically, the previous model seems to focus on the passive and active body model and a minimal control system consisting of nearest-neighbor inter-segmental proprioceptive and diffusive coupling. There, the rhythmic pattern was generated by an oscillator in the head. This model now adds neuronal detail (pattern generating neuronal populations in each segment along the VNC). Thus in this model, some patterns can be propagated along the body even in the absence of ascending (proprioceptive) or descending (brain) control, and the proprioception is then superimposed (with a demonstrated stabilizing effect). This offers an excellent framework to study the interplay between central and peripheral control in this system. It would be helpful to the readers if any differences from the previous model (regarding the biomechanics/proprioception) and head oscillator were highlighted and justified. Where there are none, this should also be stated.

Background and literature search – I was surprised the very relevant work on leech proprioception was not cited, e.g. Cang and Friesen (2002), Model for Intersegmental Coordination of Leech Swimming: Central and Sensory Mechanisms. J. Neurophys. 87(6): 2760-2769.

D) The inhibitory interneurons described in the last paragraph of section 2.1 should be related to the population of local interneurons neurons described by Kohsaka et al. 2014: the PMSIs. The same should be done for the glutamatergic interneuron GVLIs described in Itakura et al. 2015. The predictions of the model should be compared to the activity patterns reported for the PMSIs and GVLIs. The authors will note that the dynamics of GVLIs does not appear to be consistent with interactions that purely involves neighboring segments (the GVLIs lag behind the motor neurons by several segments).

Related to this point, Gjorgjieva et al. 2013 assumed a coupling of ipsilateral excitatory and inhibitory interneurons (nearest-neighbor coupling through inhibitory interneurons), which is different from that used by the authors (nearest-neighbor coupling through excitatory interneurons). Should one favor one connectivity pattern over the other one? Could the functional implications of these differences be commented on?

E) In the last paragraph of the section 3.1.2: the authors make an equivalence between the acute loss-of-function achieved through the expression of halo-rhodopsin experiments in the motorneurons (see Inada et al., 2011) and an acute increase in the activity of the inhibitory interneurons. In my view, this mathematical description of the loss-of-function is incorrect. The effect of halo-rhodopsin should be described as a dramatic reduction in the activity of the excitatory interneurons (possibly through a large increase in the value of Ei in equation 1). This perturbation might have different consequences on the network dynamics. In particular, it might explain why the authors still predict muscular contractions upon activation of halo-rhodospin, which seems incompatible with the inhibition of the motorneurons.

F) At several occasions, the length of larvae is incorrectly approximated. In the second paragraph of the Introduction, third instar larvae are assumed to be 8 mm long. I was wondering where this estimate comes from. In my experience, third instar larvae are typically 4 mm long, with a maximum length of 5 mm. In section 3.1.1, the length of the larva is assumed to be 0.7mm. Did the authors intend to write 7mm? In any case, this estimate is justified by referring to the Figure 1 in Inada et al., 2011, but the scale bar on the figure of reference shows that the larva is not larger than 4mm. The authors should correct these dimension and update their parameter estimates.

G) In the Discussion, the authors should consider aspects of the biological phenomenology that have been neglected in their model. (1) The coupling of left and right hemi-segments. Gjorgjieva et al. 2013 have discussed that the contralateral coupling of excitatory and inhibitory interneurons might be important to produce robust propagation of peristaltic waves. How would such coupling affect the present model? (2) Can the present model account for the propagation of backward waves during backing-up movements?

H) It is not quite clear to me how the approach of Mahadevan and his colleagues differs from other modeling studies investigating crawling in D. melanogaster larvae or C. elegans or other worms or snakes. The authors state in the Introduction that recent studies on C. elegans (Wen et al., 2012; Boyle, Berri and Cohen, 2012) have already focused on local sensory feedback and proprioception and show that this suffices to modulate the locomotory pattern and explain gait transitions. Boyle and his colleagues have also shown that proprioception stabilizes crawling and makes it more robust against external perturbation. Furthermore, old studies by Niebur and Erdös in the 90s have already started to emphasize the role of sensory feedback in gait generation. Or, the works on crawling in leeches by Friesen and colleagues. I think it would be really helpful for the reader if the authors could include a paragraph into the Discussion stating how their approach differs from existing ones and what the specific advantages of their approach are.

I) The second point which was not quite clear to me concerns the coupling structure (neural and proprioceptive) that the authors used in their simulations. Which parts of this structure are based on neurobiological findings in D. melanogaster larvae and which are based on physiologically reasonable assumptions, because they are known to exist in other biological systems? In line with this, the question whether different neural coupling structures, e.g. including anterior to posterior couplings between the individual segments will interfere with the results, arises.

In case that not much is known about the neurobiology of the coupling structure and that it is mainly based on assumptions, did the authors test and compare different central/neural coupling structures and their outputs?

Essential revisions:

The major weaknesses pointed out in the expert reviews and which require revision are:

1) At the moment, the model is presented for applied mathematicians but not biologists. Given that this paper targets an audience of biologists, an effort should be made to create a more intuitive understanding of the math behind the model. To help this issue, we suggest that the authors add a model overview section, preceding the current modeling section, that explains the physics and the biology of the model in a way that is accessible to a biologist, but without equations. Such a section would have to be written so that a reader who wants to struggle through the equations will have read the overview and therefore have a good intuitive understanding of what the model consists of/model assumptions etc., whereas a reader who chooses to skip the technical section can still fully follow the Results and Discussion of the paper. Particularly indicate how the parameters of the model pertain to specific interactions perhaps in reference to an expanded Figure 1. Given that the model contains a large number of parameters more effort must be made to explain how the different parameters were fitted to the model. It is important to discuss the limitations inherent in the assumptions of the model (i.e. obvious discrepancies with/simplifications of the biological process) and in the future how might these limitations be overcome?

We thank the reviewers for encouraging us to make our presentation more accessible to biologists, who are our intended audience. As suggested, we introduced a new section (section 2.1) titled Overview, which aims to give an intuitive understanding of the model. The reader should be able to read this section, skip the mathematical details, and continue with the Results. The overview section is accompanied by an expanded Figure 1. A new panel zooms in a single segment and describes the important features and parameters of the model. In choosing the parameters we did not follow and automated approach but followed a few heuristics, which we now discuss in section 2.5. In section 2, we were already careful in pointing to the limitations of the model. Now this is accompanied by a much-expanded Discussion section where shortcomings of our model and possible ways to extend it are discussed.

2) In 2013, Gjorgjieva et al. proposed a model similar to that presented by the authors. Although the conclusions of both studies do not fully overlap, the authors should discuss more thoroughly how their findings compare to those of Gjorgjieva et al. One striking difference is related to the necessary involvement of a CPG to drive peristalsis. The authors demonstrate that peristalsis can emerge from a pure neuro-mechanical coupling between neighboring segments. This result should be highlighted in the context of the CPG-centric view prevailing in the field. Perhaps part of the discussion found in supplementary information should be moved to the main text? The authors should also emphasize the fundamental differences between a CPG-based and pure neuro-mechanical coupling model.

We thank the reviewer for bringing up these important issues. While the model of Gjorgjieva et al. was purely a neural model, some of their findings indeed overlap with ours. We added a paragraph in model section 2.2 highlighting the similarities and differences between our model and that of Gjorgjieva et al. Further, we added two more references to their paper in section 3.2.1 highlighting similarities in our results.

It is true that our model can produce crawling with pure proprioceptive coupling between segments. However, because the larva can crawl with silenced proprioception, see e.g. Hughes & Thomas 2007, we do not believe such scenario is a viable model for larval crawling. Nevertheless, following the reviewer’s suggestion, we added a paragraph in Discussion highlighting this finding. We should also remind the reviewer that the possibility of sustained crawling with purely proprioceptive coupling was demonstrated and discussed in a model by two of the authors before (Paoletti and Mahadevan, 2014). Finally, the discussion in the supplementary information is not related a pure neuro-mechanical model, therefore we chose to keep it as it is.

3) Currently, the model does not integrate the results of two recent studies from the lab of Aki Nose:

A) Kohsaka H, Takasu E, Morimoto T, Nose A. A group of segmental premotor interneurons regulates the speed of axial locomotion in Drosophila larvae. Curr Biol. 2014 Nov 17;24(22):2632-42. doi: 10.1016/j.cub.2014.09.026. Epub 2014 Oct 16. PubMed PMID: 25438948.

B) Itakura Y, Kohsaka H, Ohyama T, Zlatic M, Pulver SR, Nose A. Identification of Inhibitory Premotor Interneurons Activated at a Late Phase in a Motor Cycle during Drosophila Larval Locomotion. PLoS One. 2015 Sep 3;10(9):e0136660. doi: 10.1371/journal.pone.0136660. eCollection 2015. PubMed PMID: 26335437; PubMedCentral PMCID: PMC4559423.

These papers should be considered. See B. and D. below for more detail.

We thank the reviewers for bringing these papers to our attention. Indeed, the inhibitory interneurons described in the first reference, PMSIs, seem to match some aspects of the inhibitory neurons in our model, i.e. that they get activated slightly after the motor neurons in the same segment and that their optogenetic excitation leads to a local arresting of the peristaltic wave. Now, we added a simulation showing this latter point, Figure 3D-F, and expanded the Results section 3.1.2 to discuss this simulation. Kohsaka et al. report on other results that are not consistent with our model, which we discuss in detail in section 3.1.2.

On the other hand, the activation of inhibitory interneurons described in the second reference, GLVIs, lag the motor neurons by 2-3 segments. Itakura et al. speculate that GVLIs provide a contraction termination signal when the wave reaches anterior segments. In contrast, our inhibitory neurons get only input from only the same segment. Therefore, our current model does not take into account the role of GLVIs in locomotion. We discuss this shortcoming of our model in a new paragraph in the Discussion section.

Below are excerpts from the expert reviews that expand on these needed revisions.

A) The model is simple and elegant. Necessarily, it includes a large number of assumptions and simplifications. These are in general clearly stated, but some of these lack justification (and references to the neurobiological evidence). For example, the proprioceptive coupling is only from the nearby segment (contraction of segment i+1 feeds into the neural activation of segment i). Furthermore, such coupling is subject to a highly nonlinear activation function (effectively a threshold), and is fast (the very strong nonlinearity is somewhat reminiscent of other models, e.g. in C. elegans, Bryden 2008, Boyle 2012) begging the question is such strong nonlinearity strictly necessary for the model to work and does this render the dynamics of each segment as effectively binary, up to the capacitive filtering?

We are pleased to see that the reviewer finds the model elegant. We did do a large number of assumptions and simplifications, and we tried our best to point to biological literature when for evidence when it exists. However, there are many unknowns about how the larva crawls. When faced with an unknown, we assumed the minimal mechanism. The reviewer points to two such assumptions:

1) The proprioceptive coupling is only from the nearby segment: It is true that proprioceptive coupling could extend multiple segments, however in the lack of experimental data, we chose the minimal coupling, which is the nearest neighbor coupling.

2) The nonlinearity of the relationship between strain and proprioceptive signal: We want to make two points about this. 1) No characterization of a strain-output relationship of a proprioceptor is published in the larva. The sigmoidal relationship we used between strain and proprioceptive signal is not merely chosen for convenience: A sigmoid is a strong nonlinearity, but it is also a plausible nonlinearity. One would expect neurons detecting segmental contraction to be silent for low contraction, and saturate after some point. 2) It is true that the gain of the sigmoid in our model is large (1000), and it renders the proprioceptive signal to be binary. We can lower this gain from 1000 to 5 (which is still pretty steep), with little qualitative and quantitative change in sustained crawling. Below 5, the model stops producing sustained crawling. We do not want to draw strong conclusions from this observation and “predict” the necessity of a strong nonlinearity for crawling, because there are many other parameters to adjust which may allow gains below 5 possible.

The dynamics of each segment being effectively binary, up to the capacitive filtering (by which we believe the reviewer means the filtering due to springs in equation 3) is approximately true, because there are other time constants in play. The proprioceptive signal passes through neurons and muscles, each of which has its own time constants and nonlinearities, before feeding into the dynamical equations for segments (eq. 3). However, it is obvious, and we admit, that the dynamics described here is simple and would need to be modified to explain more complicated phenomena.

B) The model assumes one excitatory and one inhibitory neuronal cell type. Recent literature has revealed more specialized cell types, e.g. excitatory PMSIs (Kohsaka et al., Curr Biol. 2014;24(22):2632-42.) and inhibitory GVLIs (Itakura et al. 2015, PLoS ONE 10(9):e0136660), with both classes implicated in peristaltic locomotion, and at least the inhibitory class also possibly consistent with the existing "mission accomplished" intuition. Is this more recent literature consistent with the model? Would it be possible to add a schematic with relevant cell types and lay out the reasoning of how these may be simplified/collapsed to the model presented?

We thank the reviewer for bringing these papers to our attention. The results reported on PMSIs (which are inhibitory) do match some aspects of the inhibitory neurons in our model, i.e. that they get activated slightly after the motor neurons in the same segment and that their optogenetic excitation leads to a local arresting of the peristaltic wave. A simulation of such local arrest is added to the Results section 3.1.2 (Figure 3D-F). Kohsaka et al. report on other results that are not consistent with our model, which we discuss in detail in a new paragraph again in section 3.1.2.

Activation of GLVIs on the other hand lag the motor neurons by 2-3 segments. We therefore believe that our current model does not take into account the role of GLVIs in locomotion. We discuss this shortcoming of our model in a new paragraph in the Discussion section. Since our model does not include GLVIs, we did not add the schematic that the reviewer requested.

C) The mathematical model builds on a 2014 paper by two of the authors (Paoletti and L Mahadevan, Proc. R. Soc. B, 2014). While this is stated, much more detail needs to be given to explain what elements of the previous model framework are built on, and what elements of the current model and results are novel (e.g. are the results in sections 3.2.1 new?). Specifically, the previous model seems to focus on the passive and active body model and a minimal control system consisting of nearest-neighbor inter-segmental proprioceptive and diffusive coupling. There, the rhythmic pattern was generated by an oscillator in the head. This model now adds neuronal detail (pattern generating neuronal populations in each segment along the VNC). Thus in this model, some patterns can be propagated along the body even in the absence of ascending (proprioceptive) or descending (brain) control, and the proprioception is then superimposed (with a demonstrated stabilizing effect). This offers an excellent framework to study the interplay between central and peripheral control in this system. It would be helpful to the readers if any differences from the previous model (regarding the biomechanics/proprioception) and head oscillator were highlighted and justified. Where there are none, this should also be stated.

Background and literature search – I was surprised the very relevant work on leech proprioception was not cited, e.g. Cang and Friesen (2002), Model for Intersegmental Coordination of Leech Swimming: Central and Sensory Mechanisms. J. Neurophys. 87(6): 2760-2769.

As the reviewer points out, and we state in the paper, our model builds on a 2014 paper by 2 of the authors (Paoletti and Mahadevan, 2014), but has many differences from it. Most importantly, the previous model was built thinking of a “generic” crawler to prove the point that crawling without a CPG is possible. Therefore, it lacked a neural coupling between segments. Besides, the neural dynamics within segments were approximated with an oscillator. The current paper aims to model larval crawling: It has neural coupling between segments because the larva is able to crawl without proprioception (Hughes and Thomas, 2007). Other main differences are: 1) Our current model does not need a start signal at the beginning of each wave as in the old model, but restarts a wave by the neural coupling between T2 and A8, 2) In the current model, friction is set to zero when muscular forces exceed a threshold, modeling lifting of the body from the ground. In the old model, friction changed by direction of motion. 3) The dynamics of neurons in the old model (phase oscillators) were different than the current one and there was only one population of neurons. 4) The proprioception only activated the next segment but did not inhibit the current segment. 5) The old model was actually numerically solved in the continuum limit assuming a large number of segments, whereas here we solve the model with discrete segments. These differences are now pointed in the mathematical model section.

Regarding novelty of our results, we claim that all are novel, except that the model is able to crawl with purely proprioceptive coupling, which is just one of the results reported in section 3.2.1. We inserted a citation to the old model, in addition to the existing one in section 2, when we discuss purely proprioceptive crawling in section 3.2.1. The rest of our results, an overwhelming majority, are novel because: 1) They involve features that did not exist in the previous model: perturbations of excitatory and inhibitory neurons (section 3.1.2), crawling with purely neural coupling (section 3.1.3), manipulations of neural coupling (section 3.2.1). 2) They report results that the previous model did not address: resemblance to D. melanogaster larva crawling (section 3.1.1) and role of friction (section 3.2.2).

We are pleased to see that the reviewer agrees that our model offers an excellent framework to study the interplay between central and peripheral control in this system. We did our best to exploit this feature in this paper.

We apologize for missing the references to the leech proprioception. We added references to it in the Introduction.

D) The inhibitory interneurons described in the last paragraph of section 2.1 should be related to the population of local interneurons neurons described by Kohsaka et al. 2014: the PMSIs. The same should be done for the glutamatergic interneuron GVLIs described in Itakura et al. 2015. The predictions of the model should be compared to the activity patterns reported for the PMSIs and GVLIs. The authors will note that the dynamics of GVLIs does not appear to be consistent with interactions that purely involves neighboring segments (the GVLIs lag behind the motor neurons by several segments).

Related to this point, Gjorgjieva et al. 2013 assumed a coupling of ipsilateral excitatory and inhibitory interneurons (nearest-neighbor coupling through inhibitory interneurons), which is different from that used by the authors (nearest-neighbor coupling through excitatory interneurons). Should one favor one connectivity pattern over the other one? Could the functional implications of these differences be commented on?

We thank the reviewer for bringing these papers to our attention. We now relate the inhibitory population of our model to PMSIs in the Results section. Experimentally, PMSIs get activated slightly after the motor neurons in the same segment and their optogenetic excitation leads to a local arresting of the peristaltic wave. A simulation of such local arrest due to extra excitation of our inhibitory neurons is added to the Results section 3.1.2 (Figure 3D-F). Kohsaka et al. report on other results that are not consistent with our model, which we discuss in detail again in section 3.1.2. As the reviewer correctly points, activation of GLVIs lag the motor neurons by 2-3 segments and hence the inhibitory neurons in our model does not describe GLVIs. In a new paragraph in the Discussion section, we discuss this inconsistency.

Gjorgieva et al. assumed two kinds of bidirectional intersegmental couplings: 1) connections between the excitatory populations of the neighboring segments and 2) connections from the inhibitory populations to the neighboring excitatory populations. Our model only has the first type and it is unidirectional. Gjorgieva et al. comment that the function of the second type of connections is “to terminate activity in the previously active segment and ensure unidirectional wave propagation.” In our model, we were able to achieve termination of activity in the previously active segment by the activity of inhibitory neurons in that segment. Unidirectional wave propagation was built into the unidirectional connectivity in the intersegmental coupling. Therefore, our model did not need connections between inhibitory and excitatory neurons in neighboring segments and hence our choice of not including them. Favoring one coupling versus the other one will ultimately depend on experiments and currently we do not know of an experimental result that can distinguish between the two cases. In a new paragraph in section 2.2, we point to the differences between the results of Gjorgieva et al. and ours.

E) In the last paragraph of the section 3.1.2: the authors make an equivalence between the acute loss-of-function achieved through the expression of halo-rhodopsin experiments in the motorneurons (see Inada et al., 2011) and an acute increase in the activity of the inhibitory interneurons. In my view, this mathematical description of the loss-of-function is incorrect. The effect of halo-rhodopsin should be described as a dramatic reduction in the activity of the excitatory interneurons (possibly through a large increase in the value of Ei in equation 1). This perturbation might have different consequences on the network dynamics. In particular, it might explain why the authors still predict muscular contractions upon activation of halo-rhodospin, which seems incompatible with the inhibition of the motorneurons.

The reviewer is completely right in that “the effect of halo-rhodopsin should be described as a dramatic reduction in the activity of the excitatory interneurons”. This is exactly what we did in the model, however our wording was mistakenly suggests that we increased the activity of inhibitory interneurons. What was done was reducing the activity of excitatory population, by setting it to zero. This current did not come from an increase in inhibitory activity (see Figure 3C). We put in some more explanation in the paragraph to make this point clear.

F) At several occasions, the length of larvae is incorrectly approximated. In the second paragraph of the Introduction, third instar larvae are assumed to be 8 mm long. I was wondering where this estimate comes from. In my experience, third instar larvae are typically 4 mm long, with a maximum length of 5 mm. In section 3.1.1, the length of the larva is assumed to be 0.7mm. Did the authors intend to write 7mm? In any case, this estimate is justified by referring to the Figure 1 in Inada et al., 2011, but the scale bar on the figure of reference shows that the larva is not larger than 4mm. The authors should correct these dimension and update their parameter estimates.

Many thanks for catching this one! Indeed, we made a mistake in our estimates. We changed our third instar larvae length assumption to 4mm. In line 229, we assumed L = 0.7 mm, which meant the length of the larva to be 10L = 7mm. We now changed it the assumption to 10L = 4mm. In addition, we rewrote the paragraph where our estimates are given, to avoid confusions. The section now reads:

“Our simulations show that the larva produces about 0.038 waves per unit time associated with the relaxation of the excitatory neurons 𝜏", where waves start when the tail moves off the ground, leading to a larval speed ≈ 0.036L/𝜏". […] This in perfect agreement with the range reported in (Heckscher, Lockery and Doe, 2012): ≈ 39.5-126.5 𝜇m/sec.”

G) In the Discussion, the authors should consider aspects of the biological phenomenology that have been neglected in their model. (1) The coupling of left and right hemi-segments. Gjorgjieva et al. 2013 have discussed that the contralateral coupling of excitatory and inhibitory interneurons might be important to produce robust propagation of peristaltic waves. How would such coupling affect the present model? (2) Can the present model account for the propagation of backward waves during backing-up movements?

Both issues brought up here are indeed phenomenology neglected in our model. We acknowledge both points (and more) in the penultimate paragraph of our Discussion. To answer in more detail:

1) Addressing the effects of contralateral coupling of excitatory and inhibitory neurons requires substantial additions to our model. In addition to building a two-sided model of VNC as in Gjorgieva et al., we need to introduce a two-sided mechanical model for body hemisegments. It is hard for us to say how the contralateral coupling would affect such model, without actually building it. We intend to do so in future work.

2) Repeating what is said in the paper, “Backward crawling, for instance, could be achieved by a separate neural circuit running from anterior-to-posterior, perhaps similar to that in C. elegans (Haspel, Donovan and Hart, 2010).” Yet another mechanism could be to introduce anterior to posterior couplings between the individual segments as in (Gjorgjieva et al. 2013) to allow propagation of activity in VNC in the opposite direction. We added this possibility to the sentence above, “…or by introducing anterior to posterior neural coupling between individual segments as in the model of (Gjorgjieva et al., 2013).”

H) It is not quite clear to me how the approach of Mahadevan and his colleagues differs from other modeling studies investigating crawling in D. melanogaster larvae or C. elegans or other worms or snakes. The authors state in the Introduction that recent studies on C. elegans (Wen et al., 2012; Boyle, Berri and Cohen, 2012) have already focused on local sensory feedback and proprioception and show that this suffices to modulate the locomotory pattern and explain gait transitions. Boyle and his colleagues have also shown that proprioception stabilizes crawling and makes it more robust against external perturbation. Furthermore, old studies by Niebur and Erdös in the 90s have already started to emphasize the role of sensory feedback in gait generation. Or, the works on crawling in leeches by Friesen and colleagues. I think it would be really helpful for the reader if the authors could include a paragraph into the Discussion stating how their approach differs from existing ones and what the specific advantages of their approach are.

The reviewer is correct in pointing that the role of sensory feedback in locomotion has been investigated before. However, our approach has novelty. First, we point that the gait we study is rectilinear crawling, which is different than the gaits studied before. As the reviewer would appreciate, it is not guaranteed that results obtained from other gaits are going to automatically generalize to all gaits, due to differences in mechanics, neural control and the interactions with the environment. The model that we present for crawling is novel and simple. Second, we study the crawling gait in Drosophila melanogaster larvae. As we discuss in the introduction, D. melanogaster is an excellent system because of especially the genetic and optogenetic tools that can be used to perturb neural and proprioceptive systems and see their effects on crawling. In the paper we reproduce many of these experiments. We don’t think any of the models cited by the reviewer have been subject to such detailed testing, except maybe that of Boyle et al. Further, we emphasize and predict how changes in friction can affect the gait, something that has not been considered in D. melanogaster literature. To summarize, the strength of our approach comes from the model system and the gait we study: we propose a simple and novel model for the crawling gait, our model reproduces many of perturbation experiments in D. melanogaster larvae and it produces clear, testable predictions.

We added the following sentences to the end of the first paragraph to address these points: “However, such attempts were mostly aimed at quantitatively replicating organisms' behaviors, without any rigorous experimental validation. Moreover, the complexity of the studied organisms prevents exploiting such models to make testable predictions that can be used to shed light on the biological mechanisms regulating locomotion.”

I) The second point which was not quite clear to me concerns the coupling structure (neural and proprioceptive) that the authors used in their simulations. Which parts of this structure are based on neurobiological findings in D. melanogaster larvae and which are based on physiologically reasonable assumptions, because they are known to exist in other biological systems? In line with this, the question whether different neural coupling structures, e.g. including anterior to posterior couplings between the individual segments will interfere with the results, arises.

In case that not much is known about the neurobiology of the coupling structure and that it is mainly based on assumptions, did the authors test and compare different central/neural coupling structures and their outputs?

There does not exist an anatomical study where the coupling structure in the VNC is mapped out. Therefore, there is no reason to choose one connectivity over the other, as long as one can reproduce experimental results. Therefore, as we mention a few times in the paper, we chose to use a minimal model that is able to reproduce experimental results and further able to make clear, testable predictions.

Different coupling structures may interfere with results, one of our contributions is to identify a coupling structure that can reproduce results of experiments where neural and proprioceptive systems are perturbed. This does not mean that our results are only specific to the particular coupling we used. For example, our results are consistent with that of (Gjorgjieva et al. 2013), where they considered a purely neural model with anterior to posterior couplings between segments (the scenario that the reviewer is inquiring about). They found that “sensory feedback improves wave propagation” as we also found, or that increasing excitatory coupling between segments makes the waves propagate faster as we also observed. The overlap between our results and that of Gjorgjieva et al. makes us believe that including anterior to posterior couplings between the individual segments will not interfere with our results. We added references to the overlap between and our results and (Gjorgjieva et al. 2013) in the section 3.2.1.

The space of all possible coupling structures is vast, and the reviewer would hopefully agree that it is an unreasonably complicated task to test and compare all such structures. Our aim here, after all, is to present a model, as minimal as it could be, to reproduce experiments and make testable predictions, as opposed to ruling out all implausible ones.

Finally, in the supplementary information, we do present an alternative coupling structure, where head and tail are driven by separate neural controls and are synchronized by an external signal. We show that such model can also produce sustained crawling gait. The reason we chose to implement this scenario was to point that the nature of head-tail synchrony, whether it is mechanical as in the main text or neural as in the supplementary information, does not change our results.

Author details

1. Cengiz Pehlevan

   1. The Swartz Program in Theoretical Neuroscience, Harvard University, Cambridge, United States
   2. Simons Center for Data Analysis, Simons Foundation, New York, United States

   Contribution

   CP, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

   "This ORCID iD identifies the author of this article:" 0000-0001-9767-6063

2. Paolo Paoletti

   School of Engineering, The University of Liverpool, Liverpool, United Kingdom

   Contribution

   PP, Conception and design, Analysis and interpretation of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

   "This ORCID iD identifies the author of this article:" 0000-0001-6131-0377

3. L Mahadevan

   1. John A Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, United States
   2. Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, United States
   3. Wyss Institute for Bioinspired Engineering, Harvard University, Cambridge, United States
   4. Kavli Institute for BioNano Science and Technology, Harvard University, Cambridge, United States
   5. Department of Physics, Harvard University, Cambridge, United States

   Contribution

   LM, Conception and design, Analysis and interpretation of data, Drafting or revising the article

   For correspondence

   lm@seas.harvard.edu

   Competing interests

   The authors declare that no competing interests exist.

   "This ORCID iD identifies the author of this article:" 0000-0002-5114-0519

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