Abstract
Descending signals from the brain play critical roles in controlling and modulating locomotion kinematics. The anatomical wiring diagram of the C. elegans nervous system suggests that the premotor interneurons AVB, the hub for sensorimotor transformation, make exclusively electrical synapses with the B-type motor neurons that activate body wall muscles and drive forward locomotion. Here, we combined genetic analysis, optogenetic manipulation, and computational modeling to elucidate the functions of AVB-B electrical couplings. First, we found that B-type motor neurons could intrinsically generate rhythmic activity, constituting distributed center pattern generators. Second, AVB-B electrical couplings provide a descending pathway to drive bifurcation of motor neuron dynamics, triggering their transition from being stationary to generating rhythmic activity. Third, directional proprioceptive couplings between neighboring B-type motor neurons entrain the undulation frequency, forcing coherent bending waves to propagate from head to tail. Together, we propose that AVB-B electrical couplings work synergistically with proprioceptive couplings to enhance sequential activation of motor activity, and to facilitate the propagation of body undulation from head to tail during C. elegans forward locomotion.
Introduction
Locomotion requires coordinated rhythmic motor activity. In vertebrate and invertebrate motor systems, oscillatory signals are generated by dedicated neural circuits with intrinsic rhythmic properties called center pattern generator (CPG) 1-6. Distributed CPG modules, which correspond to different body segments, drive muscle movement with defined spatial patterns and temporal sequences. Coordination of rhythmic movement throughout the animal’s body requires a coupling mechanism, by which the dynamics of individual CPGs are both frequency and phase-locked 5, 7.
United efforts have been carried out, by using different model systems, to define the circuit logic for driving, modulating and coordinating locomotion 8, 9. The execution and maintenance of locomotion in vertebrates require descending signals from the brainstem and other brain regions, where glutamatergic neurons make connections with spinal premotor interneurons or establish direct synaptic contacts with motor neurons 10. For example, descending reticulospinal neurons driving excitatory neurons could initiate locomotion and modulate the speed of locomotion 11-13, whereas those activating groups of inhibitory neurons in the spinal cord could terminate locomotion 14. Descending neurons directly targeting forelimb motor neurons are critical for skillful motor behaviors 15.
Besides chemical synaptic transmission, gap junction couplings are also prevalent in the invertebrate motor circuit as well as the vertebrate spinal cord 16-18. In neonatal rats, electrical coupling between motor neurons was proposed to drive synchronized inputs to muscle cells 19. In zebrafish, electrical coupling between motor neuron and premotor interneuron points to a nimbly feedback mechanism for modulating rhythmic activity 20. These findings shed new light on the roles of electrical synapses in coordinating and controlling locomotion.
With a connectome at synaptic resolution and fully identified cell-types, C. elegans offers a unique opportunity to obtain a complete system-level understanding of a locomotory circuit. Here, we study the undulatory wave coordination during forward locomotion in C. elegans, addressing how command signals from the brain control the motor circuit. C. elegans locomotion is controlled by a network of excitatory cholinergic motor neurons, the A- and B-types that control backward and forward movement respectively, and inhibitory motor neurons (D-type) innervating body wall muscle cells 21. The anatomical wiring diagram 22, 23 of the adult worm motor circuit has revealed that the premotor inter-neurons AVB, which receive numerous inputs from sensory and other inter-neurons, have long processes along the entire ventral nerve cord, and form exclusive gap junctions with most B motor neurons (Fig. 1a). What are the functions of these electrical synapses in rhythmic movement?
During forward locomotion, AVB inter-neurons exhibit elevated yet non-oscillatory neural activity 24-26. AVB-B couplings are reminiscent of a descending pathway for motor control known in vertebrate systems. Whereas reticulospinal neurons could directly drive or modulate premotor CPGs in the spinal cord, whether similar CPG elements exist in the worm nerve cord remains unclear. Our previous work has shown that directional proprioceptive couplings between adjacent body regions are responsible for transducing and propagating rhythmic head bending activity along the worm body 27, 28. This finding, however, did not preclude the presence of CPGs along the worm nerve cord 29.
Combining molecular genetics and optical neurophysiology for manipulating defined motor circuit components in freely behaving C. elegans, we found that AVB-B electrical couplings could drive a bifurcation in motor circuit dynamics, induce intrinsic oscillations in mid-body B-type motor neurons, and facilitate undulatory wave propagation. Additionally, weak electrical couplings between motor neurons allow rapid and reciprocal interaction between head and body motor activities. Our computational modeling supports that descending signals through AVB-B electrical couplings can work synergistically with proprioceptive couplings to coordinate forward locomotion in C. elegans.
RESULTS
AVB-B gap junctions facilitate bending wave propagation during forward locomotion
In the C. elegans forward motor circuit, the AVB-B electrical synapses require the expression of the UNC-7 innexin in the AVB premotor interneurons, and UNC-9 in the B motor neurons, respectively (Fig. 1a) 25, 30. To determine whether AVB-B gap junctions play a role in forward locomotion, we investigated the locomotory behaviors of unc-7 and unc-9 mutants immersed in viscous solutions. We note that in these gap junction deficient mutants, intrinsically higher level of activity in the A-type motor neurons, which control backward locomotion, would disrupt directional forward movement 25. To reduce the interference from the motor activity arising from the backward motor circuit, A-type motor neurons were persistently silenced by an active K+ channel [Punc4::TWK-18(gf)] in all examined strains (Supplementary Information).
The locomotion kinematics was visualized and quantified by curvature kymograph along the centerline of the body. During bouts of forward movement, each body segment alternates between positive (red) and negative (blue) curvature, and stripes of curvature propagate from head (head = 0) to tail (tail = 1) (Fig. 1b). In control animals, the bending amplitude was the highest near the head, it declined gradually and reached a plateau near the mid-body region (~ 40% worm body, Fig. 1c). Both unc-7(hp121) and unc-9(fc16) null mutants could propagate undulatory waves from head to tail. However, their bending amplitude diminished rapidly and monotonously towards the tail without a plateau (Fig. 1b-d and Supplementary Video 1). Similar behavioral phenotypes were observed when premotor interneurons AVB were optogenetically ablated using miniSOG 31 (Plgc-55(B)::miniSOG or Psra-11::miniSOG) (Fig. 1b-d and Supplementary Fig. 1). Rescuing innexin UNC-7 in AVB premotor interneurons (unc-7; Psra-11::UNC-7) was sufficient to restore the body bending amplitude of unc-7 mutants (Fig. 1c-d and Supplementary Video 3).
A linear proprioceptive coupling model predicts diminished bending amplitude along the body
The coordination of body undulation during C. elegans forward locomotion requires proprioceptive coupling between adjacent body regions 27. Previously, we showed that in the middle and posterior body region, curvature change induced by pneumatic microfluidic device in an anterior segment defined the curvature of the posterior neighbor, explaining the unidirectional propagation of the bending wave from from head to tail in C. elegans 27. Similarly, when we trapped the middle body region of a wild type worm in a channel with defined curvature, the unrestrained posterior body region would exhibit bending curvature in the same direction as that imposed by the microfluidic channel (Fig. 2a-b). In unc-7 and unc-9 mutants, where AVB-B gap junctions are disrupted, the posterior unrestrained body region could still follow the imposed bending of the anterior body region (Fig. 2b-c and Supplementary Fig. 2), suggesting that mechanisms for proprioceptive coupling remain largely intact in these mutants.
We therefore ask why AVB-B gap junction deficient worms cannot efficiently propagate the undulatory waves. We first adopted a simple linear coupling model 27 for transducing the bending activity, in which we asserted that the undulatory wave begins with rhythmic dorsal/ventral bends near the head. Directional proprioceptive coupling between adjacent body regions is fully described by a set of first order differential equations (see Supplementary Information). This linear coupling model, however, predicts exponential decay of bending amplitude towards the tail, independent of whether AVB and B neurons are electrically coupled (Fig. 2d and Supplementary Information). By drawing an analogy with the electrotonic length in dendritic cable theory, we define a decay length constant ξ, which, to the leading order, is given by
subject to constraint cαmax/b ≤ 1. Here l is the spatial scale for proprioceptive coupling, αmax is the maximum muscle torque, b is the bending modulus of the worm body, and c denotes the strength of proprioceptive coupling. ξ → + ∞ leads to equal bending amplitude along the worm body. Incoorporating AVB-B gap junctions into our linear model would change c to cgm/(gm + gAVB−B), where gm and gAVB−B are motor neuron membrane conductance and gap junction conductance respectively. This prefactor, however, would reduce the length constant and deteriorate the propagation of bending.
B motor neurons are nonlinear functional units
The discrepancy between theory and experimental observations suggests that some basic assumption in the linear model must be modified. We hypothesize that proprioceptive signal is used to couple nonlinear functional units within adjacent body regions, a computational ingredient that is reminiscent of incorporating voltage-dependent conductance along passive dendritic branches. In one scenario, the nonlinear functional units are intrinsic oscillators, and we set out to test whether CPGs exist along the worm nerve cord.
Our linear coupling model predicts that abolishing head bending activity would completely abolish the undulatory wave; a coupled CPGs model predicts the opposite. To distinguish these possibilities, we performed spatially selective optogenetic inhibition of B motor neurons in a defined anterior body region (Fig. 3a-b and Supplementary Video 2) of freely swimming transgenic worms (Pacr-5::Arch). Bending activity near the head was abolished during green light illumination (Fig. 3b, curvature kymograph). Interestingly, higher frequency and lower amplitude undulation emerged from the mid-body (~50% body length) (Fig. 3b, 3e). We observed similar phenomena in a different transgenic strain expressing archaerhodopsin in both B and A motor neurons (Pacr-2(s)::Arch, Supplementary Fig. 3a), or by inhibiting head muscle cells (Pmyo-3::NpHR, Supplementary Fig. 3a).
These observations suggest that the mid-body motor neurons could generate high frequency oscillations. The C. elegans premotor interneurons (AVA, AVB) do not exhibit rhythmic activity correlated with body bending 25. Moreover, while the undulation frequency decreased in more viscous solution 32, the frequency of light-induced mid-body undulation remained fixed across a wide range of viscosities (Fig. 3c and Supplementary Fig. 3b). The lack of dependence of mid-body undulation frequency on the mechanical load indicates that rhythmic activity might be generated cell-autonomously.
To determine whether B motor neurons generate mid-body oscillation when head and body were decoupled, we performed systematic optogenetic ablation of B motor neurons (Pacr-5::miniSOG) localized in worm ventral nerve cord. Ablating a subgroup of mid-body B motor neurons (DB4-5 and VB5-7) completely abolished optogenetically-induced fast oscillation (Fig. 3d-e). When fewer B neurons in this cluster were ablated, fast mid-body undulation could still be induced (Supplementary Fig. 3). Sometimes, we even observed two decoupled waves emerged from head and posterior body regions (Supplementary Fig. 3c). Together, these data strongly suggest that the motor circuit in certain body region is not simply responding to the proprioceptive signal from an anterior body region. B motor neurons are thus nonlinear functional units and constitute distributed CPG modules along the body.
AVB-B gap junctions drive a bifurcation in B neuron dynamics
We next ask whether AVB-B gap junction inputs participate in driving fast mid-body oscillation when head and body were decoupled. Indeed, in AVB-B gap junction deficient mutants, or AVB ablated worms, optogenetically-induced mid-body fast undulation was almost completely abolished (Fig. 4a-b). Inhibiting anterior B motor neurons, or inhibiting anterior body wall muscles, which eliminated head bending activity and thereby the proprioceptive signal, would cause whole-body paralysis (Fig. 4a-b and Supplementary Fig. 4). Rescuing innexin subunits UNC-7 in AVB (unc-7; Psra-11::UNC-7) significantly restored the light-induced fast undulation (Fig. 4b and Supplementary Video 3).
From a dynamic systems point of view, our data suggest that AVB-B gap junction inputs could drive a bifurcation in B neuron dynamics, leading to a transition from being stationary to rhythmic oscillation. In the presence of AVB-B gap junction inputs, the dynamics of B motor neuron loses its stability. Time-varying proprioceptive signal from anterior body region may easily trigger large change in motor neuron membrane potential and thus facilitate bending wave propagation.
This intuition was recapitulated by our simulation of a nonlinear coupling model (Supplementary Information), modified to incorporate the following two experimental observations. First, without a strong proprioceptive signal, AVB-B electrical couplings could induce high frequency rhythmic bending activity, driven by B motor neurons. Second, proprioceptive signals could entrain B motor neurons to oscillate at a normal (slower) undulation frequency. Our simulation shows that, indeed, AVB-B gap junction inputs could help equalize the bending amplitude along the body (Fig. 1b, 4c).
Electrical couplings between motor neurons allow rapid and reciprocal interactions between head and body motor activities
Having considered the functions of AVB-B gap junctions, we next ask whether local gap junctions between motor neurons (Fig. 1a), identified by the anatomical wiring diagram 22, 23, play a role in controlling worm locomotion. Interestingly, spatially selective optogenetic inhibition of B motor neurons localized in the middle or posterior body region of a freely swimming worm could reliably induce whole body paralysis within ~ 300 ms (Fig. 5a-b, Supplementary Fig. 5 and Supplementary Video 4). Electrical coupling is the best candidate for rapid redistribution of a hyperpolarization current in B motor neurons (Pacr-5::Arch). We found that in either unc-7 mutants or AVB ablated worms, whole body paralysis could still be induced (Fig. 5c), suggesting that in the absence of AVB-B gap junctions, the hyperpolarization signal could still be transduced. However, in unc-9 mutants, rhythmic bending activity in the anterior body region persisted when mid-body B motor neuron were silenced (Fig. 5a, 5c and Supplementary Video 5), and rescuing UNC-9 expression in B and A motor neurons (Pacr-2::UNC-9) was sufficient to restore light-induced whole body paralysis (Fig. 5c). These results suggest that electrical couplings between motor neurons contribute to the paralytic effect.
The probability of light-induced whole-body paralysis was dose-dependent. Lower laser intensity, which reduced the degree of hyperpolarization in mid-body B motor neurons, did not abolish forward movement, but reduced the frequency and amplitude of whole body undulation (Fig. 5d-e). Together, these data suggest that electrical couplings between motor neurons may retrogradely regulate the head bending activity, hence reconfiguring the dynamics of the body undulation.
DISCUSSION
In vertebrate motor systems, global descending signals from the hindbrain can drive premotor CPG networks in the spinal cord to generate rhythmic motor patterns 10. In C. elegans, we discovered that the B class motor neurons themselves are not passive recipients of descending command neuron inputs. Instead, AVB-B electrical couplings could induce intrinsic oscillations in at least some B motor neurons (Fig. 6b), and facilitate undulatory wave propagation. C. elegans, due to the cellular economy of its nervous system, has implemented a compressed yet converging algorithm for motor control.
We have developed a phenomenological nonlinear coupling model to explain our experimental observations (Supplementary Information). In the model, we divided whole worm into several segments, and the segment number is accordance with the number of DB motor neurons. The excitatory motor neurons were modeled as nonlinear functional units with active voltage gated ion channels. Because the detailed form of intrinsic membrane conductance in B motor neurons remains unknown, we have used the classic phenomenological model based on persistent Ca2+ and K+ conductances, which suffice to account for key experimental findings. Although a class of nonlinear neuron models could generate intrinsic rhythmic activity 33, distinguishing them requires electrophysiological recording in tiny worm motor neurons, an extremely difficult technique that has just recently become possible 34. Intriguingly, a complementary study has found that voltage dependent Ca2+ channels are required for generating intrinsic rhythmic activity in A-type motor neurons (personal communication with Shangbang Gao and Mei Zhen). Whether similar channels drive the B-type motor neuron oscillations may be addressed in future studies.
We also found a previously unknown role of electrical coupling between motor neurons, which would allow rapid and reciprocal interaction between head and body motor activities (Fig. 5-6). Strong electrical coupling between motor neurons would tend to synchronize motor activity along the whole worm body and thus deteriorate bending wave propagation (over-expressing UNC-9 in B motor neurons would paralyze worms, Supplementary Video 6). During normal locomotion, weak electrical couplings between motor neurons might help mediate the undulation frequency along the whole worm and augment the excitability of motor neurons. Consistent with this view, we found that optogenetically ablating mid-body B neurons reduced the head undulation frequency (Supplementary Fig. 1a and Supplementary Fig. 3d). Direct test of the functional contribution of local electrical couplings would require eliminating electrical synapses between motor neurons, while sparing the AVB-B ones. However, current genetic tools for manipulating gap junction expression with defined wiring specificity remain to be developed because both types of gap junctions require UNC-9 innexin subunits.
UNC-9 is also express in C. elegans body wall muscle cells 35. Could electrical coupling between muscle cells and electrical coupling between motor neurons have similar functions? We and others found that spatially selective optogenetic inhibition of mid-body muscle cells did not affect the bending activity in the anterior body region, but would abolish the bending activity in the posterior body region 36, consistent with our proprioceptive coupling model (Fig. 6). Furthermore, in unc-13 mutants, which lack motor neuron inputs, optogenetically activating targeted ventral or dorsal muscle cells could induce local body bending, but did not induce bending in neighboring regions 27. These data suggest that electrical coupling between body wall muscle cells may play a more localized role in mediating the bending activity in C. elegans.
The functional implication of gap junction in the context of neural circuit and behavior could be counterintuitive, confounding and are often underestimated 37, 38. Insight has been gained from studying small nervous systems, where individual neurons can be identified and connectivity between neurons could be mapped out. Experimental and modeling studies on crustacean stomatogastric ganglion suggested that the strength of electrical synapses can mediate the frequency of coupled oscillators 39; the interplay between electrical and chemical synapses provides degenerate circuit mechanisms for switching between fast and slow oscillatory behaviors 40. In C. elegans, electrical synapse could offer a parallel pathway for routing sensory information 41, 42, or participate in coincident detection of multiple sensory cues 43. Our findings also demonstrate that electrical couplings can have a potent influence on the motor circuit dynamics.
Our data suggest that distributed CPGs exist in the C. elegans ventral nerve cord. Because worm must adapt extreme ranges of external mechanical load imposed by a changing environment 32, directional proprioceptive couplings transduced by B motor neurons 27 play the pivotal role in entraining the rhythm of body oscillators and in propagating coherent bending waves from head to tail (Fig. 6). Despite substantial difference between worm motor circuit and those in higher animals, the neural computation for coordinating C. elegans locomotion could now also be categorized into a coupled CPG model (Fig. 6). The building blocks for CPG and the detailed coupling mechanisms between CPGs could vary across species 5, 6, yet the overarching principles underlying body coordination appear to be surprisingly similar. By integrating neuromuscular dynamics and biomechanical feedback, our findings represent one step towards a full systems model of animal locomotion.
METHODS
Worm Strains and Cultivation
Wild-type (N2), mutant, and transgenic worms were cultivated using standard methods 44. Strain information can be found in the Supplementary Experimental Procedures. Transgenic worms used in all optogenetic experiments were cultivated in dark at 20-25°C on NGM plates with Escherichia coli OP50 and all-trans retinal (ATR). We performed all experiments using young adult hermaphrodites.
Microfluidic Devices
Custom microfluidic devices were fabricated in PDMS using soft lithography techniques. We loaded each microfluidic channel with dextran solution [~25% dextran in M9 buffer (1 Pa·s viscosity)]. An individual worm was flowed into the inlet of each microfluidic channel and worm position within each channel was manually controlled by syringes connected to polyethylene tubing.
Behavioral Quantification and Optogenetic Manipulation
Experiments were performed on a Nikon inverted microscope (Ti-U) under 10X magnification with dark field illumination. Worms were immersed in viscous solution (~25% dextran in M9 buffer in most cases), sandwiched between two glass slides, and were retained within the field of view of an imaging objective by a custom tracking system. Video sequences were taken by a Basler CMOS camera (aca2000-340km), and worm body centerline were extracted in real-time. We used custom software written in MATLAB (MathWorks, Inc. Natick, MA) for post-processing the behavioral data. We used CoLBeRT system 36 to perform spatially-selective optogenetic manipulation for different motor circuit components. For optogenetic inhibition, we used a 561 nm solid-state laser with maximum intensity at 16 mW/mm2.
Optogenetic Ablation
Optogenetic ablation was carried out on transgenic strains in which mitochondrially targeted miniSOG (mini singlet oxygen generator) was specifically expressed in C. elegans neurons. Upon blue light illumination, mito-miniSOG causes rapid cell death in a cell-autonomous manner 31. To ablate AVB neurons [Plgc-55(B)::miniSOG or Psra-11::miniSOG], L3/early L4 worms cultivated on OP50 were transplanted to an unseeded NGM plate, restricted within ~1.7 cm2 area via filter paper with a hole in the center and soaked with 100 μM CUCL2. Worms were illuminated with blue LED light (M470L3-C5, Thorlabs, Inc.) with intensity 80.2 mW/cm2, measured by power meter (PM16-130, Thorlabs, Inc.). The temporal sequence was 0.5/1.5 s on/off pulses for 30 min. After illumination, worms were transplanted to newly OP50 seeded NGM plate with/without ATR for latter behavioral experiments.
For selective B motor neuron ablation (Pacr-5::miniSOG), single L3/early L4 worm was picked up from OP50 seeded NGM plate to a 3% agarose (wt/vol) coated glass slide. Worm was covered by a cover glass to be kept stationary. Spatially selective illumination pattern was generated by a digital micromirror device (DLI4130 0.7 XGA, Digital Light Innovations, Texas) to target individual neurons through a 20X objective mounted on a Nikon inverted microscope (Ti-U). Neurons were identified using mCherry fluorescence signal. We used 473 nm blue laser with intensity 29 mW/mm2. The temporal sequence was 0.5/1.5 s on/off pulses for 15 min. After illumination, the worm was first recovered by 2 μl M9 buffer, and was transplanted to OP50 seeded NGM plated with/without ATR for latter behavioral experiment.
AUTHOR CONTRIBUTIONS
T.X. and Q.W. conceived the project. T.X. and J.H. performed all experiments. T.X. performed all data analysis. S.S. and Q.W. developed the computational model. J.H., M.P., T.K., Y.L., M.W. made all reagents and transgenetic lines used in this work. Q.W. wrote the manuscript with inputs from M.Z., T.X. and S.S.
Supplementary Experimental Procedures
Strain Information
A linear proprioceptive coupling model for undulatory wave propagation in C. elegans
We first developed a linear coupling model to simulate the wave propagation within the body of C. elegans. The head motor circuit contains an oscillator that dominates the rhythmic motion of the whole body. Other segments cannot oscillate by themselves. They oscillate according to the proprioceptive input from the anterior segment. This simple model is phenomenological, for it does not simulate the detailed dynamics of individual neurons. Instead, it aims at elucidating several key properties of wave propagation based on linear proprioceptive coupling.
The dynamics of the head oscillator is dictated by the following equations.
where v describes relative neural activity of head motor neurons. It is not the real membrane potential, but a variable representing the dorsal-ventral bias of neural activity. It has a value between -1 and 1, or in a slightly broader region. When dorsal B neurons have a stronger neural activity than ventral ones, v > 0, and vice versa. αmax * u is the maximum torque that the muscle can generate, and u is a dimensionless variable. b is the bending modulus of the worm body and k is the curvature of the segment. The second equation relates neural activity and muscle torque. The nonlinear term, v − βv3, as well as the negative sensory feedback term – ak are critical for generating head oscillation. The last equation relates muscle torque and body shape change in a defined segment 1, 2. τη is a time constant proportional to the viscosity η of the medium. In the case of viscous medium, namely large η, τη ≫ τη, τη ≫ τs, the period of such a relaxation oscillator is largely determined by τη. As a result, the undulation frequency would decrease with viscosity, which has been observed experimentally 1.
Apart from the head, we divide the worm body into 5 segments (the number of segments are not essential for our theoretical argument). These segments have a different dynamics, which is described as follows.
Here ki−1 is the curvature of the anterior segment. The term −gvi denotes the contribution of AVB-B gap junction to the membrane potential of B neurons. More essentially, it should be −g(vi − vAVB).For simplicity, we set vAVB = 0. On the one hand, the membrane potential of AVB remains nearly constant during forward locomotion. On the other hand, setting it to 0 can eliminate the bias between dorsal and ventral sides and avoid unnecessary complexity in our model. Here C is the membrane capacitance, gm is the leaky membrane conductance.
Eqs. (S1) and (S2) arise naturally from a biologically more realistic model where dorsal and ventral activities are treated separately. The equations now become
If we subtract “dorsal equations” from “ventral equations”, Eqs. (S1) and (S2) will be derived: dorsal activity and ventral activity are anti-phased, or namely vd = −vi and vdi = −vvi. Simulation using Eqs. (S3) and (S4) did not show any essential difference compared to the simulation using Eqs. (S1) and (S2).
The linear proprioceptive coupling model makes two key predictions that can be tested experimentally. First, abolishing the head motor activity should abolish the undulatory wave propagation. However, we observed high frequency undulation emerged from the mid-body of the worm when head and body were decoupled in our experiments. Second, simulation suggests AVB-B gap junction coupling would deteriorate bending wave propagation (Fig. 2d and Supplementary Fig. 6a). However, experimental data revealed the opposite. Therefore, we conclude that the linear coupling model is oversimplified and must be modified.
Parameters used in simulation:
An analytical solution to the linear proprioceptive coupling model
To gain a deeper insight into the amplitude decay during wave propagation, we have developed an analytical understanding for the linear coupling model by transforming Equation (S2) into a continuous form.
The dynamic equations for wave propagation can be rewritten as follows.
In the first equation, the second term, ck(x − l), denotes how proprioceptive couplings drive the change of neural activity in motor neurons (Wen et al., 2012). l is the characteristic length of proprioception. The second equation relates the change of body shape with neural activity. Here, for simplicity, we dropped the equation that relates neural activity and muscle torque in Eqs. (S2). The delay time between neural activity and muscle torque, characterized by τη, was absorbed by the time constant τη in Eq. (S6). We shall assume a harmonic motion to describe the dynamics of the head. The v(x) and k(x) can be written as
The neural activity and curvature are not in-phase. But we can absorb the phase factor into coefficents v0 and k0, which are all complex numbers.
We postulate that the solution is also sinusoidal, namely
Substituting into Eqs. (S6), and defining τc = C/gm, we have
Replacing k(x − l) and k(x) − k′(x)l + k″(x)l2/2, and using l ≪ λ, we found
The solution is
where
Thus the decaying length constant satisfies
Expanding it to power series of ω and only preserving low order terms, we have
According to earlier works 1,
The time constant τη increases with the viscosity of the medium η, and the decay length constant ξ would decrease with η. To the leading order, the decay length constant is given by
When we incorporate electrical couplings between AVB-B neurons into this model, the equations become
where g denotes the strength of the gap junction.In this case, the decay length constant becomes
which further deteriorates the bending wave propagation. This prediction is inconsistent with the experimental observations.
A detailed nonlinear proprioceptive coupling model
The discrepancy between experimental data and our simple linear model forces us to develop a modified nonlinear coupling model. The head oscillator has the same functional form as that in the linear coupling model:
Here the subscripts ‘d’ and ‘v’ denote dorsal and ventral sides respectively, αmax * (nd − nv) is the maximal torque that the muscle can generate.
For all the other segments, our model will incorporate the experimental observations that motor neurons can generate intrinsic oscillations when head and body are decoupled, and when AVB and B neurons are electrically coupled (g > 0).The intrinsic membrane properties in B motor neuron remain unknown because of the extraordinary difficulty in performing whole-cell patch clamp recording in motor neurons (cell body ~ 1-2 μm in diameter). For simplicity, we developed a classic phenomenological model based on persistent Ca2+ and K+ conductances, which is derived from a simplified version of the INa − IK model 3.
The dynamic equations are written in the following form.
k′ is the curvature of the neighboring anterior segment; m(V), n(V) are voltage dependent gating variables; M(V) relates cell membrane potential and muscle torque.
g(V0 − V) is the current flowing through the gap junction between AVB neurons and B neurons, and V0 is the membrane potential of the AVB neurons, held at a constant value.
The terms −M(Vv) and −M(Vd) in the third and the fourth equation of (S21) reflect contralateral inhibition from D-type GABAergic motor neurons: dorsal D neurons can be activated by ventral B neurons and thereby inhibit dorsal muscle activity, and vice versa.
We were simulating 6 segments (including the head), similar to that in a linear coupling model. The parameter values are listed below.
Comparing the cases g = 0 and g = 100pS, we find that gap junction coupling between AVB and B neurons could equalize the bending amplitude (Supplementary Fig. 6b). High frequency oscillation within middle segments can also be generated when head and body are decoupled (Supplementary Fig. 6c). When we hold an anterior segment at a static curvature, the neighboring posterior segment will follow (Supplementary Fig. 6d). These properties were consistent with our experimental observations.
Finally, we incorporate electrical coupling between B neurons into our model. The dynamic equations now become
where Vd/v(+1) and Vd/v(−1) denote the membrane potentials of neighboring posterior dorsal/ventral and anterior dorsal/ventral B motor neurons.
We found that the amplitude decay became worse in the presence of strong B-B gap junctions (Supplementary Fig. 6e), consistent with experimental observation that overexpression of UNC-9 in B motor neurons would deteriorate coordinated locomotion (Supplementary Video 6). Could B-B gap junctions have other motor functions? As our optogenetic experiments suggested (Fig. 5), gap junctions between motor neurons could mediate the frequency of the head oscillator. However, a detailed model requires a deep understanding of how the head motor circuit operates. Our current head oscillator model is purely phenomenological, and a better model would be developed in a future work where all circuit components in the head motor circuit can be carefully defined and experimentally identified.
ACKNOWLEDGMENTS
The authors thank Christopher Fang-Yen and Anthony Fouad for helpful discussion. This work was funded by the CAS Hundreds Talents Plan and National Science Foundation of China (NSFC-31471051 and NSFC-91632102).