J Neurophysiol 115: 887–906, 2016. First published November 18, 2015; doi:10.1152/jn.00693.2015. A three-leg model producing tetrapod and tripod coordination patterns of ipsilateral legs in the stick insect T. I. Tóth and S. Daun-Gruhn Department of Animal Physiology, Institute of Zoology, University of Cologne, Cologne, Germany Submitted 14 July 2015; accepted in final form 10 November 2015 three-leg model; interleg coordination; locomotion; sensory feedback LOCOMOTION OF INSECTS, IN particular that of the stick insect, is a well-studied field of animal physiology. In the studies, a number of different approaches have been used. The spectrum reaches from behavioral observations (e.g., Hughes 1952; Wilson 1966; Graham 1972, 1985; Delcomyn 1981), over electrophysiological methods (e.g., Bässler 1977, 1983; Laurent and Burrows 1989a,b; Büschges 1995, 2005; Akay et al. 2007; Büschges and Gruhn 2008; Borgmann et al. 2007, 2009, 2011; Schmitz 1986a,b), to modeling work (e.g., Cruse 1980, 1990; Cruse et al. 1998; Dürr et al. 2004; Ekeberg et al. 2004; Daun-Gruhn 2011; Daun-Gruhn and Tóth 2011; Daun-Gruhn et al. 2011; von Twickel et al. 2011, 2012; Schilling et al. 2013; Tóth et al. 2015). The models cited here reflect various aspects of insect locomotion and reach from one-leg models (Ekeberg et al. 2004; Daun-Gruhn et al. 2011; von Twickel et al. 2011; Tóth et al. 2012; Knops et al. 2013) to six-leg models (Cruse 1980 1990; Cruse et al. 1998; Dürr et al. 2004; von Twickel et Address for reprint requests and other correspondence: S. Daun-Gruhn, Dept. of Animal Physiology, Institute of Zoology, Univ. of Cologne Zülpicher Strasse 47b, D-50674 Cologne, Germany (e-mail: [email protected]). www.jn.org al. 2012; Schilling et al. 2013). Thus Ekeberg et al. (2004), for example, study and simulate the mechanisms of coordination first in a restricted middle leg of the stick insect. They then examine to what extent the mechanisms found in the middle leg can, perhaps, be adapted to apply to the stepping of front and the hind leg, too. On the other end of complexity, Cruse (1980, 1990), Cruse et al. (1998), Dürr et al. (2004), and Schilling et al. (2013) have created, in a steadily evolving process, an artificial neural network that can emulate various kinds of six-leg walking of insects. Thus a large body of knowledge on this topic has been created over the years. In spite of the progress made, a number of important details of the intra- and interleg coordination during locomotion (normal walking) remain unclear. Our aim in the work presented here has been to introduce a modeling approach that is capable of grasping complex interactions of the local (neuronal) mechanisms of the legs during walking. More precisely, we endeavored to establish an intersegmental coupling mechanism between the ipsilateral legs that could produce the patterns of normal walking ipsilaterally. To achieve this, we made use of an approach that was different from the aforementioned ones. Our models consisted of local neuronal networks that controlled the function of an antagonistic muscle pair at a leg joint (e.g., protractor-retractor at the thorax-coxa joint). Even more importantly, each such network was constructed using anatomical data and experimental results, where they were available (Bässler 1983; Büschges 1995, 1998, 2005; Westmark et al. 2009), and physiologically reasonable assumptions to fill in the gaps, where data were not available. Our model was thus created by connecting these local control networks by excitatory or inhibitory sensory pathways whose existence seems to have, at least indirectly, been justified by experiments. In our earlier work, we constructed models of the neuromuscular system of an insect leg, in particular of a (middle) leg of the stick insect: Tóth et al. (2012), Knops et al. (2013), and Tóth et al. (2013a, b). The models included two or all of the three antagonistic muscle pairs (m. protractor and retractor coxae, m. levator and depressor trochanteris, and m. extensor and flexor tibiae) at the three main joints: the thorax-coxa (ThC), the coxa-trochanter (CTr), and the femur-tibia (FTi) joint. Each muscle pair in the models possessed its dedicated local neuronal control network. The structure and function of these local networks were based on experimental findings and physiologically reasonable assumptions (Borgmann et al. 2011; Daun-Gruhn et al. 2011; Tóth et al. 2012; Knops et al. 2013). Using these one-leg models, we constructed a model of three ipsilateral legs with suitable intersegmental coupling between them as described in the present work. Note that the 0022-3077/16 Copyright © 2016 the American Physiological Society 887 Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 Tóth TI, Daun-Gruhn S. A three-leg model producing tetrapod and tripod coordination patterns of ipsilateral legs in the stick insect. J Neurophysiol 115: 887–906, 2016. First published November 18, 2015; doi:10.1152/jn.00693.2015.—Insect locomotion requires the precise coordination of the movement of all six legs. Detailed investigations have revealed that the movement of the legs is controlled by local dedicated neuronal networks, which interact to produce walking of the animal. The stick insect is well suited to experimental investigations aimed at understanding the mechanisms of insect locomotion. Beside the experimental approach, models have also been constructed to elucidate those mechanisms. Here, we describe a model that replicates both the tetrapod and tripod coordination pattern of three ipsilateral legs. The model is based on an earlier insect leg model, which includes the three main leg joints, three antagonistic muscle pairs, and their local neuronal control networks. These networks are coupled via angular signals to establish intraleg coordination of the three neuromuscular systems during locomotion. In the present threeleg model, we coupled three such leg models, representing front, middle, and hind leg, in this way. The coupling was between the levator-depressor local control networks of the three legs. The model could successfully simulate tetrapod and tripod coordination patterns, as well as the transition between them. The simulations showed that for the interleg coordination during tripod, the position signals of the levator-depressor neuromuscular systems sent between the legs were sufficient, while in tetrapod, additional information on the angular velocities in the same system was necessary, and together with the position information also sufficient. We therefore suggest that, during stepping, the connections between the levator-depressor neuromuscular systems of the different legs are of primary importance. 888 INTERLEG COORDINATION Fig. 1. Schematic illustration of the main joints (segments) of a stick insect leg. The Greek letters ␣, , and ␥ denote the retraction, the levation, and the flexion angles, respectively. [Adapted with permission from M. Gruhn, unpublished observations.]. model created thus is not simply a triplication of the previous one-leg model but it reaches a higher level of complexity by the intersegmental connections that represent excitatory or inhibitory sensory pathways. In this paper, we report simulation results with this three-leg model. The model can successfully simulate the characteristic coordination patterns that occur during walking of the stick insect (so-called tripod and tetrapod coordination patterns) and the transitions between them with satisfactory accuracy. The biological relevance of intersegmental connections implemented in the model will be discussed. We will also discuss some implications of our results for the locomotion of other legged species. METHODS The three-leg model to be presented here is based on the one-leg models (Daun-Gruhn et al. 2011; Tóth et al. 2012; Knops et al. 2013). In Fig. 1, the schematic picture of a stick insect (middle) leg is Meso−thorax (ML) PR control network g g d13 d14 LD control network EF control network g g MN IN13 C49 IN14 C50 C7 C8 g MN7P C25 MN8R C26 MN g MN IN17 C53 IN18 C54 C9 C10 MN10L C28 MN g MN IN15 C51 β IN21 C57 IN22 C58 C11 C12 MN12E C30 MN CPG Lev. m. IN19 C55 d22 g MN11F C29 CPG Dep. m. Ret. m. g d21 g app11 g app12 g MN9D C27 CPG Pro. m. IN16 C52 d18 g app9 g app10 g app7 g app8 g g d17 IN20 C56 Flex. m. Ext. m. γ IN23 C59 IN24 C60 g d20 Fig. 2. The neuromuscular model of 1 (middle) leg. Local networks (PR, LD, and EF as indicated) controlling the activity of three antagonistic muscle pairs: m. protractor and retractor coxae (Pro. m. and Ret. m.), m. levator and depressor trochanteris (Lev. m. and Dep. m.), and m. extensor and flexor tibiae (Ext. m. and Flex. m.). Central pattern generator (CPG) in each of these networks: central pattern generator consisting of a pair of mutually inhibitory nonspiking neurons (e.g., C7–C8 in the PR control network); gapp,7, gapp,8, etc.: (central) input to the CPG neurons (individually variable). MN7P, MN8R, etc.: motoneurons driving the corresponding muscles; gMN: uniform excitatory input to the motoneurons. IN13-IN14, etc.: premotor interneurons; gd13, gd14, etc.: (individually variable) inhibitory inputs to these interneurons. IN15-IN16, etc.: interneurons conveying sensory signals to the corresponding CPG neurons from the other local networks. Hexagons with  or ␥ in them: sources of sensory signals encoding position, load, and ground contact corresponding to a critical value of  (see text). Other symbols: empty triangles: excitatory synapses; filled circles: inhibitory synapses. Arrows from muscles to the hexagons symbolize that the sensory signals arise because of mechanical movement due to muscle activity. Note that the neurons in this network are numbered such that they fit in the larger model that contains all 3 ipsilateral legs. J Neurophysiol • doi:10.1152/jn.00693.2015 • www.jn.org Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 displayed. It shows the three most important leg joints and segments: the thorax-coxa joint that enables pro- and retraction of the leg in the horizontal plane, the coxa-trochanter joint for leg movements in the vertical plane, and the femur-tibia joint for flexion and extension of the tibia of the leg. The three angles ␣, , and ␥ that quantitatively describe these movements are also shown in this figure. Thus ␣ is associated with the protraction-retraction movements (increasing with the retraction of the leg),  with the levation-depression movements of the leg (increasing with levation), and ␥ with the extension-flexion movements of the leg (increasing with flexion). In Fig. 2, the neuromuscular model of one (middle) leg is displayed. It consists of three interconnected local networks, as indicated, that control the activity of three antagonistic muscle pairs: m. protractor and retractor coxae (Pro. m. and Ret. m.), m. levator and depressor trochanteris (Lev. m. and Dep. m.), and m. extensor and flexor tibiae (Ext. m. and Flex. m.). Accordingly, the corresponding control networks are called PR, LD, and EF control network. The core unit of each local control network is a central pattern generator (denoted as CPG). The existence of these local control networks is experimentally well established (Büssler 1983; Büschges 1995, 2005). Each CPG consists of a pair of mutually inhibitory nonspiking neurons (e.g., C7–C8 in the PR control network). The CPG neurons receive (central) excitation, denoted by the conductances of the excitatory currents (gapp,7, gapp,8, etc.). These inputs are individually variable for each CPG neuron. Occasionally, strong inhibitory synapses suppressing the excitatory effect may become active at these site for short periods (see below). The CPG neurons also receive input from one of the other local control networks conveyed by excitatory (IN16, IN20, IN24) and inhibitory (IN15, IN19, IN23) interneurons. Motoneurons (represented by boxes) drive the corresponding muscles, e.g., MN7P means protractor motoneuron, and it drives the m. protractor coxae. The motoneurons receive uniform central drive denoted gMN by its conductance. The CPG and the motoneurons are connected via inhibitory premotor interneurons (IN13, IN14, etc.) (Büschges 1998, 2005; Westmark et al. 2009), which in turn receive individually variable central inhibitory inputs (gd13, gd14, etc.). The local control networks are connected by means of sensory signals expressing the vertical position of the femur, thus capable of signaling INTERLEG COORDINATION ground contact, and the position of the tibia relative to the femur. The former is represented by the vertical angle , the latter by the angle ␥ (hexagons with  or ␥ in them). At critical values of  (ground contact) and ␥, the conductances of IN16, IN24, and IN20, respectively, are strongly reduced or restored to their normal (high) values. This property of the model ensures that it can exhibit intrinsic oscillations, if the critical values of the  and ␥ are chosen appropriately (see RESULTS). Here, intrinsic means that the oscillations are generated by the interaction of the intraleg LD and EF local networks. Now, taking three copies of this one-leg model, we obtain a three-leg model, for the time being with no interleg connections (Fig. 3). Note that the EF local control network of the hind leg is different from those of the other two legs: the CPG neurons (C17–C18) and premotor interneurons (IN33-IN34) are “cross-connected” compared with the connection between the same type of neurons in the other two EF local control networks. This is necessary to account for the fact that the hind leg extends during the stance phase in contrast to the other two legs, which carry out flexions during the stance phase. PR control network LD control network EF control network g g g g d1 d2 MN IN1 C37 IN2 C38 C1 C2 MN2R C20 MN g MN IN5 C41 IN6 C42 C3 C4 MN β IN3 C39 MN C5 C6 g MN5F C23 MN6E C24 Flex. m. IN7 C43 IN11 C47 IN12 C48 LD control network EF control network g g g g d13 d14 g d17 d18 g app9 g app10 IN13 C49 IN14 C50 C7 C8 MN8R C26 MN g MN IN17 C53 IN18 C54 C9 C10 MN10L C28 MN g MN IN22 C58 C11 C12 g MN11F C29 IN16 C52 Flex. m. Lev. m. β IN15 C51 MN12E C30 MN CPG Dep. m. Ret. m. d22 IN21 C57 CPG CPG Pro. m. g d21 g app11 g app12 g MN9D C27 MN Ext. m. γ IN8 C44 g d8 PR control network g MN7P C25 g Lev. m. g app7 g app8 g MN IN10 C46 CPG Dep. m. Ret. m. IN4 C40 Meso−thorax (ML) MN4L C22 IN9 C45 CPG CPG Pro. m. d10 g app5 g app6 g MN3D C21 g d9 g app3 g app4 g MN1P C19 d6 Ext. m. γ IN20 C56 IN19 C55 IN23 C59 IN24 C60 g d20 PR control network g g d25 d26 g MN IN25 C61 IN26 C62 C13 C14 g MN13P C31 M14R C32 MN g MN g g d29 d30 IN29 C65 IN30 C66 C15 C16 MN16L C34 MN g MN IN27 C63 β IN33 C69 IN34 C70 C17 C18 MN18E C36 MN CPG Lev. m. IN31 C67 d34 g MN17F C35 CPG Dep. m. Ret. m. g d33 g app17 g app18 g MN15D C33 CPG Pro. m. IN28 C64 EF control network g g app15 g app16 g app13 g app14 Meta−thorax (HL) LD control network IN32 C68 Flex. m. Ext. m. IN35 C71 IN36 C72 γ g d32 Fig. 3. The 3-leg model still with no inter-leg connections in it. Note the special connection between the CPG and premotor neurons in the EF local control network of the hind leg. This connection ensures that the hind leg extends during the stance phase in contrast to the other two legs in which flexion takes place during the same phase of the stepping movement. J Neurophysiol • doi:10.1152/jn.00693.2015 • www.jn.org Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 g g d5 g app1 g app2 Pro−thorax (FL) 889 890 INTERLEG COORDINATION In the following, we will show how the three copies of the one-leg model can be connected to obtain the characteristic coordination patterns of stick insect walking (tripod and tetrapod). For the sake of simplicity, we did not make any adjustment of the mechanical properties of the legs. Thus all of them uniformly had those of a middle leg. Also, the angular ranges of the protractor-retractor movements were left to be the same for all legs. Finally, we want to emphasize that our model is a kinematic one. Gravitation therefore does not play any role in the further considerations, and the load signals to which we will occasionally refer remain abstract. Although this seems a substantial constraint on the model, several studies (e.g., Schmitz et al. 2008; Schilling et al. 2013; Knops et al. 2013) have demonstrated that a kinematic simulation of insect walking is a good approximation of the real one. RESULTS A Short Description of the Tripod and Tetrapod Coordination Patterns Figure 4 shows the tetrapod and the tripod coordination patterns as found in the experiments (Graham 1972). The diagrams show whether an individual leg (L1–R3) is lifted at any given time, time being the horizontal axis. The black spots are the swing phases, that is when the corresponding legs are lifted. During tetrapod two legs are lifted at the same time while the other four are on the ground. From the three ipsilateral legs, only one is lifted at any time. For example, the right hind leg (R3) and the left middle leg (L2) are simultaneously lifted. In another variant of tetrapod, the right hind leg (R3) is lifted together with the left front leg (L1). The phase differences between the six legs, within a step cycle, are illustrated in the in Fig. 4, right, for the former case. The legs are represented by circles, and legs having the same phase are connected. Note a characteristic property of the tetrapod coor- Intraleg Coordination During Stepping of a Single Leg As a first result, we show the angular movements of the femur and tibia in each of the three legs when there is no interleg connection between them but the intraleg connections are intact. At the same time, this result will also indicate the need for intersegmental coordination to produce walking. The simulation results show that the movements are intrinsic periodic oscillations of the angles in each of the three legs, as displayed in Fig. 5. In the absence of intersegmental coordination, all legs can, however, be simultaneously in the air, which clearly does not happen during walking. On the other hand, coordinated movements of the femur and the tibia bring about periodic stepping of each individual (simulated) leg. The special role of the hind leg can clearly be seen in Fig. 5. Here, the extension-flexion movements are complementary to those of the two other legs. Simulation results in Fig. 6 demonstrate that a permanently weak connection between the local control networks does not admit intrinsic oscillations in the leg. Interestingly, a permanently strong intraleg coupling, too, has the same effect (Fig. 6). Taking the middle leg as an example, the strength of coupling is expressed by the value of gd20, the conductance of the excitatory synaptic current from the EF control network to the LD control network of the mesothoracic ganglion. This value is, during oscillations, determined by the actual value of the angle (cf. Fig. 2). Setting gd20 permanently to a low value Tetrapod 2/3 1/3 1/3 0 0 2/3 0 1/2 1/2 0 0 1/2 Tripod Fig. 4. Experimental data on the tetrapod and tripod coordination patterns in the stick insect (Graham 1972). The horizontal axis is time. The 6 legs are listed vertically. L1 denotes the left front leg, L2 the left middle leg, and L3 the left hind leg. The notation is analogous for the right legs (R1–R3). The diagrams show the state (swing or stance phase) of the legs at any given time. The black spots represent the swing phases during walking, and the white spaces are stance phases. The main tetrapod diagram shows that during this coordination pattern two contralateral legs are lifted at the same time. The panel actually shows 2 variants of tetrapod. (They are divided by a vertical dashed line.) Note that in both variants, the swing phases of the 3 ipsilateral legs occur in the following order: hind, middle and front leg. At right, the phase differences between the leg movements are illustrated for the first variant of tetrapod. The circles represent the legs, and legs having the same phase are connected, which means that they are lifted simultaneously. The numbers are the phase differences within 1 step cycle with respect to the right middle leg (R2). The length of the cycle is normalized to 1. The phase diagram of the second variant of tetrapod can be obtained from that of the 1st variant by exchanging the sides left and right. In the tripod coordination pattern, 3 legs are simultaneously lifted at any instant of time, as it can be seen in the main panel for tripod. Note also a characteristic property of the tripod coordination pattern: the front and the hind leg on the same side of the animal are lifted simultaneously. The phase diagram is here simpler than in the tetrapod case: all phase differences are equal either to a half period of the step cycle or are 0. J Neurophysiol • doi:10.1152/jn.00693.2015 • www.jn.org Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 First, we provide a short description of the characteristic coordination patterns of stick insect walking: tetrapod and tripod. Then, we will present our simulation results that mimic them. dination pattern: the sequence of stepping on one side of the animal is hind leg, middle leg, and front leg within a step cycle. The tripod coordination pattern is somewhat simpler to describe. Here, three legs are lifted simultaneously at any time, for example, the right middle leg (R2), the left hind leg (L3), and the left front leg (L1). The phase differences between the leg movements are either zero or a half of a full step cycle (see Fig. 4, right). It is characteristic for the tripod coordination pattern that the ipsilateral front and hind leg are always lifted simultaneously and the middle leg always alone. 125 100 75 50 25 125 100 75 50 25 1000 891 Fig. 5. Autonomous oscillations of the 3 legs in the model with no interleg connection but with intact intraleg connections. FL, ML, HL: front, middle, hind leg. Red line: protraction-retraction angle ␣ in range: 28° (anterior extremal position)-128° (posterior extremal position); green line: levation-depression angle  in range: 30° (lowest position, ground contact)-60° (highest position); blue line: extension-flexion angle ␥ in range: 45° (maximal extension)-110° (maximal flexion). These coordinated oscillations produce periodic motion of the simulated leg. Note the difference in the angular movement of the 1st 2 legs and of the hind leg (bottom row). 2000 3000 4000 (gd20 ⱕ0.3 nS), independently of the value of ␥, stops the oscillations in the middle leg (Fig. 6, top right). The same kind of result is obtained when the strength of the LD¡EF connection is reduced (Fig. 6, top left). As mentioned, at a permanently strong (gd20 ⱖ4.0 nS) coupling, the oscillations also cease (Fig. 6, bottom right and left). It is thus apparent that for the intrinsic oscillations to take place in the leg, a switch between weak and strong excitatory synaptic currents (between low and high conductances) is necessary. Neither permanently strong nor weak couplings between the LD and EF systems can produce intrinsic oscillations, hence stepping movement, of the leg. These considerations are valid for the other two legs, too (with conductances gd8 and gd32, respectively). However, a closer inspection of Fig. 6, top right (weak EF¡LD), and Fig. 6, bottom left (strong LD¡EF), reveals that after the oscillations cease, all three legs remain lifted: all three -angles are at their maximum (⬇60°). This is of course a state that does not occur in stick insects. We will return to this problem later in this paper. Basic Principles of the Interleg Coordination To carry out normal locomotion, i.e., walking, insects must coordinate the movement of their legs. In the work reported here, we studied the coordination of the ipsilateral legs, only. This falls, of course, short of a full coordination of all six legs but is a necessary condition for that. Accordingly, the coordinated movement of the ipsilateral legs must be established as a precondition of the full coordination of all legs during walking. The basic constraint that determines the coordination is that if one leg is lifted some other (one or two) legs must remain on the ground. Thus sensing of ground contact, or the lack of it, in each of the legs is crucial for the leg coordination. Sensing of ground contact was implemented in our model implicitly, via the usage of the angle , which expresses the vertical position of the femur. We defined a critical value of  at which the synaptic excitation to the “retractor” CPG neuron and the inhibition of the “protractor” CPG neuron were greatly increased (cf. Fig. 2). Attaining this critical value of  signaled ground contact in the model. This started the retraction, i.e., the 5000 t (ms) stance phase. We thus succeeded in integrating the load signal into the angular signal . Accordingly, the model will use this angle as indicator to decide whether a leg is on the ground or lifted and, depending on that, impose conditions on the -angle of the other legs. In this way, a primary role in the intersegmental coordination was assigned to  in the model. Although there is, at present, no direct experimental evidence for this in the stick insect, strong evidence of this kind was found in the lobster (Ayres and Davis 1977). We therefore deem the existence of such a system in the stick insect also possible, even plausible. Beside the position signals , the corresponding angular velocities d/dt may also play a part in shaping the stepping movement. These signals provide information on whether the leg is moving upwards (d/dt ⬎ 0) or downwards (d/dt ⬍ 0) or is moving slowly (|d/dt| ⬍ ⑀, ⑀ being a small number). The actual values of the angles  and of their angular velocities supply the information needed for the desired type of interleg coordination. The activity of the motoneurons in the LD local control network, hence the muscle forces generated by it, determine the actual values of the -angles and their angular velocities. In turn, these networks will be interconnected in a suitable way, as the above considerations imply. Moreover, the CPG of the LD local control network is the ultimate source of the periodic vertical movement of the femur. It appears therefore to be reasonable to have direct synaptic connections that convey the aforementioned information on the angle  and its angular velocity d/dt to the CPG of the LD local control network of another segment. The output  (and occasionally d/dt) of the LD local control network of one segment will therefore be transmitted directly to the CPG neurons of the LD local control network of another segment in form of a synaptic current. In the behavioral studies (e.g., Graham 1972; Grabowska et al. 2012), it was found that the coordination patterns normally propagate in the posterior-to-anterior direction. Hence, there will be no intersegmental synaptic connections on the CPG of the LD local control system of the metathoracic (HL) segment (ganglion). J Neurophysiol • doi:10.1152/jn.00693.2015 • www.jn.org Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 HL α β γ (o) ML 125 100 75 50 25 α β γ (o) FL α β γ (o) INTERLEG COORDINATION 892 INTERLEG COORDINATION Weak EF −−> LD connection 3000 4000 5000 α β γ (o) 2000 α β γ (o) 125 100 75 50 25 1000 125 100 75 50 25 125 100 75 50 25 1000 t (ms) 2000 3000 4000 5000 t (ms) α β γ (o) 125 100 75 50 25 1000 125 100 75 50 25 α β γ (o ) 125 100 75 50 25 3000 4000 5000 t (ms) Strong EF −−> LD connection 125 100 75 50 25 α β γ (o) HL α β γ (o) ML α β γ (o ) FL 125 100 75 50 25 α β γ (o) Strong LD −−> EF connection 2000 125 100 75 50 25 1000 2000 3000 4000 5000 t (ms) Fig. 6. Termination of the intrinsic oscillations of the joint angles by setting the coupling strength between the local control networks to a permanently low or high value. The specific kind of the reduction or increase of the coupling strengths is indicated above each panel. Note that in 2 cases (Weak LD¡EF and Strong EF¡LD), all legs grind to a halt on the ground after the oscillations cease, whereas in the 2 other cases (Weak EF¡LD and Strong LD¡EF), all three legs remain lifted. The traces are color coded as indicated by the color of the angle symbols (␣, etc.) left of the vertical axes in each panel. The arrows in the individual panels mark the begin of the change in the connection strength. Interleg Coordination During Tripod We start with the tripod coordination pattern that has turned out to be the simpler case. Here, the ipsilateral front and hind leg move simultaneously, i.e., lift off and touch the ground in synchrony (Graham 1972). In the simulations, we found the following simple conditions to suffice for the generation of this coordination pattern. First, it is continuously monitored whether the hind leg is in the air, i.e., the angle hleg is greater than a threshold value (th ⫽ 31°). If so, the CPG of the LD local control network of the middle leg (mesothoracic ganglion) will be inhibited. This ensures that the middle leg stays on the ground. If not, the middle leg is free to lift off. Second, the same test is carried out for the middle leg. If it is lifted, the CPG of the LD local control network of the front leg is inhibited, otherwise the front leg can lift off. These simple rules are formalized as a signal flow chart in Fig. 7. It is noteworthy that these simple rules show good agreement with rules 1 and 2 of the stepping movement by Cruse (e.g., Cruse 1980; Cruse et al. 1998; Dürr et al. 2004; Schilling et al. 2007, 2013). However, our model does not use the additional rules (rules 3 and 4) by Cruse that are effective in the rostro-caudal direction (Dürr et al. 2004; Schilling et al. 2013). The corresponding synaptic pathways between the segmental LD local control networks are displayed in Fig. 8. Note that there are no connections from other segments on the LD CPG of the metathoracic (HL) segment. The actual value of the conductance (ginh3, ginh9) of the inhibitory synaptic current from the posterior segment to the next anterior one is completely determined by the actual value of  of the posterior segment. Whether ginh3 and ginh9 are large (inhibition is effective) or negligible (no inhibition) is decided according to the rules in the signal flow chart in Fig. 7. Having implemented the above conditions in the model, we carried out simulations with it. The results for the tripod coordination pattern are shown in Fig. 9. For the normal tripod, the time evolution of all three main angles are displayed in Fig. 9. These are ␣ describing the protractor-retractor (horizontal) movement of the femur,  of the vertical movement of the femur, and ␥ of the extension-flexion of the tibia. In this case, the row for the angle  clearly shows that the hind and front leg J Neurophysiol • doi:10.1152/jn.00693.2015 • www.jn.org Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 125 100 75 50 25 125 100 75 50 25 α β γ (o) HL α β γ (o) ML α β γ (o) FL 125 100 75 50 25 α β γ (o) Weak LD −−> EF connection INTERLEG COORDINATION At any instant of time t: βhleg Yes hind leg lifted? inhibit lev. CPG n. in middle leg No normal drive to lev. CPG n. in middle leg No normal drive to lev. CPG n. in front leg Fig. 7. Signal flow chart of the conditions for the tripod coordination pattern. hleg, levation angle (vertical position) of the femur of the hind leg; mleg, levation angle (vertical position) of the femur of the middle leg; lev. CPG n., levator CPG neuron. move, i.e., lift off and touch down synchronously. This synchrony can also be observed in the time evolution of the ␣-angles (Fig. 9, 1st row left). The extension-flexion movements of these two legs are, however, not synchronous but are out of phase, since, as mentioned earlier, the hind leg carries out extension during the stance phase. In addition to the angular movements, the electrical activity of the CPG of the PR local control network (Fig. 9, 4th row left) and that of the CPG of the LD local control network (Fig. 9, 5th row left) are also exhibited. As can clearly be seen in the 5th row left, the middle and the front leg are periodically strongly inhibited, the inhibition coming from the hind and the middle leg, respectively. However, as Fig. 9, 5th row left, shows, the hind leg receives no inhibition from any of the other legs. Hence, its oscillatory behavior is completely determined by the local CPGs and the coupling between them. The period, 605 ms, of the simulated tripod is in good agreement with values observed in the experiments. Also, the length of the swing phase and stance phase are about the same: about half of the whole period, as it should be (Graham 1972). In Fig. 9, right, a “failed” tripod is depicted. The failure is obvious: the front leg is permanently in the air, in the anterior extreme position, the leg maximally extended. We obtained this result by making the connections between the LD local control networks of the three segments cyclic, i.e., adding a synaptic pathway from the prothoracic segment (front leg) to the metathoracic one (hind leg). As the simulation result Next, we turn to the tetrapod coordination pattern. In this case, usually four out of six legs are simultaneously on the ground at any instant of time. Looking only at one side, i.e., at the three ipsilateral legs, a wave-like process of leg movements can be observed (Fig. 4). That is, the legs carry out basically the same kind of movement (lift-off, touch-down, protraction and retraction, extension and flexion) but with a constant phase delay within a period. The wave starts at the hind leg and propagates in the anterior direction (Graham 1972; Grabowska et al. 2012). For finding the rules and constraints for this coordination pattern, we used the same general principles outlined in Basic Principles of the Interleg Coordination. First of all, a simple reasoning shows that if the hind leg is lifted, then “both” the middle and the front leg must stay on the ground. Hence, the levator CPG neuron of both LD local control networks must be inhibited; otherwise, the central drive to the CPG would still prevail. However, this condition, although necessary, has proved to be insufficient for performing the tetrapod coordination pattern. It turned out that signals carrying information on the actual values of the angular velocities of the levation angle , i.e., the quantities d/dt at each segment, are also required. They determine whether a leg is being lifted (d/dt ⬎ 0) or is touching down (d/dt ⬍ 0). For each leg, two states are defined: “ready-tostep,” and “stepping” state. The coordination process consists of two stages: in the first one, the actual states just mentioned are determined by using the angular velocities; in the second one, it is decided whether the levator CPG neuron of the LD local control network will be inhibited. These stages are specified and formalized in the signal flow charts of Fig. 10. Thus at any instant of time, it is decided whether the hind leg is in the “ready-to-step” state, i.e., is allowed to carry out the next step. If so, the levator CPG neuron of the LD local control network of the middle leg will be inhibited to keep the middle leg on the ground. Next, it is determined whether the hind leg is being lifted (condition: dhleg/dt ⬎ 0). If it is, the hind leg’s state changes to the “stepping” state. It will only be reset to the “ready-to-step” state when the middle leg finishes its step (touches down). The hind leg thus remains in the “stepping” state until the reset happens. Exactly the same procedure is repeated with regard to the middle and front leg to determine the state of the middle leg. In the second stage, it is determined whether the hind leg is lifted. If so, the levator CPG neurons of the LD local control network of the middle and front leg will J Neurophysiol • doi:10.1152/jn.00693.2015 • www.jn.org Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 middle leg lifted? inhibit lev. CPG n. in front leg shows, this connection has destroyed the tripod coordination pattern; hence, it cannot be part of the intersegmental connections that are used to convey signals coordinating the legs’ movement during tripod. It is obvious that a too low level of connectivity precludes the occurrence of the tripod coordination pattern. This example shows that a too high level of connectivity can also lead to the disruption of the tripod coordination pattern. In summary, our simulation results show that the two intersegmental connections used in the model (Fig. 8), together with the simple rules (Fig. 7) associated with these connections, are sufficient to bring about the coordination of the leg movements during tripod. Interleg Coordination During Tetrapod βmleg Yes 893 894 INTERLEG COORDINATION PR control network LD control network EF control network g g g g d1 d2 g app1 g app2 Pro−thorax (FL) g MN IN2 C38 C1 C2 g MN1P C19 MN2R C20 MN g MN IN6 C42 C3 C4 g MN3D C21 g MN IN10 C46 C5 C6 g MN5F C23 MN6E C24 IN7 C43 Ext. m. γ IN8 C44 g d8 IN11 C47 IN12 C48 EF control network g g g d14 IN14 C50 C7 C8 MN8R C26 MN g MN d18 g app9 g app10 g inh9 IN13 C49 g d17 IN17 C53 IN18 C54 C9 C10 MN10L C28 MN g MN IN21 C57 IN22 C58 C11 C12 g MN11F C29 CPG CPG Pro. m. IN16 C52 Flex. m. Lev. m. β IN15 C51 MN12E C30 MN CPG Dep. m. Ret. m. d22 g app11 g app12 g MN9D C27 g d21 Ext. m. γ IN20 C56 IN19 C55 IN23 C59 IN24 C60 g d20 PR control network g g d25 d26 g MN IN25 C61 IN26 C62 C13 C14 g MN13P C31 M14R C32 MN g MN g g d29 d30 IN29 C65 IN30 C66 C15 C16 MN16L C34 MN g MN IN27 C63 β IN33 C69 IN34 C70 C17 C18 MN18E C36 MN CPG Lev. m. IN31 C67 d34 g MN17F C35 CPG Dep. m. Ret. m. g d33 g app17 g app18 g MN15D C33 CPG Pro. m. IN28 C64 EF control network g g app15 g app16 g app13 g app14 Meta−thorax (HL) LD control network IN32 C68 Flex. m. Ext. m. IN35 C71 IN36 C72 γ g d32 Fig. 8. The 3-leg model with the intersegmental connections that underlie the tripod coordination pattern. The intersegmental thick lines are inhibitory synaptic pathways from the posterior segment to the next anterior one. The synaptic strengths, i.e., the actual values of the conductances ginh3 (prothoracic LD CPG) and ginh9 (mesothoracic LD CPG) are completely determined by the actual values of  in the next posterior segment. Note that there is no such inhibitory synaptic connection on the LD CPG of the metathoracic (HL) segment. be inhibited, preventing both legs from lifting off. If the condition is not fulfilled then it is determined whether the hind leg is still in the “ready-to-step” state. If not, then the levator CPG neuron of the LD local control network of the middle leg receives the excitatory signal required for starting the oscillation of the CPG. In addition, it is also determined in this case whether the middle leg is lifted (using the angle mleg, as shown in Fig. 10). If the middle leg is lifted, the levator CPG neuron of the LD local control network of the front leg will be inhibited to keep the front leg on the ground; otherwise, it will be decided whether the middle leg is in the “ready-to-step” state. If not, the levator CPG neuron of the LD local control network of the front leg will receive its excitatory drive necessary to perform the next lift-off. In addition, one should bear in mind that in tetrapod the central drive to the CPGs is different from that in tripod, i.e., different gapp values apply. They bring about the change in the length of the oscillatory period and in the duty cycle. J Neurophysiol • doi:10.1152/jn.00693.2015 • www.jn.org Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 LD control network g d13 MN CPG Flex. m. PR control network g MN7P C25 MN Lev. m. β IN3 C39 d10 IN9 C45 CPG g app7 g app8 MN MN4L C22 Dep. m. Ret. m. IN4 C40 g g d9 g app5 g app6 IN5 C41 CPG Pro. m. Meso−thorax (ML) d6 g app3 g app4 g inh3 IN1 C37 g d5 INTERLEG COORDINATION 895 Failed tripod o α( ) 130 105 80 55 30 60 50 50 o β( ) 60 40 40 30 115 115 90 90 o γ ( ) 30 65 40 2000 3000 4000 5000 65 40 2000 t (ms) 3000 4000 5000 t (ms) V (mV) 10 -10 PR -30 -50 -70 hind leg middle leg V (mV) 10 front leg -10 -30 LD -50 -70 2000 3000 4000 5000 t (ms) Fig. 9. A normal and a “failed” tripod as indicated are displayed. For the normal tripod: 1st row left: shows the protractor-retractor (horizontal) movement (angle ␣); 2nd row left: the time evolution of the angle  (vertical movement) of the femur; 3rd row left: the extension-flexion movement of the tibia (angle ␥). In addition, the electrical activity of the protractor CPG neurons (4th row left) and the levator CPG neurons (5th row left) of all 3 legs are displayed. In the 5th row left, the strong inhibitory effects on the levator CPG neurons of the LD local control networks of the middle and front leg (green and red line, respectively) are clearly discernible, while there is no such effect on the hind leg (blue line). For the “failed” tripod (right), only the angular movements are shown. Note that, in this case, the front leg is permanently in the air and in protracted position (minimal value of ␣) with maximal extension of the leg (minimal value of ␥). Figure 11 displays the three-leg model with the intersegmental connections required for performing the tetrapod coordination pattern. Here, there is a direct inhibition from the metathoracic segment to both the meso- and prothoracic segment, in contrast to the intersegmental connections in tripod (cf. Fig. 8). The other important difference is that, beside the position signals , their time derivatives, the angular velocities d/dt also play an important gating role in generating tetrapod. This is symbolically indicated in Fig. 11, the gating sites are also shown as enlarged insets. (Note that the PR local networks of the three-leg model were omitted from Fig. 11 for the sake of simplicity.) Using all the aforementioned rules and conditions in the simulations, as outlined in the signal flow charts of Fig. 10, we obtained results that are satisfactorily close to the tetrapod coordination pattern. Figure 12 shows an example of a simulated tetrapod produced by the three-leg model. The tetrapod coordination pattern is clearly visible in all rows, i.e., with regard to all angle movements, and the electrical activity of the corresponding protractor and levator CPG neurons. The simu- lated tetrapod had a period of ⬇1,171 ms, which falls into the range of values observed in the experiments. Also, the relative length of ⬃0.27 of the swing phase is in good agreement with the experimental data (e.g., Graham 1972). The example of a “failed” tetrapod in the simulations (Fig. 13) emphasizes the importance of using the angular velocity signals when determining the rules and conditions for the tetrapod coordination pattern. In this example, the angular velocity signals were omitted from the conditions. They are thus indispensable sensory signals for shaping the tetrapod coordination pattern. Finally, we show that interleg coordination affects intraleg coordination. For this, we return to the simulations in which there was no interleg coordination, and the strengths of the intraleg connections were kept permanently weak or strong (cf. Fig. 6). There, we found that when the strength of the EF¡LD coupling was weak, or the LD¡EF coupling was strong, all three legs remained lifted (in the air) in the steady state, which is physiologically unrealistic. We repeated the simulations but now with interleg connections characteristic to the tetrapod J Neurophysiol • doi:10.1152/jn.00693.2015 • www.jn.org Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 γ (o) β (o) α (o) Normal tripod 130 105 80 55 30 896 INTERLEG COORDINATION At any instant of time t: βhleg d βhleg /dt hind leg ready to step? d βhleg /dt >d φcr /dt Yes βhleg d βmleg /dt hind leg ready to step? d βmleg /dt <−d φcr /dt Yes inhibit lev. CPG n. in middle leg hind leg during stepping Yes No middle leg is finishing step hind leg ready to step Repeat the same with hind leg middle leg, and middle leg front leg. βhleg Yes hind leg lifted? inhibit lev. CPG n. in middle and front leg No βmleg hind leg ready to step? No normal drive to lev. CPG n. in middle leg middle leg lifted? No Yes inhibit lev. CPG n. in front leg middle leg ready to step? No normal drive to lev. CPG n. in front leg Fig. 10. Signal flow chart of the conditions governing tetrapod. hleg Is vertical position of the femur of the hind leg; dhleg/dt is its angular velocity. For the middle leg, the variables mleg and dmleg/dt have analogous meaning. The quantity dcr/dt is a threshold value of the angular velocity used for reasons of numerical computation. The expression “hind leg¡middle leg” means that the variables and parameters relating to the middle leg are substituted in the above signal flow chart for those of the hind leg. The expression “middle leg¡front leg” has the same meaning (but using different legs). The other abbreviations are the same as in Fig. 7. coordination pattern. The results are displayed in Fig. 14. In these simulations, the same connection strengths were used as in the earlier ones with no interleg connections. While the cases “weak LD¡EF” and “strong EF¡LD” connection show essentially the same steady state as before, the two other cases do not. In the new simulations, the front and the middle leg end up touching the ground in the steady state, contrary to what was seen in the earlier simulations with no interleg connec- tions. This indicates that interleg coordination plays a part in shaping the steady state of the legs in certain conditions. By this, interleg coordination improves the physiological plausibility of the model. It does not escape, however, one’s attention that in both of the latter cases (weak EF¡LD and strong LD¡EF connection), the hind leg remains lifted. To establish ground contact of this leg, i.e., to stop it properly, a different mechanism becomes active, which has been introduced and J Neurophysiol • doi:10.1152/jn.00693.2015 • www.jn.org Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 AND INTERLEG COORDINATION LD control network EF control network g g g MN g d5 d6 IN5 C41 IN6 C42 C3 C4 MN4L C22 MN g MN IN9 C45 IN10 C46 C5 C6 g MN5F C23 CPG dβ Flex. m. Lev. m. β MN6E C24 MN CPG Dep. m. Dep. m. d10 g app5 g app6 g MN3D C21 g d9 g app3 g app4 g inh3 Pro−thorax (FL) 897 IN7 C43 Ext. m. γ IN8 C44 g d8 IN11 C47 IN12 C48 LD control network EF control network g g g MN d18 g app9 g app10 g inh9 Meso−thorax (ML) g d17 IN17 C53 IN18 C54 C9 C10 MN10L C28 MN g MN IN21 C57 IN22 C58 C11 C12 g MN11F C29 CPG dβ Flex. m. Lev. m. β MN12E C30 MN CPG Dep. m. Dep. m. d22 g app11 g app12 g MN9D C27 g d21 Ext. m. γ IN20 C56 IN19 C55 IN23 C59 IN24 C60 g d20 dβ LD control network EF control network g g g d29 d30 g app15 g app16 Meta−thorax (HL) g MN dβ dβ IN29 C65 IN30 C66 C15 C16 MN16L C34 MN g MN Lev. m. IN31 C67 IN34 C70 C17 C18 MN18E C36 MN CPG Dep. m. β IN33 C69 g MN17F C35 CPG Dep. m. d34 g app17 g app18 g MN15D C33 g d33 Flex. m. IN32 C68 Ext. m. IN35 C71 IN36 C72 γ g d32 Fig. 11. Intersegmental connections in the 3-leg model required for the tetrapod coordination pattern. The conductances ginh3 and ginh9 have the same meaning as in Fig. 8 but here, the inhibitory effect on the levator CPG neuron of the LD local control network of the prothoracic segment can directly be evoked by the -signal of the metathoracic segment (direct thick line from the metathoracic segment to the prothoracic one). The pentagons with d in them symbolize the sources of the sensory signals that convey information on the angular velocities d/dt. They are enlarged as insets for better visibility next to their location in the network. The circles on the connecting pathways express the gating function the angular velocities exert on the (inhibitory) signal flow from the meta- and mesothoracic segments, respectively. Downward directed arrows from the d pentagons pointing to these circles symbolize their influence on the next caudal leg (segment), upward pointing arrows to the next anterior one. The protractor-retractor local control networks of all segments are omitted from this figure for the sake of simplicity. J Neurophysiol • doi:10.1152/jn.00693.2015 • www.jn.org Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 dβ 898 INTERLEG COORDINATION o α( ) Normal tetrapod 130 105 80 55 30 o β( ) 60 50 40 30 o γ ( ) 65 40 2000 3000 4000 5000 t (ms) 10 V (mV) Fig. 12. Simulated tetrapod produced by the 3-leg model using the conditions in Fig. 10. The rows display the same variables, and the color codes for the legs are the same as in Fig. 9. Note again the strong inhibition of the levator CPG neurons of the LD local control networks of the middle and front leg in the bottom row. 90 -10 PR -30 -50 -70 V (mV) 10 -10 -30 LD -50 -70 2000 described in detail in Tóth et al. (2013b). This mechanism involves slow motoneurons and muscle fibers, which are omitted from the present model for the sake of simplicity and clarity. We could do so because slow muscles, although being important in maintaining steady state position of the animals, have only a small contribution to shaping the stepping movements (Bässler and Stein 1996; Gabriel et al. 2003; Guschlbauer et al. 2007; Tóth et al. 2013a). Transition Between Tripod and Tetrapod Having successfully reproduced the tripod and tetrapod coordination patterns in the simulations, the questions naturally arise: how do transitions between them take place? What mechanisms and what sensory and (possibly) central neuronal signals play a role in the transition processes? First of all, we can make some simple observations and thus formulate basic necessary conditions to be fulfilled during transitions from one type of coordination pattern to the other. 3000 4000 5000 t (ms) Such a simple observation is that no more than two legs must be lifted at the same time. Obviously, three lifted ipsilateral legs would cause the animal to lose balance and to fall on its side on which all three legs are in the air. Figure 15 shows such a “forbidden” transition. As can easily be seen, all three -angles, which reliably reflect the legs’ vertical position, have values greater than 50° at one point of time in the interval [3,000,4,000] ms. That is, at that point of time, all three ipsilateral legs are lifted. We should also bear in mind that we, in the present study, only deal with the stepping of ipsilateral legs in the two coordination patterns. The inclusion of all six legs into the considerations in some future model can, and most likely will, lead to a stricter constraint, i.e., that, occasionally, only a single leg is allowed to be lifted on one side during transition from one coordination pattern to the other. In view of this, we endeavored to adhere to this stricter constraint, with partial success. Figure 16 shows a successful transition from tetrapod J Neurophysiol • doi:10.1152/jn.00693.2015 • www.jn.org Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 115 INTERLEG COORDINATION 899 α (o) Failed tetrapod 130 105 80 55 30 Fig. 13. Example of a “failed” tetrapod. It was obtained by omitting the information carried by the angular velocities (cf. Fig. 10). The error is that the front and the hind leg lift off simultaneously, and that the middle and front leg are in the air at the same time. These anomalies are best seen in the middle row showing the time course of the -angles but they are also discernible in the time courses of the ␣ (top)and ␥ (bottom)-angles. The color codes are the same as in Fig. 9. β (o) 60 50 40 30 γ (o) 90 65 40 2000 3000 4000 5000 to tripod where this stricter constraint is fulfilled. The transition period is here virtually absent because the finishing part of the last tetrapod (front leg stepping) coincides with the starting of the tripod (concurrent lifting of the front and the hind leg). In the simulations, we systematically investigated whether all points of time within a step cycle can equally well serve as a starting point of a transition. This turned out not to be the case. The simulations showed that transition from one coordination pattern to the other one is successful only if the angle variable  and the corresponding angular velocity d/dt of the different legs obeyed specific conditions. Moreover, these conditions proved to be different depending on the direction of the transition. Transition from tetrapod to tripod. First, we treat the transition from tetrapod to tripod. Here, we found by means of simulations that the front leg or the hind leg had to be lifting off in order that the transition be successful. More precisely, the angle  had to exceed a threshold value and the corresponding angular velocity had to be positive. (Practically, it had also to be greater than a threshold value.) If these conditions were satisfied then the conductances (gapp) of the central driving currents of all nine CPGs were set to their values suitable for the tripod coordination pattern. This means that these conductance values ensured that the oscillations with the correct oscillatory period (Tp ⬇605 ms) were produced by the CPGs. It should be borne in mind that with the transition, the “tripod rules” of intersegmental coordination (Fig. 7) as well as the corresponding intersegmental connections (Fig. 8) henceforth automatically applied. Figure 17 summarizes the transition conditions. Interestingly, the middle leg plays no part in these conditions. Indeed, the simulations demonstrated that the transition from tetrapod to tripod failed, if the same kind of conditions was applied to the middle leg, i.e., to its -angle and the corresponding angular velocity d/dt. Figure 18 shows two cases when either the hind leg or the front leg satisfies the transition condition. The presumably central commands initiating the transitions are marked by arrows. These commands can be thought of as neuronal signals that activate the processes that underlie the transition, e.g., t (ms) setting the driving excitatory signals to the CPGs, once the transition conditions are fulfilled. At present, there is no direct experimental evidence for their existence. It would indeed be difficult to devise an experiment that could prove or disprove the existence of such a command. It thus remains, for the time being, hypothetical. In Fig. 18, top row, the unperturbed tetrapod coordination pattern is displayed with these arrows to show the exact instants of time when the commands appear. Figure 18, middle row, illustrates the case when the transition command arrives at the start of the lift-off of the hind leg. A transitory period of about one tetrapod oscillatory period follows during which, at some point of time, two legs, the middle and the hind leg, are simultaneously in the air. The next time the hind leg is being lifted off, it is doing so together with the front leg, i.e., they are already in tripod. In Fig. 18, bottom row, we illustrated the case when the front leg satisfies the transition condition at the arrival of the central command for it. Here, the transitory phase is virtually absent, and the tripod coordination pattern immediately commences. Transition from tripod to tetrapod. Now, we deal with the transition from tripod to tetrapod. Here, too, systematic simulations varying the command signals within one oscillatory period of the tripod coordination pattern revealed that successful transitions from tripod to tetrapod would occur if both the hind leg and the middle leg were moving slowly, or were approaching the ground. More precisely, the angular velocities in the vertical direction (Fig. 18, top row) of the legs involved were below a small positive threshold. This also includes negative velocities when the leg is approaching the ground. If this condition was fulfilled the central drives to all nine CPGs were adjusted to produce periodic oscillation suitable to tetrapod, i.e., one with a period of Tp ⬇1,171 ms. The adjustment was brought about by changing the values of all conductances (gapp) of the central drives to all CPGs. In addition, the rules of the tetrapod coordination pattern (Fig. 10) as well as the corresponding intersegmental connections (Fig. 11) automatically took effect. Figure 19 summarizes the transition conditions just described. Note that the “AND” condition is almost automatically satisfied, since if one of the legs involved is in J Neurophysiol • doi:10.1152/jn.00693.2015 • www.jn.org Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 115 900 INTERLEG COORDINATION 3000 4000 5000 α β γ (o) 2000 α β γ (o) 125 100 75 50 25 1000 125 100 75 50 25 125 100 75 50 25 125 100 75 50 25 1000 t (ms) 3000 4000 5000 t (ms) α β γ (o) 2000 α β γ ( o) 125 100 75 50 25 1000 125 100 75 50 25 125 100 75 50 25 α β γ ( o) α β γ (o) HL α β γ ( o) ML 125 100 75 50 25 α β γ ( o) FL 3000 4000 5000 t (ms) 5000 t (ms) Strong EF −−> LD connection Strong LD −−> EF connection 125 100 75 50 25 2000 125 100 75 50 25 1000 2000 3000 4000 Fig. 14. Rerunning the simulations with weak and strong intraleg coupling in the presence of interleg connection (in tetrapod). The constant phase difference between the FL, ML, and HL traces is a clear sign of its presence. Note that when the EF¡LD connection is weak (top right) or the LD¡EF connection strong (bottom left), the FL and the ML are on the ground in steady state (cf. Fig. 6). The color code for the traces and the role of the arrows in the individual panels are the same as in Fig. 6. the air and approaching the ground the other must be on the ground. However, it was easier to formulate the transition condition by using the “AND” operation. Note also that the front leg is not explicitly involved in the condition but, since it moves concurrently with the hind leg during tripod, its  angle and the corresponding angular velocity are (nearly) identical to those of the hind leg. Figure 20 illustrates a successful transition from tripod to tetrapod. In this example, the central command arrives when both the hind leg and the middle leg are moving very slowly near, or on, the ground. However, the region of admissible timing of the central command for transition is much larger. The time intervals in which the central commands are effective are of importance for the transitions in both directions. Figure 21 displays such intervals for both types of transition as indicated in the figure. They are identified by a pair of thin vertical bars. They arise from the conditions formulated for the two transition types (cf. Figs. 17 and 19). In Fig. 21, left and right, the top traces are the time courses of the angles  and the bottom traces are those of the angular velocities d/dt. The horizontal lines, parallel to the time axes, are the threshold values: 31° for  in Fig. 21, left, and 5.0 1/s for d/dt in Fig. 21, left and right. During tetrapod¡tripod transition (Fig. 21, left), the central command is effective in the time interval in which the angular velocity d/dt and  exceed their own threshold. During tripod¡tetrapod transition (Fig. 21, right), the command is effective while d/dt is below the threshold. Besides the intervals shown in Fig. 21, there is one such additional interval each in both cases, although they are not displayed in Fig. 21. During transition from tetrapod to tripod, this additional interval is about the same length commencing when the front leg starts lifting off the ground. During transition into the opposite direction, there is also an analogous condition that gives rise to an additional interval in which the command to change coordination pattern is immediately effective: the middle leg is stepping and the hind leg remains on the ground. We have thus two intervals each for the two types of transition. The intervals in the case of transition from tripod to tetrapod are much longer: 256 ms than in the opposite case: 45 ms. If the command signal arrives outside the intervals just identified, the transition will be delayed and will not start until the transition conditions are fulfilled. J Neurophysiol • doi:10.1152/jn.00693.2015 • www.jn.org Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 125 100 75 50 25 Weak EF −−> LD connection α β γ (o) HL α β γ (o) ML α β γ (o) FL 125 100 75 50 25 α β γ (o) Weak LD −−> EF connection α (o) INTERLEG COORDINATION 901 130 105 80 55 30 β (o) 60 Fig. 15. Example of a “forbidden” transition from tetrapod to tripod. The rows display the same angles, and the color codes are the same, as in Fig. 9. In the traces of the -angles (middle), all 3  are greater than 50°, meaning that all 3 legs are high in the air at the same time, at one point after the 3,000-ms mark. 50 40 30 90 65 40 2000 3000 4000 5000 DISCUSSION Building on previous models (Tóth et al. 2012; Knops et al. 2013), we constructed a three-leg model to simulate the normal locomotion (walking) of the stick insect. Our approach differed from that of other models mentioned in the Introduction (e.g., Cruse 1980, 1990; Cruse et al. 1998; Dürr et al. 2004; Ekeberg et al. 2004; von Twickel et al. 2011, 2012; Schilling et al. 2013). Those models used artificial neural networks to build the control systems that governed the muscles that move the individual leg segments. In contrast to them, we built our models of modules, called local neuronal control networks, that were constructed using anatomical data and experimental results if they were available (Bässler 1983; Büschges 1995, 1998, 2005; Westmark et al. 2009) and reasonable physiological assumptions (Daun-Gruhn 2011; Daun-Gruhn and Tóth 2011; Daun-Gruhn et al. 2011; Tóth et al. 2012; Knops et al. 2013), where no relevant data existed. On the same basis, we defined specific connections, sensory synaptic pathways, between the local control networks both intra- and intersegmentally. No such approach has, to the best of our knowledge, been applied by other models in this field. The model created thus and the simulation results produced by it can therefore be considered as novel. In the present model, for each leg, the most important antagonistic muscle pairs and their neuronal control networks have been implemented: the protractor-retractor, the levator-depressor, and the extensor-flexor neuromuscular networks. They are interconnected in a suitable way to bring about synchronized periodic oscillation of all three neuromuscular networks. The synchronization is such as to produce stepping movements of the individual legs. We kept all legs mechanically identical using data of the middle leg. This had the advantage that differing mechanical properties, which could otherwise have obscured the main functions of the local control networks of the individual legs and leg joints in the interleg coordination, were not present. At the same time, this simplification did not amount to a serious shortcoming of the model, since it is a kinematic rather than a dynamic one, as far as its mechanical properties are concerned. During walking, interleg (intersegmental) coordination of the ipsilateral legs is also required. As stated in the Introduc- t (ms) tion, the main objective of the present study has been to establish this required coordination between their movements by means of a neuronal model. Thus we looked for suitable intersegmental connections between the control networks of the individual legs. We reasoned that connections between the segmental levator-depressor control networks would play a primary role. The sensory signal expressing the vertical position  and the corresponding angular velocity d/dt completely characterize the state the leg is in (stance or swing state, and the vertical direction of movement in the latter). Ground contact of the leg can uniquely be described by a critical value of the levation angle  at which the stance phase commences. The levation angle is controlled by the neuromuscular LD system at each leg. Strong experimental evidence for its primary role in shaping locomotion (walking) was found by Ayres and Davis (1977), although not in the stick insect, but in the lobster. Using the aforementioned sensory signals, in the form of (actual) values of the levation angle  and its time derivative d/dt, we could derive conditions that determined the coordinated movements of the individual legs during the tripod and tetrapod coordination patterns. We found important differences between the conditions for tripod and tetrapod. The main qualitative difference is that the conditions for tripod only require the actual values of the angles , whereas tetrapod also requires the sensory signals encoding the angular velocities d/dt. Figures 9 and 13 demonstrate this finding. In physiological terms, this means that the tetrapod coordination pattern requires more sensory input and hence is more heavily dependent on it, to take place than the tripod coordination pattern. The intersegmental connections between the levator-depressor control networks are more complex in the former than in the latter case (cf. Figs. 8 and 11). The posterior-anterior propagation of stepping during the two coordination patterns is well known to happen in the stick insect (e.g., Graham 1972, 1985; Grabowska et al. 2012). Moreover, Grabowska et al. (2012) have also shown that the pair of middle legs plays a crucial part in the interleg coordination. If these legs are amputated, the coordination between the front and hind legs ceases. This is in good agreement J Neurophysiol • doi:10.1152/jn.00693.2015 • www.jn.org Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 γ (o) 115 INTERLEG COORDINATION o α( ) 902 130 105 80 55 30 o β( ) 60 50 40 30 o Fig. 16. Transition from tetrapod to tripod. The rows display the same variables, and the color codes are the same, as in Fig. 9. The timing of the transition command is labeled by an arrow in the row of the -traces. Note that the transition period is virtually absent. The last part of the tetrapod (step of the front leg) coincides with the start of the tripod (simultaneous lifting of the front and hind leg). 90 65 40 2000 3000 4000 5000 t (ms) V (mV) 10 -10 -30 PR -50 -70 V (mV) 10 -10 -30 LD -50 -70 2000 with the model since all coordination patterns, whether tripod or tetrapod, require sensory signals from the middle leg. Hence the interleg coordination in the model, too, is abolished if these signals are missing. As far as the transitions between tetrapod and tripod are concerned, we could also derive simple conditions for them. They are, however, dependent on the direction of the transition (cf. Figs. 17 and 19). Moreover, the direction of the transition also determines the time intervals in which a central command for transition can become effective (cf. Fig. 21). The constraints that arise show that during transition from tetrapod to tripod, these time intervals within an oscillatory period are rather small (in total ⬃90 ms), in the opposite direction, however, much larger (in total ⬃512 ms). This asymmetry can be interpreted with respect to the locomotion patterns of the stick insect that tetrapod seems to be preferred to tripod in the animal. There remains an important question to be answered: are the coordination structures used in the model for tripod and tetra- 3000 4000 5000 t (ms) pod separate? The answer simply is: no, they are not. A look at Figs. 8 and 11 suffices to see that the coordinating system that is active during tripod is a subset of that which is active during tetrapod. Hence, it is sufficient to assume the existence of a single intersegmental coordinating system in the model. This system is fully activated during tetrapod, including the sensory pathways that convey information on the angular velocities. During tripod, it is only partially activated as shown in Fig. 8. The function of the parts that are active during both tripod and tetrapod is the same, namely temporary inhibition of the corresponding CPG. The (hypothetical) command signal that initiates the switch between the coordination patterns could also activate or inactivate parts of the interleg connections according to whether they are required in the actual coordination pattern. Even though the conditions for the tetrapod and tripod coordination patterns have been implemented in the model in an abstract way illustrated by signal flow charts (cf. Figs. 7, 10, 17, and 19), they could, in principle, be converted into neuronal J Neurophysiol • doi:10.1152/jn.00693.2015 • www.jn.org Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 γ ( ) 115 INTERLEG COORDINATION 903 At any instant of time t: At any instant of time t: βfleg d βfleg /dt front leg is lifting off? βhleg d βhleg /dt d βhleg /dt hind leg is slow or approaching ground? hind leg is lifting off? Yes d βmleg /dt middle leg is slow or approaching ground? Yes Yes Yes AND set central drive of all CPGs to 3pod values Fig. 17. Conditions for the transition from tetrapod to tripod. These conditions are the same for both the front and hind leg. It suffices if one of these legs satisfies them, hence, the “OR” logical operation. Note that the middle leg plays no part in the transition conditions. networks that would produce the behavior described by those charts. However, a practical implementation would face difficult problems. The main problem is that currently, there are no data, not even hypotheses, of what such networks would look like. Would they comprise just a few dozens of neurons or hundreds of them? What kind of neurons might they be, etc.? Another problem would likely be finding suitable values for the parameters of a large number of additional neurons without knowing their anatomical and biophysical properties. Furthermore, technical difficulties would also arise from including additional model neurons into the model. The inclusion would substantially increase the size of the differential equation system to be solved in the simulations and hence reduce the numerical stability of the solutions. Moreover, it would sub- set central drive of all CPGs to 4pod values Fig. 19. Conditions for transition from tripod to tetrapod. The “AND” condition is almost trivially satisfied, since if one of the legs is lifted and approaching the ground the other must be on the ground. stantially prolong the computational time. Finally, the additional neuronal networks might distract attention from the main properties of the model, namely the connections between the LD systems of the individual legs. Hence, an extension of the kind would, at present, implant more shortcomings into the model than advantages. There exist other types of models, too, which can simulate even full insect walking, i.e., using all six legs. One example is the model by von Twickel et al. (2011, 2012). It is based by constructing appropriate artificial neural networks, which are then trained to carry out the task of controlling the movement of the leg. The single-leg controller (von Twickel et al. 2011) β (o) 60 50 40 30 Fig. 18. Two cases of successful transition from tetrapod to tripod. Top row: unperturbed tetrapod with the -angles and with the instants of time when the (central) command for transition appears. Middle row: the case when the command occurs at the beginning of the lift-off of the hind leg. A transitory phase of about one tetrapod oscillatory period can be discerned. Note also that during this period, there is a short time interval when both the middle and the hind leg are lifted (at the crossing of the green and the blue line at ⬃40°). Bottom row: the case when the front leg satisfies the transition condition, the command for it arriving at the 2nd (later) arrow as indicated at both top and bottom rows. Here, virtually no transitory phase is discernible; the tripod coordination pattern commences immediately. β (o) 60 50 40 30 β (o) 60 50 40 30 3000 4000 5000 t (ms) J Neurophysiol • doi:10.1152/jn.00693.2015 • www.jn.org Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 OR INTERLEG COORDINATION o α( ) 904 130 105 80 55 30 o β( ) 60 50 40 30 o 90 65 40 6000 Fig. 20. Transition from tripod to tetrapod. The rows display the same variables, and the color codes are the same, as in Fig. 9. The arrow indicates the timing of the central transition command. The transitory phase is about one tripod oscillatory period (⬇605 ms). 7000 8000 9000 10000 11000 t (ms) V (mV) 10 -10 -30 PR -50 -70 V (mV) 10 -10 -30 LD -50 -70 6000 uses an earlier leg model by Ekeberg et al. (2004). With the use of it, a six-legged control system (robot) (von Twickel et al. 2012) was developed. Thus the model by von Twickel et al. (2012) can emulate six-leg walking. Another example is Walknet (Cruse 1980; Cruse et al. 1998; Dürr et al. 2004; Schilling et al. 2007, 2013), which is a complex control system. Parts of it are artificial neural networks, other parts are built of the usual functional units of control systems, such as integrators, nonlinear threshold elements. Walknet uses all angle variables (␣, , and ␥) to be controlled, and several versions of it also the angular velocities, as additional control variables. The system works by implementing a number of rules, the so-called “Cruse rules” (e.g., Dürr et al. 2004; Schilling et al. 2013). These are rules that determine the interleg coordination in Walknet. As a result, Walknet is capable of mimicking six-leg walking in a number of (simulated) environmental conditions. By contrast, our model, having only three (simulated) legs, cannot replicate the full walking patterns with six legs. It can only produce the ipsilateral projection of the coordination (walking) patterns 7000 8000 9000 10000 11000 t (ms) tetrapod and tripod. However, it was built on the basis of a quite different concept: a close and direct relation to the morphological and physiological properties of its biological counterpart, the stick insect. Consequently, it can highlight new aspects of the mechanisms that generate intersegmental coordination during walking. Despite the substantial differences between these models concerning their concept and the modeling techniques applied, they implement basically the same coordination rules in their own way. This important common property therefore binds them. A model of intersegmental coordination not having mechanical variables and parameters was presented by Daun-Gruhn (2011), Daun-Gruhn and Tóth (2011), and Tóth et al. (2015). This model was obtained by using the experimental data by Borgmann et al. (2007 2009). In the majority of these experiments, all but a single front leg were amputated, in others, two ipsilateral legs (e.g., front and middle leg) were left intact. In both cases, functional connections, in particular, possible coordination between the PR systems of these legs were studied J Neurophysiol • doi:10.1152/jn.00693.2015 • www.jn.org Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 γ ( ) 115 INTERLEG COORDINATION 905 tripod −−−> tetrapod 70 60 60 o β( ) 70 50 50 40 40 30 30 dβ/dt (1/s) dβ/dt (1/s) β (o) tetrapod −−−> tripod 20 10 -10 2000 10 0 2100 2200 2300 2400 t (ms) -10 2000 2100 2200 2300 2400 t (ms) Fig. 21. Intervals in which the commands for transition are effective. Top curves: time course of the angles  of the legs; bottom curves: time course of the corresponding angular velocities d/dt. The horizontal lines, parallel to the time axes represent the threshold values for  and d/dt, respectively. In the simulations, the threshold value 5.0 1/s was used for d/dt and 31° for . (Borgmann et al. 2009). The results suggested that there were descending excitatory and inhibitory connections between the CPGs of the PR systems of the ipsilateral legs, modulated, presumably, by peripheral (load) sensory signals. Thus a descending chain of connections was identified in the experiments. In the model, we completed this descending chain by an additional connection from the meta- to the prothoracic PR system. With the use of this model, tetrapod- and tripod-like simulated electrical activity could be produced in the protractor-retractor CPGs. Also, the transition between the coordination patterns was successfully simulated. These earlier results do not contradict our present ones, since, as pointed out, in the experiments by Borgmann et al. (2007, 2009), at least one of the three ipsilateral legs were amputated; hence, the LD and EF systems of the absent legs, deprived from their vital peripheral connections, were rendered ineffective. It is therefore no surprise that no connection, let alone coordination, could be observed between the LD systems of the legs. Conversely, in the intact animal, the connections between the PR systems of the legs may well be masked by the much stronger ones between the LD systems. We conjecture that the connections between the PR systems of the different legs might play a part in the mutual adjustment of the anterior and posterior end positions of the different legs during general locomotion, including coordination patterns other than just those of normal walking. The properties of our present model and the simulation results that were achieved by using it can, in general, be directly interpreted in physiological terms (e.g., observed coordination patterns and transitions between them, neuronal mechanisms in the model and animal). However, the model makes some predictions of a basic nature. First and foremost, it predicts that the main components of the intersegmental connections are between the levator-depressor neuronal control networks of the individual segments. They are crucial in producing locomotion, more precisely walking. A further important prediction is that the tripod coordination pattern requires sensory signals represented by the angle  of the hind and middle leg, while the tetrapod one also requires the angular velocities d/dt of all three legs. Yet another prediction is how the transition between the two coordination patterns (tetrapod and tripod) takes place, including the timing of those transitions (cf. Fig. 21). These predictions have not been or could not yet be checked, i.e., confirmed or rejected, in experiments. Similarly, the details of the intersegmental connections in the model await scrutiny in suitable experiments. We thus see the main merit of our work in the predictions just mentioned. They provide further goals for the experimental investigations in a logically consistent theoretical framework. These predictions are, or will become, practicable to be tested in experiments. Finally, it is also worth briefly examining what implications our results might have for the locomotion of other legged animals. As already mentioned earlier, we adopted the idea about the primary role of the LD systems in locomotion from the lobster. It is, however, a reasonable assumption that our results would not only apply to the lobster but also to the much bigger family of crustaceans (cf. Orlovsky et al. 1999) or even arthropods. To make a further and larger jump, as far as four-legged vertebrates, and mammals in particular, are concerned, the details of the mechanisms underlying their walking are much more complex than in insects and crustaceans (Orlovsky et al. 1999). A complete description and analysis of the organization of locomotion at the level of local neuronal networks have so far not been possible. There are, however, functional relations between our results and walking in fourlegged mammals. In the latter, too, local neuronal networks control the workings of groups of antagonistic muscle groups providing for them the rhythmic excitation and inhibition needed in walking. Also, proprioceptive sensory signals can and do modify the basic walking pattern depending on the environmental conditions. Interestingly, the middle and the hind leg in our model perform “normal” walking pattern, similar to those in four-legged animals, if the front leg is isolated from the rest (Tóth and Daun-Gruhn, unpublished results). This is in very good agreement with the aforementioned experimental results by Grabowska et al. (2012). Hence, the same basic features of locomotion (walking) can be detected in a very broad spectrum of species. We would therefore J Neurophysiol • doi:10.1152/jn.00693.2015 • www.jn.org Downloaded from http://jn.physiology.org/ by 10.220.32.247 on June 17, 2017 0 20 906 INTERLEG COORDINATION venture to suggest that the primary role of the neuromuscular systems that bring about lifting and downward movement of the leg may be a general property in many kinds of animals, including legged mammals. GRANTS This work was supported by the Deutsche Forschungsgemeinschaft Grants DA1182/1-1 and GR3690/4-1 (to S. Daun-Gruhn). 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