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Department of Computer Science, Royal Institute of Technology, Stockholm, Sweden; and Department of Physiology, University of Alberta, Edmonton, Canada
Submitted 19 January 2005; accepted in final form 25 July 2005
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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; Kjaerulff and Kiehn 1996), the identification of interneurons that are involved in generating and coordinating centrally generated rhythms (Butt et al. 2002; Stokke et al. 2002), and the properties of neurons involved in pattern generation (Kiehn 1991). At the systems level, studies of the cat have revealed numerous mechanisms by which afferent signals from peripheral sensory receptors regulate the centrally generated rhythms during the step cycle (Donelan and Pearson 2004; Lam and Pearson 2002), some of which have also been identified in human infants (Pang and Yang 2000, 2002). An important unresolved issue is whether our current knowledge of the neuronal mechanisms controlling walking is sufficient to explain the production of rhythmic stepping movements in normal walking animals. Resolving this issue is not a simple matter. First, it requires an understanding of the complex interactions between the nervous system, muscles, limb and body mechanics, and the environment. Second, it cannot be done using direct experimental approaches because individual neuronal mechanisms, or sets of mechanisms, cannot be isolated in behaving animals. In addition, the contribution of other, yet unknown, mechanisms can never be ruled out. One feasible strategy to assess the role of specific mechanisms is to develop computer simulations of walking systems that incorporate known properties of neuronal systems, muscles, and body mechanics (Ekeberg et al. 2004; Ivashko et al. 2003; Taga 1998; Yakovenko et al. 2004). The great advantage of computer simulations is that they allow the isolation of individual mechanisms in a functional context. Another potential approach is to construct robot systems based on known biology (Fukuoka et al. 2003; Quinn et al. 2003; Ritzmann et al. 2004). However, to date, the results applicable to animal physiology from this approach have been limited, primarily because of the time-consuming process of design and construction and difficulty of manufacturing actuators that realistically mimic muscles.
Stepping of the cat hind legs is a behavior well suited for simulation studies because we possess considerable knowledge of the associated biology, such as the mechanical properties of the hind legs (Goslow et al. 1973; Hoy and Zernicke 1985), the kinematics of hind leg movements during stepping (Goslow et al. 1973; Miller et al. 1975), the properties of muscles in the cat hind leg (Brown et al. 1996), the patterns of electromyographic activity in many hind leg muscles (Rossignol 1996; Yakovenko et al. 2002), and the neuronal mechanisms controlling stepping (Lam and Pearson 2002; Pearson 2003). This knowledge allows reasonably realistic simulation of the mechanics of the legs and muscles and provides constraints on the mechanisms controlling muscle activation. The potential value of realistic simulation of hind leg stepping in the cat has been shown in two recent studies (Ivashko et al. 2003; Yakovenko et al. 2004). Both studies used a two-dimensional simulation of the hind legs to investigate issues related to the control of muscle activation. One led to a proposal for the neuronal architecture for the pattern-generating networks in the spinal cord (Ivashko et al. 2003), whereas the other examined the importance of stretch reflexes in controlling the level of activity in leg extensor muscles (Yakovenko et al. 2004).
Our interest has focused on using computer simulations to examine some of the sensory mechanisms involved in controlling stepping of the hind legs of walking cats. Currently, two sensory mechanisms are considered important in controlling the stance-to-swing transition in the hind legs. The first is an inhibitory influence on the system generating flexor bursts from afferents arising from Golgi tendon organs in the ankle extensor muscles. Loading the ankle extensors stops rhythmic locomotor activity in decerebrate walking cats by inhibiting flexor burst generation (Duysens and Pearson 1980), whereas electrical stimulation of group Ib afferents (arising from the Golgi tendon organs) prolongs the stance phase during walking in intact and decerebrate cats (Whelan and Pearson 1997; Whelan et al. 1995) and inhibits flexor burst generation during fictive locomotion in spinal and decerebrate cats (Conway et al. 1987; Guertin et al. 1995). These observations have led to the concept that a necessary condition for the initiation of swing is unloading of the ankle extensor muscles. This occurs naturally near the end of stance because of the transfer of load to the opposite hind leg and the shortening of the ankle extensors. The second sensory mechanism thought to be involved in regulating the stance-to-swing transition is activation of the flexor burst-generating system by sensory signals from afferents arising in the hip region. Preventing hip extension stops stepping in spinal cats (Grillner and Rossignol 1978) and imposed extension movement of the hip or stretching hip flexor muscles during stance can initiate flexor burst activity during fictive locomotion and stepping in decerebrate cats, respectively (Hiebert et al. 1996; Kriellaars et al. 1994). Thus hip extension is another necessary condition for allowing the initiation of the swing phase. One hypothesis derived from these two sets of findings is that the transition from the stance phase to the swing phase in the hind legs of intact walking animals is normally initiated by a combined reduction in an inhibitory signal from load-sensitive afferents from the ankle extensor muscles and an increase in an excitatory signal from position-sensitive afferents from the hip onto the system generating flexor bursts (McCrea 2001; Pearson 2003). If true, an important issue is to establish the relative importance of these two mechanisms. Intuitively it seems reasonable that either mechanism acting alone could be effective in controlling the stance-to-swing transition, but this suspicion may be incorrect given the difficulty of imagining the consequences of the complex mechanical interactions between the legs and the dependence of these interactions on the geometry of the legs and the state of the ankle extensor muscles. Here we have examined this issue in a simulation of the hind legs by analyzing the effects on the stepping pattern when the stance-to-swing transition was controlled by either a combination of the two mechanisms or by one or the other mechanism alone.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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Seven muscles were included in the model of each leg (Fig. 1B): five muscles acting over single joints—hip flexion [iliopsoas (IP)], hip extension [anterior biceps (AB)], knee extension [vastus lateralis (VL)], ankle extension [soleus (Sol)], and ankle flexion [tibialis anterior (TA)]—and two bifunctional muscles—hip extension and knee flexion [posterior biceps and semitendinosus (PB/St)] and knee flexion and ankle extension [gastrocnemius (Gas)]. The simulation of all muscles controlling flexion and extension movements was based on the model of Brown et al. (1996) for the soleus muscle. The parameters used for calculating the force–velocity and force–length relationships are given in the APPENDIX. The force was scaled linearly with the level of activation received from the controller. Each muscle also included an in-series elastic tendon (see APPENDIX).
The muscle torque at each joint was obtained by summing the contribution from all muscles acting at the joint. Where explicit moment arms were required (see APPENDIX), they were taken from published papers (Goslow et al. 1973). The muscle lengths were set so that they would normally operate at lengths where force increases with length (see APPENDIX). During the running of the simulation, the lengths and velocities of the muscles were calculated from the geometry of the leg segments, lever arms, and the actual angular velocities of joint movements.
Abduction and adduction movements of the hip were controlled by a passive linear spring and damper. For small movements, this would correspond to tonically activated muscles (see APPENDIX).
Simulation of leg and body mechanics
The body and leg segments were modeled as rigid segments with the center of mass located at the midpoint of each segment. The masses of the leg segments were taken from Hoy and Zernicke (1985) (see APPENDIX). The mass of the trunk and hindquarters was taken as 2 kg with the center of mass in the middle of the segment connecting hind and forelegs. The moments of inertia for all segments were calculated from estimates of the segment dimensions (see APPENDIX) assuming a constant density. In the simulation program, the location of each segment was represented by a redundant set of Cartesian coordinates specifying, individually for each segment, the location of its mid-point and its orientation in three-dimensional space. A set of constraint equations was used to specify how the different segments were allowed to move in relation to each other, thus effectively enforcing the joint restrictions. The knee and ankle allowed only one degree of freedom (hinge joints), whereas the hip allowed two (universal joint). This constrained each leg to move in a plane, but this plane could rotate around an axis parallel to the length of the trunk segment (adduction-abduction).
The ground was modeled as a flat horizontal surface with some elasticity and damping. While a paw was on the ground, any displacement 
x (mm) gave rise to a counteracting force of 
(N) and, in addition, all downward velocities v were damped by a force 
(N). The resulting ground model had the desired property of absorbing the kinetic energy of the leg at foot impact.
The time-course of the movement was simulated by repeatedly, for each time step of 0.25 ms, calculating the next location of all segments. These new locations were calculated efficiently by numerically solving the Lagrangian equations for the system in combination with the constraint equations. This calculation was done using a multidimensional Newton-Raphson method that involved computing the Jacobian matrix for the entire equation system. Because the entire system involves 180 state variables, special techniques for handling sparse equation systems where used to avoid handling the full 180 x 180 matrix. In particular, Cholesky factorization was used because it preserved the sparse structure of the system.
Simulation of the neural control system
The activation of the muscles in each leg was controlled by a system that incorporated a selected subset of the known features of the biological system controlling stepping in the hind leg of the cat. We did not attempt to incorporate all the known biological properties into the controller because our aim was to construct a simple, understandable controller that would still produce stable and realistic stepping movements. Minimizing the complexity has the advantage that the contribution of individual mechanisms can be more readily determined.
Each leg was controlled by a separate controller, each progressing sequentially and repeatedly through four states: liftoff, swing, touchdown, and stance. The switching between these states was governed by signals related to the state of the leg, and different sets of muscles were activated in each state (Fig. 2). Liftoff commenced when the force in the ankle extensor was low enough and the hip joint sufficiently extended. These two factors were combined linearly, so that a low force would initiate the swing at less extension while a higher force would require more extension. This particular rule was in some experiments replaced by alternative rules based on only the muscle force or the hip angle (see RESULTS). The ankle flexor, bifunctional knee flexor, hip extensor, and hip flexor muscles were activated simultaneously to lift the foot from the ground. Swing commenced as soon as the foot left the ground and was produced by activation of the hip and ankle flexors. Touchdown was initiated when a combination of the hip and knee angles passed a preset value. The hip extensor/knee flexor and ankle extensor/knee flexor were activated during the touchdown phase. Finally, the stance phase was initiated when the foot contacted the ground. The hip extensor, knee extensor, and both muscles extending the ankle were activated during the stance phase. Coupling between the controller for the left and right leg was implemented as an additional rule delaying the transition from stance to lift-off as long as the contralateral controller was not in its stance phase. This coupling could be removed to analyze the performance with only peripheral coupling present (see RESULTS). The exact rules used are listed in the APPENDIX.
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). The gains of the feedback pathways were adjusted manually to produce realistic movements of the leg during the stance phase. The APPENDIX lists all values used for muscle activation during stance and Fig. 3B shows the resulting movement.
Two notable features of the control scheme were 1) the possibility to remove all direct coupling between the controllers for the two legs and 2) the lack of intrinsic rhythmicity, i.e., the inability to generate rhythmic output in the absence of afferent signals. Our rationale for excluding intrinsic rhythmicity was that phase transitions during slow walking over even terrain seem to be primarily timed by sensory signals. Recent physiological data indicate that both the stance-to-swing and swing-to-stance transitions are strongly dependent on sensory signals related to unloading of the ankle extensors and hip flexion, respectively (Pearson 2003; Pearson et al. 2003). Making the model completely dependent on sensory signals excluded the possibility that an internal rhythmicity could take over during the experiments and drive the motor pattern in a different mode of operation. A central timing component could easily be added to the model by limiting the duration of flexor bursts during swing and extensor bursts during stance, but for this study, this would only have served as a complication in interpreting the simulation results. The possibility to remove the direct coupling of the controllers of the two legs allowed us to examine to what extent the mechanical linkage between the two legs may serve as a factor in coordination (see RESULTS).
Programming and running the simulation
The simulation program was developed and executed within the Debian GNU/Linux operating system. The simulation program consists of three major components: the mechanical dynamics solver, the control system, and the graphical engine. The component that numerically simulates the leg and body dynamics, including muscle properties, was written in the programming language C. C was chosen for these time critical parts to get maximal simulation performance. The model of the neural control system was written in the scripting language Python and linked to the C components. This partitioned organization allowed easy experimentation because normally only the parts written in the Python scripts were changed between trials. The resulting movements were presented graphically on the screen through the OpenGL graphics library, making it possible to directly see the consequences of various parameter changes and perturbations. The time step for the mechanical calculations was 0.25 ms, the controller was updated every 0.5 ms, and the graphics and data collection were processed once every 10 ms. When running on a 1.465-GHz processor machine (AMD Athlon XP 1700+) with a 3D-accelerated graphics card (NVidia GForce2), the rate of the simulation was 
15% of real time; for example, a 6 s experiment took about 40 s to simulate. This high rate readily allowed quick assessment of the quality of each trial.
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RESULTS |
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The initial objective of this study was to determine whether the simulation based on the control scheme shown in Fig. 2 was capable of producing stable stepping with features resembling those in normal walking cats. We began by investigating the performance of the simulation without cross-coupling between the two controllers. This was consistent with our strategy of developing the simulation with the minimum of complexity, and it gave us the opportunity to assess the extent to which mechanical linkage between the legs was involved in coordinating stepping. The parameters used for the muscles and body mechanics were set according to values based on biological measurements (see APPENDIX) and were not varied in the course of the investigation. Thus the only parameters that were changed were those in the Python script representing the neural control strategy for each leg. By systematically changing these parameters, we found that stable stepping could be easily produced. Videos of the simulations are available in the supplementary material and can also be viewed at http://sans.nada.kth.se/catstepping. Videos are referred to using the same numbers as the corresponding figure (e.g., video 3C corresponds to the data shown in Fig. 3C).
The step cycle of each leg clearly displayed the four distinct phases (F, E1, E2, and E3) that characterize the joint movements in walking cats (Engberg and Lundberg 1969), with extension at the knee and ankle during late swing (the E1 phase) and yielding at the ankle during early stance (the E2 phase; Fig. 4A). A striking and initially unexpected result of the simulation was the strong reciprocal coordination between the stepping of the two legs (Figs. 4 and 5A). This coordination must be entirely caused by mechanical coupling because there was no other linkage between the two controllers in this situation.
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The lateral movements at the hip joint were small and did not present any surprises (Fig. 4A). The neutral adduction angle of 2° gave the model a posture with the legs leaning slightly inward. During stance, the load on the leg caused an increase in the adduction up to around 4°.
Robustness of the gait
The model parameters were tuned to generate a natural walking pattern when walking over flat horizontal ground. By exposing the model to other situations, the robustness of the system could be assessed. A simple, and at the same time natural, disturbance was introduced by modifying the ground level to make the model walk in an upslope or downslope. With an upslope of 20% (11.3°), the cat slowed down but was able to continue walking (Fig. 3D). The shorter steps are mainly a consequence of the slower speed. When exposed to a downslope, the cat accelerated, forcing it to take longer steps. A 20% downslope resulted in a too high speed for the walking controller to handle, but with a 15% slope (8.5°), the model could adapt to the situation (Fig. 3E). Note that no changes or adjustments were made to the pattern of muscle activation in the different walking situations. Therefore the slowdown and speedup of forward progression when walking upslope and downslope, respectively, were caused by the gravitational force counteracting and assisting propulsive forces of the muscles. In addition to speed changes, the working range of the hip shifted backward during upslope walking and forward during downslope walking, thus matching qualitatively behavioral observations in cats (Carlson-Kuhta et al. 1998; Smith et al. 1998). Figure 3C shows a single trial where the model encountered multiple challanges: first a steep upslope and then a continuously varying ground level.
We also tested the models ability to handle external forces applied at the center of gravity of the trunk segment. With an extra load of 10 N, the model was still able to walk, taking on a more crouched posture. A lateral force of 5 N during 1 s made the model change walking direction because of the lack of friction in the front paws, but the walking pattern remained seemingly unaffected. A short (100 ms) impact of 20 N had a similar effect. A 4 N force during 1 s directed backward made the model slow down considerably, but it continued walking, much like in the upslope situation. Increasing the force even more (4.5 N) made the model stop and eventually fall backward. Similarly, a forward force of 5 N made it speed up, whereas 6 N was too much, causing it to fall by tripping (hitting the ground during swing). Videos of these experiments are available in the supplementary material.1
Finally, we tested the robustness against changes of internal parameters. Because the control system was tuned to match the particular musculo-skeletal system used, changing a critical internal parameter was expected to have an exaggerated impact when the control system was not retuned. As a critical parameter, we selected the force-length curve of the gastrocnemius muscle. When shortening the muscle by a factor of 10%, the cat was still able to walk, albeit less gracefully.
Stance termination rules
The main objective of this study was to functionally characterize the two main signals involved in determining the stance duration: 1) the sensory signal related to unloading of the ankle extensors and 2) the sensory signal related to hip extension. In particular, we wanted to find out whether a rule based on either of these signals alone would be sufficient. From hereon we refer to these two rules for regulating stance duration as the unloading rule and the hip extension rule, respectively. The unloading rule controlled stance termination by initiating liftoff when the force in the uniarticular ankle extensor muscle (homologous to soleus) fell below a fixed value of 5 N. The hip extension rule led to stance termination when the hip angle exceeded a fixed value of 107°. These threshold values correspond to the actual load and hip angle, respectively, where the switch normally occurred with the combined rule used above. To make the comparison between the two rules, we first ran the simulation using the combined rule and then bilaterally switched to one of the rules under study after stepping had stabilized (Fig. 5). The state of the simulation was thus identical at the time when the switch was made to either the unloading rule (Fig. 5B) or the hip extension rule (Fig. 5C).
After the switch, the behavior of the simulation was clearly different for the two rules. Switching to the unloading rule produced no noticeable alteration in the coordination of stepping in the two legs, and the gait pattern remained stable. On the other hand, when we switched to the hip extension rule, there was a strong tendency for the gait to become unstable and for the model to eventually either trip or fall. Figure 5C shows an example where the hip extension rule led to a progressive divergence in phase coordination of the two legs from a stable value of 0.5 (alternating gait) to a point where the left leg tripped. Tripping occurred when the end of the paw touched the ground during swing, before the condition for initiating touchdown was reached. Typically, the premature contact with the ground was brief, but it did delay significantly the completion of the swing phase. The irregular gait pattern following a trip explains the discontinuities in the phase relationships shown in Figs. 5C and 6, C and D.
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) that there is mutual inhibition between the neuronal systems generating flexor bursts in the two hind legs that prevents the two legs from entering the swing phase simultaneously. Thus to more closely simulate the physiological situation, we added a supplementary control rule that prevented the leg controller from entering the liftoff phase when the opposite leg was not in the stance phase (we refer to this rule as the cross-coupling rule). When used in combination with the unloading rule, this cross-coupling did not change anything. In fact, the result was identical to that of Fig. 5B. In contrast, when the cross-coupling rule was supplementing the hip extension rule, the simulation changed from being unstable into being stable, i.e., it produced a stationary gait pattern, but the coordination was abnormal (Fig. 5D). After the switch to this rule, the phase values drifted progressively away from 0.5 (as we observed initially when only the hip extension rule was used) to a point at which the two swing phases commenced to overlap. At this point, the cross-coupling rule became effective, and the gait entered a stable, asymmetrical mode with the onset of swing in one leg following immediately after the termination of swing of the opposite leg. This asymmetric gait was not very stable, and small changes of the initial conditions often led to a fall. Perturbation analysis
Having found that the two rules for terminating the stance phase when used alone resulted in differences is stepping behavior, we anticipated that the responses to perturbations would also differ. To examine this prediction, we perturbed the simulation by suddenly removing the ground friction for 100 ms, mid-way during the stance of the left leg (Fig. 6). The coordination of stepping in the two legs was only transiently affected by the slip perturbation when the combined rule was used (Fig. 6A) and when the unloading rule was used (Fig. 6B). In both cases, the perturbation shifted the phase values away from 0.5, but these values converged back to close to 0.5 within a few step cycles. In contrast, when the hip extension rule was used either alone (Fig. 6C) or together with the cross-coupling rule (Fig. 6D), the simulation became very unstable and either fell or collapsed quickly.
The striking difference between the behavior of the simulation using the unloading and the hip extension rules for terminating stance led us to examine the reason(s) for this difference. The basic issue was to determine why the unloading rule stabilized the coordination between the legs at a phase value close to 0.5, whereas the hip extension rule led to a progressive shift away from a phase value of 0.5. Insight into this issue came by examining the effects of slightly changing the touchdown time of the opposite leg (Fig. 7). When the unloading rule was used for controlling stance termination, slightly advancing or delaying swing termination resulted in an advance or delay, respectively, in the time of stance termination of the following step in the first leg (Fig. 7, A and B). This reaction is appropriate for maintaining a phase value of 0.5: if one leg is early, the other must be quicker to catch up. On the other hand, when the hip extension rule was used for controlling stance termination, the time of liftoff did not change appropriately. In fact, a tendency for the opposite effect was observed (Fig. 7C). This has the effect of shifting the phase values away from 0.5. The mechanisms behind these two responses can be understood by observing how delaying or advancing the opposite leg affected the ankle extensor load and hip angle around the time of stance termination (Fig. 8). The load in the ankle extensor dropped after a rather constant delay following contralateral touchdown (Fig. 8A), which is natural because the opposite leg takes part of the load. This explains the stabilizing effect of the unloading rule. The mechanism behind the reversed effect for the hip extension rule can also be understood by considering the mechanical consequences of the perturbation. Delaying the time when the swinging leg received ground contact resulted in a faster hip extension in the standing leg near the end of its stance, thus causing the hip to reach the critical angle earlier. Advancing the time of ground contact produced the opposite effects (Fig. 8B). The reason for these changes in the velocity of hip extension is that near the end of stance the force vector of the ground reaction force is directed posterior to the hip joint and therefore provides a torque supporting the hip extension. When the opposite leg becomes loaded after touchdown, this supporting torque is reduced, leading to a slower extension. Consequently, if swing termination is delayed in the opposite leg, the standing leg extends faster. If the hip extension rule is used alone for stance termination, this leads to an earlier stance termination, contrary to what is needed to restore the alternating gait. Monitoring the direction of the ground reaction force in the model confirmed that it was consistently directed posterior to the hip joint during late stance.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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; Whelan et al. 1995) and the second is excitation of the flexor burst-generating system by sensory signals related to the magnitude of activity in position-sensitive afferents from the hip (Grillner and Rossignol 1978; Hiebert et al. 1996). The combination of these two mechanisms has led to the concept that a necessary condition for the initiation of the swing phase is unloading and extension of the leg (Pearson 2003; Prochazka 1996). To date, all the data supporting this concept have come from studies on spinal or decerebrate preparations in which the mechanical properties of the preparation were often far from normal (at one extreme, data have often been collected from paralyzed preparations). Furthermore, in situations in which data were derived in a functional context (such as stepping in spinal cats; Grillner and Rossignol 1978), it has not been possible to completely isolate the contribution of specific mechanisms to the regulation of the step cycle. Ultimately, any proposal for a mechanism controlling stepping must be evaluated in a context that includes all the complex interactions between neuronal, muscular, mechanical, and environmental events. Because this has not yet been done for any proposed mechanism regulating phase transitions, we currently do not know how effectively the stance-to-swing transition can be regulated by sensory signals from the hip and/or the ankle extensor muscles. Because this issue cannot be addressed directly in normal walking animals, our approach has been to use a computer simulation that allows the study of individual mechanisms in isolation. At the outset, we recognized that the simulation could only capture some of the features of the walking system of the cat because of the absence of biological information. The mechanics were simplified by only simulating stepping in the hind legs, assuming the mass in each leg segment was uniformly distributed, and disregarding movements of the trunk. All muscles were simulated with the same model, and only a subset of all hind leg muscles was included. And finally, the controller included only a few sensory pathways and did not attempt to simulate neuronal events in the spinal cord. Our design philosophy was to keep the simulation as simple as possible by minimizing the number of elements, yet sufficiently complex to produce robust stepping on a variety of terrains and to mimic the main characteristics of stepping in the hind legs. Moreover, the simulation was specifically designed to allow an examination of the regulation of the stance phase by sensory signals from the ankle extensor muscles and the hip joint. There are undoubtedly other sensory signals, as well as central processes, involved in regulating the transition from stance to swing, but for our purposes, the absence of these processes was not a limitation because we wished to examine the characteristics of individual mechanisms in regulating the stance to swing transition independently from other mechanisms.
When doing changes to a model and observing the consequences, it is important to first ensure that the initial model is robust both with regard to its operating environment and to the precise parameter values used. We exposed the model to a number of different walking situations, such as external forces in all directions, and are quite convinced that the model is capable of producing realistic walking patterns in varying circumstances. The results in Fig. 3, C–E, show that the model is capable of continuing walking when the situation is quite different. In addition, moderate changes of internal model parameters have no serious impact on the walking behavior. Thus we believe that the general results are independent of the particular parameters we have used.
The main finding of this study was a major qualitative difference in the characteristics of coordination of stepping in the hind legs when unloading of the ankle extensor muscle was used to control the termination of stance compared to the use of hip extension for terminating stance. The unloading rule invariably led to stable alternating stepping in the hind legs, whereas the hip extension rule led to progressive shifts in the coordination (Figs. 5 and 6) and often to falling or collapse of the hindquarters. The analysis of the effects of small perturbations of one leg revealed the reason for this difference. When the unloading rule was used, a perturbation of one leg led to a modification of stepping in the opposite leg that tended to preserve a phase value of 0.5, i.e., alternating stepping (Figs. 7B and 8A). On the other hand, when the hip extension rule was used, perturbations of stepping in one leg led to changes in the opposite leg that tended to shift the phase away from 0.5 (Figs. 7C and 8B). It was the tendency to move away from an alternating gait that eventually led to instability when the hip extension rule was used alone.
It is important to note that the initial observations we made on the unloading and hip extension rules were done in simulations that did not include any cross-coupling between the controllers for the two legs. Including cross-coupling had no influence on coordination when the unloading rule was used alone because the mechanical linkage associated with this rule ensured no overlap of flexor activity in the two legs. On the other hand, the performance of the simulations using the hip extension rule was strongly influenced when cross-coupling between flexor systems was included (cf. Fig. 5, C and D). This is because the hip extension rule on its own led to overlap of flexor burst activity, and when this occurred, there was a high probability that the simulation collapsed or fell (Fig. 5C). Including cross-coupling prevented this overlap, and stable stepping resulted (Fig. 5D). However, the cross-coupling did not prevent the initial tendency of shifting the coordination away from phase values of 0.5. Thus when the simulation was started at a phase value of 0.5, the phase shifted until the activity in the flexor systems just began to overlap. At this point, the inhibitory cross-coupling prevented simultaneous flexion, and the system settled into an asymmetrical gait not normally seen in walking animals.
A significant advantage of initially excluding cross-coupling between the controllers for the two legs was that it allowed the characteristics of purely mechanical linkages between the legs to be examined. The possibility that mechanical linkages may play a significant role in coordinating stepping movements has not been considered seriously in the past. Here we have shown that mechanical linkage resulting from the unloading rule leads to robust reciprocal coordination of stepping in the two legs. Thus if the unloading rule operates in walking animals, it is likely that mechanical linkage between the legs normally functions to assist in coordination. A prediction from this conclusion is that a loss of load information from the ankle extensors would disturb the coordination of stepping in the two hindlegs. The dominance of the unloading rule over the hip extension rule (Figs. 5 and 6) also predicts that abolition of sensory feedback from the hip region would have little influence on the timing and coordination of stepping of the two hind legs. If this proves to be true, an obvious question is whether or not sensory feedback from the hip has any significant function in controlling phase transitions during walking. An alternative function for hip afferents may be to control transitions from extensor to flexor activity in situations in which the leg is unloaded or only weakly loaded, such as scratching and swimming. To our knowledge there have been no physiological studies on the role of hip afferents during swimming and scratching, so it is not possible to assess the likelihood of their involvement in regulating phase transitions during these behaviors. It is also noteworthy that lifting the hindquarters reduced stance duration in the simulation that included crossed-coupling (Fig. 9B), leading to the prediction that the same thing should happen for cats stepping on a treadmill when their hindquarters are lifted.
Another feature of our simulation was that we did not attempt to simulate intrinsic rhythmicity in the controllers (central pattern generation). Because the existence of central pattern-generating networks with intrinsic rhythmicity in the cat spinal cord is beyond doubt, it is essential that we justify excluding these networks from the simulation. Recent physiological data indicate that the phase transitions between stance and swing and between swing and stance may be switched by phasic sensory signals from peripheral receptors (Pearson 2003; Pearson et al. 2003). The finite-state simulation we used was designed to mimic the phase-switching influences of these sensory signals and thus examined whether these signals can indeed control transitions in a manner that leads to stable stepping on a horizontal terrain. This is not to say that we believe central pattern-generating networks are unimportant in the generation of locomotor activity. Indeed, the central pattern generators contribute substantially to burst generation in motoneurons, and this action is supported by sensory input to motoneurons mediated by pathways modulated by the central pattern generators (McCrea 2001). We could have easily modified our controller to produce a centrally generated rhythm in the absence of sensory input by simply limiting the duration of the swing and stance phases independent of sensory signals. However, because the focus of the study was on establishing the functional characteristics of different sensory signals that might regulate the stance to swing transition, we would have had to set the duration of stance-related activity of the central pattern generator to a value sufficiently long to allow the sensory signals to initiate swing. In this situation, limiting the duration of centrally generated bursts would have been superfluous. We acknowledge that exclusion of central pattern generators resulted in a controller that did not mimic known mechanisms for establishing the magnitude of muscle activity, but again we felt this was not necessary for examining sensory control of the swing to stance transition. The levels of muscle activation used in the controller were selected to yield stable locomotion under a variety of conditions (Fig. 3) and therefore provide a robust system for isolating the functional characteristics of the two sensory mechanisms we examined.
This study, together with a previous study by Yakovenko et al. (2004), clearly show that simulations of the interacting mechanical, muscular, and neuronal elements of the cat hind legs can yield valuable insights into mechanisms controlling stepping. Yakovenko et al. (2004) showed that feedback from muscle proprioceptors during the stance phase can stabilize stepping when the output of central pattern generating networks is low, and we have found that the ankle extensor unloading rule is able to stabilize the alternating gait when used to control the time of paw liftoff. In addition, we unexpectedly found that the unloading rule mechanically couples the two hind legs in a manner that leads to stable alternating stepping of the two legs. The use of simulations of walking in the cat will undoubtedly become more prominent in the near future. First, they provide a framework for bringing together the large body of data on the neuronal, muscular, and mechanical elements involved in producing walking. Second, they provide a powerful tool for gaining insights into complex interactions between the environment, limbs, and the neuromuscular system. Finally, the computational techniques for real-time simulations are now available and can be easily extended to simulations with more degrees of freedom, more muscles, and more complex neuronal control mechanisms.
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APPENDIX |
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The muscle force was calculated using the muscle model of Brown et al. (1996). The total force F is expressed as
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The force–length dependence is calculated as
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= 2.3, 
= 1.26, and 
= 1.62.
The force–velocity dependence is
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The passive force is given by
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Muscle insertion
The muscles were all coupled to the skeletal model so that 2° of joint motion corresponded to 1% of muscle length change, except for the gastrocnemius, where 1.5° at the ankle or 4.5° at the knee were required. In the conversion between muscle force and torque, lever arms were taken to be 15 mm for gastrocnemius and soleus insertion at the ankle, 5 mm for gastrocnemius at the knee, and 9 mm for vastus lateralis.
The muscle lengths were set so that at a neutral posture with the hip at 65°, the knee at 100°, and the ankle at 105°, all single joint muscles had a length of 85% of Lmax and all dual joint muscles were at 75%.
The maximal muscle forces (Fmax) were adjusted to give the following joint torques:
| Muscle | Joint action | Torque |
| Soleus | Ankle extension | 1.5 Nm |
| Tibialis anterior | Ankle flexion | 1.5 Nm |
| Gastrocnemius | Ankle extension | 3.0 Nm |
| Gastrocnemius | Knee flexion | 1.0 Nm |
| Vastus lateralis | Knee extension | 3.0 Nm |
| Semitendinosus | Knee flexion | 2.0 Nm |
| Semitendinosus | Hip extension | 2.0 Nm |
| Anterior biceps | Hip extension | 4.0 Nm |
| Iliopsoas | Hip flexion | 4.0 Nm |
Series elasticity
All muscles were acting through a series elasticity, primarily mimicking tendon elasticity. The stiffness for the different muscles was
| Soleus | 40 |
| Tibialis anterior | 40 |
| Gastrocnemius | 40 |
| Vastus lateralis | 80 |
| Semitendinosus | 100 |
| Anterior biceps | 200 |
| Iliopsoas | 200 |
Abduction/adduction
Abduction–adduction movements at the hip joint were controlled by a passive torque-spring with linear elasticity and damping. The neutral angle (zero torque) of the spring was set to 2° of adduction. The spring parameters were
| Stiffness | 0.2 Nm/degree |
| Damping | 0.5 Nms/degree |
Segment parameters
All segments were modeled as rectangular blocks with their mass spread out evenly throughout the volume. For the leg segments, height in the table refers to the size in the anterior-posterior direction when standing with straight legs.
Neural control
The neural control system received joint angles (hip angle, 
Hip; knee angle, Knee; both measured in degrees) as well as muscle
| Mass (g) | Length (mm) | Height (mm) | Width (mm) | |
| Trunk | 2000 | 230 | 80 | 70 |
| Thigh | 200 | 90 | 30 | 30 |
| Shank | 100 | 100 | 20 | 20 |
| Foot | 60 | 60 | 10 | 10 |
The following state transition rules were used
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When cross-coupling was active, an additional condition for the stance to lift-off transition was that the contralateral controller had to be in its stance phase.
Magnitude of muscle activation was set according to the following rules
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GRANTS |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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Address for reprint requests and other correspondence: Ö. Ekeberg, Dept. of Computer Science, Royal Inst. of Technology, S-100 44 Stockholm, Sweden (E-mail: orjan{at}nada.kth.se). 1The Supplementary Material for this article (a series of videos) is available online at http://jn.physiology.org/cgi/content/full/00065.2005/DC1
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REFERENCES |
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Butt SJ, Harris-Warrick RM, and Kiehn O. Firing properties of identified interneuron populations in the mammalian hindlimb central pattern generator. J Neurosci 22: 9961–9971, 2002.
Carlson-Kuhta P, Trank TV, and Smith JL. Forms of forward quadrupedal locomotion. II. A comparison of posture, hindlimb kinematics, and motor patterns for upslope and level walking. J Neurophysiol 79: 1687–1701, 1998.
Cazalets JR, Borde M, and Clarac F. Localization and organization of the central pattern generator for hindlimb locomotion in newborn rat. J Neurosci 16: 298–306, 1995.
Conway BA, Hultborn H, and Kiehn O. Proprioceptive input resets central locomotor rhythm in the spinal cat. Exp Brain Res 68: 643–656, 1987.[ISI][Medline]
Donelan JM and Pearson KG. Contribution of force feedback to ankle extensor activity in decerebrate walking cats. J Neurophysiol 92: 2093–2104, 2004.
