a neural network model for limb trajectory formation

Biol. Cybern. 61,417-425 (1989) Biological Cybernetics 9 1989 A Neural Network Model for Limb Trajectory Formation L. Massone and E. Bizzi Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, MIT-Building, E25-526, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Abstract. This paper deals with the problem of repre- senting and generating unconstrained aiming move- ments of a limb by means of a neural network architecture. The network produced time trajectories of a limb from a starting posture toward targets specified by sensory stimuli. Thus the network perform- ed a sensory-motor transformation. The experimen- ters trained the network using a bell-shaped velocity profile on the trajectories. This type of profile is 9 characteristic of most movements performed by bi- ological systems. We investigated the generalization capabilities of the network as well as its internal organization. Experiments performed during learning and on the trained network showed that: (i) the task could be learned by a three-layer sequential network; (ii) the network successfully generalized in trajectory space and adjusted the velocity profiles properly; (iii) the same task could not be learned by a linear network; (iv) after learning, the internal connections became organized into inhibitory and excitatory zones and encoded the main features of the training set; (v) the model was robust to noise on the input signals; (vi) the network exhibited attractor-dynamics properties; (vii) the network was able to solve the motor- equivalence problem. A key feature of this work is the fact that the neural network was coupled to a mechan- ical model of a limb in which muscles are represented as springs. With this representation the model solved the problem of motor redundancy. 1 Introduction This paper deals with the problem of representing and generating unconstrained aiming movements of a limb by means of a neural network architecture. Aiming movements are present in biological sys- tems at different levels of complexity, from accurately planned movements to reflexes (Georgopoulos 1986). The present work focuses on unconstrained limb movements elicited by sensory stimulation. They are meant to mimic the wiping movements made by the leg of spinal frogs when the frog's skin is stimulated by an irritant (Berkinblitt et al. 1986; Giszter et al. 1989). Scratch reflexes of spinal cats (Shadmehr and Lind- quist 1988) represent another example of this class of movements. Opto-electrical recordings of frogs' wiping movements (Giszter et al. 1989) show that the motor strategy remains basically the same in both intact and spinal animals. This result suggests that the basic motor programs for this particular task are generated at the spinal cord level and not explicitly planned by higher brain structures. Starting from this observation, we adopted a non-hierarchical neural network to represent such movements. The neural network's task in this work involved generating a trajectory of a limb from a starting posture toward a target specified in terms of a sensory stimulus. Hence, the network performed a sensory- motor transformation. Aiming movements were as- sumed to be planar (as in the aiming phase of the movement made by the frog when it wipes its back), but there is no theoretical limitation to the dimensionality the network could deal with. Surprisingly, a number of kinematic studies of arm movements (Morasso 1981; Abend et al. 1982; Atke- son and Hollerbach 1985; Howarth and Beggs 1981) have shown that the integrative action of thousands of sensors, neurons and skeleto-motor units result in velocity profiles whose global shape is invariantly bell- shaped "over a wide range of movements sizes and speeds. Flash and Hogan (1985) showed that a minimum-jerk model predicts both the qualitative features and the quantitative details observed experi- mentally in planar, multi-joint arm movements. Accordingly, in the present work, the experimen- ters used a bell-shaped velocity profile for the training trajectories. The duration of movements was assumed 420 of the bell-shaped velocity profile. As far as stiffness is concerned, the model does not allow, at this stage of development, direct control over the stiffness values during the transformation from end-point positions to muscle activations. We employed the inverse trans- formation to compute the output patterns necessary to train the network. The direct transformation (from muscle activation to end-point position) was used during the testing phase. To train the network we used a standard back- propagation algorithm which makes use of a momen- tum term; the learning rate was interactively lowered during the training sessions. All trajectories used during the training phase had a duration of six time steps: initial posture, target posture and four inter- mediate postures. All postures were equilibrium po- sitions as defined in 2. One of our major concerns about the training phase was how many and which sequences the network had to learn to correctly generalize the task. We started with four sequences which corresponded to sensory stimuli in the four quadrants into which the limb workspace is ideally divided by the initial end-point position. Figure 3 shows the four trajectories as they were generated by the network after learning. It is worth noting that the bottom-left trajectory contained a joint reversal on the shoulder joint. We gradually increased the number of learned trajectories with the purpose of achieving a generalization capability such that the error on the end-point position 3 for each point on the trajectory did not exceed the grid step. This requirement was equivalent to demanding that the 3 Errors were measured, for each end-point position, as the euclidean distance between the end-point position produced by the network and the expected end-point position produced by the mechanical model of muscles ri'uu iJlll uaul ii II II II I lIRi IJlllll~llllll IIIIllllliNII I I I I I I I I I I~11~11 [I II II I II I 1~11 IIIIIIIIIIIIIII IIIIIII II II II II ~ IIIIIllllllllll Fig. 3. Trajectories towards 4 stimuli in different quadrants of the limb workspace. These trajectories are generated by the network after learning. Points along the trajectories are equispaced in time but not in space because of the bell-shaped velocity profile []1111111111111 IIIIIIIIIIIIIII ilnm Ilml ilii lUll Iill lull lull Illl IIII Illl ,~nn I~II l-,in !Inj Illllllllllllll Illllllllllllll |l]llllllllllll IlJllllllllllll l|llllllllllill Itlllll]llllll IIIllllllllllll IIIIIllllllllll llllllllllltlll IIIIIIIIIIIIII I IIIIIIIdllllll ~ 1111 11111111"llI2t~lllll iiiiillllr~LIIII IIIIIIlllllllll IIIllllllllllll 111111111111111 IIIIIII1~11111 Illllllllllllll Illllllltll IIII IIl1111111~1111 IIlllllllllllll IllllJllllll|ll IIIIIIIIIIIIIII IIIIIIIklltlll Fig. 4. Generalization capability after learning 15 trajectories. The top-left trajectory contains a generalization of the joint reversal on the shoulder. The rightmost trajectory in the second row is a particular case of generalization in which the stimulus was positioned right on the limb end-point. Although the network has not been explicitly taught about the initial posture, it has "understood" how the limb is positioned at the beginning of each trajectory network behave well at the resolution imposed by the discretization of the limb workspace. This level of performance was achieved after the network was taught 15 sequences uniformly distributed over the workspace. Figure 4 shows some generalized sequenc- es: the end-point position is correct along the whole trajectory, and the velocity profile is properly adjusted. In addition, the network generated patterns of mus- cular activation which corresponded to equilibrium positions of the limb and could produce joint reversals when necessary. A more detailed account of the learning task is given in Massone and Bizzi (/989). Three further experiments were performed during the learning phase. First, the learning procedure was repeated by making use of local coding instead of coarse coding (one plan unit for each pixel for a total of 225 plan units). After learning the same 15 sequences, the network was not able to generalize and behaved like a look-up table. In the second experiment, the learning procedure was repeated for a lower resolution on the workspace, which was obtained by doubling the grid step. This doubling led to a 7 x 7 array of pixels coarse-coded by a 3 x 3 array of plan units. In this case the network could learn the task (producing errors lower than the grid step) with fewer learned trajectories -8 as compared to 15. In the third experiment we 421 repeated part of the learning procedure with a linear network obtained by removing the hidden layer. The purpose of this experiment was that of investigating the amount of non-linearity present in the input-output transformation. We tried to teach to the linear network the four trajectories shown in Fig. 3, first separately and then jointly. We observed the following behavior: 9 The linear network could learn the trajectories towards the top-left target and towards the top-right target separately. 9 The same two trajectories could not be learned jointly. This fact shows that the linear network could not handle the interferences between the two trajec- tories, while the non-linear network could. 9 The linear network could not learn the trajectories towards the bottom-left target and towards the bottom-right target, neither separately nor jointly. We concluded that the task is highly non-linear, except in a few peculiar cases. Furthermore, we observed that the trajectories that the linear network could learn were much shorter than those that could not be learned. Hence, we also investigated the existence of a possible relationship between the task linearity and the length of the trajectories. To this purpose, we tried to teach to the linear network a shorter trajectory in the direction of the bottom right target. The linear network could not learn that trajec- tory. We concluded that no relationship exists between the trajectory's length and the extent to which the linear approximation holds. 4 Internal Organization We analyzed the connections of the trained network in order to understand the organization produced during learning. Interesting patterns were found in the con- nections from hidden to output units; Table 1 shows the values of the connections after the task was learned. We observed that, whenever one hidden unit sends an excitatory connection to a flexor, the same unit sends an inhibitory connection to the corresponding ex- tensor (negative correlation.) Similarly, whenever an inhibitory connection is sent to a flexor, an excitatory connection is sent to the corresponding extensor. The network has represented in the connectivity pattern the rule of reciprocal inhibition of agonist-antagonist pairs. Inhibition and excitation were more marked for shoulder, elbow and double-joint muscles than for wrist muscles. This result agrees with the experimental data of Georgopoulos (1986), which show that aiming movements involve the wrist joint in only a very marginal way. Moreover (see again Table 1), we ob- served that: 9 units :~ 3 and ~ 10 exhibited a total positive corre- lation between all flexors and between all extensors; 9 all other units exhibited a total positive correlation between - the shoulder flexor and the double-joint flexor; - the shoulder extensor and the double-joint extensor; - the elbow flexor and the wrist flexor; - the elbow extensor and the wrist extensor; Hidden unit ~ 2 was an exception: the shoulder and double joint exhibited a negative correlation. These observations could be interpreted as follows. First of all, there is evidence for a number of synergies between all hidden units. These synergies are the necessary condition for the network to exhibit good generalization properties. Furthermore, the network seems to have represented in the connectivity pattern the main features of the set of patterns that was used as Table 1. Hidden to output connections. Each row contains the connections from all hidden units to one particular output unit Shoulder F1. 1.596938 -0.459756 0.238360 0.464764 -1.130529 1.225434 1.249900 -0.822495 0.946107 -1.160787 Shoulder Ex. 1.597220 0.458936 -0.238326 -0.464758 1.130295 -1.225155 -1.249560 0.822554 -0.946128 1.161238 Elbow F1. 2.205403 1.201014 0.990554 -0.725229 1.035587 -0.788169 -1.198095 0.046117 -1.464604 -0.762154 Elbow Ex. -4.299550 -0.824471 -2.072892 1.214777 -1.433692 0.294094 1.581221 0.024668 2.398601 0.365059 Double J. F1. -0.822154 1.664527 0.419754 0.069341 -0.511715 0.004101 0.556992 -0.617517 0.068661 -1.571512 Double J. Ex. 0.580872 -1.538881 -0.525151 -0.019188 0.490678 -0.050527 -0.561579 0.623930 0.010338 1.459015 Wrist FI. 0.260377 0.220753 0.122209 -0.092430 0.129453 -0.155435 -0.139240 -0.012050 -0.188938 -0.090522 Wrist Ex. 0.260409 -0.220748 -0.122226 0.092426 -0.129474 0.155452 0.139244 0.012054 0.188934 0.090537 422 the training set. In fact, the sign of muscle activations in the training sequences was always the same for elbow- wrist flexors and elbow-wrist extensors and almost always the same for shoulder-double joint flexors and shoulder-double joint extensors. The network encoded that "almost" by means of a negative correlation at unit ~2. Finally, the network devoted two hidden units, 4~ 3 and # 10, to encode the synergies between all flexors and between all extensors. Assuming that one hidden unit corresponds to one family ofinterneurons, these results suggest that interneurons may be orga- nized into functionally overlapping groups (Edelman 1987). 5 Experiments We performed three experiments with the trained network. The first experiment was concerned with the dura- tion of the trajectories. Pineda (1987) showed that arbitrary networks of logistic units typically have many point attractors. In other words these networks naturally exhibit certain dynamic properties. In our case, the network was instructed during training to produce certain output patterns for six time steps; no instructions were given about what should be done after the sixth time step. We tested the network for 15 time steps, and we observed that in about 80 percent of the cases (i.e. in about 80 percent of the limb work- space), the limb remained steady at the final posture which corresponded to the position of the sensory stimulus. In other words, in 80 percent of the cases, the final posture of the limb acted as a point attractor. In certain portions of the workspace, the limb became unstable after the sixth time step. The portion Varied with different learning sessions, depending on which solution the network had settled into. There were also cases in which the entire workspace was steady. The second experiment aimed at testing the robust- ness of the system when the sensory stimulus was varied. The network was trained with stimuli coded as gaussian distributions centered on the target with a certain standard deviation do; we modified the value of the standard deviation during testing as follows: dl=d0+0.1 *d o , d 2 = d o + 0.2 * d o . Both of the above cases correspond to a stimulus which is flatter and more spread over the workspace. We measured the average distance between the trajec- tories generated with standard deviation d o and the trajectories generated with standard deviation dl and d z. Given the stimulus d I the average distance was lower than 0.4; in the second case the average distance Illllllllllll IIII111111111 [llllllflllllll IIIIIIIII IJ~l I ITI 1 I'l~ll II I II I I~I II IIIIIII~lifl I I I I I I Ill '1,~311 I I I j,,,'x/ Jtiillillilill'~ [[lllllllllllll l llllIll,~%~,~lll II II II I II RiIII II IIllttt titx\attt IIIIIIITIIIIIII Illl[llllllllll II;;ii5iiiJii3i I]lllllllllllll Fig. 5. Double target experiment. The first target was turned on for two time steps, and then it was turned off. The experiment was performed on both learned and generalized trajectories Ili[lllllllllll IIIllllllllllll IIIIIIIII1111111 IIIIIIVI~M/I~III Illllllllllllll I I I I I I N~I I I I II I I I I I ~KII i II IIIJlillli~\llli IIl|t|lllllllll IIIIIIIIIIIIIll IIIIIIII]111111 IIIIII111111111 Fig. 6. Double target experiment. The first target was turned on for three time steps, and then it was turned off. The experiment was performed on both learned and generalized trajectories was higher (around 0.7), which resulted in trajectories that were "noisy", but still acceptable. This experiment showed that the architecture was reasonably robust in the face of slight changes in the stimulus representation. The third experiment was performed with the aid of a double target. A sensory stimulus was given to the network. As the limb was moving in steps toward the stimulus, the stimulus was turned off, and another stimulus at a different location was turned on. When this occurred, the limb switched direction towards the new target. The experiment was repeated in the following two cases: 1. with the duration of first stimulus correspond- ing to the first two time steps made by the limb; 2. with the duration of first stimulus correspond- ing to the first three time steps. Figure 5 shows the resulting trajectories for the first case, while Fig. 6 represents the trajectories for the 423 second case. Note that in both cases the limb reached the second target. The results of this experiment show that the network, having learned how to reach a set of targets from a fixed initial position of the limb, was also able to reach the same set of targets from a different posture, the one in which the limb was positioned when the second target was turned on. This result indicates that the network was able to solve the so-called motor- equivalence problem. Experiments on intact biological systems (Georgopoulos et al. 1981 ; Massey et al. 1986) clearly show that shape and length of the trajectories generated by a double-target experiment depend upon the duration of the first stimulus. The network in our experiments seemed to be insensitive to this parameter (compare Figs. 5 and 6). In a sense, the network "forgot" everything about the first trajectory when the second stimulus was turned on. These different behaviors may indicate that a non hierarchical system - like the neural network described in this paper- does not contain any smoothing mechanism, whereas such mechanisms are present when planning occurs. In other words, trajectory smoothing is not a direct consequence of the mechanical properties of the ac- tuator, but the result of some specialized brain func- tions. With neural networks, one could obtain a smoothing behavior by enriching the network with other blocks which somehow implement the above mentioned brain functions. Alternatively, one could add some dynamics (i.e. self-connections) at the output units (like in Jordan's flow networks [1989a]) to make the next output of the network a function of the past outputs. The latter solution would correspond to assuming that trajectory smoothing is a low-level operation handled by motorneurons. 6 Discussion In this paper we presented a model for the formation of limb trajectories, based on a neural network architec- ture. The task under consideration was that of reaching a target defined in terms of a sensory stimuli. The trajectories had a bell-shaped velocity profile. The network produced trajectories in muscle space, which were translated into end-point space by means of a model which takes into account the elastic properties of muscles. The inverse transformation, from end- point space to muscle space, was used to generate the training sequences as described in Sect. 3. The partic- ular architecture used for producing time trajectories was that proposed by Jordan (1986). We have shown that: I. The task could be learned by a three-layer sequential network trained by a standard back- propagation procedure. 2. The network successfully generalized in trajec- tory space: the error of the generalized trajectories measured at the end-point could be made lower than the discretization step of the limb workspace. Moreover, the velocity profiles generalized appropriately. 3. The same task could not be learned by a linear network. 4. The internal connections became organized, after learning, into inhibitory and excitatory zones; in particular, connections from hidden units to output units exhibited a number of positive and negative correlations that encoded the main features of the training set. Between hidden units, we observed a number of synergies which are the necessary basis for good generalization properties. 5. The model was robust with respect to the input signals: slight changes to the stimulus coding did not significantly affect the network overall performance. 6. The network spontaneously exhibited attractor- dynamics properties. Final end-point positions behaved like point attractors in the majority of the limb workspace. 7. The network could solve the motor-equivalence problem as shown by the double-target experiment. The network did not exhibit smoothing properties, and seemed to be insensitive to the duration of the first stimulus. Kawato et al. (1987) studied voluntary movements and proposed a hierarchical, structured model for generating motor commands (torques) from a desired trajectory expressed in body-centered coordinates. Moreover, Kawato et al. (1988) studied the coordinate- transformation problem and proposed an iterative control learning algorithm. Our research dealt with a sensory-motor transformation based on a non- hierarchical layered architecture which translated a sensory stimulus directly into time-varying patterns of muscular activation which corresponded to minimum jerk trajectories. We did not face the coordinate transformation problem since we made the hypothesis that both target and movement were already expressed in the same body-centered reference frame. We did address the problem of trajectory formation based on a constant sensory stimulus rather than a reference trajectory. The issue of trajectory formation was also faced, among others, by Bullock and Grossberg (1988) who presented a model called VITE which produces arm trajectories from a target position command (TPC) and a GO command which defines the movement's speed. Although VITE has nice general- ization properties, it is worth noting that trajectories are generalized in joint space. By contrast, our model could generalize trajectories in muscle space and then in end-point space through the mechanical model of 424 muscles (Mussa Ivaldi et al. 1988b). Moreover, VITE cannot be easily applied to multi-joint movements and does not address learning. In our model the information about the actual position of the end-point was not explicitly computed. (It is only implicitly available through the muscles' model.) This fact did not represent a limitation for the task under consideration, since the task was per- formed, as already pointed out, at the reflex level, without any planning. However, in human experi- ments Morasso (1981) showed that the information about the end-point position plays a crucial role at the planning level. If planning were to be incorporated in our model, its architecture ought to be expanded to include explicit computation of the end-point position. As far as task representation is concerned, we merged the kinematic problem and the velocity profile into a single three-layer network, but this was not the only possible choice; the two problems could as well have been separately addressed and represented by means of two interconnected networks. The latter possibility has been investigated by Jordan (1989b). The work described here has relevance to the robotics research since it may suggest some basic principles for designing artificial limbs whose structure is inspired by natural systems (De Rossi et al. 1988). Moreover, we plan to extend this work to cope with constrained movements, in which trajectories are affected by the surrounding environment. To this purpose, it is necessary to model and represent the environment and the interactions with it (Massone and Morasso 1986; Massone 1986, 1987). This topic raises several interesting problems which have often been addressed by the artificial intelligence research. Our future goals include building a neural-network archi- tecture capable of providing a uniform representa- tional framework for environment and movements (Hogan 1984). The analogical nature of neural net- works might provide significant insights into those problems, as well as useful suggestions for how such problems are addressed by biological systems. 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First INNS Meeting, Boston, Mass, p 359 Received: June 5, 1989 Accepted: June 6, 1989 Dr. Lina Massone Department of Brain and Cognitive Sciences Massachusetts Institute of Technology MIT-Building, E25-526 77 Massachusetts Avenue Cambridge, MA 02139 USA . coordinate- transformation problem and proposed an iterative control learning algorithm. Our research dealt with a sensory-motor transformation based on a non- hierarchical layered architecture. strategy remains basically the same in both intact and spinal animals. This result suggests that the basic motor programs for this particular task are generated at the spinal cord level and. In: Jeannerod M (ed) Attention and performance XIII. Erlbaum, Hillsdale, NJ Kawato M, Furukawa M, Suzuki R (1987) A hierarchical neural- network model for control and learning of voluntary move-