Integration of Gravitational Torques in Cerebellar Pathways Allows for the Dynamic Inverse Computation of Vertical Pointing Movements of a Robot Arm

Several authors suggested that gravitational forces are centrally represented in the brain for planning, control and sensorimotor predictions of movements. Furthermore, some studies proposed that the cerebellum computes the inverse dynamics (internal inverse model) whereas others suggested that it computes sensorimotor predictions (internal forward model).This study proposes a model of cerebellar pathways deduced from both biological and physical constraints. The model learns the dynamic inverse computation of the effect of gravitational torques from its sensorimotor predictions without calculating an explicit inverse computation. By using supervised learning, this model learns to control an anthropomorphic robot arm actuated by two antagonists McKibben artificial muscles. This was achieved by using internal parallel feedback loops containing neural networks which anticipate the sensorimotor consequences of the neural commands. The artificial neural networks architecture was similar to the large-scale connectivity of the cerebellar cortex. Movements in the sagittal plane were performed during three sessions combining different initial positions, amplitudes and directions of movements to vary the effects of the gravitational torques applied to the robotic arm. The results show that this model acquired an internal representation of the gravitational effects during vertical arm pointing movements.This is consistent with the proposal that the cerebellar cortex contains an internal representation of gravitational torques which is encoded through a learning process. Furthermore, this model suggests that the cerebellum performs the inverse dynamics computation based on sensorimotor predictions. This highlights the importance of sensorimotor predictions of gravitational torques acting on upper limb movements performed in the gravitational field

The Inactivation Principle: Mathematical Solutions Minimizing the Absolute Work and Biological Implications for the Planning of Arm Movements

An important question in the literature focusing on motor control is to determine which laws drive biological limb movements. This question has prompted numerous investigations analyzing arm movements in both humans and monkeys. Many theories assume that among all possible movements the one actually performed satisfies an optimality criterion. In the framework of optimal control theory, a first approach is to choose a cost function and test whether the proposed model fits with experimental data. A second approach (generally considered as the more difficult) is to infer the cost function from behavioral data. The cost proposed here includes a term called the absolute work of forces, reflecting the mechanical energy expenditure. Contrary to most investigations studying optimality principles of arm movements, this model has the particularity of using a cost function that is not smooth. First, a mathematical theory related to both direct and inverse optimal control approaches is presented. The first theoretical result is the Inactivation Principle, according to which minimizing a term similar to the absolute work implies simultaneous inactivation of agonistic and antagonistic muscles acting on a single joint, near the time of peak velocity. The second theoretical result is that, conversely, the presence of non-smoothness in the cost function is a necessary condition for the existence of such inactivation. Second, during an experimental study, participants were asked to perform fast vertical arm movements with one, two, and three degrees of freedom. Observed trajectories, velocity profiles, and final postures were accurately simulated by the model. In accordance, electromyographic signals showed brief simultaneous inactivation of opposing muscles during movements. Thus, assuming that human movements are optimal with respect to a certain integral cost, the minimization of an absolute-work-like cost is supported by experimental observations. Such types of optimality criteria may be applied to a large range of biological movements

Modélisation des voies réflexes et cérébelleuses, permettant le calcul des fonctions inverses (application à la commande d'un actionneur à deux muscles pneumatiques)

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Modèle des voies réflexes et cérébelleuses, permettant le calcul de fonctions inverses et la commande d'un segment mécanique mobile

14/07/2004La commande et le contrôle des mouvements des membres par les voies cérébelleuses et réflexes sont modélisées au moyen d'un circuit dont la structure est déduite de contraintes fonctionnelles. 1/ La première contrainte est que les mouvements rapides des membres doivent être précis, quoiqu'ils ne puissent pas être commandés en boucle fermée au moyen de signaux sensoriels. Les voies qui préparent les ordres moteurs doivent donc contenir des fonctions inverses approximées des fonctions biomécaniques du membre et des muscles. Or une telle fonction peut être calculée au moyen de boucles de rétroaction parallèles, dont la structure est comparable à l'anatomie des voies cérébelleuses. L'une des boucles contient un élément capable de prévoir les conséquences motrices des ordres moteurs, qui peut être modélisé par un réseau de neurones formels dont la connectivité est semblable à celle du Cortex Cérébelleux. De tels réseaux apprennent les fonctions biomécaniques directes des membres et des muscles lors d'un apprentissage, supervisé par des signaux professeurs calculés à partir des erreurs motrices. Après chaque mouvement, un signal professeur codant l'erreur est envoyé aux sites d'apprentissage, ce qui est comparable à l'envoi des signaux issus de l'Olive Inférieure, via les fibres grimpantes, aux sites d'apprentissage du Cervelet. Des règles de gradient classiques, déduites par un calcul différentiel, reproduisent la dépression à long terme (LTD) qui a lieu dans l'arborisation dendritique des cellules de Purkinje. 2/ La seconde contrainte est que les réflexes ne doivent pas contrecarrer les mouvements volontaires, mais rester à chaque instant prêts à s'opposer à des perturbations. Pour satisfaire à cette contrainte, des copies efférentes des ordres moteurs sont classiquement envoyées aux interneurones des réflexes et des réactions sensori-motrices, où elles annulent les signaux dûs aux mouvements volontaires. Après apprentissage, le modèle peut commander précisément, en vitesse et en position, les mouvements angulaires d'une tringle mobile tournant autour d'un axe vertical, et mue par deux muscles artificiels antagonistes, des muscles pneumatiques de McKibben. Des réflexes comparables aux réflexes myotatique et tendineux, ainsi que des réactions stabilisantes comparables aux réactions sensorimotrices cérébelleuses, réduisent efficacement les effets des perturbations externes. La raideur de l'articulation est déterminée par ces boucles de rétroaction, comme dans le "modèle lambda" de point d'équilibre. Ces résultats établissent un lien entre ce modèle comportemental et l'organisation anatomique : les gains des réflexes et des réactions sensori-motrices déterminent la raideur de l'articulation, et ainsi la pente de la « caractéristique invariante », tandis que les copies d'efférences fixent le « seuil du réflexe d'étirement ». Des lois mathématiques et physiques permettent ainsi de déduire la connectivité anatomique des voies cérébelleuses, et expliquent la raison d'être de la nature inhibitrice des cellules de Purkinje. Cette organisation fonctionnelle assure le calcul de fonctions inverses approximées, après apprentissage de fonctions directes, et permet de coordonner les ordre moteurs volontaires et réflexes

Daily modulation of the speed–accuracy trade-off

International audienceGoal-oriented arm movements are characterized by a balance between speed and accuracy. The relation between speed and accuracy has been formalized by Fitts’ law and predicts a linear increase in movement duration with task constraints. Up to now this relation has been investigated on a short-time scale only, that is during a single experimental session, although chronobiological studies report that the motor system is shaped by circadian rhythms. Here, we examine whether the speed–accuracy trade-off could vary during the day. Healthy adults carried out arm-pointing movements as accurately and fast as possible toward targets of different sizes at various hours of the day, and variations in Fitts’ law parameters were scrutinized. To investigate whether the potential modulation of the speed–accuracy trade-off has peripheral and/or central origins, a motor imagery paradigm was used as well. Results indicated a daily (circadian-like) variation for the durations of both executed and mentally simulated movements, in strictly controlled accuracy conditions. While Fitts’ law was held for the whole sessions of the day, the slope of the relation between movement duration and task difficulty expressed a clear modulation, with the lowest values in the afternoon. This variation of the speed–accuracy trade-off in executed and mental movements suggests that, beyond execution parameters, motor planning mechanisms are modulated during the day. Daily update of forward models is discussed as a potential mechanism

Fuzzy neuronal model of motor control inspired by cerebellar pathways to online and gradually learn inverse biomechanical functions in the presence of delay

International audienceContrary to forward biomechanical functions, which are deterministic, inverse biomechanical functions are generally not. Calculating an inverse biomechanical function is an ill-posed problem, which has no unique solution for a manipulator with several degrees of freedom. Studies of the command and control of biological movements suggest that the cerebellum takes part in the computation of approximate inverse functions, and this ability can control fast movements by predicting the consequence of current motor command. Limb movements toward a goal are defined as fast if they last less than the total duration of the processing and transmission delays in the motor and sensory pathways. Because of these delays, fast movements cannot be continuously controlled in a closed loop by use of sensory signals. Thus, fast movements must be controlled by some open loop controller, of which cerebellar pathways constitute an important part. This article presents a system-level fuzzy neuronal motor control circuit, inspired by the cerebellar pathways. The cerebellar cortex (CC) is assumed to embed internal models of the biomechanical functions of the limb segments. Such neural models are able to predict the consequences of motor commands and issue predictive signals encoding movement variables, which are sent to the controller via internal feedback loops. Differences between desired and expected values of variables of movements are calculated in the deep cerebellar nuclei (DCN). After motor learning, the whole circuit can approximate the inverse function of the biomechanical function of a limb and acts as a controller. In this research, internal models of direct biomechanical functions are learned and embedded in the connectivity of the cerebellar pathways. Two fuzzy neural networks represent the two parts of the cerebellum, and an online gradual learning drives the acquisition of the internal models in CC and the controlling rules in DCN. As during real learning, exercise and repetition increase skill and speed. The learning procedure is started by a simple and slow movement, controlled in the presence of delays by a simple closed loop controller comparable to the spinal reflexes. The speed of the movements is then increased gradually, and output error signals are used to compute teaching signals and drive learning. Repetition of movements at each speed level allows to properly set the two neural networks, and progressively learn the movement. Finally, conditions of stability of the proposed model as an inverter are identified. Next, the control of a single segment arm, moved by two muscles, is simulated. After proper setting by motor learning, the circuit is able to reject perturbations

A possible correlation between the basal ganglia motor function and the inverse kinematics calculation

International audienceThe main hypothesis of this study, based on experimental data showing the relations between the BG activities and kinematic variables, is that BG are involved in computing inverse kinematics (IK) as a part of planning and decision-making. Indeed, it is assumed that based on the desired kinematic variables (such as velocity) of a limb in the workspace, angular kinematic variables in the joint configuration space are calculated. Therefore, in this paper, a system-level computational model of BG is proposed based on geometrical rules, which is able to compute IK. Next, the functionality of each part in the presented model is interpreted as a function of a nucleus or a pathway of BG. Moreover, to overcome existing redundancy in possible trajectories, an optimization problem minimizing energy consumption is defined and solved to select an optimal movement trajectory among an infinite number of possible ones. The validity of the model is checked by simulating it to control a three-segment manipulator with rotational joints in a plane. The performance of the model is studied for different types of movement including different reaching movements, a continuous circular movement and a sequence of tracking movements. Furthermore, to demonstrate the physiological similarity of the presented model to the BG structure, the neuronal activity of each part of the model considered as a BG nucleus is verified. Some changes in model parameters, inspired by the dopamine deficiency, also allow simulating some symptoms of Parkinson's disease such as bradykinesia and akinesia

A system-level mathematical model of Basal Ganglia motor-circuit for kinematic planning of arm movements

International audienceIn this paper, a novel system-level mathematical model of the Basal Ganglia (BG) for kinematic planning, is proposed. An arm composed of several segments presents a geometric redundancy. Thus, selecting one trajectory among an infinite number of possible ones requires overcoming redundancy, according to some kinds of optimization. Solving this optimization is assumed to be the function of BG in planning. In the proposed model, first, a mathematical solution of kinematic planning is proposed for movements of a redundant arm in a plane, based on minimizing energy consumption. Next, the function of each part in the model is interpreted as a possible role of a nucleus of BG. Since the kinematic variables are considered as vectors, the proposed model is presented based on the vector calculus. This vector model predicts different neuronal populations in BG which is in accordance with some recent experimental studies. According to the proposed model, the function of the direct pathway is to calculate the necessary rotation of each joint, and the function of the indirect pathway is to control each joint rotation considering the movement of the other joints. In the proposed model, the local feedback loop between Subthalamic Nucleus and Globus Pallidus externus is interpreted as a local memory to store the previous amounts of movements of the other joints, which are utilized by the indirect pathway. In this model, activities of dopaminergic neurons would encode, at short-term, the error between the desired and actual positions of the end-effector. The short-term modulating effect of dopamine on Striatum is also modeled as cross product. The model is simulated to generate the commands of a redundant manipulator. The performance of the model is studied for different reaching movements between 8 points in a plane. Finally, some symptoms of Parkinson's disease such as bradykinesia and akinesia are simulated by modifying the model parameters, inspired by the dopamine depletion