Elsevier

Neural Networks

Volume 11, Issues 7–8, October–November 1998, Pages 1205-1218
Neural Networks

1998 Special Issue
Pseudo-inverse control in biological systems: a learning mechanism for fixation stability

https://doi.org/10.1016/S0893-6080(98)00072-0Get rights and content

Abstract

The problem of redundancy in motor control is common to both robotics and biology. Pseudo-inverse control has been proposed as a solution in robotics and appears to be used by the oculomotor system for eye position. Learning mechanisms for implementing pseudo-inverse control using a distributed system of ocular motor units were investigated by modelling integrator calibration for horizontal eye movements. Ocular motoneuron (OMN) input weights were adjusted with a gradient-descent learning rule, using a retinal-slip estimate as an error signal. Firing-rate threshold only became related to motor-unit strength when a noise term was added to OMN firing rates. The learning rule suppressed those units making the largest contribution to the noise-related error, causing the strongest units to have the highest thresholds (size principle). Because the size principle and pseudo-inverse control are related, the trained system approximated pseudo-inverse control over the central ±35° of the oculomotor range.

Introduction

Although artificial and biological motor control systems differ in obvious ways, the fundamental problems they are required to solve are often similar. One way of expressing this distinction is to say that the tasks of motor control need to be understood at two levels (cf. Marr, 1982): one is the abstract level of computational theory, the other the hardware level of its implementation. An implication of this view is that insights at the computational level which arise in one domain, be it robotics or neuroscience, may be transferable to the other. We report here an attempt to transfer computational ideas from robotics to neuroscience for a specific problem in motor control, namely that of controlling a redundant manipulator. The target domain is the control of eye position.

One reason for choosing this example is that redundancy raises a fundamental issue, namely optimisation. The controller of a redundant system has in effect to choose one of a possibly infinite number of solutions. This difficulty can be turned to advantage if evolution, like human designers, chooses solutions that enable the system to optimise some additional constraint (for example, minimum energy use). The crucial question is then how neural mechanisms are able to achieve optimal solutions.

Robinson has eloquently argued the case for choosing the oculomotor system to investigate general issues in motor control (e.g., Robinson, 1986). Reasons include both simplicity (a single joint, fixed load) and richness of anatomical and electrophysiological data. Moreover, these data indicate that the peripheral oculomotor control system is organised conveniently into separate modules for the control of different types of eye movement. The present study considers perhaps the simplest aspect of oculomotor control, namely the maintenance of steady eye position in the absence of external disturbances such as head movements. This is an important skill for the processing of visual images, and one upon which all the other eye-movement control systems depend. Its simplicity is helpful for modelling, in that dynamic factors can largely be ignored. This promotes concentration on the difficult task of applying computational concepts to complicated physiological data.

One particularly important complicating feature of oculomotor control is that its output is multidimensional. Each eye muscle is driven not by a single controller, but by several thousand. Like other muscles, the extraocular muscles (EOMs) are made up of motor units, where a motor unit consists of a motoneuron and the muscle fibres that it innervates. The multiplicity of motor units can be considered as an implementation-level feature that sharply differentiates biological and artificial motor-control systems. However, the results of the present study suggest that this idiosyncratic organisation might be exploited by the oculomotor control system to achieve computational-level optimisation.

Section snippets

Pseudo-inverse control in robotics

The redundancy problem in robotics has usually arisen in the context of controlling the position or velocity of a device located at the end of a multijoint arm (e.g., Snyder, 1985). If the arm has enough joints, it can achieve a given end-effector location by more than one configuration. This is expressed formally in the equations for the forward and inverse kinematics of the arm. The forward kinematics is given by Eq. (1).x=f(θ)This equation gives the location of the end effector (an m

Pseudo-inverse control of eye position

The redundancy problem for eye position arises because there are six extraocular muscles (EOMs) and only three rotational degrees of freedom for the eyeball. The Moore–Penrose Generalised Inverse (MPGI) seems first to have been applied to the oculomotor system by Pellionisz (Ostriker et al., 1985; Pellionisz, 1985), although in the context of a general tensorial approach which has aroused considerable controversy. Subsequently, the MPGI was used by Daunicht (Daunicht, 1988) for the linear case,

Learning pseudo-inverse control

Two ideas proved helpful for trying to answer this question. The first is that the oculomotor system adopts familiar principles of control engineering with respect to eye position, in that it appears to use a form of PID (proportional integral derivative) control. In particular, it provides the steady-state signal needed to balance the passive elastic load exerted by the EOMs and the orbital tissue when the eye is not in the primary position (cf. Eq. (8)). As Robinson has argued, this

Structure of model

The structure of the model is shown in Fig. 2. Each of the two horizontal rectus muscles is represented by a set of motor units, as shown for a single muscle in Fig. 1. The OMN pool for each muscle is driven by a set of weighted connections from two premotor units. One premotor unit conveys an excitatory drive which increases as the eye moves in the pulling direction of the muscle. The second premotor unit provides an inhibitory drive which decreases as the eye moves in the pulling direction of

Unmodified learning rule

The main result obtained by training the model shown in Fig. 2 with the learning rule of Eq. (22)was that the initial conditions determined whether the system manifested pseudo-inverse control after training.

The first example to be described is of initial conditions that did not produce pseudo-inverse control. The starting negative weights on the OMNs were chosen randomly with a mean value of −0.5, giving the distribution of OMN firing-rate thresholds illustrated in Fig. 3A (PRE condition). The

Discussion

Pseudo-inverse control has been proposed as a method of solving the redundancy problem in robotics. The main finding of the present study was that an approximation to pseudo-inverse control for horizontal eye position could be learnt by a distributed system of ocular motor units, using a method similar to that proposed for calibration of the oculomotor velocity-to-position integrator (e.g., Arnold and Robinson, 1991, Arnold and Robinson, 1997). Pseudo-inverse control was attained when the

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