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

1998 Special Issue

Pseudo-inverse control in biological systems: a learning mechanism forfixation stability

Paul Dean*, John PorrillDepartment of Psychology, University of Sheffield, Western Bank, Sheffield S10 2TP, UK

Received and accepted 5 May 1998

Abstract

The problem of redundancy in motor control is common to both robotics and biology. Pseudo-inverse control has been proposed as asolution 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 eyemovements. Ocular motoneuron (OMN) input weights were adjusted with a gradient-descent learning rule, using a retinal-slip estimate as anerror signal. Firing-rate threshold only became related to motor-unit strength when a noise term was added to OMN firing rates. The learningrule 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 controlover the central6 358 of the oculomotor range.q 1998 Elsevier Science Ltd. All rights reserved.

Keywords:Recruitment; Redundancy; Optimisation; Generalised inverse; Oculomotor integrator; Distributed model; Gradient descent; Sizeprinciple

1. Introduction

Although artificial and biological motor control systemsdiffer in obvious ways, the fundamental problems they arerequired to solve are often similar. One way of expressingthis distinction is to say that the tasks of motor control needto be understood at two levels (cf. Marr, 1982): one is theabstract level of computational theory, the other the hard-ware level of its implementation. An implication of thisview is that insights at the computational level which arisein one domain, be it robotics or neuroscience, may be trans-ferable to the other. We report here an attempt to transfercomputational ideas from robotics to neuroscience for aspecific problem in motor control, namely that of control-ling a redundant manipulator. The target domain is thecontrol of eye position.

One reason for choosing this example is that redundancyraises a fundamental issue, namely optimisation. The con-troller of a redundant system has in effect to choose one of apossibly infinite number of solutions. This difficulty can beturned 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 areable to achieve optimal solutions.

Robinson has eloquently argued the case for choosing theoculomotor system to investigate general issues in motorcontrol (e.g., Robinson, 1986). Reasons include both sim-plicity (a single joint, fixed load) and richness of anatomicaland electrophysiological data. Moreover, these data indicatethat the peripheral oculomotor control system is organisedconveniently into separate modules for the control of differ-ent types of eye movement. The present study considersperhaps the simplest aspect of oculomotor control, namelythe maintenance of steady eye position in the absence ofexternal disturbances such as head movements. This is animportant skill for the processing of visual images, and oneupon which all the other eye-movement control systemsdepend. Its simplicity is helpful for modelling, in thatdynamic factors can largely be ignored. This promotes con-centration on the difficult task of applying computationalconcepts to complicated physiological data.

One particularly important complicating feature ofoculomotor control is that its output is multidimensional.Each eye muscle is driven not by a single controller, butby several thousand. Like other muscles, the extraocularmuscles (EOMs) are made up of motor units, where a

* Requests for reprints should be sent to Paul Dean. Tel.: +44-114-222-6521; E-mail: [email protected]

0893–6080/98/$19.00q 1998 Elsevier Science Ltd. All rights reserved.PII: S0893-6080(98)00072-0

Neural Networks 11 (1998) 1205–1218PERGAMON

NeuralNetworks

motor unit consists of a motoneuron and the muscle fibresthat it innervates. The multiplicity of motor units can beconsidered as an implementation-level feature that sharplydifferentiates biological and artificial motor-controlsystems. However, the results of the present study suggestthat this idiosyncratic organisation might be exploited bythe oculomotor control system to achieve computational-level optimisation.

2. Pseudo-inverse control in robotics

The redundancy problem in robotics has usually arisen inthe context of controlling the position or velocity of a devicelocated at the end of a multijoint arm (e.g., Snyder, 1985). Ifthe arm has enough joints, it can achieve a given end-effector location by more than one configuration. This isexpressed formally in the equations for the forward andinverse kinematics of the arm. The forward kinematics isgiven by Eq. (1).

x¼ f (v) (1)

This equation gives the location of the end effector (anm-dimensional vectorx) for a given configuration of thearm joints (ann-dimensional vectorv, n $ m). Since thefunction that relates the two vectors is usually non-linear,the problem is often linearised by considering infinitesimaldisplacements (Eq. (2)).

dx¼ Jdv (2)

This shows the change in end-effector positiondx producedby a small change in joint anglesdv, related by them 3 nJacobian matrixJ in which:

Ji,k ¼]xi

]vk

From the point of view of controlling the arm, the equationof interest is the one that gives the appropriate change injoint angles for a desired change in end-effector location.This is the inverse kinematics equation, given in Eq. (3).

dv ¼ J¹ 1dx (3)

The redundancy problem arises whenm , n, so that thereare many possible changes in joint angles that will produce agiven change in the position of the end effector. In thesecircumstances the matrixJ does not have a true inverse; i.e.,J¹1 does not exist.

One solution proposed for this problem in robotics is touse the pseudo-inverseJ# of the matrixJ (e.g., Hollerbachand Suh, 1985; Klein and Huang, 1983; Whitney, 1969),where:

J ¼ JT JJTÿ �¹ 1(4)

The pseudo-inverse (also termed the Moore–PenroseGeneralised Inverse) has the important property of provid-ing a minimum-norm solution, which in this case means that

the sum of squares of thedv is is minimised. Thus, if the sumsquared changes in joint angle were thought to be a measureof effort, the use of pseudo-inverse control would minimiseeffort.

3. Pseudo-inverse control of eye position

The redundancy problem for eye position arises becausethere are six extraocular muscles (EOMs) and only threerotational degrees of freedom for the eyeball. The Moore–Penrose Generalised Inverse (MPGI) seems first to havebeen applied to the oculomotor system by Pellionisz(Ostriker et al., 1985; Pellionisz, 1985), although in thecontext of a general tensorial approach that has arousedconsiderable controversy. Subsequently the MPGI wasused by Daunicht (Daunicht, 1988) for the linear case, andlater for the non-linear case (Daunicht, 1991). The threebasic equations are given in Eqs. (5)–(7):

Mdf ¼ Lde (5)

df ¼ Zdmþ Adp (6)

dp¼ ¹ MTde (7)

Eq. (5) is the torque-balance equation for the eye. It showsthe relationship between a small change in EOM torqueproduced by a change in muscle forcesdf (a six-dimensionalvector) and the resultant small rotationde (a three-dimensional vector, expressed in Cartesian coordinates),from a given stable position of the eyef. The 33 6 matrixM is derived from the geometry of the eye muscles: itscolumns represent the six unit directions of rotation thatwould be produced by the action of an individual EOM.The 3 3 3 matrix L is the load (elasticity) matrix for thepassive orbital tissues. BothM andL vary with f. Eq. (5)states that the eye reaches a new stable position when thechange in muscle-produced torque is exactly balanced bythe change in elastic load. Eq. (6) reflects the fact that thechange in muscle forces has an active and a passive compo-nent. The active componentZdm is produced by the changein muscle commanddmacting through the muscle strengthsZ: dm is a six-dimensional vector andZ a diagonal 63 6matrix. The passive componentAdp is produced by thelength changesdp combined with the stiffness matrixA:again,dp is six-dimensional andA a diagonal 63 6 matrix.Finally, Eq. (7) shows the geometrical relationship betweenthe length changesdp and the eye rotationde.

The three equations can be combined to give the forwardkinematic equation for static eye-position control (Eq. (8)).

de¼ L þ MAMTÿ �¹ 1MZdm (8)

The change in commanddm produces a change in torqueMZdm, which is balanced by the combined elastic load ofthe EOMs and the orbital tissuesL þ MAMT. This equation

1206 P. Dean, J. Porrill / Neural Networks 11 (1998) 1205–1218

can be thought of as the EOM counterpart to Eq. (1) for theredundant robot manipulator. The inverse kinematicsequation for the eye, using the pseudo-inverse, is given byEq. (9):

dm¼ (MZ)# L þ MAMTÿ �de (9)

where (MZ)# is the pseudo-inverse of the 33 6 matrixMZ.Use of this pseudo-inverse selects, from all the possibledmsthat would produce the desiredde, the one that minimisesthe sum of squared changes in motor command. If thisquantity is regarded as constituting ‘effort’, use of thepseudo-inverse is an example of minimum-effort control.

We have recently attempted to test Daunicht’s proposalby comparing its prediction with experimental data for thesimplified one-dimensional case of horizontal eye position(Dean et al., 1996). In this simplified case, the inverse kine-matics Eq. (9) reduces to Eq. (10):

dm1 ¼z1

z21 þ z2

2

b1 þ b2 þ Lÿ �

df (10)

dm2 ¼¹ z2

z21 þ z2

2

b1 þ b2 þ Lÿ �

df

wheredm1 anddm2 are the changes in command to the twohorizontal eye muscles (lateral and medial rectus) requiredto produce the desired change in horizontal eye positiondf.The z terms are the diagonal elements of the two-dimensional matrixZ, representing muscle strength. Thebterms represent muscle stiffness andL is the elastic load fororbital tissue for horizontal stretch. All of these terms arefunctions of horizontal eye positionf, and are positive.

Eq. (10) predicts the familiar reciprocal innervation, inthat dm1 and dm2 are of opposite sign, but also gives thequantitative relationship between them, namely Eq. (11).

dm1

dm2¼ ¹

z1

z2(11)

A key issue in testing Daunicht’s prediction is therefore theinterpretation of the termsdm and z. It is here that theimportant implementational-level point, referred to inSection 1, makes its appearance. EOMs consist of manymotor units, where a motor unit refers to the muscle fibrescontrolled by an individual ocular motoneuron (OMN). Thisis shown schematically in Fig. 1. A set ofN ocular moto-neurons (whereN would be about 3000 for the human lateralrectus muscle; references in Eggers, 1988) controlsN motorunits. When all the EOMs are functioning normally, thefiring rates of the OMNs vary with eye position in the rele-vant direction (horizontal in the present case), as shown byEq. (12).

FRi ¼ k1 f ¹ vi

ÿ �for f . vi

¼ 0 otherwise(12)

The firing rateFRi of the ith OMN is linearly related to eyepositionf with slopeKi, provided the eye position is greaterthan the OMN’s threshold,v i (e.g., Keller, 1981). Thresh-olds range from more than 508 in the OFF direction of themuscle that is controlled by the OMN pool, to about 258 inits ON direction. Thus, motor units are recruited over a 758or more range of eye positions. This recruitment verymarkedly affects the size of motor-command changes, if

Fig. 1. Highly simplified diagram of motor units in a horizontal rectus muscle. Premotor commands drive ocular motoneurons (OMNs 1 toN). The ocularmotoneurons have a firing-rate thresholdv and slopeK. The firing ratesFR of the motoneurons drive their associated muscle units, each of which develops aforce that is a function ofFR, muscle lengthf and the unit’s strengthx. When summed these unit forces give the overall force developed by the muscle.(Adapted with permission from Dean, 1996.)

1207P. Dean, J. Porrill / Neural Networks 11 (1998) 1205–1218

these are interpreted (following Daunicht, 1988, 1991) aschanges in the summed firing rates of the relevant OMNs(Eq. (13)):

dm(f) ¼ d∑Ni ¼ 0

FRi

!¼

∑j

i ¼ 0Kidv (13)

wherej OMNs have been recruited at eye positionf. Thisinterpretation ofdm as a summed firing-rate gradient is adeparture from purely geometric approaches, and allowsdirect comparison with neurophysiological data.

Whether recruitment also affects muscle strengthz(Eq. (11)) has not been established directly. If the termdpin the one-dimensional version of Eq. (6) is set to 0,dfbecomes the change in force measured isometrically, sothat:

df (f) ¼ z(f)·dm(f) (14)

where all quantities are measured at eye positionf. Giventhe interpretation ofdmshown in Eq. (13), a simple-mindedand approximate interpretation ofz emerges from Fig. 1:

z(f) ¼df (f)dm(f)

<

∑i#j

xiKi∑i#j

Ki

(15)

wherexi is the strength of theith motor unit, defined as theincrease in isometric force produced by a unit increment inthe firing rate of its parent OMN (Dean, 1996; Dean et al.,1996), andj units are recruited at eye positionf. z(f) is thusa weighted mean of the strengths of the individual motorunits recruited at eye positionf. However, for the purposesof testing the predictions of pseudo-inverse control, a majordifficulty with Eq. (15) is that there are no direct measure-ments ofxi in relation tov i. It was therefore found necessaryto bypassz in testing the predictions. Eqs. (11) and (14)were combined to give Eq. (16):

df1df2

¼ ¹dFR1

dFR2

� �2

(16)

where the subscripts 1 and 2 refer to agonist and antagonistmuscles. This equation states the minimum-effort relation-ship, for a given horizontal eye positionf, between theincrements in summed firing rates for the two OMN poolsassociated with a change in eye positiondf and the corre-sponding changes in isometric force in the two horizontalrectus muscles.

When data for OMN firing rates taken from the review ofVan Gisbergen and Van Opstal (1989) were compared withdata for EOM forces from the model of Robinson (1975),the relationship shown in Eq. (16) was approximated overan oculomotor range of6 308 (Dean et al., 1996). The sizeof the range over which the approximation held dependedupon two main factors: the nature of the assumptions madeabout EOM behaviour for eye positions outside6 308 andthe source of the sample of OMN firing rates. However,

detailed consideration of currently available dataconfirmed 6 308 as the best estimate for the horizontaloculomotor range over which minimum-effort control, inthe sense defined above, is approximated. This findingraises an immediate question: how could the oculomotorsystem implement such an apparently complex mode ofcontrol?

4. Learning pseudo-inverse control

Two ideas proved helpful for trying to answer thisquestion. The first is that the oculomotor system adoptsfamiliar principles of control engineering with respect toeye position, in that it appears to use a form of PID (propor-tional integral derivative) control. In particular, it providesthe steady-state signal needed to balance the passive elasticload exerted by the EOMs and the orbital tissue when theeye is not in the primary position (cf. Eq. (8)). As Robinsonhas argued, this steady-state signal is derived from thevelocity signal that drives the eye to the eccentric positionby a process analogous to mathematical integration (forreview see, e.g., Robinson, 1989). The second idea is thatbiological control systems cannot be pre-programmed to therequisite degree of accuracy, and in any case such a pro-cedure could make no allowance for subsequent changes inthe system resulting from development, exercise, ageing ordamage. They are above all adaptive control systems (cf.Grossberg and Kuperstein, 1989), in which learning andcalibration play the central role.

…an analysis of how development and learning leadto and maintain accurate performance characteristicshas, time and again, opened a wide pathway to arapidly expanding understanding of brain mechan-isms. We believe that the unifying power of the theoryis due to the fact that principles of adaptation — suchas the laws regulating development and learning —are fundamental in determining the design ofbehavioural mechanisms (Grossberg and Kuperstein,1989, p. 1).

Applying these ideas to the problem of pseudo-inversecontrol focuses attention on the learning mechanisms thatunderlie integrator development and calibration. Theintegrator for horizontal eye position appears to have twocomponents (evidence recently reviewed by Fukushima andKaneko, 1995). One is located in the nucleus prepositushypoglossi and adjoining medial vestibular nucleus of thehindbrain (hereafter referred to as NPH). The second com-ponent is cerebellar, including the flocculus. The learningmechanisms underlying the calibration of the integratorhave only been partly characterised. It is generally supposedthat retinal image slip is the error signal: one consequence ofintegrator imprecision is drift of the eyeball at the end of asaccade, and artificial post-saccadic drift elicits adaptivechanges in the oculomotor control signal (Optican and


Miles, 1985). The adaptation requires part of the integrator,namely the flocculus (Optican et al., 1986). An additionalconsequence of integrator imprecision is movement of theimage during reflexes intended to stabilise gaze, in par-ticular the vestibulo-ocular reflex (Tiliket et al., 1994).These image movements may also be used to driveadaptive changes, again involving the flocculus (e.g., Ito,1982). However, the nature and location of the plasticsynaptic sites that mediate these behavioural adaptationsremain the subject of vigorous enquiry (e.g., Lisberger,1996).

The present study adopted a modelling approach toinvestigate the relationship between possible learningrules for calibrating the horizontal integrator and pseudo-inverse control. The problem was simplified by ignoring theeffects of head movement, so that the model was requiredonly to eliminate drift after a movement command haddriven the eye to a given location. In principle this learningcould take place in the dark, if drift were signalled by EOMproprioceptors: it is therefore a candidate process for howthe integrator might develop prenatally. The problem wasalso simplified by neglecting the subsequent maintenanceover time of accurate integrator output. The issue ofprolonging the integrator time constant has been extensivelyaddressed elsewhere (for references see Arnold andRobinson, 1997).

5. Structure of model

The structure of the model is shown in Fig. 2. Each of thetwo horizontal rectus muscles is represented by a set ofmotor units, as shown for a single muscle in Fig. 1. TheOMN pool for each muscle is driven by a set of weightedconnections from two premotor units. One premotor unitconveys an excitatory drive which increases as the eyemoves in the pulling direction of the muscle. The secondpremotor unit provides an inhibitory drive which decreasesas the eye moves in the pulling direction of the muscle. Theexcitatory premotor unit represents a lumped version of thePVP (position–vestibular–pause) cell input to the OMNs.The PVP cells are found in the rostral medial vestibularnucleus of primates, and make primarily excitatory con-nections with abducens motoneurons (e.g., Scudder andFuchs, 1992). The inhibitory premotor unit represents theburst-tonic neurons found in the nucleus prepositus hypo-glossi and adjacent medial vestibular nucleus, whose input toabducens motoneurons is thought to be primarily inhibitory(e.g., McFarland and Fuchs, 1992). An important generalpoint is that the eye-position related discharge of horizontalrectus motoneurons is derived in ‘push–pull’ fashion fromexcitatory and inhibitory inputs (cf. Baker et al., 1981).

In outline, the model worked as follows. It was assumedthat a velocity command, for example a saccadic or

Fig. 2. Model of single OMN pool and associated motor units. A notional saccade drives the eye to positionf. The premotor commandsAþ Bf andA¹ Bf arepassed to each of 100 OMNs via the excitatory weightsw1i and inhibitory weightsw2i, respectively. Each OMN has an intrinsic thresholdTi and gainGi. Itsfiring rateFRi is calculated from its premotor synaptic drive and these intrinsic properties.FRi then generates a forceFi in the OMN’s muscle unit, which is afunction of the unit’s strengthxi and the eye positionf. The unit forces are summed to give the total force in the muscle. The antagonist OMN pool is modelledsimilarly, except for the nature of the premotor synaptic weights in the OMNs:w1i is now inhibitory andw2i excitatory.


quick-phase burst, drove the eye to a particular position. Thevelocity command was also assumed to produce position-related signals in the premotor units. In principle these couldarise either from proprioceptive measurements of musclelength, or from the action of the integrator mechanism.The position-related premotor signals acted via weights toproduce a position-related discharge in the OMNs. The dis-charge of each OMN produced a force in the correspondingEOM unit. The sum of the unit forces for each EOM was theoverall force produced by the muscle at that position. If thetwo muscle forces did not balance the passive orbital force,the mismatch was used as an error signal to adjust theweights between the premotor units and the OMNs. Themanner in which this was done constituted the learningrule. The model was trained until its performanceasymptoted, and its behaviour was then examined both forconformity to pseudo-inverse control and for the relation-ships within its motor-unit population between firing-ratethreshold, firing-rate slope and unit strength.

In more detail, a learning trial began with an eye positionf randomly selected from the oculomotor range (usually6 508). The premotor signalsp1 andp2 were calculated as:

p1 ¼ Aþ Bf (17)

p2 ¼ A¹ Bf

with the constantsA and B chosen to ensure positivepvalues over the oculomotor range (see Appendix A).These equations implement the push–pull arrangementreferred to above. The synaptic drivedi to the ith OMNwas given by:

di ¼ p1w1i þ p2w2i (18)

wherew1i andw2i were the synaptic weights on theith OMNfrom the first and second premotor units respectively. Thefiring rate FRi of the ith motoneuron was then calculatedfrom Eq. (19):

FRi ¼ Gi di ¹ Ti

ÿ �for di . Ti

¼ 0 otherwise(19)

The term Ti represents the intrinsic threshold of theithOMN, andGi its intrinsic gain (further details of this wayof modelling OMNs, and of the relationship betweenintrinsic and firing-rate thresholds, are given in Dean,1997). In these simulations the intrinsic gains and thresholdswere set the same for all OMNs (see Appendix A), so thatvariation in their firing-rate thresholds was produced solelyby variation in their synaptic input (cf. Dean, 1997).

The forceFi exerted by theith motor unit in response tothis firing of the parent OMN was computed from threeequations, previously used in a distributed model of a singleEOM in Dean (1996); these are given in Appendix A. Thedistribution of motor-unit strengthsxi used in these calcula-tions was taken from the measurements given in Fig. 2C ofMeredith and Goldberg (1986) of unit tetanic tensions oncat horizontal recti. Two important features of these

measurements are that there is an approximately 50-foldrange of tetanic tensions, and that weak units outnumberstrong ones (,40% of units are 5% or less the strength ofthe strongest unit). Recent measurements of motor-unitstrengths in squirrel monkeys (Goldberg et al., 1997a)suggest that the distribution in primates is similar to thatfound in cat (Goldberg, personal communication).

Once all theFis were computed, they were summed togive the total force that each muscle produced at the eyepositionf. The desired values for the muscle forces werethose that produced no drift of the eyeball, which meant thedifference between them exactly balanced the elastic forceof the orbital tissues. If the forces were not in balance, theresultant force would act on the eyeball to produce drift,which was the source of the error signal. The simplifyingassumption was made that the motion was viscosity-dominated. The rate of drift would then be proportional tothe size of the resultant force at that position, so that theerror term actually used was the force imbalance (Eq. (20)):

E¼ Fdesired¹ Factual (20)

E¼ ¹ F1 þ F2 þ FOT

ÿ �since desiredF ¼ 0

In this equation the termsF1 and F2 represent the twomuscle forces, andFOT the force of the orbital tissues.The sign convention for both eye position and forces wasthat the pulling direction of muscle #1 was positive, so thatthe force exerted by muscle #2 was negative. The orbitaltissue force was always of opposite sign tof. It should benoted that all the terms in Eq. (20) are functions of eyepositionf.

The basic learning rule investigated in the model wasstochastic gradient descent. This rule was chosen partlybecause of its power, but also because of the possibilitythat it might implement pseudo-inverse control by makingweight adjustments on a motor unit that were proportional toits strength (see Section 6). Gradient descent is designed tominimise the sum of the squared errors over the training set,which in this case is the oculomotor range (usually6 508).On each trial the weights from the premotor units to theOMNs were adjusted according to Eq. (21):

dwij ¼ ¹ l·E·]E]wij

(21)

wheredwij is the change in the weight from theith premotorunit to thejth OMN, l is a learning rate,E is the error termfrom Eq. (20) and]E/]wij is the error gradient with respectto wij. In practice this was calculated by applying the chainrule to Eq. (21):

dwij ¼ ¹ l·E·]E]Fk

·]Fk

]FRj·]FRj

]wij

¼ ¹ l·E·sk·]Fk

]FRj·pi whenFRj . FRmin

¼ 0 otherwise

(22)


HereFk refers to the force in thekth muscle, which is themuscle that contains thejth OMN. ]E/]Fk then becomes asign termsk, positive for muscle #1 and negative for muscle#2. The term]Fk/]FRj refers to the change in isometricmuscle force produced at positionf produced by unitchange in firing rate of thejth OMN, and was calculatedby numerical differentiation.FRmin is the lowest firing rateat which a motor unit produces active tension (see AppendixA).

During learning the positive weights to the OMNs,corresponding to the input from PVP cells, were held con-stant. Only the negative weights, corresponding to inputfrom NPH cells, were affected by the learning rule.

Further details of the initial conditions used in thesimulations are given in Appendix A.

6. Results

6.1. Unmodified learning rule

The main result obtained by training the model shown inFig. 2 with the learning rule of Eq. (22) was that the initialconditions determined whether the system manifestedpseudo-inverse control after training.

The first example to be described is of initial conditionsthat did not produce pseudo-inverse control. The startingnegative weights on the OMNs were chosen randomlywith a mean value of ¹ 0.5, giving the distribution ofOMN firing-rate thresholds illustrated in Fig. 3A (PREcondition). The PRE thresholds were scattered within theexperimentally observed range of about¹50 to þ258, andthe motor units produced a force of approximately 14g inthe primary position (eye positionf ¼ 08), which is close tothe 11–12g shown in Fig. 3 of Robinson (1975). There wasno obvious relationship between the firing-rate threshold ofa motor unit and its strength. In contrast, as Fig. 3B (PREcondition) shows, there was a clear relationship between thefiring-rate thresholdv i and slopeKi (Eq. (12)). This relation-ship is broadly similar to that shown in Fig. 1 of VanGisbergen and Van Opstal (1989) for putative OMNs ingeneral, and in Fig. 5 of Fuchs et al. (1988) for identifiedOMNs in the abducens nucleus. Thus, the line of best fit forthe PRE points in Fig. 3B isK ¼ 8.7þ 0.13v (r ¼ 0.95,n ¼

100), compared withK ¼ 8.07þ 0.18v (r ¼ 0.81,n ¼ 81)for the data of Fuchs et al. (1988). Training had no effect onthe relationship between firing-rate threshold and slope (notshown), consistent with its being determined solely by the‘push–pull’ nature of the inputs to the model OMNs.

The efficacy of the training procedure is illustrated inFig. 4A, which shows a close fit between the actualdifference in muscle forces and the desired difference asrepresented by the passive orbital force. However, therelationship between the force-gradient ratio and thefiring-rate gradient ratio in the two muscles predicted bypseudo-inverse control (Eq. (16)) was not observed

(Fig. 4B: the negative sign in the equation is ignored).The reason for this failure is implicit in the POST trainingdistribution of motor-unit thresholds and strengths shown inFig. 3A. As was the case before training, there appears to beno systematic relationship between firing-rate threshold andstrength. This means that whereas the firing-rate gradientnecessarily increases as OMNs are recruited, the musclestrengthz may not increase at all (e.g., if all thexi are setequal in Eq. (15)). The matching between firing-rate

Fig. 3. (A) Plot of motor-unit strength against OMN firing-rate threshold foran OMN pool before (unfilled circles: PRE) and after (filled circles: POST)training with the unmodified gradient-descent learning rule (Eq. (22)),starting with randomised weights. (B) Plot of OMN firing-rate slope againstfiring-rate threshold before training (after training is not shown to avoidclutter).


gradient and muscle strength required by pseudo-inversecontrol (Eq. (11)) is therefore violated.

One further feature of post-training performance can benoted. The force in an individual muscle at the primaryposition, 15.5g, was close to the pretraining value of 14g.Similarly, if the initial values for the inhibitory weightswere randomised around¹0.2, the pretraining value forprimary position force was around 29g compared with 27gafter training (not shown). This is an additional example ofthe dependence of post-training performance on the initialconditions.

In the second example to be illustrated, the initial condi-tions did result in an approximation to pseudo-inversecontrol. This occurred when the size of the initial weightson the OMNs was made proportional to the strength of themotor unit that they controlled, producing the relationship

Fig. 4. (A) Forces in the simulated horizontal rectus muscles after trainingthe model with the unmodified gradient-descent learning rule, starting withrandomised weights. The forces are plotted as a function of eye position, asis the force exerted by the passive orbital tissue (plotted every 58). Com-parison of the orbital tissue force with the difference in force between thelateral and medial rectus muscles [line labelled (LRþ MR)] indicates aclose match after training. The force in an individual muscle at the primaryposition was 15.5g. (B) Comparison of (i) the ratio of isometric forcegradients in the simulated lateral and medial rectus muscles and (ii) thesquare of the ratio of summed firing-rate gradients in the correspondingOMN pools (cf. Eq. (16)). Both quantities are plotted against eye position at58 intervals. They should be equal if the system is using pseudo-inversecontrol.

Fig. 5. (A) Plot of motor-unit strength against OMN firing-rate threshold foran OMN pool before (line: PRE) and after (filled circles: POST) trainingwith the unmodified gradient-descent learning rule, starting with weightsarranged to produce the size principle. (B) Comparison of the ratio ofisometric force gradients in the simulated lateral and medial rectus muscles,and the square of the ratio of summed firing-rate gradients in thecorresponding OMN pools, after the training described in (A).


between firing-rate threshold and unit strength shown inFig. 5A (PRE curve). Training again gave a good fit betweenthe difference in muscle forces and the passive orbital force(not shown: force at primary position 14.8g), but in this casethe relationship between the force-gradient ratio and thefiring-rate gradient ratio in the two muscles was close tothat predicted by pseudo-inverse control (Fig. 5B). Thecrucial difference is in the post-training distribution offiring-rate thresholds (Fig. 5A): now they are largely relatedto the strength of the motor unit, so that the later units to berecruited are stronger than those recruited earlier. In spinalmotor units, where the relationship between recruitment andunit strength has been extensively investigated, this arrange-ment is known as the size principle (e.g., Henneman andMendell, 1981). Its importance for the pseudo-inversecontrol of eye position arises from the nature of musclestrengthz in a distributed system (Eq. (15)). As units arerecruited in order of increasing strength, soz increases. Thisis necessary for the match between muscle strength andfiring-rate gradient required by pseudo-inverse control(Eq. (11)), since the firing-rate gradient necessarilyincreases with recruitment. In addition, because there is afixed relationship between firing-rate slope and threshold forOMNs, the size principle ensures that stronger units havehigher firing-rate slopes, so that their overall contribution isweighted appropriately (cf. Eq. (15)).

In summary, application of the gradient-descent learningrule described by Eq. (22) only resulted in pseudo-inversecontrol when there was a pre-existing relationship betweenthe firing-rate threshold of a motor unit and its strength.Perhaps the most likely basis for such a relation would beif strong units had OMNs with high intrinsic thresholds(Eq. (19)). Current experimental evidence indicates thatthere is an approximately 10-fold range of intrinsicthresholds in cat OMNs (Grantyn and Grantyn, 1978;

Nelson et al., 1986), but that there are at best weakcorrelations between intrinsic OMN properties and motor-unit strength (references in Dean, 1997). A different methodof achieving pseudo-inverse control was thereforeinvestigated.

6.2. Modified learning rule

The problem for pseudo-inverse control shown in Fig. 3 isthe presence of powerful motor units with low firing-ratethresholds. Why might the control system wish to rid itselfof such units? One possible reason would be the existence ofvariability or noise within the system. If OMN firing rateswere to fluctuate, then the force exerted by individual motorunits would also fluctuate. The stronger the units involved,the greater the force fluctuation and hence the greater the

movement of the eye. In a noisy system, therefore, theinterests of image stability would favour the eliminationof strong units if these were not needed to balance thepassive orbital forces.

Electrophysiological recordings provide clear evidencefor the existence of variation in OMN firing rates. In cats,Gomez et al. (1986) found that the standard deviation ofinterspike intervals for abducens motoneurons during fixa-tion was linearly related to the mean value, with slope 4.6–16%. In rhesus monkeys, Goldstein and Robinson (1986)found that the firing rates of abducens neurons immediatelyafter identical saccades varied from trial to trial (standarddeviation about 5%), and then subsequently ‘‘waxed andwaned’’ with a standard deviation of about 10%.

The model was therefore modified to allow simulation ofnoisy OMN firing rates. First, noise was added to the firingrate calculated for each unit from Eq. (19). The additiontook the form of a fixed percentage of the firing rate, eitheradded or subtracted randomly (nFRi). The new firing rateswere then used to derive muscle forces and resultant error asbefore, only now the error had a new noise-related compo-nent En. The learning rule of Eq. (22) was thereforemodified to take account of the new error term. The ideabehind the modification is shown in Eq. (23).

dwij 9 ¼ ¹ l·En·sk·]Fk

]FRj·nFRj (23)

Here an additional noise-related adjustmentdwij 9 is made tothe inhibitory weightwij on thejth OMN. If the sign of thenoise-related increment in firing rate is the same as that ofthe noise-related error, the inhibition on that OMN isincreased: otherwise it is decreased. The stronger themotor unit, the more likely the signs are to be the same.This rule therefore reduces the contribution of strong units.The combined learning rule is given in Eq. (24):

where the weighting factora allows the relative effects ofnoise to be adjusted.

The effects of training with the modified learning rule areshown in Fig. 6. The initial conditions (Fig. 6A: PRE) werethe same as in Fig. 3A; that is, inhibitory weightsrandomised around a mean value of¹ 0.5. After trainingwith the modified learning rule (2% noise:j ¼ 1) therelationship between the force-gradient ratio and thefiring-rate gradient ratio in the two muscles was close tothat predicted by pseudo-inverse control over the range6 358 of eye position (Fig. 6B). This is close to the rangethat emerges from comparison of experimental measure-ments (Dean et al., 1996).

Similar results (not shown) were obtained when the initialweights were randomised around¹ 0.2, unlike the situationwith the unmodified learning rule. Moreover, the basic

dwij ¼ ¹ l· Eþ aEn

ÿ �·sk·

]Fk

]FRj· pi þ a·nFRj

ÿ �for FRj þ nFRj . FRmin

¼ 0 otherwise

(24)


pattern was retained when the strength of all the motor unitswas reduced by 40%; when the noise was varied from 1 to10%; when the]Fk/]FRj term in Eq. (24) was replaced bythe unit’s strengthxj; and when the cubic term in the orbital-force equation was removed. Thus, although systematicsensitivity testing has yet to be carried out, the basic findingappears to be relatively robust.

Fig. 7 shows two additional features of the trained system

whose performance is illustrated in Fig. 6. The size of theweights on the OMNs is plotted against their firing-ratethresholds in Fig. 7A. The plot makes it clear that, in thismodel, high firing-rate thresholds are the results of stronginhibition from the contralateral integrator. The secondfeature (Fig. 7B) is the relationship between firing-ratethreshold and firing-rate slope (cf. Fig. 3B). The broad

Fig. 6. (A) Plot of motor-unit strength against OMN firing-rate threshold foran OMN pool before (unfilled circles) and after (filled circles) training withthe modified gradient-descent learning rule (Eq. (24)), starting withrandomised weights. (B) Comparison of the ratio of isometric forcegradients in the simulated lateral and medial rectus muscles, and the squareof the ratio of summed firing-rate gradients in the corresponding OMNpools, after the training described in (A).

Fig. 7. Properties of the model after training with the modified gradientdescent rule, as shown in Fig. 6. (A) Size of inhibitory weight plottedagainst OMN firing-rate threshold. (B) OMN firing-rate slope plottedagainst firing-rate threshold.


relationship is in line with observation (see Section 6), butsome of the details are not. For example, Fig. 7 shows anOMN with a threshold of aboutþ408 and slope,29 Hz/deg. Experiment suggests upper limits of aboutþ258 and,20 Hz/deg (e.g., Fuchs et al., 1988; Van Gisbergen andVan Opstal, 1989). Nor is the gap in thresholds between¹20 and¹58 in accordance with experimental data. Theabsence of units with thresholds lower than 408, however,is characteristic of some experimental data sets but notothers.

7. Discussion

Pseudo-inverse control has been proposed as a method ofsolving the redundancy problem in robotics. The main find-ing of the present study was that an approximation topseudo-inverse control for horizontal eye position couldbe learnt by a distributed system of ocular motor units,using a method similar to that proposed for calibration ofthe oculomotor velocity-to-position integrator (e.g., Arnoldand Robinson, 1991, 1997). Pseudo-inverse control wasattained when the system obeyed the size principle; i.e.,when stronger motor units had higher firing-ratethresholds. The size principle itself could be achieved intwo different (but not mutually exclusive) ways. One wasto simply build it in independently of training: currentexperimental evidence does not provide strong support forthis option. The second was to add a noise term to OMNfiring rates.

As far as we know, the effects of OMN noise on inte-grator calibration have received little previous attention, andthe possible connections between noise, the size principleand pseudo-inverse control have not been suspected. Itappears that the gradient-descent learning rule deals withnoisy units by suppressing those that make the greatestcontribution to noise-induced retinal slip. Since these arethe strongest units, the effect of the noise term is thatstronger units have higher thresholds. As explained inSection 6, such a relationship is similar to the sizeprinciple observed for skeletal muscles (e.g., Hennemanand Mendell, 1981) and is necessary for the pseudo-inversecontrol of eye position given that firing-rate gradientnecessarily increases with recruitment. Also, because ofthe architecture of the model, the size principle ensuresthat stronger units have higher firing-rate slopes (as isobserved experimentally), so that their overall contributionis weighted in the manner required by pseudo-inversecontrol (cf. Eq. (15)).

Two aspects of these conclusions merit discussion. First,they apply at the level of broad principle rather than ofprecise detail. As the last paragraph of Section 6 madeexplicit, the present model is too simplified to reproducethe behaviour of OMN pools with great precision. Someof the issues which require investigation include thefollowing.

1. The possibility that in some circumstances muscle-unitforces add non-linearly (Goldberg et al., 1997b).

2. Exponential curves for passive whole-muscle tensionversus eye position (Simonsz and Spekreise, 1996).

3. A distributed representation of premotor neurons whichincorporates the observed variability in their behaviourand projections (Cullen et al., 1993; Lisberger et al.,1994; McCrea et al., 1987; McFarland and Fuchs,1992; Ohgaki et al., 1988; Scudder and Fuchs, 1992)and includes the neurons that receive input from theflocculus (‘‘flocculus target neurons’’, Lisberger et al.,1994).

4. Various forms of noise, including noise that is correlatedbetween units (Goldstein and Robinson, 1986). A relatedissue is the existence of distinguishable types of EOMunit (e.g., Spencer and Porter, 1988), in particular themultiply innervated fibre (MIF) apparently specialisedto give fused contractions at low innervation frequencies.It is not clear whether these units are recruited around theprimary position (Robinson, 1978) or far into the OFFdirection of the muscle (Dean, 1996), but in any eventtheir possible ‘low-noise’ behaviour requires exploration.

Despite these caveats, however, the fact remains that therelatively simple model used here was able to reproducepseudo-inverse control while remaining reasonably faithfulto the major features of firing-rate behaviour observedwithin OMN pools.

The second aspect to be discussed concerns the possibleneural implementation of gradient-descent learning. Thereare a number of candidate sites for plasticity in the peripheraloculomotor system.

1. There is evidence for developmental processes in whichthe firing patterns of OMNs influence the characteristicsof their motor units (e.g., Porter et al., 1998). It is pos-sible that these processes, though not directly related togradient descent learning, nonetheless help to providethe matching between OMN firing-rate threshold andslope to motor-unit strength that is required for pseudo-inverse control.

2. There is also evidence for the presence of NMDA recep-tors on the OMNs themselves (e.g., Durand, 1993),which may play a role in sculpting the synaptic driveto OMNs from premotor neurons, at least duringdevelopment (Dean, 1997). (The use of plastic synapticweights on OMNs in the present simple model wasprimarily for modelling convenience.)

3. Arnold and Robinson (1997) were able to model inte-grator calibration using a local, Hebbian-like learningalgorithm to modify synaptic weights between thepremotor neurons that formed part of a recurrentlyconnected neural network.

It may be that all of these sites are involved: investigationof another apparently simple example of motor learning,namely operant conditioning of the tendon jerk, has


revealed a remarkably complex pattern of plasticity occur-ring at multiple sites within the spinal cord (reviewed byWolpaw, 1997).

But perhaps the most interesting candidate site for imple-mentation of gradient-descent learning is the cerebellum. Asmentioned earlier, an intact cerebellar flocculus is requiredfor normal gaze holding (Zee et al., 1981) and for adaptationto artificial image drift (Optican et al., 1986). The flocculusappears to receive the information about retinal slip andOMN discharge (references in Fukushima and Kaneko,1995) that would be necessary to calculate the]Fk/]FRj

term in Eq. (24) in the gradient-descent learning rule. Acerebellar mechanism which could evaluate this quantityas a correlation (between firing-rate fluctuations andimage drift) has been proposed by Sejnowski (1977). Thereason why cerebellar involvement in gradient-descentlearning for eye-position control would be of special interestrelates to the anatomical uniformity of the cerebellum. Suchuniformity suggests that the neuronal machinery capable ofsupplying information for gradient-descent learning contextmight be available in very wide range of contexts. Thispossibility points to a quite general role for the cerebellumin motor-control optimisation.

Appendix A Calculation of force in individual motorunits

The forceFi exerted by theith motor unit in response tothe firing of the parent OMN was computed from threeequations, previously used in a distributed model of a singleEOM in Dean (1996):

Fi ¼ xi·C·k2

f þ ei

ÿ �þ

��k2

4f þ ei

ÿ �þ a2

s24 35 (A1)

ei ¼ emin þ L kemax¹ emin

ÿ �(A2)

Li ¼FRi ¹ FRmin

FRmax¹ FRminfor FRi , FRmax

¼ 0 otherwise

(A3)

Eq. (A1) is a modified form of the equation devised byRobinson (1975, Eq. 39, p. 823) to represent the behaviourof the whole muscle. Tension measured in the whole muscleis a function of muscle length, which is determined by eyeposition f. When the muscle is stretched far enough itbehaves like a spring of constant stiffness,k g/deg, of eyerotation. When it is slack, the tension approaches zero. Thecurvature terma2 ensures that the transition between the twostates is smooth, and the innervation terme determineswhere the transition occurs. As innervation increases, thechange from unstretched to stretched behaviour occurs atshorter muscle lengths. Three adaptations were made to thewhole-muscle equation to fit it to a single motor unit. Theright-hand side was multiplied by a term corresponding to

the strength of the motor unitxi (cf. Fig. 1), and also by acorrection factorC so that the sum of all the units behaved asthe whole muscle when in each case there was no innervation.Thirdly, the innervation termei for the individual motor unitwas derived from the firing rate of theith OMN.

The derivation was carried out in two steps. First, thefiring rate FRi of the ith OMN was constrained to liebetween 0 and 1 by Eq. (A3). In this equation,FRmin repre-sents the firing-frequency threshold at which active muscleforce first appears (set to 0 in the present simulations) andFRmax the frequency at which it saturates (set to 400). Inbetween these two limits, the normalised firing rateL i

varied linearly with actual firing rate. Secondly, thenormalised firing rateL i was used to produce the inner-vation parameterei by Eq. (A2). ForL i ¼ 0 this gives aninnervation parameter ofemin, which corresponds to passive(uninnervated) muscle. ForL i ¼ 1 the innervation para-meter isk·emax, whereemax is the innervation parameter forthe extreme ON position andk is a number greater than 1 toindicate the extra force produced by maximal firing in allmotor units (further details in Dean, 1996).

Appendix B Details of the model

The values ofA andB in Eq. (17) for the premotor inputswere 11 and 0.2074, respectively.

The excitatory weights (Eq. (18)) were set to 17.53, andthe intrinsic gains and thresholds (Eq. (19)) toG ¼ 1 andT ¼ 25.32 for all OMNs. This combination ensures that,without inhibition, all OMNs had firing-rate thresholdv ¼ 608 in the OFF direction and firing-rate slopeK ¼

3.64 (Eq. (12)), so that they would fire at 400 Hz (FRmax,Eq. (A2)) at the end of the oculomotor range (508 in the ONdirection).

Orbital forceFOT (Eq. (20)) was related to eye positionf

by Eq. (A4):

FOT ¼ 0:48f þ 1:563 10¹ 4f3 (A4)

which was taken from Eq. (45) in Robinson (1975).In the whole-muscle version of Eq. (A1), the parameters

k ¼ 0.76g/deg (stiffness) anda ¼ 6.24g (curvature) weretaken from Robinson (1975) and used for both horizontalrecti, assuming them to be of identical length and stiffness(cf. Clement, 1987; Simonsz and Spekreise, 1996). Thepassive curve was determined by the value ofEmin whichwas set to¹20.69, corresponding to a command of 608 inthe OFF position.Emax was set to 197.7, corresponding to amaximum command of 508 in the ON direction. The factorkwas 1.4 (Eq. (A2)).

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