Biologically Plausible Local Learning Rules for 
the Adaptation of the Vestibulo-Ocular Reflex 
Olivier Coenen* Terrence J. Sejnowski 
Computational Neurobiology Laboratory 
Howard Hughes Medical Institute 
The Salk Institute 
P.O.Box 85800 
San Diego, CA 92186-5800 
Stephen G. Lisberger 
Department of Physiology 
W.M. Keck Foundation Center 
for Integrarive Neuroscience 
University of California, 
San Fransisco, CA, 94143 
Abstract 
The vestibulo-ocular reflex (VOR) is a compensatory eye movement that 
stabilizes images on the retina during head turns. Its magnitude, or gain, 
can be modified by visual experience during head movements. Possible 
learning mechanisms for this adaptation have been explored in a model 
of the oculomotor system based on anatomical and physiological con- 
straints. The local correlational learning rules in our model reproduce the 
adaptation and behavior of the VOR under certain parameter conditions. 
From these conditions, predictions for the time course of adaptation at 
the learning sites are made. 
1 INTRODUCTION 
The primate oculomotor system is capable of maintaining the image of an object on the 
fovea even when the head and object are moving simultaneously. The vestibular organs 
provide information about the head velocity with a short delay of 14 ms but visual signals 
from the moving object are relatively slow and can take 100 ms to affect eye movements. 
The gain, G, of the VOR, defined as minus the eye velocity over the head velocity (-6/}), 
can be modified by wearing magnifying or diminishing glasses (figure 1). VOR adaptation, 
absent in the dark, is driven by the combination of image slip on the retina and head turns. 
*University of California, San Diego. Dept. of Physics. La Jolla, CA, 92037. Email address: 
olivier@ helmholtz.sdsc.edu 
961 
962 Coenen, Sejnowski, and Lisberger 
During head turns on the first day of wearing magnifying glasses, the magnified image of 
an object slips on the retina. After a few days of adaptation, the eye velocity and hence the 
gain of the VOR increases to compensate for the image magnification. 
We have constructed a model of the VOR and smooth pursuit systems that uses biologically 
plausible local learning rules that are consistent with anatomical pathways and physiological 
recordings. The learning rules in the model are local in the sense that the adaptation of a 
synapse depends solely on signals that are locally available. A similar model with different 
local learning rules has been recently proposed (Quinn et al., Neuroscience 1992). 
18 
16 
1.4 
1.2 
1.0 
x2.0 
Spectacles 
or 
0.8 
0.6 
x0.5 
Spectacles 
on 
0 1 
Spectacles 
off 
.... 72 + 0 . e-. t} 
Gain = 0.73 + 0.27 (e -o 
x0.5 
Spectacles 
off 
I I I I I I 
2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 
TIME (Days) 
Figure 1: Time course of the adapting VOR and its recovery of gain in monkeys exposed to the long- 
term influence of magnifying (upper curves) and diminishing (lower curves) spectacles. Different 
symbols obtained from different animals, demonstrating the consistency of the adaptive change. From 
Melvill Jones (1991 ), selected from Miles and Eighmy (1980). 
2 THE MODEL 
Feedforward and recurrent models of the VOR have been proposed (Fujita, 1982; Galiana, 
1986; Kawato and Gomi, 1992; Quinn et al., 1992; Arnold and Robinson, 1992; Lisberger 
and Sejnowski, 1992). In this paper we study a static and linear version of a previously 
studied recurrent network model of the VOR and smooth pursuit system (Lisberger, 1992; 
Lisberger and Sejnowski, 1992; Viola, Lisberger and Sejnowski, 1992). The time delays 
and time constants associated with nodes in the network were eliminated so that the time 
course of the VOR plasticity could be more easily analyzed (figure 2). 
The model describes the system ipsilateral to one eye. The visual error, which carries the 
image retinal slip velocity signal, is a measure of the performance of both the VOR and 
smooth pursuit system as well as the main error signal for learning. The value at each node 
represents changes in its firing rate from its resting firing rate. The transformation from the 
rate of firing of premotor signal (N) to eye velocity is represented in the model by a gain 
Biologically Plausible Local Learning Rules for Adaptation of Vestibulo-Ocular Reflex 963 
Gains 
P : Purkinje Cell  Visual error: mossy fibers 
N Vestibular Nucleus ./ - (g /a + ) 
g : Desired gai 
  lnhibRory 
head velociV eye velocity 
 % Visual error: climbin fibers 
(g fi + ) 
Figure 2: Diagram of the VOR and smooth pursuit model. The input and output of the model are, 
respectively, head velocity and eye velocity. The model has three main parts: the node P represents 
an ensemble of Purkinje cells from the ventral parafiocculus of the cerebellum, the node N represents 
an ensemble of fiocculus-target neurons in the vestibular nucleus, and the visual inputs which provide 
the visual error signals in the mossy and climbing fibers. The capital letter gains A and D, multiplying 
the input signals to the nodes, are modified according to their learning rules. The lower case letters 
b, v, and 0 are also multiplicative gains, but remain constant during adaptation. The traces represent 
head and eye velocity modulation in time. The visual error signal in the climbing fibers drives learning 
in node N but does not constitute one of its inputs in the present model. 
of - 1 The gain of the VOR in this model is given by D-A We have not modeled the 
 1-b ' 
neural integrator that converts eye velocity commands to eye position signals that drive the 
moroneurons. 
3 LEARNING RULES 
We have adopted the learning rules proposed by Marr (1969), Albus (1971) and Ito (1970) 
for adaptation in the cerebellum and by Lisberger (1988), Miles and Lisberger (1981) for 
plasticity in the brain stem (figure 3). These are variations of the delta rule and depend on 
an explicit representation of the error signal at the synapses. 
Long term depression at mossy fiber synapses on Purkinje cells has been observed in 
vitro under simultaneous stimulation of climbing fibers and mossy fibers (Ito, Sakurai and 
Tongroach, 1982). In addition, we have included a learning mechanism for potentiation 
of mossy fiber head velocity inputs under concurrent mossy fiber visual and head velocity 
inputs. Although the climbing fiber inputs to the cerebellum were not directly represented 
in this model (figure 2), the image velocity signal carried by the mossy fibers to P was used 
in the model to achieve the same result. 
There is good indirect evidence that learning also occurs in the vestibular nucleus. We 
have adopted the suggestion of Lisberger (1988) that the effectiveness of the head velocity 
input to some neurons in the vestibular nucleus may be modified by head velocity input in 
964 Coenen, Sejnowski, and Lisberger 
Learning ( Input ) ( Error ) 
Rate x Signal x Signal 
Cerebellum (P): 
Vestibular nucleus (N): 
where 
P 
( Head ) ( Mossy fiber 
= r/A X Velocity x Visual signal 
= . x x -v(g + 
= x x -v[(g - + ?] 
( Head ) ( Climbing fiber 
x Velocity x Visual signal 
x ]z x [(1 - q)(g]z + ) - qP] 
x  x [(1 - q)(g-- D)Jz + (1 - 2q)P] 
__ Purkinje 
Signal 
= A-bO-(g-O)vh 
1 -b+v 
Figure 3: Learning rules for the cerebellum and vestibular nucleus. The gains A and D change 
according to the correlation of their input signal and the error signal to the node, as shown for A at 
the top. The parameter q determines the proportion of learning from Purkinje cell inputs compared 
to learning from climbing fiber inputs. When q = 1, only Purkinje cell inputs drive the adaptation at 
node N; if q = 0, learning occurs solely from climbing fiber inputs. 
association with Purkinje cells firing. We have also added adaptation from pairing the head 
velocity input with climbing fiber firing. The relative effectiveness of these two learning 
mechanisms is controlled by the parameter q (figure 3). 
Learning for gain D depends on the interplay between several signals. If the VOR gain is 
too small, a rightward head turn P (positive value for head velocity) results in too small a 
leftward eye turn (a negative value for eye velocity). Consequently, the visual scene appears 
to move to the left (negative image slip). P then fires below its resting level (negative) and 
its inhibitory influence on N decreases so that N increases its firing rate (figure 4 bottom 
left). This corrects the VOR gain and increases gain D according to figure 3. Concurrently, 
the climbing fiber visual signal is above resting firing rate (positive) which also leads to an 
increase in gain D. 
Since the signal passing through gain A has an inhibitoryinfluence via P onto N, decreasing 
gain A has the opposite effect on the eye velocity as decreasing gain D. Hence, if the VOR 
is too small we expect gain A to decrease. This is what happens during the early phase of 
learning (figure 4 top left). 
4 RESULTS 
Finite difference equations of the learning rules were used to calculate changes in gains A 
and D at the end of each cycle during our simulations. A cycle was defined as one biphasic 
Biologically Plausible Local Learning Rules for Adaptation of Vestibulo-Ocular Reflex 965 
Desired gain 9 = 1.6 
Magnitude 
2 
1.75 
1.5 y 
1.2 
1 
0.75 
0.5 
0.25 
0 
G 
D 
A 
20 4O 60 80 
Magnitude 
2 
1.75 
1.5 
1.25 
0.75 
0.5 
O25 
2000 4000 6000 8000 
io6o TM 
A, D & VOR gain G vs time 
Amplitude 
2 
1.5  
1 
0.5 
-0.5 
-1 
N 
40 60 80 100 Time 
Amplitude 
2 
1.5 
1 
0.5 
N 
2000 4000 6000 8000 11100g ime 
P & N responses to a head turn during learning vs time 
Figure 4: Simulation of change in gain from 1.0 to 1.6. Top: Short-term (left) and long-term (right) 
adaptation of the gains A, D and G. Bottom: Changes on two time scales of P and N responses to a 
head turn of amplitude 1 during learning. The parameters were v = 1.0, b = .88, r = n__,x = 10. 
UD ' 
and q = .01. 
head velocity input as shown in figure 2. We assumed that the learning rates were so small 
that the changes in gains, and hence in the node responses, were negligibly small during 
each iteration. This allowed the replacement o.f A(t).and D(t) by their values obtained on 
the previous iteration for the calculations of A and D. The period of the iteration as well 
as the amplitude of the head velocity input were chosen so that the integral of the head 
velocity squared over one iteration equaled 1. 
For the simulations shown in figure 4 the gain G of the VOR increased monotonically from 
1 to reach the desired value 1.6 within 60 time steps. This rapid adaptation was mainly 
due to a rapid decrease in A, as expected from the local learning rule (figure 3), since the 
learning rate r/A was greater than the learning rate r/D. Over a longer time period, learning 
was transferred from A to D: D increased from 1 to reach its final value 1.6 while the VOR 
gain stayed constant. Transfer of learning occurs when P fires in conjunction with a head 
turn. P can have an elevated firing rate even though the visual error signal is zero (that is, 
even if the VOR gain G has reached the desired gain g) because of the difference between 
its two other inputs: the head velocity input through A and the eye velocity feedback input 
through b. It is only when these two inputs become equal in amplitude that P firing goes 
to zero. It can be shown that when learning settles (when D and A equal zero) D = g, 
A = bg, and P = 0. With these values for A and D, the two other inp. uts to P are indeed 
equal in amplitude: one equals Ah, while the other equals b(-1)Dh. During the later 
part of learning, gain A is driven in the opposite direction (increase) than during the earlier 
966 Coenen, Sejnowski, and Lisberger 
part (decrease). This comes from a sign reversal of the visual error input to P. After the 
first 60 time steps, the gain has reached the desired gain due to a rapid decrease in A, this 
means that any subsequent increase in D, due to transfer of learning as explained above, 
will cause the gain of the VOR G to become larger than the desired gain g, hence the visual 
error changes sign. In order to compensate for this small error, gain A increases promptly, 
keeping G very close to the desired gain. This process goes on until A and D reach their 
equilibrium values stated above. 
The short and long-term changes in P and N responses to a velocity step are also shown. 
As the firing of P decreased with the adaptation of A, the firing rate of N increased to the 
right level. 
5 OVERSHOOT OF THE VOR GAIN G 
In this section we show that for some ranges of the learning parameters, the gain G in 
the model overshoots the desired value /. Since an overshoot is not observed in animals 
(figure 1), this provides constraints on the parameters. The parameter q in the learning rule 
for the vestibular nucleus (node N, gain D), determines the proportion of learning from 
Purkinje cell inputs compared to learning from climbing fiber inputs. When q = 1, only 
Purkinje cell inputs drive the adaptation at node N; if q = 0, learning at N occurs solely 
from climbing fiber inputs. These two inputs have quite different effects on learning as 
shown in figure 5. Asymptotically, P goes to 0, and D goes to /if q = 1; and P can only 
differ from 0 if q = 0. The gain has an overshoot for any value of q different than 0, as 
shown in figure 6. Nevertheless, its amplitude is only significant for a limited extent in the 
parameter space of q and r (graph of figure 6). The overshoot is reduced with a smaller q 
and a larger r. One possibility is that q is chosen close to 0 and r >> 1, that is r/A >> r/D. 
These conditions were used to choose parameter values in the simulations (figure 4). 
6 DISCUSSION AND CONCLUSION 
The VOR model analyzed here is a static model without time delays and multiple time 
scales. We are currently studying how these factors affect the time course of learning in a 
dynamical model of the VOR and smooth pursuit. 
In our model, learning occurs in the dark if P  0, which has not been observed in animals. 
One way to avoid learning in the dark when P is firing would be to gate the learning by a 
visual input, such as that provided by climbing fibers. 
The responses of vestibular afferents to head motion can be classified into two categories: 
phase-tonic and tonic. In this model, only the tonic afferents were represented. Both 
afferent types encode head velocity, while the phasic-tonic responds to head acceleration as 
well. The steady state VOR gain can also be changed by altering the relative proportions 
of phasic and tonic afferents to the Purkinje cells (Lisberger and Sejnowski, 1992). We are 
currently investigating learning rules for which this occurs. 
The model predicts that adaptation in the cerebellum is faster than in the vestibular nucleus, 
and that learning in the vestibular nucleus is mostly driven by the climbing fiber error 
signals. 
The model shows how the dynamics of the whole system can lead to long-term adaptation 
Biologically Plausible Local Learning Rules for Adaptation of Vestibulo-Ocular Reflex 967 
Desired gain g = 1.6 
Magnitude 
2.5 
2 
1.5 
1 
0.5 
0 
q=l q=0 
 D 
100 200 300 400 $0 Time 
Magnitude 
2 
1.75 
1.5 f 
1.25 
1 
0.75 
0.25 
0 
G 
D 
A 
100 200 300 400 
Amplitude 
2.5 
1.5 
N 
1 
0.5 
A, D & VOR gain G vs time 
Amplitude 
2.5 
2 
1.!5 5 
1 
0.5 
N 
100 200 300 400 
 P 
P & N responses to a head turn during learning vs time 
Time 
500 
500 Time 
Figure 5: Effect of q on learning curves for gain increase. Left: q = 1 leads to an overshoot in 
the VOR gain G above the desired gain. D increases up to the desired gain, P starts from 0 and 
asymptotically goes back to 0; both indicate that learning is totally transferred from P to N. Right: 
With q -- 0, there is no overshoot in the VOR gain, but since A decreases to a constant value and 
D only increases very slightly, learning is not transfered. Consequently, P firing rate stays constant 
after an initial drop. 
(1-b-t-v) 
- g) q 
(2q-1)-rv 
o 
i 0.2 
' 0.4 
lO 1 
Figure 6: Overshoot e of the VOR gain G as a function ofq and r. The parameter q is the proportion 
of learning to node N (vestibular nucleus), coming from the P node (cerebellum) compared to learning 
from climbing fibers. The parameter r is the ratio of the learning rates */A and */z). No overshoot is 
seen in animals, which restricts the parameters space of q and r for the model to be valid. Note that 
the overshoot diverges for some parameter values. 
which differs from what may be expected from the local learning rules at the synapses 
because of differences in time scales and shifts of activity in the system during learning. 
This may reconcile apparently contradictory evidence between local learning rules ob- 
served in vitro (Ito, 1970) and the long-term adaptation seen in vivo in animals (Miles and 
Lisberger, 1981). 
968 Coenen, Sejnowski, and Lisberger 
Acknowledgments 
O.C. was supported by NSERC during this research. 
References 
Albus, J. S. (1971). A theory of cerebellar function. Math. Biosci., 10:25-61. 
Arnold, D. B. and Robinson, D. A. (1992). A neural network model of the vestibulo-ocular reflex using a local 
synaptic learning rule. Phil. Trans. R. Soc. Lond. B, 337:327-330. 
Fujita, M. (1982). Simulations of adaptive modification of the vestibulo-ocular reflex with an adaptive filter model 
of the cerebellum. Biological Cybernetics, 45:207-214. 
Galiana, H. L. (1986). A new approach to understanding adaptive visual-vestibular interactions in the central 
nervous system. Journal of Neurophysiology, 55:349-374. 
Ito, M. (1970). Neurophysiological aspects of the cerebellar motor control system. Int. J. Neurol., 7:162-176. 
Ito, M., Sakurai, M., and Tongroach, P. (1982). Climbing fibre induced depression of both mossy fibre respon- 
siveness and glutamate sensitivity of cerebellar purkinje cells. J. Physiol. Lond., 324:113-134. 
Kawato, M. and Gomi, H. (1992). The cerebellum and VOR/OKR learning models. Trends in Neuroscience, 
15:445-453. 
Lisberger, S. G. (1988). The neural basis for learning of simple motor skills. Science, 242:728-735. 
Lisberger, S. G. (1992). Neural basis for motor learning in the vestibulo-ocular reflex of primates:IV. The sites of 
learning. In preparation. 
Lisberger, S. G. and Sejnowski, T. J. (1992). Computational analysis suggests a new hypothesis for motor learning 
in the vestibulo-ocular reflex. Technical Report 9201, INC, Univ. of California, San Diego. 
Marr, D. (1969). A theory of cerebellar cortex. J. Physiol., 202:437-470. 
Melvill Jones, G. M. (1991). The Vestibular Contribution, volume 8 of Vision and Visual Dysfunction, chapter 2, 
pages 293-303. CRC Press, Inc., Boston. General Editor: J. R. Cronly-Dillon. 
Miles, E A. and Eighmy, B. B. (1980). Long-term adaptive changes in primate vestibulo-ocular reflex. I. Be- 
havioural observations. Journal of Neurophysiology, 43:1406-1425. 
Miles, E A. and Lisberger, S. G. (1981). Plasticity in the vestibulo-ocular reflex: A new hypothesis. Ann. Rev. 
Neurosci., 4:273-299. 
Quinn, K. J., Baker, J., and Peterson, B. (1992). Simulation of cerebellar-vestibular interactions during VOR 
adaptation. In Program 22ndAnnual Meeting. Society for Neuroscience. 
Quinn, K. J., Schmajuk, N., Jain, A., Baker, J. E, and Peterson, B. W. (1992). Vestibuloocular reflex arc analysis 
using an experimentally constrained network. Biological Cybernetics, 67:113-122. 
Viola, E A., Lisberger, S. G., and Sejnowski, T. J. (1992). Recurrent eye tracking network using a distributed 
representation of image motion. In Moody, J. E., Hansen, S. J., and Lippman, R. P., editors, Advances in 
Neural Information Processing Systems 4, San Mateo. IEEE, Morgan Kaufmann Publishers. 
