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Kawato M (2009) Cerebellum: Models. In: Squire LR (ed.) Encyclopedia of
Neuroscience, volume 2, pp. 757-767. Oxford: Academic Press.

Author's personal copy

Cerebellum: Models

757

Cerebellum: Models
M Kawato, ATR Computational Neuroscience
Laboratories, Kyoto, Japanã 2009 Elsevier Ltd. All rights reserved.

Cerebellar Learning Models
The cerebellum is one of the best- and earliest-studied
parts of the brain. In the past 50 years, a number ofconceptual models have proposed various functions,such as error correction, coordination, timing processing, and rhythm. Among these, cerebellar learningmodels have always been the most attractive, versatile, and influential. They are also the most explicit
class of theory from the computational viewpoint.
A large number of experiments and theoretical studies have been conducted to resolve the controversies
and disputes surrounding them. The learning models
are based on unique structures of the cerebellar cortex. The Purkinje cell, the only output neuron in the
cerebellar cortex, receives two major excitatory
inputs from climbing fibers and parallel fibers.
A single climbing fiber axon makes multiple synapses
on a single Purkinje cell and induces very strong
excitatory postsynaptic potentials, which lead tocomplex spikes that consist of a few Ca2þ spikes. By
contrast, 200 000 synapses from parallel fibers, which
are the axons of 1010–11 granule cells, generate simple
spikes, each of which is a single Naþ spike.
Brindley first proposed that parallel fiber–Purkinje
cell synapses undergo plastic changes if the climbing
fiber input is concomitantly activated. Around 1970,
Marr, Albus, and Ito postulated the first learning theory of the cerebellar neural circuits. Albus correctlypredicted that synaptic efficacy decreases if the parallel and climbing fibers are simultaneously activated,which was later physiologically demonstrated as longterm depression (LTD) by Ito and colleagues. Ito proposed that the climbing fiber input carries an error
signal rather than a teacher (desired output) signal andexplained adaptation of the vestibulo-ocular reflex by
an LTD-based learning model.
A general model of cerebellar learning, which is anextension of the model proposed by Ito to explain general movements with many degrees of freedom and asimplification of the model proposed later by Kawato,
is shown in Figure 1. Mossy fiber inputs carry information on the goal of a movement, sensory feedback about
the current state of the body and the external world, andthe efference copy of motor commands to granule cells.
Purkinje cells, mainly activated by parallel fiber inputs
from granule cells, project to cerebellar nuclei, whichfurther transmit information to downstream motor systems such as the oculomotor system, spinal cord, red
nucleus, and cerebral cortex via the thalamus (depicted
together as the postcerebellar network in Figure 1).
Motor commands generated by this network induce
movements, which are measured by sensory organs.
Subtracting this measured state feedback to the brain
from the desired state gives the sensory error, which isfurther transformed into the appropriate coordinates
for Purkinje cell outputs. The inferior olive neurons
send this transformed error back to Purkinje cells via
the climbing fibers.
Cerebellar Purkinje cell learning can be modeled by
the following simple supervised learning based on the
known synaptic plasticity of parallel fiber–Purkinje
cell synapses, including LTD and long-term potentiation (LTP), and the rebound potentiation of inhibitory synapses on Purkinje cells, as well as LTP
between parallel fibers and inhibitory interneurons.
For simplicity, the firing rate y of a Purkinje cell isassumed to be the weighted summation of 200 000
parallel fiber inputs xi:
y¼

200
000
X

wi xi

i¼1

Then, the inhibitory action on deep cerebellar
nuclei is negative of the Purkinje output, y. If thedesired output for the sign-inverted Purkinje cell’s
simple spike is denoted as yd, then the objective function to be minimized for the supervised learning, inwhich the synaptic weight wi is learned so that theactual output y best approximates the desired output
yd, is given as
E ¼ 1=2ðyd  yÞ2
The steepest descent direction of this error function
can be obtained by the following partial derivative(called the Widrow–Hoff rule):


@E
@y
ðyd  yÞ ¼ xi ðyd  yÞ
¼ 
wi ; / 
@wi
@wi
Þ
¼ xi ðC  C ) is the difference between the climbing
Here, (C C
fiber firing rate and its temporal average (backgroundfiring activity), and it is assumed to encode the error
signal in the last equation.
When both parallel and climbing fibers are acti ) are positive, and thus Dwi
vated, xi and (C C
is negative (LTD). In contrast, when the parallel
fiber is activated and the climbing fiber is not, thenEncyclopedia of Neuroscience (2009), vol. 2, pp. 757-767

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758 Cerebellum: Models
Cerebellar cortex
Granulecells

Purkinje
cells

Mossy fiber

Postcerebellar
network

Body and world

Sensory organ

Parallelfiber

Climbing fiber
Desired state +Sensory error−

Transformed error
Error
Inferior olivetransformation

Actual state
Figure 1 A cerebellar learning model.

 ) is negative, and thus Dwi
xi is positive but (C C
is positive (LTP). When both the climbing fiber andthe inhibitory interneuron synapse on a Purkinje cell ) are positive, and thus
are activated, xi and (C C
Dwi is negative, leading to an increase in the amplitude of wi, which is negative (LTP of inhibitory
pathway).
Cerebellar learning theory predicts that a memoryof learning is stored at parallel fiber–Purkinje cell
synapses; that LTD is an elementary cellular processof cerebellar learning; and that climbing fibers carryan error signal with some precision in the output
coordinates of Purkinje cells. These predictions haveall stimulated a vast amount of experimental examination and controversy, but they have been quite well
supported by recent data and modeling four decades
after they were first proposed. These controversial
issues are discussed below.
It is well established that when the cerebellum as awhole is lesioned, for some movements, learning
capability is lost, stored memory of learning is
destroyed, and movements are not well corrected
or coordinated. Controversies have centered mainlyon the question of whether the cerebellar cortex(Purkinje cell synapses) or the cerebellar nucleus is
the major site of memory. Recent systematic lesion
studies have shown that the cerebellar cortex is amemory site for several days at least and that the
cerebellar nuclei can contain even longer memory.
Interfering with LTD pharmacologically (e.g., by administration of hemoglobin, which is a scavenger ofNO, essential for LTD) or by genetic manipulation(knockout mice and their pharmacological rescue)are more recent and sophisticated ways of resolving
this controversy. Abolishment of LTD was shown todestroy various motor learning capabilities, such as
adaptation of walking, vestibulo-ocular reflex, optokinetic eye movements, and precise timing in eyeblink conditioning.

Computational Models of Cerebellar LTD
If LTD of parallel fiber–Purkinje cell synapses is theelementary cellular process of supervised learning, as
postulated in the cerebellar learning theory, it mustsatisfy the following three requirements. First, LTDshould be associative. That is, LTD should be induced
only when both parallel fiber inputs and climbing
fiber inputs are conjunctively activated. Second, LTDshould be maximally induced when climbing fiber
activation is delayed 50–200 ms with respect to parallel fiber activation. This is because the time window of
LTD should compensate for the delay of the reafferent
climbing fiber inputs to feedback the consequences ofthe parallel fiber–simple spike-induced movements.
Third, LTD should be synapse specific; only a parallel
fiber synapse that is conjointly activated with a climbing fiber input should be depressed, but not othersynapses. However, a considerable number of studies
have reported violations of these three requirements
under some experimental conditions; consequently,
LTD has been questioned several times as the elementary process of cerebellar supervised learning andhas been postulated for other functions, such as theEncyclopedia of Neuroscience (2009), vol. 2, pp. 757-767

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Cerebellum: Models

759

PF
Glu

NO
VGCC

AMPA-R

sGC

mGluR
PLCb

Lyn

Gq

cGMP
PKG

Raf

G-substrate

DAG

Positive
feedback AA
loop
MAPK
PLA2

MEK

PP2A

Ca2+

PKC

IP3

IP3R
Ca2+ store

CF
CRF
AMPA-R

CRHR

Figure 2 Signal transduction model of cerebellar long-term depression. AA, arachidonic acid; AMPA-R, a-amino-3-hydroxy-5-methyl4-isoxazole propionic acid receptor; CF, climbing fibers; cGMP, cyclic guanosine monophosphate; CRF, corticotropin releasing factor;
CRHR, the human corticotropin releasing factor; DAG, diacylglycerol; Glu, glutamate; Gq, G-proteins; IP3, inositol 1,4,5-triphosphate;
IP3Rs, IP3 receptors; MAPK, mitogen-activated protein kinase; MEK, MAP/extracellular signal-regulated kinase; mGluR, metabotropic
glutamate receptor; NO, nitric oxide; PF, parallel fibers; PKC, protein kinase C; PKG, cGMP-dependent protein kinase; PP2A, protein
phosphatase 2A; PLA2, cytosolic phospholipase A2; PLCb, phospholipase Cb; sGC, soluble guanylyl cyclase; VGCC, voltage-gatedcalcium channel.

normalization of total synaptic efficacies on a single
Purkinje cell. The kinetic simulation model of LTD
signal transduction pathways in a Purkinje cell dendritic spine (Figure 2) provides a coherent perspectiveon these confusing experimental data and competingtheoretical hypotheses, and it is described below.
Climbing fiber inputs activate a-amino-3-hydroxy5-methyl-4-isoxazole propionic acid (AMPA) receptors(AMPA-Rs) on dendritic shafts and induce depolarization, which opens the voltage-gated calcium channels(VGCCs), as indicated by the broken line in Figure 2,
leading to Ca2þ influx from the extracellular space into
the spine head, reaching a concentration of about
1 mmol l1. Parallel fiber inputs also induce AMPA-R
activation on dendritic spines, just as a climbing fiber
input does, but parallel fiber inputs produce smaller
depolarization and Ca2þ influx. In addition, they
simultaneously activate metabotropic glutamate receptors (mGluRs). Furthermore, they presynaptically synthesize nitric oxide (NO), which diffuses into the spine.
In the metabotropic pathway, activated mGluRs
induce production of inositol 1,4,5-trisphosphate(IP3) and diacylglycerol via G-proteins and phospholipase Cb at the postsynaptic density. IP3 then diffuses
into the cytosol and binds to IP3 receptors (IP3Rs),
which are IP3-gated Ca2þ channels on the intracellularCa2þ stores, such as endoplasmic reticulum (ER).
Cytosolic Ca2þ above some threshold triggers aregenerative cycle of IP3-dependent Ca2þ-induced
Ca2þ release (depicted as a positive feedback loop
between Ca2þ and IP3Rs in Figure 2) from the ER via
IP3Rs opened by binding of both IP3 and Ca2þ. Thismeans that the open probability of IP3Rs increases with
Ca2þ concentration, and the Ca2þ threshold of thisregenerative cycle decreases with IP3 concentration.
An increase in cytosolic IP3 concentration was
simulated to be maximal at 3 mmol l–1 about 100 ms
after the parallel fiber input. This slow rate of increasein IP3 concentration is due to the slow dynamics ofthe metabotropic pathway. Because Ca2þ influx via
VGCCs due to climbing fiber activation is an immediate
event, when the climbing fiber input is delayed about
100 ms with respect to the parallel fiber input, Ca2þincrease coincides with IP3 increase, and they together
trigger the Ca2þ-induced Ca2þ release of 10 mmol l–1. If
either climbing fiber input or parallel fiber input alone
is activated, this large Ca2þ release does not occur
because either IP3 or Ca2þ, respectively, does not reachthe threshold for the regenerative cycle. Consequently, the Ca2þ nonlinear excitable dynamics model by
Doi (right third of Figure 2) reproduced the associative
nature and the spike timing-dependent plasticity(STDP) of LTD.
In the model proposed by Kuroda, the large Ca2þincrease, which is caused by the conjunctive parallel
fiber input and the delayed climbing fiber input, induces

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760 Cerebellum: Models

the activation of linear cascades of phosphorylation ofprotein kinase C (PKC), followed by phosphorylationand then internalization of AMPA-R, which explains
the early phase of LTD up to about 10 min. On theother hand, the intermediate phase of LTD, up to several times longer, is mediated by a mitogen-activatedprotein (MAP) kinase-dependent positive feedback
loop, which consists of PKC, Raf, MAP/extracellularsignal-regulated kinase (MEK), MAP kinase, cytosolic
phospholipase A2, arachidonic acid, and back to PKC(Figure 2). This positive feedback loop has two stableequilibrium points separated by a saddle point that
determines the threshold of LTD. At the low equilibrium point, the MAP kinase positive feedback loop is
not activated, but if strong and long Ca2þ elevation
occurs, the state jumps to the higher equilibrium overthe saddle point, and AMPA-Rs continue to be phosphorylated and internalized during this prolonged activation of the positive feedback loop. Consequently, thisMAP kinase positive feedback model (middle third of
Figure 2) explains the long-term nature of LTD and is
supported by George Augustine, Keiko Tanaka, andcollaborators.
In the Kuroda model (left third of Figure 2), NO
activates soluble guanylyl cyclase (sGC), and the activated sGC catalyzes the conversion of guanosine
triphosphate into cyclic guanosine monophosphate(cGMP), which activates cGMP-dependent protein
kinase (PKG). PKG phosphorylates its substrate,
G-substrate, and phosphorylated G-substrate preferentially inhibits protein phosphatase 2A (PP2A). Activated PP2A dephosphorylates MEK and AMPA-R.
Inhibition of the dephosphorylation of MEK was
required for the activation of the MAP kinase positive
feedback loop. Therefore, by ultimately inhibiting
PP2A and activating the positive feedback loop, NO
plays a permissive role in the induction of LTD.
The Ca2þ dynamics model coherently reproduces
both the associative and the nonassociative naturesof LTD and the spike timing dependency of associative LTD as follows. Within the physiological ranges of input strength, the conjunctive stimulation of
the parallel fiber input and the delayed climbing fiber
input is essential to induce a large, regenerative Ca2þincrease, which is necessary for crossing the thresholdof the MAP kinase positive feedback loop and inducing LTD. However, if stimulation of parallel fiber
bundles is strong enough, or if uncaged Ca2þ or IP3
is abundant enough, these stimuli alone can increase
the Ca2þ concentration so that it crosses the Ca2þdynamics threshold, resulting in a large enoughCa2þ-induced Ca2þ release from the ER to reproduce
nonassociative LTD. In the hippocampal and cerebral pyramidal neurons, backpropagation of somatic
action potentials releases the magnesium block on

N-methyl-D-aspartate (NMDA) receptors and is theessential mechanism for millisecond-order STDP
there. Backpropagation of somatic action potentials
requires activity of postsynaptic cells, which is theessential condition of Hebbian learning. Neither backpropagation nor the NMDA receptors exist in Purkinjecells, suggesting cellular mechanisms for the clear
distinction of Purkinje cell-supervised learning, with
100-ms STDP, from cerebral Hebbian learning,with STDP of tens of milliseconds.
The excitable dynamics of the Ca2þ model andbistable dynamics of the MAP kinase positive feedback loop model may provide a possible molecularmechanism to resolve the ‘plasticity–stability’ dilemmaof memory stability with environmental sensitivity.
Each spine is equipped with a surprisingly small number of molecules (e.g., only tens of AMPA-Rs). Witha cascade of excitable and bistable dynamics startingfrom Ca2þ dynamics, followed by the MAP kinase
positive feedback loop, internalization of AMPA receptors, a possible change in cytoskeleton and membrane proteins, and finally a morphological change
in the spine, LTD can be induced by very delicate andlow-energy inputs but acquired memory can be maintained over a prolonged period of days, even understochasticity due to the small numbers of molecules.
The synapse specificity of LTD is also well explained by the NO model combined with an electricalcable model of Purkinje cell dendrite and spines, theCa2þ dynamics model, and the MAP kinase positive
feedback loop model. When no nearby parallel fiber isactivated and NO is low, even a conjunction of theparallel fiber and climbing fiber inputs cannot induce
LTD. If too many surrounding parallel fibers are stimulated and NO is very high, even the synapse, which
is not stimulated, exhibits LTD only by climbing fiber
stimulation. In this case, a spread of LTD occurs, andthe synapse specificity of LTD is lost. Only when amodest number of parallel fibers are activated andNO is within an intermediate range can synapsespecific LTD occur; the contextual information forstoring different motor skills might be conveyed bythis gating mechanism. This strongly predicts thesparseness of parallel fiber coding in vivo, which has
recently gained much experimental support.
Altogether, the LTD kinetic simulation models provide a comprehensive account of several seemingly
diverse, conflicting, and unrelated experimental data
within the framework of complicated nonlinear
dynamics regulated by Ca2þ, IP3, and NO concentrations. We must note that because of the nonlinear
summation effects of these signaling molecules,
nonphysiological stimuli in vitro, such as parallel
fiber bundle stimulation, uncaging of Ca2þ, and IP3,
might induce characteristics qualitatively different

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Cerebellum: Models

from those induced in vivo LTD in terms of its
associative nature, spike timing dependency, and synapse specificity.

Rhythmic versus Chaotic Firing of Inferior
Olive Neurons
The most influential model of the cerebellum other thanthe learning theory is probably the timing, rhythm, andsynchronization hypothesis. The inferior olive nucleus,
which sends out climbing fibers to Purkinje cells, possesses the highest density of gap junctions in the mammalian brain. Inferior olive neurons have several unique

ion channels distributed on the somatic and dendritic
membrane, as shown in Figure 3(a), and these exhibit
rhythmic activities in slice preparation under some conditions, especially with harmaline. However, in awake
monkeys at rest, no rhythmicity has been found forcomplex spikes. When the effective electrical conductance of gap junctions is increased by picrotoxin application to the inferior olive, which blocks inhibitory
synapses on glomeruli of inferior olive neurons from
cerebellar nuclei, the inferior olive neurons tend to bespatially synchronized in their firing.
None of these unique and well-studied anatomicaland physiological characteristics of inferior olive

2⫻2
1⫻9
3⫻3

I Ca_hI ld

I K_Cal1 (bit s–1)

80

[Ca2+]

Dendrite

761

40

gint
Isd
Soma

I ls

Ih
I K_dra

I Ca_l
I Na

0

0

0.1

0.2

0.3

0.4

0.5

g c (ms cm–2)cMutual information per spike⫻ 10−3

gc

2⫻2
1⫻9
3⫻3
3⫻9

4

3

2

1
0

b

Climbing fibers(to Purkinje cells)

d

0.1

0.2

0.3

0.4

0.5

gc (ms cm–2)

Figure 3 A network model of inferior olive (IO) neurons electrically connected by gap junctions. (a) Two-compartment biophysical modelof an IO cell. The somatic ionic currents include a low-threshold calcium inward current (ICa_l), an anomalous inward rectifier current (Ih),the Hodgkin–Huxley type inward sodium (INa) and outward delayed rectifier potassium (IK_dr) currents, and a leakage current (Ils). The
dendritic currents include a high-threshold inward calcium current (ICa_h), an outward calcium-dependent potassium current (IK_Ca), and aleakage current (Ild). As the dendritic potassium current IK_Ca is calcium dependent, Ca2+ concentration dynamics is included in the model.
gint is the coupling conductance between the two compartments and Isd is the current flowing through this conductance. The electronic
coupling between this cell and multiple other cells is not shown. (b) Between 4 and 27 IO cells are connected on a lattice by electricalconductance via gap junctions; gc, coupling strength. (c) Largest Lyapunov exponent as a function of gc for different networks ofnonidentical cells. Error bars are standard deviations obtained from 10 different simulations. When the largest exponent was large, thenetwork exhibited marked chaotic behavior. When the largest exponent was close to zero, firing was almost rhythmic. (d) Network mutual
information per spike as a function of gc for different networks of nonidentical cells. Adapted from Schweighofer N, Doya K, Fukai H, Chiron
JV, Furukawa T, and Kawato M (2004) Chaos may enhance information transmission in the inferior olive. Proceedings of the National
Academy of Sciences of the United States of America 101: 4655–4660, with permission from National Academy of Sciences, USA.

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762 Cerebellum: Models

neurons has been incorporated into or explained by
the learning theory until recently. Schweighofer andcolleagues developed a two-compartment model of aninferior olive neuron, as shown in Figure 3(a), withknown spatial distribution of ion channels. These neurons are connected by gap junctions within a planar
sheet, as shown in Figure 3(b), and the firing characteristics, as well as the input and output informationtransfer characteristics, have been examined. The network model reproduced rhythmic firing, as well assynchronized oscillation, when the coupling conductance was high. Figure 3(c) plots the largest Lyapunov
exponent of the network dynamics as a function of thecoupling conductance. The small Lyapunov exponent
at large conductance indicates that the firing is synchronized and rhythmic when electrical coupling is strong,corresponding to picrotoxin application. With small
conductance, each neuron periodically fires at its own
frequency without much interaction; thus, the Lyapunov exponent is again close to zero, indicating analmost periodic solution. However, with medium conductance, the Lyapunov exponent is much higher thanzero, indicating that neurons fire asynchronously andaperiodically. This is actually chaotic and desynchronized firing, and for this range of dynamics, input andoutput information transmission becomes the largest,as shown in Figure 3(d). These theoretical predictions
were recently supported experimentally. Consequently,
climbing fiber inputs can convey error signals mostefficiently, even with their low firing frequency (typically 1 spike s–1), when the conductance is medium andchaos is induced. Low-frequency firing usually implies
poor representation of the error signal. Chaotic anddesynchronized firing of inferior olive neurons might
be a basic neural mechanism for insuring high-fidelity
error-signal transmission even with a low firing rate.
The feedback error learning theory of the cerebellum explained in the next section theoreticallyrequires low firing rates of climbing fiber inputs.
This is because climbing fiber inputs convey timedelayed feedback motor commands, and if they were
to be strongly manifested in Purkinje cell output,
smooth and sophisticated feed-forward motor control
would be ruined. Hirano and his colleagues provided
experimental support for this prediction in AMPA-R
delta 2 knockout mouse. Climbing fiber inputs
occupy a much larger proportion of Purkinje cell output in this knockout mouse compared with a normalmouse, and optokinetic eye movements are much
more delayed than is control performance.

Feedback Error Learning Model
Although early learning models proposed that the
cerebellum conducts pattern recognition, probably

influenced by a ‘perceptron,’ they are not consistent
with physiological data showing that simple spike
firing rates of Purkinje cells temporally change andencode dynamic and kinematic features of movements. Recent studies suggest that the human cerebellum is important for sensory and cognitive functions
as well as for motor control. What general computation might be realized by the cerebellar cortex thatwould cover the sensory, cognitive, and motor
domains; would be compatible with existing hypotheses, such as timing, coordination, and error correction; and could be learned by supervised learning suchas that shown in Figure 1? Internal models seem toprovide the most versatile and compatible computational entity satisfying all these requirements. Internal
models are neural networks inside the brain (‘internal’) that can mimic (‘model’) the input–output characteristics of some dynamic process outside the brain.
In a motor control context, a forward model of abody can predict sensory consequences of movementfrom the efference copy of a motor command. This isbecause the controlled objects in movements generatea trajectory from motor commands. By contrast, theinverse model of a body and the external world cancompute the necessary motor commands from adesired movement pattern. Internal models are usefulfor predictive computations in sensory, cognitive, andmotor functions. If an inverse dynamics model for acontrolled object possessing dynamics and kinematics
with multiple degrees of freedom is destroyed, themovement becomes clumsy, slow, and not well corrected, because neither feed-forward nor feedback
control (both of which are dependent on the inverse
model) is available. Thus, control must rely on poorand crude feedback control alone. Internal models
can be learned based on supervised learning.
Around 1990, Kawato proposed that different partsof the cerebellum contain either forward or inverse
models. He also proposed a computational scheme
called feedback error learning to acquire an inverse
model in the supervised learning scheme, as shownin Figure 4(a). If supervised learning takes place inthe brain for motor control, the following difficult
computational problem called ‘distal teacher,’ formulated by Jordan, should be resolved. As shown in
Figure 1, the error between desired and actual movement patterns can be measured by sensory organs, butthese movement errors could be very different in spatial
coordinates as well as in temporal dynamics fromthe necessary error signal for the motor command.
Furthermore, it should be emphasized that there exists
no teaching signal for motor commands in supervised
motor learning, because if it were to exist, it would bedirectly utilized for motor control, and there would
be no need for learning. The distal teacher problem is

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763

Inverse modelFeed-forward
motor
Motor
command command
error
+ +
Feedback
Controlled
objectDesired + − Trajectory controller Feedback Motor
trajectory
error
motor
commandcommand

ActualtrajectoryaGranule cells
Mossy fibers

Cerebellar cortex

Parallel fiber
Simple spikes
Purkinje cells

Feed-forward
motor
commandComplex spikes
Climbing fiber

Desiredtrajectory+−

Feedback
controller+ +Feedback Motorcommandmotor
commandControlled
objectActualtrajectoryb
Figure 4 (a) The general feedback error learning model. (b) The cerebellar feedback error learning model. The ‘controlled object’ is aphysical entity, such as the eyes, hands, legs, or torso, that needs to be controlled by the central nervous system. Different colors are forcorrespondence between (a) and (b).

almost absent in simple gain control with one degree of
freedom, such as vestibulo-ocular reflex; thus, it motivated the following more recent models for generalmany-degrees-of-freedom movements with dynamics
and kinematics. Several computational models havebeen proposed, such as direct inverse modeling, backpropagation through a forward model, or decorrelation learning, but the feedback error learning model,
which transforms the sensory error into a motor error
by a preexisting feedback controller in the central nervous system (Figure 4(a)), is probably the most biologically plausible and best supported.
The feedback error learning model depicted in
Figure 4(a) is the same as the basic model shown in
Figure 1 except for the following four points. First, theinverse model transforms a desired trajectory into afeed-forward motor command. Second, a crude feedback controller generates feedback motor commandfrom the sensory error. Third, the summation of thefeed-forward and feedback motor commands is sent tothe controlled object. Finally, and most important, thefeedback motor command is used as the error signal for
motor commands in supervised learning for acquisitionof the inverse model (cf. Figures 1 and 4(a)). Stability
and convergence of the feedback error learning algorithm have been mathematically proved as well as successfully applied to several robotic demonstrations.
Figure 4(b) adapts the basic feedback error learningmodel to the cerebellar neural circuit. Here, simple
spikes represent feed-forward motor commands, andparallel fiber inputs represent the desired trajectory as
well as the sensory feedback of the current status of
the controlled object. A microzone of the cerebellar
cortex constitutes (a part of) an inverse model of a
specific controlled object such as the eye or arm. Most
importantly, climbing fiber inputs are assumed to carrya copy of the feedback motor commands generated by
the feedback control circuit. Thus, the complex spikes
are predicted to be trajectory error signals alreadyexpressed in motor command coordinates. Although

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400
300
200
100
0
Time (ms)

Cell

40 ms
100
80
60
40
20
0

N = 163

10 20 40 80 160–1
Stimulus speed (⬚ s )
U

2

C

a 3

Time (ms)

43 ms
50
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I

N = 2410
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10 20 40 80 160–1
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U
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D

D

D

DLPN neurons

Vertical cells

Horizontal cells

D

MST neurons

20

N = 87C

Time (ms)

49 ms

Cell

1

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–1

Firing rate (Imp s )764 Cerebellum: Models

Cerebellar cortex

VPFL Purkinje cells

Granule Purkinje
cells
cells

Mossy
fiber

Cerebral cortex
MT
MST
DLPN

STS
Visual cortex
LGN

Parallelfiber
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−

+

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stem

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50

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Probability

0.05

0

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c

100
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Figure 5 Continued

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Cerebellum: Models

climbing fiber inputs are delayed by 100 ms withrespect to the responsible parallel fiber inputs thatcaused the movement and the resulting error, spiketiming dependency of LTD (see the section titled‘Computational models of cerebellar LTD’) selectively
changes only the responsible parallel fiber synapses.

The Ventral Paraflocculus and OcularFollowing Responses
Kawano and his colleagues supported the cerebellar
feedback error learning model by neurophysiologicalstudies in the ventral paraflocculus of monkey cerebellum during ocular following responses. Ocularfollowing responses are tracking movements of theeyes, evoked by movements of a large visual scene,that are thought to be important for the visual stabilization of gaze. In the neural circuit controlling this
response Figure 5(b), the phylogenetically older,crude feedback circuit comprises the retina, the accessory optic system, and the brain stem. The phylogenetically newer and more sophisticated feed-forward
pathway and the inverse dynamics model correspondto the cerebral and cerebellar cortical pathway andthe cerebellum, respectively. In Figure 5(b), the sensory error signal is computed as the image motion on
the retina (retinal slip). This is different from theschemes of Figures 1 and 4 in which some neuralmechanisms explicitly compute the difference (error)
between the desired and actual states. However, inthe case of eye movement control such as ocularfollowing responses or vestibulo-ocular reflexes, thedesired state is that the eye movement is made so thatthe retinal image does not move, and thus the opticalcomputation (retinal slip detection) can replace subtraction of the desired and actual states. On the other765

hand, the desired trajectory information transmitted
to the flocculus for the vestibulo-ocular reflex is thehead motion signal detected by semicircular canals
whereas for the ocular following responses, it is theimage motion signal conveyed by the visual areas in
the cerebral cortex.
During ocular following responses, the timecourses of simple spike firing frequency show complicated patterns (Figure 5(c) top, thin line). However,they were quite accurately reconstructed by using an
inverse dynamics representation of eye movement(thick line). The model fit was good for the majorityof the neurons studied under a wide range of visual
stimulus conditions, different eye movements, anddifferent firing patterns. The same inverse dynamics
analysis for the medial superior temporal area (MST)of the cerebral cortex and dorsolateral pontine
nucleus, which provide visual mossy fiber inputs,
revealed that their firing time courses were not wellreconstructed from motor commands but rather thatthey were well reconstructed from visual retinal slip.
Taken together, these data suggest that the ventral
paraflocculus is the major site of the inverse dynamics
model of the eye for ocular following responses.
Motor commands, conveyed by simple spikes,
should be directly modified and acquired by LTD
while guided by motor command errors, which areconveyed by climbing fiber inputs. For this to work,
the climbing fiber inputs need to convey comparable
details of temporal and spatial information to themotor commands, but the ultralow discharge ratesof the latter would appear to rule out this possibility.
This apparently discrete and sporadic nature ofclimbing fiber inputs was characterized as an unexpected event detector, which suggested a reinforcement learning type theory. However, if thousands ofFigure 5 Change of neural codes and learning of inverse dynamics model in the cerebellum for ocular following responses (OFR). (a)Firing characteristics of medial superior temporal area (MST), dorsolateral pontine nucleus (DLPN), and ventral paraflocculus (VPFL)
neurons. (b) Schematic neural circuit for OFR. (c) Temporal firing patterns of VPFL Purkinje cells in upward eye movements induced by
upward visual motion. (a) The left, middle, and right columns are for MST, DLPN, and VPFL neurons, respectively. (1) Poststimulus-time
histograms of the firing rates of a typical neuron in each of the three areas. The origin of time is the onset of visual stimulus motion.
(2) Histograms of a number of cells within a given range of the optimum stimulus speeds. (3) Polar plots of optimum stimulus directions.
U, C, D, and I, upward, contralateral, downward, and ipsilateral, respectively. VPFL Purkinje cells were classified into two groups: verticalcells and horizontal cells, based on simple spike (a dotted line) and complex spike (actual line) optimum directions. (b) The circuit can bedivided into two main pathways. The upper part shows the corticocortical (the cerebral cortex to the cerebellar cortex) pathway, whichcorresponds to the feed-forward arc of feedback error learning. The lower part shows the phylogenetically older feedback pathway
containing the accessory optic system (AOS), which corresponds to a crude feedback controller in the feedback error learning scheme. (c)Temporal firing patterns of nine accumulated Purkinje cells (thin curves) and their reconstruction based on inverse dynamics model (boldcurves). The model predicts that the temporal firing patterns of simple spikes (upper pane) and complex spikes (lower pane) should be
mirror images of each other, and this was confirmed experimentally. PT, pretectum; NOT, nucleus of optic tract; MT, middle temporal area;
STS, superior temporal sulcus; LGN, lateral geniculate nucleus; EOMN, extraocular motor neurons. Adapted from Kawano K (1999)
Ocular tracking: Behavior and neurophysiology. Current Opinion in Neurobiology 9: 467–473; Takemura A, Inoue Y, Gomi H, Kawato M,
and Kawano K (2001) Change in neuronal firing patterns in the process of motor command generation for the ocular following response.
Journal of Neurophysiology 86: 1750–1763; Kobayashi Y, Kawano K, Takemura A, et al. (1998) Temporal firing patterns of Purkinje cells in
the cerebellar ventral paraflocculus during ocular following responses in monkeys. II. Complex spikes. Journal of Neurophysiology 80:
832–848. Reproduced from Kawato M (1999) Internal models for motor control and trajectory planning. Current Opinion in Neurobiology
9: 718–727, with permission from Elsevier.

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766 Cerebellum: Models

trials are averaged, the firing rates actually conveyedvery accurate and reliable time courses of motor command error within a few hundreds of milliseconds(Figure 5(c), bottom). More specifically, complex
spike time courses are better reconstructed by retinal
slip than by eye movements but in the motor command coordinates rather than in visual coordinates,
as detailed below. Because LTD is a rather slow process, taking several tens of minutes, it can conduct the
averaging over many trials. Consequently, the firing
probability of climbing fiber inputs can convey highfrequency temporal information matching information of the dynamic command signal.
The preferred directions of MST and pontine neurons were evenly distributed over 360 , as shown in
Figure 5(a)(3). Thus, the visual coordinates are uniformly distributed over all possible directions. On theother hand, projections of the three-dimensional spatial coordinates of the extraocular muscles were ineither a horizontal or vertical direction, and they wereentirely different from the visual coordinates. Preferred directions of simple spikes were either downward or ipsiversive (Figure 5(a)(3)), and at the site ofeach recording, electrical stimulation of a Purkinje
cell elicited eye movement toward the preferred
direction of the simple spike. These data indicatethat the simple spike coordinate framework is alreadythat of the motor commands. Thus, at the parallel
fiber–Purkinje cell synapse, a drastic visuomotor
coordinate transformation occurs. The neural representation dramatically changes from population coding in the MST and pontine nucleus to firing rate
coding of Purkinje cells at the parallel fiber–Purkinje
cell synapse (Figure 5(a)). What, then, is the origin of
this drastic transformation? The theory proposes thatthe complex spikes, and ultimately the accessory
optic system, are the source of this motor command
spatial framework. The preferred directions of pretectum neurons are upward, and those of the contralateral nucleus of optic tract neurons are ipsiversive(thus contraversive for the cerebellum). Their signals
are conveyed to the inferior olive and further to the
cerebellum to produce the complex spikes. In summary, the coordinate frame of the climbing fibers wasvertical and horizontal, forming the motor coordinates, not the visual coordinates of MST or pontine
nucleus. All these experimental findings were successfully simulated by the network model shown in
Figure 5(b). The coordinates of semicircular canals
and the extraocular muscles are very similar, and it israther difficult to evaluate coordinate transformationin the vestibulo-ocular reflex and flocculus comparedwith ocular following responses. However, recent physiological studies indicate that climbing fiber inputs

encode motor error rather than sensory error. This
argues against a class of model that assumes sensory–
motor transformation downstream of Purkinje cells,
including decorrelation models.

Toward a General Model of the
Cerebellum
We have a very solid computational model (feedback
error learning and inverse model) and a rich set of
supporting experimental data for the ventral paraflocculus with ocular following responses. The model
has also been well supported for other slow eye movements, such as vestibulo-ocular reflex, optokinetic
eye movements, and smooth pursuit, for the flocculus
and paraflocculus. Can we extend this to other partsof the cerebellum, other types of movements, andother computational models? Many theorists believe
so since the neural circuit is generally uniform except
for the newly found unipolar brush cells, found primarily in the flocculonodular lobe. Furthermore,
Kitazawa and colleagues have demonstrated thatcomplex spikes recorded from cerebellar hemispheres
encode reaching errors at goal points during monkey
whole-arm reaching, whereas simple spikes represent
movement dynamics such as an electromyogram.
What can we generalize about cerebellar functions
from previous studies on specific functions and specific areas? First, synaptic plasticity is unique andubiquitous, and the learning is undoubtedly supervised learning and not Hebbian or reinforcement
learning. Second, the cerebellar 100 ms STDP, in contrast to cerebral and hippocampal millisecond-order
STDP, strongly suggests that the cerebellum is better
suited for approximating some nonlinear dynamicalsystem outside the cerebellum as 100 ms is exactly thetime required for sensory signal feedback from theexternal world. Finally, the major computation isfeed-forward, with less emphasis on recurrent feedback, because spatial resources are allocated more toneurons than to their connections in the cerebellum, as
is evident from the number of neurons: more than thatof the cerebrum in a volume only one-tenth that of thecerebrum. Consequently, internal models are the mostpromising computational functions of the cerebellum.
Forward models have been theoretically postulated
several times and supported by neuroimaging experiments. Cancellation of a self-generated electrical field
by a forward model is the function of the electrosensory lobe (cerebellum-like structure) of the weakly
electric fish and might be interpreted as anothersupport. But it is actually a different story because
NMDA receptors play the essential role in antiHebbian learning there. Thus, supporting evidence

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Cerebellum: Models

for forward models is much more circumstantial thanis the evidence for inverse models, and we need moredecisive data. Computational models for how multiple internal models are integrated and used for motor
control and cognition have been proposed (e.g., themodular selection and identification for control
model), but innovative experimental methodology tofurther explore these phylogenetically newer functions of the cerebellum seems necessary.
See also: Cerebellar Microcircuitry; Cerebellar Deep
Nuclei; Cerebellum and Oculomotor Control; Cerebellum:
Clinical Pathology; Cerebellum: Evolution andComparative Anatomy; Olivocerebellar System;
Procedural Learning: Cerebellum Models; Sporadic
Degenerative Ataxias and the Dominantly Inherited
Spinocerebellar Ataxias.

Further Reading
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Daniel H, Levenes C, and Crepel F (1998) Cellular mechanisms of
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London, Series B: Biological Sciences 269: 1895–1904.
De Schutter E (1995) Cerebellar long-term depression might normalize excitation of Purkinje cells: A hypothesis. Trends in
Neurosciences 18: 291–295.
De Schutter E (1999) Using realistic models to study synaptic
integration in cerebellar Purkinje cells. Reviews in the Neurosciences 10: 233–245.
Doi T, Kuroda S, Michikawa T, and Kawato M (2005) Inositol
1,4,5-trisphosphate-dependent Ca2þ threshold dynamics detect
spike timing in cerebellar Purkinje cells. Journal of Neuroscience 25: 950–961.
Fujita M (1982) Adaptive filter model of the cerebellum. Biological
Cybernetics 45: 195–206.
Gomi H, Shidara M, Takemura A, et al. (1998) Temporal firing
patterns of Purkinje cells in the cerebellar ventral paraflocculus
during ocular following responses in monkeys. I. Simple spikes.
Journal of Neurophysiology 80: 818–831.
Imamizu H, Miyauchi S, Tamada T, et al. (2000) Human cerebellar
activity reflecting an acquired internal model of a new tool.
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Ito M (1984) The Cerebellum and Neural Control. New York:
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in Neurobiology 78: 272–303.
Ivry RB and Spencer RM (2004) The neural representation of time.
Current Opinion in Neurobiology 14: 225–232.
Jordan MI and Rumelhart DE (1992) Forward models: Supervised
learning with a distal teacher. Cognitive Science 16: 307–354.
Kawano K (1999) Ocular tracking: Behavior and neurophysiology.
Current Opinion in Neurobiology 9: 467–473.

767

Kawato M (1999) Internal models for motor control and trajectory
planning. Current Opinion in Neurobiology 9: 718–727.
Kawato M, Furukawa K, and Suzuki R (1987) A hierarchicalneural-network model for control and learning of voluntary
movement. Biological Cybernetics 57: 169–185.
Kawato M and Gomi H (1992) The cerebellum and VOR/OKR
learning models. Trends in Neurosciences 15: 445–453.
Keating JG and Thach WT (1995) Nonclock behavior of inferior
olive neurons: Interspike interval of Purkinje cell complex spike
discharge in the awake behaving monkey is random. Journal of
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611–619.
Kitazawa S, Kimura T, and Yin PB (1998) Cerebellar complex
spikes encode both destinations and errors in arm movements.
Nature 392: 494–497.
Kobayashi Y, Kawano K, Takemura A, et al. (1998) Temporal firing
patterns of Purkinje cells in the cerebellar ventral paraflocculus
during ocular following responses in monkeys. II. Complex
spikes. Journal of Neurophysiology 80: 832–848.
Kuroda S, Schweighofer N, and Kawato M (2001) Explorationof signal transduction pathways in cerebellar long-term depression by kinetic simulation. Journal of Neuroscience 21:5693–5702.
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Schweighofer N, Doya K, Fukai H, Chiron JV, Furukawa T, andKawato M (2004) Chaos may enhance information transmission in the inferior olive. Proceedings of the National Academyof Sciences of the United States of America 101: 4655–4660.
Shidara M, Kawano K, Gomi H, and Kawato M (1993) Inversedynamics model eye movement control by Purkinje cells in the
cerebellum. Nature 365: 50–52.
Shutoh F, Ohki M, Kitazawa H, Itohara S, and Nagao S (2006)
Memory trace of motor learning shifts transsynaptically from
cerebellar cortex to nuclei for consolidation. Neuroscience 139:
767–777.
Takemura A, Inoue Y, Gomi H, Kawato M, and Kawano K (2001)
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command generation for the ocular following response. Journalof Neurophysiology 86: 1750–1763.
Yamamoto K, Kobayashi Y, Takemura A, Kawano K, and Kawato M(2002) Computational studies on acquisition and adaptation ofocular following responses based on cerebellar synaptic plasticity.
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