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The Journal of Neuroscience, August 15, 1999, 19(16):7140-7151
Simulations of Cerebellar Motor Learning: Computational Analysis
of Plasticity at the Mossy Fiber to Deep Nucleus Synapse
Javier F.
Medina and
Michael D.
Mauk
W. M. Keck Center for the Neurobiology of Learning and Memory,
and Department of Neurobiology and Anatomy, University of Texas
Medical School, Houston, Texas 77030
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ABSTRACT |
We question the widely accepted assumption that a molecular
mechanism for long-term expression of synaptic plasticity is sufficient to explain the persistence of memories. Instead, we show that learning
and memory require that these cellular mechanisms be correctly
integrated within the architecture of the neural circuit. To illustrate
this general conclusion, our studies are based on the well
characterized synaptic organization of the cerebellum and its
relationship to a simple form of motor learning. Using computer
simulations of cerebellar-mediated eyelid conditioning, we examine the
ability of three forms of plasticity at mossy fiber synapses in the
cerebellar nucleus to contribute to learning and memory storage.
Results suggest that when the simulation is exposed to reasonable
patterns of "background" cerebellar activity, only one of these
three rules allows for the retention of memories. When plasticity at
the mossy fiber synapse is controlled by nucleus or climbing fiber
activity, the circuit is unable to retain memories because of
interactions within the network that produce spontaneous drift of
synaptic strength. In contrast, a plasticity rule controlled by the
activity of the Purkinje cell allows for a memory trace that is
resistant to ongoing activity in the circuit. These results suggest
specific constraints for theories of cerebellar motor learning and have
general implications regarding the mechanisms that may contribute to
the persistence of memories.
Key words:
LTP; LTD; cerebellum; eyelid conditioning; simulation; mossy fiber
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INTRODUCTION |
Analysis of the neural basis of
memory has been guided for some time by the central hypothesis that
activity-dependent changes in synapses mediate changes in behavior. As
an example that illustrates the importance of activity-dependent
plasticity, consider Pavlov's classic learning experiments (Pavlov,
1927
). With only those synapses activated by the bell eligible to be
modified by the reinforcing meat powder, the learned salivation
response would later be elicited relatively specifically when the bell
is presented. For this reason, activity-dependent forms of plasticity,
such as long-term potentiation (LTP) and long-term depression (LTD) in
the hippocampus, neocortex and cerebellar cortex, have received
particular attention (Siegelbaum and Kandel, 1991; Artola and Singer,
1993; Bear and Malenka, 1994; Linden, 1994).
Although learning requires that synaptic plasticity be limited to the
right synapses (i.e., those activated by the bell), the capacity of
memories to endure also requires that this new pattern of synaptic
weights not be erased by the abundant opportunities for
activity-dependent plasticity produced by ongoing brain activity (Sejnowski, 1977; Kenyon et al., 1998). Here we question a usually tacit hypothesis regarding the duration of memories: namely, that a
molecular mechanism for persistent expression of synaptic plasticity is
sufficient to explain the persistence of memories. With
activity-dependent plasticity, this hypothesis involves the untenable
assumption that synapses are only active during learning and are thus
ineligible for plasticity at other times. Although applicable to many
brain systems, we examine this issue with computer simulations of one type of cerebellar-mediated motor learning: Pavlovian conditioning of
eyelid responses. Our results illustrate the importance of considering
how rules for plasticity must interact with neural circuits not only to
permit the induction of plasticity during learning, but also to prevent
the subsequent induction of unwanted plasticity despite ongoing brain activity.
Using the cerebellum as an example of a brain system involved in
learning, we evaluate the ability of simulations to learn using
plasticity in the cerebellar nucleus controlled by one of the three
available signals. Evidence indicates that cerebellar-mediated motor
learning, such as adaptation of the vestibulo-ocular response (VOR) or
Pavlovian eyelid conditioning, involves plasticity at both granule to
Purkinje cell (gr 
Pkj) synapses in the cerebellar cortex and mossy fiber to nucleus (mf nuc) synapses in
the cerebellar nuclei (Robinson, 1976; Perrett and Mauk, 1995; Raymond
et al., 1996; Mauk, 1997) (see Fig. 1).
Although both LTD and LTP have been identified and characterized in the
cerebellar cortex (Sakurai, 1987; Ito, 1989; Hirano, 1990; Shibuki and
Okada, 1992; Linden, 1994; Salin et al., 1996), nothing specific is
known about the properties of the plasticity that occurs in the
nucleus. Therefore, our cerebellar simulations incorporate the known
climbing fiber-controlled (CF) form of plasticity at
gr Pkj synapses and plasticity at mf nuc synapses controlled by one of three possible
cellular signals (see Fig. 2). These
three signals are (1) the activity of the nucleus cell itself (Hebbian
rule), (2) the activity of the climbing fiber input to the nucleus cell
(climbing fiber-dependent rule), and (3) the activity of the Purkinje
cell input to the nucleus (Purkinje-dependent rule). Results suggest
that each of the three nucleus rules could support learning
with a
distribution of plasticity between cortex and nucleusif plasticity is
permitted only during the training trials. In contrast, under the more
realistic circumstance where the plasticity rule is applied at all
times, nucleus and climbing fiber-controlled rules promote spontaneous drift of the strength of synapses during background inputs, precluding the possibility for learning or retention of responses. In addition to
suggesting specific constraints on the type of plasticity that may
operate in the cerebellar nuclei during motor learning, these results
have general implications regarding the mechanisms that may contribute
to the persistence of memories.
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Figure 1.
Simulations are based on the connectivity of the
cerebellar-olivary system and its relationship to eyelid conditioning.
While the interpositus nucleus (NUC) transmits the entire
output of the cerebellum, two major excitatory afferents convey stimuli
to the cerebellum. The mossy fiber afferent (Mossy)
influences cerebellar output through direct excitatory connections onto
the nuclei cells (mf nuc synapses) and through a more
indirect projection onto a very large number of granule cells
(Granule) that ultimately results in modulation of nuclei
cells by Purkinje neurons (PURK). Granule cells
affect Purkinje cell activity through connections to inhibitory
interneurons known as basket and stellate cells (B/S) and
through a large number of direct excitatory synapses
(gr Pkj synapses) onto the Purkinje cell. In sharp
contrast to this vastly diverging input, each Purkinje cell receives
synaptic connections from a single climbing fiber
(CF), which also contacts the output cells of the
cerebellar nuclei. Evidence indicates that conditioned stimuli such as
tones are conveyed to the cerebellum via mossy fibers, the reinforcing
air puff is conveyed via climbing fibers, and paired presentation of
these stimuli leads to the expression of a conditioned eyelid response
through increases in nucleus activity. This correspondence permits a
simple representation of eyelid conditioning with a computer simulation
of the cerebellum. Increases in simulated nucleus cell output during
the conditioned stimulus are taken as a measure of the conditioned
response. Presentation of the conditioned stimulus is emulated by
altering the background activities of the mossy fibers and granule
cells, whereas a transient excitatory input to the climbing fiber
simulates the reinforcing puff. With these inputs determined, the
remainder of the circuit is simulated with stochastic neurons
(Materials and Methods). The simulations also implement the well
characterized climbing fiber-dependent plasticity at the excitatory
gr Pkj synapses and plasticity at the
mf nuc synapses controlled by one of three signals:
nucleus cell activity, climbing fiber activity, or Purkinje cell
activity.
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Figure 2.
A representation of the well characterized,
activity-dependent plasticity at the gr Pkj synapses
in the cerebellar cortex, and three possible cellular rules for
plasticity at mf nuc synapses. Black
symbols indicate that the cell is active, and gray
symbols denote inactivity. Conditions for increasing the strength
of synapses (LTP) are shown in the left column,
whereas the right column shows the signals that lead to the
induction of LTD. a, Activity-dependent
plasticity at gr Pkj synapses in the cerebellar cortex
is controlled by climbing fiber inputs. These gr Pkj
synapses undergo LTP when active in the absence of a climbing fiber
input and undergo LTD when active in the presence of a climbing fiber
input. b1, With a Hebbian rule, active
mf nuc synapses undergo LTD when the nucleus cell is
quiet and undergo LTP during periods of nucleus cell activity.
b2, With a climbing fiber-dependent rule,
mf nuc synapses that are active in the absence of a
climbing fiber input to the nucleus cell undergo LTD and undergo LTP
when the climbing fiber fires. b3, A Purkinje cell-dependent
rule assumes that active mf nuc synapses undergo LTD
during periods of Purkinje inhibition of the nucleus and LTP during
decreases in this inhibition.
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MATERIALS AND METHODS |
Network organization. The connectivity of the network
is intended to capture the basic properties of the synaptic
organization and physiology of the cerebellum (Eccles et al., 1967;
Ito, 1984; Voogd and Glickstein, 1998) (see Fig. 1). Each of 20 Purkinje cells (PURK) receives, in addition to the CF input, inhibition from an average of 10 basket/stellate cells (B/S) and excitation from
200,000 granule cells (Granule). The sole output of the simulation is
represented by a single nucleus cell (NUC) receiving a collateral climbing fiber input, inhibition from the Purkinje cells, and excitatory connections from 100 mossy fibers (Mossy). Consistent with
anatomical and physiological observations, the circuit was modeled as a
closed loop in which the Purkinje cells inhibit a nucleus cell that
inhibits the climbing fiber that provides input to the Purkinje cells
(Voogd and Bigare, 1980; Buisseret-Delmas and Angaut, 1993; Ruigrok,
1997; Miall et al., 1998; Voogd and Glickstein, 1998). The number of
presynaptic inputs received by each simulated cell is summarized in
Table 1.
Representation of neural activity. Synaptic
transmission in the nervous system is a noisy process brought on by
random fluctuations in the release of neurotransmitter and other
probabilistic causes. We have captured this inherent noisiness by
implementing the traditional method for introducing a stochastic
mechanism in the firing of neurons (Haykin, 1994). Specifically, the
probability of a neuron's firing is calculated each simulated time bin
and can be approximated by a simple, sigmoid function of the membrane
potential, V, according to the equation:
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(1)
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This standard function mimics the biological relationship
between the cell's input current and its firing rate in several ways:
the output is always non-negative, it is very small below the cell's
threshold, 
(see Table 1 for the values implemented in the different
cells that were simulated), it monotonically increases with input, and
it has an upper bound at 1. Whether a cell fires (where the term
"fires" can be applied to a single spike or alternatively a burst
of spikes) is determined each simulated time bin by comparing the newly
calculated probability with a newly generated random number taken from
a uniform distribution in the interval [0,1]. Thus:
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(2)
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(3)
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The general description given in the previous paragraph
of how activity is calculated for each of the cells that participate in
the simulation (excluding mossy fibers and granule cells; see below)
can be formalized with the following equations. The membrane potential
of each basket/stellate cell is calculated by adding the synaptic
weights of active presynaptic granule cell inputs:
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(4)
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where l is typically set to 2000 (i.e., the
number of granule synapses made onto each basket/stellate cell). The
activity of each of these cells is then obtained by applying the
described sigmoidal function:
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(5)
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with b/s set to 7.2 to allow
basket/stellate cells to discharge at their physiological spontaneous
rate of ~20 Hz (Armstrong and Rawson, 1979), which corresponds to a
0.1 value for Pb/s when using a 5 msec
time-step. Interestingly, the particular discharge rate of these or any
of the other cells were found to have no effect on the general validity
of the results because they affected all plasticity rules equally.
The membrane potential of each Purkinje cell is given by:
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(6)
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where the first term sums the weights of active granule cells
and the other term represents inhibition from active basket/stellate cells. m and k are the number of granule and
basket/stellate cell inputs to the Purkinje cell. The activity of each
Purkinje cell is always obtained by applying a threshold function
(sigmoidal) to the neuron's membrane potential except when its
associated climbing fiber fires. The empirically observed pause in
Purkinje cell activity after a climbing fiber input is simulated by
momentarily setting the probability of activity of the Purkinje cell to
0:
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(7)
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(8)
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with pkj set to 5.3 to allow
Purkinje cells to discharge at their physiological spontaneous rate of
~80 Hz (Thach, 1968), which corresponds to a 0.4 value for
Ppkj when using a 5 msec time-step. This
variable represents the total inhibitory effect of each Purkinje cell
on nucleus activity.
Similarly, the membrane potential for the nucleus cell is given by:
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(9)
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where the first term represents excitation of the
nucleus cell via active mossy fibers, the second represents inhibition from the Purkinje cells, and the third represents excitation from the
climbing fiber. j and n are the total number of
Purkinje cell and mossy fiber inputs to the nucleus cell. The activity
of the nucleus cell is then obtained by applying a threshold function (sigmoidal) to the neuron's membrane potential:
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(10)
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with nuc set to 6.0 to allow
the nucleus cell to discharge at its physiological spontaneous rate of
~40 Hz (Thach, 1968), which corresponds to a 0.2 value for
Pnuc when using a 5 msec time-step. This
variable represents the total inhibitory effect of the nucleus cell on
climbing fiber activity.
Finally, the membrane potential for the climbing fiber is given by:
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(11)
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where Eus represents the
excitatory effect of a possible unconditioned stimulus (US) and the
last term represents the inhibitory action of the nucleus cell on
climbing fibers (Knuc is typically set to 10 so
that Vcf can range from 
10 to 0 in the absence
of a US although Pnuc ranges only from 0 to 1).
The activity of the climbing fiber is then obtained by applying a
threshold function (sigmoidal) to its membrane potential:
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(12)
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with cf set to 3.3 to allow
the climbing fiber to discharge at its physiological spontaneous rate
of ~1 Hz (Keating and Thach, 1995), which corresponds to a 0.005 value for Pcf when using a 5 msec time-step.
The probabilities of activity for the input elements (i.e., granule
cells and mossy fibers) were specified and remained constant for each
simulation (Table 1). Thus, the excitatory connection between mossy
fibers and granule cells was not explicitly simulated, but rather the
probabilities of activity for these cells were independently chosen
from separate Gaussian distributions in the interval [0,1]. This
implementation allowed for independent control of the mean level of
activity of these inputs. For all the plasticity rules examined, the
rate at which gr Pkj and mf nuc
synapses changed increased with higher levels of granule cell and mossy fiber activity, respectively (data not shown). However, as long as the
mean activity was kept constant when comparing different plasticity
rules, the conclusions of this study did not depend on a particular
choice for input activity. Typically, the mean levels of activity for
both inputs were ~50 Hz (Eccles et al., 1971) (i.e., the mean of the
Gaussian distributions was 0.25 with a 5 msec time step). In contrast
to the way the activities for inputs were specified, the activities of
basket/stellate cells, Purkinje cells, the nucleus cell, and the
climbing fiber were calculated for each time step based on the strength
of their presynaptic inputs. For these cells, activity was determined
by summing the weights of all active excitatory and inhibitory synapses
for a time bin and calculating a probability of activity from these inputs.
Plasticity rules at modifiable synapses. The key
assumption of our simulations is that two sets of synapses in the
cerebellum can undergo changes in strength during motor learning. On
the basis of evidence that supports this assumption (Robinson, 1976; Perrett and Mauk, 1995; Raymond et al., 1996; Mauk, 1997), we have
implemented a plasticity rule that specifies that
gr Pkj synapses decrease in strength when active in
the presence of a climbing fiber input and increase in strength when
active in the absence of a climbing fiber input (Sakurai, 1987; Ito,
1989; Hirano, 1990; Shibuki and Okada, 1992; Linden, 1994; Salin et
al., 1996). Although these studies clearly illustrate the dependence of
LTD/LTP at gr Pkj synapses on climbing fiber activity,
there is a lack of evidence to support the model's assumption that LTD
and LTP can reverse each other. In fact, the evidence to date suggests that LTP/LTD at this synapse is not bidirectional because the expression of the former seems to be presynaptic (Salin et al., 1996),
whereas that of the later is clearly postsynaptic (Linden, 1994).
However, the fact that this reversal is a feature common to various CNS
synapses, including hippocampus (Heynen et al., 1996), visual cortex
(Kirkwood et al., 1993), motor cortex (Hess and Donoghue, 1996), and
inferotemporal cortex (Chen et al., 1996), highlights the plausibility
that gr Pkj synapses could also display the same property.
Given known anatomical constraints (Eccles et al., 1967; Ito, 1984), we
have also implemented bidirectional plasticity at mf nuc synapses controlled by activity in one of the
inputs to the nucleus cell (i.e., climbing fiber or Purkinje cells) or in the nucleus cell itself (see Fig. 2).
The presence of plasticity at mf nuc synapses is at
present an assumption of the model. Although these synapses contain
NMDA receptors (Cull-Candy et al., 1998), there is only one report of
plasticity at this location, and even then the induction protocol
required nonphysiological stimulating conditions (Racine et al., 1986).
Mathematically, the changes in strength at gr Pkj and
mf nuc synapses are implemented as follows:
(1) Plasticity at gr Pkj synapses:
(2) plasticity at mf nuc synapses:
where the different Spike terms are calculated as
shown in Equations 2 and 3, and
+gr = 0.001,
gr = 0.199,
+mf = 0.001, and
mf = 0.0015 are constants that
represent step decreases or increases in the strengths of
gr Pkj and mf nuc synapses that were
kept the same for all simulations independent of the plasticity rule implemented at the mossy fiber synapse. Under this mathematical representation, the ith gr Pkj or
mf nuc synapse is modified every time it is active
(i.e., when Spikeigr or
Spikeimf equals 1) and will undergo LTP or
LTD depending on whether the Spike term of the signal
controlling plasticity equals 0 or 1. However, the results presented
here did not change when these plasticity rules were modified to allow
a range of frequencies in the plasticity-controlling signals that left
synaptic strengths unchanged.
The precise relative timing between the Spike variables
controlling synaptic changes in the implementation of the plasticity rules outlined above deserves further consideration. For example, the
rule for LTD at gr Pkj synapses results in a decrease
of the strength of synapses that are active simultaneously with a climbing fiber input. Although it is true that only sharp temporal relations between these signals have been found to promote LTD in the
Purkinje cells of the electric fish Gnathonemus petersii (Bell et al., 1997), this is not so in most cases. The lack of temporal
constraints in protocols capable of inducing cerebellar long-term
depression is apparent in various studies in which effective timing
relations between presynaptic activation of gr Pkj
synapses and climbing fiber inputs have ranged from presynaptic first
by 250 msec (Chen and Thompson, 1995) to climbing fiber first by 125 msec (Ekerot and Kano, 1989). Our choice of a 0 msec interval (i.e.,
simultaneous activity in gr Pkj synapse and climbing fiber input) represents a consensus interval that seems to allow induction of plasticity under various experimental conditions (Ekerot
and Kano, 1989; Karachot et al., 1994; Chen and Thompson, 1995).
Simulating eyelid conditioning. During Pavlovian
conditioning, the presentation of an initially neutral conditioned
stimulus (CS) is paired with a reinforcing US. After repeated CS+US
pairings, the CS acquires the ability to elicit a conditioned response. In the case of eyelid conditioning, paired presentation of tone-CS and
puff-US results in a conditioned eyelid closure elicited by the tone
(Schneiderman et al., 1962). Converging evidence from a number of
laboratories suggests that the tone is conveyed to the cerebellum by
mossy fibers (Steinmetz et al., 1985, 1986, 1987, 1988; Solomon et al.,
1986; Lewis et al., 1987) the puff is conveyed by climbing fibers
(McCormick et al., 1985; Mauk et al., 1986), and that increases in the
activity of cerebellar output cells in the anterior interpositus
nucleus drive the expression of eyelid response (McCormick and
Thompson, 1984).
Given the straightforward manner in which these stimuli map onto the
afferent pathways to the cerebellum, eyelid conditioning can be
relatively easily represented in our simulations. Adding a constant
(Eus) to the firing probability of the climbing fiber simulates the
presence of the US during acquisition trials. This constant was chosen
so that initially, when the US is presented, the probability of
activity of the climbing fiber is 1. The CS is represented by changing
the probability of firing of the ith granule cell from
Pgri to PCSgri
and that of the ith mossy fiber from Pmfi to
PCSmfi for the duration of the stimulus.
Although the conclusions of the paper with respect to the stability of
plasticity rules did not depend on these particular activities, our
results suggest that in general, learning occurred faster as
PCS became more different from P (data not
shown). For the simulations shown, separate probabilities for mossy
fibers and granule cells were assigned from Gaussian distributions with
a mean of 0.25 and a variance of 0.20 such that
0 
Pgri, PCSgri, Pmfi, PCSmfi 1.
With a 5 msec time-step, the 0.25 mean represents a discharge rate of
50 Hz for these inputs.
Timing of eyelid conditioning stimuli. Models of
classical conditioning have been traditionally divided into three broad
categories (Gluck et al., 1990): trials-level, temporal, and real-time
models. Trials-level models treat the CS as a unitary event and
consider only the net effects of each trial. Temporal models include
factors that address the limited range of interstimulus intervals that promote conditioning (the ISI function), and real-time models also
address the timing or temporal properties of conditioned responses. In
this respect, the present simulations represent a trials-level analysis
because we did not attempt to capture the sensitivity that exists to
the temporal relationships between CS and US. Therefore, our
simulations can only describe the net effects of a training trial on
the strength of CS-US associations by presenting simultaneous CS-US
pairings lasting a single time-step.
Types of simulations. The simulations were of two
general forms. During "conditioning simulations," inputs
corresponded to the presentation of stimuli during Pavlovian
conditioning as described above without any activity between learning
trials. In contrast, "background simulations" included background
activity between occasional presentations of the CS. However, the
strengths of gr Pkj and mf nuc
synapses were not allowed to change during these occasional CS
presentations, which were necessary only to assess the retention of a
memory previously formed by a conditioning simulation.
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RESULTS |
Acquisition of conditioned eyelid responses
The architecture and events of the simulations were based on the
well characterized synaptic organization of the cerebellum (Eccles et
al., 1967; Ito, 1984; Voogd and Glickstein, 1998) and its relationship
to Pavlovian eyelid conditioning (Thompson, 1986; Thompson and Krupa,
1994; Mauk and Donegan, 1997) (Fig. 1).
Eyelid conditioning involves the paired presentation of a CS such as a
tone and a reinforcing US such as a puff of air directed at the eye.
With repeated trials, the CS acquires the ability to elicit conditioned
closure of the eyelid (Schneiderman et al., 1962). Previous studies
have demonstrated that information about the CS and US is conveyed to
the cerebellum via mossy fiber (Steinmetz et al., 1985, 1986, 1987,
1988; Solomon et al., 1986; Lewis et al., 1987) and climbing fiber
inputs (McCormick et al., 1985; Mauk et al., 1986), respectively, and
that output of the cerebellum via the interpositus nucleus is necessary
for the expression of the conditioned responses (McCormick and
Thompson, 1984). It is this wealth of knowledge about the synaptic
organization of the cerebellum, its relationship to eyelid
conditioning, and sites and mechanisms for plasticity that make this
brain system an ideal structure with which to study interactions
between network properties and forms of plasticity. Furthermore, the
correspondence between eyelid conditioning and cerebellar inputs and
outputs permits a relatively straightforward representation with
simulations. Presentation of the CS is simulated by briefly altering
the background activities of the mossy fiber (and granule cell) inputs,
and presentation of the US is simulated by applying a transient
excitatory input to the climbing fiber. Increases in simulated nucleus
cell output during the CS are then taken as an index of conditioning.
These eyelid conditioning trials were presented in three separate
simulations that incorporated the well characterized climbing
fiber-dependent plasticity at gr Pkj synapses and
one of three possible plasticity rules at mf nuc
synapses (Fig. 2).
Although under the conditions illustrated in Fig. 2 a
mf nuc synapse is modified every time it is
active, our results do not depend on this simplification. We also
implemented plasticity under the control of signals related to
rates of activity rather than single spikes. In
this case, it is possible to specify a range of activity frequencies
that will not modify the strength of active synapses. For example, a
Hebbian rule could induce LTP when the nucleus cell is firing at a high
frequency, induce LTD at low nucleus cell frequencies, and leave
synaptic strengths unchanged when the nucleus is firing at intermediate
frequencies. The nature of the results did not depend on these
different implementations as long as the frequencies that induced
plasticity could be attained, thus providing opportunities for synaptic modification.
Initially, learning was simulated with the standard modeling practice
of permitting plasticity only during the presentation of the
conditioning trial (Fujita, 1982; Moore et al., 1989; Gluck et al.,
1990; Fiala et al., 1996) and not at other times. Under these
conditions, all three simulations acquired conditioned responses; as
training proceeded, they showed increases in nucleus cell output during
the simulated CS (Fig. 3). Moreover,
these conditioned responses were produced by a combination of
plasticity in both the cerebellar cortex and cerebellar nucleus, which
is consistent with results from previous studies on eyelid conditioning
and VOR adaptation (Robinson, 1976; Perrett et al., 1993; Raymond et
al., 1996; Mauk, 1997). For all three simulations, presentation of
mossy fiber/granule cell inputs paired with a climbing fiber input led
to the induction of LTD at the CS-activated gr Pkj synapses. As training proceeded, this resulted in a learned
decrease in Purkinje cell activity during the CS input (Fig. 3,
Purkinje contribution). Although the details differ slightly
for the three rules, training also led to the induction of LTP at the
mf nuc synapses (Fig. 3, Mossy fiber
contribution). For Hebbian and Purkinje-dependent rules, the
acquired decrease in Purkinje activity during the CS aids in the
induction of LTP at the mf nuc synapses either
directly (i.e., Purkinje-dependent rule) or by disinhibiting the
nucleus (i.e., Hebbian rule). For the climbing fiber-dependent rule,
the induction of LTP at mf nuc synapses is a direct
consequence of the activation of the climbing fiber during the US and
proceeds independently of the activity of the Purkinje cell during
training. Thus, for the three simulations, LTD in the cerebellar cortex reduced the strength of the inhibitory action of Purkinje cells onto
the nucleus, whereas LTP of mossy fibers increased the strength of the
excitatory input to the nucleus. Both of these changes contributed to
the expression of the conditioned response by increasing the
probability of firing of the nucleus cell. When the CS was presented in
the absence of the US, all three simulations produced similar
extinction of the conditioned response by reversing the changes that
had occurred during acquisition (data not shown).
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Figure 3.
Acquisition of simulated conditioned responses
when a Hebbian (a), climbing fiber-dependent (b),
or Purkinje-dependent (c) plasticity rule is implemented at
mf nuc synapses and plasticity is permitted only
during the training trial. Increases in nucleus cell activity during
presentation of the CS (thick black line) provide a measure
of the conditioned response amplitude. The contributions made to the
conditioned response by plasticity at gr Pkj synapses
in the cerebellar cortex (thin black line) and by plasticity
at mf nuc synapses in the cerebellar nucleus
(gray line) are also shown. Consistent with existing data
from eyelid conditioning and VOR adaptation, conditioned responses were
produced by a combination of plasticity at these two sites.
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Retention of conditioned eyelid responses
Stopping here might lead to the conclusion that a synaptic
mechanism for the long-lasting expression of LTD and LTP at the gr Pkj and mf nuc synapses would
ensure the persistence of this memory for motor learning. Simply
continuing the simulation with a low level of background input activity
and with the plasticity rules operational shows that this is not
necessarily true. As shown in Figure 4,
two of the nucleus plasticity rules, Hebbian (dark gray line) and
climbing fiber-dependent (light gray line), produced a spontaneous
drift in the strengths of the gr Pkj and
mf nuc synapses. This rapidly caused synapses to
saturate at their maximum or minimum possible values, erasing the
learned pattern of synaptic weights, and thus abolishing the previously learned response. Although specific parameters determine whether the
synapses drift upward and saturate at maximum values or downward to
minimum values, the tendency to drift itself is parameter independent and occurs for all non-zero forms of background activity.
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Figure 4.
Retention of simulated conditioned responses when
plasticity rules are operational during background input activity.
a, The left graph shows that the amplitude of the
conditioned response decreases rapidly when either a Hebbian
(dark gray line) or climbing fiber-dependent (light
gray line) plasticity rule is implemented at
mf nuc synapses. In contrast, when a
Purkinje-dependent rule is used (black line), increases in
nucleus cell activity during presentation of the CS could be observed
for much longer periods of time (right graph). b,
The effects that implementing different rules for plasticity have on
the persistence of memory can be further illustrated by assuming that
the pattern of strengths at gr Pkj synapses
corresponds to the picture of Salvador Dali. Light pixels correspond to
weak synapses, and dark pixels correspond to strong synapses.
Implementing a Purkinje-dependent rule in the cerebellar nucleus drives
the system to a state of equilibrium where memories are retained,
because although synapses are continually changing (note that the
encoded picture slowly degrades with time), they are as likely to
increase in strength as they are to decrease (top row). In
contrast, Hebbian (data not shown) or climbing fiber-dependent
(bottom row) rules produce a spontaneous drift of synaptic
strength, which ultimately results in saturation and complete loss of
memory.
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In contrast to this rapid loss of memory, responses were retained
orders of magnitude longer when the simulations implemented plasticity
at mf nuc synapses that was directly controlled by Purkinje cell activity (Fig. 4, black line). As shown in the
top graph of Figure 5, in these
simulations the activity of the signal controlling plasticity at
gr Pkj synapses (i.e., climbing fiber activity; thick
black line) was self-regulated to an equilibrium level where
gr Pkj synapses were as likely to decrease in strength when active during a climbing fiber input as they were to increase in
strength when active in its absence. Similarly, the activity of the
signal controlling plasticity at mf nuc synapses
(i.e., Purkinje cell activity; thin black line) was maintained at an equilibrium level that balanced LTP and LTD at mf nuc
synapses such that although the strength of a synapse was modified
every time it was active, the net change was zero (Fig. 5, bottom
graph). When the system is at this equilibrium state, synapses can
still experience LTD and LTP events, thus essentially performing a
random walk that eventually erases memories. The key is that memories last much longer because this equilibrium state (see Appendix for a
formal mathematical analysis of the conditions that lead to this
equilibrium state) does not promote the directed drift toward maximum
or minimum synaptic values observed when unstable rules (i.e., Hebbian
and climbing fiber dependent) were implemented at
mf nuc synapses. Furthermore, the properties that
allow the Purkinje-dependent rule to maintain synaptic strength do not
preclude the extinction of conditioned responses when the CS is
presented in the absence of the US.
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Figure 5.
Properties of simulations that implement a
Purkinje-dependent plasticity rule in the cerebellar nucleus. As shown
in the schematic representation of the cerebellar circuitry, in these
simulations the climbing fiber signal (thick black line)
controls plasticity at gr Pkj synapses, whereas the
Purkinje cell signal (thin black line) controls plasticity
at mf nuc synapses. Under these conditions, simulated
climbing fiber and Purkinje cell activities are self-regulated to
equilibrium levels (as predicted by Eq. A2 and A9, respectively) even
when these equilibrium levels are different from each other. Although
the strength of a mf nuc or gr Pkj
synapse is modified each time the synapse is active, the bottom
graph shows that at these equilibrium levels synapses are as
likely to increase as they are to decrease in strength.
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Differences between stable and unstable rules
The fundamental differences between simulations that incorporate
the two unstable rules (Hebbian and climbing fiber dependent) and those
that implement a rule that is directly controlled by the level of
Purkinje cell activity are illustrated in Figures 5 and
6. A Purkinje-dependent rule for
mf nuc synapses allows for independent regulation of
plasticity at gr Pkj synapses controlled by the level
of climbing fiber activity and at mf nuc synapses
controlled by the level of Purkinje cell activity. The ability of the
simulated cerebellar network to independently regulate climbing fiber
and Purkinje cell activities is apparent from the synaptic organization
of the cerebellum. As shown in Figure 1, in addition to the climbing
fiber input, Purkinje cells also receive a modifiable input from
granule cells (the gr Pkj synapses). The consequence
is that although climbing fiber activity affects the
activity of the Purkinje cell, it does not determine it. Figure 1 also
illustrates that Purkinje cell activity, by itself, does not determine
the activity of its associated climbing fiber. Although the nucleus
cell functionally connects Purkinje cells to the climbing fiber, the
activity of the nucleus cell is modulated by its modifiable
mf nuc input. As a consequence, the activity of the
nucleus cell (and thus the climbing fiber) is affected but
not determined by the activity of its Purkinje cell input. Thus, it is
possible for a level of Purkinje cell activity suitable for stability
of mf nuc synapses to co-exist with a level of
climbing fiber activity suitable for stability of
gr Pkj synapses.
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Figure 6.
Properties of simulations that implement Hebbian
or climbing fiber-dependent plasticity rules in the cerebellar nucleus.
As shown in the schematic representation of the cerebellum, simulations
that implement a climbing fiber-dependent or Hebbian rule in the
nucleus automatically place both gr Pkj and
mf nuc synapses under the control of a single signal
related to the activity of the climbing fiber (thick black
line). For the parameters used in these simulations, the level of
climbing fiber activity required for equilibrium of
gr Pkj synapses is shown in the left
column. The top graph in the left column
shows that in simulations with plasticity at mf nuc
synapses turned off, climbing fiber activity was self-regulated to the
level predicted by Equation A2, and that at this equilibrium level the
mean strength of gr Pkj synapses remained constant
(left column, bottom graph). Conversely, the level of
simulated climbing fiber activity required for equilibrium of
mf nuc synapses is shown in the middle
column. The top graph in the middle column
shows that in simulations with plasticity at gr Pkj
synapses turned off, climbing fiber activity was self-regulated to the
level predicted by Equation A7, and that at this equilibrium level the
mean strength of mf nuc synapses remained constant
(middle column, bottom graph). However, as shown in the
right column, in simulations with plasticity turned on at
both sites, climbing fiber activity fell between these two equilibrium
levels (top graph), such that both sets of synapses drifted
(bottom graph). Spontaneous drift is reduced only in
simulations that use the single set of parameters that makes these two
equilibrium levels (Eq. A2 and A7) equal to each other (data not
shown).
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The key difference that prevents stability of synaptic strength with
Hebbian and climbing fiber-dependent rules is in each case a strong
coupling of the signals that control plasticity at both modifiable
sites. This coupling means that simultaneous stability of both
mf nuc and gr Pkj synapses can be
achieved only under a very specific set of parameters (see Appendix for
a formal mathematical analysis). In the case of simulations that implement a climbing fiber-dependent rule (Fig. 6), changes at mf nuc and gr Pkj synapses are
coupled because the signals controlling plasticity (i.e., climbing
fiber activity; thick black line) are identical at both sites. The
situation is similar when a Hebbian rule is implemented at
mf nuc synapses because the signal controlling
plasticity at gr Pkj synapses (i.e., climbing fiber
activity) is always completely determined by the signal controlling
plasticity at mf nuc synapses (i.e., nucleus cell activity). Therefore, for these two cases there are two sets of modifiable synapses under the control of a single signal related to the
activity of the climbing fiber. Unless a specific set of parameters is
carefully chosen so that both mf nuc and
gr Pkj synaptic strengths can be maintained constant
by the same level of climbing fiber activity, they will spontaneously
drift, erasing any previously formed memory (Fig. 6, bottom
right). This situation is analogous to controlling the temperature
of a single room with two thermostats. Unless they are set to
exactly the same temperature (analogous to severe parameter
constraints), one will always try to decrease the temperature of the
room (analogous to synaptic strengths spontaneously drifting at one
site of plasticity), whereas the other will continuously try to
increase room temperature (analogous to synaptic strengths drifting at
the other site).
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DISCUSSION |
Our results illustrate that a molecular mechanism for persistent
expression of synaptic plasticity is not sufficient to explain the
enduring retention of memories. Instead, the contribution of synaptic
plasticity to memory must be considered in the context of the circuits
in which modifiable synapses reside. This is often taken into account
when considering the relationship between the induction of plasticity
and the acquisition of memories. Debates continue, for example, as to
whether the patterns of stimulation that induce LTP or LTD are
physiological and actually occur in vivo (Otto et al., 1991;
Heynen et al., 1996; De Schutter, 1997; Mauk et al., 1997). Our results
extend this thinking by highlighting the importance of the interaction
between circuits and synapses in the stability of synaptic strengths
and the persistence of memories.
The main prediction of our results is that when the background activity
of the cerebellar network is considered, only one of three seemingly
plausible forms of plasticity in the cerebellar nucleus interacts with
network properties to produce learning. Although simulations with
Hebbian and climbing fiber-dependent rules could learn in the absence
of background inputs, under more realistic conditions their inherent
tendency to produce spontaneous drift in synaptic weights precluded
their ability to learn and to retain responses. In contrast,
Purkinje-controlled plasticity at mf nuc synapses in
the cerebellar nucleus appeared to promote an equilibrium of activity
that prevented spontaneous drift of synaptic weights and
permitted both learning and retention, independent of the amount of
background input. Importantly, the ability of this rule to prevent
synaptic weight drift did not preclude further learning because
conditioned responses could still be extinguished when the CS was
presented in the absence of the US.
The potential biological relevance of these data is enhanced by
observations that the results are not peculiar to a particular set of
parameters. The connectivity of the simulations reflects fundamental
systems-level features of the cerebellum and therefore does not depend
on exact specification of a large number of parameters. Still, it was
necessary to stipulate values for a few relatively free parameters such
as the magnitudes of change for LTP and LTD events and the rates of
background synaptic activity. However, reducing the model to its
critical essentials shows that our results do not vary significantly
over a comprehensive range of values for these parameters as long as
three fundamental constraints are satisfied: (1)
gr Pkj synapses in the cerebellar cortex and
mf nuc synapses in the cerebellar nucleus are
bidirectionally modifiable with LTP and LTD being able to reverse the
effects of each other, (2) in addition to being active during learning, these synapses can also be active during non-learning periods providing
further opportunities to modify their strength, and (3) connections in
the cerebellar-olivary system are topographically organized in a loop,
such that each Purkinje cell influences (via projections to the
nucleus) its own climbing fiber input. This observation is consistent
with mathematical analyses suggesting that the equilibrium produced by
climbing fiber-dependent plasticity in the cortex combined with
Purkinje-dependent plasticity in the nucleus is parameter independent
(Kenyon and Mauk, 1994). Similarly, we find that the spontaneous
drift of synaptic weights that occurred with Hebbian or climbing
fiber-controlled plasticity in the nucleus was an intrinsic and robust
property that arises from the interaction between those forms of
plasticity and the connectivity of the network. For these unstable
rules there exists only one set of parameters where spontaneous drift
is decreased significantly, but even in this case a second form of
instability operates and eventually causes saturation of synaptic
weights (data not shown). This robustness shows that neither our
results nor their implications depend on tweaking of free parameters,
but rather reflect basic characteristics that emerge from interactions
between cerebellar connectivity and particular forms of plasticity at
the two sites.
Although the simulations relate to cerebellar-mediated motor learning,
the implications of our results are not specific to cerebellar
synapses. A memory, for example, might be encoded by the induction of
LTP at a pattern of hippocampal or neocortical synapses. Any unchecked
tendency for systematic drift in synaptic strengths would saturate all
synapses at their maximum or minimum value, erasing the pattern of
strengths and destroying the memory. As an example of a potential
instability inherent in Hebbian plasticity, the strengthened synapses
might increase postsynaptic activity and lead to further induction of
LTP at other synapses onto the same postsynaptic cell (Sejnowski and
Tesauro, 1989; Brown et al., 1990). Recent evidence suggests that
intrinsic inhibitory circuitry may help prevent such runaway changes in
a number of systems (Artola and Singer, 1987; Steele, Mauk, 1998).
Regardless of such details, our results further illustrate how
mismatches between the properties of plasticity and the properties of
circuits can erase memories despite the existence of molecular
mechanisms that are otherwise capable of long-term expression of plasticity.
For the cerebellum, our results suggest that the stability of synaptic
weights results from a self-regulating equilibrium of climbing fiber
activity controlling LTP/LTD in the cortex and a similar equilibrium of
Purkinje activity controlling LTP/LTD in the nucleus. For climbing
fibers, this equilibrium depends on circuitry that allows the Purkinje
cell to regulate the activity of its single climbing fiber input
(Kenyon et al., 1998). In addition to the extensive anatomical tracing
investigations that support this precision of connectivity (Voogd and
Bigare, 1980; Buisseret-Delmas and Angaut, 1993; Ruigrok, 1997; Voogd
and Glickstein, 1998), recent recording studies in primates provide
more direct evidence that Purkinje cell activity can calibrate the
activity of its own climbing fiber input (Miall et al., 1998). Miall et
al. (1998) showed a small but significant relationship between
increased Purkinje activity and subsequent increases in the activity of its climbing fiber input, which would presumably induce LTD and drive
back down the activities of the Purkinje cell and climbing fiber. Our
results and previous mathematical analyses (Kenyon et al., 1998)
suggest how such self-regulation of climbing fiber activity could
combine with plasticity in the cerebellar cortex to maintain a
background equilibrium level of climbing fiber activity where the
effects of any LTP and LTD are in balance. The present results extend
this concept to Purkinje cell control of plasticity in the cerebellar
nucleus. With this rule, plasticity at the two sites is controlled by
two different signals, each of which can be at their equilibrium level.
However, with the rules found to be unstable, there is in each case a
single signal (climber fiber activity) that controls plasticity at both
sites. The resulting climbing fiber activity reflects a compromise
between the equilibrium of activity needed at each site, producing
spontaneous drift of strength at both sets of synapses.
Although the controlling signals and underlying mechanisms of
plasticity in the cerebellar nuclei have not been demonstrated explicitly, the implications of the present results are consistent both
with existing theory and data from two forms of cerebellar-mediated motor learning. In general, analysis of Pavlovian eyelid conditioning and adaptation of the VOR have yielded concordant ideas regarding cerebellar mechanisms of learning (Raymond et al., 1996). For both
behavioral paradigms, lesion studies implicate plasticity in the
cerebellar cortex and nucleus and suggest that lesions of the
cerebellar cortex block the induction of plasticity in the cerebellar
nuclei (Robinson, 1976; Perrett and Mauk, 1995; Raymond et al., 1996;
Mauk, 1997). To explain adaptation of the VOR, Miles and Lisberger
(1981; Lisberger, 1994) proposed the hypothesis that plasticity in the
cerebellar nucleus contributes to this form of motor learning and that
its induction is controlled by inputs from the Purkinje cells. The
striking similarity in results from these two different behavioral
systems suggests that the mechanism implied is not specific to eyelid
conditioning or VOR adaptation but rather is a general feature of
cerebellar processing.
The findings of Llinas and Muhlethaler (1988) suggest a concrete but
speculative mechanism for inducing LTP/LTD at mf nuc synapses that is consistent with Purkinje cell control of plasticity at
this site. Recordings from cerebellar nucleus cells in vitro have revealed a calcium conductance whose activation required depolarization from a hyperpolarized state, rather than depolarization from the resting potential (Llinas and Muhlethaler, 1988). Given that
Purkinje cells are normally active at high spontaneous rates, it
seems possible that this calcium conductance could be activated during
transient decreases in the ongoing inhibitory input that nucleus cells
receive from Purkinje cells. Drawing parallels with LTP and LTD in the
hippocampus and neocortex (Artola et al., 1990; Mulkey and Malenka,
1992; Dudek and Bear, 1993), the level of Ca2+ in
the postsynaptic nucleus cell may be an important factor in determining
whether active synapses undergo LTP or LTD. Thus, mf nuc synapses may increase in strength when coactive
during the high levels of calcium likely to exist during
transient decreases in Purkinje cell activity and decrease in strength
when active during lower levels of calcium, as may occur during strong
inhibitory input from Purkinje cells.
Although a great deal of effort has been applied to the analysis of the
molecular basis of persistent expression of synaptic plasticity, the
present results highlight the parallel importance of understanding how
these mechanisms are influenced by the networks that provide their
inputs. Our results do not obviate the importance or need to study
molecular mechanisms of persistent expression. Rather, they suggest the
additional challenge of understanding how interactions between these
plasticity mechanisms and the properties of the circuits permit
induction of plasticity when learning occurs and prevent net changes
otherwise. The cerebellum provides a clear example. The degree to which
the anatomy and physiology of the cerebellum is known has inspired many
models of how it mediates motor learning (Fujita, 1982; Moore et al.,
1989; Gluck et al., 1990; Fiala et al., 1996). However, we are not
aware of a cerebellar model or simulation that takes into account how
synaptic strength remains constant when synapses are activated
after movements have been learned. We suggest that any
realistic attempt to understand or to model learning in the brain, and
specifically motor learning in the cerebellum, must tackle the
challenge of ongoing synapse activity and its implications for the
induction of unwanted activity-dependent plasticity.
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FOOTNOTES |
Received March 16, 1999; revised May 7, 1999; accepted May 25, 1999.
This work was supported by National Institutes of Health Grant
MH 57051.
Correspondence should be addressed to Dr. Michael D. Mauk, Department
of Neurobiology and Anatomy, University of Texas Medical School, 6431 Fannin, Houston, TX 77030.
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APPENDIX |
Mathematical analysis of plasticity rules at
mf nuc synapses
The intuitive differences between the three rules for plasticity
at mf nuc synapses described above can be formalized
by considering the expected change in synaptic strength on any given time step. For gr Pkj synapses in the cerebellar
cortex, an expression for this expected change can be obtained by
combining the conditions that lead to LTD with those that lead to
LTP:
This first term in the equation simply formalizes the well
characterized rule for LTD observed at gr Pkj synapses
in the cerebellar cortex. Thus, a gr Pkj synapse is
expected to undergo LTD by an amount equal to
gr when both its probability of being
active, Pigr, and the probability of a
climbing fiber input, Pcf, are high. The
second term implements LTP by increasing the weight of active
gr Pkj synapses by +gr
when the probability of a climbing fiber input is low [i.e., when
(1 Pcf) is high.]
As originally reported by Kenyon et al. (1998), setting this equation
to zero yields an expression for the equilibrium level of climbing
fiber activity at which the expected change in gr Pkj
synaptic strength is zero. That is:
TOP
ABSTRACT
INTRODUCTION
