Computational Models of Cerebellar
Long-Term Memory
Hideaki Ogasawara and Mitsuo Kawato

Introduction
Our brain is capable of learning new things while maintaining old memory. Retentionof information requires stability, and new learning requires plasticity. As a memory
device, neurons have to meet these contradictory requirements (the “stability versus
plasticity dilemma” [1]), but stochastic noise makes this duty still more difficult. The
dendritic spine, the key unit of neuronal information processing, is very small (∼1 µm
or less in diameter) and contains only a limited number of each molecular species.
For instance, the number of α-amino-3-hydroxy-5-methylisoxazole-4-propionic acid
(AMPA)-type glutamate receptors (AMPARs) in a parallel fiber (PF)–Purkinje cell(PC) synapse is as small as 4 to 73 [2]. In such a minute environment, stochastic
fluctuations come into play and, affect the signaling pathways underlying memory
formation and maintenance. How do neurons handle the stability versus plasticity
dilemma without being overwhelmed by the noise? In this chapter, we address thisissue by reviewing several theoretical studies of cerebellar long-term depression(LTD) and simulating simple models.

Cerebellar LTD
The main neurons and wirings in the cerebellar cortex include PCs, PFs, and climbing
fibers (CFs). PCs provide the sole output from the cortex, and each PC receives two
types of excitatory inputs: one from hundreds of thousands of PFs and the other froma single CF. Marr-Albus-Ito theory [3–5] states that their neuronal circuit underlies

H. Ogasawara
National Institute of Information and Communications Technology. 2-2-2, Hikaridai,
Seika, Kyoto, 619-0288, Japan
H. Ogasawara and M. Kawato
ATR Computational Neuroscience Laboratories, 2-2-2 Hikaridai, Seika, Kyoto 619-0288, Japan

S. Nakanishi et al. (eds.), Systems Biology,
DOI:10.1007/978-4-431-87704-2_18, © Springer 2009

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supervised learning of the cerebellum. In this theory, PFs provide a sensorimotor
context to PCs, while CFs carry teaching signals that modify PF-PC synapses in an
associative manner.
In fact, the transmission efficacy of a PF–PC synapse is reduced when the CF
and PF are repetitively and synchronously activated (cerebellar LTD). The molecular mechanisms are comprehensively reviewed elsewhere [6–11]. Briefly, PF firing
induces production of inositol 1,4,5-trisphosphate (IP3) through the mGluR1
metabotropic glutamate receptor pathway. CF firing depolarizes the PC and induces
Ca2+ influx through voltage-gated calcium channels (Fig. 1a). Therefore, [IP3] and[Ca2+] represent PF and CF activities, respectively (In this chapter, [substance]
stands for the concentration of the substance.) Synergistic increase of [IP3] and[Ca2+] activates IP3 receptors (IP3Rs) located in a calcium store of the dendritic
spine and results in an enormous Ca2+ release. Ca2+ activates protein kinase C(PKC), which in turn phosphorylates the GluR2 subunit of AMPARs. Phosphorylated
receptors are removed from the postsynaptic membrane through endocytosis.

PF
NO

Glu

P

AMPAR

AMPAR

mGluR1

sGC
cGMP
PKG
G-substrate
PP2A

Gq protein
PKC
MAPKKK
MAPK-PKC
Positive
Feedback
Loop
MAPKK

AA

DAG

PLC
PIP2

Ca2+
IP3
PLA2

IP3R

MAPK
Ca2+

Calcium store

b

a
VGCC
Depolarization
CF

Fig. 1 Schematic view of signaling cascades in cerebellar long-term depression (LTD). Reactions
surrounded by the bold gray line take place inside the dendritic spine. a Parallel fiber–climbing
fiber (PF-CF) coincidence detection mechanisms (light gray area; right side) [14]. Glu, glutamate; PKG, cGMP-dependent protein kinase; PIP2, phosphatidylinositol bisphosphate. b The
mitogen-activated protein kinase (MAPK)–PKC positive feedback loop [13] and its peripherals(dark gray area; left side). AA, arachidonic acid; AMPA, α-amino-3-hydroxy-5-methylisoxazole4-propionic acid; AMPAR, AMPA-type glutamate receptor; DAG, diacyl glycerol. (From [15],with permission of S. Karger AG, Basel)Computational Models of Cerebellar Long-Term Memory

171

Essential molecules for LTD include mitogen-activated protein kinase (MAPK),MAPK kinase (MAPKK), MAPKK kinase (MAPKKK), PKC, phospholipases A2(PLA2) and C (PLC), arachidonic acid (AA), IP3, and many others [6–11].

Models to Explain Cerebellar LTD
In numerous situations such as development and cell-cycle progression, cells takediscrete states and jump from one to another, avoiding intermediate states. For
example, frog eggs are either mature or immature, and there is no half-mature egg[12]. This kind of switch-like behavior is termed “bistability.” Bistability is particularly important in neurobiology because it is implicated in the storage of cellular
information. Previous studies have shown that Ca2+-induced Ca2+ release (CICR)
and the MAPK-PKC positive feedback loop in the PC are among many bistable
dynamics of the cell (Fig. 1) [13,14] (reviewed in [15]).

Calcium Dynamics Model
Calcium release via IP3Rs is a supralinear function of calcium influx; the greater an
input stimulus (calcium influx) is, the much more greater the output signal (calcium
release) is. Doi et al. [14] simulated the molecular mechanism and demonstrated
that a positive feedback loop Ca2+ → IP3Rs → Ca2+ is responsible for the supralinearity of calcium release. They also showed that the optimal PF-CF delay (∼100 ms)
for calcium release and LTD is attributable to the slow mGluR1 pathway that takes
time to biochemically produce IP3, rather than to calcium kinetics and the binding
properties of the receptor [16] (see also comments in [17]).

MAPK-PKC Positive Feedback Loop
Kuroda et al. [13] modeled the intracellular signaling cascade of cerebellar LTD(Fig. 1b). Their model predicted that MAPK and PKC form a bistable positive feedback loop, which mediates the intermediate phase (<∼40 min) of LTD. Subsequentneurobiological experiments strongly supported their predictions [18,19].

Cascade of Excitable Dynamics
How do neurons resolve the stability versus plasticity dilemma [1] in stochastic
noise? Taking into account the fact that bistability is ubiquitous in signaling pathways [20], one of us has proposed a cascade of excitable dynamics as a likely

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Low
Low

High

High

StimuliFast
dynamics

Slower andmore stable
dynamics

Fig. 2 Threshold cascade model (see text for explanation). (From [15], with permission of
S. Karger AG, Basel)mechanism (Fig. 2) [21]. In the model (detailed in [15]), multiple bistable dynamics
with various time constants are connected in tandem, in the order of fast to slow.
Each dynamics has an energy barrier that correlates with its time constant (the fastera switch is, the lower its barrier is), and an active dynamics stimulates the next one
that is more stable and requires larger inputs for activation. In this way, when external stimuli activate the fastest system, the activity is transmitted from one to anotherslower one, and finally to the slowest one. Such a cascade of multiple bistable
dynamics will form long-term memory that is stable, still open to modifications, androbust to noise. The model is applicable to at least the first two steps of cerebellar
LTD, that is, CICR and the MAPK-PKC positive feedback loop (Fig. 1).

Protein Kinase Mζ Model
Based on these studies and an abundance of biological literature, we have proposed
a conceptual model [15] in which cerebellar LTD is induced and maintained bysuccessive activation of three bistable dynamics with different time scales: CICR,
the MAPK-PKC positive feedback loop, and protein kinase Mζ (PKMζ)-induced
PKMζ synthesis.
PKMζ is a persistently active enzyme, consisting of the catalytic domain of an
atypical PKC isoform, PKCζ. It is expressed in a brain-specific manner. A series ofexperiments performed mainly by Sacktor’s group demonstrates accumulating evidence that PKMζ plays a pivotal role in long-term maintenance of memory (exhaustively reviewed in [22]). Briefly, a procedure to induce hippocampal LTP (LTD)upregulates (downregulates) PKMζ. Injection of a PKMζ inhibitor, ZIP, to the hippocampus reverses LTP in vivo and produces persistent loss of 1-day-old spatial
information. More surprisingly, an injection of ZIP to the rat neocortex, which isregarded as the long-term repository of memory, erases associative memories as old
as 1 month [23]. These findings strongly suggest that PKMζ is the (semi-)permanent
form of memory trace. PKMζ is also expressed in the cerebellar cortex [24,25].
Because cerebellar plasticity is regarded as a mirror image of hippocampal plasticity
and they share much of their signaling networks [6,7,9,10,26], PKMζ activity islikely to be central to cerebellar LTD, as it is to hippocampal LTP.

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Sometimes hippocampal studies advance far ahead of its cerebellar counterparts.
For instance, calmodulin-dependent protein kinase II was discovered to be essential
for hippocampal plasticity and learning in 1992 [27], but it was not until 2006 thatthe kinase was shown to be also necessary for cerebellar memory [28]. One of themain purposes of this model is to shorten the interval by calling for experiments tocharacterize the role of PKMζ in the cerebellum.
Pathways
PKMζ induces PKMζ expression, and this enzyme can act in two steps; either directly
on its expression or on actin polymerization, which facilitates protein synthesis (Fig.
3a). Thus, we postulate that PKMζ forms a bistable switch by a combination of multistep ultrasensitivity and positive feedback [29,30]. Central to the model are three positive feedback loops, i.e., CICR (Fig. 3b, circle 1), the MAPK-PKC positive feedback
loop (Fig. 3b, circle 2), and PKMζ-induced PKMζ expression (Fig. 3b, circle 3). Theirdecay time constants are less than 1 s, tens of minutes, and weeks or longer, respectively.
Importantly, these excitable dynamics of different time scales are connected in tandem.
When stimuli activate the quickest dynamics (Fig. 3b, loop 1) repetitively, activity istransmitted from the quickest, to the intermediate (Fig. 3b, loop 2), and thereafter to theslowest and most stable dynamics (Fig. 3b, loop 3). As a consequence, long-term stability and plasticity of memory are established. In this sense, this model is a more concrete
representation of the cascade model of excitable dynamics [21] (Figs. 2, 3c).
Various States of Memory
This model schematizes various experimental observations of cerebellar synaptic
plasticity. In the basal state, PP2A is active and inhibits the MAPK-PKC positive
feedback loop (Fig. 3d).
LTP Induction
Postsynaptic LTP is induced in an NO-dependent manner at low [Ca2+] [31,32].
NO, whose synthesis is triggered by PF spikes, facilitates exocytosis of AMPARs
in a N-ethylmaleimide-sensitive factor (NSF)-dependent manner [33] (Fig. 3e).
Induction and Maintenance of LTD
When [Ca2+] and [NO] are increased by conjunctive PF–CF activity, NO releases the
MAPK-PKC positive feedback loop from PP2A inhibition, and Ca2+ activates PKC,
which then phosphorylates and internalizes AMPARs (Fig. 3f). In the intermediatephase (Fig. 3g), the activated MAPK-PKC positive feedback loop maintains PKC
activity and receptor endocytosis. During, and subsequent to, the late phase (Fig. 3h),PKMζDegradation

NO

IP3R

Ca

PP2A

PKMζ

MAPK

PP2A

IP3R

NO

PKC

Ca2+

AMPAR
exocytosis

AMPAR
endocytosis

AMPAR
exocytosis

AMPAR
endocytosis
NO

IP3R

Ca

2+

NO

PP2A
PKMζ

MAPK

PKC

3

PKMζ

2
MAPK
6

PP2A

IP3R

PKC

1

Ca

AMPAR
exocytosis

AMPAR
endocytosis

AMPAR
exocytosis

AMPAR
endocytosis

PF

CF

NO

IP3R

Ca2+

PP2A
PKMζ
PKM

MAPK

PKC

AMPAR
exocytosis

AMPAR
endocytosis

h Late to post-consolidation phases

PF

CF

e LTP

PF

CF

2+

b Diagram of the pathways

NO

IP3R

Ca2+

PP2A

PKMζ

MAPK

Low

PF

CF

NO

IP3R

Ca2+

PP2A

STOP

PKMζ

MAPK

PKC

AMPAR
exocytosis

AMPAR
endocytosis

inhibitor

AMPAR
exocytosis

AMPAR
endocytosis[PKMζ]

Highi Late phase of LTD disrupted by protein synth

PF

CF

PKC

MAPK-PKC positve
feedback loop

f Initial phase of LTD[Ca2+]

Low High

Inactive Active

c bistable dynamics in cascade

Fig. 3 Conceptual model of cerebellar long-term memory (see text for explanation). a Schematic diagram of long-term memory. Sharp and blunt arrows
indicate excitatory and inhibitory pathways, respectively. b The protein kinase Mζ (PKMζ) positive feedback loop. c The cascade model [21] representationof the cerebellar long-term memory model. d–i Various states of memory. Bold lines and thin lines indicate activated and inactivated pathways, respectively;
bold text and thin text indicate active or increased molecules and inactive or decreased molecules, respectively. d The basal state. e LTP. f The initial, intermediate (g), and late to post-consolidation (g) phases of LTD. i The late phase of LTD disrupted by application of a protein synthesis inhibitor. (From [15], withpermission of S. Karger AG, Basel)

PF

CF

PKMζ

MAPK

PKC

Actin polymerization

MAPK-PKC positive
feedback loop

g Intermediate phase

PF

CF

2+

d Basal stateTranslation of
PKMζ mRNANOa PKMζ positive feedback loop174
H. Ogasawara and M.Kawato

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newly synthesized PKMζ maintains AMPAR phosphorylation and internalization
for a long period.
Disruption of Late-Phase LTD by Application of a Protein Synthesis Inhibitor
The model predicts that protein synthesis inhibitor disrupts late-phase LTD by
preventing PKMζ expression (Fig. 3i), whereas it does not affect the initial andintermediate phases. Once PKMζ is sufficiently expressed, only minimal synthesisof PKMζ is required to compensate for its degradation and maintain LTD. During
this period, transient application of a protein synthesis inhibitor will fail to switch
off the PKMζ positive feedback loop, so long as the level of remaining PKMζ isabove the threshold for maintaining loop activity.

Simulation Study of Single or Interlinked Bistable Switches
We have proposed that a cascade of bistable dynamics is a possible mechanism that
resolves the stability versus plasticity dilemma in stochastic noise (Figs. 2, 3)[15,21]. To test this hypothesis computationally, we build simple simulation models.
One model contains a single loop (Fig. 4a), and the other contains two interlinked

Y

0.5
0

Y (µM)Y (µM)

1.5

b

0.5

dX/dt=0dY/dt=00
00.5
X (µM)

50

0.5Time (s)

d

1

100

0.5

50

50

Time (s)100τ = 1 (s)10.5Time (s)

f

1

100

0
1.5

Stim = 2 (µM)
τ = 10 (s)

0.5
0

0

g

0
0

1.5
Stim = 1 (µM)
τ = 10 (s)

0
1

1

1.5
Stim = 2 (µM)
τ = 1 (s)

0
0

1

e

Y (µM)

τ

1

1.5
Stim = 1 (µM)
τ = 1 (s)Y (µM)Y (µM)Xc

Y (µM)

1.5

Stim

Y (µM)

a1

Stim (µM)

2hτ = 10 (s)

1
0.5
0

0

50

Time (s)

100

0

10

Stim (µM)

20

Fig. 4 Deterministic runs of the single-loop model. a Schematic diagram of the model. X and Y
activate each other, and a stimulus (stim) is given to X. b Nullclines and stable (filled circles) andunstable (open circle) steady states. In the stable steady states, the system is either active (arrowhead)or inactive. Stim = 0 throughout. c–f Time courses of Y during and after stimulation. Stim is either
1 µM (c, d) or 2 µM (e, f) and given for 1 s in the beginning of each run. g, h Dose–response relationship. The loop with the smaller τ is more sensitive to stimulation. τ is 1 s (c, e, g) or 10 s (d, f, h)

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Y (µM)

X
τ=1s
Y

1
0.5
0

Z

W (µM)

Stim = 2 (µM)

1.5
1
0.5
0

0

τ'=10sWc

Stim = 1 (µM)

1.5

Y (µM)b
Stim

50
Time (s)

100

1.5

1.5

1

1

W (µM)

a

0.5
0

0

50
Time (s)

100

0

50
Time (s)

100

0.5
0

0

50
Time (s)100Fig. 5 Deterministic runs of the double-loop model. a Schematic diagram of the model. A stimulus (Stim) activates X of the first loop, and X and Y activate each other. Y also activates Z of thesecond loop, and Z and W activate each other. The first loop is faster than the second loop. b, cTime courses of Y and W during and after stimulation. The second loop has the same time constantas the single loop in Fig. 4d,f,h, but is more sensitive to stimuli because the first loop boosts the
inputs. τ and τ’ are 1 s and 10 s, respectively. Stim is either 1 µM (b) or 2 µM (c) and given for 1 s
in the beginning of each runloops (Fig. 5a). Note that our aim here is to capture the basic properties of isolated
or coupled switches by using least complicated models, and not to realistically represent biological signaling pathways. Each loop consists of a couple of mutuallyactivating molecules (X and Y as well as Z and W in Figs. 4a, 5a). The ordinary
differential equations (ODEs) for the single loop model and the first loop of the
double loop model are as follows:
dX
Yn
= n
Xinact − X + Stim(t )dt Y + K n
dY
Xn
t
= n
Yinact − Y
dt
X + Kntwhere n = 3 and K = 0.5 µM. The inactive forms of X and Y, Xinact and Yinact, areconstant at 1 µM and do not change in concentrations even when they are consumed(buffered molecules)1. Stim is a time-dependent stimulus that induces X. Similarly,
the ODEs for the second loop of the double loop model are as follows:1
The buffered species (e.g. Xinact) may appear to violate mass conservation, but actually do not; weassume a situation in which these molecules are abundant in the cell and freely diffuse into or outof the system, keeping their concentrations constant.

Computational Models of Cerebellar Long-Term Memory

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dZ
Wn
Z inact − Z + Y= n
dt W + K ndWZnt′Winact − W= ndt
Z + Knt′

The inactive forms of Z and W, Zinact and Winact, are constant at 1 µM. The time
constants τ and τ’ determine the rates of reactions; for instance, the bigger(smaller) τ is, the more slowly (faster) X and Y change. Numerical integration of
the ODEs yields deterministic solutions, and a Monte-Carlo method [34] serves
for stochastic simulations.
Although the interlinked loops superficially resemble the model of Brandman etal. [35], there are crucial differences between these dynamic systems. First, thestimulus acts on the first loop in our model (Fig. 5a), whereas it acts directly on bothloops in theirs. Second, in contrast to our model, the bistable property of the positive
feedback loops in [35] is not inherent but depends on the level of the stimulus.

Single-Loop ModelFirst, we need to characterize the single-loop model (Fig. 4a) by performing deterministic simulations. Figure 4b indicates steady-state solutions given that one of thetwo variables (X or Y) is kept constant (nullclines); the points of intersection are
equilibrium points of the system. The figure indicates that the loop is a bistablesystem, having two stable steady states and an unstable saddle point. In one stablesteady state, X and Y are active (∼0.8 µM; arrowhead), and in the other stable steadystate they are inactive. Figure 4c–f shows the time course of Y during and after
transient stimulation. When fast kinetics is assumed (τ = 1 s), the loop responds
only transiently to a weak stimulus (Fig. 4c), whereas it is fully and permanently
activated by a strong stimulus (Fig. 4e). At equilibrium, the loop is totally inactive
or stereotypically active depending on the strength of the stimulus (Fig. 4g), whichreflects the all-or-none nature of the bistable switch. A loop of slower kinetics(τ = 10 s) is less sensitive to stimuli (Fig. 4d,f,h) because the stimuli are too shortin duration (for 1 s) and end before the state of the loop crosses the threshold.
Next, we consider stochasticity. In deterministic simulations, X and Y do notchange once they reach the active state (Fig. 4b; arrowhead), but it does not meanthat no reaction is taking place there. Rather, each molecule is constantly being
activated and inactivated at the same rate, appearing to be static. However, in avery small system such as a dendritic spine where the number of molecules isseverely limited, stochastic fluctuations are no longer negligible. Figure 6 plots
several stochastic runs of a minute system (0.05 µm3) corresponding to a tiny spine[36]. In this system, a concentration of 1 µM is equivalent to 30 molecules.
Starting from the active state, the system is shaken by stochastic fluctuations andeventually pushed off to the inactive state (Fig. 6a–c); it would permanently

Fig. 6 Stochastic runs of the single-loop model. a–c Time courses of Y starting from the active state. The larger τ is, the longer the lifespan of the active state tends
to be. τ is either 1 s (a), 10 s (b), or 100 s (c). d Linear relationship between τ and average lifespan. Circles indicate lifespan observed in individual runs and the boldline indicates their averages. f–l Time courses of Y during and after stimulation. The loop with a small τ (1 s) is sensitive to stimulation but, it can remain at the active
state only for a short period. In contrast, the loop with a larger τ (10 s) is less sensitive to stimuli, but once activated, it maintains activity for a long time. Note thedifferent time scale of l. Stim is either 1 µM (e, i), 2 µM (f, j), 4 µM (g, k), or 16 µM (h, l) and is given for 1 s in the beginning of each run

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Computational Models of Cerebellar Long-Term Memory179

remain at the active state in deterministic simulation. The lifespan of the active
state is proportional to τ (Fig. 6d); the slower (faster) the loop is, the longer(shorter) period its activity persists for. It should be noted that no further reaction
will take place once both X and Y reach zero, as the Monte Carlo method [34]assumes that the probability of any reaction to occur in the future is proportional
to X and Y. Thus, in stochastic simulations the inactive state is practically the only
stable steady state, and the active state is no longer permanent. When τ is small(1 s), the loop is activated for a while by transient stimuli that are sufficiently
strong (Fig. 6e–h). In contrast, when τ is larger (10 s), the loop is less sensitive tostimulation (Fig. 6i–k); once activated by a very strong stimulus, however, the loop
often maintains its activity for a long time (Fig. 6l). These findings indicate that ina stochastic situation there is a trade-off between stability and sensitivity.

Interlinked Loop Model
What if two loops are coupled? We expect that a cascade of bistable dynamics
makes stable and plastic memory. In particular, we are interested in a slow loop
interlinked with and activated by a faster loop (Fig. 5a) because it mimics some of
the cerebellar LTD steps (Fig. 3c). Figure 5b,c shows deterministic runs of themodel. The faster loop (X and Y) behaves similarly as the single-loop model with
fast kinetics (Fig. 4c,e). It is activated only transiently by a weak stimulus (Fig. 5b)
and permanently by a strong stimulus (Fig. 5c). The slower loop (Z and W), which
would not be activated by the direct input of a strong stimulus (Fig. 4f), is graduallyactivated by the same size of stimulus (Fig. 5c), and eventually reaches the active
state. This increase in sensitivity is caused by amplification of the signal by thefaster loop; its time-integrated output (∫ Ydt) is infinitely larger than the time integral of the transient stimulus (∫ Stim(t)dt). The interlinked loops are concluded to
be as sensitive as the single fast loop.
When stochasticity is considered, a more interesting feature emerges. As is seenin the stochastic runs of the single loop model (Fig. 6e–h), a strong stimulus brings
the faster loop to the active state, where the loop hangs around for a while before it
eventually returns to the inactive state (Fig. 7, upper panels). Although activity of
the fast loop is only transient, in some cases it contributes sufficiently large input
to the slower loop (Fig. 7, lower panels). Once activated by the first loop, the second loop more often than not maintains its activity for a considerably long time.
If the system did not consist of two loops but rather a single slow loop, it would bemuch less sensitive to the stimuli, as is evident from Fig. 6i–k. The output of thesystem, W, indicates an important feature that the couple of fast and slow loops ismore sensitive to stimuli than a single loop with slow kinetics, whereas its activity
is more stable than that of a single loop with fast kinetics.
This simple model demonstrates that a cascade of bistable dynamics has a combination of highly desirable features for a memory device, that is, robustness tonoise (stability) and sensitivity to input stimuli (plasticity). Such switch cascades

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Stim = 1 (µM)

bStim = 4 (µM)
1.5

1

1

1

0.5

Y (µM)

1.5

0

0.5
0

0

50 100 150
Time (s)0.5
0050 100 150
Time (s)

1.5

1.5

1

1

1

0.5
0

W (µM)

1.5

W (µM)W (µM)c

Stim = 2 (µM)

1.5

Y (µM)Y (µM)

a

0.5
0

0

50 100 150
Time (s)050 100 150
Time (s)050 100 150
Time (s)0.5
0050 100Time (s)150

Fig. 7 Stochastic runs of the double-loop model. The upper and lower panels show time courses ofthe first loop and its downstream slower loop, respectively. The output of the system, W, shows that
this double loop model is more sensitive to stimuli than a single loop with slow kinetics, whereas its
activity is more stable than that of a single loop with fast kinetics. τ and τ’ are 1 s and 10 s, respectively.
Stim is either 1 µM (a), 2 µM (b), or 4 µM (c) and is given for 1 s in the beginning of each run

are biologically implementable, as is exemplified by the pairing of CICR and
MAPK-PKC positive feedback loops [13,14].

Conclusions
In this chapter, we reviewed theoretical studies on cerebellar LTD. CICR and the
MAPK-PKC positive feedback loop form bistable switches connected in tandem,
and we hypothesize that PKMζ is another switch that follows. Simulations of single
or interlinked bistable positive feedback loops supported our hypothesis that a cascade of excitable dynamics is a likely mechanism that resolves the stability versus
plasticity dilemma.
Acknowledgments We thank Dr. Sungho Hong of Okinawa Institute of Science and Technology
for his helpful comments. This work was supported by a grant from the Uehara Memorial
Foundation.

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