Review
Structural dynamics of dendritic
spines in memory and cognition
Haruo Kasai1, Masahiro Fukuda1, Satoshi Watanabe1, Akiko Hayashi-Takagi2 and
Jun Noguchi1
1
Laboratory of Structural Physiology, Center for Disease Biology and Integrative Medicine, Faculty of Medicine, and Center for
NanoBio Integration, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
2
Department of Psychiatry and Behavioral Neurosciences, Johns Hopkins University School of Medicine, Baltimore MD 21287,
USA
Recent studies show that dendritic spines are dynamic
structures. Their rapid creation, destruction and shapechanging are essential for short- and long-term plasticity
at excitatory synapses on pyramidal neurons in the
cerebral cortex. The onset of long-term potentiation,
spine-volume growth and an increase in receptor trafficking are coincident, enabling a ‘functional readout’ of
spine structure that links the age, size, strength and
lifetime of a synapse. Spine dynamics are also implicated
in long-term memory and cognition: intrinsic fluctuations in volume can explain synapse maintenance over
long periods, and rapid, activity-triggered plasticity can
relate directly to cognitive processes. Thus, spine
dynamics are cellular phenomena with important
implications for cognition and memory. Furthermore,
impaired spine dynamics can cause psychiatric and neurodevelopmental disorders.
Dendritic spines
On pyramidal neurons in the cerebral cortex, excitatory
synapses terminate at spines, which are short protrusions
joined to the main dendrite by a thin neck. Discovered in
the 19th century and intensely scrutinized in the 20th
century, dendritic spines are found in higher animals
[1,2] and some insects [3,4]. Spines exist only on certain
types of neurons, including pyramidal neurons in the
cortex, medium spiny neurons in the basal ganglia and
Purkinje cells in the cerebellum. Spines are more abundant
in higher brain regions and highly variable in shape.
Moreover, dendritic spines are the most actin-rich structures in the brain [5,6], and their morphology and density
are abnormal in several mental disorders [7].
The best-known example of input-specific, activity-dependent synaptic plasticity—Donald Hebb’s canonical
basis for learning and memory [8] —is long-term potentiation (LTP) of spine synapses in the hippocampus [9]. The
link between LTP and spine structure was suggested by
the finding that the size of the postsynaptic density (PSD)
is related to the size of the spine head [10] and the number
of AMPA-type glutamate receptors within it [11–13]. These
ultrastructural studies, however, could not determine the
functional state of a spine. This structure–function
relationship was first established in 2001 using two-photon
uncaging of a caged–glutamate compound [14–20]. Later
Corresponding author: Kasai, H. (hkasai@m.u-tokyo.ac.jp)
reports showed that spine enlargements are associated
with LTP in single identified spines (Figure 1a) [21],
indicating that Hebb’s learning rule applies even at the
level of a single synapse. Many studies have since confirmed that the induction of LTP or long-term depression
(LTD), another form of activity-dependent plasticity,
induces structural plasticity of spines in stimulated dendritic branches [22–30].
Given the apparent stability in vivo of dendritic and
axonal arbors at low magnification [31–33], the properties
that govern spine dynamics over the long-term could play a
major role in reorganizing cortical circuitry throughout life
[34,35]. In fact, spines are frequently generated and eliminated even in the adult neocortex, and these events have
been suggested as substrates for stable memory formation
[35–38]. Both formation and enlargement of spines are
important during synaptic rearrangements in the visual
cortex that follow sensory deprivation [39].
It is important to note that spine structural dynamics
include broader phenomena than LTP/LTD. Namely, they
include the generation and elimination of spines [31,32,40–
42], and long-term, activity-independent fluctuations
(described in detail below) (Figure 1b) [42]. In addition,
spines become larger in response to the force of actin
polymerization [43], which occurs within seconds [21] of
LTP induction [44]. Spines seem to display expansive force
continuously to maintain their shape and function [43].
These findings suggest that spine synapses are not just
electrochemical but also mechanical in nature.
The purpose of this article is to present the new findings
on spine dynamics that can be extrapolated to a broad
spectrum of higher-order brain functions. We summarize
the relationships between spine structural dynamics and
functional plasticity, explain the long-term maintenance of
spine structures, propose an explanation for the impairment of spine dynamics in mental disorders and introduce
the possible relationships between rapid spine dynamics
and cognitive processes.
Activity-dependent structural plasticity of dendritic
spines and receptor trafficking
At the level of the dendritic spine, structural dynamics and
receptor trafficking both contribute to functional plasticity.
For example, spine enlargement occurs within a minute
(Figure 1a) [21], a time course that matches the rapid
0166-2236/$ – see front matter ß 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.tins.2010.01.001
121
Review
Trends in Neurosciences Vol.33 No.3
We speculate that receptor trafficking does not support
the long-term maintenance of plasticity in the absence of
structural changes (Figure 1c), based on the turnover rates
for individual molecules: the lifetime of a spine is more
than a thousand times greater than that of its molecular
constituents. Even PSD-95, among the most stable molecules in the spine, redistributes within 3 h in both dissociated cultures and the intact brain in vivo [56,57]. Spine
lifetimes, by contrast, can be more than a year in vivo
[32,35]. The autophosphorylation of CaMKII, once thought
to help maintain long-term changes in synaptic strength,
has since been revealed to be rather transient [58]. Thus,
maintenance of the molecular status of the spine can
require some association with its structure. Given this
evidence, it would be interesting to test whether the functional plasticity induced without structural changes to the
spine [24,46,59,60] can persist as long as the functional
plasticity that accompanies structural alterations. We next
address how spine structure can be maintained over days
(Figure 1b).
Figure 1. Structural dynamics of dendritic spines. Rapid and slow changes in spine
shape and size result in rapid and slow synaptic plasticities. (a) Time course of
spine enlargement induced by the two-photon uncaging of a caged–glutamate
compound. Spine enlargement is specific to the stimulated spine (red circle) and
does not spread to the neighboring spine (white diamond). Enlargement occurs
within a few seconds and lasts over 2 h. (b) Slow intrinsic fluctuations in spine
sizes occur in the presence of inhibitors of NMDA receptors (NMDAR) over periods
of days. The spines show relatively small changes in size (quantified in Figure 2).
Spine generation (spines 4, 6, 7) and elimination (spines 8 and 10) still occur
frequently. (c) Relationship between structural plasticity and receptor trafficking,
the two mechanisms of synaptic plasticity. Structural changes induced by plasticity
can be maintained for a long time, enabling the functional states of spines to also
be maintained for a long time—if there are structure–function relationships. Such
relationships can be caused by the allocation of AMPA receptors in proportion to
the spine size, resulting in the ‘‘readout of structure.’’ Trafficking of receptors can
facilitate and hasten the readout processes. Figures in (a) and (b) are reproduced
with permission from the Nature publishing group [21] and the Society for
Neuroscience [42], respectively.
induction of LTP. Enlarged spines also explain the longterm maintenance of LTP, given that the number of functional AMPA receptors correlates with spine volume [14–
20] (Figure 1c). And spine enlargement [45,46], like the late
phase of LTP and long-term memory itself [47], depends on
protein synthesis. Moreover, spine structures are stable for
days in cultured hippocampal slices (Figure 1b) [42,48] and
for years in the cortex in vivo [31,32,34]. These data
support the idea that structural plasticity is the central
cellular mechanism that underlies memory formation.
Receptor trafficking also affects functional plasticity by
facilitating the readout of spine structure (Figure 1c).
During LTP induction, AMPA receptors in cytosolic
vesicles are inserted into the plasma membrane [49,50]
and diffuse laterally to supply new receptors to the PSD
[51,52]. These receptors are anchored to PSD proteins in an
actin-dependent manner [53–55]. LTD protocols also affect
the endocytosis of AMPA receptors [49]. Hence, numerous
experiments support the link between LTP/LTD induction,
receptor trafficking and the acquisition of long-term memory in behaving animals [49]. It is important to note,
however, that the rapid exchange of AMPA receptors
between synaptic and extrasynaptic regions [51] means
that trafficking mechanisms by themselves cannot account
for long-term changes in synaptic function.
122
Long-term intrinsic fluctuations and maintenance of
spines
Neuronal networks reflect the properties of spine populations, rather than those of a single spine, because the
generation of action potentials in a neuron requires the
activation of many synapses. We recently identified a key
phenomenon that affects the long-term behaviors of spine
populations. Through the observation and systematic
quantification of spine dynamics over periods of days
(Figure 1b) [42], it became apparent that spine volumes
grew and shrank spontaneously. These volumetric changes
occurred even when NMDA receptors and Na+ channels
were completely blocked to prevent activity-dependent
plasticity (Figure 2a). Such changes, termed ‘intrinsic
fluctuations,’ encompass all phenomena (other than
activity-dependent plasticity) that contribute to the structural dynamics of spines [42], and represent at least a part
of fluctuations of spine volumes reported in vivo [38,61,62].
These fluctuations reflect an inevitable lack of structural
stability in spines whose molecular components turn over
with a time constant of less than 3 h [56,57]. Despite this
constant change, the average daily volume change was
close to zero for all spine sizes (Figure 2b). Thus, spine
populations at various sizes are in equilibrium, and individual spines can act as analog memory elements despite
the stochastic changes in spine volume (up to 20% per day
in young hippocampus) (Figure 3c).
At first glance, intrinsic fluctuations would seem incompatible with long-term memory storage. However, two
additional findings reveal that this variability does not
defeat the maintenance of spine function over time
(Figure 1c). First, the volume of an average spine remains
largely the same for a certain period of time [42], consistent
with the persistence for weeks of LTP in vivo [42,63].
Second, spine lifetimes can be very long (Figure Ia in
Box 1), in line with in vivo two-photon imaging [32,34]
(for theory and examples, see Box 1).
The high proportion of small spines in the volume distribution [64] (Figure 2d) can now be explained by a positive
correlation between the size of the spine and the amplitude
Review
Trends in Neurosciences
Vol.33 No.3
Figure 2. Long-term structural dynamics of dendritic spines. (a) Fluctuations in the head volume of an individual spine as determined by the fluorescence intensity of the
spine head. There are two mechanisms controlling the long-term dynamics of spine volumes: activity-dependent plasticity (A, blue arrows) and intrinsic fluctuations (I,
black arrows). Activity-dependent plasticity is dependent on the activation of NMDA receptors, whereas intrinsic fluctuations exist even in the presence of inhibitors of
NMDA receptors and Na+ channels. Activity-dependent plasticity and intrinsic fluctuations represent learning mechanisms and maintenance processes, respectively. The
Langevin equations used to model the plasticity are shown. We assume that every spine larger than 0.02 mm3 has a presynaptic partner, although the presynaptic terminal
is drawn only for the small spine. (b) Mean values of fluctuations in spine head volume in the presence (black, I) and in the absence (blue, A) of NMDA receptor inhibitors. (c)
Standard deviations of fluctuations in spine head volume in the presence (black, I) and in the absence (green, C) of NMDA receptor inhibitors. The activity-dependent
fraction is displayed by blue (A). (d) Probability–density distributions predicted from the C and I data given in (b, c). The prediction is well fit by the actual data for spine head
volumes in slice cultures incubated for 3 days in the absence (C) or presence (I) of NMDA receptor inhibitors [42].
of its intrinsic fluctuations (Figure 2c, I). Small spines
accumulate because of smaller fluctuations, and larger
spines become less common because of their greater fluctuation amplitudes (Figure 2c) [42] (for a theoretical basis, see
Box 1). Similar spine–volume distributions have been found
in various conditions in vivo and in vitro, supporting the
interpretation that the mechanisms are the same.
In contrast to the magnitude of intrinsic fluctuations,
the frequency of activity-dependent structural changes is
greater in smaller spines (Figure Id in Box 1; for quantification of LTP, see Box 1). All spines are considered to
receive a synaptic contact [20,42], and small spines correspond approximately to the notion of ‘‘silent synapses’’ [65]
that express NMDA but not AMPA receptors [14,16,20].
Small spines display greater increases in cytosolic Ca2+
induced by NMDA receptor stimulation [16,66] and are
preferential sites for LTP induction [21,65]. The structural
changes that accompany LTP include the enlargement of
small spines and an increased number of medium-sized
spines (Figure 2d, C). Such effects on the volume distribution, however, were relatively weak (Figure 2d, compare
I and C) because intrinsic fluctuations occur constantly and
appear to be the dominant mechanism in determining
spine–volume distributions.
Large spines form only gradually, because activity-dependent enlargement acts preferentially on small, rather
than medium, spines (Figure 2a) [42]. This simple fact
predicts that older spines will tend to be larger (Figure Ib
in Box 1) [42] and have longer life expectancies (Figure Ia in
Box 1). Thus, the history of a spine is reflected in its volume,
unlike a one-bit memory element in a computer, which
cannot indicate its own history. Indeed, this history effect
explains a feature of memory first described by Ebbinghaus
in 1885—that older memories are more persistent [67].
In his seminal work, Ebbinghaus estimated the decay of
memories by memorizing a list of nonsense syllables. After
a variable period, he memorized the list again and quan-
tified memory as the decrease in memorization time. He
found a rapid loss of memory in the first day, followed by a
slower decline over the next 31 days. This non-exponential
pattern of memory decay suggested that longer-lasting
memories were more persistent. A graph of the time savings was fitted by a logarithmic curve (Figure Ic in Box 1,
red dashed line) that was later called the ‘‘savings function’’ or ‘‘forgetting curve’’ [68]. Thus, the nature of memory
depends on its history, which can be explained in turn by
the history effect of the dendritic spine.
If one posits that the creation or enlargement of a few
small spines within Broca’s area represent memory traces
for new syllables, then the idea of intrinsic fluctuations
predicts a logarithmic decay in the mean volume of small
spines (Figure Ic in Box 1, solid line) that is closely related
to the savings function (red dashed line). At the same time,
the model explains how a proportion of the memory
encoded within small spines can be saved into larger spines
as a result of intrinsic fluctuations (Figure 2a).
Intrinsic fluctuations are random forces that alternately
strengthen and destroy the smallest spines. But the stochastic elimination of existing spines (Figure 1b, spines 8
and 10), in turn, enables the spontaneous generation of new
spines (Figure 1b, spines 4, 6, 7) by clearing space and
recycling molecular resources. Indeed, the spontaneous
creation of new spines continues into adulthood and is often
detected [42] in the mature neocortex [35,37,61]. Although
new spines start small (Figure 2a), they represent functional
synapses that can readily enlarge in an activity-dependent
manner (Figure 1a). The creation of new spines accelerates
20–60 min after neuronal activity [40,69], but the sprouting
cannot be synapse-specific as there is no synapse before its
generation, in principle. Thus, activity-dependent synapse
formation seems to reflect considerable chance [48]. Perhaps
the random nature of this make-and-test process enables
animals to adapt to an unexpected environment. These new
spines could be seeds of new memory [35,37,38].
123
Review
Trends in Neurosciences Vol.33 No.3
Box 1. Population dynamics of dendritic spines
To understand the population behaviors of spines, we have
introduced a mathematical model in which spine volumes change
continuously in a semi-random manner. The properties of the
changes or fluctuations depend on spine volume (Figure 2a). Similar
approaches have been used in the study of many other biological
problems, such as inter-spike intervals [116], population genetics and
ecology [117]. The simplest mathematical model for random continuous fluctuations is Brownian motion W(t) [116,117]. In this
framework, a continuous random variable or stochastic process V(t)
that has an average change or drift of m(V) and a standard deviation of
s(V) is described as
dV ðtÞ
dW ðtÞ
¼ sðV ðtÞÞ
þ mðV ðtÞÞ;
(1)
dt
dt
where W(t) represents the standard Brownian motion with a variance
of 1/day [116,118]. We have applied this Langevin equation to the
volume fluctuations of spines V(t) using standard deviation and drift
values obtained from experiments (Figure 2b,c). We found that the
actual volume distributions of spines in control (C) treatments and in
the absence of activity-dependent plasticity (I) are well predicted by the
model (Figure 2d) [42].
This model also accounts for spine elimination by defining it as the
shrinkage of spines past a certain threshold value (0.02 mm3)
(Figure 2a). Thus, we can predict that larger spine will have a longer
life (Figure Ia in Box 1). The life expectancy of a spine with a volume of
0.3 mm3 is 57 days in the absence of activity-dependent plasticity,
assuming that the intrinsic fluctuations have a coefficient of variation
(CV=s/V) of 20%/day [42]. If the CV is reduced by a factor of v, the life
expectancy is prolonged by a factor of v2. If CV is 5%/day, we can
predict that the lifetime of a spine with a volume of 0.3 mm2 is 2 years,
consistent with in vivo measurements in mice [34]. If the CV is 1%/
day, the life expectancy is approximately 64 years [42]. Thus, if spine
volumes fluctuate by only 1%/day, they could account for the lifelong
persistence of some spines in a human being.
This model also predicts that on average, older spines will have a
larger mean volume (Figure Ib in Box 1) and a longer life expectancy.
The average time course of spine volume for spines with the initial
volume of 0.021 mm3 can be obtained using the same calculation
(Figure Ic in Box 1), which can be fitted with the same saving function
(the dashed red line in Figure Ic in Box 1) that represents the time
course of forgetting.
We can interpret our data on the activity-dependent plasticity of
spines using two opposing processes, the enlargement (LTP) and
shrinkage (LTD) of spines. If one assumes that the unitary amplitudes
All the above studies are consistent with a model in which
activity-dependent spine–volume changes regulate newmemory acquisition (by enlarging/stabilizing or eliminating
the smallest spines) and existing-memory persistence (by
changing volumes of spines) [70].Thus, activity-dependent
plasticity selects memory content and modifies memory
strength, supporting the random-generation-and-test
model of new-memory acquisition. In contrast, intrinsic
fluctuations in spine volume may change the strength of a
memory but seldom affect its content. We will next discuss
how spine dynamics can account for abnormal spine profiles
in various neurological and psychiatric conditions.
Abnormalities in spine dynamics and mental disorders:
a working model
Many clinical investigators have proposed that synapses
are major sites of pathogenesis for mental disorders such
as schizophrenia, autism and other conditions that show
normal gross anatomy [7,71–73]. Here, we summarize the
reports of dendritic spine abnormalities in these disorders
and present a hypothesis for how these changes might yield
such diverse symptoms.
124
for enlargement and shrinkage are qE and qS, respectively, and the
rates of enlargement and shrinkage are lE and lS, respectively, the
average change and standard deviation can be expressed as:
m ¼ q E lE q S lS ;
s 2 ¼ qE2 lE þ qS2 lS :
(2)
Then, the rates of enlargement and shrinkage are expressed as:
lE ¼
s 2 þ qS m
;
qE ðq E þ q S Þ
lE ¼
s2 þ qS m
:
q E ðqE þ q S Þ
(3)
If one applies this equation to the activity-dependent component of
fluctuations (Figure 2b,c, A) by assuming qE=qS=0.02, then the
frequencies of enlargement and shrinkage are predicted as shown
in Figure Id in Box 1.
Figure I. (a) Relationship between spine head volume and mean life expectancy
of a spine according to model I. (b) Dependence on spine age of the mean head
volume of a spine following model I. (c) Average time course of head volume
changes for spines that obey model I and have initial head volumes of 0.021 mm3.
The red dashed line represents the savings function, k/[(Log[at])c+k], where
k=0.13, a=480 and c=1.2. Figures reproduced, with permission from the Society
for Neuroscience, from Ref. [42]. (d) The predicted contributions of LTP and LTD
to the activity-dependent fluctuations (A) obtained using Eq. (3) in Box 1.
Dendritic spines in mental retardation
In human patients [7,74,75] and most (but not all) animal
models of mental retardations [76–79], dendritic spines
tend to be abnormally small and immature. Many types of
mental retardation have been traced to genetic defects in
scaffolding and adhesion molecules thought to maintain
synapses [80,81]. Presuming that defects in scaffolding and
adhesion proteins would affect intrinsic fluctuations, these
abnormally small spines might represent an inability to
maintain spine structure. In effect, the lack of these molecules unleashes intrinsic fluctuations (Figure 3) to erode
large spines and cause a proliferation of small spines
(Figure 2d), thereby masking the effects of activity-dependent plasticity on volume distribution [42]. The
inability to preserve large spines created by activity-dependent plasticity prevents the accumulation of proper
knowledge in the developmental period, potentially causing a deficit in general intelligence.
Dendritic spines are abnormally small in mutant mice
that lack a copy of the shank-1 gene, which encodes a major
scaffolding protein in the spine [79]. In these mice, interestingly, spatial reference learning was enhanced,
Review
Trends in Neurosciences
Vol.33 No.3
Figure 3. Various forms of spine motility and mental disorders. There are three forms of spine motility: rapid, activity-dependent plasticity; long-term, activity-dependent
plasticity; and long-term intrinsic fluctuations. Schizophrenia and mental retardation can arise from selective impairments in activity-dependent plasticity and intrinsic
fluctuations, respectively. Intrinsic fluctuations or synaptic stability can also be impaired in autism.
although the memory after 4 weeks was much worse than
controls [79]. One interpretation of these results could be
that the overproliferation of small spines augments newmemory formation, whereas the lack of large spines
indicates poor retention of older memories [80]. The
superior performance of the mutant on the spatial learning
task might represent a murine equivalent of the exceptional abilities that sometimes manifest in autistic
savants.
Dendritic spines in schizophrenia
Unlike autism or mental retardation, schizophrenia is
rarely linked to abnormalities in spine size [82]. This
observation suggests that synaptic maintenance, as influenced by intrinsic fluctuations, remains unimpaired
among schizophrenics (Figure 3). Instead, several lines
of evidence point to links between schizophrenia and
abnormal activity-dependent plasticity (Figure 3) [71,83].
The strongest such evidence is referred to as the NMDA
hypothesis [84,85]. It stems from the observation that
phencyclidine, a NMDA receptor antagonist, produces
diverse schizophrenic symptoms that are ameliorated by
NMDA receptor agonists such as D-serine. The dopamine
dysregulation model, which relates to the classical dopamine hypothesis of schizophrenia, suggests that the dopamine-fueled elevation of intracellular cAMP plays an
important role in NMDA-dependent synaptic plasticity
[86]. Furthermore, the expression products of many schizophrenia-susceptibility genes are localized near glutamatergic synapses in a statistically significant manner (and
can exist in other regions as well) [83,87] (Figure 3).
Reduced spine density in the prefrontal cortex of
schizophrenics [71,86] indicates a loss of balance between
synaptogenesis and elimination [42,88]; both processes are
strongly regulated by NMDA receptor transmission [89].
Therefore, the reduced density of dendritic spines in
schizophrenic patients must represent a reduced generation relative to elimination of synapses [42].
Adult-onset schizophrenia can also develop in response
to an inevitable post-adolescent decline in activity-dependent plasticity [90]. At this age, prodromal individuals who
exhibit some schizophrenia-associated problems [91]
already demonstrate a reduced capacity for activity-dependent synaptic plasticity. This growing deficit is often
obscured, however, by high levels of synaptic remodeling at
this age. When the decrease in activity-dependent
plasticity is uncovered in adulthood, it is often accom-
panied by evidence of abnormal neural connectivity and
the full onset of symptoms (Figure 3).
Although positive symptoms such as delusions and
hallucinations remain the major criteria for schizophrenia
diagnosis, clinicians have given greater attention in recent
decades to a range of cognitive deficits and negative symptoms [91]. Schizophrenics experience the loss of attention,
working memory, episodic memory, verbal fluency,
emotion and volition [91]—nearly every higher-order brain
function is impaired except intelligence quotient [92],
which can be spared because it is believed to reflect cognitive functions acquired during childhood and adolescence
[93].
In the context of spine dynamics, two synergic mechanisms could explain the symptoms of schizophrenia.
The first mechanism rests on the idea that normal spine
dynamics regulate experience-dependent plasticity
within neuronal networks [71,83]. By extension, abnormal spine dynamics can permit the chaotic reorganization of neuronal networks, culminating in schizophrenia.
Accordingly, the disordered thoughts and infrequent
brainwave synchrony has led to the characterization
of schizophrenia as a disease of disconnected brain
regions [94,95]. Particularly because spiny glutamatergic
neurons and their projections are so prevalent in
the affected brain regions, spine dysfunction could be
the culprit in disrupting normal connectivity. But a
second, complementary and more provocative possibility
is that spine dynamics take part directly in cognition
(Figure 4).
Rapid structural dynamics of spines in cognition: a new
hypothesis
Within the brain, the coordinated firing of neurons in space
and time underlies myriad functions, including the cognitive processes of perception, emotion and volition [8,96,97].
A major challenge in neuroscience is to delineate the
physical conditions of the conscious brain [96,98], sometimes referred to as the neuronal correlates of consciousness [99].
Unfortunately, many studies of these correlates reduce
every neuron to its action potential. This presents an incomplete picture of cognitive function, and indeed the brain is
more than its ions. Recent experiments using optogenetic
tools suggest that mechanisms other than spikes can participate in the creation of internal representations [100]. In this
section, we attempt to identify connections between the
125
Review
Figure 4. Hypothetical consequences of synchronous activities in a neuronal
network. These figures illustrate the positive feedback mechanism in which
recursive spike chains trigger the enlargement of specific spines, which in turn
facilitate the spike chains. Each neuron displays characteristic neuronal motility in
response to a spike chain, and an assembly of neurons exhibits synchronous and
complex neural motility. (a) Selective enlargement of a specific set of spines in a
neuron (yellow) by distinct spike chains 1 and 2. (b) Characteristic neuronal
motilities induced by various spike chains. (c) Possible consequences of spine
enlargement on synaptic functions. Spine enlargement enhances AMPA receptor
function, applies positive pressure on a presynaptic terminal and can enhance
neurotransmitter release.
cognitive and synaptic neurosciences to suggest a new
synaptic basis for cognitive function.
The first connection is about attention. As a physical
event, attention manifests as the synchronous firing of
a neuronal population that responds preferentially to
attended stimuli [97,101]. Synchronous firing can also
provide a solution to the binding problem [102,103], in
which the brain recognizes an object with diverse
attributes as a unitary percept. Therefore, the question
arises: How does the brain recognize synchronous firing
among spatially scattered neurons? From synaptic physiology, we know that the synchronous firing of pre- and
postsynaptic neurons induces rapid spine enlargement
[45,104]. In this way, a single spine can encode information from both neurons, and the thousands of spines
(Figure 4b) in a pyramidal neuron have the potential
to detect synchronous spikes from thousands of
other neurons scattered throughout the brain [45]
(Figure 4a). The rapid structural dynamics could even
reflect a specific spike chain (Figure 4a,b). Thus, a pyramidal neuron can detect and register the synchronous
firing of neuronal assemblies. During cognitive processes,
billions of spiny neurons show synchronous motilities in
their spines.
126
Trends in Neurosciences Vol.33 No.3
The second connection concerns speed. The speed of
thought is not instantaneous under any circumstances,
but particularly not for cognitive processes. For example,
visual perception takes at least 0.1 s of recursive network
computation [96], and tactile perception and free will each
require 0.5 s of electrical activity in the human cortex
[105]. These delays are on the order of the delay in spine
enlargement after repetitive stimulation (0.5–3 s). The
slight lag is attributed to the Ca2+ triggering of actin
polymerization; once begun, the extension of actin filaments occurs rapidly (0.03–1 mm/s) [43,106]. This rapid
enlargement does not seem to rely on the same enzymatic
reactions needed for long-term enlargement, which involve
CaMKII, Kalirin7, Rac and PAK [43,58,107].
The third connection relates to the ordering of long-term
changes. Just as cognition precedes the formation of explicit memory [108], rapid enlargement precedes long-term
enlargement of spines (Figure 1a). Thus, if cognitive processes involve spine dynamics, they use the same substrates as memory, and effectively leave their traces for
memory.
The fourth connection explores unconsciousness. General anesthetics easily and completely dispel cognition.
Spine enlargement can also be blocked by anesthesia,
because rapid spine motility has never been described in
an anesthetized animal under natural stimulation—not
even when the neurons continued to fire action potentials
[109,110]. In support of this consideration, volatile anesthetics are reported to interfere with actin organization in
resting cells [111]. In addition, the cerebellum is not a
direct source of consciousness [112], although the cerebellar Purkinje cells have prominent spines. This could be
correlated with the fact that these cells lack NMDA receptors and thus would not show rapid activity-dependent
enlargement.
The fifth connection addresses similarities between the
phenomena of cognition and structural dynamics. Cognitive processes are specific—meaning defined or individual
or particular—but their ultimate origin and subsequent
progression are stochastic [113]. Likewise, spine motilities—which represent connections between specific
neurons—are very rich cellular behaviors caused by the
complex interplay of chemical and physical forces from the
cytoskeleton, plasma membrane and surrounding cells.
They, too, are often stochastic.
The sixth and last connection recognizes self-modification. Cognitive processes are self-modifying, giving rise
to memory, emotion and executive functions [113]. Likewise, spine enlargement can dynamically alter the functions of neuronal networks. Although we know that
enlargement is associated with increased functional
expression of AMPA receptors [21], we suspect that there
are other functional consequences of spine enlargement
that await discovery. The massive actin polymerization
generates an expansive force [43] that acts on surrounding
tissues (Figure 4c), but to what effect? One possibility is
that it exerts a mechanical force on neighboring spines to
alter their functions as well [43]. Another possibility is that
the mechanical force enhances presynaptic function. This
hypothesis is based on evidence that physical forces or
chemical stimuli applied to the plasma membrane promote
Review
exocytosis [114,115]. If such forms of regulation can indeed
be exerted in presynaptic terminals, then spines can
actively maintain and trigger the firing of a particular
neuronal network or spike chain through an increase in
the release probability of presynaptic terminals. This type
of regulation must be highly complex and stochastic, as
described above. Thus, spine–volume changes could alter
synaptic connections in various ways and rapidly selfmodify neuronal–network functions.
The rapid, responsive movement of synapses shares
many features with cognition. Dendritic spines can take
part directly in cognitive processes to make them more
individual, active and stochastic—unlike a computer, in
which memory elements obey simple and deterministic
rules. Thus, cognitive processes can be easier to understand when we take account of the spine structural
dynamics. Direct imaging of spine motilities in vivo has
the ability to substantiate these possibilities.
Concluding remarks
We have described the close relationship between spine
structure and function, and introduced the hypothesis that
intrinsic fluctuations in spine volumes account for the longterm maintenance of spines. The biophysical properties of
these fluctuations could mirror the psychological properties of complex behaviors such as forgetting. Intrinsic
fluctuations also predict the spontaneous generation of
abundant new spines, leading to the random-generationand-test model of new-memory acquisition.
We propose that abnormalities in the two types of spine
structural plasticity—intrinsic fluctuations and activitydependent plasticity—are involved in the pathogenesis of
mental retardation and schizophrenia, respectively. We
also introduce the idea that rapid spine dynamics underlie
cognition. These are testable hypotheses that can be
examined quantitatively in the future. Regardless of those
future results, the ability to visualize and manipulate
spine dynamics in vivo will be useful in our investigations
of cognitive processes and the synaptic bases of psychiatric
disorders. Such investigations could provide new avenues
for studying brain functions and can lead to novel diagnostic and therapeutic approaches for psychiatric disorders.
Acknowledgements
We thank M. Fukuda, K. Kasai, A. Sawa and K. Toyama for helpful
discussions. This work was supported by Grants-in-Aids from the
Ministry of Education, Culture, Sports, Science, and Technology
(MEXT) of Japan (H.K., J.N.).
References
1 Ramon y Cajal, S. (1995) Histology of the Nervous System of Man and
Vertebrate, Oxford University Press
2 Greenough, W.T. and Bailey, C.H. (1988) The anatomy of a memory:
convergence of results across a diversity of tests. Trends Neurosci. 11,
142–147
3 Coss, R.G. and Perkel, D.H. (1985) The function of dendritic spines: a
review of theoretical issues. Behav. Neural Biol. 44, 151–185
4 Leiss, F. et al. (2009) Characterization of dendritic spines in the
Drosophila central nervous system. Dev. Neurobiol. 69, 221–234
5 Dunaevsky, A. et al. (1999) Developmental regulation of spine motility
in the mammalian central nervous system. Proc. Natl. Acad. Sci. U. S.
A. 96, 13438–13443
Trends in Neurosciences
Vol.33 No.3
6 Matus, A. (2000) Actin-based plasticity in dendritic spines. Science
290, 754–758
7 Fiala, J.C. et al. (2002) Dendritic spine pathology: cause or
consequence of neurological disorders? Brain Res. Rev. 39, 29–54
8 Hebb, D.O. (1949) The Organization of Behavior, John Wiley and Sons
9 Bliss, T.V. and Lomo, T. (1973) Long-lasting potentiation of synaptic
transmission in the dentate area of the anaesthetized rabbit following
stimulation of the perforant path. J. Physiol. 232, 331–356
10 Harris, K.M. et al. (1992) Three-dimensional structure of dendritic
spines and synapses in rat hippocampus (CA1) at postnatal day 15
and adult ages: implications for the maturation of synaptic physiology
and long-term potentiation. J. Neurosci. 12, 2685–2705
11 Nusser, Z. et al. (1998) Cell type and pathway dependence of synaptic
AMPA receptor number and variability in the hippocampus. Neuron
21, 545–559
12 Takumi, Y. et al. (1999) Different modes of expression of AMPA and
NMDA receptors in hippocampal synapses. Nat. Neurosci. 2, 618–624
13 Kharazia, V.N. and Weinberg, R.J. (1999) Immunogold localization of
AMPA and NMDA receptors in somatic sensory cortex of albino rat. J.
Comp. Neurol. 412, 292–302
14 Matsuzaki, M. et al. (2001) Dendritic spine geometry is critical for
AMPA receptor expression in hippocampal CA1 pyramidal neurons.
Nat. Neurosci. 4, 1086–1092
15 Smith, M. et al. (2003) Mechanism of the distance-dependent scaling
of Schaffer collateral synapse in CA1 pyramidal neurons. J. Physiol.
(Lond.) 548, 245–258
16 Noguchi, J. et al. (2005) Spine-neck geometry determines NMDA
receptor-dependent Ca2+ signaling in dendrites. Neuron 46, 609–622
17 Beique, J.C. et al. (2006) Synapse-specific regulation of AMPA
receptor function by PSD-95. Proc. Natl. Acad. Sci. U. S. A. 103,
19535–19540
18 Asrican, B. et al. (2007) Synaptic strength of individual spines
correlates with bound Ca2+-calmodulin-dependent kinase II. J.
Neurosci. 27, 14007–14011
19 Holbro, N. et al. (2009) Differential distribution of endoplasmic
reticulum controls metabotropic signaling and plasticity at
hippocampal synapses. Proc. Natl. Acad. Sci. U. S. A. 106, 15055–
15060
20 Zito, K. et al. (2009) Rapid functional maturation of nascent dendritic
spines. Neuron 61, 247–258
21 Matsuzaki, M. et al. (2004) Structural basis of long-term potentiation
in single dendritic spines. Nature 429, 761–766
22 Okamoto, K. et al. (2004) Rapid and persistent modulation of actin
dynamics regulates postsynaptic reorganization underlying
bidirectional plasticity. Nat. Neurosci. 7, 1104–1112
23 Otmakhov, N. et al. (2004) Persistent accumulation of calcium/
calmodulin-dependent protein kinase II in dendritic spines after
induction of NMDA receptor-dependent chemical long-term
potentiation. J. Neurosci. 24, 9324–9331
24 Lang, C. et al. (2004) Transient expansion of synaptically connected
dendritic spines upon induction of hippocampal long-term
potentiation. Proc. Natl. Acad. Sci. U. S. A. 101, 16665–16670
25 Zhou, Q. et al. (2004) Shrinkage of dendritic spines associated with
long-term depression of hippocampal synapses. Neuron 44, 749–757
26 Nagerl, U.V. et al. (2004) Bidirectional activity-dependent
morphological plasticity in hippocampal neurons. Neuron 44, 759–767
27 Kopec, C.D. et al. (2006) Glutamate receptor exocytosis and spine
enlargement during chemically induced long-term potentiation. J.
Neurosci. 26, 2000–2009
28 Bastrikova, N. et al. (2008) Synapse elimination accompanies
functional plasticity in hippocampal neurons. Proc. Natl. Acad. Sci.
U. S. A. 105, 3123–3127
29 Segal, M. (2005) Dendritic spines and long-term plasticity. Nat. Rev.
Neurosci. 6, 277–284
30 Okada, D. et al. (2009) Input-specific spine entry of soma-derived Vesl1S protein conforms to synaptic tagging. Science 324, 904–909
31 Trachtenberg, J.T. et al. (2002) Long-term in vivo imaging of
experience-dependent synaptic plasticity in adult cortex. Nature
420, 788–794
32 Grutzendler, J. et al. (2002) Long-term dendritic spine stability in the
adult cortex. Nature 420, 812–816
33 Stettler, D.D. et al. (2006) Axons and synaptic boutons are highly
dynamic in adult visual cortex. Neuron 49, 877–887
127
Review
34 Zuo, Y. et al. (2005) Development of long-term dendritic spine stability
in diverse regions of cerebral cortex. Neuron 46, 181–189
35 Yang, G. et al. (2009) Stably maintained dendritic spines are
associated with lifelong memories. Nature 462, 920–924
36 Chklovskii, D.B. et al. (2004) Cortical rewiring and information
storage. Nature 431, 782–788
37 Holtmaat, A. and Svoboda, K. (2009) Experience-dependent
structural synaptic plasticity in the mammalian brain. Nat. Rev.
Neurosci. 10, 647–658
38 Xu, T. et al. (2009) Rapid formation and selective stabilization of
synapses for enduring motor memories. Nature 462, 915–919
39 Hofer, S.B. et al. (2009) Experience leaves a lasting structural trace in
cortical circuits. Nature 457, 313–317
40 Engert, F. and Bonhoeffer, T. (1999) Dendritic spine changes
associated with hippocampal long-term synaptic plasticity. Nature
399, 66–70
41 Lendvai, B. et al. (2000) Experience-dependent plasticity of dendritic
spines in the developing rat barrel cortex in vivo. Nature 404, 876–881
42 Yasumatsu, N. et al. (2008) Principles of long-term dynamics of
dendritic spines. J. Neurosci. 28, 13592–13608
43 Honkura, N. et al. (2008) The subspine organization of actin fibers
regulates the structure and plasticity of dendritic spines. Neuron 57,
719–729
44 Gustafsson, B. and Wigstrom, H. (1990) Long-term potentiation in the
hippocampal CA1 region: its induction and early temporal
development. Prog. Brain Res. 83, 223–232
45 Tanaka, J. et al. (2008) Protein synthesis and neurotrophindependent structural plasticity of single dendritic spines. Science
319, 1683–1687
46 Yang, Y. et al. (2008) Spine expansion and stabilization associated
with long-term potentiation. J. Neurosci. 28, 5740–5751
47 Kelleher, R.J., III et al. (2004) Translational regulatory mechanisms
in persistent forms of synaptic plasticity. Neuron 44, 59–73
48 De, R.M. et al. (2008) LTP promotes a selective long-term stabilization
and clustering of dendritic spines. PLoS Biol. 6, e219
49 Kessels, H.W. and Malinow, R. (2009) Synaptic AMPA receptor
plasticity and behavior. Neuron 61, 340–350
50 Newpher, T.M. and Ehlers, M.D. (2009) Spine microdomains
for postsynaptic signaling and plasticity. Trends Cell Biol. 19, 218–
227
51 Heine, M. et al. (2008) Surface mobility of postsynaptic AMPARs
tunes synaptic transmission. Science 320, 201–205
52 Makino, H. and Malinow, R. (2009) AMPA receptor incorporation into
synapses during LTP: the role of lateral movement and exocytosis.
Neuron 64, 381–390
53 Lisman, J.E. and Zhabotinsky, A.M. (2001) A model of synaptic
memory: a CaMKII/PP1 switch that potentiates transmission by
organizing an AMPA receptor anchoring assembly. Neuron 31,
191–201
54 Nakagawa, T. et al. (2004) Quaternary structure, protein dynamics,
and synaptic function of SAP97 controlled by L27 domain
interactions. Neuron 44, 453–467
55 Kato, A.S. et al. (2008) AMPA receptor subunit-specific regulation by a
distinct family of type II TARPs. Neuron 59, 986–996
56 Okabe, S. et al. (2001) Rapid redistribution of the postsynaptic density
protein PSD-Zip45 (Homer 1c) and its differential regulation by
NMDA receptors and calcium channels. J. Neurosci. 21, 9561–9571
57 Gray, N.W. et al. (2006) Rapid redistribution of synaptic PSD-95 in the
neocortex in vivo. PLoS Biol. 4, e370
58 Lee, S.J. et al. (2009) Activation of CaMKII in single dendritic spines
during long-term potentiation. Nature 458, 299–304
59 Bagal, A.A. et al. (2005) Long-term potentiation of exogenous
glutamate responses at single dendritic spines. Proc. Natl. Acad.
Sci. U. S. A. 102, 14434–14439
60 Wang, X.B. et al. (2007) Independent expression of synaptic and
morphological plasticity associated with long-term depression. J.
Neurosci. 27, 12419–12429
61 Zuo, Y. et al. (2005) Long-term sensory deprivation prevents dendritic
spine loss in primary somatosensory cortex. Nature 436, 261–265
62 Holtmaat, A. et al. (2006) Experience-dependent and cell-type-specific
spine growth in the neocortex. Nature 441, 979–983
63 Abraham, W.C. (2003) How long will long-term potentiation last?
Philos. Trans. R. Soc. Lond. B Biol. Sci. 358, 735–744
128
Trends in Neurosciences Vol.33 No.3
64 Benavides-Piccione, R. et al. (2002) Cortical area and species
differences in dendritic spine morphology. J. Neurocytol. 31, 337–
346
65 Malenka, R.C. and Nicoll, R.A. (1999) Long-term potentiation – a
decade of progress? Science 285, 1870–1874
66 Sobczyk, A. et al. (2005) NMDA receptor subunit-dependent [Ca2+]
signaling in individual hippocampal dendritic spines. J. Neurosci. 25,
6037–6046
67 Ebbinghaus, H. (1885) Uber das Gedachtnis, Dunker & Humbolt
68 Wixted, J.T. and Ebbesen, E.B. (1997) Genuine power curves in
forgetting: a quantitative analysis of individual subject forgetting
functions. Mem. Cognit. 25, 731–739
69 Maletic-Savatic, M. et al. (1999) Rapid dendritic morphogenesis in
CA1 hippocampal dendrites induced by synaptic activity. Science 283,
1923–1927
70 Morita, S. et al. (2009) Generation, elimination and fluctuations of
synapses in the cerebral cortex. Commun. Integr. Biol. 2, 1–4
71 Lewis, D.A. and Gonzalez-Burgos, G. (2008) Neuroplasticity of
neocortical circuits in schizophrenia. Neuropsychopharmacology 33,
141–165
72 Sudhof, T.C. (2008) Neuroligins and neurexins link synaptic function
to cognitive disease. Nature 455, 903–911
73 Harrison, P.J. (1999) The neuropathology of schizophrenia. A critical
review of the data and their interpretation. Brain 122, 593–624
74 Purpura, D.P. (1974) Dendritic spine ‘‘dysgenesis’’ and mental
retardation. Science 186, 1126–1128
75 Ramakers, G.J. (2002) Rho proteins, mental retardation and the
cellular basis of cognition. Trends Neurosci. 25, 191–199
76 Nimchinsky, E.A. et al. (2001) Abnormal development of dendritic
spines in FMR1 knock-out mice. J. Neurosci. 21, 5139–5146
77 Boda, B. et al. (2004) The mental retardation protein PAK3
contributes to synapse formation and plasticity in hippocampus. J.
Neurosci. 24, 10816–10825
78 Hayashi, M.L. et al. (2004) Altered cortical synaptic morphology and
impaired memory consolidation in forebrain-specific dominantnegative PAK transgenic mice. Neuron 42, 773–787
79 Hung, A.Y. et al. (2008) Smaller dendritic spines, weaker synaptic
transmission, but enhanced spatial learning in mice lacking Shank1.
J. Neurosci. 28, 1697–1708
80 Kasai, H. et al. (2003) Structure-stability-function relationships of
dendritic spines. Trends Neurosci. 26, 360–368
81 Kasri, N.N. and Aelst, L.V. (2008) Rho-linked genes and neurological
disorders. Pflugers Arch. 455, 787–797
82 Roberts, R.C. et al. (1996) Reduced striatal spine size in
schizophrenia: a postmortem ultrastructural study. Neuroreport 7,
1214–1218
83 Hashimoto, R. et al. (2007) Postsynaptic density: a key convergent site
for schizophrenia susceptibility factors and possible target for drug
development. Drugs Today (Barc.) 43, 645–654
84 Moghaddam, B. (2003) Bringing order to the glutamate chaos in
schizophrenia. Neuron 40, 881–884
85 Goff, D. and Coyle, J. (2001) The emerging role of glutamate in the
pathophysiology and treatment of schizophrenia. Am. J. Psychiatry
158, 1367–1377
86 Keshavan, M.S. et al. (2008) Schizophrenia, ‘‘just the facts’’: what we
know in 2008. Part 3: neurobiology. Schizophr. Res. 106, 89–107
87 Harrison, P.J. and West, V.A. (2006) Six degrees of separation: on the
prior probability that schizophrenia susceptibility genes converge on
synapses, glutamate and NMDA receptors. Mol. Psychiatry 11, 981–
983
88 Alvarez, V.A. and Sabatini, B.L. (2007) Anatomical and physiological
plasticity of dendritic spines. Annu. Rev. Neurosci. 30, 79–97
89 Colonnese, M. and Constantine-Paton, M. (2006) The emerging role of
glutamate in the pathophysiology and treatment of schizophrenia.
J. Comp. Neurol. 494, 738–751
90 Holtmaat, A.J. et al. (2005) Transient and persistent dendritic spines
in the neocortex in vivo. Neuron 45, 279–291
91 Tandon, R. et al. (2009) Schizophrenia, ‘‘just the facts’’ 4. Clinical
features and conceptualization. Schizophr. Res. 110, 1–23
92 Woodberry, K.A. et al. (2008) Premorbid IQ in schizophrenia: a metaanalytic review. Am. J. Psychiatry 165, 579–587
93 Plomin, R. and Spinath, F. (2004) Intelligence: genetics, genes, and
genomics. J. Pers. Soc. Psychol. 86, 112–129
Review
94 McGlashan, T. and Hoffman, R. (2000) Schizophrenia as a disorder of
developmentally reduced synaptic connectivity. Arch. Gen. Psychiatry
57, 637–648
95 Whalley, H. et al. (2009) Connecting the brain and new drug targets
for schizophrenia. Curr. Pharm. Des. 15, 2615–2631
96 Lamme, V.A. (2003) Why visual attention and awareness are
different. Trends Cogn. Sci. 7, 12–18
97 Sakamoto, K. et al. (2008) Discharge synchrony during the transition
of behavioral goal representations encoded by discharge rates of
prefrontal neurons. Cereb. Cortex 18, 2036–2045
98 Searle, J. (2004) Mind: A Brief Introduction, Oxford University Press
99 Crick, F. and Koch, C. (1998) Consciousness and neuroscience. Cereb.
Cortex 8, 97–107
100 Airan, R.D. et al. (2009) Temporally precise in vivo control of
intracellular signalling. Nature 458, 1025–1029
101 Gregoriou, G.G. et al. (2009) High-frequency, long-range coupling
between prefrontal and visual cortex during attention. Science 324,
1207–1210
102 Gray, C.M. (1999) The temporal correlation hypothesis of visual
feature integration: still alive and well. Neuron 24, 31–47
103 von der Malsburg, C. (1999) The what and why of binding: the
modeler’s perspective. Neuron 24, 95–104
104 Harvey, C.D. and Svoboda, K. (2007) Locally dynamic synaptic
learning rules in pyramidal neuron dendrites. Nature 450, 1195–1200
105 Libet, B. (2004) Mind Time: The Temporal Factor in Consciousness,
Harvard University Press
Trends in Neurosciences
Vol.33 No.3
106 Pollard, T.D. and Borisy, G.G. (2003) Cellular motility driven by
assembly and disassembly of actin filaments. Cell 112, 453–465
107 Penzes, P. and Jones, K.A. (2008) Dendritic spine dynamics – a key
role for kalirin-7. Trends Neurosci. 31, 419–427
108 Dudai, Y. (2002) Memory from A to Z, Oxford University Press
109 Hubel, D.H. and Wiesel, T.N. (1959) Receptive fields of single
neurones in the cat’s striate cortex. J. Physiol. 148, 574–591
110 Metin, C. et al. (1988) The primary visual cortex in the mouse:
receptive field properties and functional organization. Exp. Brain
Res. 69, 594–612
111 Kaech, S. et al. (1999) Volatile anesthetics block actin-based
motility in dendritic spines. Proc. Natl. Acad. Sci. U. S. A. 96,
10433–10437
112 Penfield, W. (1975) The Mystery of Mind, Princeton University Press
113 Block, N. (1996) How can we find the neural correlate of
consciousness? Trends Neurosci. 19, 456–459
114 Rosenmund, C. and Stevens, C.F. (1996) Definition of the readily
releasable pool of vesicles at hippocampal synapses. Neuron 16, 1197–
1207
115 Kishimoto, T. et al. (2006) Vacuolar sequential exocytosis of large
dense-core vesicles in adrenal medulla. EMBO J. 25, 673–682
116 Tuckwell, H.C. (1988) In Introduction to Theoretical Neurobiology
(Vol. 2), Cambridge University Press
117 Kloeden, P.E. and Platen, E. (1999) Numerical Solution of Stochastic
Differential Equations, Springer-Verlag
118 Risken, H. (1989) The Fokker–Planck Equation, Springer
129