ORIGINAL RESEARCH
published: 13 June 2019
doi: 10.3389/fncom.2019.00038

Intrinsic Spine Dynamics Are Criticalfor Recurrent Network Learning inModels With and Without Autism
Spectrum Disorder
James Humble 1 , Kazuhiro Hiratsuka 1 , Haruo Kasai 2 and Taro Toyoizumi 1*
1

Laboratory for Neural Computation and Adaptation, RIKEN Center for Brain Science, Saitama, Japan, 2 Laboratory ofStructural Physiology, Faculty of Medicine, Center for Disease Biology and Integrative Medicine, University of Tokyo, Tokyo,Japan

It is often assumed that Hebbian synaptic plasticity forms a cell assembly, a mutuallyinteracting group of neurons that encodes memory. However, in recurrently connected
networks with pure Hebbian plasticity, cell assemblies typically diverge or fade underongoing changes of synaptic strength. Previously assumed mechanisms that stabilize
cell assemblies do not robustly reproduce the experimentally reported unimodal andlong-tailed distribution of synaptic strengths. Here, we show that augmenting Hebbian
plasticity with experimentally observed intrinsic spine dynamics can stabilize cell
assemblies and reproduce the distribution of synaptic strengths. Moreover, we posit thatstrong intrinsic spine dynamics impair learning performance. Our theory explains how
excessively strong spine dynamics, experimentally observed in several animal models ofautism spectrum disorder, impair learning associations in the brain.
Edited by:
Nicolas Brunel,
Duke University, United States
Reviewed by:Mark C. W. Van Rossum,
University of Nottingham,
United Kingdom
Gianluigi Mongillo,
Université Paris Descartes, France*Correspondence:
Taro Toyoizumi
taro.toyoizumi@riken.jp
Received: 03 March 2019
Accepted: 28 May 2019
Published: 13 June 2019
Citation:
Humble J, Hiratsuka K, Kasai H andToyoizumi T (2019) Intrinsic Spine
Dynamics Are Critical for Recurrent
Network Learning in Models With andWithout Autism Spectrum Disorder.
Front. Comput. Neurosci. 13:38.
doi: 10.3389/fncom.2019.00038

Keywords: spine dynamics, synaptic weight distribution, cell assembly, autism spectrum disorder, recurrent
network, learning

INTRODUCTION
The operation of a neural circuit is shaped by the strength of synapses that mediatesignal transduction between neurons. Activity-dependent modification of synaptic strength,
termed synaptic plasticity, is considered to be an underlying mechanism of learning and memory(Malenka and Bear, 2004; Mongillo et al., 2017). A major form of synaptic plasticity is Hebbian
plasticity (Hebb, 1949). While there are multiple molecular mechanisms (Malinow and Malenka,
2002; Matsuzaki et al., 2004; Nicoll et al., 2006) underlying Hebbian plasticity and experimental
protocols (Neves et al., 2008), it is commonly induced by coactivation of pre- and postsynaptic
neurons within a particular time window. One prominent biological mechanism for Hebbian
plasticity is activity-dependent spine volume change. Spine volume is known to be tightly
correlated with synaptic strength (Matsuzaki et al., 2001; Smith et al., 2003; Noguchi et al., 2005;
Béïque et al., 2006; Asrican et al., 2007; Holbro et al., 2009; Zito et al., 2009), and both longterm potentiation (LTP) and long-term depression (LTD) involve spine change (Lang et al.,
2004; Matsuzaki et al., 2004; Otmakhov et al., 2004; Zhou et al., 2004; Kopec et al., 2006;
Hayama et al., 2013).
It was previously proposed (Hebb, 1949; Amari S., 1977; Hopfield, 1982) that a memory can be
represented by coherent activity in a cell assembly, i.e., a group of cells mutually exciting each other,

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Learning With Intrinsic Spine Dynamics

and LTD that is set by model parameters and input to neurons.
Thus, conventional models have no mechanism to restore the
distribution of synaptic strengths.
Despite the common assumption that only synaptic plasticity
changes spines, they also dynamically change in the absence ofneural activity. Recent studies showed that spine turnover, i.e.,
generation and elimination of spines, continues even under theblockade of neural activity and calcium signaling in vivo (Kimand Nabekura, 2011; Nagaoka et al., 2016; Ziv and Brenner, 2018).
Further, spine volumes constitutively fluctuate in the absenceof neural activity, calcium signaling, and activity-dependent
plasticity in vitro (Yasumatsu et al., 2008). These intrinsic spine
dynamics are characterized by a zero drift coefficient and adiffusion coefficient proportional to the square of spine volume,v2 . Interestingly, this volume-dependent diffusion reproduces
the experimentally observed equilibrium distribution of spine
volumes with a power-law tail of exponent v−2 (Yasumatsu
et al., 2008; Ishii et al., 2018). This observation poses animportant question: How do intrinsic spine dynamics affect themaintenance of cell assemblies?
We address this question by simulating a mathematical modelof a recurrently connected neural network that implements both
multiplicative spike-timing-dependent plasticity (STDP) (vanRossum et al., 2000; Morrison et al., 2007) and experimentallyobserved intrinsic spine dynamics (Yasumatsu et al., 2008). We
also study how spine turnover and the distribution of spine
volumes are affected by these two processes. Despite a possible
perception of intrinsic spine dynamics as noise, we show that they
can help to maintain cell assemblies by preventing unnecessary
spines from growing and sustaining the physiological spine
volume distribution.
Based on the model analysis, we hypothesize that intrinsic
spine dynamics that are stronger than in wild type (WT)
conditions can explain the abnormally high spine turnover rate
observed in animal models of autism spectrum disorder (ASD)(Pan et al., 2010; Isshiki et al., 2014). By fitting model parameters
to one ASD mouse model, fmr1KO (a model of fragile X
syndrome) (Pfeiffer and Huber, 2009), we show how excessively
strong intrinsic spine dynamics may cause learning deficits in
ASD animals (Silverman et al., 2010; Padmashri et al., 2013).

and the memory can be stored in synaptic strengths between
these neurons by Hebbian plasticity. Consistently, recent
experiments have shown that the activation of a coherently active
group of cells is necessary and sufficient for the expression oflearned behavior (Liu et al., 2012; Nabavi et al., 2014). However,
how a neural circuit maintains cell assemblies stably is not wellunderstood. In some models, cell assemblies are stable because
synaptic strength is modified only during learning, and then fixed(Amari S. I., 1977; Hopfield, 1982; Vogels et al., 2011). However,these models neglect changes in synaptic strength after learningand thus do not address the maintenance of an acquired memory.
Several studies modeled ongoing Hebbian plasticity during
spontaneous activity and found that Hebbian plasticity alone
is likely not sufficient to maintain a cell assembly. In additive
Hebbian plasticity models (Gerstner et al., 1996; Song et al.,
2000; Gütig et al., 2003), in which the dependencies of the
LTP and LTD amplitudes on synaptic strength are the same,
memory tends to become unstable due to a positive feedback
during spontaneous activity (Fiete et al., 2010; Litwin-Kumar
and Doiron, 2014; Zenke et al., 2015), namely, neurons that fire
together are wired together, and then fire together more often.
This kind of a positive feedback process typically fuses assemblies
and expands the largest existing cell assembly. Some forms ofstabilizing mechanisms, such as inhibitory plasticity (LitwinKumar and Doiron, 2014), homeostatic plasticity (Zenke et al.,2013), or heterosynaptic plasticity (Zenke et al., 2015) have beensuggested to stabilize memory (Keck et al., 2017). However, evenwith these stabilizing mechanisms, the resulting distribution ofsynaptic strengths often becomes dissimilar to what has been
experimentally observed (Toyoizumi et al., 2007). For example,
while the models with positive feedback often produce a synaptic
strength distribution that is bimodal, experiments have reported
a unimodal and long-tailed distribution of synaptic strengths(Song et al., 2005; Cossell et al., 2015) and corresponding spine
volumes (Yasumatsu et al., 2008; Loewenstein et al., 2011).
An alternative proposal is multiplicative Hebbian plasticity(van Rossum et al., 2000; Gütig et al., 2003; Morrison et al., 2007),
in which the LTP amplitude is less prominent than the LTD
amplitude for large synapses, in agreement with experimental
observations (Bi and Poo, 1998; Tanaka et al., 2008; Hayama et al.,2013). This multiplicative form of Hebbian plasticity can avoid
the above instability problem, and under spontaneous activity ofneurons, synaptic strengths converge near a prefixed set point
where LTP and LTD effects balance each other, regardless of theinitial synaptic strengths (Morrison et al., 2007). This means thatmemories degrade in the presence of spontaneous neural activity.
Hence, in all the models described above, it is nontrivial tostably maintain cell assemblies and reproduce the experimentallyobserved distribution of synaptic strengths (or of spine volumes),
which has a thick tail and a peak at a rather weak strength (Songet al., 2005; Yasumatsu et al., 2008; Loewenstein et al., 2011;
Cossell et al., 2015). Interestingly, a similar distribution of spine
volumes is robustly observed even in animal models of mental
disorders (Pathania et al., 2014) and with an LTD deficiency
in calcineurin KO animals (Okazaki et al., 2018). In contrast,in the above mathematical models, the distribution of synaptic
strengths is fragile and strongly depends on the balance of LTP

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RESULTS
We study the dynamics of spine volumes using a model ofa cortical circuit. We stimulate recurrently connected spiking
neurons (Figure 1; also see Methods) to explore if the network
stores memories as cell assemblies. The cortical network consistsof 1,000 excitatory and 200 inhibitory leaky-integrate-and-firespiking neurons (Tuckwell, 1988), where the excitatory andinhibitory neurons are randomly connected by a 10% connection
probability (Figure 1A). For simplicity, the excitatory neurons
are embedded in a one-dimensional feature space that describes,
for example, orientation selectivity in V1. We assume thatsynapses can be formed between a pair of excitatory neurons
that have potential connectivity (Markram et al., 1997). Potential
connectivity (either 0 or 1) from a neuron to another is randomly

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Learning With Intrinsic Spine Dynamics

A

C

Plastic

Base

Stim

1
1

2

Inhibitory

4

10 %

3

Input

3

3s

2

Excitatory
3

Hz

10 %

4
1 /2 /3 /4

Input
B1

Inhibitory
0.3

10
Prob.

Potentialconnectivity (%)

B2Time (s)
D

0
− 0.5

0

0

0.5

Feature-space distance

1

STDP

10

Intrinsic dynamics# contact points

Spine

not functional

B4

Volume=0.02µm3

Baseline spiking activity

Neuron adaptation

Time (s)

1

0.1
10

0

0
0

10
20
Time (s)30

F

STDP window∆Spine-volume (µm3)

Firing rate (Hz)Membranepotential (mV) Groups-70
0

Excitatory
Inhibitory

20

Adaptation amount

E

4
3
2
1
-50

x10

-9

7.6

0

− 100

− 50

0

50

tpost - tpre (ms)100

Diffusion coefficient (µm 6 day -1)

B3

Intrinsic spine dynamics
-3

8 x10
6
4
2
0
0

0.2
0.4
0.6
Spine-volume (µm3)

FIGURE 1 | Recurrent network model with STDP and intrinsic spine dynamics. (A) A model of cortical circuitry. Excitatory and inhibitory neurons are modeled as
leaky-integrate-and-fire units, which are sparsely connected. We assume that only the recurrent excitatory synapses are plastic. Excitatory neurons are aligned in a
one-dimensional feature space, which is divided into four neighborhood quarters. Fourty percent of randomly chosen excitatory neurons in each quarter comprise astimulation group. The four stimulation groups are randomly stimulated one at a time during the learning period. (B1) Potential connectivity peaks at around 10% anddecays with the tuning-distance between two excitatory neurons in the feature space. Synapses can grow if two excitatory neurons are potentially connected. (B2) If
two excitatory neurons have potential connectivity, the number of contact points is randomly drawn from a truncated Poisson distribution in the range of 1 to 10. Eachcontact point can accommodate one spine. (B3) Spiking activity and membrane potential dynamics of a sample set of neurons at baseline. (B4) Excitatory neurons
have an adaptation current, which builds up with firing activity and suppresses firing rate. (C) During a learning period, one of the stimulated groups is randomly
chosen with probability ¼ and receives elevated external input for 3 s. All inhibitory neurons receive external stimulation throughout the entire learning period. (D) Spinevolumes are changed by the combination of STDP and intrinsic spine dynamics (except in Figure 2, where only STDP is considered). Arrows indicate possible
changes in spine volume and the size of the arrow represents the possible maximum change in spine volume. A threshold at 0.02 µm3 separates spines andnon-spines, and STDP only affects spines. (E) The multiplicative STDP rule used for changing spine volume. The LTD amplitude is proportional to spine volume. (F)The diffusion coefficient characterizing intrinsic spine dynamics, which is proportional to the square of spine volume.

generated and set at the beginning of a simulation. Potential
connectivity peaks at 10.4% (see Figure 7 for a systematic
exploration of this peak value) for neurons with similar selectivity
and falls off with their tuning-distance (Figure 1B1). If two
neurons have potential connectivity, the number of synaptic
contact points is drawn randomly from a truncated Poisson
distribution (Figure 1B2) (Hardingham et al., 2010). In theabsence of elevated external input, excitatory neurons in thisnetwork exhibit a background firing rate of about 0.1 Hz

Frontiers in Computational Neuroscience | www.frontiersin.org(Figure 1B3). Cortical excitatory neurons are generally adaptive
and cannot continuously fire at their maximum firing rate.
Hence, we model an adaptation current (Wang et al., 2003)
that slowly builds up with the postsynaptic spiking activity andhyperpolarizes the neuron with its characteristic time constant of
about 5 s (Figure 1B4; see Methods). Finally, only the recurrent
excitatory to excitatory synapses are assumed to change.
To model activity-dependent plasticity, we assume that spine
volume is proportional to synaptic strength (see Methods, but

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Learning With Intrinsic Spine Dynamicsµm3 (Figure 2E). A small but non-negligible number of nonstimulated synapses also formed a peak at 0.5 µm3 . These
synapses contribute to the extremely high firing rate of thegroup 3 neurons and further recruitment of non-stimulated
neurons toward the end of the simulation. During the learning
period, the mean volume of the spines connecting neurons withineach stimulated group increases due to LTP (Figure 2F). Notethat LTP is dominant over LTD for small spines because we
assume that the LTP amplitude is fixed but the LTD amplitude
is proportional to spine volume. After a few days of learning,mean spine volumes plateaued at around 0.5 µm3 , where LTP
and LTD effects roughly balance. The mean spine volume of otherspines (non-stimulated) exhibited a slow but steady increase(Figure 2F), reflecting the formation of the secondary peak of
non-stimulated spines seen in the spine volume distribution(Figure 2E). The volumes of individual spines (Figure 2G) arehomogeneous within each group and their development mirrors
the corresponding mean spine volume (Figure 2F).
The spread of activity is slow in this simulation because of thelateral inhibition that tends to shut down spikes in surrounding
neurons (Billings and van Rossum, 2009). However, the activity
will eventually spread as long as neurons fire occasionallyand spine volumes between active neurons converge near theprefixed LTP/LTD balancing point at around 0.5 µm3 . This
convergence is innate to multiplicative STDP and spontaneously
happens even in the absence of the learning period, albeit slowly(Figure S1). Hence, as expected from previous studies (Morrison
et al., 2007), a recurrently connected network with multiplicative
STDP has difficulty sustaining cell assemblies in spine volumes
in the presence of spontaneous activity. Notably, the resulting
spine volume distribution is dissimilar to an experimentallyobserved unimodal distribution with many small spines and a
fat tail.

we define a tiny protrusion of volume <0.02 µm3 as a “nonspine”) because the correlation between the synaptic strength andspine volume has been experimentally demonstrated (Matsuzaki
et al., 2004; Harvey and Svoboda, 2007; Bosch et al., 2014).
The spine volume of an excitatory to excitatory synapse ismodeled by multiplicative STDP (Figure 1E) (van Rossum et al.,2000), and thus the LTP amplitude is independent of synaptic
strength, while the LTD amplitude is proportional to synaptic
strength. Therefore, with an increase in synaptic strength, theLTD amplitude increases at a steeper rate than the LTP amplitude,
and this is consistent with experimental observations (Bi andPoo, 1998; Tanaka et al., 2008; Hayama et al., 2013). We let
the network acquire cell assemblies by providing additional
external input to subsets of neurons. We divide the feature spaceof the excitatory network into four equally sized neighboring
quarters (Figure 1A), and randomly select 40% of the neurons
in each quarter as a stimulated group. We randomly selectone of these four groups at a time during the learning period
and stimulate it with an elevated rate of Poisson spikes for
3 s (Figure 1C; see Methods). For simplicity, all inhibitory
neurons are stimulated throughout the learning period. The
spine volume, v, of each spine is initially drawn randomly from−2
a fixed distribution, proportional to v + 0.05 µm3 . Thisinitial distribution approximates an experimentally observed
spine volume distribution (see Methods). As we will see
below, changing synaptic strengths by multiplicative STDP
alone fails to sustain cell assemblies in the presence of
spontaneous activity.

Learning Without Intrinsic Spine Dynamics
Firstly, we consider the case where intrinsic spine dynamics areabsent and spine volumes are only changed by multiplicative
STDP. Figure 2 depicts the behavior of our network during andafter the learning period. The learning period (Figure 2A) is
terminated when the mean intra-group spine volume of at least
one stimulated group reaches ≥0.49 µm3 . During the learning
period, four groups of neurons were randomly stimulated one
at a time (Figure 2B, Left) with increased input, and after thelearning period, only group 3’s neurons stayed active (Figure 2B,
Right). Figure 2C plots the firing rates of all neurons in the
3rd-quarter during the entire simulation period. This showsthat the number of active neurons monotonically increased bothduring the learning period and during the maintenance period,
indicating an unstable learning outcome. Specifically, the cell
assembly initially formed among the externally stimulated group
3 neurons and spread to neighboring neurons. This spreadingof the cell assembly provided extra recurrent input from newly
recruited neurons to this cell assembly. Figure 2D summarizes
the population averaged firing rates of the four stimulatedgroups. All groups show elevated firing rates during the learning
period due to external stimulations. After the learning period,
physiological neural activity was maintained only for a day.
After that, the mean firing rate of one group (i.e., group 3)
exploded, and that of the other groups declined down to nearzero values. The spine volume distribution at the end of thesimulation included a non-physiological secondary peak at 0.5

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Learning With Normal Intrinsic Spine
Dynamics
Next, we add intrinsic spine dynamics previously observed in
hippocampal slices (Yasumatsu et al., 2008). Spine volumes
fluctuate every day, and the amplitude of these fluctuations isspine volume dependent (Figure 1F). Importantly, these intrinsic
spine dynamics were largely intact in the absence of neuralactivity and synaptic plasticity, i.e., under the pharmacologicalblockade of sodium channels, NMDA, and voltage-dependentcalcium channels in cells. In the absence of neural activity
and plasticity, the amplitude of spine volume fluctuations isroughly proportional to spine volume, ∼ αv + β, where v isspine volume with parameters α = 0.2 day−1/2 and β =0.01 µm3 day−1/2 (Yasumatsu et al., 2008). In other words, theeffect of intrinsic spine dynamics is summarized by the volumedependent diffusion coefficient D (v) = (αv + β)2 /2 with zerodrift coefficient (Yasumatsu et al., 2008) (see also Methods).
Therefore, the Fokker-Planck equation (Risken, 1989) for
describing the evolution of the spine volume distribution P(v, t)
is ∂P(v, t)/∂t = (∂ 2 /∂v2 )[D (v) P(v, t)]. It indicates that theequilibrium is reached when the diffusion intensity D (v) P(v, t)
becomes volume-independent. This gives the equilibrium spine

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FIGURE 2 | Network behavior in the absence of intrinsic spine dynamics. (A) Experimental protocol. External stimulus is provided during the learning period (blue bar)and the memory retention is studied in the maintenance period. The learning period finishes when one cell assembly becomes strong enough to sustain it’s activity. (B)Typical neural activity during (Left) and after (Right) the learning period. The panels show the firing rates of 1,000 excitatory neurons in a 50 s time window. During thelearning, neural activity is driven by external stimulus, which randomly activates one group at a time. After learning, one group of neurons is strongly active
spontaneously. (C) Firing rates of the 3rd-quarter neurons in the entire simulation period. Neurons are sorted by the first time their firing rates exceed 15 Hz. The
number of active neurons expands both during the learning and retention periods beyond the initial 40% that are stimulated. (D) Mean firing rates of the four
stimulated groups. (E) Spine volume distribution at the end of the simulation for intra-stimulated-group spines (Stimulated: pink) and other spines (Other: green). (F)Mean spine volume of intra-stimulated-group spines (pink) and other spines (green). (G) Individual volumes of a single intra-stimulated-group spine from eachstimulated group and a single non-stimulated spine. The learning period is represented by a blue bar.

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volume distribution Peq (v) ∝ 1/D(v) ∼ v−2 , which has a powerlaw tail. Note that there are two mathematical conventions forinterpreting the above equation (Gardiner, 1985), which lead todifferent semantic meanings of fluctuation. Here, we take the
Itô interpretation, in which the intrinsic spine dynamics areinterpreted as spine volume fluctuations (but see Methods foran alternative interpretation). We regard that spines smaller than
0.02 µm3 are non-spines and do not exhibit multiplicative STDP,
whereas intrinsic spine dynamics are still present even for thesesmall protrusions (cf. Figure 1D). This assumption is consistent
with the experimental observations that the baseline spine
turnover is largely activity-independent (Kim and Nabekura,
2011; Nagaoka et al., 2016).
We used the same stimulation protocol as in Figure 2 to
study cell assembly learning when both multiplicative STDP
and intrinsic spine dynamics are involved (Figures 3A–H).
Cell assembly learning was similar at the beginning of thesimulation to the case without intrinsic spine dynamics: Firing
rates increased (Figure 3D) and the intra-group spines enlarged(Figure 3F). In contrast to the previous case, a physiologicalneural activity level was maintained throughout (Figure 3D), andnone of the cell assemblies aggressively spread to neighboring
non-stimulated neurons (Figure 3C) during the maintenanceperiod. The activity-dependent formation of cell assemblies isevident from the coherent reactivation of stimulated groups(Figure 3B) during the maintenance period at much lower
spontaneous firing rates than during the learning period. This
spontaneous reactivation of cell assemblies is essential forthe maintenance of memory patterns (similarly to Mi et al.,2017). Nonetheless, these memory patterns can be successfully
maintained even if neural activity is blocked for a whole day(e.g., by tetrodotoxin; Figure S2). Notably, the mean volumeof the intra-assembly spines was stably maintained (Figure 3F),
although individual spine volumes fluctuated throughout thesimulation (Figure 3G). In contrast to the previous casewithout intrinsic spine dynamics (cf. Figure 2E), the spine
volume distribution remained unimodal after learning, with nosecondary peak around 0.5 µm3 (Figure 3E; cf. Figure 2E witha secondary peak). This is because a possible secondary peakat the tail was smeared by large fluctuations in large spines.
Despite this smearing, the memory patterns were still stored byenlarged spines located around the tail of the distribution. Finally,spine generation was increased during the learning period,
as often seen experimentally (Hofer et al., 2009; Yang et al.,2009) (Figure 3H). The contribution of each model mechanism
assumption is further explored in Figure S3.

systematically. While we simulated the spontaneous activity ofneurons, we monitored how multiplicative STDP and intrinsic
spine dynamics changed the mean intra-group spine volume,
respectively. Multiplicative STDP leads to an overall potentiationof the intra-group spines (Figure 4; orange line) if they aresmall (roughly <0.57 µm3 ) and a depression of the spines
if they are large (roughly >0.57 µm3 ). This transition isobserved for large spines because the LTD amplitude, which is
proportional to spine volume, dominates over the amplitude of
LTP, which is independent of spine volume. LTP dramaticallyincreases for spines >0.30 µm3 because strong intra-groupconnections serve as positive feedback on coincident firing
between the neurons within the group, increasing the frequency
of both LTP and LTD events. Therefore, STDP on its own
leads to one fixed point of mean spine volume at a nonphysiologically high value at around 0.57 µm3 . This fixed point
is controlled by the parameter setting the relative amplitude of
LTD (Morrison et al., 2007). Intrinsic spine dynamics, on theother hand, normalize the spine volume distribution, restoring
the distribution toward the equilibrium distribution Peq (v) ∝1/D(v) with a mean spine volume of roughly 0.15 µm3 (Figure 4;
brown line).
When the contributions from STDP and intrinsic spine
dynamics are added together with an appropriate balance(Figure 4; black line), the combination permits a bi-stability inthe mean spine volume of a population of spines. In this case,changes in the mean spine volume are dominated by the intrinsic
spine dynamics when small (roughly <0.30 µm3 ), so that thespine volumes fluctuate around 0.15 µm3 . Conversely, changes
in the mean spine volume are significantly affected by STDP
when large (roughly >0.30 µm3 ), creating a larger-volume fixed
point at approximately <0.55 µm3 . In between these stable fixed
points, there is a separation point (unstable fixed point) at around0.50 µm3 , which prevents the mean spine volume from moving
in between these two stable fixed points. Therefore, while single
spines are largely fluctuating and sporadically moving aroundthe small and large mean spine-volume fixed points, a large
cell assembly is stably maintained by the ensemble property of
intra-assembly spines: here quantified as the mean intra-assembly
spine volume.
These results suggest that intrinsic spine dynamics cannormalize the synaptic strength distribution to a stereotypicalshape in the presence of ongoing Hebbian plasticity, and at thesame time, enable the circuit to stably retain memory patterns
in the form of cell assemblies with a bi-stable mean intra-cell
assembly spine volume. The relative amplitudes of STDP and
intrinsic spine dynamics are key parameters for achieving this bistability. For example, if STDP is too strong, relative to intrinsic
spine dynamics, the small mean spine volume fixed point at 0.15
µm3 would disappear. We have already seen in our model thatin the absence of intrinsic spine dynamics, the spine volume
distribution sharply peaks at around the large mean spine volume
fixed point and activity becomes too high (Figure 1). Instead, if
STDP is too weak, relative to intrinsic spine dynamics, the large
mean spine volume fixed point around 0.55 µm3 could disappear.
Below, we explore more generally what might go wrong withexcessively strong intrinsic spine dynamics.

A Cell Assembly Is Maintained by an
Ensemble Property of Intra-Assembly
SpinesTo elucidate how intrinsic spine dynamics stabilize our
network learning and enable the storing of memories, we
examined the effects of STDP and intrinsic spine dynamics
separately (Figure 4). We initialized spine volumes randomly
as described above except for the intra-group spines of one
group, whose volumes were all set identically and changed

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Learning With Intrinsic Spine Dynamics

FIGURE 3 | Network behavior in the presence of intrinsic spine dynamics. (A–G) Conventions are as in Figure 2. (H) Spine generation and elimination.

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FIGURE 4 | Decomposition of spine volume change by STDP and intrinsic spine dynamics under spontaneous activity. We separately measured changes in mean
spine volume induced by either STDP (orange line) or intrinsic spine dynamics (brown line), by systematically initializing all intra-group spine volumes of one group to a
fixed value, and measuring any subsequent changes. The net change is separately plotted (black line). Arrows of corresponding color mark the flow of mean spine
volume change due to each mechanism, or the combination of both, at the top. STDP and intrinsic spine dynamics change the mean spine volume toward 0.57 and0.15 µm3 , respectively. When the two mechanisms are combined, the net dynamics have bistability: There are two stable fixed points at 0.55 and 0.15 µm3 , and aseparation point at roughly 0.50 µm3 , which divides the two basins of attraction. The shaded interval indicates the standard deviation.

Learning With Excess Intrinsic Spine
Dynamics as a Model of Fragile
X Syndrome
Interestingly, experimental results suggest that intrinsic spine
dynamics are abnormally high in a mouse model of fragile X
syndrome, fmr1KO (Pan et al., 2010; Nagaoka et al., 2016).
Below, we constrain the parameters α and β of the diffusion
coefficient D (v) = (αv + β)2 /2 to characterize intrinsic spine
dynamics in fmr1KO mice based on reported observations. Spine
turnover is about twice as high in fmr1KO mice as in WT
mice (Pan et al., 2010). Remarkably, this elevated spine turnoverlargely remains even when calcium activity is pharmacologically
blocked, suggesting that this is due to abnormal intrinsic spine
dynamics (Nagaoka et al., 2016). Another constraint is thespine volume distribution. Experimental reports comparing thespine volume distribution in fmr1KO and WT mice have mixed
observations – some studies detected more immature spines
in fmr1KO but others detected no significant difference (Heand Portera-Cailliau, 2013). For simplicity, we assume that any
differences in the spine volume distribution between fmr1KO
and WT mice are negligible. Based on the numerical fitting tothe observed spine turnover, these two constraints specify theparameters α = 0.43 day−1/2 and β = 0.021 µm3 day−1/2
for fmr1KO mice (Figure 5; see also Figure S4 for a systematicparameter search). Intuitively speaking, α and β are twice
as high as the corresponding WT values. This is because,
despite the presence of STDP, the spine volume distribution
is largely set by the equilibrium distribution of the intrinsic
spine dynamics, Peq (v) ∝ 1/ (v + β/α)2 . It suggests thatthe ratio β/α should be similar between WT and fmr1KO
to have similar spine volume distributions, and β in fmr1KO
is twice as large as WT to account for the doubled spine
turnover rate.

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FIGURE 5 | Spine turnover in our WT and fmr1KO models with two different
levels of intrinsic spine dynamics.

Learning and memory deficits have been reported in fmr1KO
mice (Padmashri et al., 2013). We investigated whether theselearning and memory deficits could potentially be explainedby the abnormal intrinsic spine dynamics modeled above.
Consistent with the experimental observations, storing memory
patterns was impaired in our fmr1KO model (Figures 6A–H),
where coherent spontaneous activation of cell assemblies rapidly
faded after learning (Figures 6B–D). This is because the mean
spine volumes of the stimulated groups decreased during themaintenance period (Figure 6F). This effect can be intuitively
understood from the result in Figure 4—memory cannot be
stably stored in a cell assembly if intrinsic spine dynamics are toostrong, relative to multiplicative STDP, because the stable fixed
point of the mean intra-group spine volume around 0.55 µm3
disappears. Individual spines fluctuated as in the WT model butwith a greater amount per unit time (Figure 6G). As a resultof excessively strong intrinsic spine dynamics and the decay of8June 2019 | Volume 13 | Article 38

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et al., 2008; Nagaoka et al., 2016) affect learning and memory
in recurrent circuit models. For simplicity, we assumed thatspine volume is proportional to synaptic strength (Matsuzaki
et al., 2004; Harvey and Svoboda, 2007; Bosch et al., 2014).
Contrary to a view that noisy synaptic changes are harmful tomemory, intrinsic spine dynamics in our model play a positiverole in preventing Hebbian-plasticity-driven non-specific growthof synapses. Specifically, as a result of the interaction between
STDP and intrinsic spine dynamics in our model, the mean intraassembly spine volume exhibits bistability, which is suitable tosustain memory. STDP keeps spines large within a cell assembly
due to the coherent spontaneous activity of the membershipneurons (e.g., Kenet et al., 2003; Diekelmann and Born, 2010; Wei
and Koulakov, 2014), and intrinsic spine dynamics constitutively
normalize all spines toward a physiological distribution.
In our model, memory is maintained by the total strength ofsynapses that connect neurons within an assembly. Individual
synapses fluctuate and exhibit constant turnover, but this does
not degrade the memory as long as the net strength is kept. One
property of the current model is that, given the innate Hebbian
instability of cell assemblies in spontaneously active recurrent
networks, highly overlapping cell assemblies likely merge during
maintenance, even in the presence of intrinsic spine dynamics.
In this sense, either a preprocessing mechanism that decorrelates
memory patterns (Perez-Orive et al., 2002; Leutgeb et al., 2007) ora more elaborate additional mechanism that stabilizes individual
synapses (Frey and Morris, 1997; Ziegler et al., 2015; Benna andFusi, 2016) may be helpful for improving the memory capacity(Hopfield, 1982).
There are several experimental observations that support therole of intrinsic spine dynamics in shaping the spine volume
distribution. First, the spine volume distribution in the absenceof neural activity and plasticity was well predicted by intrinsic
spine dynamics, and the distribution is similar to that in thenormal condition (Yasumatsu et al., 2008). Second, the spine
volume distribution was robust to experimental manipulations tosynaptic plasticity. Remarkably, calcineurin KO mice with little
LTD (Okazaki et al., 2018) exhibited a spine volume distributionthat was similar to WT mice. This raises an argument against the
hypothesis that the spine volume distribution is set by the balance
between LTP and LTD. We argue that previous computationalmodels that do not include intrinsic spine dynamics miss animportant component underlying synaptic organization.
We studied the interplay between activity-dependent synaptic
plasticity and intrinsic spine dynamics in the formationand maintenance of cell assemblies in recurrent networks.
Conventional studies of intrinsic spine dynamics often focus
on independent synapses (Yasumatsu et al., 2008) or learning
in a local feedforward network (Matsubara and Uehara, 2016)
instead of learning in a recurrently connected network. Anotherstudy explored the consequence of volatile synaptic strengths
in recurrently connected networks without studying how suchvolatility affects activity-dependent plasticity (Mongillo et al.,2018). Other studies model stochastic changes of synaptic
strength (Loewenstein et al., 2011) or connectivity (Degeret al., 2012; Fauth et al., 2015) without distinguishing activitydependent and -independent parts. Hence, these works domean intra-group spine volume for the stimulated groups, thestimulated spines scattered around the spine volume distribution(Figure 6E). This result is in contrast to the WT result, where
stimulated spines are localized nearer the tail of the spine volume
distribution (cf. Figure 3E). Similarly to the WT model, theexternal stimulation at the onset of learning increased the spine
generation rate by about 2.5% without significantly altering theelimination rate. Note that the baseline turnover in this fmr1KO
model was about twice as high as the WT model, consistent withthe experimental observation (Nagaoka et al., 2016).
The learning and memory deficits reproduced above are
however potentially more severe than observed experimentally
because fmr1KO mice have some learning capability, albeit
limited. Therefore, we considered whether there may be somecompensating mechanism, via which animals with excessively
strong intrinsic spine dynamics rescue some learning andmemory performance. We consider the regulation of potential
connectivity as one candidate mechanism. Figure 7 explores
how the stability of cell assemblies changes in our WT andfmr1KO models (i.e., the simulations of Figures 3, 6) if the peakpotential connectivity (Figure 1B1) is systematically altered. A
cell assembly in either the WT or fmr1KO model can exhibit
one of the following three scenarios: its mean firing rate either(1) explodes (≥100 Hz) due to unstable learning, (2) fades(≤1 Hz), or (3) is stably maintained within a physiologicalrange (>1 Hz and <100 Hz) during the maintenance period.
The peak potential connectivity of 10.4% was already optimized
for the WT model, such that nearly all cell assemblies arestable. The frequency of fading assemblies increased if potential
connectivity was too low and that of exploding assemblies
increased if potential connectivity was too high. However, theWT model stably maintained cell assemblies over a range ofthe peak potential connectivity from 9.4 to 10.8%. In contrast,
the maintenance of cell assemblies in the fmr1KO model wasmuch more sensitive to potential connectivity. As indicatedin Figure 7, most of the assemblies fade with the baseline
peak connectivity of 10.4%. The stability in the fmr1KO model
first improved up to ∼50% maintenancerate, but soon started
to decline again due to exploding assemblies, as potential
connectivity increases. Given this result, it is an interesting
possibility that fmr1KO mice with excessively strong intrinsic
spine dynamics may locally up-regulate potential connectivity,
relative to WT mice, to avoid catastrophic forgetting (Figure 7
Right; marked as compensation). This hypothesis, which predicts
an elevated frequency of exploding assemblies in fmr1KO mice,is consistent with the experimentally observed hyper neuralactivity and synchrony in fmr1KO mice (Musumeci et al., 2000;
Gonçalves et al., 2013).

DISCUSSION
In previous models of synaptic plasticity, changes in synaptic
strengths are assumed to be activity-dependent (Poo et al.,2016; Mongillo et al., 2017). Recent experimental observations
suggest that this is not the case. In this work, we modeled
how activity-independent intrinsic spine dynamics (Yasumatsu

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FIGURE 6 | Network behavior in the presence of predicted fmr1KO intrinsic spine dynamics. (A–H) Conventions are as in Figure 3.

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fmr1KO model

10Frequency

Frequency

WT model-20-16 -12 -8 -4 0

40

8 12 16 20-20-16 -12 -8 -4 0

4

8 12 16 20

Percentage change in potential connectivity

Percentage change in potential connectivity
Fade

Compensation

1

StableExplodeFIGURE 7 | Varying the probability of recurrent excitatory to excitatory connections in the WT model (Left) and the fmr1KO model (Right) permits an increased
chance of cell assembly fade or explosion at different connectivity levels. We hypothesize that fmr1KO mice may have a compensatory increase in potential
connectivity (red line) to partially rescue stable cell assemblies.

memory, the model reproduced epileptic-like neural activity thathas been reported experimentally in fmr1KO mice (Musumeci
et al., 2000; Gonçalves et al., 2013). More generally, the disrupted
cortical connectivity theory of ASD (Courchesne and Pierce,
2005; Kana et al., 2011) argues that deficiency of cortical longrange connections and compensatory strengthening of local
connections is a general feature of ASD. We contend that because
there are typically fewer long-range connections, which therefore
limits the positive feedback effect of Hebbian plasticity thatis required for maintaining cell assemblies, they are especiallysusceptible to degradation due to excessively strong intrinsic
spine dynamics, such as in our fmr1KO model. Furthermore, thelearning deficiency in our proposed local cortical circuit modelof fmr1KO, and its rescue by a compensatory increase in the local
connectivity (Figure 7), is consistent with this theory. Hence, it
is an intriguing possibility that pharmacological manipulations(Nagaoka et al., 2016), or a future neurofeedback technology(Ganguly and Poo, 2013; Yahata et al., 2016), could be used torescue memory and learning, and long-range neural association,
by reducing intrinsic spine dynamics or producing network
motifs (Watanabe and Rees, 2015) efficiently connecting target
brain areas.
A similar concept to intrinsic spine dynamics that has
been used to describe ASD is intrinsic forgetting (Davis andZhong, 2017). These two mechanisms are both related tochronic molecular signaling, which slowly degrades synapses andtherefore memories. However, important differences between the
two are how they are regulated in ASD animals and how they
could possibly be involved in producing flexible behavior. A
decrease in intrinsic forgetting has been argued to disable flexible
behavior by maintaining conflicting memories (Reaume et al.,2011; Davis and Zhong, 2017). In contrast, the excessively strong
intrinsic spine dynamics found in ASD animals (Pan et al., 2010;
Isshiki et al., 2014; Nagaoka et al., 2016) would work in the
opposite manner, by facilitating forgetting. Namely, stable cell
assemblies in our fmr1KO model, typically require more neurons
than the WT model to counter excess synaptic normalization.
Hence, given a hypothesis that the number of total neurons isroughly the same between fmr1KO and WT mice, fmr1KO mice

not distinguish their separate roles in memory and synaptic
normalization. Another line of studies (Litwin-Kumar andDoiron, 2014; Zenke et al., 2015) model activity-dependent
synaptic plasticity to account for the formation and maintenanceof cell assemblies in a recurrent network. However, because
these models do not include intrinsic spine dynamics, theirsynaptic strength distributions are typically sensitive to thefine balance between LTP and LTD and counter to theexperimental observations described above. Thus, the proposedmodel postulates how synaptic normalization, by intrinsic spine
dynamics, maintains most synapses that are not participating
in a cell assembly weak. In addition, intrinsic spine dynamics
in our model work as a homeostatic mechanism that stabilizes
Hebbian plasticity, although they are activity independent and an
explicit sensing-and-control (Davis, 2006; Shah and Crair, 2008)
mechanism is absent. For this to work, the relative magnitudeof Hebbian plasticity and intrinsic spine dynamics is important;
For example, this stabilization fails if Hebbian plasticity is too fast(see Figure 4). While we conjecture that intrinsic spine dynamics
can stabilize slow Hebbian plasticity in adults, other fast formsof homeostatic plasticity, such as inhibitory plasticity (Vogels
et al., 2011; Litwin-Kumar and Doiron, 2014), might be helpful
in the young, or once neural activity deviates beyond the levelthat intrinsic spine dynamics can compensate for.
Another main contribution of this study is the relationshipbetween intrinsic spine dynamics and ASD. The observationthat baseline spine turnover is abnormally high in various ASD
mouse models (Isshiki et al., 2014), including a model animal
for fragile X syndrome (fmr1KO) (Pan et al., 2010) suggests
abnormal intrinsic spine dynamics could be one candidate
substrate of ASD. Indeed, a recent experiment has confirmedthat high baseline spine turnover in fmr1KO mice is activityindependent (Nagaoka et al., 2016), and intrinsic spine dynamics
are stronger in fmr1KO (Ishii et al., 2018). Based on theseexperimental observations, we fitted parameters characterizing
intrinsic spine dynamics in fmr1KO mice and found that they
explain the learning deficit observed in fmr1KO mice (Padmashri
et al., 2013). Interestingly, when we included a compensatory
increase in recurrent excitatory connectivity to rescue some

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Learning With Intrinsic Spine Dynamics

with time constant τR = 3.5 ms (relative refractoriness). The
differential equations are numerically solved using the Euler
method with bin size ∆t = 0.1 ms.

may afford a smaller number of cell assemblies in the brain,
possibly reducing the number of behavioral repertoires.
Overall, our spine dynamics model provides a novel
path relating spine statistics, memory, and ASD. Notably,
it is currently difficult to block intrinsic spine dynamics
experimentally in vivo without affecting plasticity, because they
are thought to be caused by the thermal agitation of molecules.
This underscores the importance of a modeling study. Selective
manipulations of intrinsic spine dynamics are an intriguing
candidate direction to influence memory and learning, and thecurrent model serves as a guide.

Network Topology
In the network model of Figure 1A, excitatory neurons, E, andinhibitory neurons, I, are sparsely connected. The connections
from excitatory to inhibitory neurons (E → I) and thosefrom inhibitory to excitatory neurons (I → E) are randomly
generated with 10% connection probability. Each of these
directed connections is mediated by a single synapse, whosestrength is randomly drawn from a uniform distribution in therange [0, 31] mV for E → I and [−31, 0] mV for I → E,
respectively. According to our neuron model described above, atypical E → I or I → E synapse produces a postsynaptic potentialof 6 mV. For simplicity, we assume no direct connections
between inhibitory neurons. Excitatory neurons are equidistantly
placed on a one-dimensional ring of circumference 1 a.u. that
represents the feature space. Note that the tuning distancein feature space dfs does not necessarily correspond to thephysical location of a neuron, or the distance between any
two neurons. The potential connections from excitatory to
excitatory neurons (E → E) are randomly generated according
to the Gaussian probability profile (10.4%∗ exp[−0.5∗ (dfs /0.1)2 ],Figure 1B1), which peaks at 10.4% for similarly tuned neurons(dfs ≈ 0), and decays toward 0 with a length-constant of 0.1 as
dfs increases. Although this E → E peak connection probability is
optimized for memory retention, simulation outcomes are robustwith this parameter in our model (see, Figure 7, WT model).
Effectively, this allows excitatory neurons closer in the feature
space to be more interconnected, and those farther apart to beless so. If there is a potential connection from one neuron toanother, we randomly assign a fixed number K(K = 1, 2, · · · , 10)PK′′of spines from a Poisson distribution (λK /K!)/ 10
K ′ =1 (λ /K !)with parameter λ = 3 (Figure 1B2).

METHODS
Network and Neurons
A local network of NE excitatory and NI inhibitory leakyintegrate-and-fire neurons [Figure 1A; see e.g., (Tuckwell, 1988)]
is considered. We simulate a network with (N E , NI ) =(1000, 200). The dynamics of membrane potential Vi of neuron
i is described by
τm

X
dVi
wij (k)(t) = −(Vi (t) − V0 ) − Ai (t) + Ri (t)
dt
j,k
Z t
f (t − t ′ )Sj (t ′ − ∆ij )dt ′ + Ri (t)Ii ext (t)−∞with membrane time constant τm = 20 ms, resting potential
V0 = −70 mV, adaptation Ai , refractory coefficient Ri , kth(k = 1, 2, ...), synaptic strength wij (k) from neuron j to neuronP(n)i, spike train Sj (t) =n δ(t − tj ) of neuron j composed ofnth (n = 1, 2, ...) spike time tj (n) , random axonal delay ∆ij fromneuron j to neuron i uniformly sampled from interval [0.5, 5.0]
ms, and external input Ii ext to neuron i (see below). Note that δ isthe Dirac delta function. The time course of postsynaptic input ismodeled using the alpha function (Gerstner et al., 2014),
f (t) = 20 mV ·τr  −t/τfe− e−t/τr 2(t)τf − τr

Stimulation
In addition to the recurrent input, neuron i also receives
spike train Si ext (t ′ ) as external input (dashed connection in
Figure 1A). At baseline, Si ext (t ′ ) is generated by a Poisson processwith firing intensity 60 Hz. Input neurons are not modeled
explicitly here. The external input to each neuron is givenRt
by Ii ext (t) = −∞ f (t − t ′ )Si ext (t ′ )dt ′ . This sets the baseline
membrane potential and firing rate of neurons to −58.6 ±2.4 mV and 0.13 ± 0.08 Hz, respectively.
We divide the feature space of excitatory neurons into four
equally-sized consecutive parts and randomly select 40% of
the neurons from each part as a stimulated neuron group(Figure 1A). During the learning period (indicated by blue
horizontal bars below the time axes in Figures 2, 3, 6), one of the
4 stimulated neuron groups is randomly selected with probability¼ and receives additional Poisson spikes at 750 Hz for 3 s. During
the learning period, all inhibitory neurons also receive additional
Poisson spikes at 300 Hz. The learning period starts at t = 0 andends when at least one group’s mean intra-group spine volume
reaches ≥ 0.49 µm3 .

with rise time τr = 0.5 ms, fall time τf = 2.0 ms, and theHeaviside step function 2(t) that takes 1 for t ≥ 0 and 0 for t < 0.
The peak value of f (t) is about 0.39 mV. A spike is emitted when
Vi reaches a spiking threshold at −50 mV and then Vi is reset toV0 . Excitatory neurons receive adaptation input Ai that reflects aslow Na+ -activated K+ current (Wang et al., 2003) (Figure 1B4),
with dynamics described bydAi
Ai (t)(t) = −
+ 0.0017[20 mV − Ai (t)]Si (t)dtτAwith time constant τA = 13 s. In contrast, we assume no
adaptation (Ai = 0) in inhibitory neurons. Refractoriness is
imposed by Ri . Ri is fixed at 0 after each spike of neuron i for 1 ms
(absolute refractoriness), and then recovers toward 1 following
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Spine Dynamicsµm3 and such a spine produces 0.8 mV of excitatory postsynaptic
potential. We set vθ to be a threshold volume typically used inexperiments to detect spines (Yasumatsu et al., 2008). We define
spine gain and loss by a fraction of spines passing this thresholdfrom below and above per day, respectively. The exact value of vθ
does not matter for our results as long as it is sufficiently small.

Unlike E → I and I → E synapses, which have fixed weights, E→ E synapses (orange and brown dashed in Figure 1A) change
in time via two independent mechanisms: STDP and intrinsic
spine dynamics. We assume that synaptic strengths for E → E
synapses are essentially proportional to their spine volumes andtherefore model their spine volume dynamics as changes insynaptic strengths. Changes in the kth (k = 1, 2, ..., Kij ) spine
volume vij (k) on neuron i, receiving signal from neuron j, ismodeled by!
dvij (k)vij (k) (t)(t) = Ta Si (t)SSj (t) −Sj (t)SSi (t)dt
vLTD√
2(vij (k) (t) − vθ ) + S
T(αvij (k) (t) + β)ξ (t)Itô vs. Stratonovich Interpretation
The meaning of fluctuation is different under the Itô andStratonovich interpretations of intrinsic spine dynamics(Gardiner, 1985). The intrinsic spine dynamics under the Itôinterpretation that we study in the main text are described by∂P(v, t)1 ∂2=(αv + β)2 P(v, t)
∂t
2 ∂v2

where the first and second terms on the right-hand side describechanges by multiplicative STDP (van Rossum et al., 2000) andintrinsic spine dynamics (Yasumatsu et al., 2008), respectively. T
is a speed-up factor that we describe below, a = 7.6 · 10−9 µm3 isthe amplitude of STDP, S
Si is the running average of past spiking
activity of neuron i, i.e.,

In this view, changes in spine volume distribution are purely
produced by the volume-dependent fluctuation with amplitude
αv + β. In contrast, the Stratonovich interpretation represents
the same equation in a different way:∂P(v, t)∂α(αv + β)= −P(v, t)
−
∂t
∂v
2



∂ 
1 ∂
(αv + β)P(v, t)(αv + β)
+
2 ∂v
∂v

S
dS
Si
Si (t)(t) = −
+ Si (t)dt
τSTDP
with an averaging time constant τSTDP = 20 ms, and vLTD =0.5 µm3 is the scaling factor for volume-dependent LTD (vanRossum et al., 2000). STDP is assumed to be absent for small
spines of vij (k) < vθ = 0.02 µm3 . Slope parameter α = 0.2
day−½ and offset parameter β = 0.01 µm3 day−½ for intrinsic
spine dynamics are set as previously experimentally observed(Yasumatsu et al., 2008). ξ is white noise with the autocorrelation
function < ξ (t)ξ (t ′ ) >= δ(t − t ′ ). The above Langevin equation
is numerically solved by the Euler method with bin size ∆t = 0.1
ms. In addition, we set reflecting boundaries for spine volume
to enforce 0 ≤ vij (k) ≤ 1.0 µm3 for all spines. One problem
is that it is too time consuming to directly simulate the 10-day
learning period studied with the fine time resolution required tosimulate STDP and intrinsic spine dynamics. We therefore run
T = 3.3·104 times shorter simulations by speeding up both STDP√
and intrinsic spine dynamics by factors T and S
T, respectively.
(Note that volume changes v(t + ∆t) − v(t) by intrinsic spine
dynamics are diffusive and scale with the square root of time√
duration ∆t.) This way, we can extrapolate spine volume
changes happening during 10 days based on shorter simulations
up to 3,000 s. We display the time before this conversion inpanels describing neural activity in seconds, but display the timeafter this conversion in panels describing learning in days. We
initially set E → E spine volumes by randomly sampling fromthe equilibrium distribution Peq (v) ∝ (αv + β)−2 , which is set by
the intrinsic spine dynamics. Synaptic strength w of a spine withvolume v is then assumed to be

In this view, changes in spine volume distribution are describedby two terms: the first term is the drift term produced by apparent
force α(αv + β)/2 and the second term is produced by thevolume-dependent fluctuation αv+β. Hence, while the above two
equations are identical, there is a semantic difference regarding
what fluctuation means. According to the Itô interpretation, only
the fluctuation drives spine volume changes and this fluctuation
preserves mean spine volume [i.e., martingale (Øksendal, 2000)]
except for a boundary effect. According to the Stratonovichinterpretation, the apparent force shrinks and the fluctuation
enlarges mean spine volume, respectively, and the two effects
are canceled. These two interpretations become the same inthe special case of α = 0, namely, when the fluctuation isvolume independent.

Simulation Environment
Simulations were performed in custom written C code withthe forward Euler-integration method and a step size of 0.1 ms.
Code was written to efficiently run on multiple GPUs to produce
the trillions of random numbers needed to model the intrinsic
spine dynamics per simulation up to 3,000 s. Post-simulation
analysis was undertaken with MATLAB. Source code is available
upon request.

DATA AVAILABILITY

w = (43 µm−3 )v
for v ≥ vθ (a functional spine) and 0 for v < vθ (a non-functional
spine, e.g., filopodia). The median spine volume of Peq (v) is 0.047

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The datasets generated for this study are available on request to
the corresponding author.

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AUTHOR CONTRIBUTIONS

Number JP19dm0207001 (TT), and RIKEN Center for
Brain Science (JH and TT). This manuscript has beenreleased as a Pre-Print (Humble et al., 2019) at bioRxiv
525980.

HK and TT conceived the project. JH and KH performed
numerical simulations. JH, KH, HK, and TT analyzed the results
and drafted and revised the manuscript.

SUPPLEMENTARY MATERIALACKNOWLEDGMENTS

The Supplementary Material for this article can be foundonline at: https://www.frontiersin.org/articles/10.3389/fncom.
2019.00038/full#supplementary-material

This work was supported by JSPS KAKENHI Grant Number
JP18H05432 (TT), Brain/MINDS from AMED under Grant

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Conflict of Interest Statement: The authors declare that the research was
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