Neuron

ArticleModeling the Dynamic Interactionof Hebbian and Homeostatic Plasticity
Taro Toyoizumi,1,2,3,* Megumi Kaneko,4 Michael P. Stryker,4 and Kenneth D. Miller2,5
1RIKEN Brain Science Institute, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
2Center for Theoretical Neuroscience and Department of Neuroscience, College of Physicians and Surgeons, Columbia University,
1051 Riverside Drive, New York, NY 10032, USA
3Deptartment of Computational Intelligence and Systems Science, Tokyo Institute of Technology, Yokohama 226-8502, Japan
4Center for Integrative Neuroscience and Department of Physiology, University of California, 675 Nelson Rising Lane, San Francisco,
San Francisco, CA 94158, USA
5Kavli Institute for Brain Science, Columbia University, 1051 Riverside Drive, New York, NY 10032, USA*Correspondence: taro.toyoizumi@brain.riken.jp
http://dx.doi.org/10.1016/j.neuron.2014.09.036

SUMMARY

Hebbian and homeostatic plasticity together refine
neural circuitry, but their interactions are unclear. In
most existing models, each form of plasticity directly
modifies synaptic strength. Equilibrium is reachedwhen the two are inducing equal and oppositechanges. We show that such models cannot reproduce ocular dominance plasticity (ODP) because
negative feedback from the slow homeostatic plasticity observed in ODP cannot stabilize the positive
feedback of fast Hebbian plasticity. We propose amodel in which synaptic strength is the product ofa synapse-specific Hebbian factor and a postsynaptic-cell-specific homeostatic factor, with each factor separately arriving at a stable inactive state. Thismodel captures ODP dynamics and has plausible
biophysical substrates. We confirm model predictions experimentally that plasticity is inactive at
stable states and that synaptic strength overshoots
during recovery from visual deprivation. These results highlight the importance of multiple regulatory
pathways for interactions of plasticity mechanisms
operating over separate timescales.
INTRODUCTION
Hebbian plasticity and homeostatic plasticity are the two major
forms of activity-dependent plasticity that modify neuronal circuits (Turrigiano, 2008). We use ‘‘Hebbian plasticity’’ to refer toplasticity that depends on the correlations between pre- andpostsynaptic activity such that excitatory synapses that effectively drive a postsynaptic cell grow stronger while ineffective
synapses are weakened. This is a positive feedback process—strong synapses grow stronger—that in models typically leads
to synaptic instability in the absence of additional biological constraints (Miller and MacKay, 1994; Turrigiano, 2008). Synaptic
homeostasis is a negative feedback mechanism that typicallyinvolves nonspecific scaling of all excitatory or inhibitory synapses onto a cell to oppose changes in overall activity levels. This isthought to maintain activity levels within a dynamic range and,more generally, to stabilize neuronal circuit function despite
the positive feedback of Hebbian plasticity (Turrigiano, 2008). It
is not known how these two forms of plasticity interact in biological systems (Shepherd and Huganir, 2007; Turrigiano, 2011,2008).
Ocular dominance plasticity (ODP) in primary visual cortex (V1)
has been a standard system in which to study experiencedependent plasticity (Espinosa and Stryker, 2012). During thecritical period for ODP, monocular deprivation (MD)—the closureof one eye—induces rapid weakening of responses to the closed
eye and subsequent strengthening of responses to the open eye(Frenkel and Bear, 2004; Hofer et al., 2006; Mrsic-Flogel et al.,2007). A recovery period with binocular vision following MD
causes both eyes’ response levels to return to normal. Recently,
three separable processes have been identified underlying thisplasticity in mouse V1 (Kaneko et al., 2008a, 2008b):(1) Weakening of the closed eye’s responses is rapid, occurring over the first 3 days of MD, and appears to be mediated by Hebbian plasticity because of its dependence on
calcium entry through N-methyl-D-aspartate (NMDA) receptors acting on calcium calmodulin kinase type II(Taha et al., 2002). This weakening shares other molecularfeatures of Hebbian long-term depression (LTD) (Heynen
et al., 2003; Yoon et al., 2009) but differs from LTD in its
dependence on protein synthesis (Lee et al., 2003; Shepherd and Huganir, 2007; Taha and Stryker, 2002). It is notaffected by blockade of tumor necrosis factor-a (TNF-a)or the tropomyosin-related kinase B (TrkB) receptor (Kaneko et al., 2008a, 2008b).
(2) Strengthening of the open eye is slower, commencing
only after about 3 days, and appears to be mediated byhomeostatic synaptic scaling; it is specifically prevented
by blockade of TNF-a (but not of TrkB) (Kaneko et al.,2008a, 2008b), which induces a global form of homeostatic synaptic scaling (Stellwagen and Malenka, 2006).
TNF-a induces a uniform scaling up of the strengths of
excitatory synapses in response to a lowering of overall
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NeuronModeling Hebbian and Homeostatic Plasticity

activity levels. This occurs without alteration in Hebbian
plasticity as assessed by the percentage changes of synaptic strengths induced by long-term potentiation (LTP) or
LTD (Stellwagen and Malenka, 2006).
(3) Recovery from MD under binocular vision is specifically
prevented by blockade of TrkB (Kaneko et al., 2008a).
TrkB has a variety of actions on synaptic plasticity. It is
required for the growth of new synapses in neuronal cell
culture (Meyer-Franke et al., 1998) and is involved in stabilization of Hebbian LTP (Figurov et al., 1996; Kovalchuk
et al., 2002; Lai et al., 2012; Sermasi et al., 2000; Tanaka
et al., 2008).
The slow onset of homeostatic scaling, relative to the fast
onset of Hebbian plasticity, poses a problem. Synaptic dynamics under Hebbian plasticity alone are typically unstable until
synaptic strengths (‘‘weights’’) are driven to saturation near
maximum or minimum allowed values (e.g., Miller and MacKay,1994). In models that combine Hebbian with homeostatic plasticity (or with mechanisms similar to homeostatic plasticity,such as multiplicative normalization of synaptic strengths or
metaplasticity), homeostatic plasticity generally stabilizes a setof unsaturated weights that would be unstable under Hebbian
plasticity alone (von der Malsburg, 1973; Bienenstock et al.,
1982; Oja, 1982; Cooper et al., 2004; Miller and MacKay, 1994;
Toyoizumi and Miller, 2009; Toyoizumi et al., 2013). However,such stabilization fails if homeostatic plasticity is too slow
compared to unstable Hebbian plasticity (Cooper et al., 2004;
Zenke et al., 2013). This is an example of the more general resultthat slow negative feedback cannot stabilize a fast, unstable
positive feedback process. Thus, the slow onset suggests that
homeostasis cannot stabilize Hebbian ODP.
To solve this problem, Hebbian plasticity must arrive on its
own at a stable steady state, and this stability must not be
disturbed by slow, ongoing homeostatic plasticity. We showthat this solution can be instantiated and the experimental resultsrobustly reproduced by a simple model in which the total synaptic strength is the product of two factors: a synapse-nonspecificfactor, applicable to the entire postsynaptic cell, controlled byhomeostatic plasticity and a synapse-specific factor controlled
by Hebbian plasticity. In addition to demonstrating the model’sability to account for existing results, we test several key predictions of the model experimentally, including a lack of constitutive
but opposed Hebbian and homeostatic plasticity at a stablestate of the weights and a TNF-a-dependent overshoot of
formerly closed-eye weights upon reopening of the closed eye
after MD.
RESULTSMD Effects on Single-Synapse Plasticity Models
in the Monocular Cortex
In monocular cortex, which receives input only from the contralateral eye, similar dynamics of Hebbian and homeostatic plasticity are seen as in binocular cortex. During the first 3 days ofMD, the strength of the closed eye’s input decreases by about
25%–30% in an NMDA receptor-dependent manner (Frenkel
and Bear, 2004; Heynen et al., 2003; Kaneko et al., 2008a). Dur498 Neuron 84, 497–510, October 22, 2014 ª2014 Elsevier Inc.

ing the subsequent 3 days of MD, the closed-eye input strength
homeostatically increases back to near-baseline levels, an increase dependent on TNF-a (Kaneko et al., 2008b). Although
this dynamical behavior under MD is simple, existing models
cannot reproduce it, as we will show, because homeostatic plasticity that is slow enough to allow significant initial depression of
the synaptic weight is too slow to stabilize plasticity.
We consider in this section a very simple, one-synapse modelof monocular cortex: a single postynaptic cell with activity y receives a single input with synaptic strength w and activity x.
We assume the output activity is given by y = wx and take theinput activity x to be constant in each condition, x = 1 for normal
visual experience, and x<1 under MD (for simplicity we takeactivities and weights to be unitless numbers). By assuming asingle input, we are in essence assuming that the synapses projecting to the monocular cortex are relatively homogeneous sothat w and x indicate the average synaptic strength from andaverage activity of the LGN inputs from the contralateral eye.
Similarly, y indicates the average activity of monocular cortex.
Later in the paper we consider a heterogeneous population of inputs in binocular cortex and show that the conclusions reached
here are unchanged.
The BCM Rule
A learning rule that is commonly used to describe ODP and that
includes both Hebbian and homeostatic elements is the Bienenstock-Cooper-Munro (BCM) rule (Bienenstock et al., 1982;
Cooper et al., 2004). In this model, an active input ðx>0Þ undergoes LTP if the postsynaptic activity y is greater than athreshold q and undergoes LTD if y<q; this is the Hebbian
element. With the simple system considered here, this part ofthe BCM rule can be written astwdw= xyðy  qÞ;
dt(Equation 1)

where t w is a time constant that sets the Hebbian learning rate.
The homeostatic element of the BCM rule arises through the
dynamic adjustment of q over time to keep average postsynaptic
activity near a set-point activity level, y0 . q is a superlinear function of the average firing rate, typically q = y 2 =y0 , where y2 is theaverage of y2 over some averaging time. Taking this averaging
time to be t q , the dynamics of q can be written as follows:tqdqy=  q+y :dty0(Equation 2)

When the model is switched to the MD condition, the weight
initially decreases and subsequently increases. Consider starting at the fixed point for the normal ðx = 1Þ condition,y = q = w = y0 . Immediately after MD onset, input activity x, andthus y = wx, is reduced, so y<q. This causes LTD under Equation 1 and, thus, further reduction in y. However, since y<y0 , q
moves toward an equilibrium value yðy=y0 Þ that is less than y.
Once q becomes less than y, the Hebbian rule causes LTP andy rises.
The speed of the decrease in q, and thus of the switch from
LTD to LTP, is determined by the time constant t q (Equation 2).
The speed of LTD is controlled by t w (Equation 1). In order tohave significant initial LTD, t q must be sufficiently large, relative

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A

B

C

Figure 1. The Dynamics of the Synaptic
Strength under the BCM Learning Rule in
the Monocular Cortex during MD Initiatedat Time 0(A–C) The synaptic strength w (red line) and theactivity-dependent threshold q (black dashed line)
were plotted versus time for homeostatic plasticity
that is the same speed as (A: t q = t w ) or slower than
(B: t q = 3t w ) Hebbian plasticity. The initial conditionðt = 0Þ was set to the fixed point under normal
rearing (which is unstable for t q = 3t w , see C). For
faster homeostatic plasticity, there is little initialdepression of synaptic strength (A). When the homeostasis is slowed, however, the normal-rearing
fixed point becomes unstable (c.f. C), and thesynaptic strength exhibits large oscillations under
MD (B). We took the set-point activity to be y0 = 1, which sets the units of activity. We set input activity x = 1 in the normal condition, x = 1=2 under MD. We set
t w = 0:2 days; numbers on the x axis represent days. (C) A systematic study of the BCM learning rule, as a function of input activity x and speed of homeostatic
plasticity, t q =tw . We characterized the dynamics of synaptic strength under MD by two variables: the depth of synaptic depression, measured as the strength w
at the first trough of synaptic strength relative to the pre-MD ðx = 1Þ strength of w = 1 (green contour lines); and the stability of the fixed point w = 1=x, as measured
by the stability index Relt q (color plot; more positive values of the stability index indicate greater distances from instability; negative values mean the fixed
point is unstable). Here, l is the eigenvalue with greatest real part of the system linearized about the fixed point. For choices of x and t q =t w that reproducew  0:7, as in experiments, the synaptic strength dynamics are unstable for a wide range of x including the normal rearing condition x = 1. The results shown inthis figure are general and do not depend on any specific parameter choices (e.g., choosing x = 1 to represent normal rearing), as explained in Supplemental
Information Section S1.

to t w , so that significant LTD occurs before q adapts. When t q =t w
is sufficiently small, q adapts quickly, before the synaptic weightsignificantly changes, so that there is very little initial synaptic
depression (Figure 1A). However, as t q =t w is increased, the fixed
point under normal rearing becomes unstable even as the degree of synaptic depression under MD (initiated with this unstable normal-rearing fixed point as initial condition) remains small(Figure 1C; see also Supplemental Information Section S1, available online), and this MD leads to large oscillations of synaptic
strengths (Figure 1B). The oscillations arise because when y reaches y0 , q still reflects the earlier weight values, so weight change
continues in the same direction until q catches up. Instability occurs when the threshold change is so slow that the thresholdnever catches up (Cooper et al., 2004; Zenke et al., 2013)(when tq =tw >1=ðx2 y0 Þ for Equations 1 and 2).
In the BCM model, both dw=dt and dq=dt are zero at the fixed
point. However, biological weights will constantly fluctuate under ongoing changes in activity. Such fluctuations induce destabilizing Hebbian plasticity, which can only be stabilized by fast
homeostatic plasticity (i.e., fast changes in q). In this sense, theBCM model requires that the two forms of plasticity are constitutively active but opposed in the vicinity of any fixed point.
Addition of Stable Hebbian Terms and a Homeostatic
Term
To try to prevent loss of stability with slow homeostatic plasticity,
we next consider an intrinsically stable Hebbian rule. In addition,
we would like the model to be physiologically realistic in three respects in which the BCM rule is not. First, LTP and LTD should
saturate, whereas in the BCM rule there is no limit to synaptic
potentiation and LTD can reduce weights to zero. Second, in amodel with multiple synapses, homeostatic plasticity should
multiplicatively scale synaptic strengths, preserving the relative
strengths of different synapses, as suggested by homeostatic

scaling of the distribution of miniature excitatory postsynaptic
potentials (mEPSPs) (Turrigiano et al., 1998; Stellwagen andMalenka, 2006; Kaneko et al., 2008b; Turrigiano, 2011). The
sliding threshold of the BCM rule does not produce multiplicativechanges in synaptic strengths. Third, homeostatic plasticity occurs under blockade of activity through bath application of tetrodotoxin (TTX) or under blockade of NMDA receptors, both in vitro(TTX, Turrigiano et al., 1998; NMDA block, Kaneko et al., 2008b;
both, Stellwagen and Malenka, 2006) and in vivo in V1 during thecritical period (TTX, Maffei and Turrigiano, 2008; NMDA, see Figure 7A). Under the BCM rule there is no plasticity when the preor the postsynaptic firing rate is zero or when NMDA receptors
are blocked.
We again consider our one-input model of the monocular cortex. We assume that synaptic strength is modified by the sum ofan LTP term, an LTD term, and a multiplicative homeostatic
term. LTP occurs when the product of pre- and postsynaptic activities, xy, is greater than a fixed threshold q, while LTD occurs
for xy<q. To make the Hebbian plasticity intrinsically stable, the
LTP and LTD terms saturate when the weight w reaches amaximal value wmax or minimal value wmin , respectively, consistent with experiments suggesting a limited range of strengths ofindividual synapses (O’Connor et al., 2005; Petersen et al.,1998). A nonzero minimal weight value is important because
multiplicative homeostatic plasticity cannot potentiate synaptic
strength if it reaches zero. The homeostatic term changes theweight to move the time-averaged postsynaptic activity y toward the set-point value y0 . This term induces weight change
proportional to w, which—once we consider multiple inputs—
will make the homeostasis multiplicative, scaling all weights by
a common factor rather than adding or subtracting a common
amount from all weights. A parameter g determines the strength(or, equivalently, the learning speed) of homeostasis relative to
Hebbian plasticity.
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B

A

C

Figure 2. Simulation Results of Plasticity in
the Monocular Cortex during MD, Using a
Learning Rule with a Stable HebbianComponent and a Homeostatic Component(Equations 3 and 4).

(A–C) The synaptic strength, w, and the average
postsynaptic activity, y, are plotted for stronger
MD (A; x = 0:5) and weaker MD (B; x = 0:73). For themodel to reproduce the experimental result, as in(A), parameters must be tuned so that the dynamics remain relatively near the stable fixed point under Hebbian plasticity alone. When the combined Hebbian/homeostatic dynamics move sufficiently far fromthat fixed point, the Hebbian dynamics are unstable and cannot be stabilized by slow homeostatic plasticity, yielding oscillations (B) or instability. The MD fixed
point in (A), though nonoscillatory, involves constitutive but opposed LTD and homeostatic plasticity so that blockade of Hebbian plasticity beginning at day 7 of
MD yields a marked homeostatic increase in weight (C). Parameters: wmax = 1; wmin = 0:6; t w = 0:3 day; t y = 3 days; y0 = 0:8; q = 0:6; g = 0:23: x axis: time in days.

We let ½x + represent the operation of setting negative values
to zero, that is, ½x + = x when x>0, and ½x + = 0 otherwise. Thenthe equation for weight change istwdw= ½wmax  w + ½xy  q +dt
 ½w  wmin  + ½q  xy + + gwð1  y=y0 Þ:(Equation 3)

The first term in Equation 3 is the LTP term, the second is the
LTD term, and the third is the homeostatic term. This equation
does not have the problems just noted for the BCM rule: in Equation 3, LTP and LTD both saturate, homeostatic plasticity
induces a multiplicative scaling of weights, and homeostatic
plasticity will occur even in the absence of pre- or postsynaptic
activity.
The delay in the signal driving homeostatic learning is determined by the time, t y , over which the postsynaptic activity isaveraged to produce y. Assuming an exponentially weighted
average with time constant t y , the equation for y istydy
=  y + y:dt

(Equation 4)

We choose parameters to reproduce the results of MD in the
monocular cortex (Kaneko et al., 2008b) (see Supplemental Information Section S2). This model can reproduce fast LTD andslow homeostatic plasticity in the monocular cortex under MD.
However, this outcome was fragile and sensitive to parametervalues. Figure 2A shows the simulated result of the plasticity
rule under strong MD ðx = 0:5Þ. The synaptic weight rapidly
decreased and reached about 70% of the pre-MD value after
2 days of MD. Delayed homeostatic plasticity then started, due
to the loss of postsynaptic activity, and scaled up the synaptic
strength to about 90% of the pre-MD value. This result is consistent with the experimental observation. However, the result depended sensitively on the input strength under MD. Figure 2B
shows a result of a simulation under weaker MD ðx = 0:73Þ. In
this case, the synaptic strength settled into a stable oscillation(see phase plane analysis, Supplemental Information Section
S3, and Figure S1).
This behavior is similar to that of the BCM rule explored previously. The problem is a general one that does not depend on
arbitrary details of the implementation (e.g., use of a threshold
nonlinearity ½x + as opposed to more smooth nonlinearities);
for sufficiently slow homeostatic plasticity, there will be input
values yielding oscillations or instability. The saturation limits
500 Neuron 84, 497–510, October 22, 2014 ª2014 Elsevier Inc.

ensure that the Hebbian terms in Equation 3, by themselves,
have a stable fixed point. However, the system’s fixed point—
the steady-state values of w and y of the entire system, including
the homeostatic term—may be at values of w distinct from thisHebbian-only fixed point. When the two fixed points are sufficiently far apart, small perturbations of w away from the fixed
point will be amplified by fast Hebbian positive feedback, andthe slow homeostatic negative feedback through change in yeither cannot catch up (instability) or takes a long time to catchup (oscillations). In sum, system stability depends on choosing
parameters for which the Hebbian and system fixed points are
sufficiently near, but they are unlikely to remain near across multiple input values. We are once again faced with the fact that slow
homeostatic plasticity cannot stabilize fast, unstable (positive
feedback) dynamics.
Even when the MD fixed point is near enough to the Hebbian
fixed point to prevent oscillations (Figure 2A), LTD and homeostatic upscaling are constitutively active but cancelling. We
show in Section S2 of the Supplemental Information that such
constitutive plasticity always occurs at fixed points of Equations
3 and 4 for values of x outside of a finite range. As a result, if Hebbian plasticity is blocked, a large homeostatic change in the synaptic weight will occur (Figure 2C), a possibility we now test.
Experimental Test I: Lack of Constitutive Plasticity
at Steady States
As we have just seen, plasticity models in which Hebbian and homeostatic plasticity compete to directly alter a single factor, w,
make the general prediction that, for at least some rearing conditions, Hebbian and system fixed points should not coincide.
This means that Hebbian and homeostatic plasticity are bothconstitutively active at the system steady state, inducing equaland opposite weight changes that cancel. While a rule might
have both forms of plasticity separately zero at one steady state,
it would require fine-tuning of parameters for separate cancellations to occur at multiple steady states corresponding todifferent rearing conditions.
To test this prediction experimentally, we blocked Hebbian
plasticity by a partial blockade of NMDA receptors using daily
intraperitoneal injection of a low dose of the NMDA receptor
inhibitor 3-(2-carboxypiperazin-4-yl)propyl-1-phosphonic acid
(CPP). This regime was shown to block NMDA-receptor-dependent plasticity in mouse V1 (in the same strain of mice studiedhere) during the critical period and in adulthood; it blocks

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A1

A2

B1

B2

Figure 3. Lack of Constitutive Plasticity at Steady States(A and B) Hebbian plasticity and homeostatic plasticity are each separately inactive under normal rearing (A1 and B1) and after 6 days of monocular deprivation (A2and B2). (A) Experimental designs. An NMDA receptor antagonist (CPP) or vehicle solution was injected intraperitoneally for 6 days under normal rearing (A1) or forthe last 4 days of a 10-day MD period (A2). (B) In each case, responses in binocular V1 to stimulation through each eye were unchanged by the NMDA receptor
block, suggesting a lack of ongoing plasticity in these conditions. Baseline response levels were indistinguishable in CPP and control animals (Figure S7). Data arerepresented as mean ± SEM.

NMDA-receptor-dependent expression of the immediate earlygene product Zif268 and blocks ODP in response to MD but
does not block visual behavior or the responses of neurons inmouse V1 (Sato and Stryker, 2008). As we will discuss later, it
leaves homeostatic plasticity intact, while appearing to blockHebbian plasticity, when applied during days 4–7 of MD(Figure 7A). We injected CPP during each of two different,
approximately steady states during the critical period for ODP
in mice: during normal rearing and during the last 3 days of a
10-day MD (Figure 3). At each steady state, the blockade caused
no significant change in either eye’s response strength in V1, asassessed with intrinsic signal imaging, relative to control animals(daily vehicle injection), in contrast to the outcome seen in
Figure 2C. This indicates that Hebbian and homeostatic plasticity
are each separately nearly zero at each of the two steady states.
Note that this experiment does not speak against the BCM
model. In that model, all plasticity is stopped by NMDA receptor
blockade. However, we have noted other empirical problems
with the BCM model.
Our Proposed Solution: A Model with Hebbianand Homeostatic Factors that Multiply
to Produce Synaptic Strength
We have seen that the observed combination of fast synaptic
depression and slow homeostatic plasticity after MD cannot berobustly reproduced by traditional plasticity models in which
the Hebbian and homeostatic components directly compete tocontrol the same factor. Furthermore, the constitutive, equal,
and opposite forms of plasticity predicted to exist at a steadystate by models with separate, competing homeostatic andHebbian terms were not found experimentally.

This motivates our proposed solution: a model in which a synapse-specific Hebbian factor, r, and a postsynaptic-cell-specific homeostatic factor, H, each with their own learning
dynamics, are multiplied to give the synaptic strength: w = Hr
(Figure 4A). LTP and LTD rapidly modify r in an NMDA-dependent manner, while homeostatic plasticity more slowly modifies
H through the effects of TNF-a and perhaps other factors. Thismodel is consistent with the experimental result that the percentage of synaptic strength changes induced by Hebbian LTP or
LTD protocols were unchanged by modifications of homeostatic
plasticity (pretreatment with TNF-a or knockout of TNF-a receptors; Stellwagen and Malenka, 2006).
In our proposed model, the dynamics of r—the Hebbian dynamics—are stable by themselves, reaching a stable fixed point
for the given input statistics. We will assume this stabilization isachieved through saturation of r at minimal and maximal allowed values (O’Connor et al., 2005; Petersen et al., 1998),
but any stable Hebbian learning rule would suffice. The homeostatic dynamics still involve slow learning, but because of
the relationship w = Hr, such slow dynamics will scale theweights—including the minimal and maximal weights—to bringthe overall activity level toward the desired set point without
disturbing the intrinsic stability of the Hebbian dynamics. In
this model, the synaptic strength does not oscillate because
homeostatic plasticity is driven by the instantaneous activity
without the delays of slow time averaging. In the previous
models, the observed slowness of homeostatic plasticity had
to arise from slow time averaging rather than slow learning
because slow learning, which means small changes per unit
time, would have made the homeostatic term too weak to affectlearning. That is, under slow learning, the small changes in w
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A

C

Figure 4. A Schematic Description of the
Two-Factor Model(A) The ith synaptic strength is described bywi = Hri , where H is a synapse nonspecific homeostatic variable and ri is a synapse-specificHebbian variable. Hebbian plasticity rapidlychanges ri , while homeostatic plasticity slowly
changes H. The slow change in H can build up
without being overwritten by the fast Hebbian
effect. In models in which Hebbian plasticity
and homeostatic plasticity both directly change
synaptic strengths, homeostatic plasticity cannot be slow without being overwritten. In themodel, both LTP and LTD are mediated by
B
NMDA-dependent calcium influx. In addition,
BDNF/TrkB signaling is assumed to be requiredfor LTP, and homeostatic plasticity is mediatedby TNF-a.
(B) A biophysical interpretation of separable factors whose product is the synaptic strength: four
overall factors (light blue) and corresponding
biophysical elements controlling them (darkblue), including further breakup of some factors
into a product of subfactors. Our model could beinstantiated if H and r each control separate
subsets of these factors or subfactors.
(C) Simulation results of the two-factor model in the monocular cortex during MD and recovery from MD. Colored regions of plots (days 0–5) represent
periods of MD; white region (days 5–12) represents normal vision ðx = 1Þ. The plot illustrates the synaptic strength w (red line) and the maximal and minimal
possible synaptic strengths, Hrmin and Hrmax , which we term the limiting strengths, both represented by dashed lines. Changes in these limiting strengths
represent homeostatic plasticity, i.e., changes in the homeostatic factor H. When the weight w reaches a limiting strength, it indicates the Hebbian factor r
has reached its corresponding limiting value, rmax or rmin . The synaptic strength was rapidly depressed to the minimum value via the Hebbian component,
and then this minimum value was slowly potentiated by the homeostatic component. When the eye was reopened ðx = 1Þ after 5 days of MD, the synaptic
strength rapidly potentiated to the maximum value by LTP, and then this maximum value was slowly depressed by the homeostatic component. x axis: timein days. Parameters used: q = 0:6, y0 = 1, rmax = 1, rmin = 0:6, t r = 0:2 days, and t H = 8:0 days.

induced by homeostasis in any given time interval would havebeen easily reversed by the larger changes induced by thefaster Hebbian plasticity.
This two-factor model is no doubt still a simplification—aneffective model—of the complex dynamics of synaptic learning(e.g., Huganir and Nicoll, 2013), and determining the biophysicalbasis of this model awaits further experimental work. However,
we suggest the following overall framework for interpreting themodel (Figure 4B). The postsynaptic density (PSD) is thought
to have a certain number of ‘‘slots’’ for alpha-amino-3-hydroxy-5-methyl-4-isoxazole propionic acid receptors (AMPARs).
These slots are created by PSD structural proteins (Huganir andNicoll, 2013; Kessels and Malinow, 2009), some of which playroles in homeostatic plasticity (Shin et al., 2012; Steinmetz andTurrigiano, 2010; Sun and Turrigiano, 2011). The synaptic
strength, to the extent that it is determined by AMPARs, could
be understood as the product of four elements: (1) the area of
the PSD (which is increased by Hebbian LTP, which increases
spine size: Matsuzaki et al., 2004; Harvey and Svoboda, 2007;
Zhou et al., 2004; Kopec et al., 2007), (2) the density of AMPARs
bound to slots in the PSD, (3) the efficacy of the AMPARs, and (4)the presynaptic efficacy, which also undergoes activity-dependent modification (Atwood and Karunanithi, 2002; Kaneko
et al., 2010). If free AMPARs in the membrane and slots bindwith first-order kinetics, then the PSD density of bound AMPARs
can be further broken down as the product of (2a) membrane
AMPAR density, known to be increased by TNF-a (Beattie
502 Neuron 84, 497–510, October 22, 2014 ª2014 Elsevier Inc.

et al., 2002; Leonoudakis et al., 2008; Stellwagen and Malenka,
2006), (2b) PSD slot density, and (2c) slot/AMPAR binding affinity. The latter two factors and AMPAR efficacy are likely alteredby modulations (e.g., phosphorylation) and/or changes incomposition of both AMPARs and PSD structural proteins. The
presynaptic efficacy likewise breaks down to a product of subfactors. In sum, if the homeostatic factor H and Hebbian factor
r each correspond to different subsets of these biophysical factors, then the product rH would correspond to the synaptic
strength.
Such a division of mechanisms fits well with the three working
models proposed by Huganir and Nicoll (2013) for how calciumcalmodulin kinase type II (CAMKII) activation causes Hebbian
LTP. In their ‘‘PSD-centric’’ model, CaMKII alters PSD structuralproteins to increase the number of slots at the given synapse(mechanism 2b). In their ‘‘receptor-centric’’ model, CaMKII
phosphorylates AMPAR complexes to increase their binding affinity to the slots (mechanism 2c). In their ‘‘insertion’’ model,
CaMKII drives exocytosis of vesicles containing glutamate receptors, increasing the density of AMPARs in the plasma membrane (mechanism 2a). Under any of these models, Hebbian
plasticity might also control additional mechanisms and homeostasis would control one or more complementary mechanisms.
To study our proposed two-factor model, we continue toconsider the framework of Figure 2A, a single presynaptic input
in monocular cortex. The Hebbian factor is modeled as in theprevious section so that LTP and LTD occur for xy>q and xy<q,NeuronModeling Hebbian and Homeostatic Plasticity

respectively, and both are stabilized by saturation of r at
maximal or minimal values:trdr= ðrmax  rÞ½xy  q +  ðr  rmin Þ½q  xy + :dt(Equation 5)

The homeostatic term grows or decays if the activity is respectively below or above the desired set-point level y0 :dHy=H 1 
tH:(Equation 6)
dt
y0
The proportionality to H on the right side of Equation 6 is notimportant in practice but formally ensures that H remains positive. The speed of Hebbian and homeostatic learning arecontrolled by the time constants t r and t H , respectively. Notethat H depends only on postsynaptic activity y, and not directlyon the synaptic weight, so that it provides a set point for y. In a
multisynapse model, this will mean that H is a postsynapticcell-specific, rather than synapse-specific, factor.
These equations (and also those of a multisynapse model tobe studied below) can be shown to be mathematically equivalent
to a model similar to that of Equation 3 in which Hebbian and homeostatic plasticity both compete to modify a single factor w,
but in which homeostatic plasticity also scales the maximaland minimal allowed weights and, while slow, responds toinstantaneous rather than time-averaged postsynaptic activity(Supplemental Information Section S5). The former point is critical for homeostatic plasticity to multiplicatively scale synaptic
strengths without disturbing intrinsic Hebbian stability andwithout being overwritten by Hebbian plasticity. The latter pointis critical to stability: slow averaging introduces delays that canyield oscillations of synaptic strength.
We will find that this simple model, like the previous model(Figure 2), still has behavior that varies qualitatively with thestrength of MD (the value of x), although in the present casethis variation might be physiologically realistic (see Supplemental Information Section S6). The key difference, however,will be that in the present model, the behavior is always stable,
approaching a stable fixed point (at which each form of plasticity
is separately zero) without oscillations. Further issues with the
behavior can be fixed by using a more complex plasticity model,
as we will see in the next section, but the stability is intrinsic to theframework of separable factors with a stable Hebbian component (Supplemental Information Section S4). In the previous
model, instability was intrinsic to the framework for the reasons
outlined above and could not be fixed by a more complex model.
To simulate this model, we assumed that homeostatic learning
was quite slow relative to Hebbian learning to show that no instability results: tH was 40 times longer than tr (stability for any
value of tH =tr is shown in Supplemental Information Section
S4). Before MD, r was saturated at its maximal value, r = rmax .
We simulated MD ðx = 0:5Þ applied for 5 days (colored region),
followed by recovery with normal vision ðx = 1Þ for the next 7 days
(white region; Figure 4C). Behavior of the model for MD with intermediate values of x are examined in Supplemental Information Section S6. During the 5 days of MD, the synaptic strength
was first decreased through Hebbian LTD, reaching about
70% of its initial value, and then was potentiated by homeostatic

plasticity, matching experimental results well (Kaneko et al.,2008b). Restoration of normal vision at day 5 led to an overshootof synaptic strength: a rapid increase through Hebbian LTP, followed by a slower decrease toward baseline through homeostatic plasticity. This overshoot is a robust prediction of ourmodel given slow homeostatic and fast Hebbian plasticity:homeostatic scaling H had increased to compensate fordecreased activity and Hebbian synaptic depression during
MD; recovery induces a rapid LTP of r, yielding an overshootof w = rH; H then slowly decreases to return the system to the
set-point activity. We present below an experimental test of
this prediction of an initial overshoot of synaptic strength
following restoration of normal vision.
So far, we have considered the MD result only in the monocular cortex and have also neglected the heterogeneity amongsynapses, replacing multiple synaptic strengths by a representative value. In the next section, we consider a more realistic
network model with multiple presynaptic neurons and consider
binocular as well as monocular cortex.
MD Effects in Multisynapse Plasticity Models
in the Binocular Cortex
We now consider multiple presynaptic neurons ðN = 500Þ projecting to a postsynaptic neuron. The ith input contributes anamount to the postsynaptic firing rate equal to the input’s firing
rate xi times its synaptic weight wi , and the postsynaptic firing
Prate is the linear sum of these contributions: y = N
i = 1 wi xi . In
the binocular cortex, the postsynaptic neuron receives
NC = 310 synapses from the contralateral eye and NI = 190 synapses from the ipsilateral eye. This ratio was chosen to reproduce the physiological ocular-dominance index (ODI) undernormal rearing (Kaneko et al., 2008b). The ith synaptic strengthis described by wi = HAi ri , where Ai is the anatomical strengthof axonal arborization (Miller et al., 1989) (see Supplemental InP
formation Section S7.2) and is normalized as i Ai = 1 to keep
postsynaptic activity level roughly independent of the choice ofN; ri is the synapse-specific Hebbian factor, and H is the synapse-nonspecific homeostatic factor characterizing the overallscaling of synaptic strengths onto the given postsynaptic
neuron. The arborization function was introduced to reproduce
localized receptive fields with continuously decreasing magnitude toward their flanks. The presynaptic firing rates are drawn
from a Gaussian distribution with a specific mean and covariance (see Experimental Procedures). Under the normal rearing
condition, the mean firing rates hxi i are identical for all presynaptic neurons ðhxi i = 1Þ. The input covariance specifies that two inputs at a given retinotopic separation are more correlated if they
are from the same eye compared to if they are from opposite
eyes, while the correlation of two inputs with given ocular identities decreases with their retinotopic separation. MD decreases
the input firing rates within the closed eye and the covariance between closed-eye and open-eye inputs by a factor f = 0:5 and decreases the covariance between closed-eye inputs by f 2 .
To show the robustness of our learning rule to noise, we
added a symmetric Gaussian random matrix (see Experimental
Procedures) to the covariance matrix. While this noise heterogeneously perturbs individual synapses, it has only negligible influence on ODP.
Neuron 84, 497–510, October 22, 2014 ª2014 Elsevier Inc. 503

NeuronModeling Hebbian and Homeostatic Plasticity

We will use a more complex learning rule than that used previously to reproduce quantitative aspects that the simpler rule ofEquations 5 and 6 could not account for. We emphasize thatour point in using this more complex rule is not to argue that it
captures the true, complex biological rules (see also discussionof alternatives in Supplemental Information Section S7). Rather,
we only wish to demonstrate that (1) the separation of r and H
variables gives stability despite slow homeostatic plasticity,with each form of plasticity separately zero at the network stablestate and (2) suitable plasticity rules can then replicate the biological behavior without disturbing these features.
The behaviors we wish to capture with the plasticity rules are
as follows (further explained in Supplemental Information Section S7). (1) TNF-a-dependent ODP: during days 3–7 of MD,
there is continuing ODP toward the open eye in wild-type animals, but not in TNF-a knockouts (Kaneko et al., 2008b). Since
homeostatic plasticity alone would equally scale both eyes andthus cause no ODP, this suggests that TNF-a-dependent
homeostatic plasticity induces further Hebbian plasticity after
Hebbian plasticity would otherwise be saturated. (2) Saturationof homeostatic plasticity: under MD in monocular cortex,
homeostasis only returns synaptic strength to near its originallevel, which would not restore original activity levels undercontinuing MD. (3) Delay in homeostatic onset: no potentiationof open-eye responses is seen in binocular cortex until after
day 3 of MD (Kaneko et al., 2008b). Under the simple rule ofEquation 6, homeostatic plasticity, while slow, was active fromthe initiation of MD, yielding significant homeostatic plasticity
by the time of maximal LTD (evident as the rise of rmin H by day
2 in Figure 4C).
To account for TNF-a-dependent ODP, we modified the rulefor changes in the Hebbian factor by adding H dependence tormin :trdri= ðrmax  ri Þ½fi  +  ðri  rmin ðHÞÞ½fi  + :dt(Equation 7)

Here, tr = 0:2 day is a characteristic time scale; fi is the prepost covariance minus a threshold, fi = Cov½xi ; y  q withq = 0:6, and rmax = 1. We set rmin ðHÞ = 0:7 for H%1 and= 0:7H1=2 for H>1, which allows homeostatic plasticity to
induce continued LTD and thus continued ODP after LTD has
initially saturated. If rmin is constant, there would be no ODP
once LTD is saturated, while if rmin  H1 there would be no homeostatic strengthening of depressed synapses in monocularcortex (because continued LTD would cancel homeostatic upscaling). Apart from these constraints, the exact functional
form of rmin ðHÞ does not matter for the results.
To impose saturation of TNF-a-dependent homeostatic plasticity (Beattie et al., 2002), we first note that without saturation,
homeostatic dynamics that reach the set point y0 can be writtent H dH=dt =  H + Htarget with Htarget hHy0 =hyi. Here, hyi is
the instantaneous (i.e., very short timescale) mean activity(used in place of y simply to average out the Gaussian input
noise). To achieve saturation, we can modify the dynamics tot H dH=dt =  H + FðHtarget Þ with some saturating nonlinearity F.
To impose a delay in homeostasis, we assume that, after activity
decreases (e.g., due to MD), increase of the homeostatic factor
H from the baseline level of 1 is delayed until a threshold
504 Neuron 84, 497–510, October 22, 2014 ª2014 Elsevier Inc.

amount (which we take to be 1) of an underlying factor h has
accumulated:
H = Maxðh; 1Þ:(Equation 8)

h is a factor that builds to a saturating value >1 when activity is
below its set-point value. For example, h might be inversely
related to the sensing of glutamate concentrations over timeby glial cells (Stellwagen and Malenka, 2006) and would induce
a proportional increase in TNF-a release once it is above athreshold. H would reflect the resulting TNF-a concentration.
Note that Equation 8 incorporates the experimental result that
TNF-a is only involved in upscaling strengths in response to
too-low activity and not in downscaling strengths in response
to too-high activity (Stellwagen and Malenka, 2006). To ensure
adequate delay in homeostatic onset, h should decay to a baseline level well below 1 when this activity is at or above its setpoint value. We take this baseline level to be zero. Note thatthese assumptions could not solve the problems of oscillations
or instability in conventional models (Equations 1, 2, 3, and 4),
because those models would not work with delayed homeostatic onset. In those models, the dynamics yield a fixed point thatwould be unstable under Hebbian plasticity alone. Therefore,
for the baseline condition to be stable, they require ongoing
fast homeostatic responses to return weights to the fixed point
after small weight fluctuations.
Putting these assumptions together, we use the following rule(along with Equation 8) for changes in the homeostatic factor:
th



dh
=  h + F Htarget :dt(Equation 9)

Here, FðxÞ is a monotonically increasing function that is 0 for
x%1 (so that h returns to its baseline value of 0 once the homeostatic constraint is achieved), jumps from 0 to 1 when x exceeds
1, and then increases roughly linearly with x until it saturates ataround 2 (Equation S16 and Figure S5A). We take the time constant of homeostatic plasticity to be t h = 4 days.
We modeled MD of the contralateral eye initiated at day 0, followed by reopening of that eye at day 7, in binocular cortex. The
model reproduced rapid LTD of the closed-eye synapses andslow homeostatic potentiation of the open-eye synapses(Figure 5). Hebbian LTD reduced closed-eye response by about
30% after 3 days of MD by reduction of the closed eye’s Hebbian
factors ri (Figures 5A, 5B, and 5D). At the same time, the reduction of activity (Figure 5F) caused h to start increasing from thebaseline value of 0. After about 4 days of MD, h reached thethreshold value of 1, and homeostatic potentiation via increaseof H was initiated (Figure 5E). The open-eye response accordingly was unchanged for the first 4 days but was potentiatedby about 30% after 7 days of MD (Figures 5A and 5D). The homeostatic increase of H then started lowering the Hebbian saturation limit rmin and decreased the closed-eye Hebbian factors,
inducing additional ODP during MD days 3–7 (Figure 5C). After
restoration of normal visual input at day 7, the closed-eye synapses showed rapid LTP (Figures 5A, 5B, and 5D) while homeostatic plasticity decreased back to baseline slowly (Figure 5E) so thatsynaptic weights overshot their final level and then gradually
decreased back to it, as we saw in the previous model ofNeuronModeling Hebbian and Homeostatic Plasticity

A

D

B

E

C

Figure 5. Simulation Results in the Binocular Cortex before, during, and after MD

F

MD was started at day 0, and the closed eye wasreopened at day 7.
(A) Synaptic strengths wi are shown as a functionof developmental time. From bottom to top, there
are 500 rows, which we label i = 1; .; 500, with the
ith row showing the time course of the strength ofweight wi . Indices i = 1; .; 310 are from the closedeye (contralateral eye, labeled C) andi = 311; .; 500 are from the open eye (ipsilateraleye, I). See (D) for time course of the summed
weights.
(B) The Hebbian factors ri .
(C) The ocular-dominance index (the value 1means contralaterally dominated and the value 1means ipsilaterally dominated; see Experimental
Procedures).
(D) Traces of the sum of weights from the closedeye (red) and the open eye (blue), normalized tohave an initial value of 1. We refer to this as thenormalized visual response because it represents
the relative visual response through each eye thatwould be assayed by an experimenter at any giventime.
(E) The homeostatic factor H (black) and the underlying glial factor h (green).
(F) The average postsynaptic activity hyi duringongoing plasticity.

monocular cortex (Figure 4C). Simulation of MD in the monocularcortex using the present model (Supplemental Information Section S8) produced a result very similar to that of the single-synapse model presented in the previous section (Figure 4C).
In addition to explaining the MD results in wild-type mice, we
also applied the model to ODP under various forms of inactivation (Figure 6). We modeled blockade of TrkB receptors (Kaneko
et al., 2008a) as a lack of Hebbian potentiation (we removed thefirst LTP term from the right side of Equation 7), whether due to a
direct block of LTP or to block of anatomical addition of synapses. This is based on the fact that BDNF/TrkB signaling isrequired for growth of synapses in neuronal cell culture(Meyer-Franke et al., 1998) and for stabilization of LTP in hippocampus (Tanaka et al., 2008; Kovalchuk et al., 2002; Figurov
et al., 1996; Lai et al., 2012) and visual cortex (Sermasi et al.,2000). This reproduced the experimental results in TrkB-inactivated mice (Kaneko et al., 2008a): MD effects (both the initial
LTD and the subsequent homeostatic potentiation) were normal,
but the recovery from MD after reopening the closed eye wasblocked (Figure 6A).
To model partial NMDA blockade during MD days 3–7, we
assumed no Hebbian plasticity in the corresponding time interval(interval shaded green in Figure 6B). Purely homeostatic potentiation then equally scaled both eyes without changing ODI.
The elimination of closed-eye LTD led to increased closed-eye
potentiation, which in turn led to slightly less overall homeostatic
upscaling than normal, slightly reducing open-eye potentiation.
To model TNF-a blockade, we assumed that H remained constant at 1 without any homeostatic adjustment. Simulations thenshowed little or no change in open-eye synapses over 7 days of

MD (Figure 6C), as observed in TNF-a knockout (KO) mice (Kaneko et al., 2008b). Furthermore, the model predicts that theovershoot of the previously closed-eye synapses during the recovery phase in wild-type mice (c.f., Figure 5) should be absent
in TNF-a KO mice. Reopening of the previously closed eyecauses that eye’s Hebbian factors to be potentiated to the saturating value rmax . In the absence of prior homeostatically induced
increase in H, this simply returns those synapses to their originalvalues without any overshoot.
Experimental Tests II: Hebbian Depression of ClosedEye Synapses during 3–7 Days of MD and HomeostasisDependent Overshoot of Closed-Eye Responses during
Recovery from MD
As we have noted, our proposed solution makes several robust,
qualitative predictions that are independent of the model
complexities that we added to more precisely match details ofdevelopment. These predictions, which have not previously
been tested, are as follows: (1) blockade of Hebbian plasticity
during MD days 3–7 should reveal purely homeostatic plasticity,
involving equal percentage increases in each eye’s strengths
with no change in ODI (as in Figure 6B) and (2) after MD for
7 days, long enough to yield homeostatic potentiation of synapses, restoration of normal vision should cause a TNF-a-dependent overshoot of closed-eye response above the pre-MDbaseline, before it ultimately returns to the baseline (as in Figures
4C and 5D; compare Figure 6C).
We tested these predictions using intrinsic signal imaging fromthe binocular region of mouse V1 during MD and recovery, withresults as predicted by the model (Figure 7). With intraperitoneal
Neuron 84, 497–510, October 22, 2014 ª2014 Elsevier Inc. 505

NeuronModeling Hebbian and Homeostatic Plasticity

A

B

C

injections of the NMDA receptor blocker CPP during MD days 3–
7, equal potentiation of the two eyes occurred without change in
ODI and with increased closed-eye potentiation and reduced
open-eye potentiation (Figure 7A). This is consistent with theidea (Figure 6B) that CPP blocked Hebbian LTD of the closed
eye, revealing purely homeostatic plasticity, which is equal in
the two eyes, while reducing the overall strength of homeostatic
plasticity. Furthermore, normal recovery from MD produced thepredicted overshoot of closed-eye responses followed by slow
recovery to baseline, whereas if homeostatic plasticity had
been blocked during MD by cortical infusion of sTNFR1 (soluble
TNF receptor, which scavenges TNF-a), this overshoot did notoccur (Figure 7B). Note that during normal recovery, both closedand open-eye synapses showed a similar degree of potentiationfrom the pre-MD baseline at 24 hr, when the peak of the closedeye overshoot was measured, and both recovered to near the
pre-MD baseline, without further change in ODI (not shown),
over the next 24 hr. This is consistent with the idea that, at
24 hr, both eyes have saturated Hebbian LTP and are scaledby the same homeostatic factor and that over the next day thishomeostatic factor subsides to baseline (as in Figure 5D).
DISCUSSION
Hebbian and homeostatic plasticity work in tandem refining
neural circuitry, but their interactions have been unclear. In this506 Neuron 84, 497–510, October 22, 2014 ª2014 Elsevier Inc.

Figure 6. Results of MD, Days 0–7, and Recovery, Days 7–10, under Various Inactivation Conditions
Conventions are as in Figure 5.
(A) Simulation of TrkB-inactivated mice, modeled
as an absence of LTP. MD effects (both LTD andhomeostatic plasticity) were normal, but synaptic
strengths did not recover under binocular vision.
(B) Simulation of partial NMDA blockade during
MD days 3–7, modeled as no Hebbian plasticity
during colored interval. This stopped ODP,
increasing closed-eye strengthening (due to blockof LTD) and slightly decreasing open-eye
strengthening.
(C) Simulation of TNF-a inactivation, modeled as H
remaining constant at H = 1. No homeostatic
potentiation of the open-eye synapses occurred.
As a result, overshoot of synaptic strengths did notoccur during the recovery period.

paper, we have studied the dynamic interaction of fast NMDA-receptor-dependent
Hebbian plasticity and slow TNF-a-mediated homeostatic plasticity as found in
ocular dominance plasticity (ODP) (Kaneko et al., 2008b). While our studies
were based on specific, simplified
models, they revealed more general principles that we now summarize.
In most existing models, the two formsof plasticity each modify the same factor,
the synaptic strength. An equilibrium isreached when the changes induced by the two are equal andopposite, meaning that homeostatic plasticity is stabilizing a
set of strengths that would be unstable under Hebbian plasticity
alone. We have shown that these models fail to robustly reproduce ODP because slow homeostatic negative feedback cannot
stabilize fast Hebbian plasticity, which is a positive feedback
process (Cooper et al., 2004; Zenke et al., 2013).
We show instead that ODP can robustly be captured by amodel in which the two forms of plasticity modify separate factors whose product is the synaptic strength. Each form of plasticity then separately reaches its own stable state. This allows
homeostatic plasticity to scale synaptic strengths slowly without
needing to stabilize Hebbian plasticity. This model is consistent
with the apparently multiplicative scaling of weights by homeostatic plasticity (Turrigiano et al., 1998; Stellwagen and Malenka,
2006; Kaneko et al., 2008b) and the independence from TNF-alevels of the percentage change in weights induced by LTP
and LTD (Stellwagen and Malenka, 2006), and it has natural biophysical interpretations (Figure 4). The model can equivalently be
regarded as one in which both processes directly modify thesynaptic strength, but with two new features. (1) Homeostatic
plasticity responds to instantaneous activity, rather than to delayed activity signals (e.g., due to slow temporal averaging),
which lead to oscillations and instability. The slowness of homeostatic plasticity then results from slow learning (small weightchanges per unit time) and/or delayed onset, neither of whichNeuronModeling Hebbian and Homeostatic Plasticity

A

B

Figure 7. Experimental Verification of Model Predictions for MD and Recovery(A) Injection of NMDA receptor antagonist CPP during days 3–7 of contralateral-eye MD reveals pure homeostatic plasticity, as judged by equal multiplicative
upscaling of both eyes’ responses with no further shift in ODI (upper right panel). Upper left panel indicates the experimental schedule. CPP added closed-eye
potentiation and reduced open-eye potentiation (lower panels) as in Figure 6B (compare with Figure 5D). Baseline response levels were indistinguishable in CPP
and control animals (Figure S7).
(B) Upon recovery from MD, there is a TNF-a-dependent overshoot (blocked by sTNFR1, which scavenges TNF-a) of the closed-eye weight from its pre-MDbaseline level before it returns to baseline. MD, monocular deprivation; BV, binocular vision (to induce recovery of closed-eye responses). Bottom: change relative
to day 0 baseline. *p <0:05, **p <0:01, between CPP- and vehicle-injected animals in (A) between sTNFR1- and vehicle-treated animals in (B) (two-way ANOVA
followed by multiple comparisons with Bonferroni correction). Data are represented as mean ± SEM.

could stabilize fast unstable Hebbian plasticity and, thus, neitherof which could be used in existing models. (2) Homeostatic plasticity also scales the minimal and maximal strengths, allowing
multiplicative scaling of all strengths and allowing Hebbian plasticity, which ceases at these limiting strengths, to remain in its
own stable state while homeostatic plasticity proceeds. The
model can account for ODP in response to MD in both monocular and binocular cortices and under various inactivation
conditions.
The model makes several predictions that we verified experimentally. First, it predicts that both forms of plasticity should
be inactive at a steady state of the strengths. We tested this prediction in mouse V1, blocking Hebbian plasticity either during
normal rearing or during days 7–10 of MD during the critical
period for monocular deprivation. In both cases, the blockade
did not alter synaptic weights, indicating that both forms of plasticity were inactive under these conditions. Second, blocking
Hebbian plasticity during MD 3–7 days revealed pure homeostatic potentiation of both eyes without any ODP, including
closed-eye potentiation and reduced open-eye potentiation(Figure 7A), as the model predicts (Figure 6B). Finally, we verified(Figure 7B) the predictions that during binocular recovery after
6 days of MD, the previously closed eye’s synaptic strengths
will overshoot their pre-MD baseline before returning to baseline(Figure 5D) and that this overshoot will not occur if TNF-a-mediated homeostasis was blocked during MD (Figure 6C). In themodel, overshoot occurs because eye reopening causes rapid

potentiation of the Hebbian factor from its MD-depressed state,while the homeostatic factor only slowly decreases from its MDelevated state.
A similar two-factor learning rule was previously independently studied in a model of the development of V1 orientation
selectivity and its contrast invariance (Pool and Mato, 2010).
That paper motivated the rule simply as a means of modelingmultiplicative homeostatic scaling and did not address the
dynamical issues we address here.
Additional Contributions to Cortical Plasticity
Many other mechanisms are likely to contribute to cortical plasticity besides the interaction of fast Hebbian and slow homeostatic plasticity studied here. For example, here we did
not model plasticity in synapses to or from inhibitory neurons(e.g., Gandhi et al., 2008; Kuhlman et al., 2013; Maffei et al.,2006) or in intrinsic intracortical synapses (e.g., Trachtenberg
and Stryker, 2001). We have focused on physiological ODP,
ignoring mechanisms of anatomical plasticity such as branching
and retraction of axons (Antonini and Stryker, 1993; Trachtenberg and Stryker, 2001). A variety of mechanisms not considered here may contribute to closing the critical period for
MD-induced ODP (reviewed in Bavelier et al., 2010; Espinosaand Stryker, 2012). Cho et al. (2009) found in layer 4 that
open-eye potentiation after MD is NMDA dependent. Our results partially reconcile this with the result of Kaneko et al.
(2008b) that such potentiation depends on TNF-a, showing
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NeuronModeling Hebbian and Homeostatic Plasticity

that partial NMDA blockade reduces open-eye potentiation byeliminating closed-eye LTD and thus reducing homeostatic upscaling. However, Hebbian LTP, laminar differences, or otherfacts might also play roles (further discussed in Supplemental
Information Section S7.1). We have modeled only TNFa-dependent homeostatic plasticity, which mediates scaling
up of strengths in response to reduced activity, while ignoring
separate mechanisms that scale synapses down in response
to hyperactivity (Stellwagen and Malenka, 2006) and otherforms of homeostatic plasticity (Turrigiano, 2011).
Functional SignificanceThe slow activation of TNF-a may mechanistically result from thediffusive signaling interactions between neurons and glia (Stellwagen and Malenka, 2006). Is there a functional reason, acomputational benefit, for evolution to select this slow meansof maintaining an activity set point? Consider an extreme
example: if very fast homeostatic plasticity always set the postsynaptic activity to a constant set point, this would prevent coding of differing signals with differing firing rates. Similarly, for
Hebbian plasticity to learn stimulus statistics that change overminutes, hours, or a day, it may be important that homeostatic
plasticity not substantially change activity statistics over thesetimescales.
Maintaining these two processes through separable factors allows dynamic range for coding to be maintained while allowing
Hebbian mechanisms to freely learn synaptic patterns without
interference. It allows stable weight patterns to be maintained
without constitutively active but cancelling Hebbian and homeostatic plasticities, which might be metabolically costly. Finally, as
we have seen, it allows dynamic range to be maintained slowly,
relative to Hebbian learning, without oscillations or instability.
Thus, the dynamical interaction we propose here may describe
a key biological principle underlying memory and learning inneuronal circuits.
EXPERIMENTAL PROCEDURES
Mathematical Model
The single-input models are detailed in Results. For models with N = 500 inputs
we label inputs by i, i = 1; .; 500. The eye that drives a given input is indicatedby ei ˛fC; Ig (C, contralateral eye; I, ipsilateral eye). In simulations of monocularcortex, ei = C for all i. In simulations of binocular cortex, ei = C, i = 1 : 310; ei = I,i = 311 : 500, which reproduces the experimentally observed ODI of about 0.25
under normal vision (Kaneko et al., 2008b). Inputs are uniformly spaced on a
1D retinotopic axis, with zi the retinotopic position of the ith input (0%zi <1ci). In monocular cortex, zi = ði  1Þ=500. In binocular cortex, zi = ði  1Þ=310
for i = 1 : 310 and zi = ði  311Þ=190 for i = 311 : 500.
Input Statistics with Multiple Presynaptic Neurons
The input firing rates are assumed to be Gaussian random variables with mean~ij of the form:
hxi i and covariance Q
"
#
2
 
~ij = qe ;e hxi i xj exp  ðzi  zj Þ :(Equation 10)
Q
i j
2s2q
The correlation magnitudes are qC;C = qI;I = 1, qC;I = qI;C = 0:5; sq = 0:2. The
mean firing rates are hxi i = 1 for normal rearing and hxi i = 0:5 for contralateral
inputs under MD. To demonstrate the robustness of our plasticity rule to noise
and to reproduce some biological heterogeneity, we added noise to the covari~ij + 2ðxi + xj Þ, where fxi g is a set of independent random
ance matrix, i.e., Qij = Q

508 Neuron 84, 497–510, October 22, 2014 ª2014 Elsevier Inc.

Gaussian variables with unit variance. Note that, while this noise perturbs individual synapses, it does not affect ODP.
Visual Response and Ocular Dominance Index
The visual response C or I of the contralateral or ipsilateral eye, respectively,
was defined as the sum of that eye’s synaptic strengths. The ocular dominanceindex was ODI = ðC  IÞ=ðC + IÞ.
Simulation Methods
Numerical simulations in Figures 1, 2, 3, and 4 (except Figure 1C) used Mathematica (using NDSolve, which uses the Livermore Solver for Ordinary Differential Equations Adaptive [LSODA] approach). Other simulations used Matlab(a modified Rosenbrock formula of order 2).
In Vivo Experiments
All procedures were approved by the Institutional Animal Care and Use Committees at University of California San Francisco, in compliance with the NIH
guide for the care and use of laboratory animals. C57BL/6 male mice (Charles
River Laboratories) were used. Day 0 in all experiments is P25. Eyelid suture,visual stimulation, repeated optical imaging of intrinsic signals, and quantification of response amplitude and ocular dominance were performed as
described (Kaneko et al., 2008b). For the partial NMDA blockade experiments, intraperitoneal injections of 3-(2-Carboxypiperazin-4-yl)propyl-1phosphonic acid (CPP, Tocris Bioscience) (15 mg/kg) or the vehicle solution
were made every 24 hr for the duration, as indicated in Figures 3 and 7. To
inhibit TNF-a signaling, recombinant soluble TNF receptor 1 (R&D Systems)
at the rate of 8.57 mg/hr or the vehicle solution was continuously infused
into the cortex using an osmotic minipump (Alzet model 1002) during MD
as indicated in Figure 7. For details of in vivo procedures, see Supplemental
Information Section S9.
SUPPLEMENTAL INFORMATION
Supplemental Information includes Supplemental Information Sections S1–S9
and can be found with this article online at http://dx.doi.org/10.1016/j.neuron.
2014.09.036.
AUTHOR CONTRIBUTIONS
T.T. and K.D.M. developed the theory and models. T.T. conducted numericalsimulations in discussion with K.D.M. M.K. carried out the experiments withthe help of M.P.S. T.T. and K.D.M. wrote the manuscript, which all authors
revised.
ACKNOWLEDGMENTS
The authors would like to thank Yasunori Hayashi and Yukiko Goda for discussions. Supported by RIKEN Brain Science Institute (T.T.), National Institutes of
Health (NIH) T32 MH089920 (M.K.), R01 EY02874 and a Sandler Family grantfrom the UCSF Program in Breakthrough Biomedical Research (M.P.S.), andgrant NEI-EY11001 and the Gatsby Charitable Foundation (K.D.M.).
Accepted: September 24, 2014
Published: October 22, 2014
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Neuron, Volume 84

Supplemental Information
Modeling the Dynamic Interaction of Hebbian and Homeostatic Plasticity
Taro Toyoizumi, Megumi Kaneko, Michael P. Stryker, and Kenneth D. Miller

Supplemental Information
Contents
S1 BCM learning rule: Why the results of Fig. 1C are general

1

S2 Parameter choices for the model that adds stable Hebbian terms and ahomeostatic term; Related to Fig. 2
3
S3 Phase-plane analysis of two models of MD in the monocular cortex; Related to Figs. 2 and 44
S4 Stability of the two-factor, single-synapse model of plasticity; Related toFig. 46S5 Mathematical relationship between the two factor and conventional model;
Related to Figs. 2 and 4-6
9
S5.1 Single-synapse model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9S5.2 Multi-synapse model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
S6 Parameter dependency of the simple two factor model; Related to Fig. 4C 10
S7 Model modifications for the multi-synapse model; Related to Figs. 5 and 6 12
S7.1 Modification of the homeostatic and Hebbian rules . . . . . . . . . . . . . . 12
S7.2 The arbor function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
S8 MD results of the two-factor, multi-input model in the monocular cortex;
Related to Fig. 5
14
S9 Experimental Methods; Related to Figs. 3 and 7

S1

15

BCM learning rule: Why the results of Fig. 1C are
generalIn Fig. 1C, we showed the behavior of the BCM rule plotted against x, the value of the
input activity during MD, and τθ /τw , the speed of homeostatic plasticity relative to Hebbian
plasticity. Here we show that those results are general, independent of assumptions as tothe values of the model parameters.
To see this, we transform the variables of the BCM learning rule (Eqs. 1-2). Let [t] denoteunits of time. If x, y, y0 and θ all have units of firing rate or 1/[t], and w is dimensionless,then for dimensional consistency Eq. 1 should take the formdw
= kxy(y − θ)
(S1)
τw
dt
1

where k is a constant with units of [t3 ]. In Eq. 1, we implicitly worked in units in whichk = 1. Taking the BCM equations to be Eqs. S1 and 2, we define the dimensionless variables
ηx ≡ wx/y0 , ϕ ≡ θ/y0 , αx ≡ kx2 y0 τθ /τw , and s ≡ t/τθ . We have indexed ηx and αx by x
because their values will change when the input activity levels x change, as under MD. In
terms of these dimensionless variables we can rewrite the BCM equations asdηx= αx ηx (ηx − ϕ)dsdϕ
= −ϕ + ηx2 ,
ds

(S2)
(S3)

whose unique fixed point is ηx = ϕ = 1. For a fixed x, the stability of the fixed point depends
only on the dimensionless parameter αx , and the behavior of the model more generally ischaracterized entirely by αx and the initial values of ηx and ϕ at t = 0.
We assume that the input activity is x0 under normal vision and is reduced by a factorof f to xMD = f x0 , 0 < f < 1, at the onset of MD. We take the initial condition under MD
to be the normal-vision fixed point. This initial condition, ηx0 = ϕ = 1, becomes ηxMD = f ,ϕ = 1 under MD. Hence, dynamics of this rule under MD are completely characterized by fand αxMD , or equivalently by f and αxMD /f 2 = αx0 . That is, regardless of the choices of theparameters x0 , y0 , k, τθ , and τw , any models with the same f and αx0 will show exactly thesame behaviors of ηx0 and ϕ under normal rearing and of ηxMD and ϕ under MD.
In Fig. 1C, since we took x0 = 1, the y-axis value x = f x0 was equal to f for that model,
so the y-axis can more generally taken to be f . Since we also took k = 1 and y0 = 1 inthat model, the x-axis value τθ /τw was equal to αx0 for that model, so the x-axis can moregenerally taken to be αx0 . The illustrated behaviors depend only on f and αx0 and so theseplots, with the axes taken to be f and αx0 , are identical for any instantiation of the BCM
model regardless of the specific choices of the underlying parameters that give the particularvalues of αx0 .
In particular, Fig. 1C illustrates w∗ , the value of the synaptic strength w at the firsttrough of synaptic weight under MD relative to its initial value at the onset of MD. From
the definition ηxMD = wf x0 /y0 and the fact that f , x0 , and y0 are not changing as MD
proceeds, we see that the value at the first trough relative to the initial value is the samefor ηxMD as it is for w, so the plotted values of w∗ vs. f and αx0 are identical for any
model. Fig. 1C also illustrates the stability or instability of the MD fixed point, which isdetermined simply by f and αx0 , and measures the degree of stability by −Re λτθ where λis the eigenvalue with largest real part of the BCM equations linearized about the MD fixed
point. This stability measure is based on the following: if λ̃ is the corresponding eigenvalueof the dimensionless BCM equations, the amplitude of the corresponding eigenvalue develops
λ̃ tin time as eλ̃s = e τθ = eλt where the eigenvalue of the original equation is related to thatof the dimensionless equation by λ̃ = λτθ . Thus, the stability index is simply −Re λ̃, which
depends only on f and αx0 .
The BCM rule often includes saturation of Eq. S1 at high y, rather than unlimited,
quadratic-in-y growth of dw. Therefore it is important to note that our conclusions would
dt
not be affected by incorporating such saturation. First, the stability of the fixed point,2y = wx = θ = y0 , is determined by the local properties of Eq. S1 in the vicinity of the
fixed point, whereas – to allow sufficient LTP – saturation must occur at values of y well
above the fixed-point value. Thus saturation will not affect stability. Next, the value w∗ ofthe synaptic strength at the first trough is invariant to the y > θ part of Eq. S1 because,starting from the baseline fixed point, the synapse never undergoes LTP until it reaches w∗ .
Taking these points together, the conclusion of Fig. 1C, that it is impossible to make theadjustment of θ in the BCM rule slow enough to allow realistic closed-eye depression yet fastenough to avoid oscillations or instability, would not be altered by incorporating saturationof Eq. S1 at high y.

S2

Parameter choices for the model that adds stable
Hebbian terms and a homeostatic term; Related toFig. 2

We choose parameters of Eqs. 3–4 to reproduce the results of MD in the monocular cortex(Kaneko et al. 2008b). We choose input activity to be x = 1 before MD (and x < 1during MD), and set wmax = 1. These choices are arbitrary, setting the scale for the otherparameters. The minimum synaptic strength is set to wmin = 0.6 because LTD experiments
typically depress synapses by ∼30% and, more generally, a few days of MD does not eliminate
all synapses. The timescale for plasticity, τw = 0.3 day, is chosen so that the synaptic
strength reduces to about 70% of the pre-MD value within 3 days of MD in the absenceof homeostatic modulation (that is, if ȳ is held constant). The timescale for averaging thepostsynaptic activity, τȳ = 3 days, is chosen so that homeostatic plasticity significantly
potentiates the synapse after approximately 3 days of MD. Parameters y0 = 0.8 and θ = 0.6
are taken from an appropriate range, within which the results do not significantly change. If
y0  1, then homeostatic plasticity cannot be strong enough to provide sufficient homeostatic
potentiation under MD and, at the same time, weak enough to not strongly suppress synaptic
strength during normal rearing, which causes synapses to undergo LTD to saturation in
the normal condition and thus eliminates MD-induced LTD. If y0 > wmax , then Hebbianand homeostatic plasticity terms are separately zero during normal rearing, which causes
a problem when binocular cortex is considered (see discussion below of “condition 1”). Ifθ is too small, no Hebbian depression happens under a reasonable MD condition, and if θ
is too large, the synapse is strongly depressed even under the normal condition. Finally,
the magnitude of homeostatic plasticity γ = 0.23 is chosen so that, after MD and LTD,
homeostatic plasticity returns visual response to about 90% of the pre-MD value as in theexperimental observations. Larger or smaller γ lead to larger or smaller final weight values,
respectively.
While the above discussion suggests that alternative parameters cannot solve the model’s
problems, we cannot fully explore the parameter space of this model. Nonetheless, we canargue that the conclusions we have reached are general and parameter-independent. These
conclusions are: (1) for some ranges of input the Hebbian and system fixed points will not3

coincide, meaning that the two forms of plasticity are constitutively active but opposed; (2)where this is true, NMDA blockade should reveal strong weight changes due to homeostatic
plasticity; and (3) when the two fixed points are sufficiently far apart, oscillations and/or
instability should be seen in the vicinity of the system fixed point. Points (2) and (3) follow
from point (1).
The generality of point (1) can be seen from computing the range of input x that allows
a fixed point without constitutive plasticity; for the remaining x, any fixed point will haveconstitutive plasticity. To compute the fixed points, we use y = wx and the fact that at
the fixed point y = ȳ, so that, given w at the fixed point, ȳ = wx. Fixed points without
constitutive plasticity will have w = y0 /x (the condition for no homeostatic plasticity). The
condition for no Hebbian plasticity is then either
1. wx2 > θ and w ≥ wmax , i.e. y0 x > θ and y0 ≥ wmax x, or θ/y0 < x ≤ y0 /wmax .
2. wx2 < θ and w ≤ wmin , i.e. y0 x < θ and y0 ≤ wmin x, or y0 /wmin ≤ x < θ/y0 .
3. wx2 = θ, or y0 x = θ.
For values of x outside of these ranges – that is, for x > Max(θ/y0 , y0 /wmin ), x <Min(y0 /wmax , θ/y0 ), or y0 /wmax < x < y0 /wmin (the latter condition translates to wmin <w < wmax , i.e. it is true whenever the fixed-point w is strictly inside the Hebbian bounds) –any fixed point must have constitutive, opposed Hebbian and homeostatic plasticity, i.e. thesystem and Hebbian-only fixed points will not coincide. For example, for the parameters weused in the main text, fixed points will have constitutive plasticity for all values of x except
0.75 ≤ x ≤ 0.8.
Furthermore, both Hebbian and homeostatic plasticity must be constitutively active atleast under the normal rearing condition (x = 1) to reproduce ODP. If x = 1 satisfies
condition 1 or 2, Hebbian plasticity is not active in the normal condition. The summed
synaptic strength is determined by the homeostatic constraint y = y0 . Then, if we consider
inputs from two eyes in binocular cortex, any relative strengths of the two eyes – that
is, any ODI – with the given summed synaptic strength will be a fixed point so long as
Hebbian plasticity remains saturated for both eyes. But this causes a problem: so longas Hebbian plasticity remains inactive. perturbations of the ODI, e.g. due to brief MD,
will never recover but will simply persist, which is against experimental observations. In
addition, if x = 1 satisfies condition 2, there is an additional problem. In this case, Hebbian
depression is already saturated during normal rearing. Hence, MD cannot induce further
Hebbian depression.

S3

Phase-plane analysis of two models of MD in the
monocular cortex; Related to Figs. 2 and 4

Here we use phase-plane analysis to gain an intuition for the reasons that the single-factor
model of Eqs. 3-4, which involved Hebbian and homeostatic terms competing to control
4changes in w, tended to show weight oscillations. We similarly show in phase-plane analysis
why the two-factor model of Eqs. 5-6 does not oscillate, an issue we examine more rigorously
in the next section of the Supplemental Information, in which we analyze the stability andorigin of oscillations of the model of Eqs. 3-4.
We begin with the single-factor model. The differences in stability of Figs. 2A vs. 2B canbe intuitively understood from the phase plane plots of Fig. S1A,B. The arrowed lines show
the direction of change in the variables (w, ȳ) (their time derivatives) for each given valueof (w, ȳ), while line colors indicate the strength of the derivatives, from strongest (blue) to
weakest (red). The fixed point of the dynamics, where both derivatives are zero, is given by
the intersection of the two black lines: the solid black line represents the locus of points for= 0 (the w nullcline), while the dashed black line represents the locus for which
which dwdtdȳ
= 0 (the ȳ nullcline).
dt
The generalized stability of the fixed point, by which we mean whether the fixed point
is approached without significant oscillations, is largely determined by the stability of theflow in the w direction (with ȳ fixed) at the fixed point: for small perturbations in the w
direction from the fixed point, is the w-component of the flow in the opposite direction of theperturbation, back toward the fixed point (stable flow in the w direction) or is it in the same
direction of the perturbation, away from the fixed point (unstable flow in the w direction)?
x0.5

x0.73

B

1.2

1.2

1.0

1.0

0.8

0.8

0.6

0.6

y

y

A

0.4

0.4

0.2

0.2

0.0

0.0
0.0

0.5

1.0

w

1.5

2.0

2.5

0.0

0.5

1.0

w

1.5

2.0

2.5

Figure S1: Phase-plane plots of Eqs. 3-4 for x = 0.5 (A) and x = 0.73 (B). The two axes specify
values of averaged postsynaptic activity ȳ and synaptic weight w, and the lines with arrows show
the direction of change in these two variables from each given point in the plane. The colors of thelines indicate the strength of the derivatives at each point, proceeding red to yellow to green todȳ ˆblue from weakest to strongest. (Strength is assayed as the length of the gradient vector dw
dt ŵ + dt ȳˆ are unit-length vectors pointing along the w-axis and ȳ-axis respectively.) The solid
where ŵ and ȳ
black line shows the w-nullcline (the locus of points along which dw
dt = 0) and the dashed line showsdȳ
the ȳ-nullcline (the locus of points along which dt = 0).

5x  0.5

x  0.9x  0.75
4

3

3

3

H

H

4

H

4

2

2

2

1

1

1

0.4

0.6

0.8
Ρ

1.0

1.2

0.4

0.6

0.8
Ρ

1.0

1.2

0.4

0.6

0.8
Ρ

1.0

1.2

Figure S2: Phase plane analysis of the plasticity model of Eqs. 5-6.
Conventions as in Fig. S1. White regions correspond to regions of very small derivatives.
If the flow is unstable in the w direction, then, because homeostatic plasticity (changes in ȳ)are slow relative to Hebbian changes, oscillations (or instability) will result: the perturbation
will lead to large changes in w before slow changes in y lead to a reversal in the direction ofchanges in w (a crossing of the w nullcline), which in turn leads to a large change in w in the
opposite direction as y slowly changes in the opposite direction, etc. Because the derivativedw
is positive below the w nullcline and negative above the w nullcline, the flow in the w
dt
direction is unstable if the fixed point is on a positive-sloping portion of the w nullcline, as
in Fig. S1B, while it is stable if the fixed point is on a negative-sloping portion of the w
nullcline, as in Fig. S1A. The w nullcline transitions from negative to positive slope withincreasing w. As x is increased there is a range of x for which the fixed point is found on
the positive-sloping region of the w nullcline, leading to oscillations for slow homeostatic
plasticity.
In contrast, phase-plane analysis of Eqs. 5-6 (Fig. S2) shows that the fixed point always
occurs at one of the two limiting values of ρ, as is clear directly from Eq. 5, and that theflow in the ρ direction is always stable at these limiting values (as is also clear directly from
Eq. 5).

S4

Stability of the two-factor, single-synapse model ofplasticity; Related to Fig. 4Here we analytically demonstrate the stability of the fixed point and lack of oscillations inapproaching the fixed point of the two-factor, single-synapse plasticity model of Eqs. 5-6.
Before more generally analyzing stability, we consider the dynamics in the limit of infinitely
slow homeostasis (τH /τρ → ∞). For simplicity, we set τρ = 1 (which just sets the units of6

time), so our equations aredρ= (ρmax − ρ)[xy − θ]+ − (ρ − ρmin )[θ − xy]+
dt
dH
1
=
H(1 − y/y0 )
dt
τH

(S4)
(S5)

with w = ρH and y = wx. Analyzing these equations for τH → ∞ gives an intuitive pictureof why, even in the limit of extremely slow homeostasis, there are no oscillations and at mostone overshoot of the synaptic weight, as shown in Fig. 5B.
Because of the assumption τH → ∞, we can assume that ρ always moves rapidly (relative
to the slow timescale of homeostasis) to a stable fixed point of Eq. S4 for a given value ofH, before H changes appreciably. For each H, these stable fixed points of ρ are given byθρmax when H > ρmaxx2
ρ(H) =(S6)
θρmin when H < ρmin x2(Fig. S3A). Therefore, the synaptic strength converges rapidly to ρ(H)H (Fig. S3B). Noteθ< H < ρminθ x2 , so in thisthat ρ = ρmax and ρ = ρmin are both stable fixed points when ρmax
x2
case the choice between these two values of ρ depends on history. Nonetheless, there exists
a unique value of H that achieves a particular synaptic strength w = ρ(H)H (Fig. S3B).
The slow change in H can then be described as occuring with ρ always at its stable fixed
point, so that y = wx = ρ(H)Hx. The dynamics of Eq. S5 can then be written
1dH=
H[1 − ρ(H)Hx/y0 ]
dt
τH

(S7)

Thus, H evolves so that ρ(H)H approaches y0 /x (H = 0 is an unstable fixed point). Because there is only one value of H that achieves this, the final fixed point is unique andstable. Therefore, even when homeostatic plasticity becomes infinitely slow, there should
be no oscillations of weights, but only at most one rapid transition between ρmax and ρmin ,depending on the initial value of ρ, until synapses converge (described further in legend of
Fig. S3).
Now, we analyze the stability of the fixed point of Eq. S4 for finite τH . Changes in τH do
not change the locations of the fixed points, but could change their stability. However, since
we saw that the fixed points are stable even for τH → ∞, and H provides negative feedback,it would be very surprising if reductions in τ rendered the fixed point unstable, and we willfind that this does not happen.
The fixed point depends on the sign of φ0 ≡ xy0 − θ, and is given by: (i) (ρ, H) =y0
y0
) if φ0 > 0 and (ii) (ρ, H) = (ρmin , ρmin
) if φ0 < 0. We will prove that the fixed(ρmax , ρmaxx
x
point is stable and without oscillations in either case. Following
 standard procedures forδρ
calculating stability, we consider a small perturbationfrom the fixed point, andδHconsider the linearized dynamics about the fixed point:




d
δρ
δρ
=J
(S8)
δH
dt δH
7

AΡ

B ΡH

1.0
0.8
0.6
0.4
0.2

0.0

0.5

1.0

1.5

1.4
1.2
1.0
0.8
0.6
0.4
0.2

H
2.0 0.0

0.5

1.0

1.5

H
2.0

Figure S3: The two-factor model shows at most one overshoot of the synaptic weight before
it converges. (A) When the homeostatic plasticity is much slower than Hebbian plasticity, theHebbian component ρ converges rapidly for each H to its maximum value ρmax = 1 (red line) orθθminimum value ρmin = 0.6 (blue line). When ρmax
< H < ρminthose two values are bistable andx2x2
ρ depends on its history. The dashed purple line shows the separatrix (unstable fixed points). (B)The combination of ρ and H that achieves the homeostatic constraint wx = y0 , i.e. ρH = y0 /x, is
unique. Under this dynamics, the synaptic weight ρH moves monotonically along whichever stablebranch it starts on toward the stable fixed-point value y0 /x; if this value can only be reached on
the other branch, the weight jumps to the other branch (at constant H) when it reaches the end of
the first branch, and then evolves monotonically along the second branch to the fixed point. The
combination of ρ and H changes accordingly along the stable branches in (A). Thus, there is amonotonic evolution to the fixed point except for possibly one jump between the two branches.

where J is the Jacobian matrix of Eqs. S4-S5 !
(if rρ and rH are the right-hand sides of Eqs. S4and S5, respectively, then J =

∂rρ
∂ρ
∂rH
∂ρ

∂rρ
∂H
∂rH
∂H

), evaluated at the fixed point. The stability

is then determined by the eigenvalues of J. We will find that the eigenvalues are all real,indicating no oscillatory behavior around the fixed point, and negative, indicating stability.
y0(i) ρ = ρmax and H = ρmax
if φ0 > 0: In this case the Jacobian matrix is
x


J=

−φ0
0
− ρ2γy0 x −γ


,

(S9)

max

and the two eigenvalues are −φ0 and −γ, which are both negative and real.
y0(ii) ρ = ρmin and H = ρmin
if φ0 < 0: In this case the Jacobian matrix is
x


J=

φ0
0
− ργy
2 x
min

8

0
−γ


,

(S10)

and the two eigenvalues are φ0 and −γ, which are both negative and real. Hence, the fixed
point is stable and shows no oscillatory behavior around the fixed point in either case.

S5

Mathematical relationship between the two factor
and conventional model; Related to Figs. 2 and 4-6

In order to better understand this rule and to compare it to the previous rule of Eq. 3, we= H dρ
+ dH
ρ.
calculate dw
dt
dt
dt

S5.1

Single-synapse model

We use Eqs. 5-6 to obtain the following mathematically equivalent formulation of the model:


τρ
dw
y
(S11)
τρ
= (Hρmax − w)[xy − θ]+ − (w − Hρmin )[θ − xy]+ +
w 1−
dt
τH
y0


dH
y
τH
= H 1−(S12)dt
y0
Equation S11 has a form similar to Eq. 3 and is expressed as a sum of three terms, whichcorrespond to the LTP, LTD, and homeostatic terms of Eq. 3 (with the homeostatic learning
speed γ now explicitly represented as ττHρ ). We will refer to the first two terms as the Hebbian
terms and the third as the homeostatic term. Note that the factor H that scales the maximaland minimal weights is a postsynaptic-cell-specific factor that is shared across synapses if
multiple synapses are considered, while the constants ρmax and ρmin might vary from synapse
to synapse. The differences between Eq. S11 and Eq. 3 are:
1. The upper and lower limits of the synaptic strength, Hρmax and Hρmin , are not fixed
but are modified by homeostatic plasticity (Eq. S12).
2. The homeostatic term is driven by the instantaneous postsynaptic firing rate as opposedto the running-average of the postsynaptic firing rate in Eq. 3. Thus, there is no delayin the homeostatic term; as we have seen, delays can cause oscillations of synaptic
strength.
3. A minor difference is that Eq. S11 uses linear functions (Hρmax − w) and (w − Hρmin ),whereas Eq. 3 uses threshold-linear functions [wmax −w]+ and [w−wmin ]+ . Thresholding
would not alter the dynamics in Eq. S11 because the synaptic weight under Eqs. S11S12 is always restricted to Hρmin ≤ w ≤ Hρmax . This may not be obvious underEqs. S11-S12, but is obvious from the equivalent Eqs. 5-6 and the relationship w = ρH.
Thus, our proposed model can equivalently be understood as one in which Hebbian andslow-but-instantaneous homeostatic plasticity both compete to modify w, but in addition
homeostatic plasticity continually modifies the minimal and maximal weight values. With9

homeostatic modification of limiting weight values, the Hebbian terms in Eq. S11 approacha stable state of zero plasticity with weights saturated at their limiting values, and this stateremain undisturbed as homeostatic plasticity proceeds until it too reaches zero. For thisreason, Eqs. S11-S12 generically reach a steady state in which there is no constitutive plasticity (i.e., Hebbian and homeostatic plasticity are each separately zero), as is perhaps more
obvious from the equivalent Eqs. 5-6, whereas the seemingly very similar Eq. 3 genericallyreaches a steady state in which Hebbian and homeostatic plasticity are each constitutively
active but opposed (i.e. their sum is zero, but the terms individually are not zero).

S5.2

Multi-synapse model. We define the step function Θ(x) by Θ(x) = 0, x ≤ 0;
We now use Eqs. 7-9 to compute dw
dt
Θ(x) = 1, x > 0. Note that parameters ρmax and ρmin may vary across synapses though we
do not make this explicit here. Then our equations become1√dwi= (Hρmax − wi )[φi ]+ − (wi − Hρmin )[−φi ]+
dt



Hy0
τρ
1
+Θ (H − 1) wi −1 + FτhHhyi
H = Max(h, 1)


Hy0
dh
= −h + F
τh
dt
hyi

τρ

(S13)
(S14)
(S15)

Equation S13, like Eq. S11, is the sum of an LTP, an LTD, and a homeostatic term.
Equation S13√ is more complex than Eq. S11 for two reasons: the H-dependence of ρmin manifests as a H rather than H scaling of the minimum weight (with the understanding thatH ≥ 1, Eq. S14); and the dependence of H on h manifests as the complex homeostatic plasticity term. Despite this complexity, Eq. S13 has the same three properties as we describedabove for Eq. S11. Thus, the multi-synapse model can again be equivalently understoodas one in which Hebbian and slow-but-instantaneous homeostatic plasticity both compete
to modify w, but in addition homeostatic plasticity continually modifies the minimal andmaximal weight values. This again leads to a steady state in which Hebbian and homeostatic plasticity are each separately zero (no constitutive plasticity), for the same reasons
just outlined for the single-synapse model.

S6

Parameter dependency of the simple two factor model;
Related to Fig. 4C

In Fig. 5C, we considered relatively strong deprivation, x = 0.5, applied for 5 days. Here, we
considered a weaker deprivation, x = 0.75, maintained for a long period (Fig. S4A). This led
We neglect the fact that the homeostatic term in Eq. S13 is also nonzero if H = h = 1 and dh
dt > 0. Thiscondition holds only for an infinitesimal amount of time and hence contributes negligibly to Eq. S13.
1

10

A
1.8
1.6
1.4
1.2
1.0
0.8

B

x0.75
ΡmaxH

5

w

10

15

Ρmin H
20

25

t

x0.9

1.8
1.6
1.4
1.2
1.0
0.8

w
5

Ρmin H
10

ΡmaxH

15

20

25

t

Figure S4: Simulation results of the two-factor model in the monocular cortex during mild MD.
The conventions are as in Fig. 4C but input is stronger during MD. (A) For intermediate input
activity during MD, the weight was depressed by Hebbian plasticity to the minimum strength,
but in this case the slow homeostatic potentiation during continued MD ultimately brought thepre- and postsynaptic correlation xy = wx2 above the LTP threshold θ. This induced LTP to the
maximum synaptic value, followed by slow homeostatic decrease of this maximum value. (B) For
large input activity during MD, there was no LTD. The synaptic strength remained at the maximal
value and homeostatic plasticity slowly increased the maximal value to compensate for the slight
loss of input.

to a similar trajectory of initial LTD followed by slow homeostatic scaling up of the weights.
However, when we allowed the deprivation to continue for an extended period, the scalingbrought the synaptic strength across the LTP/LTD threshold. At that point, just as in thecase of recovery in Fig. 5C, rapid LTP increased ρ until it reached its maximal value, after
which homeostatic plasticity slowly reduced the synaptic strength.
Next, we considered still weaker deprivation, x = 0.9 (Fig. S4B). In this case, the decreasein the input firing rate was small enough that the LTP/LTD threshold was not crossed, so
there was no initial LTD and ρ remained at ρmax . Homeostatic plasticity slowly compensatedfor the reduction of the input by potentiating the synapse.
The strongest deprivation robustly yields the behavior we are seeking to explain: fast LTD
followed by slow homeostatic compensation (Fig. 5C). For weaker deprivation, alternative
behaviors are seen. The overshoot of synaptic strength after long periods of intermediatestrength deprivation (Fig. S4A) depends on homeostasis strengthening synapses sufficiently
to cross the Hebbian LTD/LTP threshold, and this in turn depends on model details. It
would not occur, for example, if an appropriate maximum value were imposed on H. Homeostatic strengthening in response to mild deprivation (Fig. S4B) might accurately describe
the homeostatic strengthening of excitatory synapses in response to visual deprivation that
occurs in layer 4 of rat visual cortex before the opening of the critical period for MD-induced
ODP (Maffei et al. 2004). Before the opening of the critical period, visually-driven activity
is likely to be weak relative to spontaneous activity (Toyoizumi et al. 2013), so that the11effects of deprivation on cortical activity may be relatively weak.

S7

Model modifications for the multi-synapse model;
Related to Figs. 5 and 6

S7.1

Modification of the homeostatic and Hebbian rules

For the multi-synapse plasticity models, we used a more complicated homeostatic rule than
the simpler rule of Eq. 6, and incorporated H-dependence of ρmin into the Hebbian rule, totry to solve several problems of the simpler rule:• There is continued ODP towards the open eye during 3-7 days of MD that is TNF-αdependent (it does not occur in TNF-α KO animals nor in animals in which TNF-α
receptors are blocked) (Kaneko et al. 2008b). To model this effect, we incorporated a
reduction of ρmin with increases of the homeostatic variable H, so that increases of H
allowed further LTD. The dependence on H should be weaker than a 1/H dependence
because, otherwise, homeostatic scaling could not potentiate synapses in a depressedstate.
Other factors could also contribute to TNF-α-dependent ODP, but we restricted toa single factor for simplicity. These other factors include: (1) a similar increase inρmax with increasing H could allow TNF-α-dependent LTP of the open eye; (2) there
could be a minimal, threshold level of pre-post covariance for inducing LTD, so thatLTD stops at an intermediate weight level initially as LTD lowers the covariance, but
resumes as TNF-α-dependent homeostatic plasticity increases the covariance; (3) Better representation of the stochasticity of neural activity and plasticity should allow a
broader distribution of synaptic strengths in the normal state. Then, after 3 days of
MD, a larger percentage of open-eye synapses would be in the LTD state than in ourmodel despite an overall shift toward LTP, and similarly a larger percentage of closedeye synapses would be in the LTP state despite a shift toward LTD. Homeostatic
strengthening of response might lead Hebbian plasticity to squeeze these distributions,adding to TNF-α-dependent LTP of the open eye and LTD of the closed eye. (Notethat factors like stochastic broadening of the synaptic distribution and/or a minimal
covariance threshold for LTD are also necessary to explain why LTP and LTD protocols work, e.g. why in the normal state all synapses are not already saturated at thepotentiated state before the protocol is applied, assuming that individual synapses are
generally saturated (O’Connor et al. 2005, Petersen et al. 1998). Note also that ourmodel fails to quantitatively account for the strength of decrease in open-eye potentiation induced by NMDA blockade. This could be fixed by factors (1) or (3) above or
by further modifications of the homeostatic rule, but we neglected this for simplicity.)• In the simple rule, homeostatic plasticity, while slow, was active from the initiationof MD. As a result, significant homeostatic plasticity had already occurred at the12time of maximal LTD (evident as the rise of ρmin H by day 2 in Fig. 4C). While thiscaused no problem for the single-synapse model of monocular cortex studied previously, in a binocular cortex model this would mean that some potentiation of open-eye
responses would already be visible at that time, unlike experiments (Kaneko et al.
2008b). This might be explained by LTD of the open eye that offsets homeostatic
potentiation, but this is unlikely because no depression of open eye responses is observed in TNF-α knockout mice (Kaneko et al. 2008b). To explain this, we assumed
a more complex homeostatic rule that delays the onset of homeostatic plasticity untila threshold amount of an underlying factor is accumulated. Other factors we did notconsider, such as necessity of a threshold amount of activity deviation from the set
point before homeostatic plasticity begins, could also or alternatively be involved.
• Stellwagen and Malenka (2006) showed that TNF-α signaling is required for scaling
up of synaptic weights in response to activity blockade, but not for scaling down ofsynaptic weights in response to excess activity induced by inhibitory blockade. To
capture this, we modified the rule so that H can increase from a baseline level of 1
(representing scaling up from baseline) but cannot decrease below 1 (meaning thatscaling down from baseline is not achieved by modification of H).
• Homeostatic plasticity does not appear to perfectly compensate for activity changes.
The homeostatic increase seems to be a scaling up of about 30% from the level at day 3,
both in monocular and binocular cortex, which in monocular cortex restores synaptic
weights to near the pre-MD level (Kaneko et al. 2008b); this would not be sufficient torestore activity to its pre-MD level under continuing MD. To match this, we made thegrowth of H a saturating function of the ratio of actual to set-point activity, preventing
full compensation for the activity decrease. This could also be explained by a lack ofactivation of homeostatic plasticity by activity deviations from the set point of lessthan a threshold amount.
To capture these ideas, we assumed the more complex homeostatic rule of Eqs. 8–9. In
Eq. 9, F (x) (Eq. S16) is a monotonically increasing function that is 0 for x ≤ 1, jumps
from 0 to 1 when x exceeds 1 and then increases roughly linearly with x until it saturates
at around 2. The mathematical form used was
F (x) = [1 + tanh(x − 1)]Θ(x − 1.01)(S16)

where Θ(z) is a step function, Θ(z) = 1 for z ≥ 0, = 0 otherwise. The shape of this function
is illustrated in Fig. S5A. The argument of F is chosen to be Hy0 / hyi so that, if F did notsaturate (e.g., if F (x) = x), the stable fixed point of homeostatic plasticity, starting froma condition with y0 / hyi > 1, would be hyi = y0 . That is, the homeostatic rule restores the
set-point activity for smaller decreases of activity, but saturation prevents full restoration
for larger decreases of activity.

13

6

2

4

1

A

F

1.5

2

0.5
0
0

−3

x 10

1

x

2

3

0
|

C | I |
index

Figure S5: (A) The nonlinear function F used in Eq. 8 to update h. This function is monotonicallyincreasing, jumping from 0 to 1 at x ≈ 1 and then smoothly saturating at around 2. (B) The arbor
function A used in the multi-synapse plasticity models for binocular cortex in the expression forsynaptic strength wi = HAi ρi . The value of Ai (Eq. S17) is shown vs. index value i, where indicesi = 1, . . . , 310 are from the closed eye (contralateral eye, labeled ‘C’) and i = 311, . . . , 500 are fromthe open eye (ipsilateral eye, ‘I’).

S7.2

The arbor function

The axonal arborization function Ai describes the anatomical density of axon branches tothe postsynaptic neuron from a given presynaptic neuron i. The arborization strengths are
given byAi ∝1,
1 + exp(3[(zi − 0.5)2 /(0.2)2 − 1])

(S17)

P
where the overall strengths are normalized according to N
i=1 Ai = 1. The shape of thisfunction is illustrated in Fig. S5B for the binocular case and in Fig. S6C for the monocular
case.

S8

MD results of the two-factor, multi-input model inthe monocular cortex; Related to Fig. 5

Here we examine the results of MD in monocular cortex for the multi-input model of Eqs. 79 (Fig. S6). MD to the contralateral eye was started at day 0 and the eye was re-opened
at day 7. Initial LTD induced rapid decrease of the Hebbian factors, ρ, and homeostatic
potentiation was induced by the delayed increase of the homeostatic factor H. Homeostatic
14

Figure S6: Simulation result in the monocular cortex before, during, and after MD using themodel of Eqs. 7-9. We considered 500 inputs from the contralateral eye, which was closed during
MD. MD was started at day 0 and the closed eye was re-opened at day 7. (A) Synaptic strengths
wi are shown as a function of developmental time. (B) The Hebbian factors ρi . (C) The arbor
function Ai . (D) The normalized visual response (defined as in Fig. 5). (E) The homeostatic factor
H (black) and h (green). (F) The average postsynaptic firing rate hyi.

plasticity was absent for the first few days of MD because it took time for h to reach thethreshold value of 1. After the re-opening of the contralateral eye, the model showed anovershoot of synaptic strengths (Fig. S6A,D), as in the single-synapse model of monocularcortex and in the multi-input model of binocular cortex. As in those cases, this behavior
occurred because, after restoring normal visual input, the Hebbian component ρ potentiatedrapidly and the homeostatic factor H more slowly decayed back to its baseline.

S9

Experimental Methods; Related to Figs. 3 and 7

In vivo experiments: C57BL/6 wild type breeders were purchased from Charles River
Laboratories (Hollister, CA) and bred as needed. Animals were maintained in the animal
15

facility at University of California San Francisco and used in accordance with protocols
approved by the UCSF institutional Animal Care and Use Committee. Total of 47 C57BL6
male mice were included for the analysis.
Monocular deprivation (MD) was performed by suturing shut the right eyelid (contralateral to the imaged hemisphere) at P25 as described (Kaneko et al. 2008b). The initial dayof imaging was described as day 0 and other days are numbered sequentially from there.
To produce recovery from effects of MD, vision was restored to the closed eye by simply
removing the suture. All mice were kept under standard housing conditions with free accessto food and water between recordings.
NMDA blockade: Two days prior to recording baseline responses, a custom stainless steel
plate for head fixation was attached to the skull with dental acrylic under isoflurane anesthesia as described (Kaneko and Styker, 2014). Animals were given a subcutaneous injection of
carprofen (5 mg/kg) as a post-operative analgesic. The competitive NMDA receptor antagonist (R,S)-3-(2-carboxypiperazin-4-yl)propyl-1-phosphonic acid (CPP) (Tocris Bioscience)
was dissolved in saline at a concentration of 1 mg/ml and the drug solution was injected
intraperitoneally at a dose of 15 mg/kg roughly every 24 h, as described (Sato and Stryker
2008). The first injection in each experiment was performed right after the imaging of thecorresponding day. On subsequent days, injections were made around 9AM, and imaging, if
done, occurred 1-6 hours after injection (1 hr for the first mouse imaged, 6 hr for the lastmouse imaged on given day). As expected, baseline response levels were indistinguishable in
CPP and control animals (Fig. S7).
TNF-α blockade: Intracortical infusion of soluble TNF receptor-1 (sTNFR1, R&D Systems, Inc. Minneapolis, MN) was performed as described (Kaneko et al. 2008b, except that
implantation was done on day 0). Briefly, on day 0, immediately after baseline imaging, we
implanted a cortical cannula that was connected with an Alzet osmotic minipump (model
1002) filled either with vehicle solution (PBS containing 0.1% bovine serum albumin as acarrier) or 35 mg/ml of sTNFR1. This was followed immediately by eyelid suture. Infusion
continued during days 0-5 of 6-day MD. On day 5, the cannula and minipump were removed
and response magnitudes (MD-5d) were then recorded. On day 6, vision was restored tothe closed eye by simply removing the suture to produce recovery from effects of MD, andresponses were recorded 24 and 48 hours after reopening the closed eye.
Repeated optical imaging of intrinsic signals and quantification of ocular dominance were
performed as described (Kaneko et al. 2008a). Briefly, during recording mice were anesthetized with 0.7% isoflurane in oxygen applied via a home-made nose mask, supplemented
with a single intramuscular injection of 20-25 g chlorprothixene. Intrinsic signal images wereobtained with a Dalsa 1M30 CCD camera (Dalsa, Waterloo, Canada) with a 135x50 mm
tandem lens (Nikon Inc., Melville, NY) and red interference filter (610 ± 10 nm). Frames
were acquired at a rate of 30 fps, temporally binned by 4 frames, and stored as 512x512 pixel
images after binning the 1024x1024 camera pixels by 2x2 pixels spatially. The visual stimulus for recording in the binocular zone, presented on a 40x30 cm monitor placed 25 cm infront of the mouse, consisted of 200 -wide bars, which were presented between -50 and 150 on
the stimulus monitor (00 = center of the monitor aligned to center of the mouse) and moved

16

5

4

3
vehicle (n=7)
CPP

2

0

(n=5)

3

6

days

B2

Ipsilateral
4

3

2

1

0

3

6

days

Contralateral
5vehicle (n=5)
CPP (n=5)

3

1

0

6

days

10

response amplitude (∆F/Fx104 )Contralateral

response amplitude (∆F/Fx104 )response amplitude (∆F/Fx104 )B1

response amplitude (∆F/Fx104 )Figure 3B with the absolute levels of response
Ipsilateral
5

3

1

0

6

10

days

Figure 7A with the absolute levels of response

vehicle (n = 6)
CPP (n = 7)

4
*

3

2

baseline

MD
3d

MD
7d

Ipsilateral
response amplitude (∆F/Fx104 )response amplitude (∆F/Fx104 )Contralateral

3
*

2

1

baseline

MD
3d

MD
7d

Figure S7: The CPP results in Figs. 3B and 7A remain identical when the results are shown interms of the absolute amplitude of visual response (∆F/F ∗ 104 ). Conventions as in Figs. 3B and
7A.

continuously and periodically upward or downward at a speed of 100 /sec. The phase andamplitude of cortical responses at the stimulus frequency were extracted by Fourier analysis
as described (Kalatsky and Stryker, 2003). Ocular dominance index (ODI) was computedas (R − L)/(R + L), where R and L are the peak response amplitudes through the right eye
and the left eye, respectively, as described (Kaneko et al. 2008a). Response amplitude ateach time point in individual animals was an average of at least 4 measurements.
Statistical analyses: For changes after monocular deprivation or recovery from deprivation, the response magnitude in individual animals was normalized to the baseline magnitude, followed by calculating group average and S.E.M. Response magnitudes and oculardominance index data were analyzed by a two-way ANOVA to determine the effects of pharmacological treatments (CPP or sTNFR1) and of manipulations of visual experiences (ND,
MD, or binocular recovery) in Prism 6 (GraphPad Software, CA). For multiple comparisons,Bonferroni corrections were used.
17

Supplemental References:
Kaneko, M., and Styker, M.P. (2014) Sensory experience during locomotion promotes
recovery of function in adult visual cortex. eLife, e02798.
Maffei, A., Nelson, S. B., and Turrigiano, G. G. (2004) Selective reconfiguration of layer
4 visual cortical circuitry by visual deprivation. Nature Neurosci., 7(12):1353-1359.
Sato, M., and Stryker, M.P. (2008). Distinctive features of adult ocular dominance plasticity. J. Neurosci. 28, 7520-7536.
Kalatsky, V.A., and Stryker, M.P. (2003) New paradigm for optical imaging: temporally
encoded maps of intrinsic signal. Neuron 38, 529-545.

18