Neuron, Volume 80
Supplemental Information

A Theory of the Transition to Critical
Period Plasticity: Inhibition Selectively
Suppresses Spontaneous Activity
Taro Toyoizumi, Hiroyuki Miyamoto, Yoko Yazaki‐Sugiyama, Nafiseh Atapour, Takao K. Hensch,
and Kenneth D. Miller

Contents
S1. Supplemental Analysis
S1a. A Simplified, Analytically Tractable Model
S1b. Estimation of the spontaneous‐to‐visual activity ratio

S2. Supplemental Computational and Experimental Procedures
S2a. Circuit modelS2b. Input statistics
S2c. A plasticity rule and a neuron model
S2d. Receptive field width, Response strength, Contra‐bias index, and Gain
S2e. Extracellular recording from freely behaving miceS2f. LTD Experiments

Supplemental Figures
Figure S1: Results of modeling a linear postsynaptic neuron with a simple activity‐dependent plasticity
rule
Figure S2: Baseline firing rates and LED‐evoked firing rates
Figure S3: The cumulative distributions of the bias‐corrected spontaneous‐to‐visual ratio of firing
rates
Figure S4: LTD is intact in GAD65‐KO mice

Supplemental References

S1. Supplemental Analysis
S1a. A Simplified, Analytically Tractable Model
To try to isolate the essential features that are sufficient to explain pre‐CP and CP plasticity, we simplify themodel presented in the text to a few essential ingredients. The main ideas we want to capture in this simple
model are (1) plasticity occurs through a combination of Hebbian and homeostatic plasticity; (2) before thematuration of inhibition, there is sufficient spontaneous input such that the influence of rare visual input islimited; (3) after the maturation of cortical inhibition at the onset of the critical period, the more frequent but
weaker spontaneous activity is largely suppressed by the combination of inhibition and the neuron’s nonlinear
input‐output relation (c.f. Fig. 1B), so that the visual input plays a much stronger role in determining weightdevelopment. We show that a simple, minimal implementation of these three ideas suffices to explain the mainfeatures of pre‐CP and CP plasticity, which demonstrates that these results do not depend on other, more
complex aspects of the model presented in the text. We consider a linear neuron model and a simple plasticity
rule, which together allow us to analytically describe the determinants of model behavior. This enables us topoint to the precise relationships that are needed to explain pre‐CP and CP behavior.
For a linear postsynaptic neuron, the output firing rate, y, is given by a weighted sum of input firing rates, i.e., y  w  x where x is the vector of input firing rates and w is the vector of synaptic weights. We consider asimple learning rule that combines Hebbian and homeostatic plasticity. The Hebbian term is driven by the
covariance between pre‐ and post‐synaptic activities, xy  x y , where the brackets indicate an averageover input activity patterns (Sejnowski and Tesauro, 1989). A homeostatic component, proportional to
w(y0  y ) , scales synaptic strengths depending on the difference between the average output firing rateand a desired or set‐point firing rate y0. The proportionality to w causes the homeostatic term toymultiplicatively scale synaptic strengths. Such homeostatic scaling has been experimentally observed(Turrigiano et al., 1998; Stellwagen and Malenka, 2006), including in the visual cortex of developing animals invivo (Desai et al., 2002; Mrsic‐Flogel et al., 2007; Kaneko et al., 2008). The change in synaptic weights per unit
time can then be written as

where Q  
xx T  x x T w   Qw   w(y0  y ),
T
is the input covariance matrix (here, x is a column vector and its transpose x isa row vector), =1 Hz is a constant that describes the strength of homeostasis relative to Hebbian plasticity, η=1
s2 is a learning rate, and y0 =1Hz.
As in the main text, we assume that the postsynaptic neuron receives only spontaneous input 90% of the timeand receives both spontaneous and visual input 10% of the time. In this simple model, we do not explicitly model
how inhibition suppresses the contribution of spontaneous activity to plasticity relative to that of visual activity,
but simply assume that the combination of cortical inhibition and a nonlinear plasticity rule suppresses synaptic
plasticity a fraction f (0≦f≦1) of the time in the absence of visual input. Note that f replaces the variable m fromthe main text as a quantifier of effective inhibitory strength. The spontaneous and visual inputs are Gaussian (S )

 (V )

random variables that have means  and 
input mean and covariance are given by







and covariances Q(S) and Q(V) respectively. Hence, the effective  0.1( (V )   (S ) )  0.9(1 f ) (S ) ,Q  0.1(Q(V )  Q(S ) )  0.9(1 f )Q(S ) .

The factor 0.9(1‐f) qualitatively describes the percentage of time Hebbian plasticity is induced in the absence ofvisual stimulus, which we refer to in the main text as the gain of spontaneous activity for Hebbian plasticity.
We use the same parameters to model spontaneous and visually‐driven input activity as in the main text. In
particular, we model monocular deprivation (MD) by assuming that the visual component of deprived‐eye
activity is decreased in strength by ½, while the distance over which its correlations decay is multiplied by 5
(representing the blurring effect of the closed eyelid, as has been observed in effects of eyelid closure on corticalretinotopic selectivity in vivo (Faguet et al., 2009)). We first consider two extreme values of inhibitory strength
to model the pre‐CP (f=0) and CP (f=1), which yield the final receptive fields of the two eyes shown in Fig. S1A fornormal rearing (NR) and for MD of the contralateral eye. In Fig. S1B, we more systematically examine theinfluence of inhibition on the MD effect. During the pre‐CP (weak inhibition; small f), MD suppresses both eyes’
responses while maintaining the dominance of the closed eye, with only small changes in CBI. During the CP(strong inhibition; large f), MD enhances the open‐eye and suppresses the closed‐eye response so that the open
eye becomes strongly dominant over the closed eye, with correspondingly large changes in CBI. MD widens thereceptive fields relative to NR both during the pre‐CP and the CP. This result is consistent with experiments(Smith and Trachtenberg, 2007).
We can analytically describe the model outcome as follows. The only stable fixed point of this learning rule is
proportional to the principle eigenvector (the eigenvector with largest eigenvalue), v , of the input covariancematrix Q (Oja, 1989; Miller and MacKay, 1994), which we take to be normalized to unit vector length. Assuming
the initial set of synaptic weights is not perpendicular to v , the synaptic weights converge to a steady‐state(fixed point) value wSS given byy   / 
wSS  0   v ,
v 



x and λ>0 is the largest eigenvalue of Q (the eigenvalue corresponding to v ). We further assumethat there are no negative or zero correlations between presynaptic neurons. Then the principle eigenvector, v ,where  

has no negative or zero elements (this is known as the Perron‐Frobenius theorem), which guarantees thatsteady‐state synaptic strengths are positive.

The interaction of Hebbian and homeostatic plasticity is the key to the results. As we can see from the fixed‐point
solution for the synaptic weights, the effects of MD arise through changes in the principal eigenvalue, λ, and thenormalized eigenvector, v , which are determined by the Hebbian component; and through the change in the mean presynaptic activity  and thus in v   , which contributes due to the homeostatic component. Of these,

only changes in v differentially affect the two eyes; the other changes only change the common scale factor
applied to both eyes’ weights.

The principle eigenvector of the input covariance matrix, v , which is also known as the principle component of
the input activity, represents the pattern of activity – the set of relative activities across input neurons – that has
the most variance in the input activity (the absolute activity levels are then determined by setting the vector
length of v to 1). Because of positive between‐eye correlations, this pattern involves both eyes’ inputs being
active. If the two eyes have equivalent activities, the two eyes are equally strong in v , but to the extent to which
the deprived eye has weaker activity than the non‐deprived, the deprived eye will be less active in the highest‐
variance activity pattern and so will be weaker than the non‐deprived eye in v . The highest‐variance activity
pattern is centered on the receptive field center, because the inputs at the edges of the receptive field have fewer
correlated inputs with which to be co‐active. If spatial correlations are narrow, this pattern is more tightly
peaked around the receptive field center, while broader correlations lead to a spatially broader and flatter
activity pattern. Figure S1A, shows the final receptive fields under various rearing conditions, which in each caseare proportional to v for that rearing condition. (The number of inputs is larger from the contralateral eye thanfrom the ipsilateral eye as assumed in the main text.) Thus, from Fig. S1A, we can see that, upon MD to thecontralateral‐eye, the eigenvector, v , changes its profile by suppressing the deprived eye components andfacilitating the non‐deprived eye components. At the same time, the visual input through the closed eyelid
induces more spatially uniform input correlation and flattens and spreads the eigenvector.

The eigenvalue λ is the amount of variance that v contributed to the input activity. λ decreases under MDbecause of the loss of input correlation. This decrease of λ reduces wSS and thus reduces responses to both

eyes. On the other hand, due to the loss of the deprived‐eye input activity through  , homeostatic plasticity
causes non‐selective growth of both eyes’ responses. Therefore, the deprived eye will always show less responsethan the non‐deprived eye, but, depending on three factors ‐‐ the relative strength of homeostasis, the degree of
reduction of the deprived‐eye input, and the width of the input correlations in the deprived eye ‐‐ both eyes’
responses could be suppressed, both could be enhanced, or the deprived‐eye could be suppressed while theother eye is enhanced. To cause both eyes’ response peaks to be reduced by MD during the pre‐CP, we used
relatively small  and y0, i.e. relatively weak homeostatic plasticity and a relatively low activity‐set point, whichenhances the effect of the decrease in the eigenvalue relative to the effect of homeostasis; and we used relatively
wide input correlations from the deprived eye, which flattens the eigenvector. Maturation of inhibition
suppressed the spontaneous component, which is balanced between the two eyes, and thus enhanced thedifference between the two eyes under MD via the change in the eigenvector, v .
This simple model is useful to understand how the balance of spontaneous and visually evoked activity
contributes to the activity‐dependent plasticity in the course of development. In particular, it already shows the
major effects we are trying to understand: before the maturation of inhibition, MD slows retinotopic refinement
and reduces both eyes’ responses without causing substantial OD shifts; after the maturation of inhibition, MD
causes strong OD shifts as well as a slowing of refinement. This simple model does not achieve a confined
receptive field (synaptic weights never become zero) and does not address the mechanism by which inhibition
modulates the balance of spontaneous and visually evoked activity. In the main text, we use a learning rule thatyields confined receptive fields and explicitly model how maturation of inhibition suppresses the relative
contribution of spontaneous component over the visually‐evoked component, which was described above by thefactor (1‐f).

S1b. Estimation of the spontaneous‐to‐visual activity ratio
In the main‐text, we estimated the spontaneous‐to‐visual ratio of firing rates based on extracellular recordingfrom behaving mice. We also explained in the main‐text that this ratio is likely overestimated. In this section, weshow that, under simple assumptions, the estimated spontaneous‐to‐visual ratio has a monotonic relation to thetrue underlying spontaneous‐to‐visual ratio. Assuming that the visually evoked component linearly adds tospontaneous firing rate, the true spontaneous‐to‐visual ratio (the spontaneous firing rate divided by the visually
evoked firing rate driven by natural scenes) can be written rS/(rS+rV), where rs is the spontaneous firing rate andrV is the increment of the firing rate due to an optimal visual stimulus. On the other hand the estimatedspontaneous‐to‐visual ratio (the baseline firing rate divided by the LED‐evoked firing rate) is described by(rS+q*rV)/ (rS+q*rV+b*rV), where q (0<q<1) is the frequency of visual input under the free viewing condition andthe factor b*rV with a constant, b (0<b<1), describes the increment of firing rate by the LED stimulus. Letting
k=rS/rV, the true ratio is k/(k+1), while the estimated ratio is (k+q)/(k+q+b)=k’/(k’+1).with k’=(k+q)/b. It is now
easy to see, in this case, that both the true and the estimated spontaneous‐to‐visual ratios are monotonicallyincreasing functions of k. Thus, one ratio is a monotonically increasing function of the other, and vice versa. Notethat the estimated ratio is always greater than the true ratio, because k’>k.

S2. Supplemental Computational and Experimental Procedures
S2a. Circuit model

We model a pyramidal cell in V1 that receives input from N=1184 LGN neurons. The presynaptic neuron irepresent input from either contra‐ or ipsilateral eye, ei{C,I}, and from a particular retinotopic position, zi ,uniformly spaced on a square grid across a two‐dimensional surface of  z  (z1, z2 ) | 1  z1  1, 1  z2  1 .

The first NC =28*28 presynaptic neurons represent input from the contralateral eye (ei=C) and the other NI=20*20 neurons represent input from the ipsilateral eye (ei=I). In order to model the anatomical bias toward thecontra‐lateral eye observed in mice, the number of neurons representing the contralateral eye input is about
twice larger than that from the ipsi‐lateral eye.

S2b. Input statistics
We assume that the input has both a spontaneous component, xi(S), and a visual component, xi(V). We also assume
that xi(S) and xi(V) are independent Gaussian random variables with mean i(S) and i(V) and covariance Qij(S) andQij(V), respectively. The neuron receives the spontaneous and the visual input, xi=xi(S)+xi(V), 10% of the time, andthe neuron receives only the spontaneous input, xi=xi(S), 90% of the time. This 10% figure indicates that the visual
input is sparse (Vinje, 2000) and guarantees a significant contribution of spontaneous input on synaptic
plasticity during the pre‐CP. Many of the cortical neurons in mice are orientation selective (Sato and Stryker,2008), so this may imply that the optimal visual stimulus with a preferred orientation at a preferred retinotopic
position is rare. Under the normal condition, the input means are uniform across the input neurons and are givenby i(S)=1 Hz and i(V)=2 Hz, assuming twice as strong visual input as spontaneous input. The covariance of input
depends on the corresponding eyes and also on the retinotopic distance, |zi‐zj|, of the two presynaptic neurons,

(S)i.e., Qij
(V )
ij

Q

 
 Cov  xi(S), x (Sj )   cei ,e j i(S ) (Sj )G(zi  z j , L) and  Cov  xi(V ), x (Vj )   cei ,e j i(V ) (Vj )G( zi  z j , L) , where G( z, L) 

 | z |2 
1exp   2  is a Gaussian2 L2
 2L 

distribution with isotropic standard deviation L=0.06; the within‐eye coefficients are set to ce e  2  L so that2ij

 i()i() , are invariant to the width, L, of the correlation; the between‐eyecoefficients are set to ce e  0.5ce e indicating half as strong between eye‐correlation as the within‐eyeijijcorrelation. Those correlation coefficients, ce ,e , are common to both spontaneous and visual components. Thei j()the diagonal components, Qii

spatial correlation profile is modeled as a Gaussian function of the retinotopic distance so that inputs fromnearby retinotopic positions are more strongly correlated than distant ones. This spatial correlation structure
produces localized (rather than full field) RFs under the competitive Hebbian learning rule, which will be
introduced in the next section. The interocular correlations are based on the observation of strong inter‐ocular
correlations in LGN spontaneous activity, which are induced by cortical feedback: (Ohshiro and Weliky, 2006;
Weliky and Katz, 1999).
If the between‐eye coefficient for the visual input were higher than that of spontaneous input, binocular visualdeprivation would reduce between‐eye correlations, which would tend to shift the OD toward the anatomicallystronger contralateral eye (for example, BD during the pre‐CP would shift OD towards the contralateral eye,contrary to the observations of Smith and Trachtenberg, 2007). The assumption that the spatial structure ofcovariance is the same in spontaneous and visually evoked signals was introduced for simplicity. The model
produces similar results as long as the two covariance functions have similar widths. If the spontaneous
covariance is much wider than the visually evoked covariance, the model predicts a similar level of retinotopic
refinement under MD and under monocular inactivation (MI) during the pre‐CP, because the visually evokedcovariance driven by the residual input through the closed eyelid under MD may be as broad as the spontaneous
covariance in this case. This runs counter to the findings of (Smith and Trachtenberg, 2007). If the spontaneous
covariance is much narrower than the visually evoked covariance, the model predicts that refinement of
retinotopy should slow or reverse after maturation of inhibition, which again runs counter to the experimental
findings. The parameterization of the input covariance proportional to the input mean square is consistent withthe multiplication of input by a constant factor.
Only the visual component is modulated by visual deprivation. Under MD, where one eyelid is closed by suturing,

(V )
(V )
we assume that the visual input from the closed‐eye is described by xi  AelG( zi , Lel )* xi , where theoriginal visual input, xi(V), is convolved by an eyelid filter AelG( zi , Lel ) . The closed‐eye input strength is reduced
by a factor of Ael =0.5 and is also blurred by a spatial Gaussian filter with Lel=0.3. To the extent to which theclosed eyelid passes light, it would be expected to filter out all higher spatial frequencies leaving only very
blurry, large‐scale changes in luminance, which would yield greatly broadened correlations, as we model. In ourmodel this residual visual stimulus is important to explain the restricted refinement of RF under MD but nearlynormal refinement under MI as observed during the pre‐CP (Smith and Trachtenberg, 2007). Thus, the mean
(V )
(V )
input strength from the deprived eye is reduced to  i  Aeli ; the within‐deprived‐eye covariance

 
2
)
2
 cei e j  i(V ) (Vj G( zi  z j , L  2Lel ) , where ei and ej are both the deprived eye; and the
 
2
(V )
(V ) (V )2
between‐eye covariance becomes Qij  ce e  i  j G( zi  z j , L  Lel ) , with ei the deprived eye. Theij
becomes Qij(V )between‐eye covariance and within‐deprived‐eye covariance have standard deviation of

L2  L2el andL2  2L2el because the eyelid filter is once and twice convolved with the normal covariance, respectively.
Under MI by pharmacologically inactivating retinal activity or by enucleating one eye, the visual component fromthe inactivated eye becomes zero, which corresponds to setting Ael=0 in the MD scenario. The spontaneouscomponent is assumed to be unchanged under MI for simplicity. Note that we are modeling spontaneous activity
in LGN (Ohshiro and Weliky, 2006; Weliky and Katz, 1999) rather than in retina as the source of spontaneous
input.

S2c. A plasticity rule and a neuron model

The firing rate of the postsynaptic neuron is described by y and the firing rate of the ith presynaptic neuron isN w x  my , where [] is a threshold nonlinearity that

described by xi. The output firing rate is given by y  

 i1

i i

nb



+

rectifies negative arguments to guarantee positive firing rates, wi is synaptic strength from the presynaptic
neuron i, m is the inhibitory synaptic strength from the neighboring neurons, and ynb is a long‐term averagefiring rate of the neighboring neurons. Although the average firing rate of the neighboring neurons is generally
different from the average firing rate of the output neuron y , we assume a homogeneous population of neurons
and approximate ynb by y (See the following for the definition). Thus mynb  my describes the strength of
inhibitory input to the output neuron; we model the maturation of inhibition by increasing m at the onset of theCP from m=0 to m=5 if not stated otherwise.
Learning rule and temporal averages
We introduce two kinds of averages: FirstF describes the instantaneous average with respect to the input.

Next, F describes a long‐term running average with a time constant =1 (this corresponds to 200 time steps insimulations: see, following), i.e., F(t)  et

(tt ')/F(t ')dt ' . We consider in this paper an activity dependent

learning rule for excitatory synapses within the constraint that synaptic strengths are positive, i.e., wi≧0.
Synaptic strengths are modified by the following plasticity rule:wi    x i f (y)  a x i

f (y)   wi (y0  y ), where f(y)=[y‐]+ is a threshold‐linear function withthreshold =2 Hz, =1.0*10‐3Hz‐2 is a learning rate, a=1.001 is a constant slightly larger than unity to cause
synaptic competition, =2Hz is a constant that sets the strength of homeostatic plasticity, and y0=1.2 Hz is the set
point of the output firing rate for the homeostatic plasticity. If some synaptic strengths are updated to negativevalues, they are set to 0 instead to guarantee positive strengths. The visual‐plus‐spontaneous input is presented
with probability q=0.1 (Pattern 1) and the spontaneous‐only input with probability 1‐q (Pattern 2). We update
synaptic strength by q wi  (1 q) wi at each time step of duration t=5.0*10‐3, where 1 and 2
12

describes the instantaneous averages under the two patterns. This way of updating synapses using the averagechange is justified when the learning rate is sufficiently slow compared to the typical duration of each pattern sothat synaptic strengths are relatively constant during the repeated sampling of the patterns.
This learning rule is a combination of Hebbian and homeostatic plasticity. The first term
xi f (y)  a xi f (y) is a Hebbian term that modifies synaptic strengths according to the correlation of pre‐and postsynaptic activity. The parameter a controls the size of the RF (the area with nonzero synaptic strengths)
and also determines the sensitivity to weak input correlation. If a=1, any nonzero correlation between xi(presynaptic factor) and f(y) (postsynaptic factor) can potentiate synapses and the size of the RF tends tobecome large. On the other hand, for a>1, the synaptic strength of an ineffective input is steadily decreased by
the Hebbian term, so we need to introduce a lower limit of zero for synaptic strength, wi≧0, for all i, so that such
synaptic strengths do not take negative values. If a is too large, the learning depends only on the strongest peakof the correlation and becomes insensitive to the correlation width.
The nonlinearity, f(y)=[y‐]+, suppresses Hebbian plasticity when the output firing rate is smaller than the
Hebbian threshold, . To see how inhibition suppresses the contribution of the spontaneous component alongwith the Hebbian threshold, it is useful to consider a special case of a=1. Because xi is jointly Gaussian, theHebbian term in this case is also described (the Bussgang theorem) as






Cov  xi , h  wi xi   gCov  xi ,  wi xi  , where Cov denotes covariance, h(u)   u  my    is a
 i

 i




threshold‐linear function, and the gain factor, g  h'   wi xi  , modulates the magnitude of Hebbian iplasticity. Hence, apart from the non‐trivial multiplication factor g that depends on the inhibitory strength, thislearning rule is equivalent to the standard Hebbian plasticity rule for a linear neuron model (Oja, 1989;
Sejnowski and Tesauro, 1989; Miller and MacKay, 1994). The gain factor g describes, with the current set of

parameters, the percentage of time the output firing rate exceeds the Hebbian threshold, which decreases
rapidly when the net input falls below the Hebbian threshold. Hence, weighted by the probability of receiving
visual input q, the net Hebbian change of synaptic strengths is described by  g  
 g 
qg1c1  (1 q)g2 c2  2  qc1  (1 q)c2   1 2  qc1 ,g1
 g1   where c1 and c2 are the standard linear Hebbian factors ( Cov  x, w  x  ) and g1 and g2 are the gain factors

under Pattern 1 (with visual input) and Pattern 2 (without visual input), respectively. We can see that the
spontaneous‐to‐visual ratio of gains, g1/g2, linearly interpolates two extreme conditions: the pre‐CP‐like
modification, qc1  (1 q)c2 , where both patterns with and without visual input contribute to Hebbian

plasticity, and the CP‐like modification, qc1 , where only the visual pattern contributes to Hebbian plasticity. The
switching from pre‐CP like modification to CP‐like modification robustly arises upon maturation of inhibition if
the following conditions are satisfied. During the pre‐CP, the spontaneous input should have a larger
contribution to plasticity than the sparse visually driven input. Because the input covariance is proportional to
the square of the mean input firing rate in the model, this gives the condition that, before inhibitory maturation,q ( (S ) )2  ( (V ) )2   (g2 / g1 )(1 q)( (S ) )2 . At the onset of the CP, maturation of inhibition should

significantly reduce the contribution of the spontaneous input to plasticity by suppressing the spontaneous‐to‐visual ratio of gains, g1/g2. This gives the condition that, after inhibitory maturation,q ( (S ) )2  ( (V ) )2   (g2 / g1 )(1 q)( (S) )2 . This analysis also well describes the outcome when theconstant a is greater than but close to 1 (c.f. Fig. 3F).
This analysis makes clear that the key requirement for the transition from pre‐CP to CP plasticity is that thematuration of inhibition should strongly reduce the ratio of gains,g2. This reduction in ratio will be achieved if
g1

inhibition is strong enough to largely eliminate V1 spontaneous activity. It will also be achieved if inhibition is
weaker so that weak spontaneous activity remains, but the contribution of that spontaneous activity to Hebbian
plasticity is suppressed relative to the contribution of visually‐induced activity (or equivalently, the contributionof visually‐induced activity is enhanced relative to the contribution of spontaneous activity). The function f(y) we
use, which is zero (no contribution to Hebbian plasticity) for y less than the threshold θ, yields relative
suppression of plasticity for low firing rates and thus for spontaneous activity relative to visually‐induced
activity. This could arise if low postsynaptic firing rates yield calcium entry too weak to evoke LTD or LTP as in(Artola et al., 1990)(however, the physiological conditions that induce LTD are controversial (Huang et al., 2005;
Hager and Dringenberg, 2010; Froc et al., 2000; Jiang et al., 2003)). Alternatively, relative enhancement of thecontribution of visually‐induced activity could be achieved if the function f(y) nonlinearly enhances Hebbian
synaptic changes for larger postsynaptic firing rates. This has been observed: there is an increase in LTP per
postsynaptic spike for higher postsynaptic firing rates (Sjöström et al., 2001).
The second term,  wi (y0  y ) , is a homeostatic term that scales synaptic strengths and shift the output firing
rate into an appropriate range around y0. When the average output rate, y , is greater than the set point, y0, thisterm scales down the synaptic strengths; otherwise scales up the strengths to compensate for perturbations to
the average activity level. This kind of homeostatic synaptic scaling has widely been observed (Sejnowski andTesauro, 1989). Although, the Hebbian term alone is typically unstable (strong synapses tend to drive thepostsynaptic neuron harder and grow even stronger) and can saturate all nonzero synaptic strengths to theirmaximum strength, the homeostatic term controls the overall scaling of synapses and sets the strengths
proportional to the Hebbian term, yielding a bell shaped tuning curve in the current setting. Thus, the parametervalues of the coefficient  and the set point y0 only contribute to the scaling of strengths without influencing theirrelative strength.
The induction of homeostatic plasticity is known to be relatively slow compared to the loss of responsivenessfrom the deprived eye (Turrigiano et al., 1998; Maffei et al., 2004; Mrsic‐Flogel et al., 2007; Kaneko et al., 2008;
Ibata et al., 2008). According to these experiments, we introduced a long‐term running average, y , of the output
firing rate. Because the homeostatic term compares y to the set point, it takes about the averaging time
constant  to adapt its strength after a sudden change in the input statistics caused by visual deprivations.

However, if this adaptation rate 1/ is too slow compared to the learning speed of the synaptic update rule, thisgenerally causes the oscillation of synaptic strengths – the homeostatic term drive synapses beyond the set point
until y has been updated ‐ and learning does not converge. Note that replacing the long‐term averaged firing
rate, y , by an instantaneous average, q y1 (1 q) y 2 , does not alter the fixed point of the learning rule,and, hence, does not alter main results presented in this paper.

S2d. Receptive field width, Response strength, Contra‐bias index, and Gain

In Fig. S3, we plotted several quantities, defined as follows. The width of the RF is the standard deviation of thespatial distribution of synaptic strength from each eye, i.e., Width e 

 Rw
2
i

i{i|ei e}i w , where iii{i|ei e}indexes the inputs and Ri is the retinotopic distance from the RF center and ei{C,I} the eye (either contralateralor ipsilateral) of the ith input. The response strength from each eye was measured by the maximum synaptic
strength times the density of synapses (Ne synapses in the retinotopic area of 4), i.e.,
Response e  max wi *(N e / 4) . We take response strength proportional to the peak of the RF, rather thani{i|ei e}the sum over the RF, because we are comparing principally to measurements (Smith and Trachtenberg, 2007)
from optical imaging, in which each pixel measures response over a local collection of cells, using narrow bars
(relative to RF sizes) for stimuli, which will drive only a small portion of an RF at any given time. Given nonlinear
input/output functions in which stronger inputs yield supralinearly stronger responses, due to threshold and/or
noise, the strongest responses, driven by neurons stimulated on their RF peaks, will tend to dominate the opticalresponse. The contra‐bias index was CBI=ResponseC/(ResponseC+ResponseI). The gain was the percentage of time
the output activity exceeded the Hebbian threshold, as described above in the section describing the learning
rule.

S2e. Extracellular recording from freely behaving mice

To acquire extracellular spikes, tetrodes of four nichrome wires (13 um) were stereotaxically implanted into thebinocular zone of V1 (at lambda, 3.0 mm lateral to the midline). Signals from each electrode were band‐passfiltered (0.15 – 9 KHz) and digitized at 40 KHz (MAP system, Plexon, TX). The electrode tip positions wereverified histologically using Nissl staining. Isolation of single units was performed manually using a multi‐
dimensional cluster cutting software (Off‐line sorter, Plexon). Clusters of spikes with a clear refractory period (>2 ms) were included in the analysis. Putative pyramidal neurons were discriminated by waveform features(spike width, shape, and autocorrelation (Barthó et al., 2004)). Periodic full‐field visual stimuli were provided bya light emitting diode (LED, peak 510nm, 5000 mcd) on the ceiling of the recording chamber. Peri‐event time
histograms were made using NeuroExplorer (Nex Technology, MA). Diazepam (in 50% propylene glycol/saline;
20 mg/kg i.p.) was administered acutely under halothane anesthesia. Chronic‐treated GAD65‐KO (KOc) micereceived repeated, daily intracerebroventricular injections over six consecutive days around P28 and then later
were implanted with electrodes for recording as adults (> P60).

S2f. LTD Experiments

Electrophysiological recordings in vitro were obtained at P24–35, using our standard methods (Renger et al,2002). In brief, coronal slices (350 μm) of binocular mouse visual cortex were cut and incubated (>1 h, 33°C) inequilibrated (95% O2/5% CO2) artificial cerebrospinal fluid (ACSF), containing (in mM): 119 NaCl, 2.5 KCl, 1.3
MgSO4, 10 NaH2PO4, 26.2 NaHCO3, 2.5 CaCl2, and 11 glucose. Slices were transferred to a recording chambersuperfused with the same ACSF (33°C). A two‐pathway experiment was designed in which standard coronalslices were split from white matter up to layer 4 (Renger et al, 2002). Half‐maximal field potentials evoked fromlayer 4 by a bipolar glass stimulating electrode filled with ACSF were recorded through a glass electrode filledwith 1 M NaCl (1–3 MΩ) in layer 2/3. The independence of inputs stimulated from either side of the cut was
confirmed by additive responses in layer 2/3 showing no interaction when closely paired stimuli (< 50 msec)
were presented across the two sites. Stable, baseline responses were recorded (>25 min, 0.1 Hz) before applying
low‐frequency stimulation (LFS; 900 pulses, 1 Hz) to induce LTD on the test pathway, while the other, control
pathway was switched off during conditioning. To determine saturated LTD levels, repeated LFS (3x) wasapplied to one pathway in a separate set of un‐split WT and KO slices.

Supplemental Figures

Fig. S1: Results of modeling a linear postsynaptic neuron with a simple plasticity rule, Related to Fig. 2‐3.
Results of modeling a linear postsynaptic neuron with a simple activity‐dependent plasticity rule, for which thefinal receptive field is proportional to the principle eigenvector of the input covariance matrix. (A) The final
receptive field from either eye (C: contralateral; I: ipsilateral) is shown under normal rearing (NR) and undermonocular deprivation (MD) of the contralateral eye, during the pre‐CP (f=0) and the CP (f=1). The bottom
panels show peak‐normalized one‐dimensional horizontal sections of the two‐dimensional receptive fields
shown in the upper panels (NR: black; MD: red; contra eye: solid; ipsi eye: dashed). (B) Quantifications of the MD
effect on the receptive fields, as a function of the strength of inhibition f. The left two panels plot the ratio of thevalue under MD to the value under NR of the peak response (left) and the receptive field width (middle). The
right panel shows the contralateral‐bias index (CBI) under NR (black) and under MD (red). See Supplemental
Computational and Experimental Procedures, S2d for definitions of response, width, and CBI.

Fig. S2: Baseline and LED‐evoked firing rates, Related to Fig. 4.
Baseline firing rates (A) and LED‐evoked firing rates (B) in Wild‐type mice (WT), GAD65‐KO mice (KO), andGAD65‐KO mice that were chronically treated with diazepam (KO chronic Dz). Many animals are recorded
without any acute diazepam treatment (dark‐blue dots). In order to study the effect of enhanced inhibition on
visual response ratio, a small subset of animals was recorded before (blue dots) and after (red circles) acuteadministration of diazepam. Each mark shows firing rate from one recording unit. Note that the sets of recording
units before and after the diazepam treatment were not always identical because some recording units werelost/gained during the acute administration of diazepam.

Fig. S3: The cumulative distributions of the bias‐corrected spontaneous‐to‐visual ratio of firing rates, Related toFig. 4.
The cumulative distributions of the bias‐corrected spontaneous‐to‐visual ratio of firing rates in GAD65‐KO mice(KO) and GAD65‐KO mice acutely treated with diazepam (KO+Dz), for two differerent choices of underlying
parameters. The underlying true spontaneous‐to‐visual ratio of firing rates were estimated, as described in Sec.
S1b, from the experimental data presented in Fig. 4D. The estimation was based on the assumed frequency, q=0.1,
of visual input under behaving conditions, the unknown effectiveness, b (0<b<1), of the LED stimulation incomparison to the preferred visual stimulus for individual cells, and the unknown Hebbian threshold, . The
estimation of the true spontaneous‐to‐visual ratio of firing rates for majority of cells fell in a regime that
reproduced pre‐CP and CP plasticity for KO and KO+Dz animals, respectively, if the parameters b and  werechosen appropriately. There was a range of b that achieves this for a given choice of Hebbian threshold values, ofwhich one in the middle of the range was chosen for each choice of  illustrated here (c.f. Fig. 3E), =2.0 Hz (A)and =2.5 Hz (B). Cells in the light‐blue and yellow areas showed large enough and almost no ocular dominanceshift, respectively, in the simulation of Fig. 3E.

Fig. S4: LTD is intact in GAD65‐KO mice, Related to Discussion(A) Only the test pathway was persistently depressed by LFS (black circles), revealing a clear LTD in GAD65‐KO
mouse visual cortex (n = 6 slices). Moreover, unaltered control responses (white circles; ** p < 0.01, t test 40 min
post‐LFS) verified independence of the two pathways as well as general health of the KO slices, confirming ouroriginal observations (Hensch et al, 1998). Scale bar, 500 V, 5 msec. (B) Equal saturation of LTD by repeated
LFS (3x) to one pathway in WT and GAD65‐KO mice (n = 7 slices each, p<0.6, t test). Error bars are SEM.

Supplemental References
Artola, A., Bröcher, S., and Singer, W. (1990). Different voltage‐dependent thresholds for inducing long‐term
depression and long‐term potentiation in slices of rat visual cortex. Nature 347, 69‐72.
Barthó, P., Hirase, H., Monconduit, L., Zugaro, M., Harris, K.D., and Buzsáki, G. (2004). Characterization of
neocortical principal cells and interneurons by network interactions and extracellular features. J Neurophysiol
92, 600‐608.
Desai, N.S., Cudmore, R.H., Nelson, S.B., and Turrigiano, G.G. (2002). Critical periods for experience‐dependent
synaptic scaling in visual cortex. Nat Neurosci 5, 783‐789.
Faguet, J., Maranhao, B., Smith, S.L., and Trachtenberg, J.T. (2009). Ipsilateral eye cortical maps are uniquely
sensitive to binocular plasticity. J Neurophysiol 101, 855‐861.
Froc, D.J., Chapman, C.A., Trepel, C., and Racine, R.J. (2000). Long‐term depression and depotentiation in thesensorimotor cortex of the freely moving rat. J Neurosci 20, 438‐445.
Hager, A.M., and Dringenberg, H.C. (2010). Assessment of different induction protocols to elicit long‐term
depression (LTD) in the rat visual cortex in vivo. Brain Res 1318, 33‐41.
Hensch, T.K., Fagiolini, M., Mataga, N., Stryker, M.P., Baekkeskov, S., and Kash, S.F. (1998). Local GABA circuit
control of experience‐dependent plasticity in developing visual cortex. Science 282, 1504‐1508.

Huang, Y.Z., Edwards, M.J., Rounis, E., Bhatia, K.P., and Rothwell, J.C. (2005). Theta burst stimulation of the humanmotor cortex. Neuron 45, 201‐206.
Ibata, K., Sun, Q., and Turrigiano, G.G. (2008). Rapid synaptic scaling induced by changes in postsynaptic firing.
Neuron 57, 819‐826.
Jiang, B., Akaneya, Y., Hata, Y., and Tsumoto, T. (2003). Long‐term depression is not induced by low‐frequency
stimulation in rat visual cortex in vivo: a possible preventing role of endogenous brain‐derived neurotrophic
factor. J Neurosci 23, 3761‐3770.
Kaneko, M., Stellwagen, D., Malenka, R.C., and Stryker, M.P. (2008). Tumor necrosis factor‐alpha mediates one
component of competitive, experience‐dependent plasticity in developing visual cortex. Neuron 58, 673‐680.
Maffei, A., Nelson, S.B., and Turrigiano, G.G. (2004). Selective reconfiguration of layer 4 visual cortical circuitry byvisual deprivation. Nat Neurosci 7, 1353‐1359.
Miller, K.D., and MacKay, D.J.C. (1994). The role of constraints in Hebbian learning. Neural Comput 6, 100‐126.
Mrsic‐Flogel, T.D., Hofer, S.B., Ohki, K., Reid, R.C., Bonhoeffer, T., and Hübener, M. (2007). Homeostatic regulationof eye‐specific responses in visual cortex during ocular dominance plasticity. Neuron 54, 961‐972.
Oja, E. (1989). Neural networks, principal components, and subspaces. International journal of neural systems 1,
61‐68.
Ohshiro, T., and Weliky, M. (2006). Simple fall‐off pattern of correlated neural activity in the developing lateralgeniculate nucleus. Nat Neurosci 9, 1541‐1548.
Renger, J.J., Hartman, K.N., Tsuchimoto, Y., Yokoi, M., Nakanishi, S., and Hensch, T.K. (2002). Experience‐
dependent plasticity without long‐term depression by type 2 metabotropic glutamate receptors in developing
visual cortex. Proc Natl Acad Sci U S A 99, 1041‐1046.
Sato, M., and Stryker, M.P. (2008). Distinctive features of adult ocular dominance plasticity. J Neurosci 28, 10278‐
10286.
Sejnowski, T.J., and Tesauro, G. (1989). The Hebb rule for synaptic plasticity: algorithms and implementations. In
J. Byrne and W. O. Berry, eds. Neural Models of Plasticity (New York: Academic Press), pp. 94‐103.
Sjöström, P.J., Turrigiano, G.G., and Nelson, S.B. (2001). Rate, timing, and cooperativity jointly determine corticalsynaptic plasticity. Neuron 32, 1149‐1164.
Smith, S.L., and Trachtenberg, J.T. (2007). Experience‐dependent binocular competition in the visual cortex
begins at eye opening. Nat Neurosci 10, 370‐375.
Stellwagen, D., and Malenka, R.C. (2006). Synaptic scaling mediated by glial TNF‐alpha. Nature 440, 1054‐1059.
Turrigiano, G.G., Leslie, K.R., Desai, N.S., Rutherford, L.C., and Nelson, S.B. (1998). Activity‐dependent scaling ofquantal amplitude in neocortical neurons. Nature 391, 892‐896.
Vinje, W.E. (2000). Sparse Coding and Decorrelation in Primary Visual Cortex During Natural Vision. Science
287, 1273‐1276.
Weliky, M., and Katz, L.C. (1999). Correlational Structure of Spontaneous Neuronal Activity in the Developing
Lateral Geniculate Nucleus in Vivo. Science 285, 599‐604.