A theory of the transition to critical period plasticity: inhibition selectively suppresses spontaneous activity

Modeling framework

Here we describe the main features of the model; full details and further discussion of parameter dependencies are in Supplemental Methods, S2a–c. We model a single pyramidal neuron in V1 that receives input from lateral geniculate nucleus (LGN) neurons driven by each eye (c.f. Fig. 1A). We neglect interactions between V1 cells because a simpler model suffices to understand how our theory can explain experimental observations. Each eye's inputs are uniformly spaced in a two-dimensional retinotopic space. The density of contralateral-eye input neurons is about twice that for ipsilateral-eye inputs, as in mice (e.g. Hensch et al., 1998; Hofer et al., 2006). We assume synapses from LGN are modified by Hebbian plasticity, which is driven by correlations between postsynaptic and presynaptic activity (Hebb, 1949; Markram et al., 1997; Bi and Poo, 1998) and therefore is sensitive to the correlation structure of the input (e.g., Miller 1990; Dayan and Abbott 2001; Gerstner and Kistler 2002), and by homeostatic plasticity, which regulates the overall activity level and stabilizes learning (Turrigiano et al., 1998; Maffei et al., 2004; Mrsic-Flogel et al., 2007; Ibata et al., 2008; Kaneko et al., 2008).

We model the firing rate of each input neuron as the sum of two Gaussian random variables: the spontaneous component and the visually-evoked component. The spontaneous component represents baseline input statistics in the absence of a visual stimulus, which by definition are unchanged by visual stimuli. We separately model these two components to demonstrate that acute enhancement of inhibition is sufficient to suppress the contribution to Hebbian plasticity of the spontaneous component relative to that of the visually-evoked component.

We assume that the spontaneous component is always active but relatively weak. The visually-evoked component contributes only 10% of the time, but when activated is twice as strong as the spontaneous component under normal-rearing (NR) conditions (c.f. Fig. 1B). The infrequent visual contribution instantiates the idea that visual stimuli with orientation, position, and other features appropriate to visually drive a given neuron are present only infrequently in natural stimuli (Vinje, 2000). For each component, pairs of inputs are correlated with a strength that decreases with their retinotopic separation and, for a given separation, is stronger for inputs from the same eye than from opposite eyes: (Ohshiro and Weliky, 2006; Weliky and Katz, 1999)(c.f. Fig. 1C). Under NR, spontaneous- and visually-evoked components have the same spatial and inter-ocular structure, with within-eye correlations twice as strong as between-eye. LGN spontaneous activity has strong between-eye correlations due to drive to LGN by cortical feedback (Weliky and Katz, 1999; Huberman et al., 2008). The identical spatial structure of the two components, assumed for simplicity, is supported by experiments finding almost identical shape (though not necessarily amplitude) of spatial covariance with or without visual stimulation in LGN (Ohshiro and Weliky, 2006) and V1 (Fiser et al., 2004). The structure of between-eye LGN correlations in visually-evoked activity has not been characterized and so was taken for simplicity to be the same as in spontaneous activity.

To model MD, in which the visual stimulus reaches the contralateral eye only after penetration through the closed eyelid, we assume the visual component of the closed-eye is blurred (Faguet et al., 2009) and weakened by half. Such blurring is known to influence V1 development (White et al., 2001; Akerman et al., 2002), and has been suggested to underlie the slowing of retinotopic development by MD but not monocular inactivation (MI) during the pre-CP (Smith and Trachtenberg, 2007). To model MI induced by pharmacologically inactivating retinal activity or enucleating one eye, we set the visual component from the inactivated eye to zero. We ignore MI-induced changes in spontaneous activity for simplicity, although MI alters burstiness of responses and increases correlations in LGN spontaneous activity (Weliky and Katz, 1999; Linden et al., 2009).

The experience-dependent refinement of the retinotopic map during the pre-CP was measured by intrinsic imaging as a decrease in map scatter. Scatter was defined as the average over pixels of the difference between a pixel and its near neighbors in their preferred retinotopic positions (Smith and Trachtenberg, 2007). To model this, we assume that decreased scatter corresponds to decreased RF sizes in single neurons. A larger RF will be flatter near its peak. In the presence of measurement noise, flatter RFs will produce larger map scatter. Therefore, we simply equated map scatter with RF size in the model. Of course, aligning RF peak locations among nearby cortical neurons may also reduce scatter. We ignore this factor because addressing it requires modeling multiple, recurrently connected cortical cells and perhaps plasticity of intracortical synapses. This introduces complexities and arbitrary modeling choices that are not needed to illustrate the workings of our theory. However, we note that, if intracortical interactions lead nearby RFs to receive correlated inputs, then broader spatial correlations in LGN inputs should yield more widely scattered RF peaks of nearby cells as well as larger RFs. Thus, if we were to model recurrent interactions and the movements of RF peaks, we would expect qualitatively the same effects on map scatter as we find in the simpler model examined here.

Modeling of a cortical neuron and plasticity rule

The firing rate y of the output neuron is taken to be a threshold-linear function of the inputs: $y=\left[{\sum }_{i=1}^N{w}_i{x}_i-m\stackrel{‒}{y}\right]$. Here, []₊ is a linear rectification function, [x]₊=x if x≥0, =0 otherwise; w_i≧0 is the synaptic strength from, and x_i the firing rate of, the i^th presynaptic neuron; and m is the strength of inhibition driven by a nonspecific pooling of neighboring neurons. The instantaneous activity averaged over neighboring neurons is approximated by the long-term averaged firing rate, $\stackrel{‒}{y}$, of the postsynaptic neuron, assuming spatially homogeneous average firing rates. For simplicity, we model the maturation of cortical inhibition as an immediate increase of the inhibitory strength m at the onset of the CP for OD (Di Cristo et al., 2007; Fagiolini et al., 2004; Katagiri et al., 2007; Kuhlman et al., 2010; Sugiyama et al., 2008). Instead, if a gradual increase of inhibition (Morales et al., 2002) were modeled, it would progressively open the CP by increasing the magnitude of the OD shift under MD with increasing inhibitory strength (e.g., see Fig. 3E for the steady-state OD shift vs. inhibition level).

The instantaneous change in synaptic strengths, w_i, is modeled as the sum of a Hebbian term and a homeostatic term. The Hebbian term is proportional to a measure of correlation of pre- and postsynaptic firing, ⟨x_if(y)⟩ − a⟨x_i⟩⟨f(y)⟩, where ⟨⟩ is a short-term average, a=1.001 is a constant slightly larger than 1 to force a competitive learning outcome by suppressing synapses with small input-output correlation, and f(y)= y-θ for y>θ, f(y)=0 otherwise. θ is a threshold level of postsynaptic activity required for Hebbian plasticity. We generally take θ=2 Hz but also consider varying θ. The homeostatic term is proportional to $w_i(y_0-\stackrel{‒}{y})$, with y₀=1.2 Hz. Here, $\stackrel{‒}{y}$ is the long-term average of the postsynaptic firing rate. This term scales synaptic strengths to bring $\stackrel{‒}{y}$ towards the set-point firing rate y₀. The proportionality to w_i causes the homeostatic term to multiplicatively scale synaptic strengths. Such homeostatic scaling has been experimentally observed (Turrigiano et al., 1998; Stellwagen and Malenka, 2006) including in V1 of developing animals in vivo (Desai et al., 2002; Mrsic-Flogel et al., 2007; Kaneko et al., 2008).

Our choices of a thresholded input/output function (the []₊ function in the expression for y) and subtractive inhibition ensure that inhibition suppresses spontaneous activity relative to visually-evoked activity, as illustrated in Fig. 1B. Subtraction of a fixed amount removes a greater proportion of the smaller spontaneous input than of the larger visual input and will also cause a greater fraction of spontaneous than of visual input to lie below the spiking threshold. Alternatively, inhibition might reduce all responses by an equal factor (e.g. Rothman et al., 2009). We choose a neuronal model in which inhibition changes the relative response strengths in order to explore our theory that such a change underlies CP opening. We will experimentally test the idea that inhibition acts in this way at CP opening. The function f in the plasticity rule further suppresses the contribution of weak spontaneous activity to Hebbian plasticity. We will explore how varying the threshold θ influences developmental outcome. The exact form of f is not important: e.g., f could be a smooth function that enhances Hebbian modifications at high postsynaptic firing rates instead of suppressing such modifications at low rates (see Supplemental Methods S2c).

Simulations of visual cortical development

We simulated pre-CP and CP plasticity under this model. Figure 2 shows activity-dependent development of retinotopy (as measured by RF size), OD, and response strength during the pre-CP, beginning at eye opening (time zero). The initial RF profile was chosen relatively broad. Under the NR condition (Fig. 2A Top; black lines in Figs. 2B–E), the neuron's RF narrowed (synaptic strength became more concentrated near the RF center, Fig. 2A,D) due to the spatial correlations of both the spontaneous and the visually-evoked activity components. This refinement is a typical outcome of competitive Hebbian plasticity, which tends to wire together a closely correlated group of neurons onto a given postsynaptic cell. Despite the retinotopic competition, there was relatively little between-eye competition, as reflected in the comparable strengths of the two eyes' final synaptic strengths (Fig. 2B). This was controlled by the relative magnitude of between-eye correlations vs. within-eye correlations for both the spontaneous and visually-evoked activity, with relatively weaker between-eye correlations yielding stronger inter-ocular competition. The contralateral bias index (CBI), which measures the bias toward the contralateral eye, reached slightly above 0.7 (Fig. 2C), primarily reflecting the 2X denser input from the contralateral eye (Gordon and Stryker, 2006).

Homeostatic plasticity stabilized the Hebbian plasticity, which otherwise was unstable. The multiplicative homeostatic scaling yielded a bell-shaped synaptic strength profile, because it more strongly suppressed stronger synapses. In contrast, stabilization by a hard upper bound on synaptic strengths pushes all synapses to near the maximal or minimal allowed values (e.g., Song et al., 2000; Miller and MacKay, 1994).

Under MD of the contralateral eye initiated at time zero during the pre-CP, the refinement of the RF was decreased compared to the NR condition (Fig. 2A–B,D) but with little change in OD (Fig. 2C). The refinement decreased due to the broadly correlated residual visual stimulus through the deprived eyelid (Faguet et al., 2009). Despite a dramatic change in the distance over which visual inputs were correlated, the RF widened only by about 30% (Fig. 2D), because the contribution of the frequent spontaneous input to plasticity was greater than that of the rare visually evoked input during the pre-CP.

The widening of the RF, in turn, reduced the maximum synaptic strength of each eye under the approximate conservation of total synaptic strength enforced by the homeostatic term (Fig. 2E). The total synaptic strength was roughly preserved because the reduction in input activity induced by MD was small given the dominant spontaneous input. MD did not induce much change in OD because the frequent input from spontaneous activity produced almost equal overall levels of input from both the deprived and open eyes (Fig. 2C). The CBI under MD was again about 0.7, despite similar synaptic strengths from both eyes, because of the denser input from the contralateral eye. Under MI (blockade of all visually-evoked activity) of the contralateral eye, the ipsilateral RF did not show the full widening and peak reduction seen under MD (Fig. 2D–E), qualitatively in agreement with experiments (Smith and Trachtenberg, 2007). This was due to the absence of the broadly-correlated residual visual activity that remains during MD, as suggested by (Smith and Trachtenberg, 2007).

Initiation of MD after the opening of the CP has a very different effect (Fig. 3). We continued the NR condition from Fig. 2, changing the inhibitory strength m from 0 to 5 at time 100 to represent CP opening. As discussed above (c.f., Fig. 1B), the increased inhibition suppressed the spontaneous input component more strongly than the visually-evoked component. Homeostatic plasticity compensated for the inhibition-induced loss of activity by scaling up synaptic strengths to approximately preserve the long-term average of the postsynaptic firing rate, largely maintaining the reduced spontaneous-to-visual activity ratio. The reduced ratio, together with the threshold that suppresses Hebbian modifications under small postsynaptic activity, reduced the contribution to plasticity of spontaneous activity relative to visually-evoked activity, rendering the neuron more sensitive to changes in visual statistics. As a result, MD, initiated at time 150, caused a strong OD shift toward the open (ipsilateral) eye (Fig. 3A–C), as well as greater broadening of the ipsilateral synaptic strength profile than was seen during the pre-CP (deprived-eye RF width decreased by about 50%, Fig. 3B,D; see Discussion).

The amount of OD shift depended on parameters – the strength of inhibition and the Hebbian threshold – in a seemingly complex way, but underlying this is a simple dependence of the OD shift on the degree to which visually-evoked vs. spontaneous activity contributed to Hebbian plasticity. Figure. 3E plots the final CBI after MD as a function of the spontaneous-to-visual ratio of the mean firing rate (the rate without visual input divided by the rate with visual input) at various levels of inhibition and of the Hebbian threshold. For any given threshold, a lower spontaneous-to-visual ratio of firing rates, (due to stronger inhibition) increased the OD shift. As threshold increased, the ratio of firing rates that could produce a strong OD shift increased. This is because the spontaneous component did not contribute to plasticity when its rate fell below the Hebbian threshold (c.f. Fig. 1B).

The many curves for different thresholds in Fig. 3E all collapse onto a single curve if the CBI is plotted against the spontaneous-to-visual ratio of the gains for Hebbian plasticity (Fig. 3F; see Supplemental Methods, S2c,d for a mathematical definition of the gain). Under the current set of parameters, the gain is equivalent to the frequency of events that drive Hebbian plasticity, i.e., the percentage of time the output firing rate exceeds the Hebbian threshold. That is, across parameters, the CBI is simply determined by the relative contributions of spontaneous and visual activity to Hebbian plasticity. This is the key, robust prediction of the model. A mathematical analysis of the Hebbian component also confirms that this is the key quantity for predicting the magnitude of the OD shift (see Supplemental Methods, S2c).

In Supplemental Analysis, Section S1 and Fig. S1, we present a simplified mathematical model for which we can obtain an analytic expression for the weights that develop. In the simplified model, postsynaptic activity is a simple linear sum of the weights times the input rates, and Hebbian plasticity is proportional to the covariance of pre- and postsynaptic activity without the nonlinearity used above and with a=1. The homeostatic plasticity rule and the statistics of excitatory input to the cortical neuron are unchanged. Rather than modeling subtractive inhibition, spike threshold, and Hebbian threshold as above, we simply assume that, in the absence of visual drive, postsynaptic activity is suppressed a certain fraction of the time by inhibition, where this fraction increases with inhibitory strength. This simpler model reproduces the main results of our theory: in the pre-CP, MD yields broader RFs and decreases both eyes' inputs strengths with minimal change in OD; while in the CP, MD yields strong OD shifts (and also broader RFs). The analytical expression for the weights shows that these are all robust outcomes, with one exception. The decrease in both eyes' weights induced by MD in the pre-CP involves a tradeoff between the decreased input activity, which reduces the weights overall; homeostatic plasticity, which increases the weights; and the spatially broader input activity, which spreads the weights and thus reduces the peak weight under homeostatic plasticity. Thus, relatively strong reduction in input activity, relatively weak homeostasis, and relatively broad correlations all favor reduction in the two eyes' weights, while with different sets of these three choices the open eye's weights or both eye's weights can instead be increased by MD in the pre-CP. The model also predicts that the final steady-state of the synaptic strengths is insensitive to the initial synaptic strengths, which was generally true in the previous simulations. The ability of the simple model to reproduce the results demonstrates their overall robustness, showing that the results depend on the overall framework of Hebbian plus homeostatic plasticity and an inhibition-induced reduction in the spontaneous-to-visual activity ratio, but not on the details of how inhibition causes this reduction.

Spontaneous-to-visual activity ratio in freely behaving mice

We have shown that a suppression of the spontaneous-to-visual activity ratio induced by the maturation of inhibition is sufficient to explain the transition from pre-CP to CP plasticity. To test the prediction that such suppression occurs, we recorded extracellularly with head-mounted tetrodes from putative pyramidal cells in V1 of freely behaving adult mice (Supplemental Methods, S2e). We studied both WT and GAD65-KO animals. The GAD65-KO mouse models a pre-CP state and will enter a CP even in adulthood upon chronic strengthening of inhibition by administration of diazepam (Fagiolini and Hensch, 2000). We also studied adult GAD65-KO mice that had previously been exposed to chronic diazepam for six days around P28 (KOc), which had caused them to enter and then exit the CP. These mice, like adult WT mice, are in a post-CP state in which plasticity is no longer triggered by administration of diazepam (Fagiolini and Hensch, 2000; Iwai et al., 2003).

Acute strengthening of inhibition by diazepam lowered the spontaneous-to-visual activity ratio specifically in those animals in which this treatment, when maintained over days, opens the CP. We measured both sensory-evoked responses driven by LED flashes and baseline firing rates (Fig. 4A, Fig. S2). Average firing responses to LED flashes of typical multi-unit activity from fully mature WT and GAD65-KO mice are shown in Fig. 4B before and after acute administration of diazepam (see Fig. S2 for firing rates of all isolated recorded cells under all treatment and stimulus conditions). We then computed the spontaneous-to-visual ratio of firing rates (baseline firing rate divided by LED-evoked firing rate; Fig. 4C). In agreement with the model's prediction, acute enhancement of inhibitory transmission by diazepam suppressed the spontaneous-to-visual ratio in GAD65-KO mice (p=0.002 Mann-Whitney test; 9 mice, n=68 cells), in which diazepam opens the CP (Fagiolini and Hensch, 2000). Surprisingly, diazepam did not alter the spontaneous-to-visual ratio in animals in which diazepam no longer opens plasticity (Iwai et al., 2003), neither in adult WT mice (p=0.6 Mann-Whitney test; 3 mice, n=21 cells) nor in KOc mice (p=0.1 Mann-Whitney test; 2 mice, n=18 cells).

To compare activity ratios across different animal groups, additional cohorts were recorded without acute diazepam treatment (c.f., Fig. S2, showing all firing rates for all animals in all conditions). In Fig. 4D, the cumulative distributions of the spontaneous-to-visual ratios of firing rates are shown for different mouse groups. WT mice (11 mice, n=66 cells) exhibited significantly smaller ratios than the GAD65-KO mice (p=0.003, Mann-Whitney test; KO, 21 mice, n=112 cells), consistent with the pre-CP state of the GAD65-KO mice. Acute administration of diazepam in the GAD65-KO mice (9 mice, n=68 cells) reduced their ratios to levels smaller than in adult WT mice (p=0.03, Mann-Whitney test). WT mice also showed significantly smaller ratios than in KOc mice (p=0.02, Mann-Whitney test; KOc, 21 mice, n=127 cells), whose ratios were not significantly different from those of naïve GAD65-KO mice (p=0.6, Mann-Whitney test). Thus, diazepam only transiently suppresses ratios in GAD65-KO mice to open the CP, after which the ratios in KOc animals return to their pre-CP levels. This finding was unexpected, but could reflect the enduring weak inhibition due to the persistent lack of GAD-65.

One caveat to these results is that we have somewhat overestimated the spontaneous-to-visual ratio. We over-estimated spontaneous activity, which we took as the baseline activity in the absence of LED stimulation. Animals experience natural vision under these conditions, so infrequently effective visual drive was added to the spontaneous. Moreover, an LED flash is not an optimal visual stimulus for visual cortical cells, i.e. we underestimated visually-evoked activity. However, assuming linear addition of LED-evoked and other visual responses to spontaneous firing rates, the spontaneous-to-visual ratio estimated here shows a monotonic relation to the underlying true ratio (see Supplemental Analysis, S1b). Our primary aim is to make qualitative rather than quantitative predictions, due to the simplicity of our models. Nonetheless, the measured ratio changes, corrected for overestimation using reasonable parameter values, and applied to the simple plasticity model we have presented here, account quantitatively for the changes in plasticity in the majority of cells (Fig. S3).

Mechanisms of CP initiation

No previous model of CP induction has addressed the fact that MD-induced plasticity is present before the CP, but differs in quality from CP plasticity. The alternative models that bear the closest resemblance to ours also suggest that inhibition opens the CP by changing activity patterns, but focus on changes that could open inter-ocular competition under spike-timing dependent plasticity (STDP). Kanold and Shatz (2006) argued that excess spontaneous activity of cortical origin (as opposed to activity driven by the spontaneous component of LGN input, as we consider) will decorrelate postsynaptic spiking from presynaptic inputs, causing STDP to strengthen less active inputs relative to more active inputs. Reduction of cortical spontaneous activity by inhibition causes STDP to instead strengthen more active relative to less active inputs, allowing the OD shift caused by MD. The model requires fine tuning of the level of pre-CP cortical spontaneous activity so that pre-CP MD causes no change in OD. Kuhlman et al (2010) argued that IPSPs follow EPSPs with a short time delay, so that stronger inhibition tightens the temporal window within which inputs must fire together to drive postsynaptic spikes and yield potentiation under STDP. The more temporally coherent, visually-driven open-eye inputs would therefore have an advantage over more temporally dispersed deprived-eye inputs. Inhibitory maturation would increase this advantage sufficiently that the initially weaker open eye could out-compete the deprived eye. Hensch (2005) suggested that weak inhibition could prevent interocular competition by allowing weaker as well as stronger inputs to drive backpropagating action potentials and so be strengthened by STDP.

There have been many suggestions that molecular changes in plasticity mechanisms open the CP. These proposals again would not account for the transition from pre-CP to CP plasticity, but this does not preclude contributions of molecular mechanisms. Gene and protein expression patterns develop in a manner dramatically dependent upon visual input (Tropea et al., 2006; Majdan and Shatz, 2006; Lyckman et al., 2008). Several studies suggest that plasticity rules do not change during the pre-CP to CP transition (Fox, 1995; Hensch et al., 1998; Kuhlman et al., 2010; Chun et al., 2013), as we have assumed, but others report changes in plasticity rules (Jiang et al., 2007).

One molecular proposal focuses on the switch in the predominant subunit composition of NMDA receptors from NR2B to NR2A, which occurs over the first five postnatal weeks and is boosted within hours by visual experience and attenuated or reversed by visual deprivation (Flint et al., 1997; Quinlan et al., 1999a, 1999b; Roberts and Ramoa, 1999). It has been argued that inhibitory maturation might open the CP by triggering this change in NMDA receptors: GAD65-KO mice show reduced NMDA receptor subunit NR2A levels and correspondingly slower NMDA currents, which are rescued by diazepam application (Kanold et al., 2009). However, while NR2A may modulate OD plasticity (Fagiolini et al., 2003; Cho et al., 2009), it is not required for opening or normal closing of the CP (Fagiolini et al., 2003).

There have been conflicting reports about cortical LTD in GAD65-KO mice. One study reported a defect of cortical LTD in these animals that could be rescued by chronic diazepam (Choi et al., 2002). However, our earlier studies (Hensch et al., 1998) found clear synaptic depression following low-frequency stimulation (LFS) in the same GAD65-KO mouse line of Kash et al. (1997) studied by Choi et al. 2002. Failure to observe LTD in these fragile animals could be an artifact of poor general nutrition or stress due to housing or slice recording conditions (Artola et al., 2006; Kash et al., 1997; Nishie et al., 2012).

To resolve this issue, we have revisited it in a second GAD65-KO mouse line (of Asada et al., 1996) using a two-pathway paradigm (Kirkwood and Bear, 1994; Renger et al., 2002) in which standard V1 coronal slices from GAD65-KO mice were cut vertically from white matter to layer 4 (Supplemental Methods, S2f). One `test' input is first determined to be independent of a `control' input across the cut by linearly additive responses and a lack of paired-pulse interactions when stimuli are applied at short intervals (< 50 msec) to each side. The test pathway showed clear LTD in response to LFS, while the unconditioned control input remained stable (Fig. S4A). This confirms both the independence of the two pathways and the good health of our slice preparations for the duration of the recording. In a separate series of experiments, repeated LFS to one pathway saturated LTD at comparable levels in WT and GAD65-KO mice (Fig. S4B). Thus, our data now confirm robust LTD in two lines of GAD65-KO mice.

Other factors might play a role in CP initiation. In this study we focus on plasticity of synapses from excitatory feed-forward inputs onto excitatory cells, but several recent reports also suggest a role for rapid plasticity of excitatory synapses onto inhibitory neurons (Gandhi et al., 2008; Yazaki-Sugiyama et al., 2009; Kameyama et al., 2010) and inhibitory output plasticity (Maffei et al., 2006). Further studies are required to determine the contributions made by each of these factors and the dependencies between them. While multiple mechanisms may potentially contribute to the range of phenomena in pre-CP and CP plasticity, we find it striking that the model we propose can account for both stages in a simple, unified way.

Experimental Tests

The most direct test of the mechanism we propose would involve directly manipulating the spontaneous-to-visual activity ratio. If this can switch plasticity between pre-CP and CP regimes without altering other hypothesized plasticity factors, it would demonstrate both that our proposed mechanism is sufficient and that alternative factors are not necessary to explain this shift. This might be achieved by optogenetic techniques (Knöpfel et al., 2010; Wyatt et al., 2012) or by genetically engineered reduction / increase in potassium currents to increase / decrease spontaneous activity, respectively (Mizuno et al., 2007).

Correlative tests could be performed by measuring activity patterns across experimental manipulations that change plasticity regime and / or measuring plasticity regime across manipulations that alter activity patterns. This would most directly relate to our theory if conducted before or during the animal's initial CP. While reopening of the CP in adult animals might occur by different mechanisms than initial CP opening, as further discussed below, it would also be interesting to examine spontaneous-to-visual activity ratio in a variety of paradigms of adult CP reactivation (Morishita and Hensch, 2008; Bavelier et al., 2010). These include inhibitory precursor cell transplants (Southwell et al., 2010), peri-neuronal net removal or Otx2 reduction (Beurdeley et al., 2012), Nogo receptor deletion (McGee et al., 2005), caloric restriction (Spolidoro et al., 2011), dark exposure, fluoxetine administration, and environmental enrichment (Morishita and Hensch, 2008; Bavelier et al., 2010). A particularly tractable example is the impact of cholinergic enhancement, which has long been appreciated to improve neuronal signal-to-noise (Sato et al., 1987; Parkinson et al., 1988), and which rescues amblyopia in adulthood (Morishita et al., 2010).

Our model predicts that OD shifts should be observed during the pre-CP if LGN input from one eye were mostly blocked, as this would eliminate both visually-evoked and spontaneous input from that eye. (Note that blockade of activity in one eye, rather than in LGN, is unlikely to suffice, as substantial spontaneous activity remains in the inactivated eye's LGN layers; Weliky and Katz 1999). Such OD plasticity in the pre-CP would demonstrate that spontaneous activity in the LGN is preventing an OD shift during the pre-CP and that developmental changes in other factors from pre-CP to CP are not necessary for OD plasticity. Another prediction is that dark-reared animals, in which CP onset (Fagiolini et al., 1994) and the maturation of inhibition (Morales et al., 2002; Katagiri et al., 2007) are delayed until light exposure, should have larger, pre-CP-like spontaneous-to-visual ratios that should rapidly reduce to CP-like levels upon visual experience. Consistent with this, previous experiments (Benevento et al., 1992; Gianfranceschi et al., 2003) reported larger spontaneous-to-visual activity ratios in dark-reared animals than in age-matched controls.

Points the Theory Does Not Address

We do not address the mechanisms through which inhibition suppresses the spontaneous-to-visual activity ratio in some circumstances but not others. In GAD65-KO mice, in which many neurons showed low baseline firing rates after acute diazepam administration (Fig. S2), our suggestion that subtractive inhibition and spike threshold cause suppression seems likely. It is possible that the operating mode of parvalbumin-positive basket cells that initiate the CP (Hensch, 2005) might subsequently change so that inhibition mainly divides input after the CP (Atallah et al., 2012). In any case, our theory only requires that the maturation of inhibition that starts the CP suppress the ratio, as we observed experimentally, regardless of mechanism.

Due to suppression of spontaneous input during the CP, our model predicts that the increase of RF size in MD animals relative to NR animals is much stronger during the CP than in the pre-CP. Such relative expansions of RF size are experimentally observed after long-term (Pizzorusso et al., 2006) but not brief (Gordon and Stryker, 1996) MD. The simple learning rule we used in this paper may not capture the time-course of changes in RF size, which may involve mechanisms beyond those we modeled such as anatomical changes. These may have much longer time courses than our simple model predicts.

Implications for Adult Plasticity and Disease

Our theory addresses only the opening of the CP. It does not address its ending, to which many factors may contribute (Bavelier et al., 2010; Hensch, 2005; Morishita et al., 2010; Harauzov et al., 2010; Pizzorusso et al., 2006; Maya Vetencourt et al., 2008; Southwell et al., 2010). As such, it does not make predictions about the evolution of the spontaneous-to-visual ratio after CP opening. Nonetheless, the fact that this ratio was increased in post-CP animals relative to CP animals raises the possibility that and increased ratio could be one factor in CP closure. The fact that acute inhibitory strengthening did not lower the ratio in post-CP animals, in which inhibitory strengthening does not open a CP, is also consistent with this idea.

Evidence that inhibition can alter the spontaneous-to-visual activity ratio, which in turn can alter plasticity, comes from neurological disorders. Mice experiencing early hypoxia-ischemia display reduced parvalbumin expression, increased spontaneous-to-visual ratio (due to smaller visual responses), and less OD plasticity (Failor et al., 2010). In senescent monkeys, the ratio decreases under administration of GABA or muscimol and increases under bicuculline (Leventhal et al., 2003). Rett syndrome may provide a particularly informative case study. In the MeCP2-KO mouse model, there is virtually no spontaneous activity in V1 (Dani et al., 2005) and correspondingly low spontaneous-to-visual activity ratios, relative to WT animals, as well as parvalbumin cell hyper-connectivity and a loss of visual acuity (Durand et al., 2012). These results are suggestive that hyper-inhibition and too low a spontaneous-to-visual ratio can also be deleterious to normal developmental plasticity. Notably, this ratio, parvalbumin cell connectivity and visual acuity can all be restored to CP levels by dark-rearing or by genetic deletion of NR2A receptors (Durand et al., 2012). Thus, optimization of spontaneous-to-visual activity ratio may be essential for defining different CP stages. Artificially adjusting spontaneous firing rates might then offer potential therapeutic or learning strategies when specific types or properties of developmental plasticity are desirable.

General Implications for Cortical Development

Many sensory and sensorimotor systems exhibit sequences of critical or sensitive periods that progress from simpler to increasingly more complex aspects of experience (Cang et al., 2005; Hensch, 2004; Brainard and Doupe, 2000; Werker et al., 2009; Hernandez and Li, 2007; Scott et al., 2007). In the auditory system, for example, a cascade of CPs induces plasticity for tonotopy, bandwidth tuning, binaural integration, directionality, phoneme discrimination, audio-visual matching, and semantic and syntactic development (Barkat et al., 2011; Insanally et al., 2009; Popescu and Polley, 2010; Werker et al., 2009; Robinson, 1998; Lenneberg, 1967). An appealing hypothesis is that these sequences of CPs may correspond to sequences of reductions in the spontaneous-to-evoked activity ratio along hierarchies of cortical areas (Felleman and Van Essen, 1991; Sharpee et al., 2011), switching the factors most strongly driving learning in each area from internally to externally generated (Tritsch et al., 2007; Moody and Bosma, 2005; Katz and Shatz, 1996; Hooks and Chen, 2006). Sequences of reductions in activity ratio may in turn involve sequential maturation of key GABA circuits (Condé et al., 1996), which might arise through sequential activity-dependent maturation of peri-neuronal nets. These nets induce maturation of inhibition upon binding Otx2 (Beurdeley et al. 2012), which is delivered both from retina to LGN to V1 (Sugiyama et al., 2008) and across cortex from the choroid plexus (Spatazza et al., 2013). It will be interesting to test whether a cascade of reduced spontaneous-to-evoked ratios across areas coincides with cascades of CPs, whether maturation of inhibition is implicated both in these CP onsets and in corresponding ratio changes, and whether postponement of these ratio changes postpones CPs.

In sum, the co-occurrence of a CP with the switching of the dominant learning cue from internal to external may be a general principle governing brain development.