We first describe experiments where all inhibitory neurons were optogenetically stimulated. We expressed an excitatory opsin (ChannelRhodopsin-2; ChR2) in all inhibitory neurons using a transgenic mouse line (VGAT-ChR2). We delivered blue light to the surface of the cortex, while recording activity extracellularly with multi-site silicon recording arrays (Figure 1, Figure 1—figure supplement 2). Mice were given drops of reward approximately once per minute (Materials and methods) to keep them awake and alert. Because neurons in the superficial cortical layers showed the largest responses to the stimulation light (Figure 2—figure supplement 1), we restricted our analyses to units within ≈400 µm of the cortical surface (Materials and methods), which primarily includes neurons in layer 2/3 and upper layer 4 (Harris et al., 2018). We sorted units into single- (separated from noise and other units) and multi-units (lower SNR or containing several apparent units) by waveform (Materials and methods; using a more restrictive threshold for unit inclusion does not affect our conclusions, Figure 2—figure supplement 2), and we excluded the multi-units from further analyses.

The majority of recorded units were suppressed by light (146/167 single units, 87%; Figure 2B–D; 4 animals, 7 recording days). Given that approximately 80% of cortical neurons are excitatory (DeFelipe et al., 2013; Tremblay et al., 2016), and that extracellular recordings can show several biases in sampling cortical neurons (Margrie et al., 2002; Olshausen and Field, 2005), this measured percentage could be consistent with either the presence or absence of paradoxical suppression in inhibitory neurons. Therefore, we classified recorded units as inhibitory or excitatory using in vivo pharmacology. (Our results are confirmed by other classification methods: identifying inhibitory neurons using their response at high laser intensity or by waveform width give similar results, see below and Figure 2—figure supplement 3).

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We first recorded neurons’ responses to optogenetic stimulation, and then applied blockers of excitatory and inhibitory synapses (CNQX, AP5, bicuculline; which block AMPA, kainate, NMDA, and GABA-A synapses; Methods). When recurrent synapses are substantially suppressed, inhibitory cells expressing ChR2 should increase their firing rate to optogenetic input, while excitatory cells’ firing should not increase. Indeed, by measuring responses to inhibitory stimulation in the presence of E and I blockers, we were able to classify units into two groups (Figure 2E). Units in the first class (putative excitatory) were silent for strong stimulation and units in the second class (putative inhibitory) increased their firing rate in response to stimulation. To quantitatively classify cells, we measured the change in activity induced by light when the blockers had been applied (Figure 2F–G); cells were labeled inhibitory (33%; 56/167 units) if the change compared with baseline produced at maximum laser intensity was positive (according to Welch’s t-test, $p<0.01$), and excitatory otherwise. Choosing different classification thresholds did not affect our results (Figure 2—figure supplement 4), examining a subset of cells with the most stable waveforms over time did not affect our results (Figure 2—figure supplement 5), and, finally, examining a subset of inhibitory units where waveforms had high signal-to-noise ratio also did not qualitatively change the results (Figure 2—figure supplement 2). Therefore, we used these response differences in the presence of blockers to identify recorded inhibitory units.

Pharmacologically identified inhibitory and excitatory units showed differences in waveform width and spontaneous firing rate (Figure 2H), although these two factors were insufficient alone to completely classify inhibitory cells (i.e. insufficient to predict whether units would increase firing rate in the presence of blockers). Identified excitatory cells had a wider waveform (0.58 ± 0.01 ms) and lower firing rate (6.0 ± 0.8 spk/s) than inhibitory cells (width 0.46 ± 0.02 ms, rate 10.7 ± 1.7 spk/s). However, inhibitory neurons’ widths were broadly distributed, supporting the idea that that some inhibitory classes can have broader waveforms than others (McCormick et al., 1985; Neske et al., 2015).

Once recorded units were classified as excitatory or inhibitory, we could ask whether the same units before adding blockers (i.e. with the animal in the awake state, with recurrent connections intact) showed paradoxical inhibitory effects. Indeed, inhibitory neurons showed strong suppression when stimulated (Figure 2I–J, Figure 2—figure supplement 7). The mean response over all recorded inhibitory units was suppressed (Figure 2L), and also, the large majority of inhibitory units were suppressed (distributions of response slopes shown in Figure 2J–K; responses for all inhibitory cells as power is varied shown in Figure 2—figure supplement 7). This paradoxical inhibitory suppression is one piece of evidence that an ISN model is a good description of the upper layers of primary visual cortex. (Below we show that response dynamics and responses at high laser intensity provide additional evidence for an ISN.)

These experiments were done without sensory stimulation, as animals viewed an unchanging neutral gray screen. Because visual cortical neurons’ firing rates increase with increasing contrast across a range of overall luminance levels, we expected that we would see little difference between data collected with a neutral gray screen and data collected in the dark. Confirming that expectation, in one experimental session we measured inhibitory responses to stimulation while animals were in the dark (with the blue optogenetic light shielded with light-blocking materials), and found paradoxical suppression (N = 6/6 inhibitory units with initial negative slope, median initial slope −6.9 ± 3.5 spk/s; p = 0.046, median < 0, Mann-Whitney U test).

In an ISN, paradoxical inhibitory suppression arises because exciting inhibitory cells in turn suppresses excitatory cells, which withdraw excitation from the stimulated inhibitory cells. Then, because more excitatory input from recurrent sources is withdrawn from inhibitory cells than excitatory input from optogenetic stimulation is added, the net steady-state response of inhibitory cells is suppression. For this mechanism to hold, inhibitory cells must transiently increase their firing, even if only a small number of inhibitory spikes are generated, before the paradoxical suppression (Tsodyks et al., 1997; Ozeki et al., 2009). Therefore, we looked for transient increases in inhibitory firing with short latencies after stimulation, and found a small increase in inhibitory firing rate immediately after stimulation, before the larger paradoxical suppression (Figure 3). The transient was brief (FWHM = 7.1 ms), and so the actual number of extra spikes fired in the transient was small: only 0.03 extra spikes were fired on average per inhibitory-classified unit per stimulation pulse, although because we used 100 pulse repetitions, the majority of classified inhibitory units (63%, N = 35 of 56), showed a detectable initial transient (Figure 3—figure supplement 1).

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Another ISN prediction is that inhibitory cells should increase their firing rates at high laser intensities. In a recurrent network operating as an ISN, stimulating inhibitory cells first decreases inhibitory firing rates, but stronger stimulation eventually suppresses excitatory cells until they reach a firing rate where the excitatory network is stable without reactive inhibition. Then, increasing stimulation of inhibitory cells beyond this point produces increases in inhibitory firing rates, as the network moves into a non-ISN regime (Figure 1C). Indeed, we observed this in our excitatory and inhibitory identified populations (Figure 2H).

The link between paradoxical responses and inhibition stabilization was initially derived (Tsodyks et al., 1997) assuming recurrent connections do not change their strength. Short-term synaptic plasticity (Tsodyks and Markram, 1997; Markram et al., 1998) could modify this link, as synaptic plasticity can influence network stability, for example by reducing the self-amplification of the excitatory network as excitatory rates rise. We thus set out to determine analytically how synaptic plasticity might affect both responses and dynamics in a network model of excitatory and inhibitory neurons (Appendix 1). We find that, even when short-term synaptic plasticity is included in the model, paradoxical responses imply inhibitory stabilization. Therefore, even when our data are interpreted in a modeling framework that includes short-term plasticity, our experimental observations provide evidence for inhibition stabilization.

While our extracellular recording approach has advantages, such as allowing precise measurements of spike rate dynamics, and recording a large number of E and I neurons in the same experiment, one concern might be that misclassification of excitatory waveforms into inhibitory units could mask inhibitory cell responses. However, confirming that our recording procedures accurately measured inhibitory waveforms, (1) paradoxical suppression is maintained when we make our classification thresholds more stringent (restricting analysis to units with highest waveform signal-to-noise ratio, Figure 2—figure supplement 2; varying classification threshold in the presence of blockers; Figure 2—figure supplement 4), (2) units classified by spike width, or by response at high laser power, also showed clear paradoxical effects (Figure 2—figure supplement 3), and (3) inhibitory, but not excitatory, response dynamics showed an initial increase followed by suppression (Figure 3, Figure 3—figure supplement 1), the response patterns expected for E and I neurons in an ISN. Also, as shown below, in parvalbumin-positive neuron stimulation with viral transfection, using the same waveform-sorting procedures, we observed no average paradoxical effects. Together, these observervations argue that paradoxical effects in VGAT-ChR2 animals do not arise due to waveform sorting, but instead due to the properties of the cortical inhibitory network.

Taken together, these results show that mouse V1 responses are consistent with those of a strongly-coupled network whose activity is stabilized by inhibition.

We next used these data to infer network connection parameters in a standard two-population model that describes the dynamics of population-averaged firing rates of excitatory and inhibitory neurons (see Equation 6; Materials and methods). This parameter inference is possible because we obtained E and I response measurements in three pharmacological conditions (no blockers; with E blockers; with E and I blockers; Figure 4A), and with data from these three conditions, there are more experimental observations than model parameters (Materials and methods: Model degrees of freedom). The model uses a rectified-linear single-neuron transfer function; using nonlinear transfer functions give similar results (Figure 4—figure supplement 1 and Appendix 2).

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First, we find that the model, despite being overconstrained by the data, is a good description of V1 responses (Figure 4A). The model also makes a number of predictions that are verified in the data. It predicts that the inhibitory firing rate slope for high laser intensities should be the same in the no-blocker and E-blocker conditions, as seen in the data (Figure 4A, blue: left vs. middle columns; see also Appendix 2—figure 1). The model makes three other predictions: that the reversal point of inhibitory cells ($L_0$, the point at which initial negative inhibitory slopes become positive), in all three experimental conditions, should match the point where excitatory cells change their slope to become nearly silent. These effects are also seen in the data (e.g. clearly visible in the no-blocker case, Figure 4A, left column). Note that, although these predictions have been derived in a threshold-linear model (Materials and methods), they are general features of network models which show a transition from non-ISN to ISN as the activity level of the excitatory population increases.

One prediction of balanced-state models with strong recurrent coupling (van Vreeswijk and Sompolinsky, 1996) is that the total excitatory and inhibitory currents should each be large, although they cancel to produce a small net input. To check this, we computed from the model the currents flowing into excitatory cells (Figure 4B). The excitatory and inhibitory currents are each approximately ten times larger than threshold (medians are 10.9 and −8.63, respectively), and their sum is small compared to the magnitudes of E and I currents (median net current/threshold = 2.2). The model also allows us to infer the value of the key parameter for inhibition stabilization, $W_{EE}$ (the excitatory self-amplification). The excitatory network is unstable, requiring inhibition for stabilization, and yielding an ISN, when $W_{EE}>1$. As expected when paradoxical effects are present, the fitted value is greater than 1 (Figure 4C: median 4.7, mode 2.5). This value is not much larger than unity, suggesting recurrent coupling is not as strong as some balanced-state models might predict (van Vreeswijk and Sompolinsky, 1996; van Vreeswijk and Sompolinsky, 1998; Amit and Brunel, 1997) but is more consistent with a moderately-coupled network (Ahmadian et al., 2013; Rubin et al., 2015; Hennequin et al., 2018). In Appendix 2, we extend this model analysis to examine networks of spiking neurons which feature a fixed level of input noise, and find qualitatively similar results. The model shows that measured activity is consistent with the mean input to cells being below threshold. This implies the recorded cells operate in the fluctuation-driven regime, matching expectations from cortical response variability (Shadlen and Newsome, 1994; Shadlen and Newsome, 1998; van Vreeswijk and Sompolinsky, 1996; van Vreeswijk and Sompolinsky, 1998; Amit and Brunel, 1997). Moreover, when input fluctuations are taken into account in the model, mean excitatory and inhibitory currents are found to be of order of the distance between rest and threshold, in agreement with recent estimates suggesting the cortex operates in the loosely balanced regime (Ahmadian and Miller, 2019).

Finally, we computed the parameter ranges over which the network is stable, as a function of the time constants of E and I firing rate dynamics, ${\tau }_E$ and ${\tau }_I$. We find, by computing the eigenvalues of the connection matrix (Materials and methods) that the network is stable for ${\tau }_I/{\tau }_E$ in the interval [0, 5.3], and the observed dynamics are consistent with ${\tau }_I/{\tau }_E=4.4$ and ${\tau }_E=7.8$ ms (Figure 4E); that is, with excitatory and inhibitory synaptic time constants of the same order. The values of ${\tau }_{(E,I)}$ (Figure 4D) inferred from the response dynamics are within the stable region, providing further support that the ISN model is a good description of the effects we measure. (Note that the time constants of the opsin may increase these estimates, so the values of ${\tau }_E$ and ${\tau }_I$, while small, are only upper bounds on the true dynamic response of the network to an instantaneous conductance change.) The ratio between E and I time constants is close to the point at which the network becomes unstable through an oscillatory instability (Figure 4E). The fact that the inferred network model describes the data and is stable with $W_{EE}>1$ supports the experimental observation that the underlying cortical network is inhibition-stabilized.

The data above show that the superficial layers of mouse V1 are inhibition stabilized. Another open question is how general the ISN regime is across cortical areas. We performed experiments to look for signatures of inhibitory stabilization in two other cortical areas (Figure 5): somatosensory (body-related primary somatosensory, largely medial to barrel cortex) and motor/premotor cortex. (See Figure 1—figure supplement 2 for locations of all recording sites.) As in V1, we recorded units extracellularly, restricted analysis to the superficial layers of the cortex, and examined responses to optogenetic stimulation of all inhibitory cells (as above, using the VGAT-ChR2 line).

Figure 5

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In these experiments, we classified neurons as inhibitory based on their activity at high laser intensity. In our V1 experiments, paradoxical effects are clear whether neurons are classified in this way, or with the pharmacological method (Figure 2—figure supplement 3). This high-laser-power classification is likely more stringent (i.e. may reject some inhibitory cells if they are suppressed by other inhibitory cells, due e.g. to heterogeneity of recurrent connections) than pharmacological classification. Supporting that this classification may be more stringent, some pharmacologically-classified inhibitory cells show paradoxical effects but little increase at high power (Figure 2—figure supplement 8).

In both the recorded areas (Figure 5), inhibitory neurons showed paradoxical effects, supporting that these cortical areas do also operate as an ISN. Similar to the V1 data, and as expected in an ISN, both areas showed a transition to a non-paradoxical response at large laser intensity. Compared to the somatosensory data, the motor/premotor recordings show a smaller average suppression (Figure 5B) and more inhibitory units with non-paradoxical effects (rate increases with stimulation; Figure 5D). However, in both areas, means and medians are significantly negative (see legend for statistical tests). Thus, the superficial layers of both sensory cortical areas we examined (visual and somatosensory), and one non-sensory area (motor/premotor) showed responses consistent with strong coupling and ISN operation.

Up to this point, we have studied paradoxical suppression by stimulating an opsin expressed in all inhibitory cells via the VGAT-ChR2 mouse line. A remaining question is whether stimulation of a subclass of inhibitory neurons also yields paradoxical suppression. Even in an ISN, stimulation of any single subclass of inhibitory cells need not produce paradoxical suppression (see Materials and methods and Rubin et al., 2015; Litwin-Kumar et al., 2016; Sadeh et al., 2017; Gutnisky et al., 2017).

To study this, we stimulated parvalbumin-positive (PV) neurons, which provide strong inhibitory input to other cells (Figure 6). PV basket cells are the most numerous class of cortical inhibitory cells and make strong synapses near the somata of excitatory cells (for review, see Tremblay et al., 2016), and PV stimulation effectively suppresses network firing rates (e.g. Glickfeld et al., 2013; Sparta et al., 2014).

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We first used viral methods to express an opsin in PV neurons, injecting a Cre-dependent adeno-associated virus (AAV) encoding an excitatory opsin (Chronos; Klapoetke et al., 2014), into a transgenic mouse line (PV-Cre; Hippenmeyer et al., 2005). In these experiments, the network neurons can be divided into three populations (Figure 6A): (1) excitatory, (2) Chronos-expressing PV inhibitory (PV-Chronos), and (3) remaining inhibitory neurons: non-PV (e.g. somatostatin-positive, etc.) or non-Chronos-expressing PV neurons. We identified PV-Chronos cells by measuring whether cells’ firing is increased at high laser intensity (statistically significant increase at maximum laser power, as in Figure 5; see Materials and methods). In V1 VGAT-ChR2 experiments, this classification by response at high laser power produces qualitatively similar measures of paradoxical suppression as pharmacology-based classification, Figure 2—figure supplement 3. Non PV-Chronos cells (without increase in firing at high laser intensity), are likely either excitatory cells, non-PV, or non-expressing cells. Supporting the idea that our classification approach identifies PV-Chronos cells, the majority of classified inhibitory cells have narrow waveforms (Figure 6I). While not all PV-positive cells are basket cells (Taniguchi et al., 2013), the large proportion of narrow-waveform units we found suggest our inhibitory classification detects a number of fast-spiking cells.

Stimulating the PV cells produced no significant paradoxical effect on average (Figure 6D, blue). From a theoretical point of view, this average inhibitory response is an important measure, as with the standard two-population ISN model, e.g. Tsodyks et al., 1997, it is the average response (averaged over inhibitory cells) that is paradoxically suppressed when a network is inhibition-stabilized and strongly coupled. A second important measure, e.g. for experimentalists wishing to identify inhibitory neurons, is how many individual inhibitory cells show paradoxical effects. Examining individual units showed that PV-Chronos units often showed no paradoxical effect. Initial slope was often positive and thus non-paradoxical (e.g. Figure 6H; median not sig. dif. from zero; summarized in Figure 6K–L; see legend for statistical tests).

One reason why stimulation with viral expression might produce no paradoxical effect on average is that viral expression could target only a subset of the PV cells (Mathematical methods; [Sadeh et al., 2017]). We first examined the effect of inhibitory cell subset stimulation in a model, the rate model of Figure 4, using network parameters inferred there. Stimulating a subset of inhibitory neurons in the model (60%) reduced the magnitude of the average paradoxical effect (Figure 6E, analytical derivation in Materials and methods, and Figure 6—figure supplement 1), and described the data (Figure 6D) well. We also considered a model with multiple subclasses of inhibitory neurons, using the connectivity structure measured in Pfeffer et al., 2013, and found there also that paradoxical effects were not clear when approximately half of the PV cells were stimulated (Figure 6—figure supplement 1).

Why, intuitively, can stimulation of a subset of inhibitory cells eliminate paradoxical effects? In networks of strongly-coupled excitatory and inhibitory neurons, increased inhibitory activity produced by stimulation suppresses excitatory activity and results in a withdrawal of recurrent excitation; this, in turn, drives suppression of stimulated cells (the paradoxical effect). When only a fraction of inhibitory cells is stimulated, the withdrawal of excitation coexists with other effects produced by recurrent interactions. In particular, increased activity of stimulated inhibitory cells tends to suppress non-stimulated inhibitory cells and to produce a withdrawal of inhibition to excitatory cells and to stimulated inhibitory cells. This withdrawal of inhibition increases with the fraction of non-stimulated inhibitory cells and, if large enough, can overcome the withdrawal of excitation and prevent paradoxical suppression of stimulated cells.

To test these models, and experimentally determine whether differences in number of cells expressing opsin could change the paradoxical effect, we used a transgenic approach to express a different excitatory opsin in PV cells (PV-Cre;ReaChR transgenic mice). In these mice, the opsin is expressed in most or all PV cells (Lin et al., 2013). (Since we study steady-state firing rates (Materials and methods), we do not expect differences in the onset or offset kinetics of the opsin to affect these measurements, and we verified that considering different time windows of the steady-state response does not change the results, Figure 6—figure supplement 2). Because we measured responses for a range of light intensities, differences in viral and transgenic mean levels of opsin expression would not affect paradoxical effects. On the other hand, differences in the number of expressing cells, or variability in opsin levels across cells, are predicted to change the paradoxical effect. Indeed, we found that, with transgenic expression, the average paradoxical effect was present (Figure 6F). Further, the fraction of inhibitory cells that showed paradoxical responses was significantly larger in PV-ReaChR transgenic animals than in PV-Chronos animals (Figure 6J,L), and was similar to the VGAT-ChR2 data in V1, somatosensory, and motor/premotor cortex (Figure 6M).

Changes in the number of stimulated cells could be affected by the virus infecting only a subset of neurons (Watakabe et al., 2015), or by the virus yielding different levels of opsin expression in different neurons, so that only a subset are recruited strongly at any given light intensity. To examine differences in expression pattern, we counted neurons in histological sections from both the viral (PV-AAV-Chronos) and transgenic (PV-ReaChR) animals. For these comparisons, the Cre line, expressing Cre in PV neurons, was held constant. We observed both effects. First, viral expression created some cells with strong, and some cells with weak, opsin expression (Figure 6B–C). Second, viral expression yielded opsin in about half as many cells as transgenic expression (Figure 6G); positive-opsin cell percentage significantly different between viral and transgenic cases, both observers and both thresholds, p<0.05, χ² test). The differences in the number of opsin-expressing cells roughly matched our model predictions (60% of neurons stimulated, Figure 6E).

Taken together, these data and our analysis support the idea that V1 and other areas of the mouse cortex are strongly coupled and operate as an ISN. Yet whether an average paradoxical effect is seen depends on the number of inhibitory cells stimulated. The heterogeneous responses we saw in PV-Chronos experiments (about half of inhibitory cells paradoxical, half non-paradoxical, Figure 6D,H,L, also may in part explain why paradoxical inhibitory suppression,and thus ISN operation, has not been more widely reported in optogenetic PV-stimulation experiments (also see Discussion).

To this point all the neurophysiological data we have reported is from the upper layers (L2/3 and 4) of the cortex, recorded within 400 µm of the cortical surface. We focused on the superficial layers because the blue light we used to activate ChR2 and Chronos does not penetrate more than a few hundred microns into the tissue (Yona et al., 2016), so the inhibitory neurons that receive direct optogenetic input are those in the superficial layers. However, blue light delivered to the cortical surface to stimulate inhibitory neurons has been seen to suppress activity across cortical layers, presumably due to polysynaptic effects (Li et al., 2019). Because we recorded data using silicon probes that span most of the depth of the cortex, we could also assess whether light delivered to the top of the cortex also produced paradoxical suppression in deeper inhibitory neurons.

We examined units (Figure 7) recorded ≥ 500 µm from the cortical surface (Materials and methods). In our multi-area recordings from superficial layers (e.g. Figure 5), we classified inhibitory cells in different brain areas based on increases in inhibitory response at strong light intensities, when excitatory cells are silent or largely suppressed. But in deep recordings, presumably due to attenuation of stimulation light from scattering and absorption, at our highest stimulation intensities we did not always see clear mean increases in inhibitory firing rates (e.g. Figure 7B). Therefore, for these deep recordings we classified neurons as inhibitory based on waveform width. As in superficial layers, we found bimodal distributions of waveform width in our deep-layer data (Figure 7A–C, insets). In our upper-layer V1 data, we compared classification of V1 units via pharmacology, waveform width, and responses at high power (Figure 2—figure supplement 3). Based on this comparison, in these deep-layer data, we do not expect excitatory cells to be classified as inhibitory (i.e. few or no excitatory units have narrow waveforms), even though some inhibitory cells with wider waveforms will be missed with this approach (i.e. some inhibitory units have wide waveforms).

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Examining cells recorded from deeper layers, we found that V1, motor, and somatosensory areas all showed initial paradoxical responses (Figure 7). However, the maximal suppression of inhibitory and excitatory cells was not as strong as in superficial layers (Figure 2, Figure 5). (Supporting the idea that this weaker effect of stimulation is due to attenuation of blue light, paradoxical effects in deep layers were smaller in these data than when using red light to stimulate ReaChR in V1 PV neurons, Figure 7—figure supplement 1.) In all these cases (Figure 7A–C), the mean population inhibitory responses showed initial suppression.

Thus, the deep-layer data shows no evidence for a different pattern of responses than seen in the upper layers – deep-layer inhibitory cells show paradoxical suppression to excitatory stimulation. However, because light applied to the cortical surface affects the superficial layer inhibitory neurons most directly, we do not make the claim that these data are final proof that deep layers operate in the ISN regime. On the other hand, the deep-layer data do provide evidence against alternative, non-ISN, models for L2/3 responses, such as the potential alternative where deeper inhibitory cells increase their firing and inhibit L2/3 cells. Instead, we find that most inhibitory cells we record, across layers, are paradoxically suppressed by stimulation.

Theoretical studies have pointed out that a network can switch from non-ISN to ISN when external inputs are increased (Ahmadian et al., 2013). If the cortical network did transition out of the ISN state at lower activity levels, this would be computationally important, as the way inputs sum can change depending on whether the network operates as an ISN or not (Ahmadian et al., 2013; Rubin et al., 2015). Our results above (Figure 2, Figure 3) show that superficial layers of mouse V1 operate as an ISN at rest (i.e. without sensory stimulation), and also show the transition from ISN to non-ISN is below the level of spontaneous activity, as network activity is decreased by optogenetic stimulation of inhibitory neurons. (The transition point is where the slope of I rates vs stimulation intensity switches from being negative to positive.) However, we wished to determine whether a brain state with lower activity levels, as that produced by light anesthesia, might result in a transition to a non-ISN regime.

Under light isoflurane anesthesia, we found no evidence of a transition, and instead found that paradoxical inhibitory suppression was maintained (Figure 8A). At 0.25% isoflurane (a low level, as surgical levels are often 1.0% and above), spontaneous firing rates of excitatory neurons are reduced (Figure 8C; mean 6.4 spk/s reduced to 3.5 spk/s, $p<0.02$, Wilcoxon signed-rank test; inhibitory neurons’ rates show a negative trend but are not statistically different, 14.5 spk/s awake to 11.6 spk/s anesth., $p<0.10$, Wilcoxon signed-rank tesk). Thus, the network changes induced by anesthesia do not cause the network to transition out of the ISN state. At this low level of anesthesia, we did not observe prominent up and down state slow oscillations. In one experiment, we used a higher level of anesthesia (0.5% isoflurane, Figure 8B), yielding even lower firing rates but still preserving the paradoxical effect. Further confirming the robustness of coupling to changes with anesthesia, the distribution of response slopes is roughly unchanged (Figure 8C), suggesting the network is far from a transition into a non-ISN state. Anesthesia thus preserves the paradoxical inhibitory response, leaving the network still an ISN.

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Our data and theoretical results provide the strongest evidence currently available to rule out non-ISN interpretations. Three prominent features of our data supporting a strongly-coupled recurrent origin for the results are: (1) that inhibitory response dynamics match ISN predictions (Figure 3), (2) that an ISN model describes the data well and makes predictions verified in the data (Figure 4), and (3) that many inhibitory cells show paradoxical effects at low stimulation intensity but increase their firing rate at higher stimulation intensity (e.g. Figure 2, Figure 5).

These observations allow us to rule out several potential alternatives to the ISN explanation for the paradoxical effects we observed. One potential alternative might be a model with multiple inhibitory populations and weak recurrent coupling, perhaps driven to fire spontaneously by strong external input such as thalamic or corticocortical input. In principle, paradoxical inhibitory responses could be produced in such a non-ISN model via feedforward, inhibitory-onto-inhibitory, interactions. Specifically, in this alternative, stimulation of one inhibitory subclass (e.g. PV cells) would inhibit a second set of inhibitory cells whose firing rate would decrease (e.g. somatostatin or SOM cells; the same explanation could hold with the PV and SOM classes swapped). However, in a feedforward inhibitory model, there would be little reason to predict inhibitory neurons to change from decreasing to increasing rate as stimulation intensity increases. ISN models, on the other hand, are predicted to transition into a non-ISN state for strong inhibitory stimulation where inhibitory cells begin to increase their firing, as our data show (e.g. Figure 2L).

The large fraction of inhibitory cells that we found to be suppressed (paradoxical) at low stimulation intensity (Figure 2J–K) further argues against alternative models with paradoxical effects created by multiple differentially-responding inhibitory cell populations. While excitatory opsins expressed in inhibitory neurons can sometimes cause synaptic release at terminals (Zhao et al., 2011; Babl et al., 2019), we found that many inhibitory cells fired in response to optogenetic stimulation in the presence of E and I blockers, showing that our stimulation did produce substantial neural spiking. Further, ISN-generated paradoxical effects are network effects that rely on inhibitory cells affecting other local neurons, so even if stimulation did cause terminal release, this should not affect our conclusions. Finally, ISN models predict, and we observed, that the transition point from inhibitory suppression to excitation occurs when excitatory cells are largely suppressed. It might be possible for that to occur in a non-ISN case, such as in the feedforward scenario, but it would require fine-tuning of parameters, and our ISN model’s accurate description of the pharmacology data makes such fine tuning even more unlikely.

Other non-ISN scenarios would also rely on feedforward inhibitory effects made unlikely by our data. For example, alternative models could have feedforward inhibition arising from rare, powerful inhibitory cells, perhaps so rare that we could not record them, or input from deeper-layer inhibitory cells. These feedforward inhibition scenarios are also argued against by the observations outlined above (e.g. the transition from suppression to excitation showed by our recorded inhibitory cells, etc.). And our recordings across depth showed that a majority of inhibitory cells’ firing rates were paradoxically suppressed by stimulation, independent of the depth of recording, making it unlikely that our effects were produced by feedforward inhibition arising from a deeper cortical layer. The ISN model remains a consistent explanation of all the data we observed.

In the last decade, multiple experiments have performed optogenetic stimulation of cortical interneurons, but these studies have yielded mixed results on inhibition stabilization. In layer 2/3 of mouse V1, optogenetic excitation of PV cells with viral expression was shown to generate non-paradoxical modulation of inhibition (Atallah et al., 2012). In mouse auditory cortex (A1), in contrast, Kato et al., 2017 found that suppression of inhibitory cells with viral ArchT produced paradoxical effects in intracellular currents. ISN models make similar predictions for somatic intracellular currents as for the firing rates we measured: if a network is strongly coupled and thus inhibition-stabilized, stimulating all inhibitory neurons will yield paradoxical changes in inhibitory current, as well as a transient increase in inhibitory current before suppression (Ozeki et al., 2009; Litwin-Kumar et al., 2016). Another A1 study observed mixed paradoxical effects but linked them to a non-ISN mechanism. Moore et al., 2018 found that stimulation of PV cells using viral transfection produced paradoxical effects in some PV cells and not others. and they reported evidence for feedforward, non-ISN, mechanisms in L2/3 via L4 inhibitory input. In somatosensory and motor cortex, a recent study observed paradoxical suppression for PV subclass stimulation in some layers and not others (Mahrach et al., 2020). Finally, many experiments have used excitation of inhibitory cells (e.g. Glickfeld et al., 2013) to suppress excitatory activity, without reporting paradoxical inhibitory effects, although a recent survey of such methods does report paradoxical effects (Li et al., 2019) in somatosensory and motor cortex.

Our results, combining theoretical analyses with a range of experimental conditions, suggest some explanations why paradoxical inhibitory suppression has not been previously widely reported. First, it appears a significant fraction of inhibitory cells must be stimulated to produce large paradoxical changes in firing rate (Sadeh et al., 2017). Stimulating PV cells with viral methods (which can yield subsets of cells with weaker or no expression; see also Sadeh et al., 2017; Gutnisky et al., 2017 for a similar explanation for non-paradoxical effects with viral expression), produced in our hands mixed effects on single cells and weaker mean paradoxical effects (Figure 6). Further, stimulating subtypes of inhibitory cells even with transgenic expression can produce different inhibitory responses compared to stimulating all inhibitory neurons (Mahrach et al., 2020). Another factor which can, in principle, generate non-paradoxical effects even when the network is an ISN is reducing the area of tissue illuminated by stimulation light. This reduces the number of inhibitory cells stimulated and therefore produces a situation similar to stimulating a subset of network inhibitory cells. However, this effect of reducing optogenetic light area seems to apply mainly for fairly small optogenetic light spots. Sadeh et al., 2017 found via modeling that paradoxical effects were diminished when illumination spot sizes fell to around 100 µm, well below the 500+ µm diameter spots we used here to reveal ISN-related paradoxical suppression. This effect of illumination area may, however, impact experiments using optical fibers, which can have small diameters. Beyond the fraction of neurons stimulated, a second explanation for differences, especially in experiments using extracellular recording, is that it can be difficult to definitively identify inhibitory cells. Some inhibitory cells’ waveforms are broad (Figure 2, Figure 2—figure supplement 8), and in an ISN, excitatory and inhibitory cells’ firing rate changes are often similar (inhibitory neurons may only show increased activity for a few milliseconds before paradoxical suppression begins). Third, for strong enough drive to inhibitory neurons, ISNs transition to a non-ISN state. In this non-ISN state with low excitatory activity, tracking of excitatory fluctuations is not needed to stabilize the network, and some past experiments may have used stimulation large enough for parts of the network to enter into this second phase of activity, hiding the signature of inhibition stabilization.

While paradoxical effects in a single inhibitory subclass do not imply the overall network is inhibition stabilized (Figure 6E; Mahrach et al., 2020), PV neurons may well be the principal providers of stabilizing input. In an ISN, strong bidirectional E-I synaptic coupling allows inhibitory cells to stabilize an unstable excitatory network, by tracking and responding to changes in local excitation. Thus, one major parameter that would predict which inhibitory cells would show paradoxical effects is the strength of their synaptic influence on other network neurons. PV neurons receive input from many diverse local excitatory cells (Bock et al., 2011) show responses that reflect an average of local excitatory responses (Sohya et al., 2007; Kerlin et al., 2010), cause network instability when strongly suppressed (an effect which does not result from SOM suppression) (Veit et al., 2017), and target virtually all nearby excitatory cells with strong peri-somatic synapses (Fino and Yuste, 2011; Packer and Yuste, 2011). Other cell types of the diverse inhibitory classes in the cortex (e.g. SOM or VIP+ interneurons; Tremblay et al., 2016) also could contribute to stabilization (Sadeh et al., 2017), though if their effect on the whole local network is less strong than PV cells, they may be unable to produce paradoxical suppression when stimulated alone.

Several influential models of cortical function include inhibition-stabilized parameter regimes, and our data provide new constraints on such models. The ‘balanced network’ model (Amit and Brunel, 1997; van Vreeswijk and Sompolinsky, 1996) predicts that both excitatory and inhibitory inputs to single neurons should be large but approximately cancel each other, leading to a balance between excitation and inhibition. This scenario accounts in a parsimonious way for multiple ubiquitous properties of activity in cortex, such as the highly irregular nature of neuronal firing, and the broad distributions of firing rates across neurons. Balanced network models, however, can vary in their recurrent coupling strength, and thus the size of the total excitatory and total inhibitory currents. The analytical work of van Vreeswijk and Sompolinsky (van Vreeswijk and Sompolinsky, 1996; van Vreeswijk and Sompolinsky, 1998) was performed in the limit of very large numbers of synaptic inputs per cell ($K$), leading to very large E and I currents (whose leading order in $K$ cancel each other — that is, the sum of the E and I currents is small while the sum’s variability can be substantial). Other balanced network models (Amit and Brunel, 1997; Brunel, 2000) used finite, moderate-to-large coupling strength. The network parameters that best fit our data produce moderately large (i.e. within an order of magnitude of threshold) excitatory and inhibitory inputs (Figure 4), which are on the same order of magnitude as the ones used in Brunel, 2000. This moderate recurrent coupling, or ’loose balance’, is also consistent with the SSN model (for review, see Ahmadian and Miller, 2019).

The SSN also makes a specific prediction about the ISN regime that our data constrains. The SSN predicts that cortical networks are inhibition-stabilized for strong inputs, but not an ISN for sufficiently weak inputs. Thus, the SSN predicts a transition between two operating regimes as network activity increases. Finding this transition point is important for understanding computation, as the two regimes show different (supralinear or sublinear) modes of input summation. Prior ISN studies (Ozeki et al., 2009) have left open whether this transition occurs only with sensory stimulation. A study of responses to two combined sensory stimuli (Britten and Heuer, 1999) found a linear-to-sublinear transition for increasing strength (contrast) of visual stimuli, but did not resolve whether the non-ISN to ISN transition was above or below the level of spontaneous activity. Our data do resolve this, showing several cortical networks are in the ISN state even without sensory stimulation. But a second question our data address is whether during spontaneous activity the network is on the edge of this transition point, or well into the ISN regime. Our work gives two pieces of evidence that without sensory activity the network is far above a transition into a non-ISN state: first, excitatory rates must be substantially suppressed before inhibitory responses switch from paradoxical suppression to firing rate increases (e.g. Figure 2E), and second, under light anesthesia, we find network activity is lowered but ISN behavior is preserved. Therefore, our data provides support for the idea that cortical areas generally operate above this transition point, and that the weakly-coupled, non-ISN regime predicted by the SSN is not a regime mouse cortex commonly enters during normal processing in awake behavior.

Together, the results reported here suggest that inhibition stabilization is a ubiquitous property of cortical networks. This data is consistent with a network that is in the strong coupling regime, but is not extremely strongly coupled as some balanced network models would predict (Amit and Brunel, 1997; van Vreeswijk and Sompolinsky, 1996). It is tempting to speculate that, while strong coupling allows the network to perform non-trivial computations on its inputs, increasing coupling further might not be optimal for several reasons. First, in the strong coupling limit, only linear network responses are available to a balanced network (unless synaptic non-linearities are included, Mongillo et al., 2012), while networks with moderate coupling can combine inputs in a non-linear fashion (Ahmadian et al., 2013). Second, maintaining strong connections might be metabolically expensive, so that a moderately-coupled network might represent an ideal compromise for cortical computations.

To analyze the stability of the system, we compute the eigenvalues of the dynamical matrix defined by Equation 1. For the linear transfer function we find

with $r=\frac{{\tau }_E{a}_I{W}_{IE}}{{\tau }_I{a}_E{W}_{EI}}$ This ratio cannot be inferred from the data since ${\tau }_{E,I}$ do not appear in the static solution. We thus determined ${\tau }_E$ and ${\tau }_I$ (shown in Figure 4E) from the response dynamics (Figure 4D).

To understand how the network response to PV stimulations depends on its structure, here we analyze a three-population model (Figure 6A) with one excitatory and two inhibitory populations, referred hereafter as $E$ (pyramidal cells), $P$ (ChRonos-expressing PV cells), and $I$ (non-expressing PV , SOM, VIP…). The network dynamics is described by

The model predictions in Figure 6 have been obtained from Equation 10 using

where $frac\in (0,1)$ represents the fraction of ChRonos-expressing PV cells, $W_{AB}^{VGAT}$ and $I_A^{VGAT}$ represent the parameters of Equation 8, obtained using the V1 data from the VGAT-ChR2 mouse line.

Using the rectified linear transfer function of Equation 2, and assuming $a_A=1$ to simplify expressions, the equilibrium response to stimulation of the $P$ population is given by

with

Equation 12 shows that, both for $W_{EE}<1$ and $W_{EE}>1$, depending on the network connectivity, the ratio on the r.h.s. can be either positive or negative. It follows that, in a network with two inhibitory populations, an unstable excitatory subnetwork does not imply paradoxical suppression when one of the populations is stimulated. This can be seen more easily in a simplified model in which connectivity depends only on the identity of the presynaptic unit. In particular, using $W_{EE}=W_{IE}=W_{PE}=W$, $W_{EI}=W_{II}=W_{PI}=k_{I}W$, and $W_{EP}=W_{IP}=W_{PP}=k_{P}W$, Equation 12 becomes

In the large $W$ limit, in which the excitatory population is unstable without inhibition, the network shows the paradoxical effect only if $k_I<1$ and $k_P>1-k_I$. (Note that, for the network to be stable in this condition, the matrix of the coefficients of Equation 10, linearized around the fixed point, must have eigenvalues with a negative real part, a requirement which also involves the magnitude of ${\tau }_E$, ${\tau }_I$, and ${\tau }_P$.)

In the main text, we have shown that cortical response to optogenetic stimulations is consistent with the network being stabilized by inhibition. Our conclusion was based on theoretical work linking paradoxical inhibitory responses to inhibitory stabilization of unstable excitatory activity (Tsodyks et al., 1997). This work shows that network stability is determined by two variables, strength of recurrent connections and time scale of neuronal responses, which are assumed to be constant in time. Short term plasticity (STP) (Tsodyks and Markram, 1997; Markram et al., 1998) could potentially modify the link between paradoxical response and inhibition stabilization, as it dynamically modifies synaptic efficacy, but its effects on network stability have not been analyzed yet. In this section, we show that paradoxical inhibitory response implies inhibitory stabilization also when STP is taken into account. However, the reverse is no longer guaranteed to be true, as inhibition stabilization no longer necessarily implies paradoxical response. Therefore, the conclusions of the main text remain valid also in a modeling framework which includes STP.

To investigate the effects of STP on the relationship between paradoxical effect and inhibitory stabilization, we use the phenomenological description of STP developed in Tsodyks et al., 1997; Markram et al., 1998; Tsodyks et al., 1998. This framework is accurate enough to capture quantitatively both short term depression (STD) and short term facilitation (STF), and is simple enough to allow analytical investigation. The state of each synapse is described by two variables: the fraction $x$ ($0\le x\le 1$) of vesicles available for release, and the fraction $u$ ($0\le u\le 1$) of available vesicles that release neurotransmitter after a presynaptic spike (release probability). In the model, a presynaptic spike arriving at time $t$ opens calcium channels in the presynaptic terminal, which generates an increases the value of $u(t^{-})$ by an amount $U[1-u(t^{-})]$ ($0<U<1$ is a fixed parameter) and produces the release of a fraction $u(t^{+})x(t^{-})$ of vesicles. After the spike, channels close with a time constant ${\tau }_F$ and vesicles recover with a time constant ${\tau }_D$. As shown in Tsodyks et al., 1998, when presynaptic spikes are Poisson-distributed with rate $r$, the dynamics of $x$ and $u$ is given by

In this model, synaptic efficacy and postsynaptic current are proportional to $\mathrm{\Omega }x$ and $\mathrm{\Omega }xr$, respectively. The modulation of synaptic efficacy by the presynaptic rate can be understood as follows. When $u$ is fixed, $x$ and the synaptic efficacy decrease with $r$, leading to synaptic depression. When $x$ is fixed, the increase in $u$ with $r$ augments synaptic efficacy and leads to synaptic facilitation. When both $u$ and $x$ are allowed to vary, facilitation dominates at low $r$ and depression dominates at high $r$ (Markram et al., 1998; Tsodyks et al., 1998). A stationary input rate $r(t)=r_0$ gives

and, since $U<1$, produces a synaptic current which increases monotonically with $r_0$; this monotonic increase will be important in the upcoming sections.

In what follows, we include the description of STP discussed above in our network model (Equation 1 of the main text) by adding variables $x_{AB}$ and $u_{AB}$ for all projections $W_{AB}$, so that the synaptic current $W_{AB}{r}_B$ becomes $x_{AB}{u}_{AB}{W}_{AB}{r}_B$. Therefore, the network dynamics is given by

Here $A,B\in [E,I]$ and, to simplify the analysis, we assumed threshold-linear single neuron transfer function.

In this section, we investigate conditions for paradoxical inhibitory response in networks with STP; the role of inhibition in stabilizing activity is discussed in the next section.

As discussed in Tsodyks et al., 1997; Ozeki et al., 2009 and in Figure 1—figure supplement 1, paradoxical inhibitory response can be studied using the $r_E$ and $r_I$ nullclines of the model in the $r_E/r_I$ plane. The difference with those previous studies is that we have additional dynamical variables for synaptic strengths, and these should be set to their equilibrium values. Using $d{x}_{AB}/dt=0$ and $d{u}_{AB}/dt=0$, we express ${\mathrm{\Omega }}_{AB}$ and $x_{AB}$ as a function of $r_B$ (as done in Equation A2) and define

Since each term $x_{AB}(r_B)u_{AB}(r_B)W_{AB}{r}_B$ is a monotonically increasing function of the presynaptic rate $r_B$, $f_{E,I}$ can be inverted and, in the region of rates $r_{E,I}>0$, we can write the excitatory and inhibitory nullclines as

Using the fact that $f_I^{-1}[.]$ is an increasing function of its argument, we obtain that the inhibitory nullcline $r_I^{inhibitory}(r_E,L)$ increases with $r_E$ and $L$. It follows that the response of the inhibitory population is paradoxical, that is the stationary value of $r_I$ decreases with $L$, only if the excitatory nullcline $r_I^{excitatory}(r_E)$ has positive slope, that is if

Equation A6 must be satisfied by any network of excitatory and inhibitory neurons with STP which shows paradoxical effect. In the next section, we show that this condition is violated in networks which are stable with fixed inhibition.

To conclude, we note that the condition of Equation A6 is only necessary but not sufficient, and in order for the paradoxical response to emerge, one also needs ${\displaystyle d{f}_E^{-1}(r_E)/d{r}_E<d{f}_I^{-1}(r_E)/d{r}_E}$. Without STP, this condition is met any time the network is dynamically stable (Ozeki et al., 2009). With STP, proving an analogous result is more complicated, as it involves the stability of Equation A3, that is of a ten-dimensional dynamical system.

In this section, we investigate the relation between paradoxical effect and inhibitory stabilization in networks with STP. As discussed above, plasticity can be included in all the synapses of the model (i.e. at all projections $W_{AB}$ in Equation 1) but only STP at recurrent excitatory synapses ($W_{EE}$) can influence the role of inhibition in the stabilization of dynamics. In fact, a network is said to be inhibition stabilized if instabilities emerge once inhibitory rates are forced to be constant in time (Tsodyks et al., 1997; Ozeki et al., 2009). This condition is not affected by plasticity involving inhibitory neurons (both pre- and post- synaptically), therefore we can limit our investigation to STP in $W_{EE}$. In what follows, first we study analytically networks in which only one between STD or STF is present, then we investigate numerically networks in which both STD and STF are included simultaneously. In all these cases, we find that paradoxical inhibitory response implies that the network is unstable with fixed inhibition. Therefore, if a stable network shows paradoxical response, the excitatory network must be unstable on its own and has to be stabilized by inhibition. It follows that the conclusions derived in the main text about cortical networks remain valid also when STP is taken into account in the analysis.

Effects of STD

With STD in $W_{EE}$ synapses, the network dynamics is given by

(A7) ${\displaystyle \{\begin{array}{ll}{\tau }_E\frac{d{r}_E}{dt}& =-r_E+{\left[x{W}_{EE}{r}_E-W_{EI}{r}_I+I_{EX}-x_{0E}\right]}_{+}\\ {\tau }_I\frac{d{r}_I}{dt}& =-r_I+{\left[W_{IE}{r}_E-W_{II}{r}_I+I_{IX}+\lambda L-x_{0I}\right]}_{+}\\ \frac{dx}{dt}& =\frac{1-x}{{\tau }_D}-xU{r}_E.\end{array}}$

The effect of STD on the stationary response can be seen by looking at the $r_E$ and $r_I$ nullclines of the system (Appendix 1—figure 1A) in the $r_E/r_I$ plane, setting $x$ to its steady-state value. Expressing $x$ as a function of $r_E$, we find

(A8) ${\displaystyle \{\begin{array}{ll}{r}_I^{excitatory}(r_E)& =\frac{1}{W_{EI}}\left[\left(\frac{W_{EE}}{1+{\tau }_{D}U{r}_E}-1\right)r_E+I_{EX}-x_{0E}\right],\\ r_I^{inhibitory}(r_E,L)& =\frac{1}{1+W_{II}}\left[W_{IE}{r}_E+I_{IX}+\lambda L-x_{0I}\right]\end{array}}$

The above expression shows that STD decreases the self-excitation of excitatory cells as the excitatory rate increases. The stationary solutions of the Equation A7 (here referred to as $x^{*}$, $r_E^{*}$, $r_I^{*}$) are the points at which the two nullclines of Equation A8 cross. As discussed in general terms in the previous section, increasing $L$ in Equation A8 moves the inhibitory nullcline upward and Equation A6 provides a necessary condition for the paradoxical inhibitory response to emerge; this condition is met any time that

(A9) $x^{*}>\frac{1}{\sqrt{W_{EE}}}.$

If Equation A9 is not satisfied, the network response is not paradoxical.

The stability of the stationary solutions is found with a standard perturbation analysis around the stationary state. With frozen inhibition, we can write perturbation around the stationary solutions as $x=x^{*}+{\delta }_x$ and $r_E=r_E^{*}+{\delta }_E$. In the case of $r_E^{*}>0$, the dynamics of these perturbations is given by

(A10) ${\displaystyle \{\begin{array}{ll}{\tau }_E\frac{d{\delta }_E}{dt}& =\left(-1+x^{\ast }{W}_{EE}\right){\delta }_E+W_{EE}{r}_E^{\ast }{\delta }_x\\ \frac{d{\delta }_x}{dt}& =-x^{\ast }U{\delta }_E-\left(\frac{1}{{\tau }_D}+U{r}_E^{\ast }\right){\delta }_x\end{array}.}$

Equation A10 shows that, with fixed inhibition and fixed STD (${\delta }_x=0$), the excitatory network is unstable if

(A11) $x^{*}{W}_{EE}>1$

which, since $0<x<1$, also implies that $W_{EE}>1$. With fixed inhibition and dynamic STD, the dynamics is stable if

(A12) $x^{*}<\min (\frac{1}{\sqrt{W_{EE}}},\frac{1+\sqrt{1+4\frac{{\tau }_E}{{\tau }_D}{W}_{EE}}}{2W_{EE}})$

If Equation A12 is verified, the network is stable without dynamic inhibition. Since this condition cannot be satisfied simultaneously with Equation A9, stabilization in a network with STD in $W_{EE}$ synapses which shows paradoxical effect has to be realized by inhibition.

Appendix 1—figure 1

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In the main text, using rate models without input noise, we have shown that experimental data recorded in V1 are consistent with the network operating in the inhibition stabilized regime, with mean excitatory and inhibitory inputs larger than threshold and total input of order threshold. In this section we show that similar results are obtained using a rate model which features input noise and a more biologically accurate transfer function. Furthermore, this analysis shows that inferred parameters agree with previously measured network properties, and that data are consistent with the loose balance regime (Ahmadian and Miller, 2019) and noise driven firing.

We use a network of excitatory and inhibitory neurons analogous to the one described in the main text, but with a single neuron transfer function matching that of leaky integrate and fire models. In the network, each neuron in population $A=[E,I]$ receives a Gaussian distributed input, of mean ${\mu }_A$ and variance ${\sigma }_A^2$, which is produced by recurrent and feedforward interactions. As shown in Brunel, 2000, population firing rates are found by solving:

where ${\tau }_{rp}$ is the single neuron refractory period, while $u_{max,A}$ and $u_{min,A}$ are the distance from spiking threshold $\theta$ and reset $V_r$ of the mean input ${\mu }_A$ measured in units of input noise ${\sigma }_A$, that is

In the model, using the same notation as in the linear model of the main text, means are given by

we assume noise amplitude to be fixed to ${\sigma }_E={\sigma }_I=5\text{mV}$ (Haider et al., 2013). Other model parameters are: ${\displaystyle {\tau }_{rp}=2\mathrm{m}\mathrm{s}}$, ${\displaystyle {\tau }_E=20\mathrm{m}\mathrm{s}}$, ${\displaystyle {\tau }_I=10\mathrm{m}\mathrm{s}}$, ${\displaystyle \theta =20\mathrm{m}\mathrm{V}}$, ${\displaystyle V_r=10\mathrm{m}\mathrm{V}}$. This model has a total of 9 free parameters (four connectivity parameters $W_{AB}$, two feed-forward inputs $I_{AX}$, one laser efficacy $\lambda$, two blockers efficacy ${ϵ}_A$) which will be inferred from the data.

Results of the analysis

The best model description of the data, obtained minimizing Equation B4, is shown in Appendix 2—figure 1A. The corresponding model parameters are:

(B5) ${\displaystyle \begin{array}{l}{\displaystyle W_{EE}=2.65\text{mVs},W_{EI}=2.28\text{mVs},W_{IE}=3.03\text{mVs},W_{II}=2.50\text{mVs},}\\ {\displaystyle I_{EX}=20.19\text{mV},I_{IX}=19.49\text{mV},\lambda =3.38\text{mV},{ϵ}_E=0.55,{ϵ}_I=\mathrm{0.32.}}\end{array}}$

As in the main text, we used a bootstrap approach to estimate the precision with which model parameters can be inferred from the data; results are shown in Appendix 2—figure 1B. The analysis shows that distribution of inferred parameters are localized around the optimal solution described above. Moreover, the parameters of the model can be map to known biophysical quantities by noticing that $W_{EE}={\tau }_E{K}_{EE}{J}_{EE}$ (Brunel, 2000), where $K_{EE}$ and $J_{EE}$ are the average number of recurrent excitatory projection and their efficacy (analogous expression holds for the other elements of the matrix $W$). Assuming ${\displaystyle {\tau }_E=20\mathrm{m}\mathrm{s}}$ and $K_{EE}\approx {10}^2-{10}^3$, we obtain an estimate of ${\displaystyle J_{EE}\approx 0.1-1\mathrm{m}\mathrm{V}}$, which is consistent with direct biophysical measurements (Holmgren et al., 2003).

We computed the inputs into excitatory cells; contributions coming from recurrent inhibition ($-W_{EI}{r}_I$), recurrent excitation ($W_{EE}{r}_E$), and feed-forward excitation ($I_{EX}$) are shown in Appendix 2—figure 1C. The total input is found to be below threshold, at a distance of order one in unit of input noise. This result is consistent with the balanced state model (Amit and Brunel, 1997; van Vreeswijk and Sompolinsky, 1996; van Vreeswijk and Sompolinsky, 1998), where firing is driven by input noise. However, unlike what is expected by the balanced state model, the different components of the input (e.g. feed-forward excitation) are of order threshold, consistent with the loose balance regime recently suggested to underlie cortical dynamics (Ahmadian and Miller, 2019).

In the model, the strength of self-excitation of the excitatory population is obtained from Equation B1 as

(B6) ${𝒲}_{EE}=\frac{d}{d{r}_E}{\left[{\tau }_{rp}+{\tau }_E\sqrt{\pi }{\int }_{u_{min,E}}^{u_{max,E}}{e}^{u^2}\left(1+\text{erf}(u)\right)𝑑u\right]}^{-1},$

where the integral is evaluated at the inferred baseline activity. The quantity ${𝒲}_{EE}$ is the generalization of the parameter $W_{EE}$ analyzed in the main text, which takes into account the amplification of coupling strength due to transfer function nonlinearities. Values of ${𝒲}_{EE}$ inferred with data bootstrap are shown in Appendix 2—figure 1D. Results are consistent with ${𝒲}_{EE}$ larger than one, indicating that indeed the excitatory subnetwork is unstable without inhibition, and in the range 5–15, consistent with the result obtained in the main text using a linear model.

To summarize, the analysis derived in this section shows that results obtained in the main text are preserved when a more biologically accurate, spiking, model is used to describe the data. Moreover, this approach also shows that inferred parameters are consistent with known biophysical properties of cortical networks.

Appendix 2—figure 1

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