Our understanding of sensory processing in the cortex is moving beyond simple stimulus-response analysis of single cells, and the role of networks of neurons is becoming increasingly more important in this process. Such network effects are, however, difficult to study by recording single cells in isolation or even by measuring the activity of populations of neurons, if our interaction with the circuitry is only observational. A more promising approach is to record and perturb the activity of neuronal populations at the same time. However, theory and computational modelling are needed to guide such endeavours, due to the complexity of the high-dimensional space in which experiments can be performed.

Experimentally, optical stimulation of neurons has emerged in recent years as a powerful technique to perturb neuronal activity in order to study brain function and dysfunction (Boyden et al., 2005; Fenno et al., 2011; Yizhar et al., 2011; Boyden, 2015). Optogenetic perturbations have provided us with an effective means to interrogate neuronal connectivity and dynamics in action. Despite these advances, typical perturbation paradigms are still mainly limited to simple profiles of activation or deactivation, hindering a more precise stimulation of neuronal activity which would be necessary to study specific brain functions. Although increasing number of perturbation studies are targeting specific subgroups of excitatory or inhibitory neurons with genetic markers, within each subpopulation still nonspecific perturbation patterns (either in terms of their spatiotemporal profile or functional features) are primarily used to interrogate the circuitry (Atallah et al., 2012; Lien and Scanziani, 2013; Pfeffer et al., 2013; Reinhold et al., 2015; Kato et al., 2017). However, recent advances in optogenetics are making a closed-loop, all-optical interrogation of neuronal circuitry, where neuronal activity can be measured and perturbed simultaneously by optical tools, more achievable (Emiliani et al., 2015; Ronzitti et al., 2017; Zhang et al., 2018), hence paving the way for more specific, patterned perturbations. Several recent studies have in fact used such patterned perturbations to study cortical function with more precision, for example by recording neurons involved in specific tasks (e.g. natural visual processing or memory tasks) and reactivating them in order to instigate or suppress the relevant behaviour (Chernov et al., 2018; Carrillo-Reid et al., 2019; Marshel et al., 2019; Russell, 2019; Tran et al., 2019).

Optogenetic techniques have particularly been exploited to cast light on the regime of operation of cortical networks, especially regarding the role of feedforward versus recurrent pathways. A purely feedforward regime of information processing relies on the transmission of information between layers, while a recurrent regime adds an extra level of processing within each layer. The actual architecture of the neocortex seems to favour the latter, as the recurrent connections outnumber the feedforward inputs (Peters and Payne, 1993). Strong recurrent connections between principal cells can enable them to perform many useful computations (such as signal amplification, denoising and pattern completion), but this can also lead to runaway excitation and hence unstable dynamics in the brain. It has therefore been suggested that neocortical networks operate in an inhibition-stabilized regime (inhibitory-stabilized networks, or ISNs), where the high gain of cortical recurrent excitation is stabilized, or balanced, by a potent recurrent inhibition (van Vreeswijk and Sompolinsky, 1996; van Vreeswijk and Sompolinsky, 1998; Tsodyks et al., 1997; Ozeki et al., 2009). In other words, under ISN regimes of activity, cortical networks would not be stable without inhibition and therefore undergo runaway activity.

The long-lasting question about whether cortical networks operate as inhibitory-stabilized networks has been reconsidered in the wake of new optogenetic techniques. It has been probed by measuring the presence or absence of paradoxical responses of inhibitory neurons upon perturbation (Tsodyks et al., 1997; Ozeki et al., 2009). If the network is primarily feedforward driven, increasing the input to inhibitory neurons should lead to an intuitive increase in their responses. On the other hand, in ISNs, dominated by strong recurrent interaction of excitation and inhibition, increasing the input to inhibitory neurons decreases their activity, in a non-intuitive direction (hence ‘the paradoxical effect’). Intuitively, this is because, due to the strong recurrence, increasing the input to inhibitory neurons increases inhibition to excitatory neurons, which in turn leads to an overall decrease in the drive to inhibitory neurons. Although earlier studies failed to report the presence of such paradoxical effects in mouse visual cortex (Atallah et al., 2012), more recent optogenetics experiments were able to observe them in the visual cortex of awake mice (Adesnik et al., 2012) as well as in auditory, somatosensory, and motor cortices (Kato et al., 2017; Sanzeni, 2019); but see Mahrach et al., 2020).

The conflicting results in different experiments could partly be related to simplifications in the studies of ISNs. Like optogenetic manipulations, ISN properties too have mainly been studied for simplified scenarios, both experimentally and theoretically. For example, the original theoretical model of ISNs reduced the populations of excitatory and inhibitory neurons to single nodes in the analysis (Tsodyks et al., 1997). New modelling studies have started to account for more biological realism, for example by studying the effect of including multiple subtypes of inhibitory neurons (Litwin-Kumar et al., 2016) or nonlinear neuronal transfer functions (Ahmadian et al., 2013; Rubin et al., 2015; Hennequin et al., 2018). Another extension of the original model to include more biological details (Sadeh et al., 2017; Gleeson et al., 2019) concluded that observation of the paradoxical effect may depend on the size of perturbation of the inhibitory population; a result which was in fact corroborated by subsequent experimental studies (Li et al., 2019; Sanzeni, 2019). Further theoretical and experimental exploration of ISNs can, therefore, reveal new insights about the dynamical and functional properties of cortical networks.

An important aspect that has remained understudied in the study of ISNs is the functional specificity of connections, as reported in cortical networks. For instance, a highly specific connectivity between excitatory neurons has recently been reported in mouse visual cortex, whereby neurons with similar functional properties (e.g., in terms of their receptive fields) are strongly connected (Ko et al., 2011; Ko et al., 2013; Cossell et al., 2015; Lee et al., 2016). Simulation of networks with experimentally recorded weights of specific connections within the excitatory subnetwork, however, suggests that large-scale networks with such connectivity profiles would be unstable (Znamenskiy et al., 2018). This raises the question of which mechanisms are recruited by the cortex to stabilize the unstable dynamics of functional subnetworks. One possibility is that, similar to the balancing of the nonspecific dynamics by recurrent inhibition, the specific dimension of the excitatory connectivity is also balanced by a matching specific inhibition, therefore forming subnetworks of excitatory and inhibitory neurons, all coding for similar visual features. A recent study has in fact found evidence for the presence of such functionally specific excitatory-inhibitory (E→I and I→E) connectivity in visual networks of rodents (Znamenskiy et al., 2018), suggesting that E-I subnetworks may also exist in addition to excitatory subnetworks.

The possibility of specific inhibitory feedback balancing highly specific and potentially unstable excitatory subnetworks adds another dimension to inhibitory stabilization in cortical networks. We therefore aimed to study such ‘functionally specific ISNs’ (or specific ISNs in short) here, and to characterize the dynamical signatures and functional properties arising in these networks. We found that, across model networks at multiple scales and with different degrees of biological realism, specific ISNs can be identified by a ‘specific paradoxical effect’. That is, by exciting only the inhibitory neurons within a subnetwork where all neurons respond to similar stimuli, the activity of the inhibitory neurons in the subnetwork decreases. Moreover, the inhibitory neurons that receive more perturbation (more excitation) decrease their activity more. This dynamical fingerprint would only be observed if the network is probed with patterned perturbations aligned with functional properties of neurons (i.e., perturbations are delivered within a subnetwork rather than distributed across subnetworks). Moreover, this dynamical regime was concomitant with the emergence of spontaneous transitions between selective neuronal activity patterns, a feature that has been reported experimentally in different species (Kenet et al., 2003; Miller et al., 2014). Our study thus highlights the importance of patterned perturbation in probing the operating regime of cortex and its dynamical and functional properties.

By simulating computational models at different levels of realism, and through theoretical analysis of such networks, we showed here that specific inhibitory stabilized networks (specific ISNs, where unstable specific excitatory connectivity is balanced by specific inhibition) can show ‘specific paradoxical effects’. We further showed that, to reveal such specific ISNs, patterned perturbation of inhibition is needed. Our study thus extends classical inhibitory-stabilized networks, where only the uniform paradoxical effect was studied in response to nonspecific perturbations of inhibition. Our results suggest that specific ISNs show specific paradoxical effects in addition to the uniform paradoxical effect, provided they are probed by the proper patterns of perturbation, which would effectively stimulate the specific dynamics of the network. Absent such patterned perturbation, they would behave the same as nonspecific ISNs and hence their specific inhibitory stabilization would remain undetected.

Operation of cortical networks in specific ISN regimes adds another dimension to the computational capacity of the brain. It argues for the presence of specific directions of amplification of the feedforward input by the cortical dynamics, which is stabilized by a precisely tuned inhibition along the same direction. Our study suggests that, if such specific computational dimensions existed or emerged in cortical networks, they would be manifested by the specific paradoxical effects in response to patterned perturbations. While the classic paradoxical effect is observed by perturbing and measuring the average activity of inhibitory neurons in a non-specific manner (e.g. increased average activity when the perturbation decreases the average input), the specific paradoxical effect is observed in the relation between the residual input and output response of inhibitory neurons around the mean. We provided a quantitative metric for measuring this specific paradoxical effect, by evaluating the slope of the regression line fitted to response changes as a function of input perturbations (e.g., Figure 2C). Note that, as such, the measure would be robust to different rate changes that might be induced for neurons with different firing rates (e.g., cells with high firing rates will have their absolute rate decreased more than a cell firing sparsely in response to external inhibitory current injection). Such a dependence would not change the sign of the slope, which is important to infer the specific paradoxical effect.

In reality, it is difficult to measure connectivity for every individual circuit before optogenetically perturbing it. We therefore used feature or RF specificity as a proxy for connection weights, since it has been shown to be a good predictor of connectivity (Cossell et al., 2015). Our results, however, indicated that perturbation patterns based on one-dimensional features of neurons, for example their orientation selectivity, might not be proper protocols to probe specific ISNs as they would not effectively recruit the specific dynamics of the network. Considering the full RF similarity induced a better and more effective way of unravelling specific ISNs, and our spectral analysis revealed the underlying difference between the two perturbation protocols. Based on these results, we predict that experimental protocols relying on a single one-dimensional feature to deliver patterned perturbations may not be as potent to reveal specific ISNs, compared to patterned perturbations based on full RF similarity.

Receptive field mapping can, however, be equally costly, especially given that large-scale networks of inhibitory neurons need to be imaged and perturbed to see the specific ISN effects. We therefore explored the alternative possibility of using response similarity (assayed by how similarly neurons respond to an ensemble of stimuli) to deliver patterned perturbations, since response similarity has also been suggested as a good predictor of network connectivity (Cossell et al., 2015). Patterned perturbation based on response similarity would be possible by all-optical techniques which enable reading and writing the neuronal activity at the same time (Emiliani et al., 2015; Zhang et al., 2018). Our results revealed that specific paradoxical effects can be induced with such perturbation protocols, but the choice of the ensemble of stimuli used to probe the network responses could be crucial. Stimuli with statistics similar to RFs were more powerful in revealing specific ISNs, while wide-field gratings with fixed or diverse spatial frequencies did not prove to be as effective. While we generated stimuli with the same statistics as neuronal RFs in our simulations, we expect that natural images lead to the same results. Based on these results, we therefore suggest that the most effective and feasible protocol to detect specific ISNs would be to perturb the neuronal networks based on their response similarity in response to natural images, with read/write optogenetics tools.

In order to reveal the specific paradoxical effect, patterned perturbations can be induced by either decreasing or increasing the input to inhibitory neurons (positive or negative perturbations, respectively). Negative perturbations, however, were proven to be more powerful in our study (Figure 2—figure supplement 3). This is due to the fact that the ‘floor effect’ limits positive perturbations: in sparsely active networks (like the visual cortex), potent activation of inhibitory neurons can lead to strong rectification of the network activity. Such intensity-dependent optogenetics effects were in fact implicated as a source of conflict in the results of previous studies on the functional role of different subtypes of inhibitory neurons (Lee et al., 2012; Wilson et al., 2012; El-Boustani et al., 2014). If excitatory neurons are already active at very low rates due to balance of excitation and inhibition, strong positive perturbations of inhibition would lead to silencing of the network, which can obscure the paradoxical effect by decreasing the ‘effective’ size of the network involved in excitatory-inhibitory interaction. Decreasing the input to inhibitory neurons, on the other hand, increases the activity of excitatory neurons arbitrarily. Although the specific paradoxical effect can in principle be revealed with infinitesimally small positive perturbations, in realistic experimental conditions with significant measurement noise, such small perturbations would decrease the signal-to-raise ratio, hence limiting the statistical power to infer the presence of the negative slope of response changes versus input perturbations. Our results thus suggest that experimental protocols relying on negative patterned perturbations would be more powerful to reveal specific ISNs.

Another experimental constraint for probing the presence of specific ISNs in the cortex is related to the span of recording fields. As patterned perturbations are delivered with regard to functional properties of neurons (e.g. their visual RFs), the more similar the neurons are, the more difficult it becomes to assess the slope of their response changes to input perturbations. Therefore, if the window of imaging is too small and/or perturbations are delivered too locally, the visual RFs might become too similar, which would limit the range of input perturbations based on RF dissimilarity (x-axis in Figure 2C). This, in turn, impedes the assessment of the specific paradoxical effect, by limiting the statistical power of measurements to infer a significant negative slope. Wide-field perturbation/imaging is, therefore, necessary to obtain enough heterogeneity in RFs of inhibitory neurons needed for such assessment.

Our study here has also important implications for neuronal connectivity in the cortex. We argued that networks with strong specific excitatory connections, as reported in the visual cortex of rodents (Ko et al., 2013; Cossell et al., 2015), would be unstable in the absence of specific inhibitory stabilization. Although recent experimental findings (Znamenskiy et al., 2018) have corroborated this possibility, assessing the presence or absence of specific ISNs in the cortex can be considered as a crucial experiment to investigate the functional specificity of inhibitory connectivity. This may, in turn, shed light on the long-lasting debate over the tuning of inhibitory neurons and the specificity of their connections (Liu et al., 2011; Wilson et al., 2012; Runyan and Sur, 2013). Our results thus offer detailed predictions regarding the observation of specific inhibitory stabilization in cortical networks with specific EI connectivity, while highlighting important constraints on the design of experimental protocols to effectively probe them.

To unravel their dynamical and functional properties, cortical circuits have so far been mainly probed by simple perturbation protocols. Even when optogenetic perturbations are delivered specifically to certain neuronal subtypes, for example parvalbumin positive (PV+) or somatostatin-positive (SOM+) inhibitory cells, they typically stimulate the whole, or a random fraction of the population nonspecifically. Recent advances in optogenetic techniques are enabling us - at an unprecedented pace - to design perturbation protocols where a specific pattern (e.g. a specific sequence or spatiotemporal pattern of activity) is invoked in identified neurons (Emiliani et al., 2015; Ronzitti et al., 2017; Chernov et al., 2018; Zhang et al., 2018; Carrillo-Reid et al., 2019; Marshel et al., 2019; Russell, 2019; Tran et al., 2019). Our study here proposes that such technological advances are indeed necessary to reveal fundamental principles of cortical computation. Our results in fact suggest a further level of specificity (amplitude) as a prerequisite of patterned perturbation, since not only specific neurons should be perturbed to reveal specific ISNs, but different levels of perturbation (according to specific patterns) should be delivered to achieve effective recruitment of the specific mode. Such amplitude-specific patterns of perturbation, with different levels of perturbation at identified neurons, might be instrumental in effective interrogation of cortical networks during healthy and unhealthy function (e.g. in detecting specific epileptic modes of activity).

Patterned perturbations can also shed light on the controversy over the nature of the paradoxical effects. While the observation of paradoxical effects in several studies and across different cortices (Adesnik, 2017; Kato et al., 2017; Li et al., 2019; Sanzeni, 2019) has been mainly attributed to ISN dynamics, one needs to rule out the possibility of disinhibition underlying this effect, given the complex connectivity of multiple subtypes of inhibitory neurons (Pfeffer et al., 2013). A recent study has in fact suggested that cortical circuits can exhibit paradoxical effects because of disinhibitory loops rather than inhibition-stabilization mechanisms (Mahrach et al., 2020). Patterned perturbations of the type we have proposed in this work has the potential to distinguish between these two possibilities, for the following reason. By assuming a degree of specific connectivity between pyramidal cells (PC) and PV+ interneurons (Znamenskiy et al., 2018), specific inhibitory perturbations delivered to PV+ neurons would reveal the feature-specific ISN dynamics arising within PC-PV circuitry. Disinhibition, namely inhibition of PV+ interneurons by SOM+ interneurons, can only arise from PC→SOM→PV stream (Pfeffer et al., 2013). For disinhibition to be feature-specific, we need to have similar levels of specific connectivity not only between PC→SOM, but also SOM→PV connections. Although SOM+ cells have been reported to have higher orientation selectivity than PV+ neurons (Ma et al., 2010), they are known to be involved in larger scales of spatial integration, as opposed to local processing of PV+ cells (Adesnik et al., 2012; El-Boustani and Sur, 2014). It is, therefore, likely that they have larger RFs and mediating feature-specific surround suppression, more than being involved in local processing of visual information, that shapes the type of specific connectivity we explored here. One way of testing this experimentally would be to change the spatial extent of patterned perturbations from local to global and assay the paradoxical effect. This should shed light on whether a disinhibitory mechanism governed by SOM+ neurons is involved in the paradoxical effect. However, mapping the feature-specific connectivity of SOM+ cells with pyramidal cells would ultimately be needed to constrain the computational models for the role of different subtypes of inhibition in feature-specific ISNs.

Our study of specific ISNs revealed different functional properties of these networks. We found that networks with strong specific EI connectivity can undergo spontaneous transitions between highly selective ensembles, reminiscent of what has recently been observed in mouse visual cortex (Miller et al., 2014), and similar to what had previously been reported in other species with structured cortical maps (Kenet et al., 2003). Our modelling results showed that such fast spontaneous transitions only happen in specific networks and in the absence of tuned input to neurons. Note that, in comparison, classical attractor networks show very slow dynamics (Goldberg et al., 2004), and networks with only EE specific connectivity were incapable of reproducing such fast time scales of transition (Figure 6—figure supplement 1; Sadeh et al., 2015b). This would, in turn, render such models in their simple form inconsistent with the finding that sensory-evoked activity decays rapidly in recurrent circuits after silencing the feedforward input in visual cortex (Reinhold et al., 2015). Specific ISNs, on the other hand, are equipped with a mechanism for such fast transitions due to specific EI balance (Figure 6—figure supplement 1) and, therefore, highly specific EE connectivity in these networks can be consistent with the experimentally reported time scales.

Rapid transitions between specific states, without being trapped in each attractor for long, can give the network the capacity of revisiting the functional subnetworks spontaneously. This can endow the network with intrinsic mechanism to reactivate the learned states to avoid forgetting, in the absence of specific stimulation, and it might be important for plasticity, such that reactivation during the evoked state does not lead to unlearning (Litwin-Kumar and Doiron, 2014). It can also provide a template of most likely stimulus states, thus leading to a more efficient representation of the environment and more parsimonious responses to external stimuli. Other functional benefits of specific ISNs can be in line with the computational advantages envisaged for conventional ISNs, like selective amplification, noise rejection and pattern completion. The specific ISN regime may enable the network to perform these computations specifically along the ‘right’ dynamics. In fact, the emergence of specific connectivity in developing visual cortex is correlated with an increase in the reliability of responses (Ko et al., 2013). Future studies are needed to investigate the contribution of different motifs of connectivity, including specific EI interaction, to this process.

It is also interesting to speculate on the possible mechanisms underlying the emergence of feature-specific ISNs. To obtain feature-specific connectivity of inhibition which specifically balances excitatory subnetworks, detailed balance as opposed to general balance is needed (Hennequin et al., 2014; Hennequin et al., 2017). One possibility to achieve such detailed balance is via inhibitory plasticity mechanisms, as suggested for instance by plasticity of I-to-E synapses (Vogels et al., 2011). This is in fact consistent with the specificity of I-to-E weights reported in the visual cortex (Znamenskiy et al., 2018). However, E-to-I synapses have also been reported to have a significant level of specificity (Znamenskiy et al., 2018). A large-scale model of visual processing in balanced networks equipped with a voltage- and spike-dependent plasticity rule (Clopath et al., 2010) reported the emergence of a bidirectional specificity of E-I connections (Sadeh et al., 2015a). In future studies, it would be interesting to selectivity perturb different connectivity motifs (E-to-I and I-to-E) and plasticity mechanisms and explore the role of each in the emergence of feature-specific ISNs.

In summary, our study of networks with strong and specific excitatory-inhibitory connectivity reveals new dynamical and functional properties arising from specific inhibitory stabilization. Our results suggest that specific ISNs not only show the classical paradoxical effect, reflected in the average activity of the population, but also manifest specific paradoxical effects, which is represented in the pattern of response change of inhibitory neurons in response to external perturbations. To reveal the specific paradoxical effect, however, patterned perturbations aligned with the specific mode of inhibitory stabilization in the network is necessary.

All data generated or analysed during this study are included in the manuscript and supporting files.

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Author details

1. Sadra Sadeh

   Bioengineering Department, Imperial College London, London, United Kingdom

   Contribution

   Conceptualization, Software, Methodology, Performing simulations, Writing the manuscript

   For correspondence

   s.sadeh@imperial.ac.uk

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-8159-5461

2. Claudia Clopath

   Bioengineering Department, Imperial College London, London, United Kingdom

   Contribution

   Supervision, Funding acquisition, Conceptualization, Writing the manuscript

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-4507-8648

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