Author response:
The following is the authors’ response to the original reviews.
Reviewer #1 (Public Review):
Summary:
This paper explores how diverse forms of inhibition impact firing rates in models for cortical circuits. In particular, the paper studies how the network operating point affects the balance of direct inhibition from SOM inhibitory neurons to pyramidal cells, and disinhibition from SOM inhibitory input to PV inhibitory neurons. This is an important issue as these two inhibitory pathways have largely been studies in isolation. Support for the main conclusions is generally solid, but could be strengthened by additional analyses.
Strengths:
A major strength of the paper is the systematic exploration of how circuit architecture effects the impact of inhibition. This includes scans across parameter space to determine how firing rates and stability depend on effective connectivity. This is done through linearization of the circuit about an effective operating point, and then the study of how perturbations in input effect this linear approximation.
Weaknesses:
The linearization approach means that the conclusions of the paper are valid only on the linear regime of network behavior. The paper would be substantially strengthened with a test of whether the conclusions from the linearized circuit hold over a large range of network activity. Is it possible to simulate the full network and do some targeted tests of the conclusions from linearization? Those tests could be guided by the linearization to focus on specific parameter ranges of interest.
We agree with the reviewer that it would be interesting to test if our results hold in a nonlinear regime of network behaviour (i.e. the chaotic regime, see also comment 1 by reviewer 2). As mentioned above, this requires a different type of model (either rate-based or spiking model with multiple neurons instead of modelling the mean population rate dynamics) which, in our opinion, exceeds the scope of this manuscript. Furthermore, the core measures of our study, network gain, and stability require linearization. In a chaotic regime where the linearization approach is impossible, we would need to consider/define new measures to characterize network response/activity. Therefore, while certainly being an interesting question to study, the broad scope of the studying networks in a nonlinear regime is better tackled in a separate study. We now acknowledge in the discussion of our manuscript that the linearization approach is a limitation in our study and that it would be an interesting future direction to investigate chaotic dynamics.
The results illustrated in the figures are generally well described but there is very little intuition provided for them. Are there simplified examples or explanations that could be given to help the results make sense? Here are some places such intuition would be particularly helpful:
page 6, paragraph starting ”In sum ...”
Page 8, last paragraph
Page 10, paragraph starting ”In summary ...”
Page 11, sentence starting ”In sum ...”
We agree with the reviewer that we didn’t provide enough intuition to our results. We now extended the paragraphs listed by the reviewer with additional information, providing a more intuitive understanding of the results presented in the respective chapter.
Reviewer #2 (Public Review):
Summary:
Bos and colleagues address the important question of how two major inhibitory interneuron classes in the neocortex differentially affect cortical dynamics. They address this question by studying Wilson-Cowan-type mathematical models. Using a linearized fixed point approach, they provide convincing evidence that the existence of multiple interneuron classes can explain the counterintuitive finding that inhibitory modulation can increase the gain of the excitatory cell population while also increasing the stability of the circuit’s state to minor perturbations. This effect depends on the connection strengths within their circuit model, providing valuable guidance as to when and why it arises.
Overall, I find this study to have substantial merit. I have some suggestions on how to improve the clarity and completeness of the paper.
Strengths:
(1) The thorough investigation of how changes in the connectivity structure affect the gain-stability relationship is a major strength of this work. It provides an opportunity to understand when and why gain and stability will or will not both increase together. It also provides a nice bridge to the experimental literature, where different gain-stability relationships are reported from different studies.
(2) The simplified and abstracted mathematical model has the benefit of facilitating our understanding of this puzzling phenomenon. (I have some suggestions for how the authors could push this understanding further.) It is not easy to find the right balance between biologically detailed models vs simple but mathematically tractable ones, and I think the authors struck an excellent balance in this study.
Weaknesses:
(1) The fixed-point analysis has potentially substantial limitations for understanding cortical computations away from the steady-state. I think the authors should have emphasized this limitation more strongly and possibly included some additional analyses to show that their conclusions extend to the chaotic dynamical regimes in which cortical circuits often live.
We agree with the reviewer that it would be interesting to test if our results hold in a chaotic regime of network behaviour (see also comment by reviewer 1). As mentioned above, this requires a different type of model (either rate-based or spiking model with multiple neurons instead of modelling the mean population rate dynamics) which, in our opinion, exceeds the scope of this manuscript. Furthermore, the core measures of our study, network gain, and stability require linearization. In a chaotic regime where the linearization approach is impossible, we would need to consider/define new measures to characterize network response/activity. Therefore, while certainly being an interesting question to study, the broad scope of the studying networks in a nonlinear regime is better tackled in a separate study. We now acknowledge in the discussion of our manuscript that the linearization approach is a limitation in our study and that it would be an interesting future direction to investigate chaotic dynamics.
(2) The authors could have discussed – even somewhat speculatively – how SST interneurons fit into this picture. Their absence from this modelling framework stands out as a missed opportunity.
We believe that the reviewer wanted us to speculate about VIP interneurons (and not SST interneurons, which we already do extensively in the manuscript). Previous models have included VIP neurons in the circuit (e.g. del Molino et al., 2017; Palmigiano et al., 2023; Waitzmann et al., 2024). While we do not model VIP cells explicitly, we implicitly assume that a possible source of modulation of SOM neurons comes from VIP cells. We have now added a short discussion on VIP cells in the last paragraph in our discussion section.
(3) The analysis is limited to paths within this simple E,PV,SOM circuit. This misses more extended paths (like thalamocortical loops) that involve interactions between multiple brain areas. Including those paths in the expansion in Eqs. 11-14 (Fig. 1C) may be an important consideration.
We agree with the reviewer that our framework can be extended to study many other different paths, like thalamocortical loops, cortical layer-specific connectivity motifs, or circuits with VIP or L1 inhibitory neurons. Studying these questions, however, are beyond the scope of our work. In our discussion, we now mention the possibility of using our framework to study those questions.
Reviewer #3 (Public Review):
Summary:
Bos et al study a computational model of cortical circuits with excitatory (E) and two subtypes of inhibition parvalbumin (PV) and somatostatin (SOM) expressing interneurons. They perform stability and gain analysis of simplified models with nonlinear transfer functions when SOM neurons are perturbed. Their analysis suggests that in a specific setup of connectivity, instability and gain can be untangled, such that SOM modulation leads to both increases in stability and gain. This is in contrast with the typical direction in neuronal networks where increased gain results in decreased stability.
Strengths:
- Analysis of the canonical circuit in response to SOM perturbations. Through numerical simulations and mathematical analysis, the authors have provided a rather comprehensive picture of how SOM modulation may affect response changes.
- Shedding light on two opposing circuit motifs involved in the canonical E-PV-SOM circuitry - namely, direct inhibition (SOM → E) vs disinhibition (SOM → PV → E). These two pathways can lead to opposing effects, and it is often difficult to predict which one results from modulating SOM neurons. In simplified circuits, the authors show how these two motifs can emerge and depend on parameters like connection weights.
- Suggesting potentially interesting consequences for cortical computation. The authors suggest that certain regimes of connectivity may lead to untangling of stability and gain, such that increases in network gain are not compromised by decreasing stability. They also link SOM modulation in different connectivity regimes to versatile computations in visual processing in simple models.
Weaknesses:
The computational analysis is not novel per se, and the link to biology is not direct/clear.
Computationally, the analysis is solid, but it’s very similar to previous studies (del Molino et al, 2017). Many studies in the past few years have done the perturbation analysis of a similar circuitry with or without nonlinear transfer functions (some of them listed in the references). This study applies the same framework to SOM perturbations, which is a useful and interesting computational exercise, in view of the complexity of the high-dimensional parameter space. But the mathematical framework is not novel per se, undermining the claim of providing a new framework (or ”circuit theory”).
In the introduction we acknowledge that our analysis method is not novel but is rather based on previous studies (del Molino et al., 2017; Kuchibhotla et al., 2017; Kumar et al., 2023, Litwin-Kumar et al., 2016; Mahrach et al., 2020; Palmigiano et al., 2023; Veit et al., 2023; Waitzmann et al., 2024). We now rewrote parts of the introduction to make sure that it does not sound like the computational analysis has been developed by us, but that we rather use those previously developed frameworks to dissect stability and gain via SOM modulation.
Link to biology: the most interesting result of the paper with regard to biology is the suggestion of a regime in which gain and stability can be modulated in an unconventional way - however, it is difficult to link the results to biological networks: - A general weakness of the paper is a lack of direct comparison to biological parameters or experiments. How different experiments can be reconciled by the results obtained here, and what new circuit mechanisms can be revealed? In its current form, the paper reads as a general suggestion that different combinations of gain modulation and stability can be achieved in a circuit model equipped with many parameters (12 parameters). This is potentially interesting but not surprising, given the high dimensional space of possible dynamical properties. A more interesting result would have been to relate this to biology, by providing reasoning why it might be relevant to certain circuits (and not others), or to provide some predictions or postdictions, which are currently missing in the manuscript.
- For instance, a nice motivation for the paper at the beginning of the Results section is the different results of SOM modulation in different experiments - especially between L23 (inhibition) and L4 (disinhibition). But no further explanation is provided for why such a difference should exist, in view of their results and the insights obtained from their suggested circuit mechanisms. How the parameters identified for the two regimes correspond to different properties of different layers?
As pointed out by the reviewer, the main goal of our manuscript is to provide a general understanding of how gain and stability depend on different circuit motifs (ie different connectivity parameters), and how circuit modulations via SOM neurons affect those measures. However, we agree with the reviewer that it would be useful to provide some concrete predictions or postdictions following from our study.
An interesting example of a postdiction of our model is that the firing rate change of excitatory neurons in response to a change in the stimulus (which we define as network gain, Eq. 2) depends on firing rates of the excitatory, PV, and SOM neurons at the moment of stimulus presentation (Fig. 3ii; Fig. 4Aii,Bii,Cii; Fig. 5Aii, Bii, Cii). Hence any change in input to the circuit can affect the response gain to a stimulus presentation, in line with experimental evidence which suggests that changes in inhibitory firing rates and changes in the behavioral state of the animal lead to gain modifications (Ferguson and Cardin 2020).
Another recent concrete example is the study of Tobin et al., 2023, in which the authors show that optogenetically activating SOM cells in the mouse primary auditory cortex (A1) decreases the excitatory responses to auditory stimuli. In our framework, this corresponds to the case of decreases in network gain (gE) for positive SOM modulation, as seen in the circuit with PV to SOM feedback connectivity (Suppl. Fig. S1).
Another example is the study by Phillips and Hasenstaub 2016, in which the authors study the effect of optogenetic perturbations of SOM (and PV) cells on tuning curves of pyramidal cells in mouse A1. While they find large heterogeneity in additive/subtractive or multiplicative/divisive tuning curve changes following SOM inactivation, most cells have a purely multiplicative or purely additive component (and none of the cells have a divisive component). In our study, we see that large multiplicative responses of the excitatory population follow from circuits with strong E to SOM feedback connectivity.
We note that in future computational studies, it would be useful to apply our framework with a focus on a specific brain region and add all relevant cell types (at a minimum E, PV, SOM, and VIP) plus a dendritic compartment, in order to formulate much more precise experimental predictions.
We have now added additional information to the discussion section.
- Another caveat is the range of parameters needed to obtain the unintuitive untangling as a result of SOM modulation. From Figure 4, it appears that the ”interesting” regime (with increases in both gain and stability) is only feasible for a very narrow range of SOM firing rates (before 3 Hz). This can be a problem for the computational models if the sweet spot is a very narrow region (this analysis is by the way missing, so making it difficult to know how robust the result is in terms of parameter regions). In terms of biology, it is difficult to reconcile this with the realistic firing rates in the cortex: in the mouse cortex, for instance, we know that SOM neurons can be quite active (comparable to E neurons), especially in response to stimuli. It is therefore not clear if we should expect this mechanism to be a relevant one for cortical activity regimes.
We agree with the reviewer that it’s important to test the robustness of our results. As suggested by the reviewer, we now include a new supplementary figure (Suppl. Fig. S2) which measures the percentage of data points in the respective quadrant Q1-Q4 when changing the SOM firing rates (as done in Fig. 5). We see that the quadrants in which the network gain and stability change in the same direction (Q2 and Q3) remain high in the case for E to SOM feedback (Suppl. Fig. S2A) over SOM rates ranging over 0-10 Hz (and likely beyond).
- One of the key assumptions of the model is nonlinear transfer functions for all neuron types. In terms of modelling and computational analysis, a thorough analysis of how and when this is necessary is missing (an analysis similar to what has been attempted at in Figure 6 for synaptic weights, but for cellular gains). In terms of biology, the nonlinear transfer function has experimentally been reported for excitatory neurons, so it’s not clear to what extent this may hold for different inhibitory subtypes. A discussion of this, along with the former analysis to know which nonlinearities would be necessary for the results, is needed, but currently missing from the study. The nonlinearity is assumed for all subtypes because it seems to be needed to obtain the results, but it’s not clear how the model would behave in the presence or absence of them, and whether they are relevant to biological networks with inhibitory transfer functions.
It is true that the nonlinear transfer function is a key component in our model. We chose identical transfer functions for E, PV, and SOM (; Eq. 4) to simplify our analysis. If the transfer function of one of the neuron types would be linear (β = 1), then the corresponding b terms (the slope of the nonlinearity at the steady state; b = dfX/dqX; Fig. 1B; Eq. 4) would be equal to α. Therefore, if neurons had a linear transfer function in our model, there would not be a dependence of network gain on E and PV firing rate as studied in Fig. 3-5. This is because the relationship between PV rates and their gain would be constant (bP = α) in Fig. 1B (bottom).
If all the transfer functions were linear, changes in firing rates would not have an impact on network gain or stability. Changing the nonlinear transfer function by changing the α or β terms in Eq. 4 would only scale the way a change in the rates affects the b terms and hence the results presented in Fig. 3-5. More interesting would be to study how different types of nonlinearities, like sigmoidal functions or sublinear nonlinearities (i.e. saturating nonlinearities), would change our results. However, we think that such an investigation is out of scope for this study. We now added a comment to the Methods section.
Experimentally, F-I curves have been measured also for PV and SOM neurons. For example, Romero-Sosa et al., 2021 measure the F-I curve of pyramidal, PV and SOM neurons in mouse cortical slices. They find that similar to pyramidal neurons, PV and SOM neurons show a nonlinear F-I curve. We now added the citation of Romero-Sosa et al., 2021 to our manuscript.
- Tuning curves are simulated for an individual orientation (same for all), not considering the heterogeneity of neuronal networks with multiple orientation selectivity (and other visual features) - making the model too simplistic.
The reviewer is correct that we only study changes in tuning curves in a simplistic model. In our model, the excitatory and PV populations are tuned to a single orientation (in the case of Fig. 7 to θ = 90). While this is certainly an oversimplification, it allows us to understand how additive/subtractive and multiplicative/divisive changes in the tuning curves come about in networks with different connectivity motifs. To model heterogeneity of tuning responses within a network, it requires more complex models. A natural choice would be to extend a classical ring attractor model (Rubin et al., 2015) by splitting the inhibitory population into PV and SOM neurons, or study the tuning curve heterogeneity that occurs in balanced networks (Hansel and van Vreeswijk 2012). However, this model has many more parameters, like the spatial connectivity profiles from and onto PV and SOM neurons. While highly valuable, we believe that studying such models exceeds the scope of our current manuscript. We now added a paragraph in the discussion section, mentioning this as an interesting future direction.
Reviewer #1 (Recommendations For The Authors):
The last sentence of the abstract is hard to interpret before reading the rest of the paper - suggest replacing or rephrasing.
We rephrased the sentence to make more clear what we mean.
Page 3, last full paragraph: I think this assumes that phi is positive. What is the justification for that assumption? More generally, I think you could say a bit more about phi in the main text since it is a fairly complicated term.
The reviewer is correct, for a stable system phi is always positive. We now clarify this and explain phi in more detail in the main text.
Fig 1D: It would be helpful to identify when the stimulus comes on and be clearer about what the stimulus is. I assume it’s a step increase in S input at 0.05 s or so - but that should be immediately apparent looking at the figure.
We agree with the reviewer and we added a dashed line at the time of stimulus onset in Fig. 1D.
Page 5: ”To motivate our analysis we compare ... (Fig. 2A)” - Figure 2A does not show responses without modulation, so this sentence is confusing.
The dashed lines in Fig. 2A (and Fig. 2C) actually represents the rate change without modulation.
Page 6: sentence “The central goal of our study ...” seems out of place since this is pretty far into the results, and that goal should already be clear.
We agree with the reviewer, hence we updated the sentence.
Page 10, top: the green curve in panel Aii always has a negative slope - so I am confused by the statement that increasing wSE decreases both gain and stability.
We thank the reviewer for pointing out this mistake. We now fixed it in the text.
Figure 6: in general it is hard to see what is going on in this figure (the green and blue in particular are hard to distinguish). Some additional labels would be helpful, but I would also see if the color scheme can be improved.
We added a zoom-in to the panels which were hard to distinguish.
Reviewer #2 (Recommendations For The Authors):
Major recommendations:
(1) The authors should explain early on in the results section what the key factor(s) is that differentiates SOM from PV cells in their model. E.g., in Fig. 1A, the only obvious difference is that SOM cells don’t inhibit themselves. However, later on in the paper, the difference in external stimulus drive to these interneuron classes is more heavily emphasized. Given the importance of that difference (in external stim drive), I think this should be highlighted early on.
We now mention the key factors that differentiate PV and SOM neurons already when describing Fig. 1A.
(2) The result in Figs. 5,6 demonstrate that recurrent SOM connectivity is important for achieving increases in both gain and stability. This observation could benefit from some intuitive explanation. Perhaps the authors could find this explanation by looking at their series expansion (Eqs. 11-14, Fig. 1C) and determining which term(s) are most important for this effect. The corresponding paths through the circuit – the most important ones – could then be highlighted for the reader.
We agree with the reviewer that our results benefit from more intuitive explanations. This has also been pointed out by reviewer 1 in their public review. We now extended the concluding paragraphs in the context of Fig. 4-6 with additional information, providing a more intuitive understanding of the results presented in the respective chapter. While it is possible to gain an intuitive understanding of how the network gain depends on rate and weight parameters (Eq. 2), this understanding is unfortunately missing in the case of stability. The maximum eigenvalue of the system have a complex relationship with all the parameters, and often have nonlinear dependencies on changes of a parameter (e.g. as we show in Fig. 3iv or one can see in Fig. 6). We now discuss this difficulty at the end of the section “Influence of weight strength on network gain vs stability”.
(3) I think the authors should consider including some analyses that do not rely on the system being at or near a fixed point. I admit that such analysis could be difficult, and this could of course be done in a future study. Nevertheless, I want to reiterate that this addition could add a lot of value to this body of work.
As outlined above, we decided to not include additional analysis on network behaviour in nonlinear regimes but we now acknowledge in the discussion of our manuscript that the linearization approach is a limitation in our study and that it would be an interesting future direction to investigate chaotic dynamics.
Minor recommendations:
(1) At the top of P. 6, when the authors first discuss the stability criterion involving eigenvalues, they should address the question ”eigenvalues of what?”. I suggest introducing the idea of the Jacobian matrix, and explaining that the largest eigenvalue of that matrix determines how rapidly the system will return to the fixed point after a small perturbation.
We included an additional sentence in the respective paragraph explaining the link between stability and negative eigenvalues, and we also added a sentence in the Methods section stating the the largest real eigenvalue dominates the behavior of the dynamical system.
(2) The panel labelling in Fig. 3 is unnecessarily confusing. It would be simpler (and thus better) to simply label the panels A,B,C,D, or i,ii,iii,iv, instead of the current labelling: Ai, Aii, Aiii, Aiv. (There are currently no panels ”B” in Fig. 3).
We updated the figure accordingly.
Reviewer #3 (Recommendations For The Authors):
• Suggestions for improved or additional experiments, data or analyses.
Analysis of the effect of different nonlinear transfer functions is necessary.
Please see our detailed answer to the reviewer’s comment in the public review above.
Analysis of gain modulation in models with more realistic tuning properties.
Please see our detailed answer to the reviewer’s comment in the public review above.
Mathematical analysis of the conditions to obtain ”untangled” gain and stability:
One of the promises of the paper is that it is offering a computational framework or circuit theory for understanding the effect of SOM perturbation. However, the main result, namely the untangling of gain and stability, has only been reported in numerical simulations (e.g. Fig. 6). Different parameters have been changed and the results of simulations have been reported for different conditions. Given the simplified model, which allows for rigorous mathematical analysis, isn’t it possible to treat this phenomenon more analytically? What would be the conditions for the emergence of the untangled regime? This is currently missing from the analyses and results.
We agree with the reviewer that our results benefit from more intuitive explanations. This has also been pointed out by reviewer 1 in their public review. We now extended the concluding paragraphs in the context of Fig. 4-6 with additional information, providing a more intuitive understanding of the results presented in the respective chapter. While it is possible understand analytically of how the network gain depends on rate and weight parameters (Eq. 2), this understanding is unfortunately missing in the case of stability. The maximum eigenvalue of the system have a complex relationship with all the parameters, and often have nonlinear dependencies on changes of a parameter (e.g. as we show in Fig. 3iv or one can see in Fig. 6). This doesn’t allow for a a deep analytical understanding of the entangling of gain and stability. We now discuss this difficulty at the end of the section “Influence of weight strength on network gain vs stability”.
• Recommendations for improving the writing and presentation. The Results section is well written overall, but other parts, especially the Introduction and Discussion, would benefit from proof reading - there are many typos and problems with sentence structures and wording (some mentioned below).
We have gone through the manuscript again and improved the writing.
The presentation of the dependence on weight in Figure 6 can be improved. For instance, the authors talk about the optimal range of PV connectivity, but this is difficult to appreciate in the current illustration and with the current colour scheme.
We added a zoom in to the panels which were hard to distinguish.
• Minor corrections to the text and figures. Text:
We thank the reviewer for their thorough reading of our manuscript. We fixed all the issues from below in the manuscript.
Some examples of bad structure or wording:
From the Abstract:
”We show when E - PV networks recurrently connect with SOM neurons then an SOM mediated modulation that leads to increased neuronal gain can also yield increased network stability.” From Introduction:
Sentence starting with ”This new circuit reality ...”
”Inhibition is been long identified as a physiological or circuit basis for how cortical activity changes depending upon processing or cognitive needs ...”
Sentence starting with ”Cortical models with both ...”
”... allowing SOM neurons the freedom to ..”
From Results:
”... affects of SOM neurons on E ..”
”seem in opposition to one another, with SOM neuron activity providing either a source or a relief of E neuron suppression”. The sentence after is also difficult to read and needs to be simplified.
P. 7: ”We first remark that ...”
Difficult to read/understand - long and badly structured sentence.
P. 8: ”adding a recurrent connection onto SOM neurons from the E-PV subcircuit” It’s from E (and not PV) to be more precise (Fig. 5).
Discussion:
”Firstly, E neurons and PV neurons experience very similar synaptic environments.” What does it mean?
”Fortunately, PV neurons target both the cell bodies and proximal dendrites” Fortunately for whom or what? ”in line with arge heterogeneity”
Methods:
Matrix B is never defined - the diagonal matrix of b (power law exponents) I assume.
Some of the other notations too, e.g. bs, etc (it’s implicit, but should be explained).
Structure of sentence:
”Network gain is defined as ...” (p. 17)
Figure:
The schematics in Figure 4 can be tweaked to highlight the effect of input (rather than other components of the network, which are the same and repetitive), to highlight the main difference for the reader.
