Untangling stability and gain modulation in cortical circuits with multiple interneuron classes

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.

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:
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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.

(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.

(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.

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").

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?
- 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.
- 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.
- 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.