Resonating neurons stabilize heterogeneous gridcell networks
Peer review process
This article was accepted for publication as part of eLife's original publishing model.
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Decision letter

Timothy O'LearyReviewing Editor; University of Cambridge, United Kingdom

Ronald L CalabreseSenior Editor; Emory University, United States

Alessandro TrevesReviewer; Scuola Internazionale Superiore di Studi Avanzati, Italy
Our editorial process produces two outputs: i) public reviews designed to be posted alongside the preprint for the benefit of readers; ii) feedback on the manuscript for the authors, including requests for revisions, shown below. We also include an acceptance summary that explains what the editors found interesting or important about the work.
Acceptance summary:
Grid cells in the rodent entorhinal cortex are believed to contribute to internal representations of space and other continuous quantities via periodic firing patterns. Using extensive simulations, Mittal and Narayanan show that a leading continuous attractor model of how such patterns emerge is fragile to biologically relevant heterogeneities. The authors show how this fragility is rescued by introducing intrinsic resonance in the dynamics of cells in the network. Such resonance is widely observed the entorhinal system. This work therefore shows an important potential role for single cell properties in regulating networklevel computations.
Decision letter after peer review:
Thank you for submitting your article "Resonating neurons stabilize heterogeneous gridcell networks" for consideration by eLife. Your article has been reviewed by 2 peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Ronald Calabrese as the Senior Editor. The following individual involved in review of your submission has agreed to reveal their identity: Alessandro Treves (Reviewer #2).
The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.
Essential revisions:
1) Address the reviewer's questions about the causal role of resonance in stabilising grid patterns in this specific continuous attractor model. Provide a clearer and more complete description of the singleneuron dynamics.
2) Substantiate the results with more systematic modelling or mathematical analysis, possibly in a simplified model, to provide intuition or demonstrate the mechanism underpinning the observed stabilisation of grid fields. How specific are these effects to the CAN model architecture and/or grid fields?
Reviewer #1 (Recommendations for the authors):
The authors succeed in conveying a clear and concise description of how intrinsic heterogeneity affects continuous attractor models. The main claim, namely that resonant neurons could stabilize gridcell patterns in medial entorhinal cortex, is striking.
I am intrigued by the use of a nonlinear filter composed of the product of s with its temporal derivative raised to an exponent. Why this particular choice? Or, to be more specific, would a linear bandpass filter not have served the same purpose?
The magnitude spectra are subtracted and then normalized by a sum. I have slight misgivings about the normalization, but I am more worried that , as no specific formula is given, some MATLAB function has been used. What bothers me a bit is that, depending on how the spectrogram/periodogram is computed (in particular, averaged over windows), one would naturally expect lower frequency components to be more variable. But this excess variability at low frequencies is a major point in the paper.
Which brings me to the main thesis of the manuscript: given the observation of how heterogeneities increase the variability in the low temporal frequency components, the way resonant neurons stabilize grid patterns is by suppressing these same low frequency components.
I am not entirely convinced that the observed correlation implies causality. The low temporal frequency spectra are an indirect reflection of the regularity or irregularity of the pattern formation on the network, induced by the fact that there is velocity coupling to the input and hence dynamics on the network. Heterogeneities will distort the pattern on the network, that is true, but it isn't clear how introducing a bandpass property in temporal frequency space affects spatial stability causally.
Put it this way: imagine all neurons were true oscillators, only capable of oscillating at 8 Hz. If they were to synchronize within a bump, one will have the field blinking on and off. Nothing wrong with that, and it might be that such oscillatory pattern formation on the network might be more stable than nonoscillatory pattern formation (perhaps one could even demonstrate this mathematically, for equivalent parameter settings), but this kind of causality is not what is shown in the manuscript.
Reviewer #2 (Recommendations for the authors):
I believe in selforganization and NOT in normative recommendations by reviewers: do this, don't do that. Everybody should be able to publish, in some form, what they feel is important for others to know; so I applaud the new open reviews in eLife. Besides, this manuscript is written very well, clearly, I would say equanimously, and I do not have other points to raise beyond what I observed in the open review. The figures are attractive, maybe a bit too many and too rich, but clear and engaging. My only suggestion would be, take your bandpass units, and show that they produce grids without any recurrent network. It will be fun.
https://doi.org/10.7554/eLife.66804.sa1Author response
Essential revisions:
1) Address the reviewer's questions about the causal role of resonance in stabilising grid patterns in this specific continuous attractor model. Provide a clearer and more complete description of the singleneuron dynamics.
To address this and associated concerns about the “brute force amputation of the low
frequencies” that we had employed earlier to construct resonators, we constructed a new mechanistic model for singleneuron resonance that matches the dynamical behavior of physiological resonators. Specifically, we noted that physiological resonance is elicited by a slow activitydependent negative feedback (Hutcheon and Yarom, 2000). To incorporate resonance into our ratebased model neurons, we mimicked this by introducing a slow negative feedback loop into our singleneuron dynamics (the motivations are elaborated in the new results subsection “Mechanistic model of neuronal intrinsic resonance: Incorporating a slow activitydependent negative feedback loop”). The singleneuron dynamics of mechanistic resonators were defined as follows:
Here, 𝑆 governed neuronal activity, 𝑚 defined the feedback state variable, 𝜏 represented the integration time constant, 𝐼_{𝑒} was the external current, and 𝑔 represented feedback strength. The slow kinetics of the negative feedback was controlled by the feedback time constant (𝜏_{𝑚}). In order to manifest resonance, 𝜏_{𝑚} > 𝜏 (Hutcheon and Yarom, 2000). The steadystate feedback kernel (𝑚_{∞}) of the negative feedback is sigmoidally dependent on the output of the neuron (𝑆), defined by two parameters: halfmaximal activity (𝑆_{1/2}) and slope (𝑘). The singleneuron dynamics are elaborated in detail in the methods section (new subsection: Mechanistic model for introducing intrinsic resonance in ratebased neurons).
We first demonstrate that the introduction of a slownegative feedback loop introduce
resonance into singleneuron dynamics (new Figure 9D–E). We performed systematic
sensitivity analyses associated with the parameters of the feedback loop and characterized the dependencies of intrinsic neuronal resonance on model parameters (new Figure 9F–I). We demonstrate that the incorporation of resonance through a negative feedback loop was able to generate gridpatterned activity in the 2D CAN model employed here, with clear dependencies on model parameters (new Figure 10; new Figure 10Supplements1–2). Next, we incorporated heterogeneities into the network and demonstrated that the introduction of resonance through a negative feedback loop stabilized gridpatterned generation in the heterogeneous 2D CAN model (new Figure 11).
The mechanistic route to introducing resonance allowed us to probe the basis for the
stabilization of gridpatterned activity more thoroughly. Specifically, with physiological
resonators, resonance manifests only when the feedback loop is slow (new Figure 9I;
Hutcheon and Yarom, 2000). This allowed us an additional mechanistic handle to directly probe the role of resonance in stabilizing the grid patterned activity. We assessed the emergence of gridpatterned activity in heterogeneous CAN models constructed with networks constructors with neurons with different 𝜏_{𝑚} values (new Figure 12). Strikingly, we found that when 𝜏_{𝑚} value was small (resulting in fast feedback loops), there was no stabilization of gridpatterned activity in the CAN model, especially with the highest degree of heterogeneities (new Figure 12). With progressive increase in 𝜏#, the patterns stabilized with grid score increasing with 𝜏_{𝑚} =25 ms (new Figure 12) and beyond (new Figure 11B; 𝜏_{𝑚} =75 ms). Finally, our spectral analyses comparing frequency components of homogeneous vs. heterogeneous resonator networks (new Figure panels 13D–F) showed the suppression of lowfrequency perturbations in heterogeneous CAN networks.
Together, the central hypothesis in our study was that intrinsic neuronal resonance could stabilize heterogeneous gridcell networks through targeted suppression of lowfrequency perturbations. In the revised manuscript, we present the following lines of evidence in support of this hypothesis (mentioned now in the first paragraph of the Discussion section of the revised manuscript):
1. Neuralcircuit heterogeneities destabilized gridpatterned activity generation in a 2D CAN model (Figures 2–3).
2. Neuralcircuit heterogeneities predominantly introduced perturbations in the lowfrequency components of neural activity (Figure 4).
3. Targeted suppression of lowfrequency components through phenomenological (Figure 5C) or through mechanistic (new Figure 9D) resonators resulted in stabilization of the heterogeneous CAN models (Figure 8 and new Figure 11). We note that the stabilization was achieved irrespective of the means employed to suppress lowfrequency components: an activityindependent suppression of lowfrequencies (Figure 5) or an activitydependent slow negative feedback loop (new Figure 9).
4. Changing the feedback time constant 𝜏# in mechanistic resonators, without changes to neural gain or the feedback strength allowed us to control the specific range of frequencies that would be suppressed. Our analyses showed that a slow negative feedback loop, which results in targeted suppression of lowfrequency components, was essential in stabilizing gridpatterned activity (new Figure 12). As the slow negative feedback loop and the resultant suppression of low frequencies mediates intrinsic resonance, these analyses provide important lines of evidence for the role of targeted suppression of low frequencies in stabilizing grid patterned activity.
5. We demonstrate that the incorporation of phenomenological (Figure 13A–C) or
mechanistic (new Figure panels 13D–F) resonators specifically suppressed lower
frequencies of activity in the 2D CAN model.
6. Finally, the incorporation of resonance through a negative feedback loop allowed us to link our analyses to the wellestablished role of network motifs involving negative feedback loops in inducing stability and suppressing external/internal noise in engineering and biological systems. We envisage intrinsic neuronal resonance as a cellularscale activity dependent negative feedback mechanism, a specific instance of a wellestablished network motif that effectuates stability and suppresses perturbations across different networks (Savageau, 1974; Becskei and Serrano, 2000; Thattai and van Oudenaarden, 2001; Austin et al., 2006; Dublanche et al., 2006; Raj and van Oudenaarden, 2008; Lestas et al., 2010; Cheong et al., 2011; Voliotis et al., 2014). A detailed discussion on this important link to the stabilizing role of this network motif, with appropriate references to the literature is included in the new discussion subsection “Slow negative feedback: Stability, noise suppression, and robustness”.
We thank the reviewers and the editors for their comments, as it allowed us to introduce a physiologicallyrooted model for resonance and provide more lines of evidence in support of our hypothesis. We have provided complete descriptions of single neuron dynamics associated with the three kinds of singleneuron models (integrators, phenomenological resonators and mechanistic resonators) in the methods section (equations 6–13), also providing detailed sensitivity analyses with reference to their frequencydependent response properties (Figure 8 and new Figure 11).
2) Substantiate the results with more systematic modelling or mathematical analysis, possibly in a simplified model, to provide intuition or demonstrate the mechanism underpinning the observed stabilisation of grid fields. How specific are these effects to the CAN model architecture and/or grid fields?
As mentioned above (Q1), in the revised manuscript, we have provided additional systematic modelling and the additional lines of analyses with a physiologicallyrooted mechanistic model for resonance (new Figures 9–12, 13D–F; four new Figure supplements). In probing the mechanism behind stabilization, we perturbed the time constant associated with the feedback loop, directly demonstrating the critical role of targeted suppression of neural activity in stabilizing gridpatterned activity in heterogeneous CAN models (new Figure 12). The incorporation of resonance through a negative feedback loop provides further intuition through the wellestablished role of negative feedback loops in stabilizing systems and reducing the impact of perturbations. This intuition, with appropriate references to engineering and biology literature, is now elaborated in the new discussion subsection “Slow negative feedback: Stability, noise suppression, and robustness”.
With reference to the question on the specificity of these effects to the CAN model architecture and/or grid fields, we submit that our conclusions are limited to the 2D ratebased CAN model presented here (Discussion subsection: “Future directions and considerations in model interpretation”). We postulate that mechanisms that suppress lowfrequency components could be a generalized route to stabilize heterogeneous biological networks, based on our analyses and based on the wellestablished stabilizing role of negative feedback loops. However, we do not claim generalization to other networks or to other models for generating gridpatterned neural activity. As several theoretical frameworks and computational models do not explicitly incorporate the several heterogeneities in afferent, intrinsic and synaptic properties of biological networks, it is essential that the stability of these networks are first assessed in the presence of heterogeneities. The next step would be to assess if there are perturbations in lowfrequency components as a consequence of introducing heterogeneities. Finally, if the introduction of heterogeneities destabilized network function and resulted in lowfrequency perturbations, then the role of intrinsic resonators in stabilizing these heterogeneities could be probed. As heterogeneities could manifest in different ways, and as their impact on networks could be very different in other networks compared to the network assessed here, we believe that such detailed network and heterogeneityspecific analyses would be essential before any generalization.
Finally, as our analyses are limited to the stabilizing role of resonating neurons in the
heterogeneous CAN network analyzed here, we have not explored the role of adaptation or resonance in the generation of gridpatterned neural activity. We have however noted with appropriate citations (Discussion subsection: “Future directions and considerations in model interpretation”, first paragraph) that there are other models for the generation of gridpatterned neural activity where resonance has distinct roles compared the stabilizing role proposed here. As the impact of neural heterogeneities have not been assessed in these other models, future studies could assess if heterogeneities could indeed destabilize these models and if resonance could act as a stabilizing mechanism there as well.
Reviewer #1 (Recommendations for the authors):
The authors succeed in conveying a clear and concise description of how intrinsic heterogeneity affects continuous attractor models. The main claim, namely that resonant neurons could stabilize gridcell patterns in medial entorhinal cortex, is striking.
We thank the reviewer for their time and effort in evaluating our manuscript, and for their rigorous evaluation and positive comments on our study.
I am intrigued by the use of a nonlinear filter composed of the product of s with its temporal derivative raised to an exponent. Why this particular choice? Or, to be more specific, would a linear bandpass filter not have served the same purpose?
Please note that the exponent was merely a mechanism to effectively tune the resonance frequency of the resonating neuron. In the revised manuscript, we have introduced a new physiologically rooted means to introduce intrinsic neuronal resonance, thereby confirming that network stabilization achieved was independent of the formulation employed to achieve resonance.
The magnitude spectra are subtracted and then normalized by a sum. I have slight misgivings about the normalization, but I am more worried that , as no specific formula is given, some MATLAB function has been used. What bothers me a bit is that, depending on how the spectrogram/periodogram is computed (in particular, averaged over windows), one would naturally expect lower frequency components to be more variable. But this excess variability at low frequencies is a major point in the paper.
We have now provided the specific formula employed for normalization as equation (16) of the revised manuscript. We have also noted that this was performed to account for potential differences in the maximum value of the homogeneous vs. heterogeneous spectra. The details are provided in the Methods subsection “Quantitative analysis of grid cell temporal activity in the spectral domain” of the revised manuscript. Please note that what is computed is the spectra of the entire activity pattern, and not a periodogram or a scalogram. There was no tiling of the timefrequency plane involved, thus eliminating potential roles of variables there on the computation here.
In addition to using variances of normalized differences to quantify spectral distributions, we have also independently employed octavebased analyses (which doesn’t involve normalized differences) to strengthen our claims about the impact of heterogeneities and resonance on different bands of frequency. These octavebased analyses also confirm our conclusions on the impact of heterogeneities and neuronal resonance on lowfrequency components.
Finally, we would like to emphasize that spectral computations are the same for different networks, with networks designed in such a way that there was only one component that was different. For instance, in introducing heterogeneities, all other parameters of the network (the specific trajectory, the seed values, the neural and network parameters, the connectivity, etc.) remained exactly the same with the only difference introduced being confined to the heterogeneities. Computation of the spectral properties followed identical procedures with activity from individual neurons in the two networks, and comparison was with reference to identically placed neurons in the two networks. Together, based on the several routes to quantifying spectral signatures, based on the experimental design involved, and based on the absence of any signalspecific tiling of the timefrequency plane, we argue that the impact of heterogeneities or the resonators on lowfrequency components is not an artifact of the analysis procedures.
We thank the reviewer for raising this issue, as it helped us to elaborate on the analysis procedures employed in our study.
Which brings me to the main thesis of the manuscript: given the observation of how heterogeneities increase the variability in the low temporal frequency components, the way resonant neurons stabilize grid patterns is by suppressing these same low frequency components.
I am not entirely convinced that the observed correlation implies causality. The low temporal frequeny spectra are an indirect reflection of the regularity or irregularity of the pattern formation on the network, induced by the fact that there is velocity coupling to the input and hence dynamics on the network. Heterogeneities will distort the pattern on the network, that is true, but it isn't clear how introducing a bandpass property in temporal frequency space affects spatial stability causally.
Put it this way: imagine all neurons were true oscillators, only capable of oscillating at 8 Hz. If they were to synchronize within a bump, one will have the field blinking on and off. Nothing wrong with that, and it might be that such oscillatory pattern formation on the network might be more stable than nonoscillatory pattern formation (perhaps one could even demonstrate this mathematically, for equivalent parameter settings), but this kind of causality is not what is shown in the manuscript.
The central hypothesis of our study was that intrinsic neuronal resonance could stabilize heterogeneous gridcell networks through targeted suppression of lowfrequency perturbations. In the revised manuscript, we present the following lines of evidence in support of this hypothesis (mentioned now in the first paragraph of the Discussion section of the revised manuscript):
1. Neuralcircuit heterogeneities destabilized gridpatterned activity generation in a 2D CAN model (Figures 2–3).
2. Neuralcircuit heterogeneities predominantly introduced perturbations in the low frequency components of neural activity (Figure 4).
3. Targeted suppression of lowfrequency components through phenomenological (Figure 5C) or through mechanistic (new Figure 9D) resonators resulted in stabilization of the heterogeneous CAN models (Figure 8 and new Figure 11). We note that the stabilization was achieved irrespective of the means employed to suppress lowfrequency components: an activityindependent suppression of lowfrequencies (Figure 5) or an activitydependent slow negative feedback loop (new Figure 9).
4. Changing the feedback time constant 𝜏_{#} in mechanistic resonators, without changes to neural gain or the feedback strength allowed us to control the specific range of frequencies that would be suppressed. Our analyses showed that a slow negative feedback loop, which results in targeted suppression of lowfrequency components, was essential in stabilizing gridpatterned activity (new Figure 12). As the slow negative feedback loop and the resultant suppression of low frequencies mediates intrinsic resonance, these analyses provide important lines of evidence for the role of targeted suppression of low frequencies in stabilizing grid patterned activity.
5. We demonstrate that the incorporation of phenomenological (Figure 13A–C) or mechanistic (new Figure panels 13D–F) resonators specifically suppressed lower frequencies of activity in the 2D CAN model.
6. Finally, the incorporation of resonance through a negative feedback loop allowed us to link our analyses to the wellestablished role of network motifs involving negative feedback loops in inducing stability and suppressing external/internal noise in engineering and biological systems. We envisage intrinsic neuronal resonance as a cellularscale activitydependent negative feedback mechanism, a specific instance of a wellestablished network motif that effectuates stability and suppresses perturbations across different networks (Savageau, 1974; Becskei and Serrano, 2000; Thattai and van Oudenaarden, 2001; Austin et al., 2006; Dublanche et al., 2006; Raj and van Oudenaarden, 2008; Lestas et al., 2010; Cheong et al., 2011; Voliotis et al., 2014). A detailed discussion on this important link to the stabilizing role of this network motif, with appropriate references to the literature is included in the new discussion subsection “Slow negative feedback: Stability, noise suppression, and robustness”.
We thank the reviewer for their detailed comments. These comments helped us to introducing a more physiologically rooted mechanistic form of resonance, where we were able to assess the impact of slow kinetics of negative feedback on network stability, thereby providing more direct lines of evidence for our hypothesis. This also allowed us to link resonance to the wellestablished stability motif: the negative feedback loop. We also note that our analyses don’t employ resonance as a route to introducing oscillations in the network, but as a means for targeted suppression of lowfrequency perturbations through a negative feedback loop. Given the strong quantitative links of negative feedback loops to introducing stability and suppressing the impact of perturbations in engineering applications and biological networks, we envisage intrinsic neuronal resonance as a stabilityinducing cellularscale activitydependent negative feedback mechanism.
Reviewer #2 (Recommendations for the authors):
I believe in selforganization and NOT in normative recommendations by reviewers: do this, don't do that. Everybody should be able to publish, in some form, what they feel is important for others to know; so I applaud the new open reviews in eLife. Besides, this manuscript is written very well, clearly, I would say equanimously, and I do not have other points to raise beyond what I observed in the open review. The figures are attractive, maybe a bit too many and too rich, but clear and engaging. My only suggestion would be, take your bandpass units, and show that they produce grids without any recurrent network. It will be fun.
We thank the reviewer for their detailed and rigorous review, which helped us very much in strengthening the lines of evidence in support of our central conclusions in the manuscript. We thank the reviewer for their belief in selforganization in terms of how the review process should evolve.
We believe that the new resonator model, the demonstration of network stability with the mechanistic resonators and the strong theoretical link to the stabilitythroughnegative feedback literature provides stronger support for our original conclusions. With reference to the gridcell model without the recurrent network, we agree that it is an important direction to pursue. However, we also respectfully submit that the direction would be a digression from the central hypothesis of our analyses about the stabilizing role of resonating neurons in heterogeneous CANbased gridcell networks. We note that addressing the question on gridpatterned activity with individual resonator neurons, in conjunction placecell inputs and intrinsic/synaptic plasticity, would answer the question of generation of gridlike patterns. However, as our central question relates to heterogeneous networks and stability therein, we instead focused on the reviewer’s comments on our resonator model and built a more realistic model for neural resonators and demonstrated stability with these resonator models.
The generalizability of our conclusions to other grid cells models is an important question, but would require exhaustive analyses of the impact of heterogeneities and resonance on the other models in terms of their ability to generating stable gridpatterned activity. We have stated the limitations of our analyses in the Discussion section, also suggesting directions for future research involving other models for gridpatterned activity generation.
We believe that the new lines of evidence presented in the revised manuscript with the new resonator model, along with the theoretical link to the stability literature involving negative feedbacks have considerably strengthened our central conclusions. We gratefully thank the reviewer for their rigorous and stepbystep review identifying the specific problems in our analyses; this enabled us to rectify these issues in the revised manuscript. We sincerely hope that the revised manuscript is stronger in terms of the lines of evidence presented to support the central conclusions of the overall analyses.
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https://doi.org/10.7554/eLife.66804.sa2