Stability of network function, defined as the network’s ability to elicit robust functional outcomes despite perturbations to or widespread variability in its constitutive components, is a central theme that governs the functional design of several biological networks. Biological systems exhibit ubiquitous parametric variability spanning different scales of organization, quantified through statistical heterogeneities in the underlying parameters. Strikingly, in spite of such large-scale heterogeneities, outputs of biological networks are stable and are precisely tuned to meet physiological demands. A central question that spans different scales of organization is on the ability of biological networks to achieve physiological stability in the face of ubiquitous parametric variability (Turrigiano and Nelson, 2000; Edelman and Gally, 2001; Maslov and Sneppen, 2002; Stelling et al., 2004; Marder and Goaillard, 2006; Barkai and Shilo, 2007; Kitano, 2007; Félix and Barkoulas, 2015).

Biological heterogeneities are known to play critical roles in governing stability of network function, through intricate and complex interactions among mechanisms underlying functional emergence (Edelman and Gally, 2001; Renart et al., 2003; Marder and Goaillard, 2006; Tikidji-Hamburyan et al., 2015; Mishra and Narayanan, 2019; Rathour and Narayanan, 2019). However, an overwhelming majority of theoretical and modeling frameworks lack the foundation to evaluate the impact of such heterogeneities on network output, as they employ unnatural homogeneous networks in assessing network function. The paucity of heterogeneous network frameworks is partly attributable to the enormous analytical or computational costs involved in assessing heterogeneous networks. In this study, we quantitatively address questions on the impact of distinct forms of biological heterogeneities on the functional stability of a two-dimensional continuous attractor network (CAN), which has been implicated in the generation of patterned neuronal activity in grid cells of the medial entorhinal cortex (Burak and Fiete, 2009; Knierim and Zhang, 2012; Couey et al., 2013; Domnisoru et al., 2013; Schmidt-Hieber and Häusser, 2013; Yoon et al., 2013; Tukker et al., 2021). Although the continuous attractor framework has offered insights about information encoding across several neural circuits (Samsonovich and McNaughton, 1997; Seung et al., 2000; Renart et al., 2003; Wills et al., 2005; Burak and Fiete, 2009; Knierim and Zhang, 2012; Schmidt-Hieber and Häusser, 2013; Yoon et al., 2013; Kim et al., 2017), the fundamental question on the stability of 2D CAN models in the presence of biological heterogeneities remains unexplored. Here, we systematically assessed the impact of biological heterogeneities on stability of emergent spatial representations in a 2D CAN model and unveiled a physiologically plausible neural mechanism that promotes stability despite the expression of heterogeneities.

We first developed an algorithm to generate virtual trajectories that closely mimicked animal traversals in an open arena, to provide better computational efficiency in terms of covering the entire arena within shorter time duration. We employed these virtual trajectories to drive a rate-based homogeneous CAN model that elicited grid-patterned neural activity (Burak and Fiete, 2009) and systematically introduced different degrees of three distinct forms of biological heterogeneities. The three distinct forms of biological heterogeneities that we introduced, either individually or together, were in neuronal intrinsic properties, in afferent inputs carrying behavioral information and in local-circuit synaptic connectivity. We found that the incorporation of these different forms of biological heterogeneities disrupted the emergence of grid-patterned activity by introducing perturbations in neural activity, predominantly in low-frequency components. In the default model where neurons were integrators, grid patterns and spatial information in neural activity were progressively lost with increasing degrees of biological heterogeneities and were accompanied by progressive increases in low-frequency perturbations.

As heterogeneity-induced perturbations to neural activity were predominantly in the lower frequencies, we postulated that suppressing low-frequency perturbations could ameliorate the disruptive impact of biological heterogeneities on grid-patterned activity. We recognized intrinsic neuronal resonance as an established biological mechanism that suppresses low-frequency components, effectuated by the expression of resonating conductances endowed with specific biophysical characteristics (Hutcheon and Yarom, 2000; Narayanan and Johnston, 2008). Consequently, we hypothesized that intrinsic neuronal resonance could stabilize the heterogeneous grid-cell network through targetted suppression of low-frequency perturbations. To test this hypothesis, we developed two distinct strategies to introduce intrinsic resonance in our rate-based neuronal model to mimic the function of resonating conductances in biological neurons: (1) a phenomenological approach where an additional tunable high-pass filter (HPF) was incorporated into single-neuron dynamics and (2) a mechanistic approach where resonance was realized through a slow negative feedback loop akin to the physiological mechanism behind neuronal intrinsic resonance (Hutcheon and Yarom, 2000). We confirmed that the emergence of grid-patterned activity was not affected by replacing all the integrator neurons in the homogeneous CAN model with theta-frequency resonators (either phenomenological or mechanistic). We systematically incorporated different forms of biological heterogeneities into the 2D resonator CAN model and found that intrinsic neuronal resonance stabilized heterogeneous neural networks, through suppression of low-frequency components of neural activity. Although this stabilization was observed with both phenomenological and mechanistic resonator networks, the mechanistic resonator was extremely effective in suppressing low-frequency activity without introducing spurious high-frequency components into neural activity. Importantly, we found that the slow kinetics of the negative feedback loop was essential in stabilizing CAN networks built with mechanistic resonators.

Together, our study unveils an important role for intrinsic neuronal resonance in stabilizing network physiology through the suppression of heterogeneity-induced perturbations in low-frequency components of network activity. Our analyses suggest that intrinsic neuronal resonance constitutes a cellular-scale activity-dependent negative feedback mechanism, a specific instance of a well-established network motif that effectuates stability and suppresses perturbations across different biological 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). As the dominance of low-frequency perturbations is pervasive across biological networks (Hausdorff and Peng, 1996; Gilden, 2001; Gisiger, 2001; Ward, 2001; Buzsaki, 2006), we postulate that mechanisms that suppress low-frequency components could be a generalized route to stabilize heterogeneous biological networks.

The rate-based CAN model consisting of a 2D neural sheet (default size: 60 × 60 = 3600 neurons) with the default integrator neurons for eliciting grid-cell activity was adopted from Burak and Fiete, 2009. The sides of this neural sheet were connected, yielding a toroidal (or periodic) network configuration. Each neuron in the CAN model received two distinct sets of synaptic inputs, one from other neurons within the network and another feed-forward afferent input that was dependent on the velocity of the virtual animal. The movement of the virtual animal was modeled to occur in a circular 2D spatial arena (Figure 1A; Real trajectory) and was simulated employing recordings of rat movement (Hafting et al., 2005) to replicate earlier results (Burak and Fiete, 2009). However, the real trajectory spanned 590 s of real-world time and required considerable simulation time to sufficiently explore the entire arena, essential for computing high-resolution spatial activity maps. Consequently, the computational costs required for exploring the parametric space of the CAN model, with different forms of network heterogeneities in networks of different sizes and endowed with distinct kinds of neurons, were exorbitant. Therefore, we developed a virtual trajectory, mimicking smooth animal movement within the arena, but with sharp turns at the borders.

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Our analyses demonstrated the ability of virtual trajectories to cover the entire arena with lesser time (Figure 1A; Virtual trajectory), while not compromising on the accuracy of the spatial activity maps constructed from this trajectory in comparison to the maps obtained with the real trajectory (Figure 1B, Figure 1—figure supplement 1). Specifically, the correlation values between the spatial autocorrelation of rate maps of individual grid cells for real trajectory and that from the virtual trajectory at different rat runtimes ($T_{\mathrm{r}\mathrm{u}\mathrm{n}}$) were high across all measured runtimes (Figure 1B). These correlation values showed a saturating increase with increase in $T_{\mathrm{r}\mathrm{u}\mathrm{n}}$. As there was no substantial improvement in accuracy beyond $T_{\mathrm{r}\mathrm{u}\mathrm{n}}$ = 100 s, we employed $T_{\mathrm{r}\mathrm{u}\mathrm{n}}$ = 100 s for all simulations (compared to the 590 s of the real trajectory). Therefore, our virtual trajectory covered similar areas in approximately six times lesser duration (Figure 1A), which reduced the simulation duration by a factor of ~10 times when compared to the real rat trajectory, while yielding similar spatial activity maps (Figure 1B, Figure 1—figure supplement 1). Our algorithm also allowed us to generate fast virtual trajectories mimicking rat trajectories in open arenas of different shapes (Circle: Figure 1A; Square: Figure 2E).

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Biologically prevalent network heterogeneities disrupted the emergence of grid-cell activity in CAN models

The simulated CAN model (Figure 1) is an idealized homogeneous network of intrinsically identical neurons, endowed with precise local connectivity patterns and identical response properties for afferent inputs. Although this idealized CAN model provides an effective phenomenological framework to understand and simulate grid-cell activity, the underlying network does not account for the ubiquitous biological heterogeneities that span neuronal intrinsic properties and synaptic connectivity patterns. Would the CAN model sustain grid-cell activity in a network endowed with different degrees of biological heterogeneities in neuronal intrinsic properties and strengths of afferent/local synaptic inputs?

To address this, we systematically introduced three distinct forms of biological heterogeneities into the rate-based CAN model. We introduced intrinsic heterogeneity by randomizing the value of the neuronal integration time constant $\tau$ across neurons in the network. Within the rate-based framework employed here, randomization of $\tau$ reflected physiological heterogeneities in neuronal excitability properties, and larger spans of $\tau$ defined higher degrees of intrinsic heterogeneity (Table 1; Figure 2A). Afferent heterogeneity referred to heterogeneities in the coupling of afferent velocity inputs onto individual neurons. Different degrees of afferent heterogeneities were introduced by randomizing the velocity scaling factor ($\alpha$) in each neuron through uniform distributions of different ranges (Table 1; Figure 2B). Synaptic heterogeneity involved the local connectivity matrix, and was introduced as additive jitter to the default center-surround connectivity matrix. Synaptic jitter was independently sampled for each connection, from a uniform distribution whose differential span regulated the associated higher degree of heterogeneity (Table 1; Figure 2C). In homogeneous networks, $\tau$ (=10 ms), $\alpha$ (=45) and $W_{ij}$ (center-shifted Mexican hat connectivity without jitter) were identical for all neurons in the network. We assessed four distinct sets of heterogeneous networks: three sets endowed with one of intrinsic, afferent and synaptic heterogeneities, and a fourth where all three forms of heterogeneities were co-expressed. When all heterogeneities were present together, all three sets of parameters were set randomly with the degree of heterogeneity defining the bounds of the associated distribution (Table 1).

We found that the incorporation of any of the three forms of heterogeneities into the CAN model resulted in the disruption of grid pattern formation, with the deleterious impact increasing with increasing degree of heterogeneity (Figures 2E and 3). Quantitatively, we employed grid score (Fyhn et al., 2004; Hafting et al., 2005) to measure the emergence of pattern formation in the CAN model activity, and found a reduction in grid score with increase in the degree of each form of heterogeneity (Figure 2F). We found a hierarchy in the disruptive impact of different types of heterogeneities, with synaptic heterogeneity producing the largest reduction to grid score, followed by afferent heterogeneity. Intrinsic heterogeneity was the least disruptive in the emergence of grid-cell activity, whereby a high degree of heterogeneity was required to hamper grid pattern formation (Figure 2E,F). Simultaneous progressive introduction of all three forms of heterogeneities at multiple degrees resulted in a similar progression of grid-pattern disruption (Figure 2E,F). The introduction of the highest degree of all heterogeneities into the CAN model resulted in a complete loss of patterned activity and a marked reduction in the grid score values of all neurons in the network (Figure 2E,F). We confirmed that these results were not artifacts of specific network initialization choices by observing similar grid score reductions in five additional trials with distinct initializations of CAN models endowed with all three forms of heterogeneities (Figure 3—figure supplement 1F). In addition, to rule out the possibility that these conclusions were specific to the choice of the virtual trajectory employed, we repeated our simulations and analyses with a different virtual trajectory (Figure 3—figure supplement 2A) and found similar reductions in grid score with increase in the degree of all heterogeneities (Figure 3—figure supplement 2B).

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Detailed quantitative analysis of grid-cell activity showed that the introduction of heterogeneities did not result in marked population-wide changes to broad measures such as average firing rate, mean size of grid fields and average grid spacing (Figure 3, Figure 3—figure supplements 1–2). Instead, the introduction of parametric heterogeneities resulted in a loss of spatial selectivity and disruption of patterned activity, reflected as progressive reductions in grid score, peak firing rate, information rate and sparsity (Figures 2–3, Figure 3—figure supplements 1–2). Our results showed that in networks with intrinsic or afferent heterogeneities, grid score of individual neurons was not dependent on the actual value of the associated parameter ($\tau$ or α, respectively), but was critically reliant on the degree of heterogeneity (Figure 3H). Specifically, for a given degree of intrinsic or afferent heterogeneity, the grid score value spanned similar ranges for the low or high value of the associated parameter; but grid score reduced with increased degree of heterogeneities. In addition, grid score of cells reduced with increase in local synaptic jitter, which increased with the degree of synaptic heterogeneity (Figure 3H).

Thus far, our analyses involved a CAN model with a specific size (60×60). Would our results on heterogeneity-driven disruption of grid-patterned activity extend to CAN models of other sizes? To address this, we repeated our simulations with networks of size 40×40, 50×50, 80×80 and 120×120, and introduced different degrees of all three forms of heterogeneities into the network. We found that the disruptive impact of heterogeneities on grid-patterned activity was consistent across all tested networks (Figure 3—figure supplement 3A), manifesting as a reduction in grid score with increased degree of heterogeneities (Figure 3—figure supplement 3B). Our results also showed that the disruptive impact of heterogeneities on grid-patterned activity increased with increase in network size. This size dependence specifically manifested as a complete loss of spatially selective firing patterns (Figure 3—figure supplement 3A) and the consistent drop in the grid score value to zero (i.e., %change = –100 in Figure 3—figure supplement 3B) across all neurons in the 120 × 120 network with high degree of heterogeneities. We also observed a progressive reduction in the average and the peak firing rates with increase in the degree of heterogeneities across all network sizes, although with a pronounced impact in networks with higher size (Figure 3—figure supplement 3C,D).

Together, our results demonstrated that the introduction of physiologically relevant heterogeneities into the CAN model resulted in a marked disruption of grid-patterned activity. The disruption manifested as a loss in spatial selectivity in the activity of individual cells in the network, progressively linked to the degree of heterogeneity. Although all forms of heterogeneities resulted in disruption of patterned activity, there was differential sensitivity to different heterogeneities, with heterogeneity in local network connectivity playing a dominant role in hampering spatial selectivity and patterned activity.

Incorporation of biological heterogeneities predominantly altered neural activity in low frequencies

How does the presence of heterogeneities affect activity patterns of neurons in the network as they respond to the movement of the virtual animal? A systematic way to explore patterns of neural activity is to assess the relative power of specific frequency components in neural activity. Therefore, we subjected the temporal outputs of individual neurons (across the entire period of the simulation) to spectral analysis and found neural activity to follow a typical response curve that was dominated by lower frequency components, with little power in the higher frequencies (e.g., Figure 4A, HN). This is to be expected as a consequence of the low-pass filtering inherent to the neuronal model, reflective of membrane filtering.

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We performed spectral analyses of temporal activity patterns of neurons from networks endowed with different forms of heterogeneities at various degrees (Figure 4). As biological heterogeneities were incorporated either by reducing or by increasing default parameter values (Table 1), there was considerable variability in how individual neurons responded to such heterogeneities (Figure 4). Specifically, when compared against the respective neuron in the homogeneous network, some neurons showed increases in activity magnitude at certain frequencies and others showed a reduction in magnitude (e.g., Figure 4A). To quantify this variability in neuronal responses, we first computed the difference between frequency-dependent activity profiles of individual neurons obtained in the presence of specific heterogeneities and of the same neuron in the homogeneous network (with identical initial conditions and subjected to identical afferent inputs). We computed activity differences for all the 3600 neurons in the network and plotted the variance of these differences as a function of frequency (e.g., Figure 4B). Strikingly, we found that the impact of introducing biological heterogeneities predominantly altered lower frequency components of neural activity, with little to no impact on higher frequencies. In addition, the variance in the deviation of neural activity from the homogenous network progressively increased with higher degrees of heterogeneities, with large deviations occurring in the lower frequencies (Figure 4).

Table 1

Degree of heterogeneity	Intrinsic heterogeneity (${\displaystyle \tau }$)		Afferent heterogeneity (${\displaystyle \alpha }$)		Synaptic heterogeneity (${\displaystyle W_{ij}}$)
	Lower bound	Upper bound	Lower bound	Upper bound	Lower bound	Upper bound
1	8	12	35	55	0	300
2	6	14	25	65	0	600
3	4	16	15	75	0	900
4	2	18	5	85	0	1200
5	1	20	0	100	0	1500

Together, these observations provided an important insight that the presence of physiologically relevant network heterogeneities predominantly affected lower frequency components of neural activity, with higher degrees of heterogeneity yielding larger variance in the deviation of low-frequency activity from the homogeneous network.

Introducing intrinsic resonance in rate-based neurons: phenomenological model

Several cortical and hippocampal neuronal structures exhibit intrinsic resonance, which allows these structures to maximally respond to a specific input frequency, with neural response falling on either side of this resonance frequency (Hutcheon and Yarom, 2000; Narayanan and Johnston, 2007; Narayanan and Johnston, 2008; Das et al., 2017). More specifically, excitatory (Erchova et al., 2004; Giocomo et al., 2007; Nolan et al., 2007; Garden et al., 2008; Pastoll et al., 2012) and inhibitory neurons (Boehlen et al., 2016) in the superficial layers of the medial entorhinal cortex manifest resonance in the theta frequency range (4–10 Hz). Therefore, the model of individual neurons in the CAN model as integrators of afferent activity is inconsistent with the resonating structure intrinsic to their physiological counterparts. Intrinsic resonance in neurons is mediated by the expression of slow restorative ion channels, such as the HCN or the M-type potassium, which mediate resonating conductances that suppress low-frequency inputs by virtue of their kinetics and voltage-dependent properties (Hutcheon and Yarom, 2000; Narayanan and Johnston, 2008; Hu et al., 2009). As the incorporation of biological heterogeneities predominantly altered low-frequency neural activity (Figure 4), we hypothesized that the expression of intrinsic neuronal resonance (especially the associated suppression of low-frequency components) could counteract the disruptive impact of biological heterogeneities on network activity, thereby stabilizing grid-like activity patterns.

An essential requirement in testing this hypothesis was to introduce intrinsic resonance in the rate-based neuronal models in our network. To do this, we noted that the integrative properties of the integrator model neuron are mediated by the low-pass filtering kinetics associated with the parameter $\tau$. We confirmed the low-pass filter (LPF) characteristics of the integrator neurons by recording the response of individual neurons to a chirp stimulus (Figure 5A,B). As this provides an equivalent to the LPF associated with the membrane time constant, we needed a HPF to mimic resonating conductances that suppress low-frequency activity in biological neurons. A simple approach to suppress low-frequency components is to introduce a differentiator, which we confirmed by passing a chirp stimulus through a differentiator (Figure 5A,C). We therefore passed the outcome of the integrator (endowed with LPF characteristics) through a first-order differentiator (HPF), with the postulate that the net transfer function would manifest resonance. We tested this by first subjecting a chirp stimulus to the integrator neuron dynamics and feeding that output to the differentiator. We found that the net output expressed resonance, acting as a band-pass filter (Figure 5A,C).

Figure 5

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Physiologically, tuning of intrinsic resonance to specific frequencies could be achieved by altering the characteristics of the HPF or the LPF (Hutcheon and Yarom, 2000; Narayanan and Johnston, 2008; Das et al., 2017; Mittal and Narayanan, 2018; Rathour and Narayanan, 2019). Matching this physiological tuning procedure, we tuned resonance frequency ($f_{\mathrm{R}}$) in our rate-based resonator model either by changing $\tau$ that governs the LPF (Figure 5D,E) or by altering an exponent $\epsilon$ (Equation 9) that regulated the slope of the HPF on the frequency axis (Figure 5F–G). We found $f_{\mathrm{R}}$ to decrease with an increase in $\tau$ (Figure 5E) or a reduction in $\epsilon$ (Figure 5G). In summary, we mimicked the biophysical mechanisms governing resonance in neural structures to develop a phenomenological methodology to introduce and tune intrinsic resonance, yielding a tunable band-pass filtering structure in rate-based model neurons. In this model, as resonance was introduced through a HPF, a formulation that does not follow the mechanistic basis of resonance in physiological systems, we refer this as a phenomenological model for resonating neurons.

Homogeneous CAN models constructed with phenomenological resonator neurons exhibited stable grid-patterned neural activity

How does the expression of intrinsic resonance in individual neurons of CAN models alter their grid-patterned neural activity? How do grid patterns respond to changes in resonance frequency, realized either by altering $\tau$ or $\epsilon$? To address these, we replaced all neurons in the homogeneous CAN model with theta-frequency resonators and presented the network with the same virtual trajectory (Figure 2E). We found that CAN models with resonators were able to reliably and robustly produce grid-patterned neural activity, which were qualitatively (Figure 6A,B) and quantitatively (Figure 6C–F, Figure 6—figure supplement 1) similar to patterns produced by networks of integrator neurons across different values of $\tau$. Importantly, increase in $\tau$ markedly increased spacing between grid fields (Figure 6D) and their average size (Figure 6E), consequently reducing the number of grid fields within the arena (Figure 6F). It should be noted that the reduction in grid score with increased $\tau$ (Figure 6C) is merely a reflection of the reduction in the number of grid fields within the arena, and not indicative of loss of grid-patterned activity. Although the average firing rates were tuned to be similar across the resonator and the integrator networks (Figure 6—figure supplement 1A), the peak-firing rate in the resonator network was significantly higher (Figure 6—figure supplement 1B). In addition, for both resonator and integrator networks, consistent with increases in grid-field size and spacing, there were marked increases in information rate (Figure 6—figure supplement 1C) and reductions in sparsity (Figure 6—figure supplement 1D), with increase in $\tau$.

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Within our model framework, it is important to emphasize that although increasing $\tau$ reduced $f_{\mathrm{R}}$ in resonator neurons (Figures 5E and 6B), the changes in grid spacing and size are not a result of change in $f_{\mathrm{R}}$, but a consequence of altered $\tau$. This inference follows from the observation that altering $\tau$ has qualitatively and quantitatively similar outcomes on grid spacing and size in both integrator and resonator networks (Figure 6). Further confirmation for the absence of a direct role for $f_{\mathrm{R}}$ in regulating grid spacing or size (within our modeling framework) came from the invariance of grid spacing and size to change in $\epsilon$, which altered $f_{\mathrm{R}}$ (Figure 7). Specifically, when we fixed $\tau$, and altered $\epsilon$ across resonator neurons in the homogeneous CAN model, $f_{\mathrm{R}}$ of individual neurons changed (Figure 7A), but did not markedly change grid field patterns (Figure 7A), grid score, average grid spacing, mean size, number, or sparsity of grid fields (Figure 7B). However, increasing $\epsilon$ decreased grid-field sizes, the average and the peak firing rates, consequently reducing the information rate in the activity pattern (Figure 7B).

Figure 7

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Together, these results demonstrated that homogeneous CAN models with phenomenological resonators reliably and robustly produce grid-patterned neural activity. In these models, the LPF regulated the size and spacing of the grid fields and the HPF governed the magnitude of activity and the suppression of low-frequency inputs.

Phenomenological resonators stabilized the emergence of grid-patterned activity in heterogeneous CAN models

Having incorporated intrinsic resonance into the CAN model, we were now equipped to directly test our hypothesis that the expression of intrinsic neuronal resonance could stabilize grid-like activity patterns in heterogeneous networks. We incorporated heterogeneities, retaining the same forms/degrees of heterogeneities (Table 1; Figure 2A–D) and the same virtual trajectory (Figure 2E), in a CAN model endowed with resonators to obtain the spatial activity maps for individual neurons (Figure 8A). Strikingly, we observed that the presence of resonating neurons in the CAN model stabilized grid-patterned activity in individual neurons, despite the introduction of the highest degrees of all forms of biological heterogeneities (Figure 8A). Importantly, outputs produced by the homogeneous CAN model manifested spatially precise circular grid fields, exhibiting regular triangular firing patterns across trials (e.g., Figure 1A), which constituted forms of precision that are not observed in electrophysiologically obtained grid-field patterns (Fyhn et al., 2004; Hafting et al., 2005). However, that with the incorporation of biological heterogeneities in resonator neuron networks, the imprecise shapes and stable patterns of grid-patterned firing (e.g., Figure 8A) tend closer to those of biologically observed grid cells. Quantitatively, we computed grid-cell measurements across all the 3600 neurons in the network and found that all quantitative measurements (Figure 8B–I, Figure 8—figure supplement 1), including the grid score (Figure 8B) and information rate (Figure 8H), were remarkably robust to the introduction of heterogeneities (cf. Figures 2–3 for CAN models with integrator neurons).

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To ensure that the robust emergence of stable grid-patterned activity in heterogeneous CAN models with resonator neurons was not an artifact of the specific network under consideration, we repeated our analyses with two additional sets of integrator-resonator comparisons (Figure 8—figure supplement 2). In these networks, we employed a baseline $\tau$ of either 14 ms (Figure 8—figure supplement 2A) or 8 ms (Figure 8—figure supplement 2B) instead of the 10 ms value employed in the default network (Figures 2 and 8). A pairwise comparison of all grid-cell measurements between networks with integrator vs. resonator neurons demonstrated that the resonator networks were resilient to the introduction of heterogeneities compared to integrator networks (Figure 8—figure supplement 2).

Together, these results demonstrated that phenomenologically incorporating intrinsic resonance into neurons of the CAN model imparts resilience to the network, facilitating the robust emergence of stable grid-cell activity patterns, despite the expression of biological heterogeneities in neuronal and synaptic properties.

Mechanistic model of neuronal intrinsic resonance: incorporating a slow activity-dependent negative feedback loop

In the previous sections, we had phenomenologically introduced intrinsic resonance in rate-based neurons, by incorporating an artificial high-pass filter to the low-pass filtering property of an integrator neuron. Physiologically, however, intrinsic resonance results from the presence of resonating conductances in the neural system, which introduce a slow activity-dependent negative feedback into the neuron. To elaborate, there are two requirements for any conductance to act as a resonating conductance (Hutcheon and Yarom, 2000). First, the current mediated by the conductance should actively suppress any changes in membrane voltage of neurons. In resonating conductances, this is implemented by the voltage-dependent gating properties of the ion channel mediating the resonating conductance whereby any change in membrane voltage (de)activates the channel in such a way that the channel current suppresses the change in membrane voltage. This constitutes an activity-dependent negative feedback loop. The second requirement for a resonating conductance is that it has to be slower compared to the membrane time constant. This requirement ensures that cutoff frequency of the lowpass filter (associated with the membrane time constant) is greater than that of the high-pass filter (mediated by the resonating conductance). Together these two requirements mechanistically translate to a slow activity-dependent negative feedback loop, a standard network motif in biological systems for yielding damped oscillations and resonance (Alon, 2019). In intrinsically resonating neurons, resonance is achieved because the resonating conductance actively suppresses low-frequency signals through this slow negative feedback loop, in conjunction with the suppression of higher frequency signals by the low-pass filter mediated by the membrane time constant. Resonating conductances do not suppress high frequencies because they are slow and would not activate/deactivate with faster high-frequency signals (Hutcheon and Yarom, 2000; Narayanan and Johnston, 2008).

In the single-neuron integrator model employed in this study, the LPF is implemented by the integration time constant τ. Inspired by the mechanisms behind the physiological emergence of neuronal intrinsic resonance, we mechanistically incorporated intrinsic resonance into this integrator neuronal model by introducing a slow negative feedback to the single neuron dynamics (Figure 9A). The dynamics of the coupled evolution of neuronal activity ($S$) and the feedback state variable ($m$) are provided in Equations (11–13). A sigmoidal feedback kernel ($m_{\infty }$) regulated the activity dependence, and the feedback time constant (${\mathrm{\tau }}_m$) determined the kinetics of the negative feedback loop (Equations 11 and 12). The response of this single-neuronal model, endowed with the slow-negative feedback loop, to a pulse input manifested typical sag associated with the expression of intrinsic resonance (Hutcheon and Yarom, 2000), owing to the slow evolution of the feedback state variable $m$ (Figure 9B,C). To directly test whether the introduction of the slow negative feedback loop yielded a resonating neuronal model, we assessed single-neuron output to a chirp stimulus (Figure 9D) and found the manifestation of intrinsic resonance when the feedback loop was activated (Figure 9E). As this resonator neuron model involving a slow negative feedback loop was derived from the mechanistic origins of intrinsic neuronal resonance, we refer to this as the mechanistic resonator model.

Figure 9

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Resonance frequency in the mechanistic resonator model was tunable by altering the parameters associated with the feedback loop. Specifically, increasing the half-maximal activity of the feedback kernel ($S_{1/2}$) reduced resonance frequency within the tested range of values (Figure 9F), whereas increasing the slope of the feedback kernel ($k$) enhanced $f_{\mathrm{R}}$ initially, but reduced $f_{\mathrm{R}}$ with further increases in $k$ (Figure 9G). Furthermore, enhancing the strength of the feedback ($g$) increased $f_{\mathrm{R}}$, and is reminiscent of increased $f_{\mathrm{R}}$ with increase in resonating conductance density (Narayanan and Johnston, 2008), the parameter that defines the strength of the negative feedback loop in intrinsic resonating neurons. An important requirement for the expression of resonance through the mechanistic route that we chose is that the feedback time constant (${\tau }_m$) has to be slower than the integration time constant ($\tau$ = 10 ms). To test this in our model, we altered ${\tau }_m$ of the feedback loop while maintaining all other values to be constant (Figure 9I) and found that resonance did not manifest with low values of ${\tau }_m$. With increase in ${\tau }_m$, there was an initial rapid increase in $f_{\mathrm{R}}$, which then reduced upon further increase in ${\tau }_m$ (Figure 9I). It may be noted that low values of ${\tau }_m$ translate to a fast negative feedback loop, thus emphasizing the need for a slow negative feedback loop for the expression of resonance. These observations are analogous to the dependence of resonance frequency in intrinsically resonating neurons on the activation time constant of the resonating conductance (Narayanan and Johnston, 2008). Together, we have proposed a tunable mechanistic model for intrinsic resonance in rate-based neurons through the incorporation of a slow negative feedback loop to the neuronal dynamics.

Mechanistic resonators stabilized the emergence of grid-patterned activity in heterogeneous CAN models

How was grid-patterned neural activity in a homogeneous CAN model affected by the expression of intrinsic neuronal resonance through a mechanistic framework? To address this, we replaced all the integrator neurons in the homogeneous CAN model (Figure 2E; homogeneous network) with mechanistic theta-frequency resonator neurons, while maintaining identical connectivity patterns and velocity dependent inputs. We observed that neurons within this homogeneous CAN model built with mechanistic resonators were able to produce distinct grid-patterned neural activity (Figure 10A). Sensitivity analyses showed that grid-field size and spacing were affected by the parameters associated with the slow negative feedback loop, consequently affecting the grid score (Figure 10B) and other measurements associated with the grid-patterned activity (Figure 10—figure supplements 1–2). Within the tested ranges, the strength of feedback ($g$) had limited effect on grid-patterned neural activity, whereas the slope ($k$) and half-maximal acivity ($S_{1/2}$) of the feedback kernel had strong impact on grid-field size and spacing (Figure 10). Although the grid fields were well-defined across all values of $k$ and $S_{1/2}$, the grid scores were lower for some cases (e.g., $k$=0.5; $S_{1/2}$=0.1 in Figure 10B) due to the smaller number of grid fields within the arena (Figure 10A). Importantly, the feedback time constant (${\tau }_m$) had very little impact on grid pattern neural activity (Figure 10A) and on all quantified measurements (Figure 10B, Figure 10—figure supplements 1–2). Together, these analyses demonstrated that a homogenous CAN model built with mechanistic resonators yielded stable and robust grid pattern neural activity whose grid field size and spacing can be tuned by parameters of slow negative feedback that governed resonance.

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Next, we systematically incorporated the different degrees of heterogeneities into the CAN model with mechanistic resonator neurons to test if resonance could stabilize grid-patterned activity in the heterogeneous CAN models. We employed 5 degrees of all heterogeneities with the same virtual trajectory (Figure 2E) in a CAN model endowed with mechanistic resonators to compute the spatial activity maps for individual neurons (Figure 11A). As with the network built with phenomenological resonators (Figure 8), we observed stable and robust grid-like spatial maps emerge even with highest degree of heterogeneities in the CAN model with mechanistic resonator neurons (Figure 11). Furthermore, compared to homogeneous model, the grid pattern activity obtained by the heterogeneous CAN model closely mimicked the grid-cell activity, endowed with imprecise grid field shapes observed in neurons in MEC region of the brain (Figure 11). Quantitatively, the grid score (Figure 11B), average spacing (Figure 11C), and mean grid field size (Figure 11G) were remarkably robust to the introduction of heterogeneities. However, the average (Figure 11E) and peak firing (Figure 11D) reduced with increase in degree of heterogeneities (Figure 11D) and consequently affected measurements that were dependent on firing rate (i.e., information rate and sparsity; Figure 11H–I). Together, these results demonstrate that intrinsic neuronal resonance, introduced either through phenomenological (Figure 8) or through mechanistic (Figure 11) single-neuron models, yielded stable and robust grid-like pattern in heterogeneous CAN models.

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The grid-cell activity on a 2D plane represents a spatially repetitive periodic pattern. Consequently, a phase plane constructed from activity along the diagonal of the 2D plane on one axis (Figure 11—figure supplement 1A), and the spatial derivative of this activity on the other should provide a visual representation of the impact of heterogeneities and resonance on this periodic pattern of activity. To visualize the specific changes in the periodic orbits obtained with homogeneous and heterogeneous networks constructed with integrators or resonators, we plotted the diagonal activity profiles of five randomly picked neurons using the phase plane representation (Figure 11—figure supplement 1B). This visual representation confirmed that homogenous CAN models built of integrators or resonators yielded stable closed orbit trajectories, representing similarly robust grid-patterned periodic activity (Figure 11—figure supplement 1B). The disparate sizes of different orbits are merely a reflection of the intersection of the diagonal with a given grid field. Specifically, if the grid fields were considered to be ideal circles, the orbital size is the largest if diagonal passes through the diameter, with orbital size reducing for any other chord. Upon introduction of heterogeneities, the phase-plane plots of diagonal activity profiles from the heterogeneous integrator network lacked closed orbits (Figure 11—figure supplement 1B). This is consistent with drastic reductions in grid score and information rate for these heterogeneous network models (Figures 2–3). In striking contrast, the phase-plane plots from the heterogeneous resonator network manifested closed orbits even with the highest degree of heterogeneities introduced, irrespective of whether resonance was introduced through a phenomenological or a mechanistic model (Figure 11—figure supplement 1B). Although these phase-plane trajectories were noisy compared to those from the homogeneous resonator network, the presence of closed orbits indicated the manifestation of spatial periodicity in these activity patterns (Figure 11—figure supplement 1B). These observations visually demonstrated that resonator neurons stabilized the heterogeneous network and maintained spatial periodicity in grid-cell activity, irrespective of whether resonance was introduced through a phenomenological or a mechanistic model.

The slow kinetics of the negative feedback loop in mechanistic resonators is a critical requirement for stabilizing heterogeneous CAN models

Our rationale behind the introduction of intrinsic resonance into the neuron was that it would suppress the low-frequency perturbations introduced by biological heterogeneities (Figure 4). Consequently, our hypothesis is that the targeted suppression of low-frequency components resulted in the stabilization of the CAN model. Two lines of evidence for this hypothesis were that the suppression of low-frequency components through phenomenological (Figure 8) or through mechanistic (Figure 11) resonators resulted in stabilization of the heterogeneous CAN models. An advantage of recruiting a mechanistic model for introducing resonance is that sensitivity analyses on its parameters could provide valuable mechanistic insights about how such stabilization is achieved. With reference to specific hypothesis, the ability to tune resonance by altering the feedback time constant ${\tau }_m$ (Figure 9I), without altering the feedback kernel or the feedback strength, provides an efficient route to understand the mechanistic origins of the stabilization. Specifically, the value of ${\tau }_m$ (default value = 75 ms) governs the slow kinetics of the feedback loop and is the source for the targeted suppression of low-frequency components. Reducing ${\tau }_m$ would imply a faster negative feedback loop, thereby suppressing even higher-frequency components. As a further test of our hypothesis on the role of suppressing low-frequency components in stabilizing the CAN network, we asked if mechanistic resonators with lower values of ${\tau }_m$ would be able to stabilize heterogeneous CAN models (Figure 12). If fast negative feedback loops (i.e., low values of ${\tau }_m$) were sufficient to stabilize heterogeneous CAN models, that would counter our hypothesis on the requirement of targeted suppression of low-frequency components. To the contrary, we found that heterogeneous CAN networks with neurons endowed with fast negative feedback loops were incapable of stabilizing the grid-cell network (Figure 12). With progressive increase in ${\tau }_m$, the grid-patterned firing stabilized even for high degrees of heterogeneities (Figure 12B,C for low values of ${\tau }_m$; Figure 11B for ${\tau }_m$ = 75 ms), thus providing a direct additional line of evidence supporting our hypothesis on the need for targeted suppression of low-frequency components in stabilizing the network. We noted that the impact of altering ${\tau }_m$ was specifically related to stabilizing heterogeneous networks by suppressing heterogeneities-driven low-frequency perturbations because altering ${\tau }_m$ did not alter grid-patterned activity in homogeneous resonator networks (Figure 9A,B). Together, our results provide multiple lines of evidence that the slow kinetics of the negative feedback loop in single-neuron dynamics (Figure 9A) mediates targeted suppression of low-frequency signals (Figure 9D,E), thereby yielding intrinsic neuronal resonance (Figure 9I) and stabilizing grid-patterned activity in heterogeneous CAN models (Figures 11 and 12).

Figure 12

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Resonating neurons suppressed the impact of biological heterogeneities on low-frequency components of neural activity

As detailed above, our hypothesis on a potential role for intrinsic neuronal resonance in stabilizing grid-patterned firing in heterogeneous CAN models was centered on the ability of resonators in specifically suppressing low-frequency components. Although our analyses provided evidence for stabilization of grid-patterned firing with phenomenological (Figure 8) or mechanistic (Figure 11) resonators, did resonance specifically suppress the impact of biological heterogeneities of low-frequency components of neural activity? To assess this, we performed frequency-dependent analyses of neural activity in CAN models with phenomenological or mechanistic resonators (Figure 13, Figure 13—figure supplements 1–3) and compared them with integrator-based CAN models (Figure 4). First, we found that the variance in the deviations of neural activity in heterogeneous networks with reference to the respective homogeneous models were considerably lower with resonator neurons (both phenomenological and mechanistic models), compared to their integrator counterparts (Figure 13A,B, Figure 13D,E, Figure 13—figure supplements 1–3). Second, comparing the relative power of neural activity across different octaves, we found that network with resonators suppressed lower frequencies (predominantly the 0–2 Hz band) and enhanced power in the range of neuronal resonance frequencies, when compared with their integrator counterparts (Figure 13C, Figure 13F, Figure 13—figure supplements 1–3). This relative suppression of low-frequency power accompanied by the relative enhancement of high-frequency power was observed across all networks with resonator, either homogeneous (Figure 13—figure supplements 1–2) or heterogeneous with distinct forms of heterogeneities (Figure 13—figure supplement 3G–I). Importantly, given the slow activity-dependent negative feedback loop involved in mechanistic resonators, the low-frequency suppression was found to be extremely effective across all degrees of heterogeneities (Figure 13D,E) with minimal increases of power in high-frequency bands (Figure 13F) compared to their phenomenological counterparts (Figure 13A–C). The phenomenological resonators were built with a simple HPF that is not endowed with activity-dependent filtering capabilities. In addition, the derivative-based implementation of the phenomenological resonator model yielded spurious high-frequency power, which was averted with the slow activity-dependent negative feedback loop incorporated into the mechanistic resonator model (Figure 13D–F).

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Together, our results demonstrated that biological heterogeneities predominantly altered low-frequency components of neural activity, and provided strong quantitative lines of evidence that intrinsic neuronal resonance plays a stabilizing role in heterogeneous networks by targeted suppression of low-frequency inputs, thereby counteracting the disruptive impact of biological heterogeneities on low-frequency components.

The principal conclusion of this study is that intrinsic neuronal resonance stabilizes heterogeneous 2D CAN models, by suppressing heterogeneity-induced perturbations in low-frequency components of neural activity. Our analyses provided the following lines of evidence in support of this conclusion:

1. Neural-circuit heterogeneities destabilized grid-patterned activity generation in a 2D CAN model (Figures 2 and 3).

2. Neural-circuit heterogeneities predominantly introduced perturbations in the low-frequency components of neural activity (Figure 4).

3. Targeted suppression of low-frequency components through phenomenological (Figure 5C) or through mechanistic (Figure 9D) resonators resulted in stabilization of the heterogeneous CAN models (Figures 8 and 11). Thus, stabilization was achieved irrespective of the means employed to suppress low-frequency components: an activity-independent suppression of low frequencies (Figure 5) or an activity-dependent slow negative feedback loop (Figure 9).

4. Changing the feedback time constant ${\tau }_m$ 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 low-frequency components, was essential in stabilizing grid-patterned activity (Figure 12). As the slow negative feedback loop and the resultant suppression of low frequencies mediates intrinsic neuronal 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 (Figure 13D–F) resonators specifically suppressed lower frequencies of activity in the 2D CAN model.

All data generated or analyzed during this study are included in the manuscript and supporting files. The custom-written simulations and analyses code (in MATLAB) employed for simulations are available as source code.

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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 network-level computations.

Decision letter after peer review:

Thank you for submitting your article "Resonating neurons stabilize heterogeneous grid-cell 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 single-neuron 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 grid-cell 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 non-oscillatory 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 self-organization 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 band-pass units, and show that they produce grids without any recurrent network. It will be fun.

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 single-neuron 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 single-neuron resonance that matches the dynamical behavior of physiological resonators. Specifically, we noted that physiological resonance is elicited by a slow activity-dependent negative feedback (Hutcheon and Yarom, 2000). To incorporate resonance into our rate-based model neurons, we mimicked this by introducing a slow negative feedback loop into our single-neuron dynamics (the motivations are elaborated in the new results subsection “Mechanistic model of neuronal intrinsic resonance: Incorporating a slow activity-dependent negative feedback loop”). The single-neuron 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 steady-state feedback kernel (𝑚_∞) of the negative feedback is sigmoidally dependent on the output of the neuron (𝑆), defined by two parameters: half-maximal activity (𝑆_1/2) and slope (𝑘). The single-neuron dynamics are elaborated in detail in the methods section (new subsection: Mechanistic model for introducing intrinsic resonance in rate-based neurons).

We first demonstrate that the introduction of a slow-negative feedback loop introduce

resonance into single-neuron 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 grid-patterned activity in the 2D CAN model employed here, with clear dependencies on model parameters (new Figure 10; new Figure 10-Supplements1–2). Next, we incorporated heterogeneities into the network and demonstrated that the introduction of resonance through a negative feedback loop stabilized grid-patterned 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 grid-patterned 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 grid-patterned 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 grid-patterned 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 low-frequency perturbations in heterogeneous CAN networks.

Together, the central hypothesis in our study was that intrinsic neuronal resonance could stabilize heterogeneous grid-cell networks through targeted suppression of low-frequency 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. Neural-circuit heterogeneities destabilized grid-patterned activity generation in a 2D CAN model (Figures 2–3).

2. Neural-circuit heterogeneities predominantly introduced perturbations in the low-frequency components of neural activity (Figure 4).

3. Targeted suppression of low-frequency 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 low-frequency components: an activity-independent suppression of low-frequencies (Figure 5) or an activity-dependent 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 low-frequency components, was essential in stabilizing grid-patterned 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 well-established 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 cellular-scale activity dependent negative feedback mechanism, a specific instance of a well-established 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 physiologically-rooted 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 single-neuron models (integrators, phenomenological resonators and mechanistic resonators) in the methods section (equations 6–13), also providing detailed sensitivity analyses with reference to their frequency-dependent 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 physiologically-rooted 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 grid-patterned activity in heterogeneous CAN models (new Figure 12). The incorporation of resonance through a negative feedback loop provides further intuition through the well-established 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 rate-based CAN model presented here (Discussion subsection: “Future directions and considerations in model interpretation”). We postulate that mechanisms that suppress low-frequency components could be a generalized route to stabilize heterogeneous biological networks, based on our analyses and based on the well-established stabilizing role of negative feedback loops. However, we do not claim generalization to other networks or to other models for generating grid-patterned 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 low-frequency components as a consequence of introducing heterogeneities. Finally, if the introduction of heterogeneities destabilized network function and resulted in low-frequency 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 heterogeneity-specific 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 grid-patterned 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 grid-patterned 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 grid-cell 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 time-frequency 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 octave-based analyses (which doesn’t involve normalized differences) to strengthen our claims about the impact of heterogeneities and resonance on different bands of frequency. These octave-based analyses also confirm our conclusions on the impact of heterogeneities and neuronal resonance on low-frequency 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 signal-specific tiling of the time-frequency plane, we argue that the impact of heterogeneities or the resonators on low-frequency 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 non-oscillatory 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 grid-cell networks through targeted suppression of low-frequency 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. Neural-circuit heterogeneities destabilized grid-patterned activity generation in a 2D CAN model (Figures 2–3).

2. Neural-circuit heterogeneities predominantly introduced perturbations in the low frequency components of neural activity (Figure 4).

3. Targeted suppression of low-frequency 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 low-frequency components: an activity-independent suppression of low-frequencies (Figure 5) or an activity-dependent 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 low-frequency components, was essential in stabilizing grid-patterned 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 well-established 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 cellular-scale activity-dependent negative feedback mechanism, a specific instance of a well-established 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 well-established 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 low-frequency 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 stability-inducing cellular-scale activity-dependent negative feedback mechanism.

Reviewer #2 (Recommendations for the authors):

I believe in self-organization 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 band-pass 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 self-organization 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 stability-through-negative feedback literature provides stronger support for our original conclusions. With reference to the grid-cell 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 CAN-based grid-cell networks. We note that addressing the question on grid-patterned activity with individual resonator neurons, in conjunction place-cell inputs and intrinsic/synaptic plasticity, would answer the question of generation of grid-like 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 grid-patterned activity. We have stated the limitations of our analyses in the Discussion section, also suggesting directions for future research involving other models for grid-patterned 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 step-by-step 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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Author details

1. Divyansh Mittal

   Cellular Neurophysiology Laboratory, Molecular Biophysics Unit, Indian Institute of Science, Bangalore, India

   Contribution

   Conceptualization, Software, Formal analysis, Validation, Investigation, Visualization, Methodology, Writing - original draft, Writing - review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-4233-8176

2. Rishikesh Narayanan

   Cellular Neurophysiology Laboratory, Molecular Biophysics Unit, Indian Institute of Science, Bangalore, India

   Contribution

   Conceptualization, Resources, Formal analysis, Supervision, Funding acquisition, Investigation, Visualization, Methodology, Writing - original draft, Project administration, Writing - review and editing

   For correspondence

   rishi@iisc.ac.in

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-1362-4635

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