Structure, dynamics, coding and optimal biophysical parameters of efficient excitatory-inhibitory spiking networks

Reviewer #1 (Public Review):

Koren et al. derive and analyse a spiking network model optimised to represent external signals using the minimum number of spikes. Unlike most prior work using a similar setup, the network includes separate populations of excitatory and inhibitory neurons. The authors show that the optimised connectivity has a like-to-like structure, leading to the experimentally observed phenomenon of feature competition. They also characterise the impact of various (hyper)parameters, such as adaptation timescale, ratio of excitatory to inhibitory cells, regularisation strength, and background current. These results add useful biological realism to a particular model of efficient coding. However, not all claims seem fully supported by the evidence. Specifically, several biological features, such as the ratio of excitatory to inhibitory neurons, which the authors claim to explain through efficient coding, might be contingent on arbitrary modelling choices. In addition, earlier work has already established the importance of structured connectivity for feature competition. A clearer presentation of modelling choices, limitations, and prior work could improve the manuscript.

Major comments:

(1) Much is made of the 4:1 ratio between excitatory and inhibitory neurons, which the authors claim to explain through efficient coding. I see two issues with this conclusion: (i) The 4:1 ratio is specific to rodents; humans have an approximate 2:1 ratio (see Fang & Xia et al., Science 2022 and references therein); (ii) the optimal ratio in the model depends on a seemingly arbitrary choice of hyperparameters, particularly the weighting of encoding error versus metabolic cost. This second concern applies to several other results, including the strength of inhibitory versus excitatory synapses. While the model can, therefore, be made consistent with biological data, this requires auxiliary assumptions.

(2) A growing body of evidence supports the importance of structured E-I and I-E connectivity for feature selectivity and response to perturbations. For example, this is a major conclusion from the Oldenburg paper (reference 62 in the manuscript), which includes extensive modelling work. Similar conclusions can be found in work from Znamenskiy and colleagues (experiments and spiking network model; bioRxiv 2018, Neuron 2023 (ref. 82)), Sadeh & Clopath (rate network; eLife, 2020), and Mackwood et al. (rate network with plasticity; eLife, 2021). The current manuscript adds to this evidence by showing that (a particular implementation of) efficient coding in spiking networks leads to structured connectivity. The fact that this structured connectivity then explains perturbation responses is, in the light of earlier findings, not new.

(3) The model's limitations are hard to discern, being relegated to the manuscript's last and rather equivocal paragraph. For instance, the lack of recurrent excitation, crucial in neural dynamics and computation, likely influences the results: neuronal time constants must be as large as the target readout (Figure 4), presumably because the network cannot integrate the signal without recurrent excitation. However, this and other results are not presented in tandem with relevant caveats.

(4) On repeated occasions, results from the model are referred to as predictions claimed to match the data. A prediction is a statement about what will happen in the future - but most of the "predictions" from the model are actually findings that broadly match earlier experimental results, making them "postdictions". This distinction is important: compared to postdictions, predictions are a much stronger test because they are falsifiable. This is especially relevant given (my impression) that key parameters of the model were tweaked to match the data.

Reviewer #2 (Public Review):

Summary:

In this work, the authors present a biologically plausible, efficient E-I spiking network model and study various aspects of the model and its relation to experimental observations. This includes a derivation of the network into two (E-I) populations, the study of single-neuron perturbations and lateral-inhibition, the study of the effects of adaptation and metabolic cost, and considerations of optimal parameters. From this, they conclude that their work puts forth a plausible implementation of efficient coding that matches several experimental findings, including feature-specific inhibition, tight instantaneous balance, a 4 to 1 ratio of excitatory to inhibitory neurons, and a 3 to 1 ratio of I-I to E-I connectivity strength. It thus argues that some of these observations may come as a direct consequence of efficient coding.

Strengths:

While many network implementations of efficient coding have been developed, such normative models are often abstract and lacking sufficient detail to compare directly to experiments. The intention of this work to produce a more plausible and efficient spiking model and compare it with experimental data is important and necessary in order to test these models.

In rigorously deriving the model with real physical units, this work maps efficient spiking networks onto other more classical biophysical spiking neuron models. It also attempts to compare the model to recent single-neuron perturbation experiments, as well as some long-standing puzzles about neural circuits, such as the presence of separate excitatory and inhibitory neurons, the ratio of excitatory to inhibitory neurons, and E/I balance. One of the primary goals of this paper, to determine if these are merely biological constraints or come from some normative efficient coding objective, is also important.

Though several of the observations have been reported and studied before (see below), this work arguably studies them in more depth, which could be useful for comparing more directly to experiments.

Weaknesses:

Though the text of the paper may suggest otherwise, many of the modeling choices and observations found in the paper have been introduced in previous work on efficient spiking models, thereby making this work somewhat repetitive and incremental at times. This includes the derivation of the network into separate excitatory and inhibitory populations, discussion of physical units, comparison of voltage versus spike-timing correlations, and instantaneous E/I balance, all of which can be found in one of the first efficient spiking network papers (Boerlin et al. 2013), as well as in subsequent papers. Metabolic cost and slow adaptation currents were also presented in a previous study (Gutierrez & Deneve 2019). Though it is perfectly fine and reasonable to build upon these previous studies, the language of the text gives them insufficient credit.

Furthermore, the paper makes several claims of optimality that are not convincing enough, as they are only verified by a limited parameter sweep of single parameters at a time, are unintuitive and may be in conflict with previous findings of efficient spiking networks. This includes the following. Coding error (RMSE) has a minimum at intermediate metabolic cost (Figure 5B), despite the fact that intuitively, zero metabolic cost would indicate that the network is solely minimizing coding error and that previous work has suggested that additional costs bias the output. Coding error also appears to have a minimum at intermediate values of the ratio of E to I neurons (effectively the number of I neurons) and the number of encoded variables (Figures 6D, 7B). These both have to do with the redundancy in the network (number of neurons for each encoded variable), and previous work suggests that networks can code for arbitrary numbers of variables provided the redundancy is high enough (e.g., Calaim et al. 2022). Lastly, the performance of the E-I variant of the network is shown to be better than that of a single cell type (1CT: Figure 7C, D). Given that the E-I network is performing a similar computation as to the 1CT model but with more neurons (i.e., instead of an E neuron directly providing lateral inhibition to its neighbor, it goes through an interneuron), this is unintuitive and again not supported by previous work. These may be valid emergent properties of the E-I spiking network derived here, but their presentation and description are not sufficient to determine this.

Alternatively, the methodology of the model suggests that ad hoc modeling choices may be playing a role. For example, an arbitrary weighting of coding error and metabolic cost of 0.7 to 0.3, respectively, is chosen without mention of how this affects the results. Furthermore, the scaling of synaptic weights appears to be controlled separately for each connection type in the network (Table 1), despite the fact that some of these quantities are likely linked in the optimal network derivation. Finally, the optimal threshold and metabolic constants are an order of magnitude larger than the synaptic weights (Table 1). All of these considerations suggest one of the following two possibilities. One, the model has a substantial number of unconstrained parameters to tune, in which case more parameter sweeps would be necessary to definitively make claims of optimality. Or two, parameters are being decoupled from those constrained by the optimal derivation, and the optima simply corresponds to the values that should come out of the derivation.

Reviewer #3 (Public Review):

Summary:

In their paper the authors tackle three things at once in a theoretical model: how can spiking neural networks perform efficient coding, how can such networks limit the energy use at the same time, and how can this be done in a more biologically realistic way than previous work?

They start by working from a long-running theory on how networks operating in a precisely balanced state can perform efficient coding. First, they assume split networks of excitatory (E) and inhibitory (I) neurons. The E neurons have the task to represent some lower dimensional input signal, and the I neurons have the task to represent the signal represented by the E neurons. Additionally, the E and I populations should minimize an energy cost represented by the sum of all spikes. All this results in two loss functions for the E and I populations, and the networks are then derived by assuming E and I neurons should only spike if this improves their respective loss. This results in networks of spiking neurons that live in a balanced state, and can accurately represent the network inputs.

They then investigate in-depth different aspects of the resulting networks, such as responses to perturbations, the effect of following Dale's law, spiking statistics, the excitation (E)/inhibition (I) balance, optimal E/I cell ratios, and others. Overall, they expand on previous work by taking a more biological angle on the theory and showing the networks can operate in a biologically realistic regime.

Strengths:

(1) The authors take a much more biological angle on the efficient spiking networks theory than previous work, which is an essential contribution to the field.

(2) They make a very extensive investigation of many aspects of the network in this context, and do so thoroughly.

(3) They put sensible constraints on their networks, while still maintaining the good properties these networks should have.

Weaknesses:

(1) The paper has somewhat overstated the significance of their theoretical contributions, and should make much clearer what aspects of the derivations are novel. Large parts were done in very similar ways in previous papers. Specifically: the split into E and I neurons was also done in Boerlin et al (2008) and in Barrett et al (2016). Defining the networks in terms of realistic units was already done by Boerlin et al (2008). It would also be worth it to discuss Barrett et al (2016) specifically more, as there they also use split E/I networks and perform biologically relevant experiments.

(2) It is not clear from an optimization perspective why the split into E and I neurons and following Dale's law would be beneficial. While the constraints of Dale's law are sensible (splitting the population in E and I neurons, and removing any non-Dalian connection), they are imposed from biology and not from any coding principles. A discussion of how this could be done would be much appreciated, and in the main text, this should be made clear.

(3) Related to the previous point, the claim that the network with split E and I neurons has a lower average loss than a 1 cell-type (1-CT) network seems incorrect to me. Only the E population coding error should be compared to the 1-CT network loss, or the sum of the E and I populations (not their average). In my author recommendations, I go more in-depth on this point.

(4) While the paper is supposed to bring the balanced spiking networks they consider in a more experimentally relevant context, for experimental audiences I don't think it is easy to follow how the model works, and I recommend reworking both the main text and methods to improve on that aspect.

Assessment and context:

Overall, although much of the underlying theory is not necessarily new, the work provides an important addition to the field. The authors succeeded well in their goal of making the networks more biologically realistic, and incorporating aspects of energy efficiency. For computational neuroscientists, this paper is a good example of how to build models that link well to experimental knowledge and constraints, while still being computationally and mathematically tractable. For experimental readers, the model provides a clearer link between efficient coding spiking networks to known experimental constraints and provides a few predictions.

Author response:

eLife assessment

This study offers a useful treatment of how the population of excitatory and inhibitory neurons integrates principles of energy efficiency in their coding strategies. The analysis provides a comprehensive characterisation of the model, highlighting the structured connectivity between excitatory and inhibitory neurons. However, the manuscript provides an incomplete motivation for parameter choices. Furthermore, the work is insufficiently contextualized within the literature, and some of the findings appear overlapping and incremental given previous work.

We thank the Reviewers and the Reviewing Editor for taking time to provide extremely valuable suggestions and comments, which will help us to substantially improve our paper. In what follows we summarize our current plan to improve the paper taking up on their suggestions.

Public Reviews:

Reviewer #1 (Public Review):

Summary: Koren et al. derive and analyse a spiking network model optimised to represent external signals using the minimum number of spikes. Unlike most prior work using a similar setup, the network includes separate populations of excitatory and inhibitory neurons. The authors show that the optimised connectivity has a like-to-like structure, leading to the experimentally observed phenomenon of feature competition. They also characterise the impact of various (hyper)parameters, such as adaptation timescale, ratio of excitatory to inhibitory cells, regularisation strength, and background current. These results add useful biological realism to a particular model of efficient coding. However, not all claims seem fully supported by the evidence. Specifically, several biological features, such as the ratio of excitatory to inhibitory neurons, which the authors claim to explain through efficient coding, might be contingent on arbitrary modelling choices. In addition, earlier work has already established the importance of structured connectivity for feature competition. A clearer presentation of modelling choices, limitations, and prior work could improve the manuscript.

Thanks for these insights and for this summary of our work.

Major comments:

(1) Much is made of the 4:1 ratio between excitatory and inhibitory neurons, which the authors claim to explain through efficient coding. I see two issues with this conclusion: (i) The 4:1 ratio is specific to rodents; humans have an approximate 2:1 ratio (see Fang & Xia et al., Science 2022 and references therein); (ii) the optimal ratio in the model depends on a seemingly arbitrary choice of hyperparameters, particularly the weighting of encoding error versus metabolic cost. This second concern applies to several other results, including the strength of inhibitory versus excitatory synapses. While the model can, therefore, be made consistent with biological data, this requires auxiliary assumptions.

We will describe better the ratio of numbers of E and I neurons found in real data, as suggested. The first submission already contained an analysis of how this ratio of neuron numbers depends on the weighting of the loss of E and I neurons and on the relative weighting of the encoding error vs the metabolic cost in the loss function (see Fig 6E). We will make sure that these results are suitably expanded and better emphasized in revision. We will also include new analysis of dependence of optimal parameters on the relative weighting of encoding error vs metabolic cost in the loss function when studying other parameters (namely: noise intensity, metabolic constant, ratio of mean I-I to E-I connectivity, time constants of single E and I neurons).

(2) A growing body of evidence supports the importance of structured E-I and I-E connectivity for feature selectivity and response to perturbations. For example, this is a major conclusion from the Oldenburg paper (reference 62 in the manuscript), which includes extensive modelling work. Similar conclusions can be found in work from Znamenskiy and colleagues (experiments and spiking network model; bioRxiv 2018, Neuron 2023 (ref. 82)), Sadeh & Clopath (rate network; eLife, 2020), and Mackwood et al. (rate network with plasticity; eLife, 2021). The current manuscript adds to this evidence by showing that (a particular implementation of) efficient coding in spiking networks leads to structured connectivity. The fact that this structured connectivity then explains perturbation responses is, in the light of earlier findings, not new.

We agree that the main contribution of our manuscript in this respect is to show how efficient coding in spiking networks can lead to structured connectivity similar to those proposed in the above papers. We apologize if this was not clear enough in the previous version. We will make it clearer in revision. We nevertheless think it useful to report the effects of perturbations within this network because the structure derived in our network is not identical to those studied in the above paper, and because these results give information about how lateral inhibition works in this network. Thus, we will keep presenting it in the revised version, although we will de-emphasize and simplify its presentation to give more emphasis to the novelty of the derivation of this connectivity rule from the principles of efficient coding.

(3) The model's limitations are hard to discern, being relegated to the manuscript's last and rather equivocal paragraph. For instance, the lack of recurrent excitation, crucial in neural dynamics and computation, likely influences the results: neuronal time constants must be as large as the target readout (Figure 4), presumably because the network cannot integrate the signal without recurrent excitation. However, this and other results are not presented in tandem with relevant caveats.

We will improve the Limitations paragraph in Discussion, and also anticipate caveats in tandem with results when needed, as suggested.

(4) On repeated occasions, results from the model are referred to as predictions claimed to match the data. A prediction is a statement about what will happen in the future - but most of the "predictions" from the model are actually findings that broadly match earlier experimental results, making them "postdictions".

This distinction is important: compared to postdictions, predictions are a much stronger test because they are falsifiable. This is especially relevant given (my impression) that key parameters of the model were tweaked to match the data.

We will better distinguish between pre- and post-dictions in revision.

Reviewer #2 (Public Review):

Summary: In this work, the authors present a biologically plausible, efficient E-I spiking network model and study various aspects of the model and its relation to experimental observations. This includes a derivation of the network into two (E-I) populations, the study of single-neuron perturbations and lateral-inhibition, the study of the effects of adaptation and metabolic cost, and considerations of optimal parameters. From this, they conclude that their work puts forth a plausible implementation of efficient coding that matches several experimental findings, including feature-specific inhibition, tight instantaneous balance, a 4 to 1 ratio of excitatory to inhibitory neurons, and a 3 to 1 ratio of I-I to E-I connectivity strength. It thus argues that some of these observations may come as a direct consequence of efficient coding.

Strengths:

While many network implementations of efficient coding have been developed, such normative models are often abstract and lacking sufficient detail to compare directly to experiments. The intention of this work to produce a more plausible and efficient spiking model and compare it with experimental data is important and necessary in order to test these models.

In rigorously deriving the model with real physical units, this work maps efficient spiking networks onto other more classical biophysical spiking neuron models. It also attempts to compare the model to recent single-neuron perturbation experiments, as well as some long-standing puzzles about neural circuits, such as the presence of separate excitatory and inhibitory neurons, the ratio of excitatory to inhibitory neurons, and E/I balance. One of the primary goals of this paper, to determine if these are merely biological constraints or come from some normative efficient coding objective, is also important.

Though several of the observations have been reported and studied before (see below), this work arguably studies them in more depth, which could be useful for comparing more directly to experiments.

Thanks for these insights and for the kind words of appreciation of the strengths of our work.

Weaknesses:

Though the text of the paper may suggest otherwise, many of the modeling choices and observations found in the paper have been introduced in previous work on efficient spiking models, thereby making this work somewhat repetitive and incremental at times. This includes the derivation of the network into separate excitatory and inhibitory populations, discussion of physical units, comparison of voltage versus spike-timing correlations, and instantaneous E/I balance, all of which can be found in one of the first efficient spiking network papers (Boerlin et al. 2013), as well as in subsequent papers. Metabolic cost and slow adaptation currents were also presented in a previous study (Gutierrez & Deneve 2019). Though it is perfectly fine and reasonable to build upon these previous studies, the language of the text gives them insufficient credit.

We will improve the text to make sure that credit to previous studies is more precisely and more clearly given.

Furthermore, the paper makes several claims of optimality that are not convincing enough, as they are only verified by a limited parameter sweep of single parameters at a time, are unintuitive and may be in conflict with previous findings of efficient spiking networks. This includes the following. Coding error (RMSE) has a minimum at intermediate metabolic cost (Figure 5B), despite the fact that intuitively, zero metabolic cost would indicate that the network is solely minimizing coding error and that previous work has suggested that additional costs bias the output. Coding error also appears to have a minimum at intermediate values of the ratio of E to I neurons (effectively the number of I neurons) and the number of encoded variables (Figures 6D, 7B). These both have to do with the redundancy in the network (number of neurons for each encoded variable), and previous work suggests that networks can code for arbitrary numbers of variables provided the redundancy is high enough (e.g., Calaim et al. 2022). Lastly, the performance of the E-I variant of the network is shown to be better than that of a single cell type (1CT: Figure 7C, D). Given that the E-I network is performing a similar computation as to the 1CT model but with more neurons (i.e., instead of an E neuron directly providing lateral inhibition to its neighbor, it goes through an interneuron), this is unintuitive and again not supported by previous work. These may be valid emergent properties of the E-I spiking network derived here, but their presentation and description are not sufficient to determine this.

We are addressing this issue in two ways. First, we will present results of joint sweeps of variations of pairs of parameters whose joint variations are expected to influence optimality in a way that cannot be understood varying one parameter at a time. Namely we plan to vary jointly the noise intensity and the metabolic constant, as well as the ratio of E to I neuron numbers and the ratio of mean I-I to E-I connectivity. Second, we will individuate a reasonable/realistic range of possible variations of each individual parameter and then perform a Monte Carlo search for the optimal point within this range, and compare the so-obtained results with those obtained from the understanding gained from varying one or two parameters at a time. We will also add the suggested citation to Calaim et al. 2022 in regard to the points discussed above.

We will improve the comparison between the Excitatory-Inhibitory and the 1-Cell-Type model (see reply to the suggestions of Referee 3 for more details).

Alternatively, the methodology of the model suggests that ad hoc modeling choices may be playing a role. For example, an arbitrary weighting of coding error and metabolic cost of 0.7 to 0.3, respectively, is chosen without mention of how this affects the results. Furthermore, the scaling of synaptic weights appears to be controlled separately for each connection type in the network (Table 1), despite the fact that some of these quantities are likely linked in the optimal network derivation. Finally, the optimal threshold and metabolic constants are an order of magnitude larger than the synaptic weights (Table 1). All of these considerations suggest one of the following two possibilities. One, the model has a substantial number of unconstrained parameters to tune, in which case more parameter sweeps would be necessary to definitively make claims of optimality. Or two, parameters are being decoupled from those constrained by the optimal derivation, and the optima simply corresponds to the values that should come out of the derivation.

In the previously submitted manuscript we presented both the encoding error and the metabolic cost separately as a function of the parameters, so that readers could get an understanding of how stable optimal parameters would be to the change of the relative weighting of encoding error and metabolic cost. We will improve this work by adding the suggested calculations to provide quantitative measures of the dependence of the optimal network parameters and configurations on this relative weighting.

Reviewer #3 (Public Review):

Summary: In their paper the authors tackle three things at once in a theoretical model: how can spiking neural networks perform efficient coding, how can such networks limit the energy use at the same time, and how can this be done in a more biologically realistic way than previous work?

They start by working from a long-running theory on how networks operating in a precisely balanced state can perform efficient coding. First, they assume split networks of excitatory (E) and inhibitory (I) neurons. The E neurons have the task to represent some lower dimensional input signal, and the I neurons have the task to represent the signal represented by the E neurons. Additionally, the E and I populations should minimize an energy cost represented by the sum of all spikes. All this results in two loss functions for the E and I populations, and the networks are then derived by assuming E and I neurons should only spike if this improves their respective loss. This results in networks of spiking neurons that live in a balanced state, and can accurately represent the network inputs.

They then investigate in-depth different aspects of the resulting networks, such as responses to perturbations, the effect of following Dale's law, spiking statistics, the excitation (E)/inhibition (I) balance, optimal E/I cell ratios, and others. Overall, they expand on previous work by taking a more biological angle on the theory and showing the networks can operate in a biologically realistic regime.

Strengths:

(1) The authors take a much more biological angle on the efficient spiking networks theory than previous work, which is an essential contribution to the field.

(2) They make a very extensive investigation of many aspects of the network in this context, and do so thoroughly.

(3) They put sensible constraints on their networks, while still maintaining the good properties these networks should have.

Thanks for this summary and for these kind words of appreciation of the strengths of our work.

Weaknesses:

(1) The paper has somewhat overstated the significance of their theoretical contributions, and should make much clearer what aspects of the derivations are novel. Large parts were done in very similar ways in previous papers. Specifically: the split into E and I neurons was also done in Boerlin et al (2008) and in Barrett et al (2016). Defining the networks in terms of realistic units was already done by Boerlin et al (2008). It would also be worth it to discuss Barrett et al (2016) specifically more, as there they also use split E/I networks and perform biologically relevant experiments.

We will improve the text to make sure that credit to previous studies is more precisely and more clearly given.

(2) It is not clear from an optimization perspective why the split into E and I neurons and following Dale's law would be beneficial. While the constraints of Dale's law are sensible (splitting the population in E and I neurons, and removing any non-Dalian connection), they are imposed from biology and not from any coding principles. A discussion of how this could be done would be much appreciated, and in the main text, this should be made clear.

We indeed removed non-Dalian connections because having only connections respecting Dale’s law is a major constraint for biological plausibility. Our logic was to consider efficient coding within the space of networks that satisfy this (and other) biological plausibility constraints. We did not intend to claim that removing the non-Dalian connections was the result of an analytical optimization. However, to get better insights into how Dale’s Law constrains or influences the design of efficient networks, we added a comparison of the coding properties of networks that either do or do not satisfy Dale’s law. We apologize if this was not sufficiently clear in the previous version and we will clarify this in revision.

(3) Related to the previous point, the claim that the network with split E and I neurons has a lower average loss than a 1 cell-type (1-CT) network seems incorrect to me. Only the E population coding error should be compared to the 1-CT network loss, or the sum of the E and I populations (not their average). In my author recommendations, I go more in-depth on this point.

We will perform the suggested detailed comparisons between the network loss in the 1CT-model and E-I model and then revise or refine conclusions if and as needed, according to the results we will obtain.

(4) While the paper is supposed to bring the balanced spiking networks they consider in a more experimentally relevant context, for experimental audiences I don't think it is easy to follow how the model works, and I recommend reworking both the main text and methods to improve on that aspect.

We will try to make the presentation of the model more accessible to a non-computational audience.

Assessment and context: Overall, although much of the underlying theory is not necessarily new, the work provides an important addition to the field. The authors succeeded well in their goal of making the networks more biologically realistic, and incorporating aspects of energy efficiency. For computational neuroscientists, this paper is a good example of how to build models that link well to experimental knowledge and constraints, while still being computationally and mathematically tractable. For experimental readers, the model provides a clearer link between efficient coding spiking networks to known experimental constraints and provides a few predictions.

Thanks for these kind words. We will make sure that these points emerge more clearly and in a more accessible way from the revised paper.