Information gain at the onset of habituation to repeated stimuli

Reviewer #1 (Public review):

Summary:

The manuscript by Nicoletti et al. presents a minimal model of habituation, a basic form of non-associative learning, addressing both from dynamical and information theory aspects of how habituation can be realized. The authors identify that negative feedback provided with a slow storage mechanism is sufficient to explain habituation.

Strengths:

The authors combine the identification of the dynamical mechanism with information-theoretic measures to determine the onset of habituation and provide a description of how the system can gain maximum information about the environment.

Weaknesses:

I have several main concerns/questions about the proposed model for habituation and its plausibility. In general, habituation does not only refer to a decrease in the responsiveness upon repeated stimulation but as Thompson and Spencer discussed in Psych. Rev. 73, 16-43 (1966), there are 10 main characteristics of habituation, including (i) spontaneous recovery when the stimulus is withheld after response decrement; dependence on the frequency of stimulation such that (ii) more frequent stimulation results in more rapid and/or more pronounced response decrement and more rapid spontaneous recovery; (iii) within a stimulus modality, the less intense the stimulus, the more rapid and/or more pronounced the behavioral response decrement; (iv) the effects of repeated stimulation may continue to accumulate even after the response has reached an asymptotic level (which may or may not be zero, or no response). This effect of stimulation beyond asymptotic levels can alter subsequent behavior, for example, by delaying the onset of spontaneous recovery.

These are only a subset of the conditions that have been experimentally observed and therefore a mechanistic model of habituation, in my understanding, should capture the majority of these features and/or discuss the absence of such features from the proposed model.

Furthermore, the habituated response in steady-state is approximately 20% less than the initial response, which seems to be achieved already after 3-4 pulses, the subsequent change in response amplitude seems to be negligible, although the authors however state "after a large number of inputs, the system reaches a time-periodic steady-state". How do the authors justify these minimal decreases in the response amplitude? Does this come from the model parametrization and is there a parameter range where more pronounced habituation responses can be observed?

The same is true for the information content (Figure 2f) - already at the first pulse, IU, H ~ 0.7 and only negligibly increases afterwards. In my understanding, during learning, the mutual information between the input and the internal state increases over time and the system extracts from these predictions about its responses. In the model presented by the authors, it seems the system already carries information about the environment which hardly changes with repeated stimulus presentation. The complexity of the signal is also limited, and it is very hard to clarify from the presented results, whether the proposed model can actually explain basic features of habituation, as mentioned above.
Additionally, there have been two recent models on habituation and I strongly suggest that the authors discuss their work in relation to recent works (bioRxiv 2024.08.04.606534; arXiv:2407.18204).

Reviewer #2 (Public review):

In this study, the authors aim to investigate habituation, the phenomenon of increasing reduction in activity following repeated stimuli, in the context of its information-theoretic advantage. To this end, they consider a highly simplified three-species reaction network where habituation is encoded by a slow memory variable that suppresses the receptor and therefore the readout activity. Using analytical and numerical methods, they show that in their model the information gain, the difference between the mutual information between the signal and readout after and before habituation, is maximal for intermediate habituation strength. Furthermore, they demonstrate that the Pareto front corresponds to an optimization strategy that maximizes the mutual information between signal and readout in the steady state, minimizes some form of dissipation, and also exhibits similar intermediate habituation strength. Finally, they briefly compare predictions of their model to whole-brain recordings of zebrafish larvae under visual stimulation.

The author's simplified model might serve as a solid starting point for understanding habituation in different biological contexts as the model is simple enough to allow for some analytic understanding but at the same time exhibits all basic properties of habituation in sensory systems. Furthermore, the author's finding of maximal information gain for intermediate habituation strength via an optimization principle is, in general, interesting. However, the following points remain unclear or are weakly explained:

(1) Is it unclear what the meaning of the finding of maximal information gain for intermediate habituation strength is for biological systems? Why is information gain as defined in the paper a relevant quantity for an organism/cell? For instance, why is a system with low mutual information after the first stimulus and intermediate mutual information after habituation better than one with consistently intermediate mutual information? Or, in other words, couldn't the system try to maximize the mutual information acquired over the whole time series, e.g., the time series mutual information between the stimulus and readout?

(2) The model is very similar to (or a simplification of previous models) for adaptation in living systems, e.g., for adaptation in chemotaxis via activity-dependent methylation and demethylation. This should be made clearer.

(3) It remains unclear why this optimization principle is the most relevant one. While it makes sense to maximize the mutual information between stimulus and readout, there are various choices for what kind of dissipation is minimized. Why was \delta Q_R chosen and not, for instance, \dot{\Sigma}_int or the sum of both? How would the results change in that case? And how different are the results if the mutual information is not calculated for the strong stimulation input statistics but for the background one?

(4) The comparison to the experimental data is not too strong of an argument in favor of the model. Is the agreement between the model and the experimental data surprising? What other behavior in the PCA space could one have expected in the data? Shouldn't the 1st PC mostly reflect the "features", by construction, and other variability should be due to progressively reduced activity levels?

Reviewer #3 (Public review):

The authors use a generic model framework to study the emergence of habituation and its functional role from information-theoretic and energetic perspectives. Their model features a receptor, readout molecules, and a storage unit, and as such, can be applied to a wide range of biological systems. Through theoretical studies, the authors find that habituation (reduction in average activity) upon exposure to repeated stimuli should occur at intermediate degrees to achieve maximal information gain. Parameter regimes that enable these properties also result in low dissipation, suggesting that intermediate habituation is advantageous both energetically and for the purpose of retaining information about the environment.

A major strength of the work is the generality of the studied model. The presence of three units (receptor, readout, storage) operating at different time scales and executing negative feedback can be found in many domains of biology, with representative examples well discussed by the authors (e.g. Figure 1b). A key takeaway demonstrated by the authors that has wide relevance is that large information gain and large habituation cannot be attained simultaneously. When energetic considerations are accounted for, large information gain and intermediate habituation appear to be a favorable combination.

While the generic approach of coarse-graining most biological detail is appealing and the results are of broad relevance, some aspects of the conducted studies, the problem setup, and the writing lack clarity and should be addressed:

(1) The abstract can be further sharpened. Specifically, the "functional role" mentioned at the end can be made more explicit, as it was done in the second-to-last paragraph of the Introduction section ("its functional advantages in terms of information gain and energy dissipation"). In addition, the abstract mentions the testing against experimental measurements of neural responses but does not specify the main takeaways. I suggest the authors briefly describe the main conclusions of their experimental study in the abstract.

(2) Several clarifications are needed on the treatment of energy dissipation.
- When substituting the rates in Eq. (1) into the definition of δQ_R above Eq. (10), "σ" does not appear on the right-hand side. Does this mean that one of the rates in the lower pathway must include σ in its definition? Please clarify.
- I understand that the production of storage molecules has an associated cost σ and hence contributes to dissipation. The dependence of receptor dissipation on , however, is not fully clear. If the environment were static and the memory block was absent, the term with would still contribute to dissipation. What would be the nature of this dissipation?
- Similarly, in Eq. (9) the authors use the ratio of the rates Γ_{s → s+1} and Γ_{s+1 → s} in their expression for internal dissipation. The first-rate corresponds to the synthesis reaction of memory molecules, while the second corresponds to a degradation reaction. Since the second reaction is not the microscopic reverse of the first, what would be the physical interpretation of the log of their ratio? Since the authors already use σ as the energy cost per storage unit, why not use σ times the rate of producing S as a metric for the dissipation rate?

(3) Impact of the pre-stimulus state. The plots in Figure 2 suggest that the environment was static before the application of repeated stimuli. Can the authors comment on the impact of the pre-stimulus state on the degree of habituation and its optimality properties? Specifically, would the conclusions stay the same if the prior environment had stochastic but aperiodic dynamics?

(4) Clarification about the memory requirement for habituation. Figure 4 and the associated section argue for the essential role that the storage mechanism plays in habituation. Indeed, Figure 4a shows that the degree of habituation decreases with decreasing memory. The graph also shows that in the limit of vanishingly small Δ⟨S⟩, the system can still exhibit a finite degree of habituation. Can the authors explain this limiting behavior; specifically, why does habituation not vanish in the limit Δ⟨S⟩ -> 0?

Author response:

Reviewer #1 (Public review):

Summary:

The manuscript by Nicoletti et al. presents a minimal model of habituation, a basic form of non-associative learning, addressing both from dynamical and information theory aspects of how habituation can be realized. The authors identify that negative feedback provided with a slow storage mechanism is sufficient to explain habituation.

Strengths:

The authors combine the identification of the dynamical mechanism with information-theoretic measures to determine the onset of habituation and provide a description of how the system can gain maximum information about the environment.

We thank the reviewer for highlighting the strength of our work.

Weaknesses:

I have several main concerns/questions about the proposed model for habituation and its plausibility. In general, habituation does not only refer to a decrease in the responsiveness upon repeated stimulation but as Thompson and Spencer discussed in Psych. Rev. 73, 16-43 (1966), there are 10 main characteristics of habituation, including (i) spontaneous recovery when the stimulus is withheld after response decrement; dependence on the frequency of stimulation such that (ii) more frequent stimulation results in more rapid and/or more pronounced response decrement and more rapid spontaneous recovery; (iii) within a stimulus modality, the less intense the stimulus, the more rapid and/or more pronounced the behavioral response decrement; (iv) the effects of repeated stimulation may continue to accumulate even after the response has reached an asymptotic level (which may or may not be zero, or no response). This effect of stimulation beyond asymptotic levels can alter subsequent behavior, for example, by delaying the onset of spontaneous recovery.

These are only a subset of the conditions that have been experimentally observed and therefore a mechanistic model of habituation, in my understanding, should capture the majority of these features and/or discuss the absence of such features from the proposed model.

We are really grateful to the reviewer for pointing out these aspects of habituation that we overlooked in the previous version of our manuscript. Indeed, our model is able to capture most of these 10 observed behaviors, specifically: 1) habituation; 2) spontaneous recovery; 3) potentiation of habituation; 4) frequency sensitivity; and 5) intensity sensitivity. Here, we are following the same terminology employed in bioRxiv 2024.08.04.606534, the paper highlighted by the referee. Regarding the hallmark 6) subliminal accumulation, we also believe that our model can capture it as well, but more analyses are needed to substantiate this claim. We will include the discussion of these points in the revised version.

Notably, in line with the discussion in bioRxiv 2024.08.04.606534, we also think that feature 10) long-term habituation, is ambiguous and its appearance might be simply related to the other features discussed above. In the revised version, we will detail our take on this aspect in relation to the presented model.

All other hallmarks require the presence of multiple stimuli and, as a consequence, they cannot be observed within our model, but are interesting lines of research for future investigations. We believe that this addition will help clarify the validity of the model and the relevance of our result, consequently improving the quality of our manuscript.

Furthermore, the habituated response in steady-state is approximately 20% less than the initial response, which seems to be achieved already after 3-4 pulses, the subsequent change in response amplitude seems to be negligible, although the authors however state "after a large number of inputs, the system reaches a time-periodic steady-state". How do the authors justify these minimal decreases in the response amplitude? Does this come from the model parametrization and is there a parameter range where more pronounced habituation responses can be observed?

The referee is correct, but this is solely a consequence of the specific set of parameters we selected. We made this choice solely for visualization purposes. In the next version, when different emerging behaviors characterizing habituation are discussed, we will also present a set of parameters for which habituation can be better appreciated, justifying our new choice.

We stated that the time-periodic steady-state is reached “after a large number of stimuli” from a mathematical perspective. However, by using a habituation threshold, as defined in bioRxiv 2024.08.04.606534 for example, we can say that the system is habituated after a few stimuli for the set of parameters selected in the first version of the manuscript. We will also discuss this aspect in the Supplemental Material of the revised version, as it will also be important to appreciate the hallmarks of habituation listed above.

The same is true for the information content (Figure 2f) - already at the first pulse, IU, H ~ 0.7 and only negligibly increases afterwards. In my understanding, during learning, the mutual information between the input and the internal state increases over time and the system extracts from these predictions about its responses. In the model presented by the authors, it seems the system already carries information about the environment which hardly changes with repeated stimulus presentation. The complexity of the signal is also limited, and it is very hard to clarify from the presented results, whether the proposed model can actually explain basic features of habituation, as mentioned above.

The point about information is more subtle. We can definitely choose a set of parameters for which the information gain is higher and we will show it in the Supplemental Material of the revised version. However, as the reviewer correctly points out, it is difficult to give an interpretation of the specific value of I_U,H for such a minimal model.

We also remark that, since the readout population and the receptor both undergo a fast dynamics (with appropriate timescales as discussed in the text), we are not observing the transient gain of information associated with the first stimulus and, as such, the mutual information presents a discontinuous behavior resembling the dynamics of the readout.

Additionally, there have been two recent models on habituation and I strongly suggest that the authors discuss their work in relation to recent works (bioRxiv 2024.08.04.606534; arXiv:2407.18204).

We thank the reviewer for pointing out these relevant references. We will discuss analogies and differences in the revised version of the main text. The main difference is the fact that information-theoretic aspects of habituation are not discussed in the presented references, while the idea of this work is to elucidate exactly the interplay between information gain and habituation dynamics.

Reviewer #2 (Public review):

In this study, the authors aim to investigate habituation, the phenomenon of increasing reduction in activity following repeated stimuli, in the context of its information-theoretic advantage. To this end, they consider a highly simplified three-species reaction network where habituation is encoded by a slow memory variable that suppresses the receptor and therefore the readout activity. Using analytical and numerical methods, they show that in their model the information gain, the difference between the mutual information between the signal and readout after and before habituation, is maximal for intermediate habituation strength. Furthermore, they demonstrate that the Pareto front corresponds to an optimization strategy that maximizes the mutual information between signal and readout in the steady state, minimizes some form of dissipation, and also exhibits similar intermediate habituation strength. Finally, they briefly compare predictions of their model to whole-brain recordings of zebrafish larvae under visual stimulation.

The author's simplified model might serve as a solid starting point for understanding habituation in different biological contexts as the model is simple enough to allow for some analytic understanding but at the same time exhibits all basic properties of habituation in sensory systems. Furthermore, the author's finding of maximal information gain for intermediate habituation strength via an optimization principle is, in general, interesting. However, the following points remain unclear or are weakly explained:

We thank the reviewer for deeming our work interesting and for considering it a solid starting point for understanding habituation in biological systems.

(1) Is it unclear what the meaning of the finding of maximal information gain for intermediate habituation strength is for biological systems? Why is information gain as defined in the paper a relevant quantity for an organism/cell? For instance, why is a system with low mutual information after the first stimulus and intermediate mutual information after habituation better than one with consistently intermediate mutual information? Or, in other words, couldn't the system try to maximize the mutual information acquired over the whole time series, e.g., the time series mutual information between the stimulus and readout?

This is an important and delicate aspect to discuss. We considered the mutual information with a prolonged stimulation when building the Pareto front, by maximizing this quantity while minimizing the dissipation. The observation that the Pareto front lies in the vicinity of the maximum of the information gain hints at the fact that reducing the information gain by increasing the mutual information at each stimulation will require more energy. However, we did not thoroughly explore this aspect by considering all sources of dissipation and the fact that habituation is, anyway, a dynamical phenomenon. In the revised version, we will clarify this point, extending our analyses.

We would like to add that, from a naive perspective, while the first stimulation will necessarily trigger a certain mutual information, multiple observations of the same stimulus have to reflect into accumulated infor

mation that consequently drives the onset of observed dynamical behaviors, such as habituation.

(2) The model is very similar to (or a simplification of previous models) for adaptation in living systems, e.g., for adaptation in chemotaxis via activity-dependent methylation and demethylation. This should be made clearer.

We apologize for having missed this point. Our choice has been motivated by the fact that we wanted to avoid any confusion between the usual definition of (perfect) adaptation and habituation. At any rate, we will add this clarification in the revised version.

(3) It remains unclear why this optimization principle is the most relevant one. While it makes sense to maximize the mutual information between stimulus and readout, there are various choices for what kind of dissipation is minimized. Why was \delta Q_R chosen and not, for instance, \dot{\Sigma}_int or the sum of both? How would the results change in that case? And how different are the results if the mutual information is not calculated for the strong stimulation input statistics but for the background one?

We thank the referee for giving us the opportunity to deepen this aspect of the manuscript. We decided to minimize \delta Q_R since this dissipation is unavoidable. In fact, considering the existence of two different pathways implementing sensing and feedback, the presence of any input will result in a dissipation produced by the receptor. This energy consumption is reflected in \delta Q_R. Conversely, the dissipation associated with the storage is always zero in the limit of a fast memory. However, we know that such a limit is pathological and leads to no habituation. As a consequence, in the revised version we will discuss other choices for our optimization approach, along with their potentialities and limitations.

The dependence of the Pareto front on the stimulus strength is shown in the Supplemental Material, but not in relation to habituation and information gain. We will strengthen this part in the revised version of the manuscript, elaborating more on the connection between optimality, information gain, and dynamical behavior.

(4) The comparison to the experimental data is not too strong of an argument in favor of the model. Is the agreement between the model and the experimental data surprising? What other behavior in the PCA space could one have expected in the data? Shouldn't the 1st PC mostly reflect the "features", by construction, and other variability should be due to progressively reduced activity levels?

The agreement between data and model is not surprising - we agree on this - since the data exhibit habituation. However, the fact that, without any explicit biological details, our minimal model is able to capture the features of a complex neural system just by looking at the PCs is non-trivial. The 1st PC only reflects the feature that captures most of the variance of the data and, as such, it is difficult to have a-priori expectations on what it should represent. Depending on the behavior of higher-order PCs, we may include them in the revised version if any interesting results arise.

Reviewer #3 (Public review):

The authors use a generic model framework to study the emergence of habituation and its functional role from information-theoretic and energetic perspectives. Their model features a receptor, readout molecules, and a storage unit, and as such, can be applied to a wide range of biological systems. Through theoretical studies, the authors find that habituation (reduction in average activity) upon exposure to repeated stimuli should occur at intermediate degrees to achieve maximal information gain. Parameter regimes that enable these properties also result in low dissipation, suggesting that intermediate habituation is advantageous both energetically and for the purpose of retaining information about the environment.

A major strength of the work is the generality of the studied model. The presence of three units (receptor, readout, storage) operating at different time scales and executing negative feedback can be found in many domains of biology, with representative examples well discussed by the authors (e.g. Figure 1b). A key takeaway demonstrated by the authors that has wide relevance is that large information gain and large habituation cannot be attained simultaneously. When energetic considerations are accounted for, large information gain and intermediate habituation appear to be a favorable combination.

We thank the referee for this positive assessment of our work and its generality.

While the generic approach of coarse-graining most biological detail is appealing and the results are of broad relevance, some aspects of the conducted studies, the problem setup, and the writing lack clarity and should be addressed:

(1) The abstract can be further sharpened. Specifically, the "functional role" mentioned at the end can be made more explicit, as it was done in the second-to-last paragraph of the Introduction section ("its functional advantages in terms of information gain and energy dissipation"). In addition, the abstract mentions the testing against experimental measurements of neural responses but does not specify the main takeaways. I suggest the authors briefly describe the main conclusions of their experimental study in the abstract.

We thank the referee for this suggestion. The revised version will present a modified abstract in line with the reviewer’s proposal.

(2) Several clarifications are needed on the treatment of energy dissipation.

- When substituting the rates in Eq. (1) into the definition of δQ_R above Eq. (10), "σ" does not appear on the right-hand side. Does this mean that one of the rates in the lower pathway must include σ in its definition? Please clarify.

We apologize to the referee for this typo. Indeed, \sigma sets the energy scale of the feedback and, as such, it appears in the energetic driving given by the feedback on the receptor, i.e., together with \kappa in Eq. (1). We will fix this issue in the revised version. Moreover, we will check the entire manuscript to be sure that all formulas are consistent.

- I understand that the production of storage molecules has an associated cost σ and hence contributes to dissipation. The dependence of receptor dissipation on , however, is not fully clear. If the environment were static and the memory block was absent, the term with would still contribute to dissipation. What would be the nature of this dissipation?

In the spirit of building a paradigmatic minimal model with a thermodynamic meaning, we considered H to act as an external thermodynamic driving. Since this driving acts on a different pathway with respect to the one affected by the storage, the receptor is driven out of equilibrium by its presence. By eliminating the memory block, we would also be necessarily eliminating the presence of the pathway associated with the storage effect (“internal pathway” in the manuscript). In this case, the receptor is a 2-state, 1-pathway system and, as such, it always satisfies an effective detailed balance. As a consequence, the definition of \delta Q_R reported in the manuscript does not hold anymore and the receptor does not exhibit any dissipation. Our choice to model two different pathways has been biologically motivated. We will make this crucial aspect clearer in the revised manuscript.

- Similarly, in Eq. (9) the authors use the ratio of the rates Γ_{s → s+1} and Γ_{s+1 → s} in their expression for internal dissipation. The first-rate corresponds to the synthesis reaction of memory molecules, while the second corresponds to a degradation reaction. Since the second reaction is not the microscopic reverse of the first, what would be the physical interpretation of the log of their ratio? Since the authors already use σ as the energy cost per storage unit, why not use σ times the rate of producing S as a metric for the dissipation rate?

In the current version of the manuscript, we employed the scheme of a controlled birth and death process to model the coupled process of readout and storage production. Since we are not dealing with a detailed biochemical underlying network, we used this coarse-grained description to capture the main features of the dynamics. In this sense, the considered reactions produce and destroy a molecule from a certain pool even if they are controlled in different ways by the readout. However, we completely agree with the point of view of the referee and will analyze our results following their suggestion.

(3) Impact of the pre-stimulus state. The plots in Figure 2 suggest that the environment was static before the application of repeated stimuli. Can the authors comment on the impact of the pre-stimulus state on the degree of habituation and its optimality properties? Specifically, would the conclusions stay the same if the prior environment had stochastic but aperiodic dynamics?

The initial stimulus is indeed stochastic with an average constant in time. Model response depends on the pre-stimulus level, since it also sets the stationary storage concentration before the first “strong” stimulation arrives. This dependence is not crucial for our result but deserves proper discussion, as the referee correctly pointed out. We will clarify this point in the revised version of this study.

(4) Clarification about the memory requirement for habituation. Figure 4 and the associated section argue for the essential role that the storage mechanism plays in habituation. Indeed, Figure 4a shows that the degree of habituation decreases with decreasing memory. The graph also shows that in the limit of vanishingly small Δ⟨S⟩, the system can still exhibit a finite degree of habituation. Can the authors explain this limiting behavior; specifically, why does habituation not vanish in the limit Δ⟨S⟩ -> 0?

We apologize for the lack of clarity here. Actually, Δ⟨S⟩ is not strictly zero, but equal to 0.15% at the final point. However, due to rounding this appears as 0% in the plot, and we will fix it in the revised version. Let us note that the fact that Δ⟨S⟩ is small signals a nonlinear dependence of Δ⟨U⟩ from Δ⟨S⟩, but no contradiction. We will clarify this aspect in the revised version.