Testing the state-dependent model of subsecond time perception against experimental evidence
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eLife assessment
This useful paper explores a mathematical model of subsecond time perception, testing potential neural mechanisms behind the linear psychophysical law, Weber's law, and dopaminergic modulation of subjective durations. The model employed readout units to decode an interval. Nevertheless, the work is incomplete and presented as data-driven, but there is no analysis of empirical data.
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Abstract
Coordinated movements, speech, and other actions are impossible without precise timing. Computational models of interval timing are expected to provide key insights into the underlying mechanisms of timing, which are currently largely unknown. So far, existing models have only been partially replicating key experimental observations, such as the linear psychophysical law, the linear increase of the standard deviation (the scalar property or Weber’s law), and the modulation of subjective duration via dopamine. Here, we incorporate the state-dependent model for subsecond timing as proposed by Buonomano (2000) into a strongly data-driven computational network model of PFC We show that this model variant, the state-dependent PFC model, successfully encodes time up to 750 milliseconds and reproduces all key experimental observations mentioned above, including many of its details. Investigating the underlying mechanisms, we find that the representations of different intervals are based on the natural heterogeneity in the parameters of the network, leading to stereotypic responses of subsets of neurons. Furthermore, we propose a theory for the mechanism underlying subsecond timing in this model based on correlation and ablation analyses as well as mathematical analyses explaining the emergence of the scalar property and Vierordt law. The state-dependent PFC model proposed here constitutes the first data-driven model of subsecond timing in the range of hundreds of milliseconds that has been thoroughly tested against a variety of experimental data, providing an ideal starting point for further investigations of subsecond timing.
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Author response:
Reviewer #1
[…] it seems that the readout units are not operating in continuous time, and that interval discrimination relies in part on external information. Specifically, the readout units only look at the spike counts during the window delta_t_w.
In the first version of the review, the reviewer implied that each readout unit only receives input during a small window around the interval it represents. However, this is not the case. The small window that is depicted in Fig. 16 is a sliding window that is used to compute the states (i.e., an estimate of the instantaneous firing rate) at each point in time. The fact that the readout units indeed do operate in continuous time is apparent from Fig. 2A, showing the activity of all output units as a function of time: There is gradually changing activity with a peak at …
Author response:
Reviewer #1
[…] it seems that the readout units are not operating in continuous time, and that interval discrimination relies in part on external information. Specifically, the readout units only look at the spike counts during the window delta_t_w.
In the first version of the review, the reviewer implied that each readout unit only receives input during a small window around the interval it represents. However, this is not the case. The small window that is depicted in Fig. 16 is a sliding window that is used to compute the states (i.e., an estimate of the instantaneous firing rate) at each point in time. The fact that the readout units indeed do operate in continuous time is apparent from Fig. 2A, showing the activity of all output units as a function of time: There is gradually changing activity with a peak at the represented interval. If each unit would only receive input during a window of a couple milliseconds, there would be a single peak of activity at the represented interval, and near-zero activity at any other time.
This misunderstanding has been cleared out in the current version of the review (see last paragraph of review #1).
Stimulus onset occurs at 1500 ms in order to allow the network to stabilize. Ideally, this value should be randomized across trials to ensure performance generalizes across initial states.
This is a valid point which we will address in the revision. However, we note that experimentation with different onset values did not change the dynamics of the network systematically in previous studies (i.e., Hass et al., 2022).
Why does StDev saturate? Is that because subjective time saturates as well?
Indeed, the two phenomena are closely related. In section “Deviations from the scalar property and the origin on Vierordt’s law”, we discuss that both is caused by the broadening of the tuning curves of the readout units (Fig 1A) as the longest time constants of the network are exceeded.
In the discussion, it would be nice to explain that dopaminergic modulation of subjective timing is not as universally observed as the linear psychophysical law or the scalar property, and I believe somewhat controversial (e.g., Ward, ..., Balsam, 2009).
We are thankful for this advice and will adapt the discussion accordingly in the revision. Still, we note that dopaminergic modulation of subjective timing is one of the more robust effects observed in several time perception experiments.
Reviewer #2:
(1) Lack of Empirical Data: […] The paper would benefit from quantitative and qualitative simulations of results from specific, large-sample studies to anchor the model's predictions in concrete empirical evidence.
While it is correct that this study does not attempt the replicate a concrete empirical study, we note that do compare the model's results with specific studies wherever possible. The comparison is done on the level of parameters of functional relationships: For the linear psychophysical law, we compare the slope and the indifference point of the model with those from experimental studies. For the scalar property, we compare the Weber fraction of the model to those computed from experiments. For dopaminergic modulation of subjective duration, no direct comparison with experimental data is possible, as the levels of modulation are estimated from in vitro experiments and cannot be directly compared with modulations in vivo. However, we discuss a range of qualitative observations in experiments that are reproduced (and explained) by the model.
The above arguments notwithstanding, one can discuss whether the presentation of the experimental results and the comparison with the simulations is appropriate, and we do plan to extend this presentation in a revision.
(2) Methodological Ambiguities: The training and testing procedures lack robust checks for generalization, leading to potential overfitting issues.
It is correct that formal checks for generalization, such as cross-validation protocols, are missing, and we will include them in the revision. However, as we obtained a mechanistic understanding of how the model tells time, we are confident that our results are not due to overfitting.
(3) Inadequate Visualization of Empirical Data: References to empirical data are vague and not directly visualized alongside model outputs. Future iterations should include empirical data, not general trends from psychophysics, in figures for a clear comparison.
As mentioned above, the comparison between simulation and empirical data will be extended in a revision. However, we argue that the “general trends”, namely adherence of the model to the often-reported psychophysical regularities, are of greater importance compared to the replication of, e.g. one specific slope of the linear psychophysical law, which does vary a lot between experiments.
(4) Limitations in Model Scope and Dynamics: […] Expanding the model limitations to consider isochronous pulse processing and the emergence of limit-cycle behaviors after prolonged stimulation would provide a more comprehensive understanding of the model's capabilities and limitations.
The current research focuses on the estimation of a single duration rather than the processing of sequences of durations. Sequence processing is a vast field, and it has been argued that it comprises different mechanisms compared to duration estimation. Thus, we feel that including sequences processing would be beyond the scope of the already quite extensive paper. However, we will discuss a possible extension of the model to sequence processing in the revision.
Additionally, the justification for using(N_{Poisson}) as a proxy for more connections is unclear and warrants a more direct approach.
We considered different means to vary the noise input into the network, including changes in the number of connections. We ultimately chose to vary the firing rate of a fixed number of Poisson input neurons. As the sum of the firing rates of N independent Poisson neurons with the same f is simply N*f and the synaptic contributions from each spike also linearly add up, this is equivalent to adding more Poisson neurons and thus, more connections.
(5) Omissions and Redundancies: Certain omissions, such as the lack of a condition in Figure 7A or missing references to relevant models and reviews, detract from the paper's thoroughness.
The reviewer refers to a condition where everything is ablated except NMDA. We will include such a condition in the revision. Regarding missing references, the reviewer requests including references that focus on sequence processing. While the focus of the current work is on estimating a single duration rather than a sequence of durations (see above), we will include a review on this topic as an outlook on this possible extension of the model.
Moreover, some statements and terms like "internal clock" are used without a clear mechanistic definition within the model.
We are thankful for this advice and will adapt the revision accordingly.
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eLife assessment
This useful paper explores a mathematical model of subsecond time perception, testing potential neural mechanisms behind the linear psychophysical law, Weber's law, and dopaminergic modulation of subjective durations. The model employed readout units to decode an interval. Nevertheless, the work is incomplete and presented as data-driven, but there is no analysis of empirical data.
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Reviewer #1 (Public Review):
Summary:
This paper addresses the important question of the neural mechanisms underlying interval discrimination. The authors develop a detailed and biologically plausible model based on a previously proposed theory of timing. The model proposes that the interval between two stimuli can be encoded in the state of the neuronal and synaptic properties, specifically those with time constants on the order of hundreds of milliseconds, such as short-term synaptic plasticity and GABAb currents. Based on biological parameters in the PFC the authors show that the model can account for interval discrimination for up to 750 ms. Furthermore, the model accounts for three well-established psychophysical properties of interval timing: the linear relation between objective and neural time, the scalar property/Weber's law, …Reviewer #1 (Public Review):
Summary:
This paper addresses the important question of the neural mechanisms underlying interval discrimination. The authors develop a detailed and biologically plausible model based on a previously proposed theory of timing. The model proposes that the interval between two stimuli can be encoded in the state of the neuronal and synaptic properties, specifically those with time constants on the order of hundreds of milliseconds, such as short-term synaptic plasticity and GABAb currents. Based on biological parameters in the PFC the authors show that the model can account for interval discrimination for up to 750 ms. Furthermore, the model accounts for three well-established psychophysical properties of interval timing: the linear relation between objective and neural time, the scalar property/Weber's law, and dopaminergic modulation of timing (although this property is less robust). Of particular novelty is the demonstration of Weber's law, and an explanation of how many complex and nonlinear neuronal properties produce a linear relationship between the standard deviation of interval estimates and their mean.This is an interesting paper that addresses a significant gap in the field. However, I have one major concern. As I understood the methods (and I may have misunderstood) it seems that the readout units are not operating in continuous time, and that interval discrimination relies in part on external information. Specifically, the readout units only look at the spike counts during the window delta_t_w. Thus, discrimination between 100 and 200 ms looks only at the spikes at 120-145 and 220-245, respectively, meaning that the experimenters are providing interval information for the readout of the intervals being discriminated. If this is indeed the case the model is fairly limited in biological plausibility and significantly dampens my enthusiasm for the paper.
Stimulus onset occurs at 1500 ms in order to allow the network to stabilize. Ideally, this value should be randomized across trials to ensure performance generalizes across initial states.
Why does StDev saturate? Is that because subjective time saturates as well?
The model captures the effect of D2 receptors observed in some timing studies, specifically and DR2 activation increases "clock" speed. In the discussion, it would be nice to explain that dopaminergic modulation of subjective timing is not as universally observed as the linear psychophysical law or the scalar property, and I believe somewhat controversial (e.g., Ward, ..., Balsam, 2009).
(NB: Regarding my potential concern that that the decoding was performed in discontinuous time, the authors have clarified that decoding was done in continuous time--i.e., each output unit was trained to respond to a given time bin of the target interval but exposed to all time bins of all intervals during testing. Thus confirming the robustness of their decoding procedure and model.)
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Reviewer #2 (Public Review):
Summary:
The paper explores a mathematical model of subsecond time perception, engaging with established theories such as the linear psychophysical law, Weber's law, and dopaminergic modulation of subjective durations. While it ambitiously attempts to confirm specific mechanisms of time perception and presents a comprehensive description of these mechanisms, the work is presented as data-driven but its empirical backing and model generalization capabilities are questionable. The title's implication of a robust empirical foundation is misleading, as the main figures do not reflect empirical data directly but rather model outputs aligned with general trends in psychophysical studies. This disjunction raises concerns about the model's applicability and the strength of the claims made regarding time perception …Reviewer #2 (Public Review):
Summary:
The paper explores a mathematical model of subsecond time perception, engaging with established theories such as the linear psychophysical law, Weber's law, and dopaminergic modulation of subjective durations. While it ambitiously attempts to confirm specific mechanisms of time perception and presents a comprehensive description of these mechanisms, the work is presented as data-driven but its empirical backing and model generalization capabilities are questionable. The title's implication of a robust empirical foundation is misleading, as the main figures do not reflect empirical data directly but rather model outputs aligned with general trends in psychophysical studies. This disjunction raises concerns about the model's applicability and the strength of the claims made regarding time perception mechanisms.Strengths:
(1) The paper describes specific mechanisms of time perception, providing a theoretical examination of linear psychophysical law, Weber's law, and dopaminergic modulation. This aspect is valuable for readers seeking a theoretical understanding of temporal perception.(2) The authors describe a range of psychophysical studies and theories, attempting to position their model within the broader scientific discourse on time perception.
Weaknesses:
(1) Lack of Empirical Data: The absence of two things: 1) quantification of error between model and empirical data with interpretation of what this degree of error means, and 2) clear comparisons between model and empirical data in all figures and tables, to substantiate the model's predictions stands out. The reliance on general trends rather than specific empirical studies undermines the strength and reliability of the model's claims. The paper would benefit from quantitative and qualitative simulations of results from specific, large-sample studies to anchor the model's predictions in concrete empirical evidence.(2) Methodological Ambiguities: The training and testing procedures lack robust checks for generalization, leading to potential overfitting issues. Clarifications are needed on whether and how the model reaches a steady state before stimulation and the implications of the chosen model time constants in the absence of stimulation. The overlap between training (50ms) and testing (25ms) steps and the implications for model generalization need validation with "traditional" parameter fitting protocols, such as formal model cross-validation across well-defined datasets and splits, as well as evaluations to understand and assess potential overfitting.
(3) Inadequate Visualization of Empirical Data: References to empirical data are vague and not directly visualized alongside model outputs. Future iterations should include empirical data, not general trends from psychophysics, in figures for a clear comparison.
(4) Limitations in Model Scope and Dynamics: The exploration of limitations is narrowly focused on interval length and noise. Expanding the model limitations to consider isochronous pulse processing and the emergence of limit-cycle behaviors after prolonged stimulation would provide a more comprehensive understanding of the model's capabilities and limitations. Additionally, the justification for using \(N_{Poisson}\) as a proxy for more connections is unclear and warrants a more direct approach. Adding more units to a truly data-driven model should be trivial.
(5) Omissions and Redundancies: Certain omissions, such as the lack of a condition in Figure 7A or missing references to relevant models and reviews, detract from the paper's thoroughness. Moreover, some statements and terms like "internal clock" are used without a clear mechanistic definition within the model.
Guidance for Readers
Readers should approach this paper as a theoretical exploration into the mechanisms of subsecond-time perception. The model offers a detailed theoretical framework that engages with established laws and theories in time perception. However, it's crucial to note the model's reliance on general trends and its lack of direct empirical backing. The findings should be interpreted as a hypothesis-generating exercise rather than conclusive evidence. -
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