Fitting bifurcation structure, not voltage traces: A biophysically inspired derivation of reduced neuron models exemplified by potassium dynamics
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eLife Assessment
This work demonstrates an objective way to select parameter values for a quadratic integrate-and-fire model so that its bifurcation diagram matches a specific target diagram, generated from the Wang-Buzsaki model. The method is useful for the field and is presented with convincing evidence. The method is currently limited in its ability to be applied to data, but improves our mathematical tools to treat a rarely studied type of bifurcation.
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Abstract
Abstract
In conductance-based models, spiking-induced ion concentrations fluctuations can modify single neurons’ excitability. What are the consequences in networks? To study this, simple models capturing ion concentration dynamics realistically are needed. We propose a method to derive a phenomenological model capturing the coupled extracellular potassium and voltage dynamics from a given class I conductance-based model. Rather than fitting voltage traces, we fit the bifurcation structure of the target model, thereby capturing parameter heterogeneity and rich dynamics. The resulting model extends the quadratic integrate-and-fire model, with extracellular potassium accumulation altering voltage dynamics by increasing the reset voltage. We apply our systematic reduction procedure to the Wang-Buzsáki model. Its phenomenological version exhibits quantitatively comparable dynamics and replicates the reshaping of the phase-response curve associated with the transition from SNIC to HOM spikes at elevated potassium. To illustrate the derived model’s applicability, we explore how changes in potassium concentration influence synchronization in networks.
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eLife Assessment
This work demonstrates an objective way to select parameter values for a quadratic integrate-and-fire model so that its bifurcation diagram matches a specific target diagram, generated from the Wang-Buzsaki model. The method is useful for the field and is presented with convincing evidence. The method is currently limited in its ability to be applied to data, but improves our mathematical tools to treat a rarely studied type of bifurcation.
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Reviewer #1 (Public review):
Summary:
From a big picture viewpoint, this work aims to provide a method to fit parameters of reduced models for neural dynamics so that the resulting tuned model has a bifurcation diagram that matches that of a more complex, computationally expensive model. The matching of bifurcation diagrams ensures that the model dynamics agree on a region of parameter space, rather than just at specially tuned values, and that the models share properties such as qualitative features of their phase response curves, as the authors demonstrate. A notable point is the inclusion of extracellular potassium concentration dynamics into the reduced model - here, the quadratic integrate-and-fire model; this is straightforward but nonetheless useful for studying certain phenomena.
Strengths:
The paper demonstrates the method …
Reviewer #1 (Public review):
Summary:
From a big picture viewpoint, this work aims to provide a method to fit parameters of reduced models for neural dynamics so that the resulting tuned model has a bifurcation diagram that matches that of a more complex, computationally expensive model. The matching of bifurcation diagrams ensures that the model dynamics agree on a region of parameter space, rather than just at specially tuned values, and that the models share properties such as qualitative features of their phase response curves, as the authors demonstrate. A notable point is the inclusion of extracellular potassium concentration dynamics into the reduced model - here, the quadratic integrate-and-fire model; this is straightforward but nonetheless useful for studying certain phenomena.
Strengths:
The paper demonstrates the method specifically on the fitting of the quadratic integrate-and-fire model, with potassium concentration dynamics included, to the Wang-Buzsaki model extended to include the potassium component. The method works very well overall in this instance. The resulting model is thoroughly compared with the original, in terms of bifurcation diagrams, production of various activity patterns, phase response curves, and associated phase-locking and synchronization properties.
Weaknesses:
It is important to note that the proposed method requires that a target bifurcation diagram be known. In practical terms, this means that the method may be well suited to fitting a reduced model to another, more complicated model, but is not likely to be useful for fitting the model to data. Certainly, the authors did not illustrate any such application. Secondly, the authors do not provide any sort of general algorithm but rather give a demonstration of a single example of fitting one specific reduced model to one specific conductance-based model. Finally, the main idea of the paper seems to me to be a natural descendant of the chain of reasoning, starting from Rinzel - continuing through Bertram; Golubitsky/Kaper/Josic; Izhikevich; and others - that a fundamental way to think about neuronal models, especially those involving bursting dynamics, is in terms of their bifurcation structure. According to this line of reasoning, two models are "the same" if they have the same bifurcation structure. Thus, it becomes natural to fit a reduced model to a more complicated model based on the bifurcation structure. The authors deserve credit for recognizing and implementing this step, and their work may be a useful example to the community. But the manuscript should have described and cited this chain of works to put the current study in the correct context.
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Reviewer #2 (Public review):
Summary:
The authors derive an integrate-and-fire model to describe the dynamics of a more complex Wang-Buzsaki model and compare the two models. A detailed discussion of bifurcation schemes in both models is convincing and allows us to evaluate the simpler model.
Strengths:
The idea is interesting, and the mathematical approach appears to be convincing. In addition, differences between the simple and original models are also discussed.
Weaknesses:
A comparison to experimental data is necessary to support the theoretical work.
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Author response:
We thank the reviewers for their constructive feedback on the article’s strengths and weaknesses. In response, we plan to strengthen our work in a revised version by (i) providing an additional example of our method’s implementation and (ii) framing our contribution more clearly as a continuation of the line of research that characterises neuronal models in terms of their bifurcation structure.
Experimental validation, however, is beyond the scope of this study. Constructing experimental bifurcation diagrams remains a major challenge, particularly for unstable branches. Although some techniques exist to approximate branches of unstable steady states, unstable limit cycles are far more difficult to capture. Additionally, in practice, many factors vary during recordings, and generating reliable diagrams would require a …
Author response:
We thank the reviewers for their constructive feedback on the article’s strengths and weaknesses. In response, we plan to strengthen our work in a revised version by (i) providing an additional example of our method’s implementation and (ii) framing our contribution more clearly as a continuation of the line of research that characterises neuronal models in terms of their bifurcation structure.
Experimental validation, however, is beyond the scope of this study. Constructing experimental bifurcation diagrams remains a major challenge, particularly for unstable branches. Although some techniques exist to approximate branches of unstable steady states, unstable limit cycles are far more difficult to capture. Additionally, in practice, many factors vary during recordings, and generating reliable diagrams would require a large number of tightly controlled experimental repetitions whose stability often cannot be ensured. Two-dimensional bifurcation diagrams, as needed for the analysis in our manuscript, are even more challenging to obtain because the extensive and stable recordings would have to be available from the same cell at different values of the second parameter (such as different extracellular potassium concentrations). At this stage, our method can be applied to the reduction of detailed conductance-based models, which themselves are constrained by experimental data (for example, gating functions fitted to voltage-clamp recordings). This way, simple yet dynamically faithful phenomenological models for efficient use in network analysis and simulation can be derived from more complex, biophysical models. In contrast to the traditional voltage fitting approach, these models can also capture changes in additional parameters (such as extracellular potassium concentration).
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