Approaches to Parameter Estimation from Model Neurons and Biological Neurons
Round 1
Reviewer 1 Report
The paper focusses on the problem of estimating neuron parameters from experimental data. These parameters are then used to propose predictive neuron models. The review is well written and organized.
I have only one concern about it. Since the journal Algorithms addresses to a multidisciplinary scientific community, I suggest the authors to include more details regarding section 2.1, where they present the general neuron models’ choices and is the starting point of the review. Equation 1 describes the general form for the current, what about the equation for the membrane voltage V?. The authors could present in summarized way (maybe in a supplementary section?) some more details about the system model used to obtain the voltage time series shown in figure 1 (I am not able to identify the black trace). I consider this could be useful for a better comprehension of the rest of the paper. How many parameters were estimated in this case? How many ionic channels were considered? A brief description of the conductance model taken into account can be very helpful for the general reader to follow the article.
Author Response
The paper focusses on the problem of estimating neuron parameters from experimental data. These parameters are then used to propose predictive neuron models. The review is well written and organized.
REPLY: Thank you.
I have only one concern about it. Since the journal Algorithms addresses to a multidisciplinary scientific community, I suggest the authors to include more details regarding section 2.1, where they present the general neuron models’ choices and is the starting point of the review. Equation 1 describes the general form for the current, what about the equation for the membrane voltage V?. The authors could present in summarized way (maybe in a supplementary section?) some more details about the system model used to obtain the voltage time series shown in figure 1
REPLY: We have now provided the details of the exemplar model used in Fig.1 in the supplementary information section.
(I am not able to identify the black trace).
REPLY: This is how excellent the fit of the model to the data is!
We have now explicitly noted in the figure caption that the green trace (model) is overlaid directly on top of the black trace (data). The black trace becomes more easily noticeable in the predicted interval where it departs from the predicted data in a few places.
I consider this could be useful for a better comprehension of the rest of the paper. How many parameters were estimated in this case? How many ionic channels were considered? A brief description of the conductance model taken into account can be very helpful for the general reader to follow the article.
REPLY: The model of Fig.1 had 9 ion channels, 12 coupled first order equations and 71 parameters. This is now stated in the supplementary information section.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
In the present manuscript, the author writes a thorough review focusing on parameter estimation for neurons. It is indeed a theoretical challenge that is mainly put forward by theoretical/computational neuroscientists that deserves special attention. The author discusses the current set of the field and some of the issues behind parameter estimation. I find it particularly important the description of protocols that elucidate the full spectrum of ion channels, so their parameters can be estimated. Similarly, how to deal with noise in experiments is also a recurrent and important discussion in this manuscript.
The manuscript is well written, and I do think it will be of interest to the community.
Since this is a review, I have a couple of suggestions for the author which I would like him to address. My specific comments should be used so that the quality of the work can be improved.
- One particular aspect that I think could be better discussed is parameter degeneracy. The author discusses this point very briefly (e.g. line 194). This is in my view of extreme importance, especially in the era of artificial neural networks where many parameters can be estimated for a specific neuronal behavior and how to overcome this problem (if we can call it a problem).
- The entire manuscript is focused on parameter estimation of ionic currents. Combining ionic currents in the Hodgkin-Huxley formalism can be used to study neurons but can also be used to study the ionic currents alone. The behavior of a single ionic current is still interest by its own, and I suppose the author could include a discussion about the usefulness of parameter estimation here. An example is the IMI current found in Pyloric and gastric networks in the Cancer borealisSTG which is currently being studied (it is also found in some of the citations provided by the author).
- I think the author should provide at least a few examples of software (and links) where parameter estimation can be performed nowadays. The reader may wish to follow up with those.
- Lines 75-77: rebound mechanisms could also emerge from potassium currents (e.g. the hyperpolarization-activated Ih current).
- Figure 1: Please improve the quality of the labels in the figure. Also, in the caption, “Blue arrows show 3 action potentials missing in the real neuron (stochastic threshold fluctuations) which are correctly predicted by the model”, if they are missing they are not correctly predicted.
- Line 111: many conductance-based models do not have a hard threshold, but they fire due to the ionic dynamics. Please clarify.
- Line 125: repetition of the word “the”.
- Line 172: “the data in fully known”, -> the data is fully known?
- Section 2.4: Typo in “effeect”, and in the first line type in “volage”.
- Line 250: Indeed, noise can be helpful to shift the position of local and global minima. An example of that is stochastic gradient descent and the author could include this information.
- Section 3.1: The author could mention other cost functions.
- Line 313: “may ‘be’ added “
- Lines 340-342: This example could be supported by citations.
- LASSO for parameter estimation where a penalty is used could be briefly mentioned in the review.
Author Response
In the present manuscript, the author writes a thorough review focusing on parameter estimation for neurons. It is indeed a theoretical challenge that is mainly put forward by theoretical/computational neuroscientists that deserves special attention. The author discusses the current set of the field and some of the issues behind parameter estimation. I find it particularly important the description of protocols that elucidate the full spectrum of ion channels, so their parameters can be estimated. Similarly, how to deal with noise in experiments is also a recurrent and important discussion in this manuscript.
The manuscript is well written, and I do think it will be of interest to the community.
REPLY: Thank you.
Since this is a review, I have a couple of suggestions for the author which I would like him to address. My specific comments should be used so that the quality of the work can be improved.
- One particular aspect that I think could be better discussed is parameter degeneracy. The author discusses this point very briefly (e.g. line 194). This is in my view of extreme importance, especially in the era of artificial neural networks where many parameters can be estimated for a specific neuronal behavior and how to overcome this problem (if we can call it a problem).
REPLY: In well-posed problems, where the model is known, and provided the criteria listed in section 2 are fulfilled, interior point optimization will recover the original parameters of the typical conductance model with 100% certainty and to within 0.1%. This was observed half a dozen model variants to the model in SI. This model has 9 ion channels (12 differential equations) and 71 parameters.
Parameter degeneracy would occur if two ion channels happen to have the same mathematical form. Such a model would fail the identifiability criterion. It is also a theoretical possibility that identifiability may also fail if the data do not contain enough information to constrain the model parameters. This is very much a theoretical possibility because in practice the data set (10,000 points) is much greater than the number of parameters in the model (<100). In practice, and which the above proviso, parameter degeneracy is never an issue in assimilating well-posed problems.
ill-posed problems where the model is unknown, are a different matter. Parameters often become correlated if the cost function fails to properly include model error (this is still an active area of research). If the cost function only includes a data error term, the parameter search algorithm will assign erroneous parameter values to compensate for model error. In that case parameters will appear to be degenerate. Besides the cost function never reach the low values seen in the well-posed case. The potential surface to minimize develops deep narrow valleys along which parameter degeneracies/correlations develop. The answer is to include model error in the cost function and compute the degree of confidence on has in the model by computing the model error covariance matrix.
The issue of parameter degeneracy has been clarified in the manuscript in lies 129-132.
- The entire manuscript is focused on parameter estimation of ionic currents. Combining ionic currents in the Hodgkin-Huxley formalism can be used to study neurons but can also be used to study the ionic currents alone. The behavior of a single ionic current is still interest by its own, and I suppose the author could include a discussion about the usefulness of parameter estimation here. An example is the IMI current found in Pyloric and gastric networks in the Cancer borealisSTG which is currently being studied (it is also found in some of the citations provided by the author).
REPLY: Yes, we agree. Currents can of course be reconstructed to be studied once the fundamental parameters of the model are obtained.
- I think the author should provide at least a few examples of software (and links) where parameter estimation can be performed nowadays. The reader may wish to follow up with those.
REPLY: I have included references to IPOPT, 4DVar, Gradient descent and LASSO. in the second section of the supplementary information.
- Lines 75-77: rebound mechanisms could also emerge from potassium currents (e.g. the hyperpolarization-activated Ih current).
REPLY: Now corrected in the text.
- Figure 1: Please improve the quality of the labels in the figure. Also, in the caption, “Blue arrows show 3 action potentials missing in the real neuron (stochastic threshold fluctuations) which are correctly predicted by the model”, if they are missing they are not correctly predicted.
REPLY: The missing spikes are due to the spurious coupling of the neuron to other neurons in the slice which in spite of precautions taken (inhibitors) cannot be entirely suppressed. Compare with the action potential burst at 1650ms with the two preceding bursts at 1350ms and 1550ms. Stimuli are identical but responses are different. This is now clarified in the caption. The figure has now been reformatted with a higher resolution to make the subscripts more readable in the labels.
- Line 111: many conductance-based models do not have a hard threshold, but they fire due to the ionic dynamics. Please clarify.
REPLY: The stochasticity of the ion channel population is already included in the voltage width of the transition region from the open to closed state of an ionic gate (see model in the supplementary information). This width accounts for the variance of this distribution as the centre of which is the voltage threshold.
- Line 125: repetition of the word “the”.
REPLY: Now corrected, thank you.
- Line 172: “the data in fully known”, -> the data is fully known?
REPLY: Now corrected.
- Section 2.4: Typo in “effeect”, and in the first line type in “volage”.
REPLY: Now corrected.
- Line 250: Indeed, noise can be helpful to shift the position of local and global minima. An example of that is stochastic gradient descent and the author could include this information.
REPLY: Now inserted.
- Section 3.1: The author could mention other cost functions.
REPLY: We are aware of alternative costs functions used in meteorology (4DVar) which are of similar form but with have one data error component and one estimation error component (e.g. Eugenia Kalnay’s book in our references). In meteorology, the model is completely known but the data are measured to different degrees of confidence. In neuroscience, it is the other way round: the data are readily accessible whereas the models are guessed. This new situation is focusing research efforts on model error covariance matrices. Other cost functions may or may not be relevant to this problem. We want to avoid giving the reader the false impression that by citing various cost function the problem of quantifying model error is a solved.
- Line 313: “may ‘be’ added “
REPLY: Now corrected.
- Lines 340-342: This example could be supported by citations.
REPLY: As mentioned in the text this is very much an active research area. The closest reference on the state of the art is the paper by Rey et al which is already cited.
- LASSO for parameter estimation where a penalty is used could be briefly mentioned in the review.
REPLY: We have referenced this method in the manuscript and SI as requested although linear regression is indirectly applicable to the convex problems at hand here.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The authors have satisfactorily addressed all my comments and made the necessary changes to the manuscript.
