Author response:
The following is the authors’ response to the original reviews.
Public Reviews:
Reviewer 1 (Public Review):
The authors explain that an action potential that reaches an axon terminal emits a small electrical field as it ”annihilates”. This happens even though there is no gap junction, at chemical synapses. The generated electrical field is simulated to show that it can affect a nearby, disconnected target membrane by tens of microvolts for tenths of a microsecond. Longer effects are simulated for target locations a few microns away.
To simulate action potentials (APs), the paper does not use the standard Hodgkin-Huxley formalism because it fails to explain AP collision. Instead, it uses the Tasaki and Matsumoto (TM) model which is simplified to only model APs with three parameters and as a membrane transition between two states of resting versus excited. The authors expand the strictly binary, discrete TM method to a Relaxing Tasaki Model (RTM) that models the relaxation of the membrane potential after an AP. They find that the membrane leak can be neglected in determining AP propagation and that the capacitive currents dominate the process.
The strength of the work is that the authors identified an important interaction between neurons that is neglected by the standard models. A weakness of the proposed approach is the assumptions that it makes. For instance, the external medium is modeled as a homogeneous conductive medium, which may be further explored to properly account for biological processes.
The authors provide convincing evidence by performing experiments to record action potential propagation and collision properties and then developing a theoretical framework to simulate the effect of their annihilation on nearby membranes. They provide both experimental evidence and rigorous mathematical and computer simulation findings to support their claims. The work has the potential of explaining significant electrical interaction between nerve centers that are connected via a large number of parallel fibers.
We thank the reviewer for the distinct analysis of our work and the assessment that we ’identified an important interaction between neurons that is neglected by standard models’.
Indeed, we modeled the external (extracellular) medium as homogeneous conductive medium and, compared to real biological systems, this is a simplification. Our intention is to keep our formal model as general as possible, however, it can be extended to account for specific properties. Accessory structures at axon terminals (such as the pinceau at Purkinje cells) most likely evolved to shape ephaptic coupling. In addition, the extracellular medium is neither homogeneous nor isotropic, and to fully mimic a particular neural connection this has to be implemented in a model as well. We agree and look forward to see how specific modification of the external medium in biological systems will affect ephaptic coupling. We hope to facilitate progress on this question by providing our source code for further exploration. Using the tools that have been developed by the BRIAN community one can generate or import arbitrary complex cell morphologies (e.g. NeuroML files). Our source code adds the TM- and RTM model, which allows exploring the direct impact of extracellular properties on target neurons.
Reviewer 2 (Public Review):
In this study, the authors measured extracellular electrical features of colliding APs travelling in different directions down an isolated earthworm axon. They then used these features to build a model of the potential ephaptic effects of AP annihilation, i.e. the electrical signals produced by colliding/annihilating APs that may influence neighbouring tissue. The model was then applied to some different hypothetical scenarios involving synaptic connections. The conclusion was that an annihilating AP at a presynaptic terminal can ephaptically influence the voltage of a postsynaptic cell (this is, presumably, the ’electrical coupling between neurons’ of the title), and that the nature of this influence depends on the physical configuration of the synapse.
As an experimental neuroscientist who has never used computational approaches, I am unable to comment on the rigour of the analytical approaches that form the bulk of this paper. The experimental approaches appear very well carried out, and here I just have one query - an important assumption made is that the conduction velocity of anti- and orthodromically propagating APs is identical in every preparation, but this is never empirically/statistically demonstrated.
My major concern is with the conclusions drawn from the synaptic modelling, which, disappointingly, is never benchmarked against any synaptic data. The authors state in their Introduction that a ’quantitative physical description’ of ephaptic coupling is ’missing’, however, they do not provide such a description in this manuscript. Instead, modelled predictions are presented of possible ephaptic interactions at different types of synapses, and these are then partially and qualitatively compared to previous published results in the Discussion. To support the authors’ assertion that AP annihilation induces electrical coupling between neurons, I think they need to show that their model of ephaptic effects can quantitatively explain key features of experimental data pertaining to synaptic function. Without this, the paper contains some useful high-precision quantitative measurements of axonal AP collisions, some (I assume) high-quality modelling of these collisions, and some interesting theoretical predictions pertaining to synaptic interactions, but it does not support the highly significant implications suggested for synaptic function.
We thank the reviewer for highlighting the potential and the limitation of our model. We demonstrated with empirical data that measured conduction velocities of anti- and orthodromic propagating APs are indeed very similar and values are provided in Appendix 3 – table 1.
In order to address how our model ’of ephaptic effects can quantitatively explain key features of experimental data’, we used the measured modulation of AP rates in Purkinje fibers by Blot and Babour (2014) and our results are now included in the manuscript. In our model, we implemented the ephaptic coupling of the Basket cell (with an annihilating AP) and predicted the modulation of AP rate in the Purkinje cell. Our model predictions are compared to the measured modulation of AP-rates in Purkinje cells and is added as Fig. 5 to the main manuscript (line 264 to 284 ). With this example, we show that ephaptic coupling as described with our RTM model can quantitatively describe key features of experimental data. Both, the rapid inhibition and the rebound activity is described by our model with implementation of non-excitable parts at the pinceau of the Basket cell. Future, experimental research can use the provided formalism to investigate in more detail the ephaptic coupling in systems like the Mauthner cell and the Purkinje cell by exploring how accessory structures and concomitant physical parameters, e.g. the extracellular properties impact ephaptic coupling.
Reviewer 3 (Public Review):
This manuscript aims to exploit experimental measurements of the extracellular voltages produced by colliding action potentials to adjust a simplified model of action potential propagation that is then used to predict the extracellular fields at axon terminals. The overall rationale is that when solving the cable equation (which forms the substrate for models of action potential propagation in axons), the solution for a cable with a closed end can be obtained by a technique of superposition: a spatially reflected solution is added to that for an infinite cable and this ensures by symmetry that no axial current flows at the closed boundary. By this method, the authors calculate the expected extracellular fields for axon terminals in different situations. These fields are of potential interest because, according to the authors, their magnitude can be larger than that of a propagating action potential and may be involved in ephaptic signalling. The authors perform direct measurements of colliding action potentials, in the earthworm giant axon, to parameterise and test their model.
Although simplified models can be useful and the trick of exploiting the collision condition is interesting, I believe there are several significant problems with the rationale, presentation, and application, such that the validity and potential utility of the approach is not established.
Simplified model vs. Hogdkin and Huxley
The authors employ a simplified model that incorporates a two-state membrane (in essence resting and excited states) and adds a recovery mechanism. This generates a propagating wave of excitation and key observables such as propagation speed and action potential width (in space) can be adjusted using a small number of parameters. However, even if a Hodgkin-Huxley model does contain a much larger number of parameters that may be less easy to adjust directly, the basic formalism is known to be accurate and typical modifications of the kinetic parameters are very well understood, even if no direct characterisations already exist or cannot be obtained. I am therefore unconvinced by the utility of abandoning the HodgkinHuxley version.
In several places in the manuscript, the simplified model fits the data well whereas the Hodgkin-Huxley model deviates strongly (e.g. Fig. 3CD). This is unsatisfying because it seems unlikely that the phenomenon could not be modelled accurately using the HH formulation. If the authors really wish to assert that it is ”not suitable to predict the effects caused by AP [collision]” (p9) they need to provide a good deal more analysis to establish the mechanism of failure.
We are not as convinced as the reviewer that, at the current state of parameter estimation, the HH model is suited for predicting ephaptic coupling after ’adjusting’ parameters. There are strong arguments against such an approach. A major function of a model is to make testable predictions rather than to just mimic a biological phenomenon. The predictive power of a model heavily depends on how reasonable model parameters can be estimated or measured. As the reviewer correctly points out in the specific comments (”... the parameters adjusted to fit the model are the membrane capacitance and intracellular resistance. These have a physical reality and could easily be measured or estimated quite accurately...”), our model contains only parameters that can be assessed experimentally, thus it has a better predictive power compared to the HH model with a multitude of parameters for which ”no direct characterisations already exist or cannot be obtained” (citing reviewer from above).
Already the founders of the HH model were well aware of the limitations, as stated by Hodgkin and Huxley in 1952 (J Physiol 117:500–544):
An equally satisfactory description of the voltage clamp data could no doubt have been achieved with equations of very different form ... The success of the equations is no evidence in favour of the mechanism of permeability change that we tentatively had in mind when formulating them.
A catchy but sloppy description for the problem of overfitting with too many parameters is given by the quote of John von Neumann: With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.
We do not rule out the possibility that the HH model eventually can be used to predict ephaptic coupling. However, at the moment, parameter estimation for the HH model prevents its usability for predicting ephaptic coupling.
(In)applicability of the superposition principle
The reflecting boundary at the terminal is implemented using the symmetry of the collision of action potentials. However, at a closed cable there is no reflecting boundary in the extracellular space and this implied assumption is particularly inappropriate where the extracellular field is one objective of the modelling, as here. I believe this assumption is not problematic for the calculation of the intracellular voltage, because extracellular voltage gradients can usually be neglected1, but the authors need to explain how the issue was dealt with for the calculation of the extracellular fields of terminals. I assume they were calculated from the membrane currents of one-half of the collision solution, but this does not seem to be explained. It might be worth showing a spatial profile of the calculated field.
We disagree with the reviewer’s statement ’...at a closed cable there is no reflecting boundary in the extracellular space and this implied assumption is particularly inappropriate...’. We do not imply this assumption in our model! We do not assume any symmetry or boundary condition in the extracellular space. Instead, the extracellular field is calculated for an infinite homogeneous volume conductor (Eq.
6).
We conduct separate calculations for (1) source membrane current, (2) resulting extracellular field, and (3) impact upon a target neuron. The boundary condition used for our calculations only refers to the axial current being zero at the axon terminal. Consequently all the internal current that enters the last compartment must leave the last compartment as membrane current and contributes to the extracellular current and field.
The extracellular field around the axon terminal is not symmetric, as can be seen by it’s impact upon a target in Figure 4—figure supplement 1 which is also not symmetric. The symmetry of the extracellular field when APs are colliding (Cf. symmetry in Fig 1C) is merly the result of the symmetric stimulation and counterpropagation of two APs. We now are describing more specifically the bounday condition for colliding and terminating APs already in the introduction: ’A suitable boundary condition (intracellular, axial current equals zero) can be generated experimentally by a collision of two counter-propagating APs ... Within any cable model, the very same boundary condition also exists within the axon at the synaptic terminal due to the broken translation symmetry for the current loops ...’ Later, at the result section (Discharge of colliding APs), we continue with ’AP propagation is blocked when the axial current is shut down at a boundary condition, e.g. by reaching the axon terminal or by AP collision....’ and implement this condition in our calculations for the axon terminals.
Missing demonstrations
Central analytical results are stated rather brusquely, notably equations (3) and (4) and the relation between them. These merit an expanded explanation at the least. A better explanation of the need for the collision measurements in parameterising the models should also be provided.
We thank the reviewer for pointing out the insufficient explanation of the equations 3 and 4. We rephrased the paragraph ’Discharge of colliding APs’ in order to clarify the origin and the function of the two equations (eq. 3: how much charge is expelled and eq. 4: the resulting extracellular potential that is used for model validation).
Later, in the Discussion, we rephrased the paragraph where we describe the annihilation process and explain further that one term of eq. 4 sometimes is refered to ’activating function’ when using microelectrodes for stimulation.
With respect to the ’explanation of the need for the collision measurement’, we think that the explanations we give at several locations in the manuscript are sufficient as is. We explain and elaborate in the introduction: ’We explore the behaviour of APs at boundaries ... In this study, we first focus on collisions of APs. Our experimental observation of colliding APs provides unique access to the spatial profile of the extracellular potential around APs that are blocked by collisions and thus annihilate..... Recording propagating APs allows to determine both the propagation velocity and the amplitude of the extracellular electric potentials. The collision experiment provides additional information ... In the results we recall: ’The width of the collision is a measure of the characteristic length λ⋆ of the AP and is uniquely revealed by a collision sweep experiment.’
Adjusted parameters
I am uncomfortable that the parameters adjusted to fit the model are the membrane capacitance and intracellular resistance. These have a physical reality and could easily be measured or estimated quite accurately. With a variation of more than 20-fold reported between the different models in Appendix 2 we can be sure that some of the models are based upon quite unrealistic physical assumptions, which in turn undermines confidence in their generality.
The fact that the parameters of our model have physical realities is clearly in favor of our models. We rephrased the legend of the table, now explaining the procedure for the model fitting and the rational behind. Although the values of g⋆ can differ by a factor of 15 and the resulting amplitude is very different, the relationship ri cm = vpλ⋆ is very similar, independently of the model used and this confirms our analytical framework.
p8 - the values of both the extracellular (100 Ohm m) and intracellular resistivity (1 Ohm m) appear to be in error, especially the former.
We have the following justification for the resistivity values we used. For the intracellular resistivity, literature values range from 0.4 - 1.5 Ohm m, and therefore we selected 1 Ohm m. See: Carpenter et al (1975) doi: 10.1085/jgp.66.2.139; Cole et al (1975) doi: 10.1085/jgp.66.2.133; Bekkers (2014) doi: 10.1007/978-1-46147320-6 35-2.
Estimating extracellular resistivity is less straight forward, since it depends crucially on the structure around the synapse which consists of conducting saline and insulating fatty tissue. Ranges from 3 to 600 Ohm m are reported (Linden et al (2011) doi: 10.1016/j.neuron.2011.11.006) and Bakiri et al (2011) doi: 10.1113/jphysiol.2010.201376). Weiss et al (2008; doi: 10.1073/pnas.0806145105) report extracellular resistivities in the Mauthner Cap between 50-600 Ohm m in SI. Since the pinceau is structurally similar to the Mauthner cells axon cap, we argue that a value of 100 Ohm m is a reasonable choice for our calculations. Additionally, we derived a value from Blot and Barbour (doi:c10.1038/nn.3624), rephrased the paragraph in the main text and added our calculation to the supplementary material (Appendix 1).
(In)applicability to axon terminals
The rationale of the application of the collision formalism to axon terminals is somewhat undermined by the fact that they tend not to be excitable. There is experimental evidence for this in the Calyx of Held and the cerebellar pinceau.
The solution found via collision is therefore not directly applicable in these cases.
We do not agree with the reviewer’s statement that ’the solution found via collision is (therefore) not directly applicable...’. Our model is well suited for application on axon terminals that are not excitable, e.g. the pinceau of the basket cell, as the reviewer points out. We have included a calculation for this case and present the results in the new Fig. 5 (main text line 264 to 284 ).
Comparison with experimental data
More effort should be made to compare the modelling with the extracellular terminal fields that have been reported in the literature.
As outlined above (see: Reponse to reviewer 2), we now compare directly the predictions of our models with measured modulation of AP rates in Purkinje fibers (Blot and Babour 2014) and our results are included in the manuscript (Fig. 5 and main text line 264 to 284). See also our response to reviewer 2 in which we address how our model ’of ephaptic effects can quantitatively explain key features of experimental data’.
Choice of term ”annihilation”
The term annihilation does not seem wholly appropriate to me. The dictionary definitions are something along the lines of complete destruction by an external force or mutual destruction, for example of an electron and a positron. I don’t think either applies exactly here. I suggest retaining the notion of collision which is well understood in this context.
Experimentally, we generated a collision of APs and showed that colliding APs dissapear and do not pass each other. For this process the term annihilation is used in our and in other studies (see e.g. Berg et al (2017) doi: 10.1103/PhysRevX.7.028001; Johnson et al (2018) doi: 10.3389/fphys.2018.00779; Follmann (2015) doi: 10.1103/PhysRevE.92.032707; Shrivastava et al (2018) doi: 10.1098/rsif.2017.0803). The physical processes involved in the termination of an AP at a closed end are essentially identical to those of two colliding APs. This we think justifies using the term annihilation for those processes.
Recommendations for the authors:
We believe the work is of high quality and should motivate future experimental work. We are including the review comments here for your information. The main piece of feedback we are offering is that the broad claims need to be adjusted to the strength of evidence provided: as is, the manuscript provides compelling predictions but the claim that these predictions are in full agreement with data remains to be substantiated. A technical concern raised by the reviewers is that the reflecting boundary condition may need further justification. The authors may wish to respond to this issue in a rebuttal and/or adjust the manuscript as necessary.
We substantiated our claim that our predictions are in full agreement with experimental data. We added to the manuscript a section in which we compare our models’ predictions to published, experimental data. To this aim, we extracted date from the publication of Blot and Babour (2014), we elaborated on the parameters used and run our model accordingly. We added to the Results/Model of ephaptic coupling a paragraph on ’The modulation of activity in Purkinje cells...’ (line 264), where we describe our results and we also included another figure to the main text for illustration (Fig. 5).
We clarified the term ’boundary condition’ by rephrasing parts of the introduction and we explain the rational behind in ’Discharge of colliding APs (...AP propagation is blocked when axial current is shut down...) and in ’Model of ephaptic coupling (Within any cable model, the same boundary...). See also our response to the general comments of reviewer 3 above.
Reviewer 1 (Recommendations For The Authors):
Major:
Accessing data and code requires signing in, which should not be required. The link provided also seems to be not accessible yet - could be pending review.
The repository is now publicly availible. We did provide an access code within the letter to the editor, this code is no longer required.
Line 74: how about morphology? Authors should clarify and emphasize in the introduction that the TM model is a spatially continuous model with partial differential equations as opposed to discrete morphological models to simulate HH equations.
The reviewer is correct that the TM model is continous. However, so is the HH model. The difference between HH and TM is only that the TM model can be solved analytically, which yields a spatially homogeneous analytical solution. It should be noted that this analytical solution can only be valid for a homogeneous (therefore infinite) nerve. Every numerical computation, be it HH or TM, requires a finite number of discrete compartments. In our calculations, we used identical compartment models for HH, TM and RTM model. In each compartment, the differential equations are solved numerically. Since there is no fundamental difference between these models, we obstain from changing the text.
Minor:
Major typo: ventral nerve cord, not ”chord”. Repeated in several places.
Thank you for indicating this typo to us.
Line 25: inhibition, excitation, and modulation?
We changed the line to: ... leads to modulation, e.g. excitation or inhibition
Line 70: better term for ”length” of AP would be ”duration”. Also, the sentence could be simplified to use either ”its” or ”of the AP”
Space and time are not interchangable. Thus, the term lenght can not be replaced by duration. We simplified the structure of the sentence as suggested.
Fig 1A/B: it’s strange that panel B precedes panel A.
Exchanged
Fig 1C: don’t see the ”horizontal line”; also regarding ”The recording was at a medial position”, the caption is not clear until one reads the main text.
We changed the legend to: ... The collision is captured in the recording line at y-position 0 mm, while orthodromic propagation is at the top and antidromic propagation is at the bottom. (D) The peak amplitude as a function of the distance to the collision. Examples of four sweeps at three positions along the nerve cord....
Line 127: the per distance measures could be named as ”specific” conductivity, etc.
We explicitly provide the units thereby defining the quantities unambigously.
Line 176: typo ”ad-hoc”.
Thank you.
Fig 4B: should clarify that the circle in the schematic is not the soma but a synaptic bouton.
We rephrased to ’...(B,C) when the AP is annihilating at a bouton of a neuron terminal (upper neuron in end-to-shaft geometry, similar to the Basket cell–Purkinje cell synapse)...’, and we added a label to Fig 4B.
Reviewer 2 (Recommendations For The Authors):
Can the authors’ model be quantitatively compared with experimental data of ephaptic interactions at synapses (e.g. the Blot & Barbour study described in the Discussion)?
We did so as outlined in our response to the reviewer above.
Can statistical evidence be provided that the velocities of anti- and orthodromic APs are indeed identical in the earthworm nerve recordings?
These data and statistics are available in Appendix 2, now 3 – table 1
Why not reorder ABCD in Fig1 so the subpanels run from left to right?
We adjusted the labels accordingly.
