Annihilation of action potentials induces electrical coupling between neurons

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.

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.

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 Hodgkin-Huxley 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.

(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 neglected, 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.

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.

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.

p8 the values of both the extracellular (100 Ohm m) and intracellular resistivity (1 Ohm m) appear to be in error, especially the former.

(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.

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.

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.