One equation, two unknowns

A paper from the group of David Holcman investigates the biophysics of dendritic spines by analysing fluorescence measurements of voltage-sensitive dyes during focal uncaging of glutamate and by electrodiffusion modelling. Complex analysis and optimisation procedures are reportedly used to extract an estimate of the spine neck resistance. However, examination of the procedures reveals that the resistance value is wholly determined by fixed parameter values: there is no extraction. The results are also potentially affected by errors in the modelling and unrealistic parameter choices. Finally, the paper highlights a potential dilemma for authors who share data—should they sign the resulting paper if they disagree with it?

Spines

Before digging into the paper [1], it will be helpful to justify an approximate and very simple model of the spine, in which it is reduced to just the neck resistance. Spine experts will know this and can skip to the next section.

spine
Fig. 1. Equivalent electrical circuit of a spine, with a spherical head and a cylindrical neck. The membrane resistance and neck capacitance have already been neglected, and we’ll see that the head capacitance Csp can be, too. Only the neck resistance Rn has any influence on single-spine biophysics. I, synaptic current; Vsp, spine head voltage; Vd, dendritic voltage.

Although spines display a degree of variability, we’ll consider as typical a spine with a head of radius 0.3 µm, a neck of diameter 0.15 µm and length 1 µm. The key to attaining a useful intuitive understanding of spine behaviour is to clarify which electrical elements can be neglected.

It is generally assumed that a spherical conductor can be well approximated as isopotential, unless particularly concentrated currents flow. So for now we’ll consider the voltage throughout the spine head to be uniform (we’ll re-examine this assumption below, in the light of the authors’ results).

The narrowness of the neck means that most of the membrane is found in the head, so let’s begin by calculating its surface area (ignoring the neck attachment): 4πr2 ≈ 1 µm2. Given the specific membrane capacitance of 1 µF cm-2, we obtain an estimate of the spine capacitance of ~10-14 F. This is small and, as we shall see, can be neglected from most points of view. (Similar arguments also lead us to neglect the membrane resistance, which isn’t shown, but not the synaptic conductance of the spine.)

How much charge would be required to change the voltage of the spine head capacitance? For a round maximum of 100 mV, Q = CV gives 10-15 C, equivalent to 1 pA flowing for 1 ms. In other words, the current flowing through about one AMPA receptor channel is sufficient to charge the spine capacitance; typically there are tens to hundreds of receptors in a spine.

The highest estimates of spine neck resistance so far reported are about 1 GΩ. This would give an RC time constant of 10 µs. Thus, if the voltage in the parent dendrite changed, the spine would follow with this time constant. Conversely, if a constant synaptic current flows across the spine neck to the dendrite, the spine voltage will equilibrate to its new value with the same time constant. These relaxations are all quite fast, close to negligibly fast, on biologically relevant time scales. From this we can conclude that the only electrical parameter of any significance to spine behaviour is the neck resistance. In the absence of a synaptic current, the spine voltage follows the dendritic voltage. When there is a synaptic current, the voltage across the neck resistance is determined by Ohm’s law. (It should be noted, however, that the collective contribution of spine capacitance to dendritic and cellular capacitance can be very significant; for instance, spines contribute about 80% of the total capacitance of cerebellar Purkinje cells.)

profile
Fig. 2. When a synaptic current flows, the approximate voltage profile expected in a spine is uniform across the head and linear down the neck.

Armed with the uniformity of the head voltage and the fact that the neck resistance is the only significant electrical component of the spine, a very usable approximation for the voltage profile in the spine during the synaptic current is shown in Fig. 2: the head will be at a uniform voltage, more depolarised than the dendrite, and there will be a linear decline of voltage between head and base.

It is often necessary to take the voltage change in the dendrite into account. We shall therefore consider the voltage divider formed by the neck resistance and input impedance of the dendrite.

divider
Fig. 3. The spine, reduced to its neck resistance, is of course attached to a dendrite. The absolute voltages in the spine head will depend upon the dendritic impedance Zd. In the paper being analysed, the complex dendritic impedance is assumed to be purely resistive.

With rather unnecessary complexity, the authors call Rn the effective neck resistance and Rn + Zd the intrinsic resistance (they neglect the dendritic and cell capacitances).

Data and processing

The data comes from the Yuste lab, but, notably, the author contribution statement carefully limits their involvement to supplying this data; they had no other involvement in this paper. The data have already been published once by the Yuste lab and a spine neck resistance of 90–100 MΩ reported [7]. It is also worth pointing to a theoretical preprint from the Yuste lab that covers much of the same modelling ground as the present paper.

The data are voltage dye (“ArcLight”) fluorescence measurements of the simultaneous voltages in spine heads and parent dendrites during focal uncaging of glutamate or backpropagating action potentials. As a general comment, the fluorescence signals are unavoidably small, noisy and slow. A very complex deconvolution procedure is applied to work back to the original voltages: filtering, fitting with constrained waveforms, deconvolution. The deconvolution appears to work for somatic signals, but anybody who has tried signal deconvolution will retain a healthy scepticism about the robustness of the procedure as applied to the very noisy spine signals. All the deconvolved signals are still slow—the response to uncaging lasts 100 ms (perhaps calling into question the synaptic specificity), while backpropagating action potentials are an eye-catching 100 ms in duration (it turns out that some of the current-clamp recordings were made using a Cs-based internal solution).

The authors thus have at their disposal time courses of deconvolved voltages at the head and base of the spine during uncaging. Referring to Fig. 3, they have estimates of Vsp and Vd.

One equation, two unknowns

We now see the benefit of our initial analysis simplifying the spine to its neck resistance. The voltage across the spine neck is given by the following relation:

(Vsp – Vd) = IsynRn.

This is of course Ohm’s law, although the resistance may not be perfectly Ohmic. The authors have a problem. There is only one equation, with two unknowns: the desired resistance and the synaptic current induced by the focal uncaging of glutamate. There is no way of splitting IsynRn without additional information. Although the authors don’t present the problem in this way, the additional complexity of their formulation does nothing to get around the underlying biophysics or fact that they do not know the current at any point in time.

There are a few methods in the literature for resolving this problem. In one elegant recent method, Popovic et al (2015) [2] integrate the voltage difference to obtain QRn and estimate the synaptic charge Q from a simultaneous somatic recording, which, despite filtering, is able to recover much of the synaptic charge (and the loss can be estimated for greater accuracy). The previous Yuste lab analysis of the present data estimated Zd, allowing them to estimate the current and then the resistance [7]. Various other groups have monitored the activation of calcium entry in spines via voltage-dependent channels or NMDA receptors to determine the spine voltage indirectly [3, 4, 5]. Here, the authors do none of these things, instead they use electrodiffusion modelling…

Spine models

The authors employ a number of models. One is equivalent to the capacitor and resistor of Fig. 1 (although we know that the capacitance should be neglected), attached when necessary to a dendritic resistance (Fig. 3). They also examine more geometrically detailed models. Finally, they sometimes use full electrodiffusion models, in which the concentrations and fluxes of ionic species are represented explicitly. These can be particularly useful to track changes of the ionic concentrations, but are often unnecessarily complex if only electrical behaviour is of interest.

The optimisation procedure by which the authors claim to extract the resistance while knowing only the voltage (i.e. not the current) is particularly complicated. It combines the simple RC spine model and an electrodiffusion model. No rationale is given for this combination, although one consequence is that the procedure would have appeared mightily complex to referees. A summary of the method described in the paper is as follows. To keep things simple, we’ll ignore the capacitance and focus on the neck resistance/conductance.

  1. The authors initialise the neck conductance G in the simple model. (Note, this appears to be the intrinsic conductance, which includes the unknown dendritic impedance.)
  2. From the voltage data, they generate a current trace using the simple model.
  3. They feed this current trace into the electrodiffusion model of the spine neck to generate a voltage trace. (Whereas here the model is of the effective neck resistance, which does not include the dendritic impedance.)
  4. By comparing this voltage trace with the data, they adjust the neck conductance G. Return to step 2.

It’s not clear how the translation from intrinsic to effective conductance at each iteration was implemented (if it was); the dendritic impedance is unknown. Anyway, if the optimisation converges, a conductance value for the simple model should have been obtained such that the voltage output from the electrodiffusion model matches the data. However, no part of the electrodiffusion model is altered in the optimisation, which means that G in the simple model should converge to (approximately) the conductance set by the fixed parameters of the electrodiffusion model (these are the geometry, ionic concentrations and diffusion coefficients). In other words, there is no optimisation of the neck conductance! The value of 100 MΩ in the abstract was not “extracted”, but chosen a priori. Thus, unsurprisingly, the authors have not managed to determine two unknowns from a single equation. Moreover, had the authors really attempted to fit the electrodiffusion model, they would have had an even greater number of unknowns to determine. They might also have had to analyse the system to decide in which regimes the different parameters could be separated. If the information is unavailable, throwing a complex optimisation routine at the problem won’t help.

The diffusion coefficient Dp is that for potassium ions taken from Chen & Nicholson (2000). There, it is given as 2.2 x 10-5 cm2/s. That is equivalent to 2200 µm2/s, not the 200 µm2/s given in Table 2, an 11-fold difference. What happened there? An error while converting units? (As well as rounding?)

The capacitance values obtained through the optimisation (Table 1) are complete nonsense for 2/5 recordings. 18 pF is about 1000-fold greater than the approximate real capacitance calculated above. In reality, that trivial calculation could have shown the authors that the spine capacitance would be completely negligible and undetectable in their recording situation.

The electrodiffusion models appear to have boundary conditions that are inconsistent with the biophysics under investigation. Thus, Eq. 39 has ∂V/∂x = 0, whereas any current flow through a resistor would give a non-zero voltage gradient (Ohm’s law again). Additionally, the ∂Cm/∂x = 0 condition is probably intended to reflect the fact that the synaptic current is purely cationic. However, the anions are not independent of the cations. If there is a synaptic flux of cations that tends to establish a concentration gradient (as the authors will suggest), then electroneutrality will impose a corresponding anion gradient, including at the boundary.

Similarly inconsistent boundary conditions are applied in the full 3d model of the spine head and neck (Eqs. 58; the injection boundary is Ωi). In apparent contradiction with the condition of zero voltage gradient, we can see a very strong voltage gradient at the site of current injection in Fig. 3. In Fig. S7 there is an analogous gradient for Cp at the site of injection, which by electroneutrality must be mirrored by a non-zero Cm gradient, which would also contradict a boundary condition. Quite how the solution has been affected by these inconsistent boundary conditions is difficult to predict.

(The problem of separating I and R, the confusion over the role of the capacitance, the lack of knowledge of Zd and the questions regarding the boundary conditions have been the subject of inconclusive prior discussions with the authors.)

What use is electrodiffusion modelling?

Putting aside for now the above doubts about the accuracy of the electrodiffusion modelling, what new biophysical behaviour have the authors discovered? If we compare the intuitive prediction for the voltage profile (Fig. 2) with the authors’ Fig. 3B,D, we see that the main deviation is a strong voltage gradient near the site of current injection. Beyond that, there are less striking deviations from voltage uniformity across the head and from a linear decline of voltage down the neck. The relation between current and voltage across the neck also becomes nonlinear.

The voltage gradient at the site of injection is probably strongly exaggerated, for at least two reasons:

  1. The currents are modelled as entering the spine head through a postsynaptic density (PSD) of radius 10 nm. Ref [6] allows calculation of a mean spine PSD area of 0.11µm2, which yields a radius of 0.18 µm if a circular shape is assumed. It can be shown that the peak voltage is approximately inversely proportional to the PSD radius, so this parameter choice alone accounts for a factor of 15–20.
  2. If an error of the diffusion coefficient is confirmed, the intracellular resistivity and therefore the peak voltage may have been overestimated by an additional factor.

It is therefore likely that under more realistic conditions there is no meaningful deviation from voltage uniformity across the head in the spine, including under the PSD. The peak sub-PSD voltage caused by the synaptic current can also be estimated directly by modelling a circular disk current source in a semi-infinite medium. With a radius of 180 nm, a 100 pA current and an intracellular resistivity of 150 Ωcm, I calculate a peak voltage deviation of 0.26 mV, which is much smaller than the deviations predicted by the authors.

The deviations from Ohmic linearity in the neck result from another mechanism. The authors point out that, as positive ions enter, their concentration at the point of entry increases, attracting anions. Over time a spatial concentration gradient is established (Fig. S7). The concentration gradient causes a gradient of resistivity and thus a nonlinear voltage gradient. This proposed mechanism seems sound, but the magnitude of the effect is uncertain, for several reasons:

  1. The effect is evaluated in the steady state, which allows ionic gradients to accumulate. Conversely, synaptic currents are brief, especially at physiological temperature.
  2. The possible diffusion coefficient error may affect these gradients.
  3. The modelling includes very mobile anions. Most anions inside cells are somewhat larger, less mobile molecules. This reduced mobility will impede the accumulation of anions and, through electroneutrality, oppose accumulation of cations also. This will reduce all of the effects somewhat. An extreme example of this was reported by Qian & Sejnowski (1989), who simply ignored anions in their modelling, in essence assuming they were all immobile. In consequence, they predicted only the tiniest variations of total ion concentration.

I would expect more careful parameter choices (and, if required, a corrected model) to show that the electrical approximation of Fig. 3 remains adequate for most uses. The Yuste lab preprint estimates that the maximum reduction of resistance during a synaptic current is about 20%, and that reduction will only be attained sometime after the peak of the synaptic current. Not a completely negligible effect, but maybe not one of great physiological significance either. It would probably be difficult to verify experimentally with current measurement techniques.

On a positive note, I did find it interesting to realise that a typical synaptic current could transiently replace quite a significant fraction of the potassium ions in the spine with sodium ions (Qian & Sejnowski, 1989). We can calculate that a spine contains about 10 million charges, so about 5 million potassium ions. A 100 pA x 1 ms synaptic current injects 100 fC which is equivalent to about 0.5 million sodium ions.

a + b > a

The authors’ complex neologisms “intrinsic conductance” and “effective neck resistance” were explained with respect to Fig. 3. The supplementary information contains a section to show that Rn < Rn + Zd, where the dendritic impedance is assumed to be purely resistive. In other words, after 4 lines of equations, we discover that the sum of two strictly positive numbers (a, b) is greater than one of them: a + b > a.

Limitations of the cable equation?

Throughout the manuscript the authors inflate the importance of electrodiffusion modelling. The whipping boy is the old-fashioned cable theory. Amongst the hype, towards the end of the supplementary information, there is a surprising error. In the section entitled “Limitation of the cable theory”, the authors compare the ability of electrodiffusion and cable models of the spine neck to reproduce the attenuation of voltage from spine head to base. The results are shown in Fig. S6. For the electrodiffusion model there is a head-to-base voltage attenuation of about 50%. For the cable model, there is essentially none (the head and base traces superimpose). In order to recover the observed attenuation in the cable model, it proved necessary to increase the intracellular resistivity by a factor of greater than 105! Who knew the cable equation was that bad?

Inspection of the actual equations offers an alternative explanation. The boundary condition of Eq. 61 implies no current flow. This is a cable with a closed end that is not terminated by a dendritic impedance. This is illustrated graphically in Fig. 4. It seems not to have crossed the authors’ minds that if the standard approaches really were in error by a factor of 105, somebody might just have had the wit to notice before.

cable.jpg
Fig. 4. In comparing their electrodiffusion model and a cable model of voltage attenuation down the spine neck, the authors mistakenly compare two quite different configurations. The cable (right) is not attached to a dendrite.

Conclusion

The headline figure of 100 MΩ for the spine neck resistance was selected in specifying the electrodiffusion model, not extracted from the experimental data as reported. To have done as they claimed, the authors would have had to determine two unknowns from a single equation in which only their product appears. In the electrodiffusion modelling, an error appears to have been introduced while converting the units of the diffusion coefficient. The authors use boundary conditions that are inconsistent with the biophysical model, with unknown effects on the results.  Unrealistic parameter choices are likely to have exaggerated the reported effects, particularly regarding voltage non-uniformity in the spine head. Finally, criticism of the cable equation is wildly misplaced, the result of another mix-up involving boundary conditions.

This paper also raises an interesting question of principle. These days, authors are encouraged, indeed obliged, to share data. I don’t think it is unreasonable for them to receive credit for that in the form of authorship, as long as the author contributions are specific, as they are in this case. However, what should they do if they do not agree with the conclusions drawn from their data? (I don’t know how Kwon and Yuste view this paper.)

I welcome discussion, either below or on PubPeer.

Update

It turns out that the authors have posted a reply that I only recently discovered. This pdf is linked on this news page. For now, I think their document can largely speak for itself, but I will highlight that the authors have not even been able to admit their error regarding the application of the cable equation to the spine neck. They are effectively still asserting that the cable equation would be in error by a factor of 100000.

References

1. Cartailler J, Kwon T, Yuste R, Holcman D. (2018) Deconvolution of Voltage Sensor Time Series and Electro-diffusion Modeling Reveal the Role of Spine Geometry in Controlling Synaptic Strength. Neuron. 97:1126-1136.e10.
doi: 10.1016/j.neuron.2018.01.034.

2. Popovic MA, Carnevale N, Rozsa B, Zecevic D. (2015) Electrical behaviour of dendritic spines as revealed by voltage imaging. Nat Commun. 6:8436.
doi: 10.1038/ncomms9436.

3. Grunditz A, Holbro N, Tian L, Zuo Y, Oertner TG. (2008)
Spine neck plasticity controls postsynaptic calcium signals through electrical compartmentalization. J Neurosci. 28:13457-66.
doi: 10.1523/JNEUROSCI.2702-08.2008.

4. Beaulieu-Laroche L, Harnett MT. (2018) Dendritic Spines Prevent Synaptic Voltage Clamp. Neuron. 97:75-82.e3.
doi: 10.1016/j.neuron.2017.11.016.

5. Ly R, Bouvier G, Szapiro G, Prosser HM, Randall AD, Kano M, Sakimura K, Isope P, Barbour B, Feltz A. (2016) Contribution of postsynaptic T-type calcium channels to parallel fibre-Purkinje cell synaptic responses. J Physiol. 594:915-36.
doi: 10.1113/JP271623.

6. Harris KM, Jensen FE, Tsao B. (1992) Three-Dimensional Structure of Dendritic Spines and Synapses in Rat Hippocampus (CA1) at Postnatal Day 15 and Adult Ages: Implications for the Maturation of Synaptic  Physiology and Long-term Potentiation. J Neurosci. 12:2665-2705.
doi:

7. Kwon T, Sakamoto M, Peterka DS, & Yuste R. (2017) Attenuation of Synaptic Potentials in Dendritic Spines. Cell Reports 20, 1100–1110.
doi: 10.1016/j.celrep.2017.07.012

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