On 2017 Feb 10, Romain Brette commented:

Dear authors,

I am very disappointed and rather surprised by the tone of your reply, and even more surprised by your final reflections. You consider that one should not post comments publicly. But to post comments and let authors reply is the whole point of PubMed Commons. It is not clear to me what harm is done to science since the authors's responses are published; on the contrary. There is no reason to be aggressive when discussing with a peer.

Best regards, Romain Brette

On 2017 Feb 10, David Attwell commented:

The answers to Dr Brette’s new points are as follows.

(1a) Ohmic vs Goldman dependence of current on voltage

For a cell as assumed in Attwell & Laughlin (2001) (i.e. 200 Megohm membrane measured with a 10mV hyperpolarizing step, VNa=+50mV, VK=-100mV, Vrp=-70mV, [Na]o=[K]i=140mM, T=37C), assuming a Goldman voltage dependence for the Na⁺ and K⁺ fluxes through ion channels leads to PNa/PK=0.0746 and a Na⁺ influx at the resting potential of 130pA, corresponding to an ATP consumption of 2.7x10⁸ molecules/sec. This is 21% less than the 3.42x10⁸ molecules/sec we calculated using an ohmic dependence of the currents on voltage. This difference is negligible given the variation of measured input resistances and the range of other assumptions that we needed to make. Furthermore, there are no data establishing whether the voltage-dependence of Na+ influx is better described by an ohmic or a Goldman equation.

(In a later post Dr Brette claims that the error arising is 40%. We suspect that his value arises from forgetting the contribution of the Na/K pump current to setting the resting potential, which leads erroneously to PNa/PK=0.05, and a 41.3% lower value than the ohmic dependence predicts.)

(1b) Cl^- permeability

Our point was that the Cl^- permeability was negligible. Which equation was used to derive that fact is therefore irrelevant.

(2) Pumps

We still disagree with the notion that, for a membrane with just Na⁺ and K⁺ fluxes, the system is unstable if it only has a Na/K pump. The Na pump rate is adjusted to match activity via its dependence on [Na⁺ ]i and by the insertion of more pumps when needed.

(3) Cost of Na⁺ extrusion at the mean potential

Tonic synaptic activity may depolarize cells by a mean value of ~4-8mV (Paré et al., 1998, J Neurophysiol 79, 1450). This will affect the calculation of “resting” Na+ influx negligibly (e.g. by ~6mV/120mV = 5% for a 6mV depolarization with Vrp=-70mV and VNa=+50mV). As stated in our earlier comment, this depolarization does not affect the ATP used per Na⁺ pumped by the Na/K pump. Finally the ATP used on extruding synaptic ion entry is considered separately in the calculations.

(4) What input resistance tells us

For cortical L2/3 pyramidal cells the majority of the membrane area is in the basal dendrites, which have an electrotonic length of ~0.24 space constants, while the apical dendrites have an electrotonic length of ~0.69 space constants (mean data at body temperature from Trevelyan & Jack, 2002, J Physiol 539, 623). Larkman et al. (1992, J Comp Neurol 323, 137) similarly concluded that most of the dendrites of L2/3 and L5 pyramidal cells were within 0.5 space constants of the soma.

Elementary cable theory shows that, for a cable (dendrite or local axon) with a sealed end, with current injection at one end, the ratio of the apparent conductance to the real conductance, and thus the ratio of our calculated ATP usage (on Na⁺ pumping to maintain the cable’s resting potential) to the real ATP usage, is given by (1/L).(exp(2L) - 1)/(exp(2L) + 1) where L is the electrotonic length (cable length/space constant). For L=0.24, 0.5 and 0.69, respectively, this predicts errors in the calculated ATP use of 1.9%, 7.6% and 13.3%, which are all completely negligible in the context of the other assumptions that we had to make.

For the axon collaterals near the soma, there is less information on electrotonic length, but the few measurements of axon space constant that exist (Alle & Geiger, 2006, Science 311, 1290; Shu et al., 2006, Nature 441, 761) suggest that the axon collaterals near the soma will similarly be electrically compact and thus that their conductance will be largely reflected in measurements of input resistance at the soma. We excluded the part of the axon in the white matter from our analysis, but did include the terminal axon segments in the grey matter (where the white matter axon rises back into a different cortical area. Re-reading after 16 years the source (Braitenberg & Schüz, 1991, Anatomy of the Cortex, Chapter 17) of the dimensions of these axons, it is clear that those authors were uncertain about the contribution of the terminal axon segments to the total axon length, but assumed that they contributed a similar length to that found near the soma in order to account for the total axon length they observed in cortex. It is unlikely that these distant axon segments will contribute much to the conductance of the cell measured at the soma but, partly compensating for this, part of the axon in the white matter will. This, along with the electrical compactness of the dendrites and proximal axons discussed above, implies that our calculated ATP use on the resting potential is likely to be correct to within a factor of 1/f = 1.57 (where f=0.64 is the fraction of the cell area that is electrically compact [ignoring the minor voltage non-uniformity quantified above], i.e. the soma, dendrites and proximal axons, calculated from the capacitances in Attwell & Laughlin and assuming that the proximal axons provide half of the total axon capacitance in the grey matter).

In fact the situation is likely to be better than this, because this estimate is based on membrane area, but ATP use is proportional to membrane conductance. Estimated values of the conductance of axons (Alle & Geiger, 2006, Science 311, 1290) suggest that the specific membrane conductance per unit area in axons is significantly lower than that in the soma and dendrites (see the Supplementary Information section on Granule Cells in Howarth et al. (2010) JCBFM 30, 403), which reduces the ATP used on maintaining the resting potential of axons.

General reflections on what people expect from the Attwell & Laughlin paper

Our paper tried to introduce a new way of thinking about the brain, based on energetics. Given the large number of assumptions involved it would be a mistake to expect individual values of ATP consumption to be highly accurate. Remarkably, the total energy use that we predicted for the grey matter turned out to be pretty well exactly what is measured experimentally. Nevertheless, constant updating of the assumptions and values is, of course, essential. It is interesting that the value we derived for the ATP used per cell on the action potential (3.84x10⁸ ATP) was initially revised downwards nearly 4-fold in the light of papers showing less temporal overlap of the voltage-gated Na⁺ and K⁺ currents than occurs in squid axon (Alle et al., 2009, Science 325, 1405), but has increased with more recent estimates back to be close to our original estimate (3.77-8.00x10⁸ ATP, Hallermann et al., 2012, Nature Neuroscience 15, 1007).

The most important assumption that we made was that all cells were identical, which immediately implies that this can only be an approximate analysis. We were very happy that the total energy use that we predicted from measured ionic currents, cell anatomy and cell densities was so close to the correct value.

General reflections on post-publication peer comments

We believe that if someone has questions about a paper then the most productive way to get them answered is: (i) to think about the issues; if that fails (ii) to write to the authors and ask them about the questions, rather than posting some vague and erroneous comments that will forever be linked to the paper, regardless of their validity; and if that fails (iii) to write a paper or review which goes through peer review, pointing out the problems. Peer review is crucial for determining whether the points are valid or not - it potentially saves many readers the time needed to read possibly erroneous comments.

It takes a long time to reply to such comments, and we feel that Dr Brette could have done the calculations that we have provided in our two sets of responses. We will not be posting further responses therefore.

David Attwell & Simon Laughlin, 09-02-17

On 2017 Feb 08, Romain Brette commented:

A qualification about point (2): I agree it is theoretically possible to use a single 3:2 Na/K pump and have a stable system, if firing rate is lower than a limit set by the minimum voltage at which the pump can work (from thermodynamic considerations). In this case, it is probably ok to neglect the other pumps (assuming chloride does not play a role); at least the model is consistent. As I wrote in my initial comment, point (4) seems to be the more problematic point. Point (1a) leads to a correction of about 40%.

On 2017 Feb 01, Romain Brette commented:

Thank you very much for your detailed reply. I obviously sympathize with the idea that theoretical work requires simplifying assumptions. The question is to what extent these simplifications lead to accurate estimates, and I disagree that the number of citations is a good indicator of this point.

Here is a more detailed analysis. I will respond starting from the last point, as it will hopefully be clearer.

(4) What input resistance tells us

Thank you for pointing to your more recent studies. It still remains that the assumption “that input resistance measured at the soma can give us a rough estimate of ATP use on the resting potential” cannot be correct here. I am simply using the numbers given in the paper (Attwell & Laughlin 2001), in particular in section “Energy needed for action potentials”, where it is considered that the typical unmyelinated axon has length 4 cm and diameter 0.3 µm. This means that most of the axon’s length is beyond the characteristic length of voltage attenuation (not more than a few hundred µm) and so the input resistance at the soma does not include the axon. But the membrane area of the axon is many times that of the soma, again according to the numbers used in this paper. One can simply divide the charge needed to depolarize the axon by the charge needed to depolarize the soma, as given in the text: 3.77 x 10^-11 / 1.96 x 10^-12 = about 20. So the membrane resistance (1/total conductance) should be about 20 times lower than measured at the soma. One may argue that the input resistance at the soma also includes part of that of the dendrites. But the total dendritic area is (following the numbers given in the paper) 1/3 of that of the axon, so even if we include all of it (very conservative), we still find that the membrane resistance is overestimated by a factor 3. So one needs to apply a correction factor of 3-20 (3 being very conservative).

(3) Cost of Na+ extrusion at the mean potential

I am afraid I was not clear enough. I was referring to a much more elementary point, which is that the fluxes of Na+ and K+ depend on the membrane potential, since the ionic currents depend on the membrane potential (at least through the driving force). With synaptic activity, the membrane potential is typically depolarized because of the mean synaptic current. So the balance of currents used in the methods should include the total current Isyn + INa + IK, in addition to the pump current. Unfortunately we have now three unknowns for two observables (input resistance and mean potential).

(2) Pumps

Again I apologize that my comment was apparently not clear. In a system with a single pump with 3:2 stoechiometry, equilibrium can only be achieved if the mean flux of Na+ through the channels exactly equals 3/2 of the mean flux of K+. This can only happen at a very specific potential; so the pump can only do its job around one particular membrane potential value. If the neuron is depolarized by synaptic activity, for example, the pump fails to maintain equilibrium (too much K+ leaking out). Finally it can also be seen that, because each action potential produces an equivalent flux of Na+ and K+ (electroneutrality), i.e. with 1:1 stoechiometry, a single 3:2 pump cannot maintain equilibrium at high activity.

To compensate for arbitrary fluxes of n ions, one needs to modulate the activity of at least n pumps. In a Na/K system, theoretically a second Na/K pump with a different stoechiometry would work. Alternatively, one can use two additional cotransporters which carry Cl- in opposite directions together with K+ and/or Na+ (eg NKCC1 + KCC2). In such a system, the Na/K pump is used not only to compensate for Na+ flowing into the ionic channels, but also to provide energy for the other pumps in the form of the Na+ gradient. In any case, ensuring ionic equilibrium independently of activity (input or output) requires at least two pumps, and unfortunately this makes it impossible to estimate their activity from just the knowledge of input resistance and mean/resting potential.

(1a) Ohmic vs Goldman-Hodgkin-Katz dependence of current on voltage.

I agree that using a linear model is more convenient. My point was rather that it is not accurate. It should be recalled that the linear (or “ohmic”) model of ionic currents has no biophysical basis (in contrast with GHK, which is based on a biophysical model, although indeed a simple one). It was derived empirically by Hodgkin & Huxley in the squid axon. However in the same preparation, Hodgkin and Katz (HODGKIN AL, 1949) have shown that the resting potential is better predicted by the GHK equation.

(1b) Cl- permeability.

Thank you for these references, it is very helpful. Note however that to show that chloride permeability is low, Xu and Adams used indeed the GHK equation, not the ohmic equation. Note also that all the points I have made above still apply to a Na/K system.

Summary

In summary, these different points make it unlikely that it is possible to estimate Na+ fluxes from just the knowledge of input resistance (especially at the soma) and resting (or mean) potential. One would rather need estimates of K+ and Na+ currents at the mean potential (eg from patch-clamp measurements).

On 2017 Jan 31, David Attwell commented:

Our answers to Dr Brette’s comments are as follows.

(1a) Ohmic vs Goldman-Hodgkin-Katz dependence of current on voltage

We assumed an ohmic dependence on voltage of the ‘leak’ currents for Na⁺ and K⁺ for simplicity. The maths can also be done for a GHK dependence of Na⁺ and K⁺ current on voltage (e.g. see Johnston & Wu (1995), Foundations of Cellular Neurophysiology, Chapter 2, Example 2.3), but it is more convenient to assume an ohmic dependence because that then allows easier conversion of input resistances (1/(GNa+GK)) reported in the literature into ATP consumption (without the need to assume a value for [Na⁺ ]i). In any case, the GHK equation makes assumptions (https://en.wikipedia.org/wiki/GHK_flux_equation) that are unlikely to all be correct.

(1b) Cl^- permeability

Our paper was for the grey matter of the mammalian neocortex. The permeability of the membrane to chloride is reported to be very low in mammalian cortical neurons (see Fig 5c and p121 of the Discussion of Thompson, Deisz & Prince, 1988, J Neurophysiol 60, 105, available at http://jn.physiology.org/content/jn/60/1/105.full.pdf). Similarly, for rat parasympathetic neurons, Xu & Adams (1992, J Physiol 456, 405, https://www.ncbi.nlm.nih.gov/pubmed/1284080) found that for the resting membrane PCl/PK<0.001.

Nevertheless, we extended our analysis slightly, to include the Cl^- component of the membrane permeability, here: Howarth, Peppiatt-Wildman & Attwell (2010) JCBFM 30, 403 http://journals.sagepub.com/doi/pdf/10.1038/jcbfm.2009.231

(2) Pumps

Dr Brette states “The model considers only the Na/K pump. However, such a system cannot be stable; there has to be at least another pump”.

This is simply incorrect, and the system is stable. The pump exports 3 Na⁺ and imports 2 K⁺ for each ATP consumed. The equations in our paper are set up so that there is no net current across the membrane (the sum of the Na⁺ , K⁺ and pump currents are zero), and so that the magnitude of the pump current is 1/3 of the Na⁺ charge entry (so d[Na⁺ ]i/dt=0) and 1/2 of the K⁺ charge exit through the resting conductance (so d[K⁺ ]i/dt=0). This can be seen by evaluating the resting potential for zero net current (following equations 1-3 of Attwell & Laughlin, 2001), and then calculating the ion fluxes.

(3) Cost of Na⁺ extrusion at the mean potential

We assume this point is based on the notion that the energy needed to extrude Na⁺ should be voltage-dependent, so that the ATP needed would be smaller at a more depolarised potential. In fact the stoichiometry of the Na/K pump is apparently not significantly voltage-dependent (between 0 and -60mV), so that the ATP used is always 1/3 of the Na⁺ pumped (see Rakowski, Gadsby & de Weer, 1989, J Gen Physiol 93, 903, https://www.ncbi.nlm.nih.gov/pubmed/2544655), as we assumed.

(4) What input resistance tells us

The Attwell & Laughlin (2001) paper attempted to provide order of magnitude estimates for the ATP used on different subcellular processes in neurons, and this involved assuming that input resistance measured at the soma can give us a rough estimate of ATP use on the resting potential. The analysis addressed energy use only in the grey matter, excluding the majority of the cortical neuron axon in the white matter (which we dealt with here: https://www.ncbi.nlm.nih.gov/pubmed/22219296). Nevertheless, of course voltage is non-uniform in spatially distributed neurons, and input resistance measured at the soma will then not precisely define the resting influx of Na⁺ measured all over the cell. In later work (Howarth, Peppiatt-Wildman & Attwell (2010) JCBFM 30, 403 http://journals.sagepub.com/doi/pdf/10.1038/jcbfm.2009.231; Howarth, Gleeson & Attwell (2012) JCBFM 32, 1222, http://journals.sagepub.com/doi/pdf/10.1038/jcbfm.2012.35) we estimated resting Na⁺ influx in different cellular locations. Thus, there is plenty of scope for improving estimates of the energy consumed on resting potentials, as more data become available.

Summary

Broadly, most of these points reflect the fact that theoretical work often requires simplifying assumptions. Clearly the assumptions that we made have been useful, because (according to Web of Science or Google Scholar respectively) the paper has been cited 1067 or 1752 times. However, there is always room for improvement and we look forward to seeing Dr Brette’s own detailed analysis.

David Attwell & Simon Laughlin, 31-1-17

On 2017 Jan 18, Romain Brette commented:

This is a seminal paper on the energetics of the brain, where the authors derive estimates of the energy (ATP) consumption of various components of the nervous system, and in particular that due to excitatory neural activity, not including myelinated axons. The methods contain theoretical reasoning to calculate the energy cost of an action potential and of maintaining the resting potential, in particular. Thus I would highly recommend anyone interested in the theory of neural energetics to read this paper. It is also a good source of relevant empirical data.

There are two theoretical approaches to derive energetic estimates. One is to estimate ATP consumption for various intracellular processes, for example using the stoechiometry of the Na/K pump (3 Na+ out, 2 K+ in, 1 ATP consumed). The problem with this approach is metabolic processes are very complex and only partially known, so any theoretical approach is bound to make more or less drastic approximations. The other is rather based on thermodynamic theory, where one calculates the change in free energy due to transmembrane ion movements. From this, one can then derive the minimal rate of ATP consumption. The disavantage of this approach is it only gives a lower bound on ATP consumption. However, if we assume that metabolic processes are quite efficient, then it should still give a correct order of magnitude. This paper chooses the first approach.

I have a few criticisms however about the calculation of the energy consumption of the resting potential (first part of the methods).

1) The calculation is based on a model with two linear currents (Na+ and K+), following the Hodgkin-Huxley formalism. However, even in the HH model, the leak current (represented here by the K+ current) is carried by several ions. It is a mixture of currents (Na+, K+, Cl-). Implicitly, the approach of the authors assumes that the neuron’s resting potential is above the reversal potential of K+ because there is a Na+ flux at rest. But according to H&H, the leak current (at least in the squid axon) is mostly carried by chloride, which has a higher reversal potential than potassium. Therefore, the fact that the resting potential is above EK tells us little about the flux of Na+ at rest. In fact, the resting potential is more accurately predicted by the GHK voltage equation, as a function of the permeabilities to Na+, K+ and Cl-. Unfortunately, the sole knowledge of resting potential and input resistance (2 observations) does not allow us to deduce the fluxes of 3 different ions.

2) The model considers only the Na/K pump. However, such a system cannot be stable; there has to be at least another pump (or three pumps if we include Cl-). Presumably, this simplification was made because the Na/K pump is the major source of energy, and because other relevant pumps are not electrogenic. Indeed there are co-transporters such as NKCC1 (moves Na+, K+, Cl- into the cell, with 1:1:2 stoechiometry) but these do not consume energy. However, including these in the model (which as mentioned is necessary for stability of ionic concentrations) changes the balance of fluxes and therefore the activity of the Na/K pump. This relates to the comment above on the approach based on the stoechiometry of a few selected processes.

3) The authors estimate the cost of Na+ extrusion at the resting potential. However, it would be more relevant to apply the calculation to the mean potential, not the resting potential. This is typically quite a bit higher.

4) This final point is in my view more problematic. The theory relates ATP consumption rate to the input resistance; that is, it is inversely proportional to Rin. This is indeed logical since the ionic fluxes are proportional to the total membrane conductance. However, the relevant quantity here is total conductance summed over the entire membrane area of the neuron (everywhere where there are ionic fluxes), but the quantity that is used in the paper is the input resistance measured at the soma, which corresponds essentially to the soma and proximal dendrites. It does not, for example, include the area of the axonal membrane, which is a least an order of magnitude larger than the soma. So the result will be incorrect by at least an order of magnitude.

Point (1) leads to substantial overestimation of ATP consumption rate; points (3) and especially (4) lead to substantial underestimation of ATP consumption rate. Together, this makes the energy consumption due to resting potentials very uncertain.

Regarding the calculation of the cost of propagated action potentials, I believe the approach is correct, except the 1/4 efficiency factor (Na+ flux is 4 times larger than necessary to charge the membrane capacitance) comes from the Hodgkin-Huxley model of the squid axon, while vertebrate axons are more efficient. But this has been corrected in subsequent studies.

On 2017 Jan 18, Romain Brette commented:

This is a seminal paper on the energetics of the brain, where the authors derive estimates of the energy (ATP) consumption of various components of the nervous system, and in particular that due to excitatory neural activity, not including myelinated axons. The methods contain theoretical reasoning to calculate the energy cost of an action potential and of maintaining the resting potential, in particular. Thus I would highly recommend anyone interested in the theory of neural energetics to read this paper. It is also a good source of relevant empirical data.

There are two theoretical approaches to derive energetic estimates. One is to estimate ATP consumption for various intracellular processes, for example using the stoechiometry of the Na/K pump (3 Na+ out, 2 K+ in, 1 ATP consumed). The problem with this approach is metabolic processes are very complex and only partially known, so any theoretical approach is bound to make more or less drastic approximations. The other is rather based on thermodynamic theory, where one calculates the change in free energy due to transmembrane ion movements. From this, one can then derive the minimal rate of ATP consumption. The disavantage of this approach is it only gives a lower bound on ATP consumption. However, if we assume that metabolic processes are quite efficient, then it should still give a correct order of magnitude. This paper chooses the first approach.

I have a few criticisms however about the calculation of the energy consumption of the resting potential (first part of the methods).

1) The calculation is based on a model with two linear currents (Na+ and K+), following the Hodgkin-Huxley formalism. However, even in the HH model, the leak current (represented here by the K+ current) is carried by several ions. It is a mixture of currents (Na+, K+, Cl-). Implicitly, the approach of the authors assumes that the neuron’s resting potential is above the reversal potential of K+ because there is a Na+ flux at rest. But according to H&H, the leak current (at least in the squid axon) is mostly carried by chloride, which has a higher reversal potential than potassium. Therefore, the fact that the resting potential is above EK tells us little about the flux of Na+ at rest. In fact, the resting potential is more accurately predicted by the GHK voltage equation, as a function of the permeabilities to Na+, K+ and Cl-. Unfortunately, the sole knowledge of resting potential and input resistance (2 observations) does not allow us to deduce the fluxes of 3 different ions.

2) The model considers only the Na/K pump. However, such a system cannot be stable; there has to be at least another pump (or three pumps if we include Cl-). Presumably, this simplification was made because the Na/K pump is the major source of energy, and because other relevant pumps are not electrogenic. Indeed there are co-transporters such as NKCC1 (moves Na+, K+, Cl- into the cell, with 1:1:2 stoechiometry) but these do not consume energy. However, including these in the model (which as mentioned is necessary for stability of ionic concentrations) changes the balance of fluxes and therefore the activity of the Na/K pump. This relates to the comment above on the approach based on the stoechiometry of a few selected processes.

3) The authors estimate the cost of Na+ extrusion at the resting potential. However, it would be more relevant to apply the calculation to the mean potential, not the resting potential. This is typically quite a bit higher.

4) This final point is in my view more problematic. The theory relates ATP consumption rate to the input resistance; that is, it is inversely proportional to Rin. This is indeed logical since the ionic fluxes are proportional to the total membrane conductance. However, the relevant quantity here is total conductance summed over the entire membrane area of the neuron (everywhere where there are ionic fluxes), but the quantity that is used in the paper is the input resistance measured at the soma, which corresponds essentially to the soma and proximal dendrites. It does not, for example, include the area of the axonal membrane, which is a least an order of magnitude larger than the soma. So the result will be incorrect by at least an order of magnitude.

Point (1) leads to substantial overestimation of ATP consumption rate; points (3) and especially (4) lead to substantial underestimation of ATP consumption rate. Together, this makes the energy consumption due to resting potentials very uncertain.

Regarding the calculation of the cost of propagated action potentials, I believe the approach is correct, except the 1/4 efficiency factor (Na+ flux is 4 times larger than necessary to charge the membrane capacitance) comes from the Hodgkin-Huxley model of the squid axon, while vertebrate axons are more efficient. But this has been corrected in subsequent studies.

On 2017 Jan 31, David Attwell commented:

Our answers to Dr Brette’s comments are as follows.

(1a) Ohmic vs Goldman-Hodgkin-Katz dependence of current on voltage

We assumed an ohmic dependence on voltage of the ‘leak’ currents for Na⁺ and K⁺ for simplicity. The maths can also be done for a GHK dependence of Na⁺ and K⁺ current on voltage (e.g. see Johnston & Wu (1995), Foundations of Cellular Neurophysiology, Chapter 2, Example 2.3), but it is more convenient to assume an ohmic dependence because that then allows easier conversion of input resistances (1/(GNa+GK)) reported in the literature into ATP consumption (without the need to assume a value for [Na⁺ ]i). In any case, the GHK equation makes assumptions (https://en.wikipedia.org/wiki/GHK_flux_equation) that are unlikely to all be correct.

(1b) Cl^- permeability

Our paper was for the grey matter of the mammalian neocortex. The permeability of the membrane to chloride is reported to be very low in mammalian cortical neurons (see Fig 5c and p121 of the Discussion of Thompson, Deisz & Prince, 1988, J Neurophysiol 60, 105, available at http://jn.physiology.org/content/jn/60/1/105.full.pdf). Similarly, for rat parasympathetic neurons, Xu & Adams (1992, J Physiol 456, 405, https://www.ncbi.nlm.nih.gov/pubmed/1284080) found that for the resting membrane PCl/PK<0.001.

Nevertheless, we extended our analysis slightly, to include the Cl^- component of the membrane permeability, here: Howarth, Peppiatt-Wildman & Attwell (2010) JCBFM 30, 403 http://journals.sagepub.com/doi/pdf/10.1038/jcbfm.2009.231

(2) Pumps

Dr Brette states “The model considers only the Na/K pump. However, such a system cannot be stable; there has to be at least another pump”.

This is simply incorrect, and the system is stable. The pump exports 3 Na⁺ and imports 2 K⁺ for each ATP consumed. The equations in our paper are set up so that there is no net current across the membrane (the sum of the Na⁺ , K⁺ and pump currents are zero), and so that the magnitude of the pump current is 1/3 of the Na⁺ charge entry (so d[Na⁺ ]i/dt=0) and 1/2 of the K⁺ charge exit through the resting conductance (so d[K⁺ ]i/dt=0). This can be seen by evaluating the resting potential for zero net current (following equations 1-3 of Attwell & Laughlin, 2001), and then calculating the ion fluxes.

(3) Cost of Na⁺ extrusion at the mean potential

We assume this point is based on the notion that the energy needed to extrude Na⁺ should be voltage-dependent, so that the ATP needed would be smaller at a more depolarised potential. In fact the stoichiometry of the Na/K pump is apparently not significantly voltage-dependent (between 0 and -60mV), so that the ATP used is always 1/3 of the Na⁺ pumped (see Rakowski, Gadsby & de Weer, 1989, J Gen Physiol 93, 903, https://www.ncbi.nlm.nih.gov/pubmed/2544655), as we assumed.

(4) What input resistance tells us

The Attwell & Laughlin (2001) paper attempted to provide order of magnitude estimates for the ATP used on different subcellular processes in neurons, and this involved assuming that input resistance measured at the soma can give us a rough estimate of ATP use on the resting potential. The analysis addressed energy use only in the grey matter, excluding the majority of the cortical neuron axon in the white matter (which we dealt with here: https://www.ncbi.nlm.nih.gov/pubmed/22219296). Nevertheless, of course voltage is non-uniform in spatially distributed neurons, and input resistance measured at the soma will then not precisely define the resting influx of Na⁺ measured all over the cell. In later work (Howarth, Peppiatt-Wildman & Attwell (2010) JCBFM 30, 403 http://journals.sagepub.com/doi/pdf/10.1038/jcbfm.2009.231; Howarth, Gleeson & Attwell (2012) JCBFM 32, 1222, http://journals.sagepub.com/doi/pdf/10.1038/jcbfm.2012.35) we estimated resting Na⁺ influx in different cellular locations. Thus, there is plenty of scope for improving estimates of the energy consumed on resting potentials, as more data become available.

Summary

Broadly, most of these points reflect the fact that theoretical work often requires simplifying assumptions. Clearly the assumptions that we made have been useful, because (according to Web of Science or Google Scholar respectively) the paper has been cited 1067 or 1752 times. However, there is always room for improvement and we look forward to seeing Dr Brette’s own detailed analysis.

David Attwell & Simon Laughlin, 31-1-17

On 2017 Feb 10, David Attwell commented:

The answers to Dr Brette’s new points are as follows.

(1a) Ohmic vs Goldman dependence of current on voltage

For a cell as assumed in Attwell & Laughlin (2001) (i.e. 200 Megohm membrane measured with a 10mV hyperpolarizing step, VNa=+50mV, VK=-100mV, Vrp=-70mV, [Na]o=[K]i=140mM, T=37C), assuming a Goldman voltage dependence for the Na⁺ and K⁺ fluxes through ion channels leads to PNa/PK=0.0746 and a Na⁺ influx at the resting potential of 130pA, corresponding to an ATP consumption of 2.7x10⁸ molecules/sec. This is 21% less than the 3.42x10⁸ molecules/sec we calculated using an ohmic dependence of the currents on voltage. This difference is negligible given the variation of measured input resistances and the range of other assumptions that we needed to make. Furthermore, there are no data establishing whether the voltage-dependence of Na+ influx is better described by an ohmic or a Goldman equation.

(In a later post Dr Brette claims that the error arising is 40%. We suspect that his value arises from forgetting the contribution of the Na/K pump current to setting the resting potential, which leads erroneously to PNa/PK=0.05, and a 41.3% lower value than the ohmic dependence predicts.)

(1b) Cl^- permeability

Our point was that the Cl^- permeability was negligible. Which equation was used to derive that fact is therefore irrelevant.

(2) Pumps

We still disagree with the notion that, for a membrane with just Na⁺ and K⁺ fluxes, the system is unstable if it only has a Na/K pump. The Na pump rate is adjusted to match activity via its dependence on [Na⁺ ]i and by the insertion of more pumps when needed.

(3) Cost of Na⁺ extrusion at the mean potential

Tonic synaptic activity may depolarize cells by a mean value of ~4-8mV (Paré et al., 1998, J Neurophysiol 79, 1450). This will affect the calculation of “resting” Na+ influx negligibly (e.g. by ~6mV/120mV = 5% for a 6mV depolarization with Vrp=-70mV and VNa=+50mV). As stated in our earlier comment, this depolarization does not affect the ATP used per Na⁺ pumped by the Na/K pump. Finally the ATP used on extruding synaptic ion entry is considered separately in the calculations.

(4) What input resistance tells us

For cortical L2/3 pyramidal cells the majority of the membrane area is in the basal dendrites, which have an electrotonic length of ~0.24 space constants, while the apical dendrites have an electrotonic length of ~0.69 space constants (mean data at body temperature from Trevelyan & Jack, 2002, J Physiol 539, 623). Larkman et al. (1992, J Comp Neurol 323, 137) similarly concluded that most of the dendrites of L2/3 and L5 pyramidal cells were within 0.5 space constants of the soma.

Elementary cable theory shows that, for a cable (dendrite or local axon) with a sealed end, with current injection at one end, the ratio of the apparent conductance to the real conductance, and thus the ratio of our calculated ATP usage (on Na⁺ pumping to maintain the cable’s resting potential) to the real ATP usage, is given by (1/L).(exp(2L) - 1)/(exp(2L) + 1) where L is the electrotonic length (cable length/space constant). For L=0.24, 0.5 and 0.69, respectively, this predicts errors in the calculated ATP use of 1.9%, 7.6% and 13.3%, which are all completely negligible in the context of the other assumptions that we had to make.

For the axon collaterals near the soma, there is less information on electrotonic length, but the few measurements of axon space constant that exist (Alle & Geiger, 2006, Science 311, 1290; Shu et al., 2006, Nature 441, 761) suggest that the axon collaterals near the soma will similarly be electrically compact and thus that their conductance will be largely reflected in measurements of input resistance at the soma. We excluded the part of the axon in the white matter from our analysis, but did include the terminal axon segments in the grey matter (where the white matter axon rises back into a different cortical area. Re-reading after 16 years the source (Braitenberg & Schüz, 1991, Anatomy of the Cortex, Chapter 17) of the dimensions of these axons, it is clear that those authors were uncertain about the contribution of the terminal axon segments to the total axon length, but assumed that they contributed a similar length to that found near the soma in order to account for the total axon length they observed in cortex. It is unlikely that these distant axon segments will contribute much to the conductance of the cell measured at the soma but, partly compensating for this, part of the axon in the white matter will. This, along with the electrical compactness of the dendrites and proximal axons discussed above, implies that our calculated ATP use on the resting potential is likely to be correct to within a factor of 1/f = 1.57 (where f=0.64 is the fraction of the cell area that is electrically compact [ignoring the minor voltage non-uniformity quantified above], i.e. the soma, dendrites and proximal axons, calculated from the capacitances in Attwell & Laughlin and assuming that the proximal axons provide half of the total axon capacitance in the grey matter).

In fact the situation is likely to be better than this, because this estimate is based on membrane area, but ATP use is proportional to membrane conductance. Estimated values of the conductance of axons (Alle & Geiger, 2006, Science 311, 1290) suggest that the specific membrane conductance per unit area in axons is significantly lower than that in the soma and dendrites (see the Supplementary Information section on Granule Cells in Howarth et al. (2010) JCBFM 30, 403), which reduces the ATP used on maintaining the resting potential of axons.

General reflections on what people expect from the Attwell & Laughlin paper

Our paper tried to introduce a new way of thinking about the brain, based on energetics. Given the large number of assumptions involved it would be a mistake to expect individual values of ATP consumption to be highly accurate. Remarkably, the total energy use that we predicted for the grey matter turned out to be pretty well exactly what is measured experimentally. Nevertheless, constant updating of the assumptions and values is, of course, essential. It is interesting that the value we derived for the ATP used per cell on the action potential (3.84x10⁸ ATP) was initially revised downwards nearly 4-fold in the light of papers showing less temporal overlap of the voltage-gated Na⁺ and K⁺ currents than occurs in squid axon (Alle et al., 2009, Science 325, 1405), but has increased with more recent estimates back to be close to our original estimate (3.77-8.00x10⁸ ATP, Hallermann et al., 2012, Nature Neuroscience 15, 1007).

The most important assumption that we made was that all cells were identical, which immediately implies that this can only be an approximate analysis. We were very happy that the total energy use that we predicted from measured ionic currents, cell anatomy and cell densities was so close to the correct value.

General reflections on post-publication peer comments

We believe that if someone has questions about a paper then the most productive way to get them answered is: (i) to think about the issues; if that fails (ii) to write to the authors and ask them about the questions, rather than posting some vague and erroneous comments that will forever be linked to the paper, regardless of their validity; and if that fails (iii) to write a paper or review which goes through peer review, pointing out the problems. Peer review is crucial for determining whether the points are valid or not - it potentially saves many readers the time needed to read possibly erroneous comments.

It takes a long time to reply to such comments, and we feel that Dr Brette could have done the calculations that we have provided in our two sets of responses. We will not be posting further responses therefore.

David Attwell & Simon Laughlin, 09-02-17