Metabolic efficiency with fast spiking in the squid axon
aa r X i v : . [ q - b i o . N C ] N ov Metabolic efficiency with fast spiking in the squid axon
Abdelmalik Moujahid and Alicia d’Anjou
Computational Intelligence Group, Department of Computer Science, University of theBasque Country UPV/EHU, 20018 San Sebastian, Spain
Abstract
Fundamentally, action potentials in the squid axon are consequence of theentrance of sodium ions during the depolarization of the rising phase of thespike mediated by the outflow of potassium ions during the hyperpolariza-tion of the falling phase. Perfect metabolic efficiency with a minimum chargeneeded for the change in voltage during the action potential would confinesodium entry to the rising phase and potassium efflux to the falling phase.However, because sodium channels remain open to a significant extent dur-ing the falling phase, a certain overlap of inward and outward currents isobserved. In this work we investigate the impact of ion overlap on the num-ber of the adenosine triphosphate (ATP) molecules and energy cost requiredper action potential as a function of the temperature in a Hodgkin-Huxleymodel. Based on a recent approach to computing the energy cost of neuronalAP generation not based on ion counting, we show that increased firing fre-quencies induced by higher temperatures imply more efficient use of sodiumentry, and then a decrease in the metabolic energy cost required to restorethe concentration gradients after an action potential. Also, we determinevalues of sodium conductance at which the hydrolysis efficiency presents aclear minimum.
Keywords:
HodgkinHuxley model, action potential, neuron metabolicenergy, sodium entry, overlap load, regular-spiking cells
Preprint submitted to Frontiers in Computational Neuroscience November 15, 2012 etabolic efficiency with fast spiking in the squid axon
Abdelmalik Moujahid and Alicia d’Anjou
Computational Intelligence Group, Department of Computer Science, University of theBasque Country UPV/EHU, 20018 San Sebastian, Spain
1. Introduction
The generation of action potentials in mammalian neurons involves theflux of different ions such as sodium, potassium and calcium across the cellmembrane. In this process, the electrochemical gradients are partially al-tered and must be restored by ion pumps which move ions from one side ofthe membrane to the other at the expense of energy. Reestablishing the con-centration gradients after electrical discharges demands most of the energyused for neuronal metabolism [1, 2, 3, 4]. This requirement for metabolicenergy has important implications for the brains evolution and function [5],and the availability of energy may impose a limit on neural activity takinginto account that the brain has very small energy reserves [6]. It is, however,generally accepted that energy metabolism is highly organized within cellsresulting in energetically efficient mechanisms that transfer energy from thesite of generation to the processes that require it [6, 7].On other hand, all energy used for neural metabolism is finally trans-formed into heat [6], and the metabolic brain activation appears to be theprimary cause of heat production. Because neural properties are tempera-ture dependent, potential imbalance between heat production and dissipationcould lead to overheating and aberrant functioning [8, 9, 10]. Studying therelationship between temperature, firing frequency, sodium entry and the en-ergy cost required to generate an action potential using neuron models likethe Hodgkin-Huxley model [11] will provides a useful framework for address-ing these issues.The Hodgkin-Huxley model representing the dynamics of the squid giantaxon continues to be the most frequently used model to study the dynamicsand other properties of actual neurons. Based on biophysical considerationsabout the nature of the Hodgkin-Huxley model, we have recently found an
Preprint submitted to Frontiers in Computational Neuroscience November 15, 2012 nalytical expression of the electrochemical energy involved in the dynamicsof the model, which provides a new approach for estimating the energy con-sumption during the resting and active states of neurons [12]. This energyfunction was used as a measure to evaluate the metabolic energy consumptionof a neuron to maintain its signalling activity and to estimate the metaboliccost of transmitting information between neurons.This approach, contrary to other methods [5, 13], does not require ioncounting for estimating the metabolic energy consumption of the generationof action potentials, and gives us the opportunity to check in the Hodgkin-Huxley model which ion counting gives the correct metabolic energy con-sumption. In this work we investigate the impact of ion currents overlappingon the number of ATP molecules required to restore the concentration gra-dients after an action potential in the Hodgkin-Huxley model. Because theobserved overlap is temperature dependent, we have computed the number ofATP molecules per action potential and its corresponding energy cost at dif-ferent values of temperature. Both the classic study by Hodgkin and Huxleyof the squid axon [11], and other recent works [5, 13] assume that the actionpotential requires four times Na + charge compared to the charge needed forthe change in voltage. This waste of Na + charge, and accordingly metabolicenergy, is the result of extensive overlap between inward Na + and outwardK + during the generation of action potentials.However, it has been demonstrated that mammalian central neurons,characterized by action potentials similar to those of the squid giant axon,are significantly more efficient in generating action potentials [14].We show in this work that increased firing frequencies induced by highertemperatures in the Hodgkin and Huxley model imply more efficient useof sodium entry and metabolic energy. The paper is organized as follows.In section 2, the dynamics and electrochemical energy of the Hodgkin andHuxley model are introduced. In section 3 we discuss the overlap of ioncurrents and energy efficiency as a function of temperature in the squid axon.Finally, conclusions are drawn in section 4.3 . Materials and Methods In the original Hodgkin-Huxley model [15], the dynamics governing themembrane potential is given by: C ˙ V = − g Na m h ( V − E Na ) − g K n ( V − E K ) − g l ( V − E l ) + I, (1)where V is the membrane potential in mV, C the membrane capacitancedensity in µ F/cm , I is the total membrane current density in µ A/cm . g N a , g K and g l are the maximal conductances per unit area for ion and leakagechannels, and E Na , E K and E l are the corresponding reversal potentials.The gating variables m , h and n , representing respectively sodium chan-nels activation and deactivation variables, and potassium channels activa-tion variable, obey the standard kinetic equation ˙ x = α x (1 − x ) − β x x ,( x = m, h, n ), where α x and β x are voltage-dependent variables. For sodiumchannels, the activation and deactivation rates are given by, α m ( V ) = (2 . − . V ) / (exp (2 . − . V ) − ,β m ( V ) = 4 exp ( − V / ,α h ( V ) = 0 .
07 exp ( − V / ,β h ( V ) = 1 / (exp (3 − . V ) + 1) . and for potassium channels, α n ( V ) = (0 . − . V ) / (exp (1 − . V ) − ,β n ( V ) = 0 .
125 exp ( − V / . In this work we have used for these parameters the standard constantvalues given in Table (1) [16].The ion currents of sodium, potassium and leakage (mainly chloride)correspond respectively to the three first terms in the right hand of theEq. (1), and are generated in response to a change in the respective ionconductances.Figure (1) shows in part (a) the shape of the sodium and potassium cur-rents corresponding to a particular action potential. The sodium current isnegative but has been depicted with a positive sign for a better appreciation4 g x (mS / cm ) E x (mV) N a
120 115K 36 -12l 0.3 10.6
Table 1: The parameters of the Hodgkin-Huxley equations. The membrane capacitancedensity is C = 1 µ F / cm . The voltage scale is shifted so that the resting potential vanishes. of the great extent of its overlapping with the potassium current. Note thatas sodium and potassium currents are both of positive charges but movingin opposite directions of the cell’s membrane they neutralize each other tothe extent of their mutual overlap. The sodium charge that is not counter-balanced by simultaneously flowing potassium charge is much smaller for agreater overlap.The unbalanced current load, represented in Fig. 1(b), consists of twocomponents which occur respectively during the depolarizing and hyperpo-larizing phases of the membrane potential action. The integral of the firstcomponent of this unbalanced load gives the net Na + ion charge that is notcounterbalanced by simultaneously flowing K + crossing into the membraneduring the rising of the action potential. The integral of the total unbalancedionic current is directly proportional to the the number of ATP moleculesrequired to restore the resting potential.For the action potential represented in Fig. 1(b) generated for an externalstimulus I = 13 µA/cm , the total sodium charge transfer computed as theintegral of Na + current was 1168 nC/cm which agree with the estimate of1098 nC/cm reported recently in [17]. Neutralized currents that accountfor the overlapping of Sodium and potassium fluxes give rise to an excessiveoverlap charge of about 1092 nC/cm . This overlap has been calculated asthe difference between the total Na + current and the depolarizing unbalancedcomponent of Na + current. As stated in the work of Hodgkin, the squidaction potential is very inefficient in the sense that it requires a fourfoldNa + charge compared to the minimum charge necessary to depolarize a purecapacitor [11]. The efficiency of sodium entry during the generation of actionpotential in the squid axon at different temperatures is discussed in Section3. The values of sodium and overlap load reported above correspond to atemperature of 6.3 ◦ C. 5 I on c u rr en t s ( µ A / c m )
28 30 32 34−250−200−150−100−50050100 (b)Time (ms) U nba l an c ed i on c ha r ge ( µ A / c m ) − Na + K + Membranepotential (mV)
Figure 1: (a) The currents of sodium and potassium ions during an action potential gen-erated for an external stimulus I = 13 µA/cm . (b) The unbalanced ion charge resultingfrom the overlapping and offsetting of Na + and K + flux (the membrane potential is repre-sented in dashed line). The rising and falling phases are separated by the vertical dashedline. The area under the curve in the rising phase region corresponds to the net unbalancedsodium ion charge crossing into the membrane during the rising of the action potential.
6o estimate the energy consumption necessary to restore the resting po-tential in the Hodgkin-Huxley model, we have used a new approach not basedon the ion counting method. Following previous works of finding energy func-tions of neuron models of chaotic dynamics [18, 19, 20], we have deduced forthe model given by Eq. (1) an energy function representing the analyticalexpression of the electrochemical energy involved in its dynamics. The pro-cedure followed to find this energy has been reported in detail in [12], and issummarized below.It is well known that the Hodgkin-Huxley equation given by Eq.(1) ex-presses an electrical circuit consisting of capacitor C and three Na, K andL ionic channels, where g N a , g K and g l are the maximal conductances, andbatteries stand for the Nernst potentials of their corresponding ions. If V is the membrane potential, the total electrical energy accumulated in thecircuit at a given moment in time is, H ( t ) = 12 CV + H N a + H K + H l , (2)where the first term in the summation gives the electrical energy accumulatedin the capacitor and represents the energy needed to create the membranepotential V of the neuron. The other three terms are the respective energiesin the batteries needed to create the concentration jumps in sodium, potas-sium, and chloride. The electrochemical energy accumulated in the batteriesis unknown. Nevertheless, the rate of electrical energy provided to the circuitby a battery is known to be the electrical current through the battery timesits electromotive force. Thus, the total derivative with respect to time of theabove energy will be,˙ H ( t ) = CV ˙ V + i N a E N a + i K E K + i l E l . (3)where E Na , E K and E l are the Nernst potentials of the sodium, potassiumand leakage ions in the resting state of the neuron. And i N a , i K , and i l arethe ion currents of sodium, potassium and leakage, given by, i N a = g Na m h ( V − E Na ) ,i K = g K n ( V − E K ) ,i l = g l ( V − E l ) , (4)If we substitute Eqs. (1) and (4) in Eq. (3), we have for the energy rate7 Time (ms) S od i u m c on s u m p t i on ( n J / s ) (a) 28 30 32 340246810 x 10 Time (ms) T o t a l m e t abo li c c on s u m p t i on ( n J / s ) (b) V (mV)
Figure 2: (a) Electrochemical energy consumption corresponding to sodium ions at anaction potential (dashed line) generated for an external stimulus I=13 µA/cm . Valuesof membrane potential are scaled with a factor of 200. (b) Total metabolic consumptioncorresponding to three ion channels at the same action potential. Magnitudes refer tonJ/s. Vertical dashed line in panels (a) and (b) separates between the rising and fallingphase regions. in the circuit, ˙ H = V I − g N a m h ( V − E N a ) − g K n ( V − E K ) − g l ( V − E l ) , (5)which provides the total derivative of the electrochemical energy in the neu-ron as a function of its state variables. The first term in the right handsummation represents the electrical power given to the neuron via the differ-ent junctions reaching the neuron and the other three terms of the summationrepresent the energy per second consumed by the ion channels. This equa-tion permits evaluation of the total energy consumed by the neuron and alsogives information about the consumption associated to each of the sodium,potassium and leaking channels.Figure 2(a) reports the time course of the electrochemical energy con-sumption in nJ/s corresponding to sodium ion channel for the particular ac-tion potential (dashed line) generated at an external constant stimulus I=13 µA/cm . This energy consumption is given by the term g N a m h ( V − E N a ) I = 13 µ A/cm which is the value used to generate the particular action potential an-alyzed previously, the average of the total metabolic consumption depicted inFig. 2(b) is about 11.4 µ J/s per membrane unit area. This consumption mustbe replenished by metabolic ATP supply. The number of ATP molecules permembrane unit area hydrolyzed by the Na + /K + ATPase pump to extrudethe Na + load can be deduced from the amount of Na + ions crossing the mem-brane during an action potential, operating with a ratio of 3Na + per ATP[5, 21].The ratio of the total metabolic consumption (in J/s) to one third of thenumber of Na + load through the membrane expressed in electronvolts perATP represents the efficiency of the ATP hydrolysis measured as the freeenergy provide by the hydrolysis of one molecule of ATP. We will show thatour calculation of the actual energy consumption and the number of ATPmolecules involved in the generation of an action potential are consistent withrelevant data in the literature and that the Hodgkin-Huxley model producesaccurate estimates of energy consumption.
3. Results
The overlap of ion currents in the Hodgkin-Huxley model decreases asthe temperature increases and the impact of overlap on the number of ATPmolecules required per action potential can be analyzed rescaling the modelequations to include the temperature dependence. In this work, we haveadopted the original assumption of Hodgkin and Huxley multiplying therates of change of the activation m , n and inactivation h gating variables bya factor k = 3 ( T − . / , T [ ◦ C] [22].To illustrate the overlap decrease with increasing temperature, Fig. 3shows instantaneous Na + and K + currents elicited by a single spike at re-9 I on c u rr en t s ( µ A / c m )
44 44.5 450200400600800 (b)Time (ms) I on c u rr en t s ( µ A / c m )
20 40 60020406080 20 40 60020406080 214 Hz75 Hz
Figure 3: Simulated sodium (solid line) and potassium (dashed line) currents at 6.3 ◦ C (a),and 18.5 ◦ C (b), showing different degree of overlap. Firing frequencies are 75 Hz and 214Hz respectively. Sodium current is reversed for comparison. ) spectively 6.3 ◦ C and 18.5 ◦ C. To perform the simulation we have rescaled thecurrent equations to different temperatures between 6.3 ◦ C and 18.5 ◦ C whichis the range of temperatures in the original study of the squid giant axon byHodgkin and Huxley. It should be noticed that for higher temperature thefiring regime in the Hodgkin-Huxley model can only be maintained at largevalues of the injected current.To quantify the current overlap we have adopted two different measures.Following [5], we calculated the dimensionless charge separation as the Na + charge that is not counterbalanced by simultaneously flowing K + charge (de-polarizing component of the unbalanced load depicted in Fig. 1(b) dividedby total Na + charge per action potential. The relationship between chargeseparation and temperature is illustrated in the inset of Fig. 4(a).As it can be appreciated, charge separation shows a 2.98-fold increasewith increasing temperature and varies from 0.0652 at 6.3 ◦ C to 0.1942 at18.5 ◦ C. At this temperature Alle et al. [23] studying mossy fibres of the rathippocampus report an average charge separation of 0.769. The respectiveconsumptions per action potential reflect this different overlap. At 18.5 ◦ Cwith an injection current I = 13 µA/cm , the Hodgkin-Huxley model of thesquid giant axon demands 0.68 × AT P/cm to produce one action poten-tial (see Table 2), while according to [23], mossy fibers of the rat hippocampus10
00 600 800 100040060080010001200 overlap Na+ load (nC/cm ) T o t a l N a + l oad ( n C / c m ) (b)0.06 0.08 0.1 0.12 0.14 0.16 0.1811.52 x 10 A T P m o l e c u l e c m − Charge separation(a)
Temperature (ºC) C ha r ge s epa r a t i on Figure 4: (a) The number of ATP molecules required per action potential versus chargeseparation. In the inset charge separation measured as Na + charge that is not counterbal-anced by simultaneously flowing K + charge divided by total charge per action potential.(b) The relation between the total Na + load and the overlap charge, calculated as thedifference between the total Na + current and its depolarizing component. demand only 0.32 × AT P/cm per action potential.The collected results for the number of ATP molecules per unit membranearea to produce one action potential related to charge separation are depictedin Fig. 4(a) . As it can be seen the increase in separation implies a 3.54-folddecrease in the number of ATP molecule/cm . At 6.3 ◦ C the Na + chargetransfer per unit membrane area of an action potential in the squid axonis about 1168 nC/cm , consuming 2.43 10 ATP molecules/cm . While at18 ◦ C, the Na + load is 346 nC/cm consuming 0.72 10 ATP molecules/cm .So, high frequency firing induced by high temperature appears to be moreefficient in the use of Na + entry. Table 2 reports details of values achievedfor theses measures at different values of temperature.The other measure used in this work to quantify the current overlaphas dimension of charge and is computed following [21] as the differencebetween the total Na + load and the depolarizing component of the Na + load. Figure 4(b) shows the relationship between the total Na + load and theoverlap load measured in nC/cm . As it can be seen the total Na + chargeincreases linearly with overlap load with a slope close to unity. That is, theoverlap load is positively correlated with the total Na + charge that crosses11
00 400 600 800 1000 120060 D epo l a r i z i ng unba l an c ed N a + l oad ( n C / c m )
200 400 600 800 1000 120075
Overlap (nC/cm ) T o t a l unba l an c ed l oad ( n C / c m ) Figure 5: Hodgkin-Huxley model scaled for different temperatures between 6.3 ◦ C and18.5 ◦ C . In square markers, the unbalanced Na + load crossing the membrane during therising phase of the action potential as a function of overlap load. In circle markers, isrepresented the total unbalanced load calculated as the sum of sodium and potassiumcurrents related to the overlap load.) the membrane resulting in a decisive factor when analyzing the efficiency.The same behavior is observed when considering the relationship betweenthe total unbalanced load computed as the sum of Na + and K + currents,and the overlap load.Figure 5 reports, in the right axis, the collected values of the total unbal-anced load related to the overlap load. And, in the left axis, the unbalancedcomponent of Na + load crossing the membrane during the rising phase asa function of overlap. At 6.3 ◦ C the total unbalanced load is 175 nC/cm which is 2.3 times the depolarizing unbalanced Na + load, while at 18 ◦ C bothmeasures provide close values around 65 nC/cm . We observe that for adecrease of overlap load between its maximum value achieved at 6.3 ◦ C andvalues around 560 nC/cm corresponding to a temperature of about 12 ◦ C,the depolarized unbalanced load undergoes only a slight decrease. However,further increase of temperature causes the overlap to decrease by 2.12-foldresulting in a 1.13-fold decrease of the depolarizing unbalanced Na + load.12 .2. Energy efficiency The calculated values of the electrochemical energy involved in the dy-namics of the Hodkin-Huxley model according to our method are reportedin Fig. 6. Both the total metabolic consumption and the metabolic con-sumption in the ionic channels show a decreasing pattern as the temperatureincreases. At 6.3 ◦ C, the total action potential energy cost is around 152nJ/cm , 45% of which is consumed in the sodium channel, and the the rest ismainly consumed in the potassium channel. While at 18 ◦ C the energy con-sumed in both channels represents 49% of the total metabolic consumption.At this temperature the total metabolic consumption experiences a 3.35-foldsignificant decrease.The total cost of one pump’s cycle, that pumps three Na + ions out ofthe cell and two K + ion in, is computed as the ratio of the total metabolicconsumption (given by the last three terms of Eq. 5) to one third of the totalnumber of Na + load. According to our calculations, the liberated free energyby hydrolyzing one ATP molecule, defined as hydrolysis efficiency, seem tobe independent of temperature and shows values around 0.39 eV which isclose to other estimates in the literature [24, 25].However, if we consider only the depolarizing metabolic consumption inthe Na + channel associated to the depolarizing unbalanced Na + load, thehydrolysis efficiency seem to be affected by temperature showing a parabolicshape with a minimum around 0.38 eV (See Fig. 7(a)). This minimum isobserved for a temperature around 13 ◦ C corresponding to a firing frequencyof about 138 Hz.The same behavior (See Fig. 7(b)) was observed when studying the de-polarizing efficiency as a function of the sodium channel conductance. Therange of variation of the ion channel conductances was obtained multiplyingthe maximum conductances g Na , g K and g l in the Hodgkin-Huxley model bythe expression k = 1 . ( T − . / for temperatures ranging between 6.3 ◦ C and18.5 ◦ C according to [22]. The maximum depolarizing efficiency occurs for asodium channel conductance value around 160 mS/cm which is close to thebiological density conductance. Crotty et al. [21], studying the energetic costthat arise from an action potential using computational models of the squidaxon as a function of sodium channel densities, showed that the energy costassociated with the action potential produce a convex curve with a minimumaround the same value reported above.We have seen that the fast-spiking regime in the Hodgkin-Huxley modelinduced by higher temperatures implies more efficient use of sodium entry,13 T o t a l ene r g y c on s u m p t i on ( n J / c m ) (a) 10 15 202030405060708090 E ne r g y c on s u m p t i on i n t he N a + c hanne l ( n J / c m ) Temperature (ºC)(b) K + Na + Figure 6: The Hodgkin-Huxley model scaled for different temperatures between 6.3 ◦ Cand 18.5 ◦ C. (a) The total metabolic consumption required per action potential versustemperature. (b) The metabolic consumption in the Na + (circle marker) and K + (squaremarker) channels related to temperature. All magnitudes refer to cm of membrane. and gives rise to energy efficient action potentials characterized by less over-lap between sodium and potassium currents. Since this firing regime could bealso induced by raising the external stimulus, it would be interesting to inves-tigate whether the same fast-spiking regime achieved, for one hand, by raisingthe temperature, and for the other hand, by increasing the external current,contributes in the same way in reducing both the energy consumption andthe overlap load. To do so, we have computed the energy consumption re-quired to generate an action potential and the corresponding firing frequencyfor different values of the external stimulus I. We have considered values of Iranging between 13 µ A/cm and 40 µ A/cm .As it can be appreciated in Fig. 8 (left panel), the influence of higherstimulus on the firing frequency in the squid giant axon is more significantonly for higher temperatures. In fact, at a given temperature, raising theexternal current causes the firing frequency to increase by 1.46-fold, whileat a given external stimulus, the firing frequency experiences a 2.96-foldincrease with increasing temperature. On the other hand, according to ourresults, the fast-spiking regime induced by higher stimuli is less efficient ingenerating action potentials expending more energy compared with the samefast-spiking regime induced by higher temperatures. For example, at T=8 ◦ C14
10 12 14 16 180.370.380.390.40.41 Temperature (ºC) H y d r o l ys i s e ff i c i en cy ( e V / A T P ) (a) 120 140 160 180 2000.370.380.390.40.41 g Na mS/cm H y d r o l ys i s e ff i c i en cy ( e V / A T P ) (b) g Na =120 mS/cm Figure 7: (a) The ratio of the depolarizing Na + energy consumption to one third ofthe depolarizing unbalanced Na + load as a function of temperature. (b) The ratio ofthe depolarizing Na + energy consumption to one third of the depolarizing unbalancedNa + load as a function of sodium channel conductance. The range of variation of theNa + channel conductance was obtained multiplying the maximum conductances in theHodgkin-Huxley model by the factor k = 1 . ( T − . / for temperatures ranging between6.3 ◦ C and 18.5 ◦ C . requency (Hz)I ( µ A/cm ) T ( º C )
20 30 4081012141618 100150200250300 Energy consumption (nJ/cm )I ( µ A/cm ) T ( º C )
20 30 4081012141618 406080100120140
At these pointsF=127 HzAt these pointsF=127 Hz
Figure 8: The squid giant axon. (Left) The spiking regime as a function of temperature andexternal stimulus. (Right) The total metabolic consumption required per action potentialrelated to temperature and external stimulus. The square and circle markers in the figurecorrespond to two configurations with the same firing regime. and a relatively high external current I=39 µ A/cm (see circle markers inFig. 8), the firing frequency of the squid axon is of about 127 Hz, andare necessary 106.75 nJ/cm to generate one action potential. While at ahigher temperature T=12 ◦ C and a low current I=13 µ A/cm (see squaremarkers in Fig. 8) corresponding to the same firing regime (i.e, F=127 Hz),the energy consumption is only 83.24 nJ/cm . This 0.78-fold decrease inenergy consumption is accompanied by a 0.76-fold decrease in overlap loadbetween the spike-generating Na + current and delayed rectifier K + current.The corresponding values of overlap load are 740.83 nC/cm and 563.92nC/cm respectively. This confirms again that the overlap load of voltage-gated currents of Na + and K + dominates energy efficiency, and efficient actionpotentials have little overlap.
4. Discussions
Our results with simulated spikes of the squid axon show that increasedfiring frequencies induced by higher temperatures imply more efficient use ofsodium entry due mainly to the reduced overlap load between inward Na + current and outward K + current. This, corroborates what has been reported16 requency (Hz)I ( µ A/cm ) T ( º C ) )I ( µ A/cm ) T ( º C ) Figure 9: The regular-spiking model reproducing the typical firing characteristics of regularspiking neurons in ferret visual cortex in vivo. (Left) The firing regime as a function oftemperature and external stimulus. (Right) The total metabolic consumption requiredper action potential related to temperature and external stimulus. recently in [14, 17, 23], i.e, the most energy efficient action potentials arethose generated by Na + and K + currents that have substantially reducedoverlap.The values of sodium entry are close to the original values calculated byHodgkin and Huxley [15], and are in nice agreement with values reportedrecently by Sengupta et al. [17]. At 6.3 ◦ C corresponding to 75 Hz we obtaina sodium influx of 12.12 pmole/cm per spike (1168 nC/cm ) while at 18 ◦ Cand 206 Hz the sodium influx is 3.58 pmole/cm per spike (346 nC/cm )which means a 3.38-fold decrease in sodium entry corresponding to a 2.75-fold increase in firing frequency.Regarding the energy consumption associated to the generation of actionpotentials in the squid axon, we have found that the hydrolysis of one ATPmolecule liberates a free energy with optimum values that range from 0.37eV achieved for a sodium conductance around 160 mS/cm and when onlythe depolarizing components of both energy consumption and Na + load areconsidered. To a value of about 0.39 eV produced when considering thetotal metabolic consumption associated to the total Na + load. As statedbefore, these values of hydrolysis efficiency are in nice agreement with other17stimates, which confirms that our method of calculation of the actual energyconsumption by the pump and the number of ATP molecules involved areconsistent with other data in the literature.Also, we have found that the fast-spiking regime in the Hodgkin-Huxleymodel induced by a higher stimulus appears to be less efficient in generatingaction potentials expending more energy compared with the same fast-spikingregime induced by a higher temperature. Accordingly, the reduction in theoverlap of the Na + and K + currents is less when the firing frequency israised by rising the external stimulus. Carter and Bean [14] have suggestedthat the primary determinant of differences in Na + entry efficiency amongneurons is their different action potential shapes. Indeed, the shape of theaction potential in the squid axon at a given frequency is different dependingon whether it has been generated by raising the temperature or the externalstimulus.The findings of this work were validated using others Hodgkin-Huxley-likemodel neurons, in particular, we have considered the simplest model of regu-lar spiking cells in neocortex which consists of sodium and potassium currentsresponsible for generating spikes, and an additional slow voltage-dependentpotassium current responsible for spike-frequency adaptation. This modelgenerates action potentials which capture the typical firing characteristicsof regular spiking neurons in ferret visual cortex in vivo [26]. To carry outthe comparison, we performed the same experiment as for the squid giantaxon, i.e, rescaling the model equations given in Ref. [26] to include thetemperature dependence of the membrane ionic conductances, using valuesthat range between 20 ◦ C and 40 ◦ C which corresponds to the normal rangeof temperature in these cells. The external current was varied between 1.5 µ A/cm to 5 µ A/cm . The results (see Fig. 9) show a behavior qualitativelysimilar to that we observed in the squid axon. i.e, less energy is spent peraction potential at higher temperatures than at lower ones due mainly tothe reduced overlap between sodium and potassium currents. And, the theefficiency of action potentials is more dependent on temperature than on theexternal stimulus. At T=20 ◦ C and an external high stimulus of about 4.75 µ A/cm , the regular spiking cell fires at a frequency of about 46.5 Hz, andrequires 90.5 nJ/cm to generate one action potential. While, for a highertemperature T=36.5 ◦ C and relatively a small current I=2.75 µ A/cm , mak-ing the cell to fire at the same firing regime (i.e, frequency of about 46.5Hz), the energy consumption is 29 nJ/cm representing only one third ofthe energy expended when considering higher stimulus. This difference in18 emperature( ◦ C) 6.3 8 10 12 14 16 18 18.5Firing rate(Hz) 75 88 106 127 150 177 206 214Metabolicconsumption (nJ/cm ) 152.3 126.9 102.6 83.2 67.7 55.3 45.4 43.2Total Na + load (nC/cm ) 1168 973 786 637 518 422 346 329Overlap + load (nC/cm ) 1092 897 712 564 447 354 281 265Na + (pmole/spike) 12.12 10.09 8.15 6.6 5.37 4.38 3.58 3.41ATP × (molecule/spike) 2.43 2.02 1.63 1.32 1.07 0.87 0.72 0.68 Table 2: Simulation values at different temperatures between 6.3 ◦ C and 18.5 ◦ C of theoverlap between N + and K + ion currents in the squid giant axon. The degree of overlapis measured as the difference between the total Na + load and the depolarizing componentof the Na + load per action potential. Their corresponding values of firing frequency,consumption rate, ATP molecules and Na + entry per spike are also given. All extensivemagnitudes refer to cm of membrane. energy consumption is due to a significant reduction of the overlap load thatdecreases from 506 nC/cm to 88 nC/cm .Our principal findings were that the energy consumption required to gen-erate action potentials in the squid giant axon as well as in the regular-spikingmodel of cells in neocortex is lower at higher temperatures. Also, we foundthat for these cells, the fast-spiking regimes induced by higher temperaturesare more energy efficient than those induced by higher stimuli. 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