{"title": "Sodium entry efficiency during action potentials: A novel single-parameter family of Hodgkin-Huxley models", "book": "Advances in Neural Information Processing Systems", "page_first": 2173, "page_last": 2180, "abstract": "Sodium entry during an action potential determines the energy efficiency of a neuron. The classic Hodgkin-Huxley model of action potential generation is notoriously inefficient in that regard with about 4 times more charges flowing through the membrane than the theoretical minimum required to achieve the observed depolarization. Yet, recent experimental results show that mammalian neurons are close to the optimal metabolic efficiency and that the dynamics of their voltage-gated channels is significantly different than the one exhibited by the classic Hodgkin-Huxley model during the action potential. Nevertheless, the original Hodgkin-Huxley model is still widely used and rarely to model the squid giant axon from which it was extracted. Here, we introduce a novel family of Hodgkin-Huxley models that correctly account for sodium entry, action potential width and whose voltage-gated channels display a dynamics very similar to the most recent experimental observations in mammalian neurons. We speak here about a family of models because the model is parameterized by a unique parameter the variations of which allow to reproduce the entire range of experimental observations from cortical pyramidal neurons to Purkinje cells, yielding a very economical framework to model a wide range of different central neurons. The present paper demonstrates the performances and discuss the properties of this new family of models.", "full_text": "Sodium entry ef\ufb01ciency during action potentials: A\nnovel single-parameter family of Hodgkin-Huxley\n\nmodels\n\nAnand Singh\n\nInstitute of Pharmacology and Toxicology\nUniversity of Z\u00a8urich, Z\u00a8urich, Switzerland\n\nanands@pharma.uzh.ch\n\nRenaud Jolivet\u2217\n\nInstitute of Pharmacology and Toxicology\nUniversity of Z\u00a8urich, Z\u00a8urich, Switzerland\n\nrenaud.jolivet@a3.epfl.ch\n\nPierre J. Magistretti\u2020\nBrain Mind Institute\n\nEPFL, Lausanne, Switzerland\n\npierre.magistretti@epfl.ch\n\nBruno Weber\n\nInstitute of Pharmacology and Toxicology\nUniversity of Z\u00a8urich, Z\u00a8urich, Switzerland\n\nbweber@pharma.uzh.ch\n\nAbstract\n\nSodium entry during an action potential determines the energy ef\ufb01ciency of a\nneuron. The classic Hodgkin-Huxley model of action potential generation is\nnotoriously inef\ufb01cient in that regard with about 4 times more charges \ufb02owing\nthrough the membrane than the theoretical minimum required to achieve the ob-\nserved depolarization. Yet, recent experimental results show that mammalian neu-\nrons are close to the optimal metabolic ef\ufb01ciency and that the dynamics of their\nvoltage-gated channels is signi\ufb01cantly different than the one exhibited by the clas-\nsic Hodgkin-Huxley model during the action potential. Nevertheless, the original\nHodgkin-Huxley model is still widely used and rarely to model the squid giant\naxon from which it was extracted. Here, we introduce a novel family of Hodgkin-\nHuxley models that correctly account for sodium entry, action potential width\nand whose voltage-gated channels display a dynamics very similar to the most\nrecent experimental observations in mammalian neurons. We speak here about a\nfamily of models because the model is parameterized by a unique parameter the\nvariations of which allow to reproduce the entire range of experimental observa-\ntions from cortical pyramidal neurons to Purkinje cells, yielding a very economical\nframework to model a wide range of different central neurons. The present paper\ndemonstrates the performances and discuss the properties of this new family of\nmodels.\n\n1\n\nIntroduction\n\nAction potentials play the central role in neuron-to-neuron communication. At the onset of an action\npotential, the change in the membrane potential leads to opening of voltage-gated sodium channels,\nleading to in\ufb02ux of sodium ions. Once the membrane is suf\ufb01ciently depolarized, the opening of\nvoltage-gated potassium channels leads to an ef\ufb02ux of potassium ions and brings the membrane\nback to the resting potential. During and after this process, the ionic gradients are restored by the\nNa,K-ATPase electrogenic pump which extrudes 3 sodium ions in exchange for 2 potassium ions\nand requires 1 ATP molecule per cycle.\n\n\u2217Contact author.\n\u2020Second af\ufb01liation: Center for Psychiatric Neuroscience, University of Lausanne, Lausanne, Switzerland.\n\n1\n\n\fThere is thus a metabolic cost in terms of ATP molecules to be spent associated with every action\npotential. This metabolic cost can be roughly estimated to be 1/3 of the sodium entry into the\nneuron. A metabolically ef\ufb01cient action potential would have sodium entry restricted to the rising\nphase of the action potential so that a minimal number of charges is transported to produce the\nobserved voltage change. This can be encapsulated into a measure called Sodium Entry Ratio (SER)\nde\ufb01ned as the integral of the sodium current during the action potential divided by the product of\nthe membrane capacitance by the observed change in membrane voltage. A metabolically optimally\nef\ufb01cient neuron would have a SER of 1 or close to 1.\nThe metabolic ef\ufb01ciency critically depends on the gating kinetics of the voltage-dependent channels\nand on their interaction during the action potential. All biophysical models of action potential gener-\nation rely on the framework originally established by Hodgkin and Huxley [1] and certain models in\nuse today still rely on their parameters for the voltage-gated sodium and potassium channels respon-\nsible for the action potential generation, even though parameterization of the Hodgkin-Huxley model\noptimized for certain mammalian neurons have been available and used for years [2,3]. Analyzing\nthe squid giant axon action potential, Hodgkin and Huxley established that the SER is approxi-\nmately 4, owing to the fact that the sodium channels remain open during the falling phase of the\naction potential [1]. This has led to the idea that action potentials are metabolically inef\ufb01cient and\nthese numbers were used as key input in a number of studies aiming at establishing an energy budget\nfor brain tissue (see e.g. [4]). However, two recent studies have demonstrated that mammalian neu-\nrons, having fundamentally similar action potentials as the squid giant axon, are signi\ufb01cantly more\nef\ufb01cient owing to lesser sodium entry during the falling phase of the action potential [5,6].\nIn the \ufb01rst study, Alle and colleagues observed that action potentials in mossy \ufb01ber boutons of hip-\npocampal granule neurons have about 30% extra sodium entry than the theoretical minimum [5]\n(SER (cid:39) 1.3). In the second study, Carter and Bean expanded this \ufb01nding, showing that different\ncentral neurons have different SERs [6]. More speci\ufb01cally, they measured that cortical pyramidal\nneurons are the most ef\ufb01cient with a SER (cid:39) 1.2 while pyramidal neurons from the CA1 hippocam-\npus region have a SER (cid:39) 1.6. On the other hand, inhibitory neurons were found to have less\nef\ufb01cient action potentials with cerebellar Purkinje neurons having a SER (cid:39) 2 and cortical basket\ncell interneurons having a SER (cid:39) 2.\nInterestingly, this is postulated to originate in the type or\ndistribution of voltage-gated potassium channels present in each of these cell types. Even the less\nef\ufb01cient neurons are twice more metabolically ef\ufb01cient than the original Hodgkin-Huxley neuron.\nThese recent \ufb01ndings call for a revision of the original Hodgkin-Huxley model which fails on several\naccounts to describe accurately central mammalian neurons.\nThe aim of the present work is to formulate an in silico model for an accurate description of the\nsodium and potassium currents underlying the generation of action potentials in central mammalian\nneurons. To this end, we introduce a novel family of Hodgkin-Huxley models HH\u03be parameterized\nby a single parameter \u03be. Varying \u03be in a meaningful range allows reproducing the whole range of\nobservations of Carter and Bean [6] providing a very economic modeling strategy that can be used\nto model a wide range of central neurons from cortical pyramidal neurons to Purkinje cells.\nThe next section provides a brief description of the model, of the strategy to design it as well as a\nformal de\ufb01nition of the key parameters like the Sodium Entry Ratio against which the predictions\nof our family of models is compared. The third section demonstrates the performances of the novel\nfamily of models and characterize its properties. Finally the last section discusses the implications\nof our results.\n\n2 Model and methods\n\n2.1 Hodgkin-Huxley model family\n\nIn order to develop a novel family of Hodgkin-Huxley models, we started from the original Hodgkin-\nHuxley formalism [1]. In this formalism, the evolution of the membrane voltage V is governed by\n\nC\n\ndV\ndt\n\nIk + Iext\n\n(1)\n\n= \u2212(cid:88)\n\nk\n\n2\n\n\fwith C the membrane capacitance and Iext an externally applied current. The currents Ik are trans-\nmembrane ionic currents. Following the credo, they are described by\n\nIk = gNa m3 h (V \u2212 ENa) + gK n4 (V \u2212 EK) + gL (V \u2212 EL)\n\n(2)\n\n\u2212(cid:88)\n\nk\n\nwith gNa, gK and gL the ionic conductances and ENa, EK and EL the reversal potentials associated\nwith the sodium current iNa = gNa m3 h (V \u2212 ENa), the potassium current iK = gK n4 (V \u2212 EK)\nand the uncharacterized leak current iL = gL (V \u2212EL). All three gating variables m, n and h follow\nthe generic equation\n\n= \u03b1x(V ) (1 \u2212 x) \u2212 \u03b2x(V ) x\n\ndx\ndt\n\nwith x standing alternatively for m, n or h. The terms \u03b1x and \u03b2x are non-trivial functions of the\nvoltage V . It is sometimes useful to reformulate Eq. 3 as\n\ndx\ndt\n\n= \u2212 1\n\n\u03c4x(V )\n\n(x \u2212 x\u221e(V ))\n\n(4)\n\nin which the equilibrium value x\u221e = \u03b1x/(\u03b1x +\u03b2x) is reached with the time constant \u03c4x = 1/(\u03b1x +\n\u03b2x) which has units of [ms].\nSpeci\ufb01c values for the constants (C, gx and Ex) and for the functions \u03b1x and \u03b2x were originally\nchosen to match those introduced in [7] with the exception that the model introduced in [7] includes\na secondary potassium channel that was abandoned here, thus retaining only the channels originally\ndescribed by Hodgkin and Huxley. The reversal potentials Ex were then adjusted to match known\nconcentrations of the respective ions in and around mammalian cells.\nWe then proceeded to explore the behavior of the model and observed that the speci\ufb01c dynamics of\niNa and iK during an action potential is critically dependent on the exact de\ufb01nition of \u03b1n. In our\ncase, \u03b1n is de\ufb01ned by\n\n\u03b1n(V ) =\n\np1 V \u2212 p2\n\n1 \u2212 e\u2212(p3 V \u2212p4)/p5\n\n(3)\n\n(5)\n\nwith p1, . . ., p5 some parameters. More speci\ufb01cally, we observed that by varying p5 in a meaning-\nful range, we could reproduce qualitatively the observations of Carter and Bean [6] regarding the\ndynamics of the sodium current iNa during individual action potentials.\nBuilding on these premises, we set p5 = \u03be with \u03be varying in the range 10.5 \u2264 \u03be \u2264 16. These\nboundary values were chosen relatively arbitrarily by exploring the range in which the models stay\nclose to experimental observations. All the other parameters appearing in the \u03b1x and \u03b2x functions\nwere then optimized using a standard optimization algorithm so that the model reproduces as closely\nas possible the values characterizing action potential dynamics as reported in [6].\nThe \ufb01nal values for parameters of the novel family of Hodgkin-Huxley models are reported in Table\n1. The values of other parameters used in the model are: C = 1.0 \u00b5F/cm2, gL = 0.25 mS/cm2,\nEL = \u221270 mV.\n\nTable 1: The novel family of Hodgkin-Huxley models HH\u03be\n\nchannel\n\nvariable\n\n\u03b1x\n\n\u03b2x\n\ngx (mS/cm2) Ex (mV)\n\nNa\n\nK\n\nm\n\nh\n\nn\n\n41.3 V \u22123051\n1\u2212exp(\u2212 V \u221277.46\n13.27 )\n\n1.2499\n\nexp(V /42.129)\n\n112.7\n\n50\n\n0.0036\nexp( V\n\n24.965 )\n0.992 V \u221296.73\n\n1\u2212exp(\u2212 1.042 V \u221297.517\n\n\u03be\n\n10.405\n\nexp(\u2212 1.024 V \u221226.181\n\n15.488\n\n)+1\n\n0.0159\n\n)\n\nexp(V /21.964)\n\n224.6\n\n-85.0\n\nThe voltage V is expressed in mV.\n\n3\n\n\f(a)\n\n(b)\n\n(c)\n\nFigure 1: Dynamics of the membrane voltage V (top; black line), of the sodium current iNa (bottom;\ngreen line), of the potassium current iK (bottom; blue line) and of the total current C dV /dt (bottom;\nred line; see Eqs. 1-2) upon stimulation by a superthreshold pulse of current (cyan area; Iext =\n25.5 \u00b5A/cm2 for 1 ms). In each panel, SER stands for Sodium Entry Ratio (see Eq. 6) and \u201cwidth\u201d\nindicates the width of the action potential measured at the position indicated by the cyan arrow (see\n\u201cSodium entry ratio and numerics\u201d subsection). (a) \u03be = 10.5. (b) \u03be = 13.5. (c) \u03be = 16.0.\n\n2.2 Sodium entry ratio and numerics\n\nThe relevant parameters to compare the novel family of Hodgkin-Huxley models HH\u03be to the exper-\nimental dataset under consideration are: (i) the action potential peak, (ii) the action potential width\nand (iii) the sodium entry ratio (SER). The action potential peak is simply de\ufb01ned as the maximal\ndepolarization reached during the action potential. Following [6], the action potential width is mea-\nsured at half the action potential height, measured as the difference in membrane potential from the\npeak to the resting potential. Finally, still following [6], the SER is de\ufb01ned for an isolated action\npotential by\n\n(cid:90)\n\nSER =\n\niNa/C\u2206V\n\n(6)\n\nwith \u2206V the change in voltage during the action potential measured from the action potential thresh-\nold \u03d1 to its peak. The action potential threshold \u03d1 was de\ufb01ned as 1% of the maximal dV /dt.\nAll simulations were implemented in MATLAB (The Mathworks, Natick MA). The system of equa-\ntions was integrated using a solver for stiff problems and a time step of 0.05 ms.\n\n3 Results\n\nRecent experimental results suggest that the dynamics of the action potential generating voltage-\ngated channels in the classical Hodgkin-Huxley model do not correctly reproduce what is observed\nin mammalian neurons [5,6]. More speci\ufb01cally, the Hodgkin-Huxley equations generate a sodium\ncurrent with a characteristic secondary peak during the action potential decaying phase, leading to\na very important in\ufb02ux of sodium ions that counter the effect of potassium ions making the model\nmetabolically inef\ufb01cient [1]. Mammalian neurons display a sodium current with a unique sharp peak\nor at most a low amplitude secondary peak [5,6].\n\n4\n\n303540\u221280040voltage [mV]width=1.30[ms]SER=1.55303540\u22125000500time [ms]currents [\u03bcA/cm(cid:31)]303540\u221280040voltage [mV]width=0.60[ms]SER=1.91303540\u22125000500time [ms]currents [\u03bcA/cm(cid:31)]303540\u221280040voltage [mV]width=0.40[ms]SER=2.67303540\u22125000500time [ms]currents [\u03bcA/cm(cid:31)]\fFigure 2: Predictions of our model family are compared to the experimentally observed correlation\nbetween the action potential width and the SER. Experimental observations (red squares) are adapted\nfrom [4]. Data were collected for (from left to right): Purkinje cells, cortical interneurons, CA1\npyramidal neurons and cortical pyramidal neurons. Error bars stand for the standard deviation. The\nred line is a simple linear regression through the experimental data (R2 = 0.99). The predictions\nof our model (black squares) are indicated for decreasing values of \u03be from left (\u03be = 16) to right\n(\u03be = 10.5).\n\nIn the precedent section, we have introduced a novel family of models HH\u03be parameterized by the\nunique parameter \u03be (see Table 1). We will now show how varying \u03be allows reproducing the wide\nrange of dynamics observed experimentally. Figure 1 shows the behavior of HH\u03be during an isolated\naction potential for three different values of \u03be. In all three cases, the action potential is triggered\nby the same unique square pulse of current generating an isolated action potential with roughly the\nsame latency about 4 s after the end of the stimulating pulse. Yet the behavior of the model is very\ndifferent in each case. For low values of \u03be, the sodium current iNa exhibits a single very sharp\npeak, being almost null after the action potential has peaked. At high values of \u03be, iNa exhibits a\ndistinctive secondary peak after the action potential has peaked. The potassium current iK is also\nmuch bigger in the latter case. As a consequence, the model has a low Sodium Entry Ratio (SER) at\nlow values of \u03be and a high SER at high values of \u03be (see Eq. 6). We also observe a negative correlation\nbetween \u03be and the width of the action potential. The width of action potentials decreases when \u03be\nincreases. Finally action potentials generated at low \u03be values return to the resting potential from\nabove while action potentials generated at high \u03be values exhibit an after-hyperpolarization. These\ndifferent instances of our family of models HH\u03be cover all the experimentally observed behaviors as\nreported in [6] (compare with Figures 1-3 therein).\nIndeed, Carter and Bean observed neurons with low SER, broad action potentials and a single sharp\npeak in the sodium current dynamics (cortical and CA1 pyramidal neurons). They also observed\nneurons with high SER, narrow action potentials and a distinctive secondary peak in the sodium\ncurrent dynamics during the action potential decaying phase (cortical interneurons and cerebellar\nPurkinje cells). Figure 2 compares the predictions of our model family with the observations re-\nported in [6]. It clearly demonstrates that by varying \u03be, our model family is able to capture the\nwhole range of observed behaviors and quantitatively \ufb01ts the measured SER and action potential\nwidths. We also observe a faint positive correlation between the action potential width and its peak\nlike in [6] (not shown).\nWhile the dynamics of gating variables is traditionally formulated in terms of \u03b1x and \u03b2x functions\n(see Eq. 3), it is convenient to reformulate the governing equation in the form of Eq. 4, yielding for\n\n5\n\n00.511.50.511.522.53AP width [ms]SER \fFigure 3: Equilibrium function x\u221e (top) and time constant \u03c4x (bottom) as a function of the mem-\nbrane voltage for different values of \u03be for the gating variables m (red line), h (green line) and n\n(dotted blue lines).\n\neach gating variable an equilibrium value x\u221e(V ) and a time constant \u03c4x(V ). Figure 3 shows x\u221e and\n\u03c4x for all three gating variables of the model as a function of the membrane voltage V , the variable\nopening the sodium channel m, the variable closing the sodium channel h and the variable associated\nwith the potassium channel n. With increment in the value of \u03be, the asymptotic value n\u221e shifts\ntowards lower membrane potentials, in other words for the same membrane voltage, the equilibrium\nvalue is higher. On the opposite, with increment in the value of \u03be, the time constant \u03c4x is reduced in\nthe range [\u221240; +40] mV. In summary, at low \u03be values, the potassium current iK is only activated\nwhen the membrane potential is high and it kicks in slowly. At high \u03be values, iK is activated earlier\nin the action potential and kicks in faster. This supports remarkably well the arguments of Carter\nand Bean to explain the relative metabolic inef\ufb01ciency of GABAergic neurons. Indeed, fast-spiking\nneurons with narrow action potentials use fast-activating Kv3 channels to repolarize the membrane.\nIt is postulated that, in these cells, recovery begins sooner and from more hyperpolarized voltages\nin remarkable agreement with the evolution of n\u221e and \u03c4n in our modeling framework. It is also\ninteresting to note that Kv3 channels enable fast spiking [8]. This is supposedly due to incomplete\nsodium channel inactivation and to earlier recovery, in effect speeding recovery and reducing the\nrefractory period.\nFinally, Figure 4 shows the membrane voltage V when the model is subjected to a constant input\nas well as the corresponding gain functions or frequency versus current curves. The f \u2212 I curve\nhas the typical saturating pro\ufb01le observed for many neurons [9] and all the models start spiking at a\nnon-zero frequency. In line with the idea that neurons with a sharp action potential and incomplete\ninactivation of sodium channels can spike faster, the discharge frequency increases with the value of\n\u03be for a given input current.\n\n4 Discussion\n\nRecent experimental results have highlighted that the original Hodgkin-Huxley model [1] is not\nparticularly well suited to describe the dynamics of sodium and potassium voltage-gated channels\nduring the course of an action potential in mammalian neurons. The Hodgkin-Huxley model is also a\npoor foundation for studies dedicated to computing an energy budget for the mammalian brain since\nit severely overestimates the metabolic cost associated with action potentials by at least a factor of\n2. Despite that, the Hodgkin-Huxley model is still widely used and often for modeling projects\nspeci\ufb01cally targeting the mammalian brain.\n\n6\n\nxin(cid:31)nity\u03c4x[ms]membranepotential[mV]-1000+10005001\u03be = 14\u03be = 16\u03be = 16\u03be = 12\u03be = 10\u03be = 14\u03be = 10\u03be = 12mhn\f(a)\n\n(b)\n\nFigure 4: Gain functions and spike trains elicited by constant input. (a) The gain function (f \u2212 I\ncurve) is plotted for different values of the parameter \u03be. The models were stimulated with a constant\ncurrent input of 5 sec after an initial 30 ms pulse. (b) Sample spike trains for \u03be = 14 for different\nvalues of the externally applied current Iext.\n\nHere we have introduced a novel instance of the Hodgkin-Huxley model aimed at correcting these\nissues. The proposed family of models uses the original equations of Hodgkin and Huxley as they\nwere formulated originally but introduces new expressions for the functions \u03b1x and \u03b2x that char-\nacterize the dynamics of the gating variables m, n and h. Moreover, the speci\ufb01c expression for\n\u03b1n depends on an extra parameter \u03be. By varying \u03be in a speci\ufb01c range, our family of models is\nable to quantitatively reproduce a wide range of dynamics for the voltage-gated sodium and potas-\nsium channels during individual action potentials. Our family of models is able to generate broad,\nmetabolically ef\ufb01cient action potentials with a sharp single peak dynamics of the sodium current as\nwell as narrow, metabolically inef\ufb01cient action potentials with incomplete inactivation of the sodium\nchannels during the decaying phase of the action potential. These different behaviors cover neuron\ntypes as different as cortical pyramidal neurons, cortical interneurons or Purkinje cells.\nFor this study we chose a single-compartment Hodgkin-Huxley-type model because it is well suited\nto compare with the experimental conditions of Carter and Bean [6]. However, when comparing the\nparticular parameterization of the model that is achieved here and experimental data (see Figure 2), it\nsuggests that other changes, e.g. in sodium channel inactivation, may help to explain the differences\nbetween different cell types. It should also be noted that action potentials as narrow as 250 \u00b5s can be\nas energy-ef\ufb01cient (SER = 1.3) [10] as the widest action potentials measured by Carter and Bean [6],\nsuggesting that sodium channel kinetics, in addition to potassium channel kinetics, is also different\nfor different cell types and subcellular compartments.\nNumerous studies have been dedicated to study the energy constraints of the brain from the coding\nand network design perspective [4,11] or from the channel kinetic perspective [3,5,6,12]. Recently it\nhas been argued that energy minimization under functional constraints could be the unifying princi-\nple governing the speci\ufb01c combination of ion channels that each individual neuron expresses [12]. In\nsupport of this hypothesis, it was demonstrated that some mammalian neurons generate their action\npotentials with currents that almost reach optimal metabolic ef\ufb01ciency [5]. So far, these studies have\nmostly addressed the question of metabolic ef\ufb01ciency considering isolated action potentials. More-\nover, it can be dif\ufb01cult to compare neurons with very different properties. Here, we have introduced\na new family of biophysical models able to reproduce different action potentials relevant to this de-\nbate and their underlying currents [6]. We believe that our approach is very valuable in providing\nmechanistic insights into the speci\ufb01c properties of different types of neurons. It also suggests that\nit could be possible to design a generic Hodgkin-Huxley-type model family that could encompass\na very broad range of different observed behaviors in a similar way than the Izhikevich model does\n\n7\n\n0510152025303500.020.040.060.080.10.120.140.160.180.2 xsi = 12xsi = 13xsi = 14xsi = 15xsi = 16f [kHz]I [\u03bcA/cm(cid:31)]app01020304050607080010203040506070800102030405060708001020304050607080time [ms]I = 4appI = 12appI = 20appI = 28app\ffor integrate-and-\ufb01re type model neurons [13]. Finally we believe that our model family will prove\ninvaluable in studying metabolic questions and in particular in addressing the speci\ufb01c question: why\nare inhibitory neurons less metabolically ef\ufb01cient than excitatory neurons?\n\nAcknowledgements\n\nRJ is supported by grants from the Olga Mayen\ufb01sch Foundation and from the Hartmann M\u00a8uller\nFoundation. The authors would like to thank Dr Arnd Roth for helpful discussions.\n\nReferences\n\n[1] Hodgkin AL, Huxley AF. J Physiol 1952; 116: 449\u2013472.\n[2] Destexhe A, Par\u00b4e D. J Neurophysiol 1999; 81: 1531\u20131547.\n[3] Sengupta B, Stemmler M, Laughlin SB, Niven JE. PLoS Comp. Biol. 2010; 6: e1000840.\n[4] Attwell D, Laughlin SB. J Cereb Blood Flow Metab 2001; 21: 1133\u20131145.\n[5] Alle H, Roth A, Geiger J. Science 2009; 325: 1405\u20131408.\n[6] Carter BC, Bean BP. Neuron 2009; 64: 898\u2013909.\n[7] Jolivet R, Lewis TJ, Gerstner W. J Neurophysiol 2004; 92: 959\u2013976.\n[8] Lien CC, Jonas P. J Neurosci 2003; 23: 2058\u20132068.\n[9] Rauch A, La Camera G, L\u00a8uscher HR, Senn W, Fusi S. J Neurophysiol 2003; 90: 1598\u20131612.\n[10] Alle H and Geiger J. Science 2006; 311: 1290\u20131293.\n[11] Laughlin SB, Sejnowski T. Science 2003; 301: 1870\u20131874.\n[12] Hasenstaub A, Otte S, Callaway E, Sejnowski TJ. PNAS 2010; 107: 12329\u201312334.\n[13] Izhikevich E. IEEE Trans Neural Net 2003; 14: 1569- 1572.\n\n8\n\n\f", "award": [], "sourceid": 964, "authors": [{"given_name": "Anand", "family_name": "Singh", "institution": null}, {"given_name": "Renaud", "family_name": "Jolivet", "institution": null}, {"given_name": "Pierre", "family_name": "Magistretti", "institution": null}, {"given_name": "Bruno", "family_name": "Weber", "institution": null}]}