{"title": "Energetically Optimal Action Potentials", "book": "Advances in Neural Information Processing Systems", "page_first": 1566, "page_last": 1574, "abstract": "Most action potentials in the nervous system take on the form of strong, rapid, and brief voltage deflections known as spikes, in stark contrast to other action potentials, such as in the heart, that are characterized by broad voltage plateaus. We derive the shape of the neuronal action potential from first principles, by postulating that action potential generation is strongly constrained by the brain's need to minimize energy expenditure. For a given height of an action potential, the least energy is consumed when the underlying currents obey the bang-bang principle: the currents giving rise to the spike should be intense, yet short-lived, yielding spikes with sharp onsets and offsets. Energy optimality predicts features in the biophysics that are not per se required for producing the characteristic neuronal action potential: sodium currents should be extraordinarily powerful and inactivate with voltage; both potassium and sodium currents should have kinetics that have a bell-shaped voltage-dependence; and the cooperative action of multiple `gates' should start the flow of current.", "full_text": "Energetically Optimal Action Potentials\n\nMartin Stemmler\n\nBCCN and LMU Munich\n\nGrosshadernerstr. 2,\n\nPlanegg, 82125 Germany\n\nBiswa Sengupta, Simon Laughlin, Jeremy Niven\n\nDepartment of Zoology,\nUniversity of Cambridge,\n\nDowning Street, Cambridge CB2 3EJ, UK\n\nAbstract\n\nMost action potentials in the nervous system take on the form of strong, rapid, and\nbrief voltage de\ufb02ections known as spikes, in stark contrast to other action poten-\ntials, such as in the heart, that are characterized by broad voltage plateaus. We\nderive the shape of the neuronal action potential from \ufb01rst principles, by postulat-\ning that action potential generation is strongly constrained by the brain\u2019s need to\nminimize energy expenditure. For a given height of an action potential, the least\nenergy is consumed when the underlying currents obey the bang-bang principle:\nthe currents giving rise to the spike should be intense, yet short-lived, yielding\nspikes with sharp onsets and offsets. Energy optimality predicts features in the\nbiophysics that are not per se required for producing the characteristic neuronal\naction potential: sodium currents should be extraordinarily powerful and inacti-\nvate with voltage; both potassium and sodium currents should have kinetics that\nhave a bell-shaped voltage-dependence; and the cooperative action of multiple\n\u2018gates\u2019 should start the \ufb02ow of current.\n\n1 The paradox\n\n\u2206E = RTV(cid:88)\n\n\u03b1\n\nNerve cells communicate with each other over long distances using spike-like action potentials,\nwhich are brief electrical events traveling rapidly down axons and dendrites. Each action potential is\ncaused by an accelerating in\ufb02ux of sodium or calcium ions, depolarizing the cell membrane by forty\nmillivolts or more, followed by repolarization of the cell membrane caused by an ef\ufb02ux of potassium\nions. As different species of ions are swapped across the membrane during the action potential, ion\npumps shuttle the excess ions back and restore the ionic concentration gradients.\nIf we label each ionic species by \u03b1, the work \u2206E done to restore the ionic concentration gradients\nis\n\n[\u03b1]out\n[\u03b1]in\n\n,\n\n\u2206[\u03b1]in ln\n\n(1)\nwhere R is the gas constant, T is the temperature in Kelvin, V is the cell volume, [\u03b1]in|out is the\nconcentration of ion \u03b1 inside or outside the cell, and \u2206[\u03b1]in is the concentration change inside the\n\u03b1 z\u03b1\u2206[\u03b1] = 0,\nwhere z\u03b1 is the charge on ion \u03b1, as no net charge accumulates during the action potential and no\nnet work is done by or on the electric \ufb01eld. Often, sodium (Na+) and potassium (K+) play the\ndominant role in generating action potentials, in which case \u2206E = \u2206[Na]inFV(ENa \u2212 EK), where\n\ncell, which is assumed to be small relative to the total concentration. The sum(cid:80)\nF is Faraday\u2019s constant, ENa = RT /F ln(cid:0)[Na]out/[Na]in\nno net sodium current \ufb02ows, and EK = RT /F ln(cid:0)[K]out/[K]in\n\n(cid:1) is the reversal potential for Na+, at which\n(cid:1). This estimate of the work done\n\ndoes not include heat (due to loss through the membrane resistance) or the work done by the ion\nchannel proteins in changing their conformational state during the action potential.\nHence, the action potential\u2019s energetic cost to the cell is directly proportional to \u2206[Na]in; taking\ninto account that each Na+ ion carries one elementary charge, the cost is also proportional to the\n\n1\n\n\fcharge QNa that accumulates inside the cell. A maximally ef\ufb01cient cell reduces the charge per spike\nto a minimum. If a cell \ufb01res action potentials at an average rate f, the cell\u2019s Na/K pumps must\nmove Na+ and K+ ions in opposite directions, against their respective concentration gradients, to\ncounteract an average inward Na+ current of f QNa. Exhaustive measurements on myocytes in the\nheart, which expend tremendous amounts of energy to keep the heart beating, indicate that Na/K\npumps expel \u223c 0.5 \u00b5A/cm2 of Na+ current at membrane potentials close to rest [1]. Most excitable\ncells, even when spiking, spend most of their time close to resting potential, and yet standard models\nfor action potentials can easily lead to accumulating an ionic charge of up to 5 \u00b5C/cm2 [2]; most of\nthis accumulation occurs during a very brief time interval. If one were to take an isopotential nerve\ncell with the same density of ion pumps as in the heart, then such a cell would not be able to produce\nmore than an action potential once every ten seconds on average. The brain should be effectively\nsilent.\nClearly, this con\ufb02icts with what is known about the average \ufb01ring rates of neurons in the brainstem\nor even the neocortex, which can sustain spiking up to at least 7 Hz [3]. Part of the discrepancy can\nbe resolved by noting that nerve cells are not isopotential and that action potential generation occurs\nwithin a highly restricted area of the membrane. Even so, standard models of action potential gener-\nation waste extraordinary amounts of energy; recent evidence [4] points out that many mammalian\ncortical neurons are much more ef\ufb01cient.\nAs nature places a premium on energy consumption, we will argue that one can predict both the\nshape of the action potential and the underlying biophysics of the nonlinear, voltage-dependent\nionic conductances from the principle of minimal energy consumption. After reviewing the ionic\nbasis of action potentials, we \ufb01rst sketch how to compute the minimal energy cost for an arbitrary\nspike shape, and then solve for the optimal action potential shape with a given height. Finally, we\nshow how minimal energy consumption explains all the dynamical features in the standard Hodgkin-\nHuxley (HH) model for neuronal dynamics that distinguish the brain\u2019s action potentials from other\nhighly nonlinear oscillations in physics and chemistry.\n\n2\n\nIonic basis of the action potential\n\nIn an excitable cell, synaptic drive forces the membrane permeability to different ions to change\nrapidly in time, producing the dynamics of the action potential. The current density I\u03b1 carried by\nan ion species \u03b1 is given by the Goldman-Hodgkin-Katz (GHK) current equation[5, 6, 2], which\nassumes that ions are driven independently across the membrane under the in\ufb02uence of a constant\nelectric \ufb01eld. I\u03b1 depends upon the ions membrane permeability, P\u03b1, its concentrations on either\nside of the membrane [\u03b1]out and [\u03b1]in and the voltage across the membrane, V , according to:\n\nI\u03b1 = P\u03b1\n\n\u03b1V F 2\nz2\nRT\n\n[\u03b1]out \u2212 [\u03b1]in exp (z\u03b1V F/RT )\n\n1 \u2212 exp(z\u03b1V F/RT )\n\n,\n\n(2)\n\nTo produce the fast currents that generate APs, a subset of the membranes ionic permeabilities P\u03b1 are\ngated by voltage. Changes in the permeability P\u03b1 are not instantaneous; the voltage-gated perme-\nability is scaled mathematically by gating variables m(t) and h(t) with their own time dependence.\nAfter separating constant from time-dependent components in the permeability, the voltage-gated\npermeability obeys\n\nP\u03b1(t) = m(t)rh(t)s\n\nsuch that\n\n0 \u2264 P\u03b1(t) \u2264 \u00afP\u03b1,\n\nwhere r and s are positive, and \u00afP\u03b1 is the peak permeability to ion \u03b1 when all channels for ion \u03b1 are\nopen. Gating is also referred to as activation, and the associated nonlinear permeabilities are called\nactive. There are also passive, voltage-insensitive permeabilities that maintain the resting potential\nand depolarise the membrane to trigger action potentials.\nThe simplest possible kinetics for the gating variables are \ufb01rst order, involving only a single deriva-\ntive in time. The steady state of each gating variable at a given voltage is determined by a Boltzmann\nfunction, to which the gating variables evolve:\n\n\u03c4m\n\ndm\ndt\ndh\ndt\n\nand\n\n\u03c4h\n\n= r(cid:112) \u00afP\u03b1m\u221e(V ) \u2212 m(t)\n\n=h\u221e(V ) \u2212 h(t),\n\n2\n\n\fwith m\u221e(V ) = {1 + exp ((V \u2212 Vm)/sm)}\u22121 the Boltzmann function described by the slope sm >\n0 and the midpoint Vm; similarly, h\u221e(V ) = {1 + exp ((V \u2212 Vh)/sh)}\u22121, but with sh < 0. Scaling\nm\u221e(V ) by the rth root of the peak permeability \u00afP\u03b1 is a matter of mathematical convenience.\nWe will consider both voltage-independent and voltage-dependent time constants, either setting \u03c4j =\n\u03c4j,0 to be constant, where j \u2208 {m(t), h(t)}, or imposing a bell-shaped voltage dependence \u03c4j(V ) =\n\u03c4j,0 sech [sj (V \u2212 Vj)]\nThe synaptic, leak, and voltage-dependent currents drive the rate of change in the voltage across the\nmembrane\n\nC\n\ndV\ndt\n\n= Isyn + Ileak +\n\nI\u03b1,\n\n(cid:88)\n\n\u03b1\n\nwhere the synaptic permeability and leak permeability are held constant.\n\n3 Resistive and capacitive components of the energy cost\n\nBy treating the action potential as the charging and discharging of the cell membrane capacitance,\nthe action potentials measured at the mossy \ufb01bre synapse in rats [4] or in mouse thalamocortical\nneurons [7] were found to be highly energy-ef\ufb01cient: the nonlinear, active conductances inject only\nslightly more current than is needed to charge a capacitor to the peak voltage of the action poten-\ntial. The implicit assumption made here is that one can neglect the passive loss of current through\nthe membrane resistance, known as the leak. Any passive loss must be compensated by additional\ncharge, making this loss the primary target of the selection pressure that has shaped the dynamics\nof action potentials. On the other hand, the membrane capacitance at the site of AP initiation is\ngenerally modelled and experimentally con\ufb01rmed [8] as being fairly constant around 1 \u00b5F/cm2; in\ncontrast, the propagation, but not generation, of AP\u2019s can be assisted by a reduction in the capac-\nitance achieved by the myelin sheath that wraps some axons. As myelin would block the \ufb02ow of\nions, we posit that the speci\ufb01c capacitance cannot yield to selection pressure to minimise the work\nW = QNa(ENa \u2212 EK) needed for AP generation.\nTo address how the shape and dynamics of action potentials might have evolved to consume less\nenergy, we \ufb01rst \ufb01x the action potential\u2019s shape and solve for the minimum charge QNa ab initio,\nwithout treating the cell membrane as a pure capacitor. Regardless of the action potential\u2019s partic-\nular time-course V (t), voltage-dependent ionic conductances must transfer Na+ and K+ charge to\nelicit an action potential. Figure 1 shows a generic action potential and the associated ionic currents,\ncomparing the latter to the minimal currents required. The passive equivalent circuit for the neuron\nconsists of a resistor in parallel with a capacitor, driven by a synaptic current. To charge the mem-\nbrane to the peak voltage, a neuron in a high-conductance state [9, 10] may well lose more charge\nthrough the resistor than is stored on the capacitor. For neurons in a low-conductance state and\nfor rapid voltage de\ufb02ections from the resting potential, membrane capacitance will be the primary\ndeterminant of the charge.\n\n4 The norm of spikes\n\nHow close can voltage-gated channels with realistic properties come to the minimal currents? What\ntime-course for the action potential leads to the smallest minimal currents?\nTo answer these questions, we must solve a constrained optimization problem on the solutions to the\nnonlinear differential equations for the neuronal dynamics. To separate action potentials from mere\nsmall-amplitude oscillations in the voltage, we need to introduce a metric. Smaller action potentials\nconsume less energy, provided the underlying currents are optimal, yet signalling between neurons\ndepends on the action potential\u2019s voltage de\ufb02ection reaching a minimum amplitude. Given the\nimportance of the action potential\u2019s amplitude, we de\ufb01ne an Lp norm on the voltage wave-form\nV (t) to emphasize the maximal voltage de\ufb02ection:\n\n(cid:107)V (t) \u2212 (cid:104)V (cid:105)(cid:107)p =\n\n(cid:107)V (t) \u2212 (cid:104)V (cid:105)(cid:107)p dt\n\n,\n\n(cid:41) 1\n\np\n\n(cid:40)(cid:90) T\n\n0\n\n3\n\n\fFor a \ufb01xed action potential waveform V (t):\nMinimum INa(t) = \u2212LV (t)\u03b8(LV (t))\nMinimum IK(t) = \u2212LV (t)\u03b8(\u2212LV (t))\n\nwith LV (t) \u2261 C \u02d9V (t) + Ileak[V (t)] + Isyn[V (t)].\n\nFigure 1: To generate an action potential with an arbitrary time-course V (t), the nonlinear, time-\ndependent permeabilities must deliver more charge than just to load the membrane capacitance\u2014\n(a) The action potential\u2019s time-course in a generic HH\nresistive losses must be compensated.\nmodel for a neuron, represented by the circuit diagram on the right. The peak of the action po-\ntential is \u223c 50 mV above the average potential.\n(b) The inward Na+ current, shown in green\ngoing in the negative direction, rapidly depolarizes the potential V (t) and yields the upstroke of\nthe action potential. Concurrently, the K+ current activates, displayed as a positive de\ufb02ection,\nand leads to the downstroke in the potential V (t). Inward and outward currents overlap signi\ufb01-\ncantly in time. The dotted lines within the region bounded by the solid lines represent the minimal\nNa+ current and the minimal K+ current needed to produce the V (t) spike waveform in (a). By\nthe law of current conservation, the sum of capacitive, resistive, and synaptic currents, denoted by\nLV (t) \u2261 C \u02d9V (t) + Ileak[V (t)] + Isyn[V (t)], must be balanced by the active currents. If the cell\u2019s\npassive properties, namely its capacitance and (leak) resistance, and the synaptic conductance are\nconstant, we can deduce the minimal active currents needed to generate a speci\ufb01ed V (t). The mini-\nmal currents, by de\ufb01nition, do not overlap in time. Taking into account passive current \ufb02ow, restoring\nthe concentration gradients after the action potential requires 29 nJ/cm2. By contrast, if the active\ncurrents were optimal, the cost would be 8.9 nJ/cm2. (c) To depolarize from the minimum to the\nmaximum of the AP, the synaptic voltage-gated currents must deliver a charge Qcapacitive to charge\nthe membrane capacitance and a charge Qresistive to compensate for the loss of current through leak\nchannels. For a large leak conductance in the cell membrane, Qresistive can be larger than Qcapacitive.\n\n4\n\nGeneric Action PotentialActive and Minimal CurrentsResistive vs. CapacitiveMinimum Charge0.2 0.4 0.6 0.8 1.0 1.2 1.410.50 10080604020 0-20-40-60-80-100current [\u00b5A/cm2]Qresistive/Qcapacitivet [ms]leak conductance [mS/cm2]Minimum IKActive IK 0 2 4 6 8 10 12 14 16Minimum INaActive INa-10-20-30-40-50-60V [mV]t [ms] 0 2 4 6 8 10 12 14 16abcgNaCgsyn++gK+gleak+\fwhere (cid:104)V (cid:105) is the average voltage. In the limit as p \u2192 \u221e, the norm simply becomes the difference\nbetween the action potential\u2019s peak voltage and the mean voltage, whereas a \ufb01nite p ensures that\nthe norm is differentiable. In parameter space, we will focus our attention to the manifold of action\npotentials with constant Lp norm with 2 (cid:28) p < \u221e, which entails that the optimal action potential\nwill have a \ufb01nite, though possibly narrow width. To be close to the supremum norm, yet still have a\nnorm that is well-behaved under differentiation, we decided to use p = 16.\n\n5 Poincar\u00b4e-Lindstedt perturbation of periodic dynamical orbits\n\nStandard (secular) perturbation theory diverges for periodic orbits, so we apply the Poincar-\nLindstedt technique of expanding both in the period and the dynamics of the asymptotic orbit and\nthen derive a set of adjoint sensitivity equations for the differential-algebraic system. Solving once\nfor the adjoint functions, we can easily compute the parameter gradient of any functional on the\norbit, even for thousands of parameters.\nWe start with a set of ordinary differential equations \u02d9x = F(x; p) for the neuron\u2019s dynamics, an\nasymptotically periodic orbit x\u03b3(t) that describes the action potential, and a functional G(x; p) on\nthe orbit, representing the energy consumption, for instance. The functional can be written as an\nintegral\n\n(cid:90) \u03c9(p)\u22121\n\nG(x\u03b3; p) =\n\ng(x\u03b3(t); p) dt,\n\n0\n\nover some source term g(x\u03b3(t); p). Assume that locally perturbing a parameter p \u2208 p induces a\nsmooth change in the stable limit cycle, preserving its existence. Generally, a perturbation changes\nnot only the limit cycle\u2019s path in state space, but also the average speed with which this orbit is\ntraversed; as a consequence, the value of the functional depends on this change in speed, to lowest\norder. For simplicity, consider a single, scalar parameter p. G(x\u03b3; p) is the solution to\n\nwhere we have normalised time via \u03c4 = \u03c9(p)t. Denoting partial derivatives by subscripts, we\n\n\u03c9(p)\u2202\u03c4 [G(x\u03b3; p)] = g(x\u03b3; p),\n\nexpand p (cid:55)\u2192 p + \u0001 to get the O(cid:0)\u00011(cid:1) equation\n(cid:90) \u03c9\u22121\n\nd\u03c4 [Gp(x\u03b3; p)] + \u03c9pg(x\u03b3; p) = gx(x\u03b3; p)xp + gp(x\u03b3; p)\n\nin a procedure known as the Poincar\u00b4e-Lindstedt method. Hence,\n\ndG\ndp\n\n=\n\n0\n\n(gp + gxxp \u2212 \u03c9pg) dt,\n\nwhere, once again by the Poincar\u00b4e-Lindstedt method, xp is the solution to\n\n\u02d9xp =Fx(x\u03b3)xp + Fp(x\u03b3) \u2212 \u03c9pF (x\u03b3) .\n\nFollowing the approach described by Cao, Li, Petzold, and Serban (2003), introduce a Lagrange\nvector AG(x) and consider the augmented objective function\n\n(cid:90) \u03c9\u22121\n(cid:90) \u03c9\u22121\n\n0\n\n(cid:90) \u03c9\u22121\n\n(cid:90) \u03c9\u22121\n\nI(x\u03b3; p) = G(x\u03b3; p) \u2212\n\nAG(x\u03b3). (F(x\u03b3) \u2212 \u02d9x\u03b3) dt,\n\nwhich is identical to G(x\u03b3; p) as F(x) \u2212 \u02d9x = 0. Then\n\ndI(x\u03b3; p)\n\ndp\n\n=\n\n(gp + gxxp \u2212 \u03c9pg) dt \u2212\n\nAG. (Fp + Fxxp \u2212 \u03c9pF \u2212 \u02d9xp) dt.\n\n0\n\n0\n\nIntegrating the AG(x). \u02d9xp term by parts and using periodicity, we get\n\n(cid:2)gp \u2212 \u03c9pg \u2212 AG. (Fp \u2212 \u03c9pF)(cid:3) dt \u2212\n\n(cid:90) \u03c9\u22121\n\n(cid:104)\u2212gx + \u02d9AG + AG.F\n\n(cid:105)\n\nxp dt.\n\ndI(x\u03b3; p)\n\ndp\n\n=\n\n0\n\n0\n\n5\n\n\fParameter\npeak permeability \u00afPNa\npeak permeability \u00afPK\nmidpoint voltage Vm \u2228 Vh\nslope sm \u2228 (\u2212sh)\ntime constant \u03c4m,0 \u2228 \u03c4h,0\ngating exponent r \u2228 s\n\nminimum maximum\n0.15 \u00b5m/s\n0.24 fm/s\n11 \u00b5m/s\n6.6 fm/s\n70 mV\n- 72 mV\n3.33 mV\n200 mV\n200 ms\n5 \u00b5s\n0.2\n5.0\n\nTable 1: Parameter limits.\n\nWe can let the second term vanish by making the vector AG(x) obey\n\n\u02d9AG(x) = \u2212FT\n\nthe term \u03c9p is given by \u03c9p = \u03c9(cid:82) \u03c9\u22121\n\nLabel the homogeneous solution (obtained by setting gx(x\u03b3; p) = 0) as Z(x). It is known that\nZ(x).Fp(x) dt, provided Z(x) is normalised to satisfy\nZ(x).F(x) = 1. We can add any multiple of the homogeneous solution Z(x) to the inhomoge-\nneous solution, so we can always make\n\nx (x; p) AG(x) + gx(x; p).\n\n0\n\nAG(x).F(x) dt = G\n\nAG(x) (cid:55)\u2192 AG(x) \u2212 Z(x)\n\n(cid:33)\n\nAG(x).F(x) dt \u2212 \u03c9G\n\n.\n\n(3)\n\n(cid:90) \u03c9\u22121\n\n0\n\nby taking\n\n(cid:32)(cid:90) \u03c9\u22121\n(cid:90) \u03c9\u22121\n\n0\n\n0\n\nThis condition will make AG(x) unique. Finally, with eq. (3) we get\n\ndG(x\u03b3; p)\n\ndp\n\n=\n\ndI(x\u03b3; p)\n\ndp\n\n=\n\n(cid:0)gp \u2212 AG. Fp\n\n(cid:1) dt.\n\nThe \ufb01rst term in the integral gives rise to the partial derivative \u2202G(x\u03b3; p)/ \u2202p. In many cases, this\nterm is either zero, can be made zero, or at least made independent of the dynamical variables.\nThe parameters for the neuron models are listed in Table 1 together with their minimum and maxi-\nmum allowed values.\nFor each parameter in the neuron model, an auxiliary parameter on the entire real line is introduced,\nand a mapping from the real line onto the \ufb01nite range set by the biophysical limits is de\ufb01ned. Gradi-\nent descent on this auxiliary parameter space is performed by orthogonalizing the gradient dQ\u03b1/dp\nto the gradient dL/dp of the norm. To correct for drift off the constraint manifold of constant norm,\nillustrated in Fig. 3, steps of gradient ascent or descent on the Lp norm are performed while keeping\nQ\u03b1 constant. The step size during gradient descent is adjusted to assure that \u2206Q\u03b1 < 0 and that a\nperiodic solution x\u03b3 exists after adapting the parameters. The energy landscape is locally convex\n(Fig. 3).\n\n6 Predicting the Hodgkin-Huxley model\n\nWe start with a single-compartment Goldman-Hodgkin-Katz model neuron containing voltage-gated\nNa+ and leak conductances (Figure 1). A tonic synaptic input to the model evokes repetitive \ufb01ring\nof action potentials. We seek those parameters that minimize the ionic load for an action potential of\nconstant norm\u2014in other words, spikes whose height relative to the average voltage is fairly constant,\nsubject to a trade-off with the spike width. The ionic load is directly proportional to the work W\nperformed by the ion \ufb02ux. All parameters governing the ion channels\u2019 voltage dependence and\nkinetics, including their time constants, mid-points, slopes, and peak values, are subject to change.\nThe simplest model capable of generating an action potential must have two dynamical variables and\ntwo time scales: one for the upstroke and another for the downstroke. If both Na+ and K+ currents\n\n6\n\n\fFigure 2: Optimal spike shapes and currents for neuron models with different biophysical features.\nDuring optimization, the spikes were constrained to have constant norm (cid:107)V (t) \u2212 (cid:104)V (cid:105)(cid:107)16 = 92 mV,\nwhich controls the height of the spike. Insets in the left column display the voltage-dependence of\nthe optimized time constants for sodium inactivation and potassium activation; sodium activation is\nmodeled as occurring instantaneously. (a) Model with voltage-dependent inactivation of Na+; time\nconstants for the \ufb01rst order permeability kinetics are voltage-independent (inset). Inactivation turns\noff the Na+ current on the downstroke, but not completely: as the K+ current activates to repolarize\nthe membrane, the inward Na+ current reactivates and counteracts the K+ current; the peak of the\nresurgent Na+ current is marked by a triangle. (b) Model with voltage-dependent time constants\nfor the \ufb01rst order kinetics of activation and inactivation. The voltage dependence minimizes the\nresurgence of the Na+ current. (c) Power-law gating model with an inwardly rectifying potassium\ncurrent replacing the leak current. The power law dependence introduces an effective delay in the\nonset of the K+ current, which further minimizes the overlap of Na+ and K+ currents in time.\n\n7\n\nct [ms]Optimal Action PotentialCooperative Gating ModelFalling Phase CurrentsV [mV]-2 -1 0 1 26040200-20-40-600 0.2 0.47505002500current [\u03bcA/cm2]t [ms]IK[V]Excess INa[V]Peak Resurgence51\u03c4 [ms]t [ms]\u03c4n\u03c4hV [mV]Optimal Action PotentialV [mV]a51t [ms]\u03c4 [ms]\u03c4n\u03c4hV [mV]Optimal Action PotentialV [mV]bTransient Na Current ModelVoltage-dependent (In)activation ModelFalling Phase CurrentsFalling Phase Currents300200100t [ms]current [\u03bcA/cm2]current [\u03bcA/cm2]0 0.25 0.5 0.752001000 0.25 0.5 0.75t [ms]-4 -2 0 2 4-4 -2 0 2 4-60 0 6040200-20-40-60-60 0 606040200-20-40-60IK[V]Excess INa[V]Peak ResurgenceIK[V]Excess INa[V]Peak ResurgenceQ = 239 nC/cm2PNa = m(t)h(t)PK = n(t)Q = 169 nC/cm2PNa = m(t)h(t)PK = n(t)\u03c4i = \u03c4i(V) Q = 156 nC/cm2PNa = m(t)h(t)PK = n(t)s\u03c4i = \u03c4i(V)\u03c4 [ms]V [mV]51\u03c4n\u03c4h-60 0 60delay\fFigure 3: The energy required for an action potential three parameters governing potassium activa-\ntion: the midpoint voltage VK, the slope sK, and the (maximum) time constant \u03c4K. The energy is\nthe minimum work required to restore the ionic concentration gradients, as given by Eq. (1). Note\nthat the energy within the constrained manifold of constant norm spikes is locally convex.\n\nare persistent, current \ufb02ows in opposite directions at the same time, so that, even at the optimum, the\nionic load is 1200 nC/cm2. On the other hand, no voltage-gated K+ channels are even required for\na spike, as long as Na+ channels activate on a fast time scale and inactivate on a slower time scale\nand the leak is powerful enough to repolarize the neuron. Even so, the load is still 520 nC/cm2.\nWhile spikes require dynamics on two time scales, suppressing the overlap between inward and\noutward currents calls for a third time scale. The resulting dynamics are higher-dimensional and\nreduce the load to to 239 nC/cm2.\nMaking the activation and inactivation time constants voltage-dependent permits ion channels to\nlatch to an open or closed state during the rising and falling phase of the spike, reducing the ionic\nload to 189 nC/cm2 (Fig. 2) . The minimal Na+ and K+ currents are separated in time, yet dynamics\nthat are linear in the activation variables cannot enforce a true delay between the offset of the Na+\ncurrent and the onset of the K+ current. If current \ufb02ow depends on multiple gates that need to be\nactivated simultaneously, optimization can use the nonlinearity of multiplication to introduce a delay\nin the rise of the K+ current that abolishes the overlap, and the ionic load drops to 156 nC/cm2.\nAny number of kinetic schemes for the nonlinear permeabilities P\u03b1 can give rise to the same spike\nwaveform V (t), including the simplest two-dimensional one. Yet only the full Hodgkin-Huxley\n(HH) model, with its voltage-dependent kinetics that prevent the premature resurgence of inward\ncurrent and cooperative gating that delays the onset of the outward current, minimizes the energetic\ncost. More complex models, in which voltage-dependent ion channels make transitions between\nmultiple closed, inactivated, and open states, instantiate the energy-conserving features of the HH\nsystem at the molecular level. Furthermore, features that are eliminated during optimization, such as\na voltage-dependent inactivation of the outward potassium current, are also not part of the delayed\nrecti\ufb01er potassium current in the Hodgkin-Huxley framework.\n\n8\n\n12141618101214161020ybycyaSurface of Constant Norm SpikesVK [mV]sK [mV]\u03c4K [ms]yaybyc Energy per SpikeVK [mV]sK [mV]121416181012141616.316.416.5VE [nJ/cm2]16.3 nJ/cm2 \u2265 16.5 1000-2 0 2V [mV]t [ms] 1000-2 0 2V [mV]t [ms] 1000-2 0 2V [mV]t [ms]TaTbTc\fReferences\n[1] Paul De Weer, David C. Gadsby, and R. F. Rakowski. Voltage dependence of the na-k pump.\n\nAnn. Rev. Physiol., 50:225\u2013241, 1988.\n\n[2] B. Frankenhaeuser and A. F. Huxley. The action potential in the myelinated nerve \ufb01bre of\nxenopus laevis as computed on the basis of voltage clamp data. J. Physiol., 171:302\u2013315,\n1964.\n\n[3] Samuel S.-H. Wang, Jennifer R. Shultz, Mark J. Burish, Kimberly H. Harrison, Patrick R. Hof,\nLex C. Towns, Matthew W. Wagers, and Krysta D. Wyatt. Functional trade-offs in white matter\naxonal scaling. J. Neurosci., 28(15):4047\u20134056, 2008.\n\n[4] Henrik Alle, Arnd Roth, and J\u00a8org R. P. Geiger. Energy-ef\ufb01cient action potentials in hippocam-\n\npal mossy \ufb01bers. Science, 325(5946):1405\u20131408, 2009.\n\n[5] D. E. Goldman. Potential, impedance and recti\ufb01cation in membranes. J. Gen. Physiol., 27:37\u2013\n\n60, 1943.\n\n[6] A. L. Hodgkin and B. Katz. The effect of sodium ions on the electrical activity of the giant\n\naxon of the squid. J. Physiol., 108:37\u201377, 1949.\n\n[7] Brett C. Carter and Bruce P. Bean. Sodium entry during action potentials of mammalian neu-\nrons: Incomplete inactivation and reduced metabolic ef\ufb01ciency in fast-spiking neurons. Neu-\nron, 64(6):898\u2013909, 2009.\n\n[8] Luc J. Gentet, Greg J. Stuart, and John D. Clements. Direct measurement of speci\ufb01c membrane\n\ncapacitance in neurons. Biophys. J., 79:314\u2013320, 2000.\n\n[9] Alain Destexhe, Michael Rudolph, and Denis Par\u00b4e. The high-conductance state of neocortical\n\nneurons in vivo. Nature Neurosci. Rev., 4:739\u2013751, 2003.\n\n[10] Bilal Haider and David A. McCormick. Rapid neocortical dynamics: Cellular and network\n\nmechanisms. Neuron, 62:171\u2013189, 2009.\n\n9\n\n\f", "award": [], "sourceid": 902, "authors": [{"given_name": "Martin", "family_name": "Stemmler", "institution": null}, {"given_name": "Biswa", "family_name": "Sengupta", "institution": null}, {"given_name": "Simon", "family_name": "Laughlin", "institution": null}, {"given_name": "Jeremy", "family_name": "Niven", "institution": null}]}