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87 nips-2011-Energetically Optimal Action Potentials

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Author: Martin B. Stemmler, Biswa Sengupta, Simon Laughlin, Jeremy Niven

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. 1 The paradox Nerve cells communicate with each other over long distances using spike-like action potentials, which are brief electrical events traveling rapidly down axons and dendrites. Each action potential is caused by an accelerating inﬂux of sodium or calcium ions, depolarizing the cell membrane by forty millivolts or more, followed by repolarization of the cell membrane caused by an efﬂux of potassium ions. As different species of ions are swapped across the membrane during the action potential, ion pumps shuttle the excess ions back and restore the ionic concentration gradients. If we label each ionic species by α, the work ∆E done to restore the ionic concentration gradients is [α] ∆E = RT V ∆[α]in ln out , (1) [α]in α where R is the gas constant, T is the temperature in Kelvin, V is the cell volume, [α]in|out is the concentration of ion α inside or outside the cell, and ∆[α]in is the concentration change inside the cell, which is assumed to be small relative to the total concentration. The sum α zα ∆[α] = 0, where zα is the charge on ion α, as no net charge accumulates during the action potential and no net work is done by or on the electric ﬁeld. Often, sodium (Na+ ) and potassium (K+ ) play the dominant role in generating action potentials, in which case ∆E = ∆[Na]in F V(ENa − EK ), where F is Faraday’s constant, ENa = RT /F ln [Na]out /[Na]in is the reversal potential for Na+ , at which no net sodium current ﬂows, and EK = RT /F ln [K]out /[K]in . This estimate of the work done does not include heat (due to loss through the membrane resistance) or the work done by the ion channel proteins in changing their conformational state during the action potential. Hence, the action potential’s energetic cost to the cell is directly proportional to ∆[Na]in ; taking into account that each Na+ ion carries one elementary charge, the cost is also proportional to the 1 charge QNa that accumulates inside the cell. A maximally efﬁcient cell reduces the charge per spike to a minimum. If a cell ﬁres action potentials at an average rate f , the cell’s Na/K pumps must move Na+ and K+ ions in opposite directions, against their respective concentration gradients, to counteract an average inward Na+ current of f QNa . Exhaustive measurements on myocytes in the heart, which expend tremendous amounts of energy to keep the heart beating, indicate that Na/K pumps expel ∼ 0.5 µA/cm2 of Na+ current at membrane potentials close to rest [1]. Most excitable cells, even when spiking, spend most of their time close to resting potential, and yet standard models for action potentials can easily lead to accumulating an ionic charge of up to 5 µC/cm2 [2]; most of this accumulation occurs during a very brief time interval. If one were to take an isopotential nerve cell with the same density of ion pumps as in the heart, then such a cell would not be able to produce more than an action potential once every ten seconds on average. The brain should be effectively silent. Clearly, this conﬂicts with what is known about the average ﬁring rates of neurons in the brainstem or even the neocortex, which can sustain spiking up to at least 7 Hz [3]. Part of the discrepancy can be resolved by noting that nerve cells are not isopotential and that action potential generation occurs within a highly restricted area of the membrane. Even so, standard models of action potential generation waste extraordinary amounts of energy; recent evidence [4] points out that many mammalian cortical neurons are much more efﬁcient. As nature places a premium on energy consumption, we will argue that one can predict both the shape of the action potential and the underlying biophysics of the nonlinear, voltage-dependent ionic conductances from the principle of minimal energy consumption. After reviewing the ionic basis of action potentials, we ﬁrst sketch how to compute the minimal energy cost for an arbitrary spike shape, and then solve for the optimal action potential shape with a given height. Finally, we show how minimal energy consumption explains all the dynamical features in the standard HodgkinHuxley (HH) model for neuronal dynamics that distinguish the brain’s action potentials from other highly nonlinear oscillations in physics and chemistry. 2 Ionic basis of the action potential In an excitable cell, synaptic drive forces the membrane permeability to different ions to change rapidly in time, producing the dynamics of the action potential. The current density Iα carried by an ion species α is given by the Goldman-Hodgkin-Katz (GHK) current equation[5, 6, 2], which assumes that ions are driven independently across the membrane under the inﬂuence of a constant electric ﬁeld. Iα depends upon the ions membrane permeability, Pα , its concentrations on either side of the membrane [α]out and [α]in and the voltage across the membrane, V , according to: Iα = Pα 2 zα V F 2 [α]out − [α]in exp (zα V F/RT ) , RT 1 − exp(zα V F/RT ) (2) To produce the fast currents that generate APs, a subset of the membranes ionic permeabilities Pα are gated by voltage. Changes in the permeability Pα are not instantaneous; the voltage-gated permeability is scaled mathematically by gating variables m(t) and h(t) with their own time dependence. After separating constant from time-dependent components in the permeability, the voltage-gated permeability obeys ¯ Pα (t) = m(t)r h(t)s such that 0 ≤ Pα (t) ≤ Pα , ¯ where r and s are positive, and Pα is the peak permeability to ion α when all channels for ion α are open. Gating is also referred to as activation, and the associated nonlinear permeabilities are called active. There are also passive, voltage-insensitive permeabilities that maintain the resting potential and depolarise the membrane to trigger action potentials. The simplest possible kinetics for the gating variables are ﬁrst order, involving only a single derivative in time. The steady state of each gating variable at a given voltage is determined by a Boltzmann function, to which the gating variables evolve: dm r ¯ τm = Pα m∞ (V ) − m(t) dt dh and τh =h∞ (V ) − h(t), dt 2 −1 with m∞ (V ) = {1 + exp ((V − Vm )/sm )} the Boltzmann function described by the slope sm > −1 0 and the midpoint Vm ; similarly, h∞ (V ) = {1 + exp ((V − Vh )/sh )} , but with sh < 0. Scaling ¯ m∞ (V ) by the rth root of the peak permeability Pα is a matter of mathematical convenience. We will consider both voltage-independent and voltage-dependent time constants, either setting τj = τj,0 to be constant, where j ∈ {m(t), h(t)}, or imposing a bell-shaped voltage dependence τj (V ) = τj,0 sech [sj (V − Vj )] The synaptic, leak, and voltage-dependent currents drive the rate of change in the voltage across the membrane dV C = Isyn + Ileak + Iα , dt α where the synaptic permeability and leak permeability are held constant. 3 Resistive and capacitive components of the energy cost By treating the action potential as the charging and discharging of the cell membrane capacitance, the action potentials measured at the mossy ﬁbre synapse in rats [4] or in mouse thalamocortical neurons [7] were found to be highly energy-efﬁcient: the nonlinear, active conductances inject only slightly more current than is needed to charge a capacitor to the peak voltage of the action potential. The implicit assumption made here is that one can neglect the passive loss of current through the membrane resistance, known as the leak. Any passive loss must be compensated by additional charge, making this loss the primary target of the selection pressure that has shaped the dynamics of action potentials. On the other hand, the membrane capacitance at the site of AP initiation is generally modelled and experimentally conﬁrmed [8] as being fairly constant around 1 µF/cm2 ; in contrast, the propagation, but not generation, of AP’s can be assisted by a reduction in the capacitance achieved by the myelin sheath that wraps some axons. As myelin would block the ﬂow of ions, we posit that the speciﬁc capacitance cannot yield to selection pressure to minimise the work W = QNa (ENa − EK ) needed for AP generation. To address how the shape and dynamics of action potentials might have evolved to consume less energy, we ﬁrst ﬁx the action potential’s shape and solve for the minimum charge QNa ab initio, without treating the cell membrane as a pure capacitor. Regardless of the action potential’s particular time-course V (t), voltage-dependent ionic conductances must transfer Na+ and K+ charge to elicit an action potential. Figure 1 shows a generic action potential and the associated ionic currents, comparing the latter to the minimal currents required. The passive equivalent circuit for the neuron consists of a resistor in parallel with a capacitor, driven by a synaptic current. To charge the membrane to the peak voltage, a neuron in a high-conductance state [9, 10] may well lose more charge through the resistor than is stored on the capacitor. For neurons in a low-conductance state and for rapid voltage deﬂections from the resting potential, membrane capacitance will be the primary determinant of the charge. 4 The norm of spikes How close can voltage-gated channels with realistic properties come to the minimal currents? What time-course for the action potential leads to the smallest minimal currents? To answer these questions, we must solve a constrained optimization problem on the solutions to the nonlinear differential equations for the neuronal dynamics. To separate action potentials from mere small-amplitude oscillations in the voltage, we need to introduce a metric. Smaller action potentials consume less energy, provided the underlying currents are optimal, yet signalling between neurons depends on the action potential’s voltage deﬂection reaching a minimum amplitude. Given the importance of the action potential’s amplitude, we deﬁne an Lp norm on the voltage wave-form V (t) to emphasize the maximal voltage deﬂection: 1 p T V (t) − V p V (t) − V = 0 3 p dt , Generic Action Potential -10 + a V [mV] -20 -30 gsyn -40 -50 -60 0 2 4 6 8 t [ms] 10 12 14 16 gNa Active and Minimal Currents 100 gK + gleak C + + 80 2 current [µA/cm ] 60 b Active IK Minimum IK 40 20 0 -20 For a ﬁxed action potential waveform V (t): Active INa Minimum INa -40 -60 Minimum INa (t) = −LV (t)θ(LV (t)) Minimum IK (t) = −LV (t)θ(−LV (t)) -80 -100 0 2 4 6 8 10 t [ms] 12 14 ˙ with LV (t) ≡ C V (t) + Ileak [V (t)] + Isyn [V (t)]. 16 c Qresistive/Qcapacitive Resistive vs. Capacitive Minimum Charge 1 0.5 0 0.2 0.4 0.6 0.8 1.0 1.2 leak conductance [mS/cm2] 1.4 Figure 1: To generate an action potential with an arbitrary time-course V (t), the nonlinear, timedependent permeabilities must deliver more charge than just to load the membrane capacitance— resistive losses must be compensated. (a) The action potential’s time-course in a generic HH model for a neuron, represented by the circuit diagram on the right. The peak of the action potential is ∼ 50 mV above the average potential. (b) The inward Na+ current, shown in green going in the negative direction, rapidly depolarizes the potential V (t) and yields the upstroke of the action potential. Concurrently, the K+ current activates, displayed as a positive deﬂection, and leads to the downstroke in the potential V (t). Inward and outward currents overlap signiﬁcantly in time. The dotted lines within the region bounded by the solid lines represent the minimal Na+ current and the minimal K+ current needed to produce the V (t) spike waveform in (a). By the law of current conservation, the sum of capacitive, resistive, and synaptic currents, denoted by ˙ LV (t) ≡ C V (t) + Ileak [V (t)] + Isyn [V (t)], must be balanced by the active currents. If the cell’s passive properties, namely its capacitance and (leak) resistance, and the synaptic conductance are constant, we can deduce the minimal active currents needed to generate a speciﬁed V (t). The minimal currents, by deﬁnition, do not overlap in time. Taking into account passive current ﬂow, restoring the concentration gradients after the action potential requires 29 nJ/cm2 . By contrast, if the active currents were optimal, the cost would be 8.9 nJ/cm2 . (c) To depolarize from the minimum to the maximum of the AP, the synaptic voltage-gated currents must deliver a charge Qcapacitive to charge the membrane capacitance and a charge Qresistive to compensate for the loss of current through leak channels. For a large leak conductance in the cell membrane, Qresistive can be larger than Qcapacitive . 4 where V is the average voltage. In the limit as p → ∞, the norm simply becomes the difference between the action potential’s peak voltage and the mean voltage, whereas a ﬁnite p ensures that the norm is differentiable. In parameter space, we will focus our attention to the manifold of action potentials with constant Lp norm with 2 p < ∞, which entails that the optimal action potential will have a ﬁnite, though possibly narrow width. To be close to the supremum norm, yet still have a norm that is well-behaved under differentiation, we decided to use p = 16. 5 Poincar´ -Lindstedt perturbation of periodic dynamical orbits e Standard (secular) perturbation theory diverges for periodic orbits, so we apply the PoincarLindstedt technique of expanding both in the period and the dynamics of the asymptotic orbit and then derive a set of adjoint sensitivity equations for the differential-algebraic system. Solving once for the adjoint functions, we can easily compute the parameter gradient of any functional on the orbit, even for thousands of parameters. ˙ We start with a set of ordinary differential equations x = F(x; p) for the neuron’s dynamics, an asymptotically periodic orbit xγ (t) that describes the action potential, and a functional G(x; p) on the orbit, representing the energy consumption, for instance. The functional can be written as an integral ω(p)−1 G(xγ ; p) = g(xγ (t); p) dt, 0 over some source term g(xγ (t); p). Assume that locally perturbing a parameter p ∈ p induces a smooth change in the stable limit cycle, preserving its existence. Generally, a perturbation changes not only the limit cycle’s path in state space, but also the average speed with which this orbit is traversed; as a consequence, the value of the functional depends on this change in speed, to lowest order. For simplicity, consider a single, scalar parameter p. G(xγ ; p) is the solution to ω(p)∂τ [G(xγ ; p)] = g(xγ ; p), where we have normalised time via τ = ω(p)t. Denoting partial derivatives by subscripts, we expand p → p + to get the O 1 equation dτ [Gp (xγ ; p)] + ωp g(xγ ; p) = gx (xγ ; p)xp + gp (xγ ; p) in a procedure known as the Poincar´ -Lindstedt method. Hence, e dG = dp ω −1 (gp + gx xp − ωp g) dt, 0 where, once again by the Poincar´ -Lindstedt method, xp is the solution to e ˙ xp =Fx (xγ )xp + Fp (xγ ) − ωp F (xγ ) . Following the approach described by Cao, Li, Petzold, and Serban (2003), introduce a Lagrange vector AG (x) and consider the augmented objective function ω −1 I(xγ ; p) = G(xγ ; p) − ˙ AG (xγ ). (F(xγ ) − xγ ) dt, 0 γ ˙ which is identical to G(x ; p) as F(x) − x = 0. Then dI(xγ ; p) = dp ω −1 ω −1 ˙ AG . (Fp + Fx xp − ωp F − xp ) dt. (gp + gx xp − ωp g) dt − 0 0 ˙ Integrating the AG (x).xp term by parts and using periodicity, we get dI(xγ ; p) = dp ω −1 ω −1 ˙ −gx + AG + AG .F xp dt. G gp − ωp g − A . (Fp − ωp F) dt − 0 0 5 Parameter ¯ peak permeability PNa ¯ peak permeability PK midpoint voltage Vm ∨ Vh slope sm ∨ (−sh ) time constant τm,0 ∨ τh,0 gating exponent r ∨ s minimum 0.24 fm/s 6.6 fm/s - 72 mV 3.33 mV 5 µs 0.2 maximum 0.15 µm/s 11 µm/s 70 mV 200 mV 200 ms 5.0 Table 1: Parameter limits. We can let the second term vanish by making the vector AG (x) obey ˙ AG (x) = −FT (x; p) AG (x) + gx (x; p). x Label the homogeneous solution (obtained by setting gx (xγ ; p) = 0) as Z(x). It is known that ω −1 the term ωp is given by ωp = ω 0 Z(x).Fp (x) dt, provided Z(x) is normalised to satisfy Z(x).F(x) = 1. We can add any multiple of the homogeneous solution Z(x) to the inhomogeneous solution, so we can always make ω −1 AG (x).F(x) dt = G 0 by taking ω −1 G G AG (x).F(x) dt − ωG . A (x) → A (x) − Z(x) (3) 0 This condition will make AG (x) unique. Finally, with eq. (3) we get dI(xγ ; p) dG(xγ ; p) = = dp dp ω −1 gp − AG . Fp dt. 0 The ﬁrst term in the integral gives rise to the partial derivative ∂G(xγ ; p)/ ∂p. In many cases, this term is either zero, can be made zero, or at least made independent of the dynamical variables. The parameters for the neuron models are listed in Table 1 together with their minimum and maximum allowed values. For each parameter in the neuron model, an auxiliary parameter on the entire real line is introduced, and a mapping from the real line onto the ﬁnite range set by the biophysical limits is deﬁned. Gradient descent on this auxiliary parameter space is performed by orthogonalizing the gradient dQα /dp to the gradient dL/dp of the norm. To correct for drift off the constraint manifold of constant norm, illustrated in Fig. 3, steps of gradient ascent or descent on the Lp norm are performed while keeping Qα constant. The step size during gradient descent is adjusted to assure that ∆Qα < 0 and that a periodic solution xγ exists after adapting the parameters. The energy landscape is locally convex (Fig. 3). 6 Predicting the Hodgkin-Huxley model We start with a single-compartment Goldman-Hodgkin-Katz model neuron containing voltage-gated Na+ and leak conductances (Figure 1). A tonic synaptic input to the model evokes repetitive ﬁring of action potentials. We seek those parameters that minimize the ionic load for an action potential of constant norm—in other words, spikes whose height relative to the average voltage is fairly constant, subject to a trade-off with the spike width. The ionic load is directly proportional to the work W performed by the ion ﬂux. All parameters governing the ion channels’ voltage dependence and kinetics, including their time constants, mid-points, slopes, and peak values, are subject to change. The simplest model capable of generating an action potential must have two dynamical variables and two time scales: one for the upstroke and another for the downstroke. If both Na+ and K+ currents 6 Transient Na Current Model Optimal Action Potential Falling Phase Currents 40 20 a τ [ms] 5 1 2 Q = 239 nC/cm PNa = m(t)h(t) PK = n(t) 0 -60 0 -20 τh τn 60 current [μA/cm2] V [mV] V [mV] -40 IK[V] Excess INa[V] Peak Resurgence 300 200 100 -60 -4 -2 0 0 4 40 20 τ [ms] 5 1 Q = 169 nC/cm2 PNa = m(t)h(t) PK = n(t) τi = τi(V) 0 -60 0 -20 τh τn 60 current [μA/cm2] 60 V [mV] -40 -60 -4 -2 0 2 t [ms] 0.5 0.75 IK[V] Excess INa[V] Peak Resurgence 200 100 0 4 0.25 40 5 1 PNa = m(t)h(t) s PK = n(t) τi = τi(V) 20 0 delay τ [ms] Q = 156 nC/cm2 current [μA/cm2] 60 -60 0 -20 τh τn 60 V [mV] -40 -60 t [ms] 0.5 t [ms] 0.75 Cooperative Gating Model Optimal Action Potential Falling Phase Currents V [mV] c 0.25 Voltage-dependent (In)activation Model Falling Phase Currents Optimal Action Potential V [mV] b 2 t [ms] -2 -1 0 t [ms] 1 750 500 250 0 0 2 IK[V] Excess INa[V] Peak Resurgence 0.2 t [ms] 0.4 Figure 2: Optimal spike shapes and currents for neuron models with different biophysical features. During optimization, the spikes were constrained to have constant norm V (t) − V 16 = 92 mV, which controls the height of the spike. Insets in the left column display the voltage-dependence of the optimized time constants for sodium inactivation and potassium activation; sodium activation is modeled as occurring instantaneously. (a) Model with voltage-dependent inactivation of Na+ ; time constants for the ﬁrst order permeability kinetics are voltage-independent (inset). Inactivation turns off the Na+ current on the downstroke, but not completely: as the K+ current activates to repolarize the membrane, the inward Na+ current reactivates and counteracts the K+ current; the peak of the resurgent Na+ current is marked by a triangle. (b) Model with voltage-dependent time constants for the ﬁrst order kinetics of activation and inactivation. The voltage dependence minimizes the resurgence of the Na+ current. (c) Power-law gating model with an inwardly rectifying potassium current replacing the leak current. The power law dependence introduces an effective delay in the onset of the K+ current, which further minimizes the overlap of Na+ and K+ currents in time. 7 Energy per Spike Surface of Constant Norm Spikes ya 10 16 V [mV] K 18 10 12 14 14 16 T b V [mV] 0 t [ms] 2 10 18 16 12 s [mV] K V [mV] K T a V [mV] 12 nJ/cm2 ≥ 16.5 16.3 16.3 yc 16 sK [mV] 100 0 -2 16.4 T c 100 0 -2 V [mV] 14 14 VE [nJ/cm2] yc ya τK [ms] yb 12 16.5 yb 20 0 t [ms] 2 100 0 -2 0 t [ms] 2 Figure 3: The energy required for an action potential three parameters governing potassium activation: the midpoint voltage VK , the slope sK , and the (maximum) time constant τK . The energy is the minimum work required to restore the ionic concentration gradients, as given by Eq. (1). Note that the energy within the constrained manifold of constant norm spikes is locally convex. are persistent, current ﬂows in opposite directions at the same time, so that, even at the optimum, the ionic load is 1200 nC/cm2 . On the other hand, no voltage-gated K+ channels are even required for a spike, as long as Na+ channels activate on a fast time scale and inactivate on a slower time scale and the leak is powerful enough to repolarize the neuron. Even so, the load is still 520 nC/cm2 . While spikes require dynamics on two time scales, suppressing the overlap between inward and outward currents calls for a third time scale. The resulting dynamics are higher-dimensional and reduce the load to to 239 nC/cm2 . Making the activation and inactivation time constants voltage-dependent permits ion channels to latch to an open or closed state during the rising and falling phase of the spike, reducing the ionic load to 189 nC/cm2 (Fig. 2) . The minimal Na+ and K+ currents are separated in time, yet dynamics that are linear in the activation variables cannot enforce a true delay between the offset of the Na+ current and the onset of the K+ current. If current ﬂow depends on multiple gates that need to be activated simultaneously, optimization can use the nonlinearity of multiplication to introduce a delay in the rise of the K+ current that abolishes the overlap, and the ionic load drops to 156 nC/cm2 . Any number of kinetic schemes for the nonlinear permeabilities Pα can give rise to the same spike waveform V (t), including the simplest two-dimensional one. Yet only the full Hodgkin-Huxley (HH) model, with its voltage-dependent kinetics that prevent the premature resurgence of inward current and cooperative gating that delays the onset of the outward current, minimizes the energetic cost. More complex models, in which voltage-dependent ion channels make transitions between multiple closed, inactivated, and open states, instantiate the energy-conserving features of the HH system at the molecular level. Furthermore, features that are eliminated during optimization, such as a voltage-dependent inactivation of the outward potassium current, are also not part of the delayed rectiﬁer potassium current in the Hodgkin-Huxley framework. 8 References [1] Paul De Weer, David C. Gadsby, and R. F. Rakowski. Voltage dependence of the na-k pump. Ann. Rev. Physiol., 50:225–241, 1988. [2] B. Frankenhaeuser and A. F. Huxley. The action potential in the myelinated nerve ﬁbre of xenopus laevis as computed on the basis of voltage clamp data. J. Physiol., 171:302–315, 1964. [3] Samuel S.-H. Wang, Jennifer R. Shultz, Mark J. Burish, Kimberly H. Harrison, Patrick R. Hof, Lex C. Towns, Matthew W. Wagers, and Krysta D. Wyatt. Functional trade-offs in white matter axonal scaling. J. Neurosci., 28(15):4047–4056, 2008. [4] Henrik Alle, Arnd Roth, and J¨ rg R. P. Geiger. Energy-efﬁcient action potentials in hippocamo pal mossy ﬁbers. Science, 325(5946):1405–1408, 2009. [5] D. E. Goldman. Potential, impedance and rectiﬁcation in membranes. J. Gen. Physiol., 27:37– 60, 1943. [6] A. L. Hodgkin and B. Katz. The effect of sodium ions on the electrical activity of the giant axon of the squid. J. Physiol., 108:37–77, 1949. [7] Brett C. Carter and Bruce P. Bean. Sodium entry during action potentials of mammalian neurons: Incomplete inactivation and reduced metabolic efﬁciency in fast-spiking neurons. Neuron, 64(6):898–909, 2009. [8] Luc J. Gentet, Greg J. Stuart, and John D. Clements. Direct measurement of speciﬁc membrane capacitance in neurons. Biophys. J., 79:314–320, 2000. [9] Alain Destexhe, Michael Rudolph, and Denis Par´ . The high-conductance state of neocortical e neurons in vivo. Nature Neurosci. Rev., 4:739–751, 2003. [10] Bilal Haider and David A. McCormick. Rapid neocortical dynamics: Cellular and network mechanisms. Neuron, 62:171–189, 2009. 9

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 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. [sent-3, score-0.951]

2 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. [sent-4, score-1.126]

3 1 The paradox Nerve cells communicate with each other over long distances using spike-like action potentials, which are brief electrical events traveling rapidly down axons and dendrites. [sent-6, score-0.291]

4 Each action potential is caused by an accelerating inﬂux of sodium or calcium ions, depolarizing the cell membrane by forty millivolts or more, followed by repolarization of the cell membrane caused by an efﬂux of potassium ions. [sent-7, score-1.313]

5 As different species of ions are swapped across the membrane during the action potential, ion pumps shuttle the excess ions back and restore the ionic concentration gradients. [sent-8, score-1.357]

6 The sum α zα ∆[α] = 0, where zα is the charge on ion α, as no net charge accumulates during the action potential and no net work is done by or on the electric ﬁeld. [sent-10, score-1.047]

7 This estimate of the work done does not include heat (due to loss through the membrane resistance) or the work done by the ion channel proteins in changing their conformational state during the action potential. [sent-12, score-0.696]

8 Hence, the action potential’s energetic cost to the cell is directly proportional to ∆[Na]in ; taking into account that each Na+ ion carries one elementary charge, the cost is also proportional to the 1 charge QNa that accumulates inside the cell. [sent-13, score-0.807]

9 A maximally efﬁcient cell reduces the charge per spike to a minimum. [sent-14, score-0.385]

10 If a cell ﬁres action potentials at an average rate f , the cell’s Na/K pumps must move Na+ and K+ ions in opposite directions, against their respective concentration gradients, to counteract an average inward Na+ current of f QNa . [sent-15, score-0.909]

11 Exhaustive measurements on myocytes in the heart, which expend tremendous amounts of energy to keep the heart beating, indicate that Na/K pumps expel ∼ 0. [sent-16, score-0.19]

12 5 µA/cm2 of Na+ current at membrane potentials close to rest [1]. [sent-17, score-0.425]

13 Most excitable cells, even when spiking, spend most of their time close to resting potential, and yet standard models for action potentials can easily lead to accumulating an ionic charge of up to 5 µC/cm2 [2]; most of this accumulation occurs during a very brief time interval. [sent-18, score-0.921]

14 If one were to take an isopotential nerve cell with the same density of ion pumps as in the heart, then such a cell would not be able to produce more than an action potential once every ten seconds on average. [sent-19, score-0.925]

15 Part of the discrepancy can be resolved by noting that nerve cells are not isopotential and that action potential generation occurs within a highly restricted area of the membrane. [sent-22, score-0.51]

16 Even so, standard models of action potential generation waste extraordinary amounts of energy; recent evidence [4] points out that many mammalian cortical neurons are much more efﬁcient. [sent-23, score-0.501]

17 As nature places a premium on energy consumption, we will argue that one can predict both the shape of the action potential and the underlying biophysics of the nonlinear, voltage-dependent ionic conductances from the principle of minimal energy consumption. [sent-24, score-0.936]

18 After reviewing the ionic basis of action potentials, we ﬁrst sketch how to compute the minimal energy cost for an arbitrary spike shape, and then solve for the optimal action potential shape with a given height. [sent-25, score-1.148]

19 Finally, we show how minimal energy consumption explains all the dynamical features in the standard HodgkinHuxley (HH) model for neuronal dynamics that distinguish the brain’s action potentials from other highly nonlinear oscillations in physics and chemistry. [sent-26, score-0.65]

20 2 Ionic basis of the action potential In an excitable cell, synaptic drive forces the membrane permeability to different ions to change rapidly in time, producing the dynamics of the action potential. [sent-27, score-1.45]

21 The current density Iα carried by an ion species α is given by the Goldman-Hodgkin-Katz (GHK) current equation[5, 6, 2], which assumes that ions are driven independently across the membrane under the inﬂuence of a constant electric ﬁeld. [sent-28, score-0.648]

22 Changes in the permeability Pα are not instantaneous; the voltage-gated permeability is scaled mathematically by gating variables m(t) and h(t) with their own time dependence. [sent-30, score-0.525]

23 After separating constant from time-dependent components in the permeability, the voltage-gated permeability obeys ¯ Pα (t) = m(t)r h(t)s such that 0 ≤ Pα (t) ≤ Pα , ¯ where r and s are positive, and Pα is the peak permeability to ion α when all channels for ion α are open. [sent-31, score-0.932]

24 Gating is also referred to as activation, and the associated nonlinear permeabilities are called active. [sent-32, score-0.083]

25 There are also passive, voltage-insensitive permeabilities that maintain the resting potential and depolarise the membrane to trigger action potentials. [sent-33, score-0.789]

26 The simplest possible kinetics for the gating variables are ﬁrst order, involving only a single derivative in time. [sent-34, score-0.194]

27 Scaling ¯ m∞ (V ) by the rth root of the peak permeability Pα is a matter of mathematical convenience. [sent-36, score-0.333]

28 The implicit assumption made here is that one can neglect the passive loss of current through the membrane resistance, known as the leak. [sent-39, score-0.366]

29 Any passive loss must be compensated by additional charge, making this loss the primary target of the selection pressure that has shaped the dynamics of action potentials. [sent-40, score-0.406]

30 As myelin would block the ﬂow of ions, we posit that the speciﬁc capacitance cannot yield to selection pressure to minimise the work W = QNa (ENa − EK ) needed for AP generation. [sent-42, score-0.164]

31 To address how the shape and dynamics of action potentials might have evolved to consume less energy, we ﬁrst ﬁx the action potential’s shape and solve for the minimum charge QNa ab initio, without treating the cell membrane as a pure capacitor. [sent-43, score-1.318]

32 Regardless of the action potential’s particular time-course V (t), voltage-dependent ionic conductances must transfer Na+ and K+ charge to elicit an action potential. [sent-44, score-1.075]

33 Figure 1 shows a generic action potential and the associated ionic currents, comparing the latter to the minimal currents required. [sent-45, score-1.009]

34 The passive equivalent circuit for the neuron consists of a resistor in parallel with a capacitor, driven by a synaptic current. [sent-46, score-0.227]

35 To charge the membrane to the peak voltage, a neuron in a high-conductance state [9, 10] may well lose more charge through the resistor than is stored on the capacitor. [sent-47, score-0.908]

36 For neurons in a low-conductance state and for rapid voltage deﬂections from the resting potential, membrane capacitance will be the primary determinant of the charge. [sent-48, score-0.676]

37 4 The norm of spikes How close can voltage-gated channels with realistic properties come to the minimal currents? [sent-49, score-0.219]

38 What time-course for the action potential leads to the smallest minimal currents? [sent-50, score-0.488]

39 To separate action potentials from mere small-amplitude oscillations in the voltage, we need to introduce a metric. [sent-52, score-0.418]

40 Smaller action potentials consume less energy, provided the underlying currents are optimal, yet signalling between neurons depends on the action potential’s voltage deﬂection reaching a minimum amplitude. [sent-53, score-1.286]

41 4 Figure 1: To generate an action potential with an arbitrary time-course V (t), the nonlinear, timedependent permeabilities must deliver more charge than just to load the membrane capacitance— resistive losses must be compensated. [sent-64, score-1.129]

42 (a) The action potential’s time-course in a generic HH model for a neuron, represented by the circuit diagram on the right. [sent-65, score-0.291]

43 The peak of the action potential is ∼ 50 mV above the average potential. [sent-66, score-0.548]

44 (b) The inward Na+ current, shown in green going in the negative direction, rapidly depolarizes the potential V (t) and yields the upstroke of the action potential. [sent-67, score-0.562]

45 Concurrently, the K+ current activates, displayed as a positive deﬂection, and leads to the downstroke in the potential V (t). [sent-68, score-0.228]

46 The dotted lines within the region bounded by the solid lines represent the minimal Na+ current and the minimal K+ current needed to produce the V (t) spike waveform in (a). [sent-70, score-0.327]

47 By the law of current conservation, the sum of capacitive, resistive, and synaptic currents, denoted by ˙ LV (t) ≡ C V (t) + Ileak [V (t)] + Isyn [V (t)], must be balanced by the active currents. [sent-71, score-0.152]

48 If the cell’s passive properties, namely its capacitance and (leak) resistance, and the synaptic conductance are constant, we can deduce the minimal active currents needed to generate a speciﬁed V (t). [sent-72, score-0.701]

49 The minimal currents, by deﬁnition, do not overlap in time. [sent-73, score-0.058]

50 Taking into account passive current ﬂow, restoring the concentration gradients after the action potential requires 29 nJ/cm2 . [sent-74, score-0.6]

51 By contrast, if the active currents were optimal, the cost would be 8. [sent-75, score-0.339]

52 (c) To depolarize from the minimum to the maximum of the AP, the synaptic voltage-gated currents must deliver a charge Qcapacitive to charge the membrane capacitance and a charge Qresistive to compensate for the loss of current through leak channels. [sent-77, score-1.603]

53 For a large leak conductance in the cell membrane, Qresistive can be larger than Qcapacitive . [sent-78, score-0.253]

54 In the limit as p → ∞, the norm simply becomes the difference between the action potential’s peak voltage and the mean voltage, whereas a ﬁnite p ensures that the norm is differentiable. [sent-80, score-0.716]

55 In parameter space, we will focus our attention to the manifold of action potentials with constant Lp norm with 2 p < ∞, which entails that the optimal action potential will have a ﬁnite, though possibly narrow width. [sent-81, score-0.888]

56 ˙ We start with a set of ordinary differential equations x = F(x; p) for the neuron’s dynamics, an asymptotically periodic orbit xγ (t) that describes the action potential, and a functional G(x; p) on the orbit, representing the energy consumption, for instance. [sent-85, score-0.499]

57 Generally, a perturbation changes not only the limit cycle’s path in state space, but also the average speed with which this orbit is traversed; as a consequence, the value of the functional depends on this change in speed, to lowest order. [sent-88, score-0.073]

58 Denoting partial derivatives by subscripts, we expand p → p + to get the O 1 equation dτ [Gp (xγ ; p)] + ωp g(xγ ; p) = gx (xγ ; p)xp + gp (xγ ; p) in a procedure known as the Poincar´ -Lindstedt method. [sent-91, score-0.12]

59 Hence, e dG = dp ω −1 (gp + gx xp − ωp g) dt, 0 where, once again by the Poincar´ -Lindstedt method, xp is the solution to e ˙ xp =Fx (xγ )xp + Fp (xγ ) − ωp F (xγ ) . [sent-92, score-0.324]

60 (gp + gx xp − ωp g) dt − 0 0 ˙ Integrating the AG (x). [sent-97, score-0.209]

61 xp term by parts and using periodicity, we get dI(xγ ; p) = dp ω −1 ω −1 ˙ −gx + AG + AG . [sent-98, score-0.109]

62 (Fp − ωp F) dt − 0 0 5 Parameter ¯ peak permeability PNa ¯ peak permeability PK midpoint voltage Vm ∨ Vh slope sm ∨ (−sh ) time constant τm,0 ∨ τh,0 gating exponent r ∨ s minimum 0. [sent-101, score-1.092]

63 We can let the second term vanish by making the vector AG (x) obey ˙ AG (x) = −FT (x; p) AG (x) + gx (x; p). [sent-108, score-0.071]

64 x Label the homogeneous solution (obtained by setting gx (xγ ; p) = 0) as Z(x). [sent-109, score-0.071]

65 (3) we get dI(xγ ; p) dG(xγ ; p) = = dp dp ω −1 gp − AG . [sent-118, score-0.123]

66 The parameters for the neuron models are listed in Table 1 together with their minimum and maximum allowed values. [sent-122, score-0.061]

67 For each parameter in the neuron model, an auxiliary parameter on the entire real line is introduced, and a mapping from the real line onto the ﬁnite range set by the biophysical limits is deﬁned. [sent-123, score-0.061]

68 6 Predicting the Hodgkin-Huxley model We start with a single-compartment Goldman-Hodgkin-Katz model neuron containing voltage-gated Na+ and leak conductances (Figure 1). [sent-130, score-0.234]

69 A tonic synaptic input to the model evokes repetitive ﬁring of action potentials. [sent-131, score-0.356]

70 We seek those parameters that minimize the ionic load for an action potential of constant norm—in other words, spikes whose height relative to the average voltage is fairly constant, subject to a trade-off with the spike width. [sent-132, score-1.079]

71 The ionic load is directly proportional to the work W performed by the ion ﬂux. [sent-133, score-0.457]

72 All parameters governing the ion channels’ voltage dependence and kinetics, including their time constants, mid-points, slopes, and peak values, are subject to change. [sent-134, score-0.508]

73 The simplest model capable of generating an action potential must have two dynamical variables and two time scales: one for the upstroke and another for the downstroke. [sent-135, score-0.463]

74 25 40 5 1 PNa = m(t)h(t) s PK = n(t) τi = τi(V) 20 0 delay τ [ms] Q = 156 nC/cm2 current [μA/cm2] 60 -60 0 -20 τh τn 60 V [mV] -40 -60 t [ms] 0. [sent-139, score-0.089]

75 4 Figure 2: Optimal spike shapes and currents for neuron models with different biophysical features. [sent-144, score-0.434]

76 During optimization, the spikes were constrained to have constant norm V (t) − V 16 = 92 mV, which controls the height of the spike. [sent-145, score-0.103]

77 Insets in the left column display the voltage-dependence of the optimized time constants for sodium inactivation and potassium activation; sodium activation is modeled as occurring instantaneously. [sent-146, score-0.476]

78 (a) Model with voltage-dependent inactivation of Na+ ; time constants for the ﬁrst order permeability kinetics are voltage-independent (inset). [sent-147, score-0.401]

79 Inactivation turns off the Na+ current on the downstroke, but not completely: as the K+ current activates to repolarize the membrane, the inward Na+ current reactivates and counteracts the K+ current; the peak of the resurgent Na+ current is marked by a triangle. [sent-148, score-0.503]

80 (b) Model with voltage-dependent time constants for the ﬁrst order kinetics of activation and inactivation. [sent-149, score-0.162]

81 The voltage dependence minimizes the resurgence of the Na+ current. [sent-150, score-0.286]

82 (c) Power-law gating model with an inwardly rectifying potassium current replacing the leak current. [sent-151, score-0.371]

83 The power law dependence introduces an effective delay in the onset of the K+ current, which further minimizes the overlap of Na+ and K+ currents in time. [sent-152, score-0.373]

84 5 yb 20 0 t [ms] 2 100 0 -2 0 t [ms] 2 Figure 3: The energy required for an action potential three parameters governing potassium activation: the midpoint voltage VK , the slope sK , and the (maximum) time constant τK . [sent-158, score-0.919]

85 The energy is the minimum work required to restore the ionic concentration gradients, as given by Eq. [sent-159, score-0.388]

86 Note that the energy within the constrained manifold of constant norm spikes is locally convex. [sent-161, score-0.194]

87 are persistent, current ﬂows in opposite directions at the same time, so that, even at the optimum, the ionic load is 1200 nC/cm2 . [sent-162, score-0.35]

88 On the other hand, no voltage-gated K+ channels are even required for a spike, as long as Na+ channels activate on a fast time scale and inactivate on a slower time scale and the leak is powerful enough to repolarize the neuron. [sent-163, score-0.302]

89 While spikes require dynamics on two time scales, suppressing the overlap between inward and outward currents calls for a third time scale. [sent-165, score-0.583]

90 The resulting dynamics are higher-dimensional and reduce the load to to 239 nC/cm2 . [sent-166, score-0.128]

91 Making the activation and inactivation time constants voltage-dependent permits ion channels to latch to an open or closed state during the rising and falling phase of the spike, reducing the ionic load to 189 nC/cm2 (Fig. [sent-167, score-0.709]

92 The minimal Na+ and K+ currents are separated in time, yet dynamics that are linear in the activation variables cannot enforce a true delay between the offset of the Na+ current and the onset of the K+ current. [sent-169, score-0.597]

93 If current ﬂow depends on multiple gates that need to be activated simultaneously, optimization can use the nonlinearity of multiplication to introduce a delay in the rise of the K+ current that abolishes the overlap, and the ionic load drops to 156 nC/cm2 . [sent-170, score-0.439]

94 Any number of kinetic schemes for the nonlinear permeabilities Pα can give rise to the same spike waveform V (t), including the simplest two-dimensional one. [sent-171, score-0.182]

95 Yet only the full Hodgkin-Huxley (HH) model, with its voltage-dependent kinetics that prevent the premature resurgence of inward current and cooperative gating that delays the onset of the outward current, minimizes the energetic cost. [sent-172, score-0.539]

96 More complex models, in which voltage-dependent ion channels make transitions between multiple closed, inactivated, and open states, instantiate the energy-conserving features of the HH system at the molecular level. [sent-173, score-0.221]

97 Furthermore, features that are eliminated during optimization, such as a voltage-dependent inactivation of the outward potassium current, are also not part of the delayed rectiﬁer potassium current in the Hodgkin-Huxley framework. [sent-174, score-0.409]

98 The action potential in the myelinated nerve ﬁbre of xenopus laevis as computed on the basis of voltage clamp data. [sent-188, score-0.704]

99 The effect of sodium ions on the electrical activity of the giant axon of the squid. [sent-223, score-0.244]

100 Sodium entry during action potentials of mammalian neurons: Incomplete inactivation and reduced metabolic efﬁciency in fast-spiking neurons. [sent-230, score-0.534]

similar papers computed by tfidf model

tfidf for this paper:

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Note that the diffusive white-noise does not imply white noise ﬂuctuations of the voltage V (t), the probability distribution of V (t) will depend on ∆T and Θ. The second variable, w, describes the subthreshold as well as the spike-triggered adaptation both ˆ parametrized by the coupling strength a and the time constant τw . Each time tj the voltage goes to inﬁnity, we assumed that a spike is emitted. 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The most common link-function used to model single neuron activities are the linear-rectiﬁer and the exponential function. 3 Mapping In order to map the stochastic AdEx to the SRM we follow a two-step procedure. First we derive the ﬁlter κ(t) and the kernels ηv (t) and ηw (t) analytically as a function of AdEx parameters. Second, we derive the link-function of the SRM from the stochastic spike emission of the AdEx. Figure 1: Mapping of the subthreshold dynamics of an AdEx to an equivalent SRM. A. Membrane ﬁlter κ(t) for three different sets of parameters of the AdEx leading to over-damped, critically damped and under-damped cases (upper, middle and lower panel, respectively). B. Spike-Triggered η(t) (black), ηv (t) (light gray) and ηw (gray) for the three cases. C. Example of voltage trace produced when an AdEx is stimulated with a step of colored noise (black). The corresponding voltage from a SRM stimulated with the same current and where we forced the spikes to match those of the AdEx (red). D. Error in the subthreshold voltage (VAdEx − VGLM ) as a function of the mean voltage of the AdEx, for the three different cases: over-, critically and under-damped (light gray, gray and black, respectively) with ∆T = 1 mV. Red line represents the voltage threshold Θ. E. Root Mean Square Error (RMSE) ratio for the three cases with ∆T = 1 mV. The RMSE ratio is the RMSE between the deterministic VSRM and the stochastic VAdEx divided by the RMSE between repetitions of the stochastic AdEx voltage. The error bar shows a single standard deviation as the RMSE ratio is averaged accross multiple value of σ. 3.1 Subthreshold voltage dynamics We start by assuming that the non-linearity for spike initiation does not affect the mean subthreshold voltage of the stochastic AdEx (see Figure 1 D). This assumption is motivated by the small ∆T 3 observed in in-vitro recordings (from 0.5 to 2 mV [8, 9]) which suggest that the subthreshold dynamics are mainly linear except very close to Θ. Also, we expect that the non-linear link-function will capture some of the dynamics due to the non-linearity for spike initiation. Thus it is possible to rewrite the deterministic subthreshold part of the AdEx (Eq. 1-2 without and without ∆T exp((V − Θ)/∆T )) using matrices: ˙ x = Ax (5) with x = V w and A = − τ1 m a τw − gl1m τ − τ1 w (6) In this form, the dynamics of the deterministic AdEx voltage is a damped oscillator with a driving force. Depending on the eigenvalues of A the system could be over-damped, critically damped or under-damped. The ﬁlter κ(t) of the GLM is given by the impulse response of the system of coupled differential equations of the AdEx, described by Eq. 5 and 6. In other words, one has to derive the response of the system when stimulating with a Dirac-delta function. The type of damping gives three different qualitative shapes of the kernel κ(t), which are summarized in Table 3.1 and Figure 1 A. Since the three different ﬁlters also affect the nature of the stochastic voltage ﬂuctuations, we will keep the distinction between over-damped, critically damped and under-damped scenarios throughout the paper. This means that our approach is valid for at least 3 types of diffusive voltage-noise (i.e. the white noise in Eq. 1 ﬁltered by 3 different membrane ﬁlters κ(t)). To complete the description of the deterministic voltage, we need an expression for the spiketriggered kernels. The voltage reset at each spike brings a spike-triggered jump in voltage of magˆ nitude ∆ = Vr − V (t). This perturbation is superposed to the current ﬂuctuations due to I(t) and can be mediated by a Delta-diract pulse of current. Thus we can write the voltage reset kernel by: ηv (t) = ∆ ∆ [δ ∗ κ] (t) = κ(t) κ(0) κ(0) (7) where δ(t) is the Dirac-delta function. The shape of this kernel depends on κ(t) and can be computed from Table 3.1 (see Figure 1 B). Finally, the AdEx mediates spike-frequency adaptation by the jump of the second variables w. From Eq. 2 we can see that this produces a current wspike (t) = b exp (−t/τw ) that can cumulate over subsequent spikes. The effect of this current on voltage is then given by the convolution of wspike (t) with the membrane ﬁlter κ(t). Thus in the SRM framework the spike-frequency adaptation is taken into account by: ηw (t) = [wspike ∗ κ](t) (8) Again the precise form of ηw (t) depends on κ(t) and can be computed from Table 3.1 (see Figure 1 B). At this point, we would like to verify our assumption that the non-linearity for spike emission can be neglected. Fig. 1 C and D shows that the error between the voltage from Eq. 3 and the voltage from the stochastic AdEx is generally small. Moreover, we see that the main contribution to the voltage prediction error is due to the mismatch close to the spikes. However the non-linearity for spike initiation may change the probability distribution of the voltage ﬂuctuations, which in turn inﬂuences the probability of spiking. This will inﬂuence the choice of the link-function, as we will see in the next section. 3.2 Spike Generation Using κ(t), ηv (t) and ηw (t), we must relate the spiking probability of the stochastic AdEx as a function of its deterministic voltage. According to [2] the probability of spiking in time bin dt given the deterministic voltage V (t) is given by: p(V ) = prob{spike in [t, t + dt]} = 1 − exp (−f (V (t))dt) (9) where f (·) gives the ﬁring intensity as a function of the deterministic V (t) (Eq. 3). Thus to extract the link-function f we have to compute the probability of spiking given V (t) for our SRM. To do so we apply the method proposed by Jolivet et al. (2004) [18], where the probability of spiking is simply given by the distribution of the deterministic voltage estimated at the spike times divided by the distribution of the SRM voltage when there is no spike (see ﬁgure 2 A). One can numerically compute these two quantities for our models using N repetitions of the same stimulus. 4 Table 1: Analytical expressions for the membrane ﬁlter κ(t) in terms of the parameters of the AdEx for over-, critically-, and under-damped cases. Membrane Filter: κ(t) over-damped if: (τm + τw )2 > 4τm τw (gl +a) gl κ(t) = k1 eλ1 t + k2 eλ2 t λ1 = 1 2τm τw (−(τm + τw ) + critically-damped if: (τm + τw )2 = 4τm τw (gl +a) gl κ(t) = (αt + β)eλt λ= under-damped if: (τm + τw )2 < 4τm τw (gl +a) gl κ(t) = (k1 cos (ωt) + k2 sin (ωt)) eλt −(τm +τw ) 2τm τw λ= −(τm +τw ) 2τm τw (τm + τw )2 − 4 τm τw (gl + a) gl λ2 = 1 2τm τw (−(τm + τw ) − α= τm −τw 2Cτm τw ω= τw −τm 2τm τw 2 − a g l τm τw (τm + τw )2 − 4 τm τw (gl + a) gl k1 = −(1+(τm λ2 )) Cτm (λ1 −λ2 ) k2 = 1+(τm λ1 ) Cτm (λ1 −λ2 ) β= 1 C k1 = k2 = 1 C −(1+τm λ) Cωτm The standard deviation σ of the noise and the parameter ∆T of the AdEx non-linearity may affect the shape of the link-function. We thus extract p(V ) for different σ and ∆T (Fig. 2 B). Then using visual heuristics and previous knowledge about the potential analytical expression of the link-funtion, we try to ﬁnd a simple analytical function that captures p(V ) for a large range of combinations of σ and ∆T . We observed that the log(− log(p)) is close to linear in most studied conditions Fig. 2 B suggesting the following two distributions of p(V ): V − VT (10) p(V ) = 1 − exp − exp ∆V V − VT p(V ) = exp − exp − (11) ∆V Once we have p(V ), we can use Eq. 4 to obtain the equivalent SRM link-function, which leads to: −1 f (V ) = log (1 − p(V )) (12) dt Then the two potential link-functions of the SRM can be derived from Eq. 10 and Eq. 11 (respectively): V − VT f (V ) = λ0 exp (13) ∆V V − VT (14) f (V ) = −λ0 log 1 − exp − exp − ∆V 1 with λ0 = dt , VT the threshold of the SRM and ∆V the sharpness of the link-function (i.e. the parameters that governs the degree of the stochasticity). Note that the exact value of λ0 has no importance since it is redundant with VT . Eq. 13 is the standard exponential link-function, but we call Eq. 14 the log-exp-exp link-function. 3.3 Prediction The next point is to evaluate the ﬁt quality of each link-function. To do this, we ﬁrst estimate the parameters VT and ∆V of the GLM link-function that maximize the likelihood of observing a spike 5 Figure 2: SRM link-function. A. Histogram of the SRM voltage at the AdEx ﬁring times (red) and at non-ﬁring times (gray). The ratio of the two distributions gives p(V ) (Eq. 9, dashed lines). Inset, zoom to see the voltage histogram evaluated at the ﬁring time (red). B. log(− log(p)) as a function of the SRM voltage for three different noise levels σ = 0.07, 0.14, 0.18 nA (pale gray, gray, black dots, respectively) and ∆T = 1 mV. The line is a linear ﬁt corresponding to the log-exp-exp linkfunction and the dashed line corresponds to a ﬁt with the exponential link-function. C. Same data and labeling scheme as B, but plotting f (V ) according to Eq. 12. The lines are produced with Eq. 14 with parameters ﬁtted as described in B. and the dashed lines are produced with Eq. 13. Inset, same plot but on a semi-log(y) axis. train generated with an AdEx. Second we look at the predictive power of the resulting SRM in terms of Peri-Stimulus Time Histogram (PSTH). In other words we ask how close the spike trains generated with a GLM are from the spike train generated with a stochastic AdEx when both models are stimulated with the same input current. For any GLM with link-function f (V ) ≡ f (t|I, θ) and parameters θ regulating the shape of κ(t), ˆ ηv (t) and ηw (t), the Negative Log-Likelihood (NLL) of observing a spike-train {t} is given by:   NLL = − log(f (t|I, θ)) − f (t|I, θ) (15) t ˆ t It has been shown that the negative log-likelihood is convex in the parameters if f is convex and logconcave [19]. It is easy to show that a linear-rectiﬁer link-function, the exponential link-function and the log-exp-exp link-function all satisfy these conditions. This allows efﬁcient estimation of ˆ ˆ the optimal parameters VT and ∆V using a simple gradient descent. One can thus estimate from a single AdEx spike train the optimal parameters of a given link-function, which is more efﬁcient than the method used in Sect. 3.2. The minimal NLL resulting from the gradient descent gives an estimation of the ﬁt quality. A better estimate of the ﬁt quality is given by the distance between the PSTHs in response to stimuli not used for parameter ﬁtting . Let ν1 (t) be the PSTH of the AdEx, and ν2 (t) be the PSTH of the ﬁtted SRM, 6 Figure 3: PSTH prediction. A. Injected current. B. Voltage traces produced by an AdEx (black) and the equivalent SRM (red), when stimulated with the current in A. C. Raster plot for 20 realizations of AdEx (black tick marks) and equivalent SRM (red tick marks). D. PSTH of the AdEx (black) and the SRM (red) obtained by averaging 10,000 repetitions. E. Optimal log-likelihood for the three cases of the AdEx, using three different link-functions, a linear-rectiﬁer (light gray), an exponential link-function (gray) and the link-function deﬁned by Eq. 14 (dark gray), these values are obtained by averaging over 40 different combinations σ and ∆T (see Fig. 4). Error bars are one standard deviation, the stars denote a signiﬁcant difference, two-sample t-test with α = 0.01. F. same as E. but for Md (Eq. 16). then we use Md ∈ [0, 1] as a measure of match: Md = 2 2 (ν1 (t) − ν2 (t)) dt ν1 (t)2 dt + ν2 (t)2 dt (16) Md = 1 means that it is impossible to differentiate the SRM from the AdEx in terms of their PSTHs, whereas a Md of 0 means that the two PSTHs are completely different. Thus Md is a normalized similarity measure between two PSTHs. In practice, Md is estimated from the smoothed (boxcar average of 1 ms half-width) averaged spike train of 1 000 repetitions for each models. We use both the NLL and Md to quantify the ﬁt quality for each of the three damping cases and each of the three link-functions. Figure 3 shows the match between the stochastic AdEx used as a reference and the derived GLM when both are stimulated with the same input current (Fig. 3 A). The resulting voltage traces are almost identical (Fig. 3 B) and both models predict almost the same spike trains and so the same PSTHs (Fig. 3 C and D). More quantitalively, we see on Fig. 3 E and F, that the linear-rectiﬁer ﬁts signiﬁcantly worse than both the exponential and log-exp-exp link-functions, both in terms of NLL and of Md . The exponential link-function performs as well as the log-exp-exp link-function, with a spike train similarity measure Md being almost 1 for both. Finally the likelihood-based method described above gives us the opportunity to look at the relationship between the AdEx parameters σ and ∆T that governs its spike emission and the parameters VT and ∆V of the link-function (Fig. 4). We observe that an increase of the noise level produces a ﬂatter link-function (greater ∆V ) while an increase in ∆T also produces an increase in ∆V and VT (note that Fig. 4 shows ∆V and VT for the exponential link-function only, but equivalent results are obtained with the log-exp-exp link-function). 4 Discussion In Sect. 3.3 we have shown that it is possible to predict with almost perfect accuracy the PSTH of a stochastic AdEx model using an appropriate set of parameters in the SRM. Moreover, since 7 Figure 4: Inﬂuence of the AdEx parameters on the parameters of the exponential link-function. A. VT as a function of ∆T and σ. B. ∆V as a function of ∆T and σ. the subthreshold voltage of the AdEx also gives a good match with the deterministic voltage of the SRM, we expect that the AdEx and the SRM will not differ in higher moments of the spike train probability distributions beyond the PSTH. We therefore conclude that diffusive noise models of the type of Eq. 1-2 are equivalent to GLM of the type of Eq. 3-4. Once combined with similar results on other types of stochastic LIF (e.g. correlated noise), we could bridge the gap between the literature on GLM and the literature on diffusive noise models. Another noteworthy observation pertains to the nature of the link-function. The link-function has been hypothesized to be a linear-rectiﬁer, an exponential, a sigmoidal or a Gaussian [16]. We have observed that for the AdEx the link-function follows Eq. 14 that we called the log-exp-exp linkfunction. Although the link-function is log-exp-exp for most of the AdEx parameters, the exponential link-function gives an equivalently good prediction of the PSTH. This can be explained by the fact that the difference between log-exp-exp and exponential link-functions happens mainly at low voltage (i.e. far from the threshold), where the probability of emitting a spike is so low (Figure 2 C, until -50 mv). Therefore, even if the exponential link-function overestimates the ﬁring probability at these low voltages it rarely produces extra spikes. At voltages closer to the threshold, where most of the spikes are emitted, the two link-functions behave almost identically and hence produce the same PSTH. The Gaussian link-function can be seen as lying in-between the exponential link-function and the log-exp-exp link-function in Fig. 2. This means that the work of Plesser and Gerstner (2000) [16] is in agreement with the results presented here. The importance of the time-derivative of the ˙ voltage stressed by Plesser and Gerstner (leading to a two-dimensional link-function f (V, V )) was not studied here to remain consistent with the typical usage of GLM in neural systems [14]. Finally we restricted our study to exponential non-linearity for spike initiation and do not consider other cases such as the Quadratic Integrate-and-ﬁre (QIF, [5]) or other polynomial functional shapes. We overlooked these cases for two reasons. First, there are many evidences that the non-linearity in neurons (estimated from in-vitro recordings of Pyramidal neurons) is well approximated by a single exponential [9]. Second, the exponential non-linearity of the AdEx only affects the subthreshold voltage at high voltage (close to threshold) and thus can be neglected to derive the ﬁlters κ(t) and η(t). Polynomial non-linearities on the other hand affect a larger range of the subthreshold voltage so that it would be difﬁcult to justify the linearization of subthreshold dynamics essential to the method presented here. References [1] R. B. Stein, “Some models of neuronal variability,” Biophys J, vol. 7, no. 1, pp. 37–68, 1967. [2] W. Gerstner and W. Kistler, Spiking neuron models. Cambridge University Press New York, 2002. [3] E. Izhikevich, “Resonate-and-ﬁre neurons,” Neural Networks, vol. 14, no. 883-894, 2001. [4] M. J. E. Richardson, N. Brunel, and V. Hakim, “From subthreshold to ﬁring-rate resonance,” Journal of Neurophysiology, vol. 89, pp. 2538–2554, 2003. 8 [5] E. Izhikevich, “Simple model of spiking neurons,” IEEE Transactions on Neural Networks, vol. 14, pp. 1569–1572, 2003. [6] S. Mensi, R. Naud, M. Avermann, C. C. H. Petersen, and W. Gerstner, “Parameter extraction and classiﬁcation of three neuron types reveals two different adaptation mechanisms,” Under review. [7] N. Fourcaud-Trocme, D. Hansel, C. V. Vreeswijk, and N. Brunel, “How spike generation mechanisms determine the neuronal response to ﬂuctuating inputs,” Journal of Neuroscience, vol. 23, no. 37, pp. 11 628–11 640, 2003. [8] R. Brette and W. Gerstner, “Adaptive exponential integrate-and-ﬁre model as an effective description of neuronal activity,” Journal of Neurophysiology, vol. 94, pp. 3637–3642, 2005. [9] L. Badel, W. Gerstner, and M. Richardson, “Dependence of the spike-triggered average voltage on membrane response properties,” Neurocomputing, vol. 69, pp. 1062–1065, 2007. [10] P. McCullagh and J. A. Nelder, Generalized linear models, 2nd ed. Chapman & Hall/CRC, 1998, vol. 37. [11] W. Gerstner, J. van Hemmen, and J. Cowan, “What matters in neuronal locking?” Neural computation, vol. 8, pp. 1653–1676, 1996. [12] D. Hubel and T. Wiesel, “Receptive ﬁelds and functional architecture of monkey striate cortex,” Journal of Physiology, vol. 195, pp. 215–243, 1968. [13] J. Pillow, L. Paninski, V. Uzzell, E. Simoncelli, and E. Chichilnisky, “Prediction and decoding of retinal ganglion cell responses with a probabilistic spiking model,” Journal of Neuroscience, vol. 25, no. 47, pp. 11 003–11 013, 2005. [14] K. Doya, S. Ishii, A. Pouget, and R. P. N. Rao, Bayesian brain: Probabilistic approaches to neural coding. The MIT Press, 2007. [15] S. Gerwinn, J. H. Macke, M. Seeger, and M. Bethge, “Bayesian inference for spiking neuron models with a sparsity prior,” in Advances in Neural Information Processing Systems, 2007. [16] H. Plesser and W. Gerstner, “Noise in integrate-and-ﬁre neurons: From stochastic input to escape rates,” Neural Computation, vol. 12, pp. 367–384, 2000. [17] J. Schemmel, J. Fieres, and K. Meier, “Wafer-scale integration of analog neural networks,” in Neural Networks, 2008. IJCNN 2008. (IEEE World Congress on Computational Intelligence). IEEE International Joint Conference on, june 2008, pp. 431 –438. [18] R. Jolivet, T. Lewis, and W. Gerstner, “Generalized integrate-and-ﬁre models of neuronal activity approximate spike trains of a detailed model to a high degree of accuracy,” Journal of Neurophysiology, vol. 92, pp. 959–976, 2004. [19] L. Paninski, “Maximum likelihood estimation of cascade point-process neural encoding models,” Network: Computation in Neural Systems, vol. 15, pp. 243–262, 2004. 9

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Shunting inhibition lurks “silently” because it gets effective only if the neuron is driven away from its resting potential. With synaptic input, the neuron decays into its equilibrium state Vrest β + Vexc gexc + Vinh ginh V∞ ≡ (2) β + gexc + ginh according to V (t) = V∞ (1 − exp(−t/τm )). Without external input, V (t 1) → Vrest . The time scale is set by τm . Without synaptic input τm ≡ Cm /β. Slowly varying inputs gexc , ginh > 0 modify the time scale to approximately τm /(1 + (gexc + ginh )/β). For highly dynamic inputs, such as in late phase of the object approach, the time scale gets dynamical as well. The ψ-model assigns synaptic inputs5 ˙ ˙ ˙ ˙ gexc (t) = ϑ(t), ϑ(t) = ζ1 ϑ(t − ∆tstim ) + (1 − ζ1 )Θ(t) (3a) e ginh (t) = [γϑ(t)] , ϑ(t) = ζ0 ϑ(t − ∆tstim ) + (1 − ζ0 )Θ(t) 1 (3b) This linear approximation gets worse with increasing Θ, but turns out to work well until short before ttc (τ adopts a minimum at tc − 0.428978 · l/|v|). 2 LGMD activity is usually monitored via its postsynaptic neuron, the Descending Contralateral Movement Detector (DCMD) neuron. This represents no problem as LGMD spikes follow DCMD spikes 1:1 under visual stimulation [22] from 300Hz [21] to at least 400Hz [24]. 3 Here we assume that the membrane potential serves as a predictor for the LGMD’s mean ﬁring rate. 4 Set to unity for all simulations. 5 LGMD receives also inhibition from a laterally acting network [21]. The η-function considers only direct feedforward inhibition [22, 6], and so do we. 2 Θ ∈ [7.63Â°, 180.00Â°[ temporal resolution ∆ tstim=1.0ms l/|v|=20.00ms, β=1.00, γ=7.50, e=3.00, ζ0=0.90, ζ1=0.99, nrelax=25 0.04 scaled dΘ/dt continuous discretized 0.035 0.03 Θ(t) (input) ϑ(t) (filtered) voltage V(t) (output) t = 56ms max t =300ms c 0.025 0 10 2 η(t): α=3.29, R =1.00 n =10 → t =37ms log Θ(t) amplitude relax max 0.02 0.015 0.01 0.005 0 −0.005 0 50 100 150 200 250 300 −0.01 0 350 time [ms] 50 100 150 200 250 300 350 time [ms] (b) ψ versus η (a) discretized optical variables Figure 1: (a) The continuous visual angle of an approaching object is shown along with its discretized version. Discretization transforms angular velocity from a continuous variable into a series of “spikes” (rescaled). (b) The ψ function with the inputs shown in a, with nrelax = 25 relaxation time steps. Its peak occurs tmax = 56ms before ttc (tc = 300ms). An η function (α = 3.29) that was ﬁtted to ψ shows good agreement. For continuous optical variables, the peak would occur 4ms earlier, and η would have α = 4.44 with R2 = 1. For nrelax = 10, ψ is farther away from its equilibrium at V∞ , and its peak moves 19ms closer to ttc. t =500ms, dia=12.0cm, ∆t c =1.00ms, dt=10.00µs, discrete=1 stim 250 n relax = 50 2 200 α=4.66, R =0.99 [normal] n = 25 relax 2 α=3.91, R =1.00 [normal] n =0 relax tmax [ms] 150 2 α=1.15, R =0.99 [normal] 100 50 0 β=1.00, γ=7.50, e=3.00, V =−0.001, ζ =0.90, ζ =0.99 inh −50 5 10 15 20 25 30 0 35 1 40 45 50 l/|v| [ms] (a) different nrelax (b) different ∆tstim ˆ ˆ Figure 2: The ﬁgures plot the relative time tmax ≡ tc − t of the response peak of ψ, V (t), as a function of half-size-to-velocity ratio (points). Line ﬁts with slope α and intercept δ were added (lines). The predicted linear relationship in all cases is consistent with experimental evidence [9]. (a) The stimulus time scale is held constant at ∆tstim = 1ms, and several LGMD time scales are deﬁned by nrelax (= number of intercalated relaxation steps for each integration time step). Bigger values of nrelax move V (t) closer to its equilibrium V∞ (t), implying higher slopes α in turn. (b) LGMD time scale is ﬁxed at nrelax = 25, and ∆tstim is manipulated. Because of the discretization of optical variables (OVs) in our simulation, increasing ∆tstim translates to an overall smaller number of jumps in OVs, but each with higher amplitude. Thus, we say ψ(t) ≡ V (t) if and only if gexc and ginh are deﬁned with the last equation. The time ˙ scale of stimulation is deﬁned by ∆tstim (by default 1ms). The variables ϑ and ϑ are lowpass ﬁltered angular size and rate of expansion, respectively. The amount of ﬁltering is deﬁned by memory constants ζ0 and ζ1 (no ﬁltering if zero). The idea is to continue with generating synaptic input ˙ after ttc, where Θ(t > tc ) = const and thus Θ(t > tc ) = 0. Inhibition is ﬁrst weighted by γ, and then potentiated by the exponent e. Hodgkin-Huxley potentiates gating variables n, m ∈ [0, 1] instead (potassium ∝ n4 , sodium ∝ m3 , [12]) and multiplies them with conductances. Gabbiani and co-workers found that the function which transforms membrane potential to ﬁring rate is better described by a power function with e = 3 than by exp(·) (Figure 4d in [8]). 3 Dynamics of the ψ-function 3 Discretization. In a typical experiment, a monitor is placed a short distance away from the insect’s eye, and an approaching object is displayed. Computer screens have a ﬁxed spatial resolution, and as a consequence size increments of the displayed object proceed in discrete jumps. The locust retina is furthermore composed of a discrete array of ommatidia units. We therefore can expect a corresponding step-wise increment of Θ with time, although optical and neuronal ﬁltering may ˙ smooth Θ to some extent again, resulting in ϑ (ﬁgure 1). Discretization renders Θ discontinuous, ˙ For simulating the dynamics of ψ, we discretized angular size what again will be alleviated in ϑ. ˙ with ﬂoor(Θ), and Θ(t) ≈ [Θ(t + ∆tstim ) − Θ(t)]/∆tstim . Discretized optical variables (OVs) were re-normalized to match the range of original (i.e. continuous) OVs. To peak, or not to peak? Rind & Simmons reject the hypothesis that the activity peak signals impending collision on grounds of two arguments [28]: (i) If Θ(t + ∆tstim ) − Θ(t) 3o in consecutively displayed stimulus frames, the illusion of an object approach would be lost. Such stimulation would rather be perceived as a sequence of rapidly appearing (but static) objects, causing reduced responses. (ii) After the last stimulation frame has been displayed (that is Θ = const), LGMD responses keep on building up beyond ttc. This behavior clearly depends on l/|v|, also according to their own data (e.g. Figure 4 in [26]): Response build up after ttc is typically observed for sufﬁ˙ ciently small values of l/|v|. Input into ψ in situations where Θ = const and Θ = 0, respectively, ˙ is accommodated by ϑ and ϑ, respectively. We simulated (i) by setting ∆tstim = 5ms, thus producing larger and more infrequent jumps in discrete OVs than with ∆tstim = 1ms (default). As a consequence, ϑ(t) grows more slowly (deˆ layed build up of inhibition), and the peak occurs later (tmax ≡ tc − t = 10ms with everything else ˆ ˆ identical with ﬁgure 1b). The peak amplitude V = V (t) decreases nearly sixfold with respect to default. Our model thus predicts the reduced responses observed by Rind & Simmons [28]. Linearity. Time of peak ﬁring rate is linearly related to l/|v| [10, 9]. The η-function is consistent ˆ with this experimental evidence: t = tc − αl/|v| + δ (e.g. α = 4.7, δ = −27ms). The ψ-function reproduces this relationship as well (ﬁgure 2), where α depends critically on the time scale of biophysical processes in the LGMD. We studied the impact of this time scale by choosing 10µs for the numerical integration of equation 1 (algorithm: 4th order Runge-Kutta). Apart from improving the numerical stability of the integration algorithm, ψ is far from its equilibrium V∞ (t) in every moment ˙ t, given the stimulation time scale ∆tstim = 1ms 6 . Now, at each value of Θ(t) and Θ(t), respectively, we intercalated nrelax iterations for integrating ψ. Each iteration takes V (t) asymptotically closer to V∞ (t), and limnrelax 1 V (t) = V∞ (t). If the internal processes in the LGMD cannot keep up with stimulation (nrelax = 0), we obtain slopes values that underestimate experimentally found values (ﬁgure 2a). In contrast, for nrelax 25 we get an excellent agreement with the experimentally determined α. This means that – under the reported experimental stimulation conditions (e.g. [9]) – the LGMD would operate relatively close to its steady state7 . Now we ﬁx nrelax at 25 and manipulate ∆tstim instead (ﬁgure 2b). The default value ∆tstim = 1ms corresponds to α = 3.91. Slightly bigger values of ∆tstim (2.5ms and 5ms) underestimate the experimental α. In addition, the line ﬁts also return smaller intercept values then. We see tmax < 0 up to l/|v| ≈ 13.5ms – LGMD activity peaks after ttc! Or, in other words, LGMD activity continues to increase after ttc. In the limit, where stimulus dynamics is extremely fast, and LGMD processes are kept far from equilibrium at each instant of the approach, α gets very small. As a consequence, tmax gets largely independent of l/|v|: The activity peak would cling to tmax although we varied l/|v|. 4 Freeze! Experimental data versus steady state of “psi” In the previous section, experimentally plausible values for α were obtained if ψ is close to equilibrium at each instant of time during stimulation. In this section we will thus introduce a steady-state 6 Assuming one ∆tstim for each integration time step. This means that by default stimulation and biophysical dynamics will proceed at identical time scales. 7 Notice that in this moment we can only make relative statements - we do not have data at hand for deﬁning absolute time scales 4 tc=500ms, v=2.00m/s ψ∞ → (β varies), γ=3.50, e=3.00, Vinh=−0.001 tc=500ms, v=2.00m/s ψ∞ → β=2.50, γ=3.50, (e varies), Vinh=−0.001 300 tc=500ms, v=2.00m/s ψ∞ → β=2.50, (γ varies), e=3.00, Vinh=−0.001 350 300 β=10.00 β=5.00 norm. rmse = 0.058...0.153 correlation (β,α)=−0.90 (n=4) ∞ β=1.00 e=4.00 norm. |η−ψ | = 0.009...0.114 e=3.00 300 norm. rmse = 0.014...0.160 correlation (e,α)=0.98 (n=4) ∞ e=2.50 250 250 norm. |η−ψ | = 0.043...0.241 ∞ norm. rmse = 0.085...0.315 correlation (γ,α)=1.00 (n=5) 150 tmax [ms] 200 tmax [ms] 200 tmax [ms] γ=5.00 γ=2.50 γ=1.00 γ=0.50 γ=0.25 e=5.00 norm. |η−ψ | = 0.020...0.128 β=2.50 250 200 150 100 150 100 100 50 50 50 0 5 10 15 20 25 30 35 40 45 0 5 50 10 15 20 l/|v| [ms] 25 30 35 40 45 0 5 50 10 15 20 l/|v| [ms] (a) β varies 25 30 35 40 45 50 l/|v| [ms] (b) e varies (c) γ varies ˆ ˆ Figure 3: Each curve shows how the peak ψ∞ ≡ ψ∞ (t) depends on the half-size-to-velocity ratio. In each display, one parameter of ψ∞ is varied (legend), while the others are held constant (ﬁgure title). Line slopes vary according to parameter values. Symbol sizes are scaled according to rmse (see also ﬁgure 4). Rmse was calculated between normalized ψ∞ (t) & normalized η(t) (i.e. both functions ∈ [0, 1] with original minimum and maximum indicated by the textbox). To this end, the ˆ peak of the η-function was placed at tc , by choosing, at each parameter value, α = |v| · (tc − t)/l (for determining correlation, the mean value of α was taken across l/|v|). tc=500ms, v=2.00m/s ψ∞ → (β varies), γ=3.50, e=3.00, Vinh=−0.001 tc=500ms, v=2.00m/s ψ∞ → β=2.50, γ=3.50, (e varies), Vinh=−0.001 tc=500ms, v=2.00m/s ψ∞ → β=2.50, (γ varies), e=3.00, Vinh=−0.001 0.25 β=5.00 0.12 β=2.50 β=1.00 0.1 0.08 (normalized η, ψ∞) 0.12 β=10.00 (normalized η, ψ∞) (normalized η, ψ∞) 0.14 0.1 0.08 γ=5.00 γ=2.50 0.2 γ=1.00 γ=0.50 γ=0.25 0.15 0.06 0.04 0.02 0 5 10 15 20 25 30 35 40 45 50 meant |η(t)−ψ∞(t)| meant |η(t)−ψ∞(t)| meant |η(t)−ψ∞(t)| 0.06 0.04 e=5.00 e=4.00 e=3.00 0.02 e=2.50 10 l/|v| [ms] 15 20 25 30 35 40 45 50 l/|v| [ms] (a) β varies (b) e varies 0.1 0.05 0 5 10 15 20 25 30 35 40 45 50 l/|v| [ms] (c) γ varies Figure 4: This ﬁgure complements ﬁgure 3. It visualizes the time averaged absolute difference between normalized ψ∞ (t) & normalized η(t). For η, its value of α was chosen such that the maxima of both functions coincide. Although not being a ﬁt, it gives a rough estimate on how the shape of both curves deviate from each other. The maximum possible difference would be one. version of ψ (i.e. equation 2 with Vrest = 0, Vexc = 1, and equations 3 plugged in), ψ∞ (t) ≡ e ˙ Θ(t) + Vinh [γΘ(t)] e ˙ β + Θ(t) + [γΘ(t)] (4) (Here we use continuous versions of angular size and rate of expansion). The ψ∞ -function makes life easier when it comes to ﬁtting experimental data. However, it has its limitations, because we brushed the whole dynamic of ψ under the carpet. Figure 3 illustrates how the linˆ ear relationship (=“linearity”) between tmax ≡ tc − t and l/|v| is inﬂuenced by changes in parameter values. Changing any of the values of e, β, γ predominantly causes variation in line slopes. The smallest slope changes are obtained by varying Vinh (data not shown; we checked Vinh = 0, −0.001, −0.01, −0.1). For Vinh −0.01, linearity is getting slightly compromised, as slope increases with l/|v| (e.g. Vinh = −1 α ∈ [4.2, 4.7]). In order to get a notion about how well the shape of ψ∞ (t) matches η(t), we computed timeaveraged difference measures between normalized versions of both functions (details: ﬁgure 3 & 4). Bigger values of β match η better at smaller, but worse at bigger values of l/|v| (ﬁgure 4a). Smaller β cause less variation across l/|v|. As to variation of e, overall curve shapes seem to be best aligned with e = 3 to e = 4 (ﬁgure 4b). Furthermore, better matches between ψ∞ (t) and η(t) correspond to bigger values of γ (ﬁgure 4c). And ﬁnally, Vinh marches again to a different tune (data not shown). Vinh = −0.1 leads to the best agreement (≈ 0.04 across l/|v|) of all Vinh , quite different from the other considered values. For the rest, ψ∞ (t) and η(t) align the same (all have maximum 0.094), 5 ˙ (a) Θ = 126o /s ˙ (b) Θ = 63o /s Figure 5: The original data (legend label “HaGaLa95”) were resampled from ref. [10] and show ˙ DCMD responses to an object approach with Θ = const. Thus, Θ increases linearly with time. The η-function (ﬁtting function: Aη(t+δ)+o) and ψ∞ (ﬁtting function: Aψ∞ (t)+o) were ﬁtted to these data: (a) (Figure 3 Di in [10]) Good ﬁts for ψ∞ are obtained with e = 5 or higher (e = 3 R2 = 0.35 and rmse = 0.644; e = 4 R2 = 0.45 and rmse = 0.592). “Psi” adopts a sigmoid-like curve form which (subjectively) appears to ﬁt the original data better than η. (b) (Figure 3 Dii in [10]) “Psi” yields an excellent ﬁt for e = 3. RoHaTo10 gregarious locust LV=0.03s Θ(t), lv=30ms e011pos014 sgolay with 100 t =107ms max ttc=5.00s ψ adj.R2 0.95 (LM:3) ∞ η(t) adj.R2 1 (TR::1) 2 ψ : R =0.95, rmse=0.004, 3 coefficients ∞ → β=2.22, γ=0.70, e=3.00, V =−0.001, A=0.07, o=0.02, δ=0.00ms inh η: R2=1.00, rmse=0.001 → α=3.30, A=0.08, o=0.0, δ=−10.5ms 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 time [s] (b) α versus β (a) spike trace Figure 6: (a) DCMD activity in response to a black square (l/|v| = 30ms, legend label “e011pos14”, ref. [30]) approaching to the eye center of a gregarious locust (ﬁnal visual angle 50o ). Data show the ﬁrst stimulation so habituation is minimal. The spike trace (sampled at 104 Hz) was full wave rectiﬁed, lowpass ﬁltered, and sub-sampled to 1ms resolution. Firing rate was estimated with Savitzky-Golay ﬁltering (“sgolay”). The ﬁts of the η-function (Aη(t + δ) + o; 4 coefﬁcients) and ψ∞ -function (Aψ∞ (t) with ﬁxed e, o, δ, Vinh ; 3 coefﬁcients) provide both excellent ﬁts to ﬁring rate. (b) Fitting coefﬁcient α (→ η-function) inversely correlates with β (→ ψ∞ ) when ﬁtting ﬁring rates of another 5 trials as just described (continuous line = line ﬁt to the data points). Similar correlation values would be obtained if e is ﬁxed at values e = 2.5, 4, 5 c = −0.95, −0.96, −0.91. If o was determined by the ﬁtting algorithm, then c = −0.70. No clear correlations with α were obtained for γ. despite of covering different orders of magnitude with Vinh = 0, −0.001, −0.01. Decelerating approach. Hatsopoulos et al. [10] recorded DCMD activity in response to an ap˙ proaching object which projected image edges on the retina moving at constant velocity: Θ = const. ˙ This “linear approach” is perceived as if the object is getting increasingly implies Θ(t) = Θ0 + Θt. slower. But what appears a relatively unnatural movement pattern serves as a test for the functions η & ψ∞ . Figure 5 illustrates that ψ∞ passes the test, and consistently predicts that activity sharply rises in the initial approach phase, and subsequently declines (η passed this test already in the year 1995). 6 Spike traces. We re-sampled about 30 curves obtained from LGMD recordings from a variety of publications, and ﬁtted η & ψ∞ -functions. We cannot show the results here, but in terms of goodness of ﬁt measures, both functions are in the same ballbark. Rather, ﬁgure 6a shows a representative example [30]. When α and β are plotted against each other for ﬁve trials, we see a strong inverse correlation (ﬁgure 6b). Although ﬁve data points are by no means a ﬁrm statistical sample, the strong correlation could indicate that β and α play similar roles in both functions. Biophysically, β is the leakage conductance, which determines the (passive) membrane time constant τm ∝ 1/β of the neuron. Voltage drops within τm to exp(−1) times its initial value. Bigger values of β mean shorter τm (i.e., “faster neurons”). Getting back to η, this would suggest α ∝ τm , such that higher (absolute) values for α would possibly indicate a slower dynamic of the underlying processes. 5 Discussion (“The Good, the Bad, and the Ugly”) Up to now, mainly two classes of LGMD models existed: The phenomenological η-function on the one hand, and computational models with neuronal layers presynaptic to the LGMD on the other (e.g. [25, 15]; real-world video sequences & robotics: e.g. [3, 14, 32, 2]). Computational models predict that LGMD response features originate from excitatory and inhibitory interactions in – and between – presynaptic neuronal layers. Put differently, non-linear operations are generated in the presynaptic network, and can be a function of many (model) parameters (e.g. synaptic weights, time constants, etc.). In contrast, the η-function assigns concrete nonlinear operations to the LGMD [7]. The η-function is accessible to mathematical analysis, whereas computational models have to be probed with videos or artiﬁcial stimulus sequences. The η-function is vague about biophysical parameters, whereas (good) computational models need to be precise at each (model) parameter value. The η-function establishes a clear link between physical stimulus attributes and LGMD activity: It postulates what is to be computed from the optical variables (OVs). But in computational models, such a clear understanding of LGMD inputs cannot always be expected: Presynaptic processing may strongly transform OVs. The ψ function thus represents an intermediate model class: It takes OVs as input, and connects them with biophysical parameters of the LGMD. For the neurophysiologist, the situation could hardly be any better. Psi implements the multiplicative operation of the η-function by shunting inhibition (equation 1: Vexc ≈ Vrest and Vinh ≈ Vrest ). The η-function ﬁts ψ very well according to our dynamical simulations (ﬁgure 1), and satisfactory by the approximate criterion of ﬁgure 4. We can conclude that ψ implements the η-function in a biophysically plausible way. However, ψ does neither explicitly specify η’s multiplicative operation, nor its exponential function exp(·). Instead we have an interaction between shunting inhibition and a power law (·)e , with e ≈ 3. So what about power laws in neurons? Because of e > 1, we have an expansive nonlinearity. Expansive power-law nonlinearities are well established in phenomenological models of simple cells of the primate visual cortex [1, 11]. Such models approximate a simple cell’s instantaneous ﬁring rate r from linear ﬁltering of a stimulus (say Y ) by r ∝ ([Y ]+ )e , where [·]+ sets all negative values to zero and lets all positive pass. Although experimental evidence favors linear thresholding operations like r ∝ [Y − Ythres ]+ , neuronal responses can behave according to power law functions if Y includes stimulus-independent noise [19]. Given this evidence, the power-law function of the inhibitory input into ψ could possibly be interpreted as a phenomenological description of presynaptic processes. The power law would also be the critical feature by means of which the neurophysiologist could distinguish between the η function and ψ. A study of Gabbiani et al. aimed to provide direct evidence for a neuronal implementation of the η-function [8]. Consequently, the study would be an evidence ˙ for a biophysical implementation of “direct” multiplication via log Θ − αΘ. Their experimental evidence fell somewhat short in the last part, where “exponentation through active membrane conductances” should invert logarithmic encoding. Speciﬁcally, the authors observed that “In 7 out of 10 neurons, a third-order power law best described the data” (sixth-order in one animal). Alea iacta est. Acknowledgments MSK likes to thank Stephen M. Rogers for kindly providing the recording data for compiling ﬁgure 6. MSK furthermore acknowledges support from the Spanish Government, by the Ramon and Cajal program and the research grant DPI2010-21513. 7 References [1] D.G. Albrecht and D.B. Hamilton, Striate cortex of monkey and cat: contrast response function, Journal of Neurophysiology 48 (1982), 217–237. [2] S. Bermudez i Badia, U. Bernardet, and P.F.M.J. Verschure, Non-linear neuronal responses as an emergent property of afferent networks: A case study of the locust lobula giant movemement detector, PLoS Computational Biology 6 (2010), no. 3, e1000701. [3] M. Blanchard, F.C. Rind, and F.M.J. Verschure, Collision avoidance using a model of locust LGMD neuron, Robotics and Autonomous Systems 30 (2000), 17–38. [4] D.F. Cooke and M.S.A. Graziano, Super-ﬂinchers and nerves of steel: Defensive movements altered by chemical manipulation of a cortical motor area, Neuron 43 (2004), no. 4, 585–593. [5] L. Fogassi, V. Gallese, L. Fadiga, G. Luppino, M. Matelli, and G. Rizzolatti, Coding of peripersonal space in inferior premotor cortex (area f4), Journal of Neurophysiology 76 (1996), 141–157. [6] F. Gabbiani, I. Cohen, and G. Laurent, Time-dependent activation of feed-forward inhibition in a looming sensitive neuron, Journal of Neurophysiology 94 (2005), 2150–2161. [7] F. Gabbiani, H.G. Krapp, N. Hatsopolous, C.H. Mo, C. Koch, and G. Laurent, Multiplication and stimulus invariance in a looming-sensitive neuron, Journal of Physiology - Paris 98 (2004), 19–34. [8] F. Gabbiani, H.G. Krapp, C. Koch, and G. Laurent, Multiplicative computation in a visual neuron sensitive to looming, Nature 420 (2002), 320–324. [9] F. Gabbiani, H.G. Krapp, and G. Laurent, Computation of object approach by a wide-ﬁeld, motionsensitive neuron, Journal of Neuroscience 19 (1999), no. 3, 1122–1141. [10] N. Hatsopoulos, F. Gabbiani, and G. Laurent, Elementary computation of object approach by a wide-ﬁeld visual neuron, Science 270 (1995), 1000–1003. [11] D.J. Heeger, Modeling simple-cell direction selectivity with normalized, half-squared, linear operators, Journal of Neurophysiology 70 (1993), 1885–1898. [12] A.L. Hodkin and A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, Journal of Physiology 117 (1952), 500–544. [13] F. Hoyle, The black cloud, Pinguin Books, London, 1957. [14] M.S. Keil, E. Roca-Morena, and A. Rodr´guez-V´ zquez, A neural model of the locust visual system for ı a detection of object approaches with real-world scenes, Proceedings of the Fourth IASTED International Conference (Marbella, Spain), vol. 5119, 6-8 September 2004, pp. 340–345. [15] M.S. Keil and A. Rodr´guez-V´ zquez, Towards a computational approach for collision avoidance with ı a real-world scenes, Proceedings of SPIE: Bioengineered and Bioinspired Systems (Maspalomas, Gran Canaria, Canary Islands, Spain) (A. Rodr´guez-V´ zquez, D. Abbot, and R. Carmona, eds.), vol. 5119, ı a SPIE - The International Society for Optical Engineering, 19-21 May 2003, pp. 285–296. [16] J.G. King, J.Y. Lettvin, and E.R. Gruberg, Selective, unilateral, reversible loss of behavioral responses to looming stimuli after injection of tetrodotoxin or cadmium chloride into the frog optic nerve, Brain Research 841 (1999), no. 1-2, 20–26. [17] C. Koch, Biophysics of computation: information processing in single neurons, Oxford University Press, New York, 1999. [18] D.N. Lee, A theory of visual control of braking based on information about time-to-collision, Perception 5 (1976), 437–459. [19] K.D. Miller and T.W. Troyer, Neural noise can explain expansive, power-law nonlinearities in neuronal response functions, Journal of Neurophysiology 87 (2002), 653–659. [20] Hideki Nakagawa and Kang Hongjian, Collision-sensitive neurons in the optic tectum of the bullfrog, rana catesbeiana, Journal of Neurophysiology 104 (2010), no. 5, 2487–2499. [21] M. O’Shea and C.H.F. Rowell, Projection from habituation by lateral inhibition, Nature 254 (1975), 53– 55. [22] M. O’Shea and J.L.D. Williams, The anatomy and output connection of a locust visual interneurone: the lobula giant movement detector (lgmd) neurone, Journal of Comparative Physiology 91 (1974), 257–266. [23] S. Peron and F. Gabbiani, Spike frequency adaptation mediates looming stimulus selectivity, Nature Neuroscience 12 (2009), no. 3, 318–326. [24] F.C. Rind, A chemical synapse between two motion detecting neurones in the locust brain, Journal of Experimental Biology 110 (1984), 143–167. [25] F.C. Rind and D.I. Bramwell, Neural network based on the input organization of an identiﬁed neuron signaling implending collision, Journal of Neurophysiology 75 (1996), no. 3, 967–985. 8 [26] F.C. Rind and P.J. Simmons, Orthopteran DCMD neuron: a reevaluation of responses to moving objects. I. Selective responses to approaching objects, Journal of Neurophysiology 68 (1992), no. 5, 1654–1666. [27] , Orthopteran DCMD neuron: a reevaluation of responses to moving objects. II. Critical cues for detecting approaching objects, Journal of Neurophysiology 68 (1992), no. 5, 1667–1682. [28] , Signaling of object approach by the dcmd neuron of the locust, Journal of Neurophysiology 77 (1997), 1029–1033. [29] , Reply, Trends in Neuroscience 22 (1999), no. 5, 438. [30] S.M. Roger, G.W.J. Harston, F. Kilburn-Toppin, T. Matheson, M. Burrows, F. Gabbiani, and H.G. Krapp, Spatiotemporal receptive ﬁeld properties of a looming-sensitive neuron in solitarious and gregarious phases of desert locust, Journal of Neurophysiology 103 (2010), 779–792. [31] S.K. Rushton and J.P. Wann, Weighted combination of size and disparity: a computational model for timing ball catch, Nature Neuroscience 2 (1999), no. 2, 186–190. [32] Yue. S., Rind. F.C., M.S. Keil, J. Cuadri, and R. Stafford, A bio-inspired visual collision detection mechanism for cars: Optimisation of a model of a locust neuron to a novel environment, Neurocomputing 69 (2006), 1591–1598. [33] G.R. Schlotterer, Response of the locust descending movement detector neuron to rapidly approaching and withdrawing visual stimuli, Canadian Journal of Zoology 55 (1977), 1372–1376. [34] H. Sun and B.J. Frost, Computation of different optical variables of looming objects in pigeon nucleus rotundus neurons, Nature Neuroscience 1 (1998), no. 4, 296–303. [35] J.R. Tresilian, Visually timed action: time-out for ’tau’?, Trends in Cognitive Sciences 3 (1999), no. 8, 1999. [36] Y. Wang and B.J. Frost, Time to collision is signalled by neurons in the nucleus rotundus of pigeons, Nature 356 (1992), 236–238. [37] J.P. Wann, Anticipating arrival: is the tau-margin a specious theory?, Journal of Experimental Psychology and Human Perceptual Performance 22 (1979), 1031–1048. [38] M. Wicklein and N.J. Strausfeld, Organization and signiﬁcance of neurons that detect change of visual depth in the hawk moth manduca sexta, The Journal of Comparative Neurology 424 (2000), no. 2, 356– 376. 9

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