{"title": "Channel Noise in Excitable Neural Membranes", "book": "Advances in Neural Information Processing Systems", "page_first": 143, "page_last": 149, "abstract": null, "full_text": "Channel Noise in Excitable Neuronal \n\nMembranes \n\nAmit Manwani; Peter N. Steinmetz and Christof Koch \nComputation and Neural Systems Program, M-S 139-74 \nCalifornia Institute of Technology Pasadena, CA 91125 \n\n{ quixote,peter,koch } @klab.caltech.edu \n\nAbstract \n\nStochastic fluctuations of voltage-gated ion channels generate current \nand voltage noise in neuronal membranes. This noise may be a criti(cid:173)\ncal determinant of the efficacy of information processing within neural \nsystems. Using Monte-Carlo simulations, we carry out a systematic in(cid:173)\nvestigation of the relationship between channel kinetics and the result(cid:173)\ning membrane voltage noise using a stochastic Markov version of the \nMainen-Sejnowski model of dendritic excitability in cortical neurons. \nOur simulations show that kinetic parameters which lead to an increase \nin membrane excitability (increasing channel densities, decreasing tem(cid:173)\nperature) also lead to an increase in the magnitude of the sub-threshold \nvoltage noise. Noise also increases as the membrane is depolarized from \nrest towards threshold. This suggests that channel fluctuations may in(cid:173)\nterfere with a neuron's ability to function as an integrator of its synaptic \ninputs and may limit the reliability and precision of neural information \nprocessing. \n\n1 Introduction \n\nVoltage-gated ion channels undergo random transitions between different conformational \nstates due to thermal agitation. Generally, these states differ in their ionic permeabilities \nand the stochastic transitions between them give rise to conductance fluctuations which \nare a source of membrane noise [1]. In excitable cells, voltage-gated channel noise can \ncontribute to the generation of spontaneous action potentials [2, 3], and the variability of \nspike timing [4] . Channel fluctuations can also give rise to bursting and chaotic spiking \ndynamics in neurons [5, 6] . \n\nOur interest in studying membrane noise is based on the thesis that noise ultimately limits \nthe ability of neurons to transmit and process information. To study this problem, we com(cid:173)\nbine methods from information theory, membrane biophysics and compartmental neuronal \nmodeling to evaluate ability of different biophysical components of a neuron, such as the \nsynapse, the dendritic tree, the soma and so on, to transmit information [7, 8, 9]. These \nneuronal components differ in the type, density, and kinetic properties of their constituent \nion channels. Thus, measuring the impact of these differences on membrane noise rep-\n\n\u2022 http://www.klab.caltech.edwquixote \n\n\f144 \n\nA. Manwani, P. N. Steinmetz and C. Koch \n\nresents a fundamental step in our overall program of evaluating information transmission \nwithin and between neurons. \n\nAlthough information in the nervous system is mostly communicated in the form of action \npotentials, we first direct our attention to the study of sub-threshold voltage fluctuations for \nthree reasons. Firstly, voltage fluctuations near threshold can cause variability in spike tim(cid:173)\ning and thus directly influence the reliability and precision of neuronal activity. Secondly, \nmany computations putatively performed in the dendritic tree (coincidence detection, mul(cid:173)\ntiplication, synaptic integration and so on) occur in the sub-threshold regime and thus are \nlikely to be influenced by sub-threshold voltage noise. Lastly, several sensory neurons in \nvertebrates and invertebrates are non-spiking and an analysis of voltage fluctuations can be \nused to study information processing in these systems as well. \n\nExtensive investigations of channel noise were carried out prior to the advent of the patch(cid:173)\nclamp technique in order to provide indirect evidence for the existence of single ion chan(cid:173)\nnels (see [1] for an excellentreview). More recently, theoretical studies have focused on the \neffect of random channel fluctuations on spike timing and reliability of individual neurons \n[4], as well as their effect on the dynamics of interconnected networks of neurons [5, 6). \nIn this paper, we determine the effect of varying the kinetic parameters, such as channel \ndensity and the rate of channel transitions, on the magnitude of sub-threshold voltage noise \nin an iso-potential membrane patches containing stochastic voltage-gated ion channels us(cid:173)\ning Monte-Carlo simulations. The simulations are based on the Mainen-Sejnowski (MS) \nkinetic model of active channels in the dendrites of cortical pyramidal neurons [10). By \nvarying two model parameters (channel densities and temperature), we investigate the rela(cid:173)\ntionship between excitability and noise in neuronal membranes. By linearizing the channel \nkinetics, we derive analytical expressions which provide closed-form estimates of noise \nmagnitudes; we contrast the results of the simulations with the linearized expressions to \ndetermine the parameter range over which they can be used. \n\n2 Monte-Carlo Simulations \n\nConsider an iso-potential membrane patch containing voltage-gated K+and Na+channels \nand leak channels, \n\n-c dt = 9K (Vm - EK) + 9Na (Vm - ENa + 9L Vm - Ed + Iinj \n\ndVrn \n\n) \n\n( \n\n(1) \n\nwhere C is the membrane capacitance and 9K (9Na, 9L) and EK (ENa, EL) denote the \nK+(Na+, leak) conductance and the K+(Na+, leak) reversal potential respectively. Current \ninjected into the patch is denoted by Iinj , with the convention that inward current is nega(cid:173)\ntive. The channels which give rise to potassium and sodium conductances switch randomly \nbetween different conformational states with voltage-dependent transition rates. Thus,9K \nand 9Na are voltage-dependent random processes and eq. 1 is a non-linear stochastic differ(cid:173)\nential equation. Generally, ion channel transitions are assumed to be Markovian [11] and \nthe stochastic dynamics of eq. 1 can be studied using Monte-Carlo simulations of finite(cid:173)\nstate Markov models of channel kinetics. \n\nEarlier studies have carried out simulations of stochastic versions of the classical Hodgkin(cid:173)\nHuxley kinetic model [12] to study the effect of conductance fluctuations on neuronal \nspiking [13, 2, 4]. Since we are interested in sub-threshold voltage noise, we consider \na stochastic Markov version of a less excitable kinetic model used to describe dendrites of \ncortical neurons [10]. We shall refer to it as the Mainen-Sejnowski (MS) kinetic scheme. \nThe K+conductance is modeled by a single activation sub-unit (denoted by n) whereas \nthe Na+conductance is comprised of three identical activation sub-units (denoted by m) \nand one inactivation sub-unit (denoted by h). Thus, the stochastic discrete-state Markov \nmodels of the K+and Na+channel have 2 and 8 states respectively (shown in Fig. 1). The \n\n\fChannel Noise in Excitable Neural Membranes \n\n145 \n\nsingle channel conductances and the densities of the ion channels (K+ ,Na+) are denoted \nby (,K,''(Na) and ('TJK,'f)Na) respectively. Thus, 9K and 9Na) are given by the products of \nthe respective single channel conductances and the corresponding numbers of channels in \nthe conducting states. \n\nA \n\nWe performed Monte-Carlo simulations \nof the MS kinetic scheme using a fixed \ntime step of i).t = 10 J.tsec. During each \nstep, the number of sub-units undergo(cid:173)\ning transitions between states i and j \nwas determined by drawing a pseudo(cid:173)\nrandom binomial deviate (bnldev sub(cid:173)\nroutine [14] driven by the ran2 subrou(cid:173)\ntine of the 2nd edition) with N equal \nto the number of sub-units in state i \nand p given by the conditional proba(cid:173)\nbility of the transition between i and j. \nAfter updating the number of channels \nin each state, eq. 1 was integrated us(cid:173)\ning fourth order Runge-Kutta integration \nwith adaptive step size control [14]. Dur-\ning each step, the channel conductances \nwere held at the fixed value corresponding to the new numbers of open channels. (See [4] \nfor details ofthis procedure). \n\nFigure I: Kinetic scheme for the voltage-gated \nMainen-Sejnowski K+(A) and Na+(B) channels. \nno and nl represent the closed and open states of \nK+channel. mO-2hl represent the 3 closed states, \nmO-3ho the four inactivated states and m3hl the \nopen state of the N a + channel. \n\n6 \n\n4 \n\n-4 \n\n-6 \n\nDue to random channel transitions, the \nmembrane voltage fluctuates around the \nsteady-state resting membrane voltage \nVrest . By varying the magnitude of \nthe constant injected current linj, the \nsteady-state voltage can be varied over a \nbroad range, which depends on the chan(cid:173)\nnel densities. The current required to \nmaintain the membrane at a holding volt(cid:173)\nage Vhold can be determined from the \nsteady-state I-V curve of the system, as \nshown in Fig. 2. Voltages for which \nthe slope of the I-V curve is negative \ncannot be maintained as steady-states. \nBy injecting an external current to offset \nthe total membrane current, a fixed point \nin the negative slope region can be ob(cid:173)\ntained but since the fixed point is unsta(cid:173)\nble, any perturbation, such as a stochastic \nion channel opening or closing, causes \nthe system to be driven to the closest sta(cid:173)\nble fixed point. We measured sub-threshold voltage noise only for stable steady-state hold(cid:173)\ning voltages. A typical voltage trace from our simulations is shown in Fig. 3. To estimate \nthe standard deviation of the voltage noise accurately, simulations were performed for a \ntotal of 492 seconds, divided into 60 blocks of 8.2 seconds each, for each steady-state \nvalue. \n\nFigure 2: Steady-state I-V curves for different \nmultiples (f\\,Na) of the nominal MS Na+channel \ndensity. Circles indicate locations of fixed-points \nin the absence of current injection. \n\n_8L-~------~----~------~~ \n\n-60V \nm \n\n-70 \n\n(mV)50 \n\n-40 \n\n\f146 \n\nA. Manwani, P. N. Steinmetz and C. Koch \n\n5r---~----~--~----~---. -~ \n\n'\" ~ 4 \nffi \n1:; o \n~ 3 \n\n! \n\n'0 2 \n~ \nE \n:0 \nZ 1 \n\n-65 :> \n.\u00a7. \n~ \n-66 ~ \nQ) \nc: \n~ \nD \n\n- 67 ~ \n\n100 \n\n200 \n\n300 \n\nTime (msec) \n\n400 \n\n3 Linearized Analysis \n\nFigure 3: Monte-Carlo simulations \nof a 1000 j.Lm2 membrane patch with \nstochastic Na+ and deterministic K+ \nchannels with MS kinetics. Bottom \nrecord shows the number of open Na+ \nchannels as a function of time. Top \ntrace shows the corresponding fluctua(cid:173)\ntions of the membrane voltage. Sum(cid:173)\nmary of nominal MS parameters: em = \n0.75 j.LF/cm2 , 11K = 1.5 channels/j.Lm2 , \n11Na = 2 channelslj.Lm2 , EK = -90 mY, \nENa = 60 mY, EL = -70 mY, gL = 0.25 \npSlj.Lm2 , \"IK = \"INa = 20 pS. \n\nThe non-linear stochastic differential equation (eq. 1) cannot be solved analytically. How(cid:173)\never, one can linearize it by expressing the ionic conductances and the membrane voltage \nas small perturbations (8) around their steady-state values: \n\n-c d~~m = (9~ + 9Na + 9L) 8Vm + (V~ - EK) 89K + (V~ - ENa) 89Na \n\n(2) \n\nwhere 9~ and 9Na denote the values of the ionic conductances at the steady-state voltage \nva. G = 9K + 9N a + 9 L is the total steady-state patch conductance. Since the leak channel \nconductance is constant , 89 L = o. On the other hand, 89 K and 89 N a depend on 8V and t. \nIt is known that, to first order, the voltage- and time-dependence of active ion channels can \nbe modeled as phenomenological impedances [15, 16]. Fig. 4 shows the linearized equiv(cid:173)\nalent circuit of a membrane patch, given by the parallel combination of the capacitance C, \nthe conductance G and three series RL branches representing phenomenological models of \nK+activation, Na+activation and Na+inactivation. \n\nIn = 9K(EK - V~) + 9Na(ENa - V~) \n\n(3) \nrepresents the current noise due to fluctuations in the channel conductances (denoted by \n9K and 9Na) at the membrane voltage V~ (also referred to as holding voltage Vhald) . \nThe details of the linearization are provided [16]. The complex admittance (inverse of the \nimpedance) of Fig. 4 is given by, \nY(J) = G + j27r fC + \n\nI I I \n. \n\n+ \n\n(4) \n\n+\n\n. \n\nTm + J27rf l m \nThe variance of the voltage fluctuations O\"~ can be computed as, \n\nTn + J27rf ln \n/ 00 \nSIn(J) \n-00 df IY(J)12 \n\nO\"v = \n\n2 \n\n. \n\nTh + J27rf l h \n\n(5) \n\nwhere the power spectral density of In is given by the sum of the individual channel current \nnoise spectra, SIn(J) = SIK(J) + SINa(J). \nFor the MS scheme, the autocovariance of the K+ current noise for patch of area A, clamped \nat a voltage V~, can be derived using [1, 11], \n\nCIK (t) = A 'f/K \"Ik ( V~ - EK)2 noo (1 - noo) e- Itl/rn \n\n(6) \nwhere n oo and Tn respectively denote the steady-state probability and time constant of the \nK+ activation sub-unit at V~ . The power spectral density of the K+ current noise S I K (J) \ncan be obtained from the Fourier transform of C I K (t), \n\nS \n\n(f) = 2 A 'f/K \"Ik (V~ - EK )2noo Tn \n\n1 + (21ffTn)2 \n\nIK \n\n(7) \n\n\fChannel Noise in Excitable Neural Membranes \n\n147 \n\nc \n\nG \n\nFigure 4: Linearized circuit of the \nmembrane patch containing stochastic \nvoltage-gated ion channels. C denotes \nthe membrane capacitance, G is the sum \nof the steady-state conductances of the \nchannels and the leak. ri's and li'S de(cid:173)\nnote the phenomenological resistances \nand inductances due to the voltage- and \ntime-dependent ionic conductances. \n\nThus, SIK(J) is a single Lorentzian spectrum with cut-off frequency determined by Tn. \nSimilarly, the auto-covariance of the MS Na+ current noise can be written as [1], \nCINa(t) = A rJNa ,iva (V~ - ENa)2 m~ hoo [m 3 (t) h(t) - m~ hoo] \n\n(8) \n\nwhere \n\nm(t) = moo + (1 - moo) e- t / Tm , h(t) = hoo + (1 - hoo) e-t / Th \n\n(9) \nAs before, moo (hoo ) and Tm (Th) are the open probability and the time constant of \nNa+activation (inactivation) sub-unit. The Na+current noise spectrum SINa(J) can be \nexpressed as a sum of Lorentzian spectra with cut-off frequencies corresponding to the \nseven time constants T m, Th, 2 T m, 3 T m, T m + Th, 2 T m + Th and 3 T m + Th. The details of \nthe derivations can be found in [8]. \n\nA \n\n5 \n\n4 \n\n1 \n\n+ + \n\nB \n\n3 \n\n1 \n\n0~~~--~~----~4~0----~-20 \n\nVh01d(mV) \n\no~o:s:-=----::-:---~:------! \n-20 \n\n-60 \n\n-40 \n\nVhOId(mv) \n\nFigure 5: Standard deviation of the voltage noise av in a 1000 f..\u00a3m 2 patch as a function of the \nholding voltage Vho1d . Circles denote results of the Monte-Carlo simulations for the nominal MS \nparameter values (see Fig. 3). The solid curve corresponds to the theoretical expression obtained \nby linearizing the channel kinetics. (A) Effect of increasing the sodium channel density by a factor \n(compared to the nominal value) of 2 (pluses), 3 (asterisks) and 4 (squares) on the magnitude of \nvoltage noise. (B) Effect of increasing both the sodium and potassium channel densities by a factor \nof two (pluses). \n\n4 Effect of Varying Channel Densities \n\nFig. 5 shows the voltage noise for a 1000 J.im2 patch as a function of the holding voltage \nfor different values of the channel densities. Noise increases as the membrane is depolar(cid:173)\nized from rest towards -50 mV and the rate of increase is higher for higher Na+densities. \nThe range of Vho1d for sub-threshold behavior extends up to -20 m V for nominal densities, \n\n\f148 \n\nA. Manwani, P N Steinmetz and C. Koch \n\nbut does not exceed -60 m V for higher N a + densities. For moderate levels of depolariza(cid:173)\ntion, an increase in the magnitude of the ionic current noise with voltage is the dominant \nfactor which leads to an increase in voltage noise; for higher voltages phenomenological \nimpedances are large and shunt away the current noise. Increasing Na+density increases \nvoltage noise, whereas, increasing K+density causes a decrease in noise magnitude (com(cid:173)\npare Fig. SA and SB). We linearized closed-form expressions provide accurate estimates \nof the noise magnitudes when the noise is small (of the order 3 m V). \n\n5 Effect of Varying Temperature \n\nFig. 6 shows that voltage noise decreases with \ntemperature. To model the effect of temperature, \ntransition rates were scaled by a factor Q':oT/lO \n(QlO = 2.3 for K+, QlO = 3 for Na+). Tem(cid:173)\nperature increases the rates of channel transitions \nand thus the bandwidth of the ionic current noise \nfluctuations. The magnitude of the current noise, \non the other hand, is independent of temperature. \nSince the membrane acts as a low-pass RC fil(cid:173)\nter (at moderately depolarized voltages, the phe(cid:173)\nnomenological inductances are small), increasing \nthe bandwidth of the noise results in lower volt(cid:173)\nage noise as the high frequency components are \nfiltered out. \n\n6 Conclusions \n\n1.5 .---~---~---...-, \n\n0_5 \n\no~--~---~----~ \n20 \n\n35 \n\n25 \n\n30 \n\nT (CelsiuS) \n\nFigure 6: ay as a function of tem(cid:173)\nperature for a 1000 J-Lm2 patch with \nMS kinetics (V hold = -60 m V). Circles \ndenote Monte-Carlo simulations. solid \ncurve denotes linearized approximation. \n\nWe studied sub-threshold voltage noise due to stochastic ion channel fluctuations in an iso(cid:173)\npotential membrane patch with Mainen-Sejnowski kinetics. For the MS kinetic scheme, \nnoise increases as the membrane is depolarized from rest, up to the point where the phe(cid:173)\nnomenological impedances due to the voltage- and time-dependence of the ion channels \nbecome large and shunt away the noise. Increasing Na+channel density increases both the \nmagnitude of the noise and its rate of increase with membrane voltage. On the other hand, \nincreasing the rates of channel transitions by increasing temperature, leads to a decrease \nin noise. It has previously been shown that neural excitability increases with Na+channel \ndensity [17] and decreases with temperature [IS] . Thus, our findings suggest that an in(cid:173)\ncrease in membrane excitability is inevitably accompanied by an increase in the magnitude \nof sub-threshold voltage noise fluctuations . The magnitude and the rapid increase of volt(cid:173)\nage noise with depolarization suggests that channel fluctuations can contribute significantly \nto the variability in spike timing [4] and the stochastic nature of ion channels may have a \nsignificant impact on information processing within individual neurons. It also potentially \nargues against the conventional role of a neuron as integrator of synaptic inputs [18], as \nthe the slow depolarization associated with integration of small synaptic inputs would be \naccompanied by noise. making the membrane voltage a very unreliable indicator of the \nintegrated inputs. We are actively investigating this issue more carefully. \n\nWhen the magnitudes of the noise and the phenomenological impedances are small, the \nnon-linear kinetic schemes are well-modeled by their linearized approximations. We have \nfound this to be valid for other kinetic schemes as well [19] . These analytical approxi(cid:173)\nmations can be used to study noise in more sophisticated neuronal models incorporating \nrealistic dendritic geometries, where Monte-Carlo simulations may be too computationally \nintensive to use. \n\n\fChannel Noise in Excitable Neural Membranes \n\n149 \n\nAcknowledgments \n\nThis work was funded by NSF, NIMH and the Sloan Center for Theoretical Neuroscience. \nWe thank our collaborators Michael London, Idan Segev and YosefYarom for their invalu(cid:173)\nable suggestions. \n\nReferences \n[1] DeFelice LJ. (1981). Introduction to Membrane Noise. Plenum Press: New York, New York. \n[2] Strassberg A.F. & DeFelice LJ. (1993). Limitations of the Hodgkin-Huxley formalism: effect \nof single channel kinetics on transmembrane voltage dynamics. Neural Computation, 5:843-\n855 . \n\n[3] Chow C. & White l (1996). Spontaneous action potentials due to channel fluctuations. Biophy. \n\n1.,71 :3013-3021. \n\n[4] Schneidman E., Freedman B. & Segev I. (1998). 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Cable theory in neurons with active, linearized membranes. BioI. Cybem., \n\n50:15-33. \n\n[17] Sabah N.H. & Leibovic K.N. (1972). The effect of membrane parameters on the properties of \n\nthe nerve impulse. Biophys. 1., 12:1132-44. \n\n[18] Shadlen M.N. & Newsome w.T. (1998). The variable discharge of cortical neurons: implica(cid:173)\n\ntions for connectivity, computation, and information coding. 1. Neurosci., 18:3870-3896. \n\n[19] P. N. Steinmetz A. Manwani M.L. & Koch C. (1999). Sub-threshold voltage noise due to \n\nchannel fluctuations in active neuronal membranes. In preparation. \n\n\f", "award": [], "sourceid": 1758, "authors": [{"given_name": "Amit", "family_name": "Manwani", "institution": null}, {"given_name": "Peter", "family_name": "Steinmetz", "institution": null}, {"given_name": "Christof", "family_name": "Koch", "institution": null}]}