{"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).  Ion-channel  stochasticity  may be critical in \ndetermining the reliability and precision of spike timing.  Neural Computation,  10:1679-1703. \n[5]  DeFelice LJ. &  Isaac A.  (1992).  Chaotic states in a random world.  1.  Stat.  Phys., 70:339-352. \n[6]  White lA.,  Budde  T.  &  Kay  A.R.  (1995).  A  bifurcation  analysis  of neuronal  subthreshold \n\noscillations.  Biophy.  J., 69:1203-1217. \n\n[7]  Manwani A.  &  Koch C.  (1998).  Synaptic transmission:  An information-theoretic perspective. \nIn:  Jordan  M.,  Kearns  M.S. &  SoBa S.A.,  eds. , Advances  in  Neural  Information  Processing \nSystems 10. pp 201-207. MIT Press: Cambridge, Massachusetts. \n\n[8]  Manwani  A.  &  Koch C.  (1999).  Detecting and estimating signals in noisy cable structures:  I. \n\nNeuronal noise sources.  Neural Computation.  In press. \n\n[9]  Manwani  A.  &  Koch C.  (1999).  Detecting and estimating signals in noisy cable structures:  II. \n\nInformation-theoretic analysis.  Neural Computation.  In press. \n\n[10]  Mainen Z.F.  &  Sejnowski TJ. (1995).  Reliability of spike timing in neocortical neurons.  Sci(cid:173)\n\nence,  268: 1503-1506. \n\n[11]  Johnston D.  &  Wu S.M.  (1995) .  Foundations of Cellular Neurophysiology.  MIT Press:  Cam(cid:173)\n\nbridge, Massachusetts. \n\n[12]  Hodgkin A.L.  &  Huxley A.F.  (1952).  A  quantitative description of membrane current and its \n\napplication to conduction and excitation in nerve.  1.  Physiol.  (London),  117: 500-544. \n\n[13]  Skaugen E.  &  Wallre L.  (1979).  Firing behavior in a stochastic nerve membrane model based \n\nupon the Hodgkin-Huxley equations.  Acta Physiol. Scand., 107:343-363. \n\n[14]  Press W.H., Teukolsky S.A., Vetterling w.T. &  Flannery B.P. (1992).  Numerical Recipes in C: \n\nThe An of Scientific Computing.  Cambridge University Press, second edn. \n\n[15]  Mauro A., Conti F.,  Dodge F.  & Schor R.  (1970).  Subthreshold behavior and phenomenological \n\nimpedance of the squid giant axon.  1.  Gen. Physiol., 55:497-523. \n\n[16]  Koch C.  (1984).  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}]}