{"title": "Information Capacity and Robustness of Stochastic Neuron Models", "book": "Advances in Neural Information Processing Systems", "page_first": 178, "page_last": 184, "abstract": null, "full_text": "Information Capacity and Robustness of \n\nStochastic Neuron Models \n\nElad Schneidman \n\nIdan Segev  N aftali Tishby \n\nInstitute of Computer Science, \nDepartment of Neurobiology and \nCenter for  Neural  Computation, \n\nHebrew University \n\nJerusalem 91904, Israel \n\n{ elads, tishby} @cs.huji.ac.il,  idan@lobster.ls.huji.ac.il \n\nAbstract \n\nThe  reliability  and  accuracy  of  spike  trains  have  been  shown  to \ndepend  on  the  nature  of  the  stimulus  that  the  neuron  encodes. \nAdding  ion  channel  stochasticity  to  neuronal  models  results  in  a \nmacroscopic behavior that replicates the input-dependent reliabili(cid:173)\nty and precision of real neurons.  We  calculate the amount of infor(cid:173)\nmation that an ion channel based stochastic Hodgkin-Huxley (HH) \nneuron model can encode about a wide set of stimuli.  We show that \nboth the information  rate and the information  per  spike  of the s(cid:173)\ntochastic model are similar to the values  reported experimentally. \nMoreover,  the  amount  of information  that  the  neuron  encodes  is \ncorrelated with the amplitude of fluctuations in the input, and less \nso with the average firing rate of the neuron.  We also show that for \nthe HH  ion  channel density,  the information capacity is  robust  to \nchanges  in  the density  of ion  channels  in  the  membrane,  whereas \nchanging  the  ratio  between  the  Na+  and  K+  ion  channels  has  a \nconsiderable effect on the information that the neuron can encode. \nFinally,  we  suggest  that neurons  may maximize  their information \ncapacity by appropriately balancing the density of the different ion \nchannels that underlie neuronal excitability. \n\n1 \n\nIntroduction \n\nThe capacity of neurons to encode information is  directly connected  to the nature \nof spike  trains as  a  code.  Namely,  whether the fine  temporal structure of the spik~ \ntrain carries information or whether  the fine  structure of the train is  mainly  noise \n(see e.g.  [1,  2]).  Experimental studies show that neurons in  vitro  [3,  4]  and  in  vivo \n[5,  6,  7],  respond  to fluctuating  inputs  with  repeatable  and  accurate spike  trains, \nwhereas slowly varying inputs result  in lower  repeatability and 'jitter' in the spike \ntiming.  Hence,  it seems that the nature of the code utilized by the neuron depends \non the input that it encodes  [3,  6]. \nRecently,  we  suggested that  the biophysical  origin  of this  behavior is  the stochas-\n\n\fCapacity and Robustness oJStochastic Neuron Models \n\n179 \n\nticity  of single  ion  channels.  Replacing  the  average  conductance dynamics  in  the \nHodgkin-Huxley  (HH)  model  [8],  with  a  stochastic  channel  population  dynamics \n[9,  10,  11],  yields  a  stochastic neuron model  which  replicates rather well  the spike \ntrains' reliability and precision of real neurons [12].  The stochastic model also shows \nsubthreshold oscillations, spontaneous and missing spikes,  all observed experimen(cid:173)\ntally.  Direct measurement of membranal noise has also been replicated successfully \nby such stochastic models [13].  Neurons use many tens of thousands of ion channels \nto encode the synaptic current that reaches the soma into trains of spikes  [14].  The \nnumber  of ion  channels  that underlies  the spike  generation  mechanism,  and  their \ntypes,  depend  on  the  activity  of the  neuron  [15,  16].  It is  yet  unclear  how  such \nchanges may affect the amount and nature of the information that neurons encode. \nHere we  ask what is  the information encoding capacity of the stochastic HH  mod(cid:173)\nel  neuron  and  how  does  this  capacity  depend  on  the  densities  of  different  of ion \nchannel types in  the membrane.  We  show  that both the information rate  and  the \ninformation per spike of the stochastic HH  model are similar to the values reported \nexperimentally  and  that  neurons  encode  more  information  about  highly  fluctuat(cid:173)\ning  inputs.  The information encoding  capacity is  rather robust  to  changes  in  the \nchannel densities of the HH  model.  Interestingly,  we  show that there is  an optimal \nchannel population size, around the natural channel density of the HH  model.  The \nencoding capacity is rather sensitive to changes in the distribution of channel types, \nsuggesting  that  changes  in  the  population  ratios  and  adaptation through  channel \ninactivation may change the information content of neurons. \n\n2  The Stochastic HH Model \n\nThe stochastic HH (SHH) model expands the classic HH model [8],  by incorporating \nthe  stochastic  nature  of  single  ion  channels  [9,  17].  Specifically,  the  membrane \nvoltage dynamics is given by the HH  description, namely, \n\ndV \n\ncmTt =  -gLCV - VL)  - gK(V, t)(V - VK)  - gNa(V, t)(V - VNa) + I \n\n(1) \n\nwhere V  is the membrane potential,  VL,  VK  and VNa  are the reversal potentials of \nthe leakage, potassium and sodium currents, respectively, gL,  gK(V, t) and gNa(V, t) \nare the corresponding ion  conductances,  Cm  is  the membrane capacitance and I  is \nthe injected  current.  The ion  channel stochasticity is  introduced  by  replacing  the \nequations describing the ion channel conductances with explicit  voltage-dependent \nMarkovian  kinetic  models  for  single  ion  channels  [9,  10].  Based  on  the  activation \nand inactivation variables of the deterministic HH model, each K+  channel can be \nin  one of five  different  states, and the rates for  transition between these states are \ngiven in the following  diagram, \n\n[ \n~  ~  ~  ~  ~  ~  ~  ~  ~ \n\n] \n\n[ \n\n] \n\n[ \n\n] \n\n[ \n\n] \n\n[ \n\n] \n\nan \n4f3n \n\n4Qn \nf3n \n\n3an \n2f3n \n\n2an \n3f3n \n\n(2) \nwhere  [nj]  refers  to  the  number  of  channels  which  are  currently  in  the  state  nj. \nHere  [n4]  labels  the single open state of a  potassium channel,  and an,  i3n,  are the \nvoltage-dependent rate-functions in the HH  formalism.  A similar_model is  used for \nthe  Na+  channel  (The  Na+  kinetic  model  has  8 states,  with  only one  open  state, \nsee  [12]  for  details). \nThe potassium and sodium membrane conductances are given by, \ngNa(V, t) =  ,Na [mahl] \n\nwhere ,K and ,Na are the conductances of an ion channel for  the K+  and  Na+  re(cid:173)\n\ngK(V, t) =  ,K [114] \n\nspectively.  We take the conductance of a single channel to be 20pS [14]  for both the \n\n(3) \n\n\f180 \n\nE.  Schneidman.  l.  Segev and N.  Tishby \n\nK+  and  Na+  channel types  1.  Each of the ion  channels will  thus respond stochas(cid:173)\ntically by  closing or opening its  'gates'  according to the kinetic  model,  fluctuating \naround the average expected behavior.  Figure 1 demonstrates the effect  of the ion \n\nA \n\nB \n\nFigure  1:  Reliability  of  firing  patterns  in  a  model  of  an  isopotential  Hodgkin-Huxley \nmembrane patch in response  to different  current inputs.  (A)  Injecting  a  slowly  changing \ncurrent input  (low-pass  Gaussian  white  noise  with  a  mean  TJ  = 8I1A/cm 2 ,  and standard \ndeviation a  = 1 p,A/ cm2  which was convolved with an 'alpha-function' with a time constant \nTo  =  3 msec,  top frame),  results  in  high  'jitter'  in  the timing of the  spikes  (raster  plots \nof spike responses,  bottom frame).  (B)  The same patch was  again  stimulated repeatedly, \nwith  a  highly  fluctuating  stimulus  (TJ  = 8 p,A/cm2 ,  a  = 7 p,A/cm2  and  To  = 3 msec,  top \nframe)  The  'jitter'  in spike  timing  is  significantly  smaller  in  B  than in  A  (i.e.  increased \nreliability for the fluctuating current input).  Patch area used was 200 p,m2 ,  with 3,600 K+ \nchannels  and  12,000 Na+  channels. \n(C)  Average  firing \nrate in response to DC  current input of both the HH and the stochastic  HH model.  (D) \nCoefficient  of variation  of the  inter  spike  interval  of the  SHH  model  in  response  to  DC \ninputs, giving values which  are comparable to those observed in real neurons \n\n(Compare  to  Fig.l  in  see  [3]). \n\nchannel  stochasticity,  showing  the  response  of a  200 J.Lm2  SHH  isopotential  mem(cid:173)\nbrane patch  (with the  'standard'  SHH  channel  densities)  to  repeated presentation \nof supra threshold current input.  When the same slowly varying input is  repeatedly \npresented  (Fig.  lA) , the spike  trains  are very different  from  each other,  i.e. , spike \nfiring  time  is  unreliable.  On  the  other  hand,  when  the  input  is  highly  fluctuat(cid:173)\ning  (Fig.  IB),  the  reliability of the spike timing is  relatively  high.  The stochastic \nmodel  thus  replicates  the  input-dependent  reliability  and precision  of spike  trains \nobserved in pyramidal cortical neurons [3] .  As for cortical neurons, the Repeatability \nand Precision of the spike trains of the stochastic model (defined in [3])  are strongly \ncorrelated with the fluctuations in the current input and may get to sub-millisecond \nprecision [12].  The f-I  curve of the stochastic model  (Fig. lC)  and the coefficient of \nvariation (CV)  of the inter-spike intervals (lSI) distribution for DC inputs (Fig.  ID) \nare both similar to the behavior of cortical neurons in vivo [18],  in clear contrast to \nthe deterministic model  2 \n\n1 The number of channels is thus the ratio between the total conductance of a single type \nof ion  channels and the single  channel conductance,  and so  the 'standard'  SHH densities \nwill  be 60  Na+  and 18  Na+  channels per p,m 2 . \n\n2 Although the total number of channels in the model is very large, the microscopic level \nion channel  noise has  a  macroscopic effect  on the spike train reliability,  since the number \n\n\fCapacity and Robustness of Stochastic Neuron Models \n\n181 \n\n3  The Information Capacity of the SHH Neuron \n\nExpanding the  Repeatability and  Precision  measures  [3],  we  turn  to  quantify  how \nmuch information the neuron model encodes about the stimuli it receives.  We  thus \npresent the model with a set of 'representative' input current traces, and the amount \nof information that the respective spike trains encode is  calculated. \nFollowing Mainen and Sejnowski [3],  we use a set of input current traces which imi(cid:173)\ntate the synaptic current that reaches the soma from the dendritic tree.  We convolve \na Gaussian white noise trace (with a mean current 1}  and standard deviation 0')  with \nan alpha function  (with a To:  =  3 msec).  Six different mean current values  are used \n(1}  = 0,2,4,6,8,10 pA/cm2 )  ,  and five different std values (0'  = 1,3,5,7, 9pA/cm2 ), \nyielding a set of 30 input current traces (each is  10 seconds long).  This set of inputs \nis representative of the wide variety of current traces that neurons might encounter \nunder  in  vivo  conditions  in  the  sense  that  the  average  firing  rates  for  this  set  of \ninputs which range between 2 - 70  Hz  (not shown). \nWe present these input traces to the model, and calculate the amount of information \nthat  the  resulting  spike  trains  convey  about  each  input,  following  [6,  19].  Each \ninput  is  presented  repeatedly  and  the  resulting  spike  trains  are  discretized  in  D..T \nbins,  using a  sliding  'window'  of size T  along the discretized  sequence.  Each  train \nof  spikes  is  thus  transformed  into  a  sequence  of  K-letter  'words'  (K  =  T/D..T) , \nconsisting of O's  (no spike)  and  l's (spike).  We  estimate  P(W),  the probability of \nthe word W  to appear in the spike trains, and then compute the entropy rate of its \ntotal word  distribution, \n\nHtotal  =  - L P(W) log2 P(W) \n\nW \n\nbits/word \n\n(4) \n\nwhich measures the capacity of information that the neuron spike trains hold [20,  6, \n19].  We  then examine the set of words  that the neuron model  used  at a  particular \ntime  t  over  all  the  repeated  presentations  of  the  stimulus,  and  estimate  P(Wlt), \nthe time-dependent  word  probability distribution.  At each time t  we  calculate the \ntime-dependent entropy rate, and then take the average of these entropies \n\nHnoise  =  (- LP(Wlt)lOg2 P(Wlt))t \n\nw \n\nbits/word \n\n(5) \n\nwhere  ( .. . )t  denotes  the average over  all  times t.  Hnoise  is  the noise entropy rate, \nwhich measures how  much of the fine  structure of the spike trains of the neuron is \njust noise.  After  performing the calculation for  each  of the inputs,  using different \nword  sizes  3,  we  estimate the limit  of the total entropy and noise entropy rates  at \nT  --*  00,  where the entropies converge to their real values  (see  [19]  for  details)  . \nFigure 2A shows the total entropy rate of the responses to the set of stimuli, ranging \nfrom  10 to 170 bits/sec.  The total entropy rate is correlated with the firing rates of \nthe neuron (not shown).  The noise entropy rate however, depends in a different way \non the input  parameters:  Figure 2B  shows  the noise entropy rate of the responses \nto the set of stimuli, which may get up to 100 bits/sec.  Specifically, for  inputs with \nhigh  mean  current  values  and  low  fluctuation  amplitude,  many  of the  spikes  are \n\nof ion  channels  which  are  open  near  the spike  firing  threshold  is  rather  small  [12).  The \nfluctuations  in  this  small  number of open  channels  near  firing  threshold  give  rise  to the \ninput-dependent reliability of the spike timing. \n\n3t he  bin  size  T  =  2  msec  has  been  set  to  be  small  enough  to  keep  the  fine  tem(cid:173)\n\nporal  structure of the spike  train within  the word  sizes  used,  yet  large  enough  to  avoid \nundersampling problems \n\n\f182 \n\nE.  Schneidman,  /.  Segev and N.  Tishby \n\njust noise, even if the mean firing  rate is high.  The difference between the neuron's \nentropy rate  (the total capacity of information of the neuron's spike train)  and the \nnoise entropy rate, is exactly the average rate of information that the neuron's spike \ntrains encode  about  the  input,  I(stimulus , spike train)  =  Htotal  - Hnoise  [20, 6], \nthis  is  shown  in  Figure  2C.  The  information  rate  is  more  sensitive  to  the  size  of \n\nA \n\n200  B \n\n.~: \n\n150 \n\n100 \n\n50 \n\n0 \n\n10 \n\n0  0 \n\na fl.LA/cm2 ) \n\n10 \n\n'1  fl.LA/cm 2) \n\n0  0 \n\na fl.LA/cm 2) \n\n200 \n\n;f. ':'~ \n\n150 \n\n100 \n\n50 \n\n0 \n\n3 \n\n~;) \n\n2 .5 \n\n:~ \n\n,.  100  D \n;;; \n80  a  3 \n\n. \n\n)~r\n\n. \n\n.~ \n\n2 \n\n~ \n\n40  ~  1 \n\n~  ~ 60 \n20  i  0 \n\n0 .5 \n\n2 \n\n1.5 \n\n10 \n\n10 \n\n5 \n\n10 \n\n5 \na lllAIcm2 ) \n\n0  0 \n\n0 \n\n'1  [llAIcm 2 ) \n\n0  0 \n\n5 \na lllAIcm2) \n\nFigure  2:  Information  capacity  of  the  SHH  model.  (A)  The  total  spike  train  entropy \nrate of the SHH model  as  a function  of 'TI,  the current  input  mean,  and  a,  the standard \ndeviation  (see  text  for  details).  Error  bar  values  of  this  surface  as  well  as  for  the other \nframes  range  between  1 - 6%  (not  shown).  (B)  Noise  entropy  rate  as  a  function  of the \ncurrent input parameters.  (C) The information rate about the stimulus in the spike trains, \nas  a  function  of the input  parameters,  calculated  by  subtracting noise  entropy  from  the \ntotal  entropy  (note the change  in  grayscale  in  C  and  D).  (D)  Information  per spike  as  a \nfunction  of the input parameters, which  is  calculated by  normalizing the results  shown  in \nC by  the average firing rate of the responses  to each  of the inputs. \n\nfluctuations  in the input than to the mean value of the current trace (as expected, \nfrom  the  reliability  and  precision of spike  timing  observed  in  vitro  [3]  and  in  vivo \n[6]  as well  as in simulations  [12]).  The dependence of the neural code on the input \nparameters is  better reflected when  calculating the average amount of information \nper spike  that the model gives for  each of the inputs  (Fig.  2D)  (see for  comparison \nthe values for  the Fly's HI neuron  [6]). \n\n4  The effect of Changing the Neuron Parameters on the \n\nInformation Capacity \n\nIncreasing the density of ion channels in the membrane compared to the 'standard' \nSHH  densities,  while  keeping  the  ratio  between  the  K+  and  Na+  channels  fixed, \nonly  diminishes  the  amount  of information  that  the  neuron encodes  about  any  of \nthe inputs in  the set.  However, the  change  is  rather small:  Doubling  the  channel \ndensity  decreases  the  amount  of information  by  5 - 25%  (Fig.  3A),  depending  on \nthe  specific  input.  Decreasing  the  channel  densities  of  both  types,  results  in  en(cid:173)\ncoding  more  information  about  certain  stimuli  and  less  about  others.  Figure  3B \nshows  that having half the  channel  densities  would  result  with in  10%  changes in \nthe information in both directions.  Thus, the information rates conveyed by the s(cid:173)\ntochastic model are robust to changes in the ion channel density.  Similar robustness \n(not shown)  has been observed for  changes in the membrane area (keeping channel \n\n\fCapacity and Robustness of Stochastic Neuron Models \n\n183 \n\ndensity fixed)  and in the temperature (which effects the channel kinetics).  However, \n\nA \njl.2 \n:5  1 \n\n~ ..s0.8 \n\n10 \n\nc \n\n10 \n\n5 \n\no  0 \n\na[~em2) \n\n1.2  B \n\n....... , \n\n1 .1  .21.2 \n\n1  \"1  \n\n~ \n\n'1 \n\n0.9  ~o.s \n10 \n\nO.S \n\n5 \n'1  [pAIem2) \n\n$A  D \n\n3 .5 \n\n3 \n\n2.5 \n\n2 \n1.5 \n\n0 .5 \n\n1.2 \n\n1.1 \n\n0 .9 \n\nO.S \n\n0.4 \n\n>~ \n\n0.3 \n\n0.2 \n\n0 .1 \n\n0 \n\n10 \n\n5 \n\no  0 \n\na[~Alem2) \n\n10 \n\no  0 \n\na[~Alem2) \n\nFigure  3:  The effect  of changing  the  ion  channel  densities  on the information  capacity. \n(A)  The  ratio  of the  information  rate  of the  SHH  model  with  twice  the  density  of  the \n'standard'  SHH  densities  divided  by  the  information  rate  of  the  mode  with  'standard' \nSHH  densities.  (B)  As  in  A,  only  for  the  SHH model  with  half the  'standard'  densities. \n(C)  The ratio of the info rate of the SHH model with twice as many Na+  channels, divided \nby the info rate of the standard SHH Na+  channel density,  where the K+  channel density \nremains  untouched  (note the change  in graycale  in  C  and  D).  (D)  As  in  C,  only  for  the \nSHH model with the number of Na+  channels reduced by half. \n\nchanging the density of the Na+  channels alone has a larger impact on the amount \nof information that the neuron conveys  about the stimuli.  Increasing Na+  channel \ndensity by a factor of two results in less information about most of the stimuli,  and \na gain in a few  others (Fig.  3C). However, reducing the number of Na+  channels by \nhalf results in drastic loss of information for  all of the inputs  (Fig.  3D). \n\n5  Discussion \n\nWe  have  shown that the amount of information  that the stochastic HH  model en(cid:173)\ncodes about its current input is highly correlated with the amplitude of fluctuations \nin  the  input  and  less  so  with  the  mean  value  of  the  input.  The  stochastic  HH \nmodel,  which incorporates ion channel noise,  closely replicates the input-dependent \nreliability  and  precision of spike trains observed in  cortical neurons.  The informa(cid:173)\ntion  rates  and  information  per spike  are  also  similar  to those  of real  neurons.  As \nin  other biological  systems  (e.g.,  [21]),  we  demonstrate robustness  of macroscopic \nperformance to changes in the cellular properties - the information coding rates of \nthe  SHH  model  are  robust  to  changes  in  the  ion  channels  densities  as  well  as  in \nthe  area of the excitable membrane patch and in the temperature  (kinetics)  of the \nchannel  dynamics.  However,  the  information  coding  rates  are  rather  sensitive  to \nchanges  in  the  ratio  between  the  densities  of different  ion  channel  types,  suggests \nthat the ratio between the density of the K+  channels and the Na+  channels in the \n'standard' SHH  model  may  be optimal in terms of the information capacity.  This \nmay have important implications on the nature of the neural code under adaptation \nand  learning.  We  suggest  that these notions  of optimality and  robustness may  be \na  key  biophysical principle of the operation of real  neurons.  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How voltage-dependent conductances can adapt to maxi(cid:173)\nmize the information encoded by neuronal firing  rate.  Nat.  Neurosci.,  2:521-7,  1999. \n\n\f", "award": [], "sourceid": 1657, "authors": [{"given_name": "Elad", "family_name": "Schneidman", "institution": null}, {"given_name": "Idan", "family_name": "Segev", "institution": null}, {"given_name": "Naftali", "family_name": "Tishby", "institution": null}]}