{"title": "Synaptic Transmission: An Information-Theoretic Perspective", "book": "Advances in Neural Information Processing Systems", "page_first": 201, "page_last": 207, "abstract": null, "full_text": "Synaptic Transmission:  An \n\nInformation-Theoretic Perspective \n\nAmit Manwani and Christof Koch \n\nComputation and Neural Systems Program \n\nCalifornia Institute of Technology \n\nPasadena, CA 91125 \n\nemail:  quixote@klab.caltech.edu \n\nkoch@klab.caltech.edu \n\nAbstract \n\nHere  we  analyze  synaptic  transmission  from  an  infonnation-theoretic \nperspective. We derive c1osed-fonn expressions for the lower-bounds on \nthe capacity of a simple model of a cortical synapse under two explicit \ncoding paradigms.  Under the \"signal estimation\" paradigm, we assume \nthe signal to be encoded in the mean firing rate of a Poisson neuron.  The \nperfonnance of an optimal linear estimator of the  signal  then  provides \na lower bound on the  capacity for signal estimation.  Under the  \"signal \ndetection\" paradigm, the presence or absence of the signal has to be de(cid:173)\ntected.  Perfonnance of the optimal spike detector allows us to compute \na lower bound on the capacity for signal detection.  We  find  that single \nsynapses (for empirically measured parameter values) transmit infonna(cid:173)\ntion poorly but  significant  improvement can be  achieved  with  a  small \namount of redundancy. \n\n1  Introduction \n\nTools from  estimation and infonnation theory have  recently been applied by researchers \n(Bialek et.  ai,  1991) to quantify how well neurons transmit infonnation about their random \ninputs in their spike outputs.  In these approaches, the neuron is  treated like  a black-box, \ncharacterized empirically by a set of input-output records.  This ignores the specific nature \nof neuronal processing in tenns of its known biophysical properties. However, a systematic \nstudy of processing at various stages in a biophysically faithful model of a single neuron \nshould be able to identify the role of each stage in infonnation transfer in tenns of the pa(cid:173)\nrameters relating to the neuron's dendritic structure, its spiking mechanism, etc.  Employing \nthis reductionist approach, we  focus  on a important component of neural processing,  the \nsynapse, and analyze a simple model of a cortical synapse under two different representa(cid:173)\n\u00b7tional paradigms. Under the \"signal estimation\" paradigm, we assume that the input signal \n\n\f202 \n\nA. Manwani and C.  Koch \n\nis linearly encoded in the mean firing rate of a Poisson neuron and the mean-square error \nin the  reconstruction of the  signal  from  the post-synaptic  voltage  quantifies  system  per(cid:173)\nformance.  From the  performance of the  optimal  linear estimator of the  signal,  a  lower \nbound on the capacity for signal estimation can be computed. Under the \"signal detection\" \nparadigm, we assume that information is encoded in an all-or-none format and the error in \ndeciding whether or not a presynaptic spike occurred by observing the post-synaptic voltage \nquantifies system performance. This is similar to the conventional absentipresent(Yes-No) \ndecision paradigm used  in psychophysics.  Performance of the optimal spike  detector in \nthis case allows us to compute a lower bound on the capacity for signal detection. \n\n0x- 0 \n\nNoR \n\nRelease \n1 \n\nNoSpI<o\n\n!'-. \n\nSpoke \n1 \n\n&, \n1-\n\nh{l) \n\nStochastic \n\nVariable EPSC \n\nVesicle Release \n\nAmplitude \n\nEPSP \nShape \n\nPoisson \nEncoding \n\nStimulys \n\nSpike I \nNo Spike \n\nOptimal \nEstimator \n\nOptimal \nDetector \n\nSpike I \nNo Spike \n\nEncodlna \n\nSynaptic Channel \n\nDecoding \n\nFigure  1:  Schematic  block diagram  for  the  signal  detection  and  estimation  tasks.  The \nsynapse is modeled as a binary channel followed by a filter h(t) =  at exp( -tits). where \na  is a random variable with probability density,  P(a)  =  a  (aa)k- 1exp( -aa)/(k - 1)!. \nThe binary channel, (inset, EO  = Pr[spontaneous release],  E1  = Pr [release failure]) models \nprobabilistic vesicle release  and h(t)  models the  variable epsp  size  observed for cortical \nsynapses. n( t)  denotes additive post-synaptic voltage noise and is assumed to be Gaussian \nand  white  over a  bandwidth  En.  Performance  of the  optimal  linear  estimator (Wiener \nFilter) and the optimal spike detector (Matched Filter) quantify synaptic efficacy for signal \nestimation and detection respectively. \n\n2  The Synaptic Channel \n\nSynaptic transmission in cortical neurons is known to be highly random though the role of \nthis variability in neural computation and coding is still unclear.  In central synapses, each \nsynaptic bouton contains only a single active release zone, as opposed to the hundreds or \nthousands found at the much more reliable neuromuscular junction. Thus, in response to an \naction potential in the presynaptic terminal at most one vesicle is released (Kom and Faber, \n1991).  Moreover,  the  probability of vesicle  release p  is  known to  be  generally low  (0.1 \nto 0.4) from in vitro  studies in some vertebrate and invertebrate systems (Stevens,  1994). \nThis unreliability is further compounded by the trial-to-trial variability in the amplitude of \nthe post-synaptic response to a vesicular release (Bekkers et.  ai,  1990). In some cases, the \nvariance in the size of EPSP is as large as the mean.  The empirically measured distribution \nof amplitudes is usually skewed to the right (possibly biased due the inability of measuring \nvery small events) and can be modeled by a Gamma distribution. \nIn light of the above, we model the synapse as a binary channel cascaded by a random am(cid:173)\nplitude filter (Fig.  1).  The binary channel accounts for the probabilistic vesicle release.  EO \n\n\fSynaptic Trarumission:  An Information-Theoretic Perspective \n\n203 \n\nand \u20acl  denote the probabilities of spontaneous vesicle release and failure respectively.  We \nfollow the binary channel convention used in digital communications (\u20ac  1  = 1-p), whereas, \np is more commonly used in neurobiology.  The filter h(t)  is chosen to  correspond to the \nepsp profile of a fast AMPA-like synapse.  The amplitude of the filter a is modeled as ran(cid:173)\ndom variable with density Pea),  mean J.la  and standard deviation aa.  The  CV (standard \ndeviation/mean) of the distribution is denoted by eVa. We also assume that additive Gaus(cid:173)\nsian voltage noise net)  at the post-synaptic site further corrupts the epsp response.  net)  is \nassumed to  white with variance a~ and a bandwidth En  corresponding to the  membrane \ntime constant T.  One can define an effective signal-to-noise ratio,  SN R  =  Ea/No\u2022 given \nby the ratio of the energy in the epsp pulse, Eh  = 1000 h2 (t) dt to the noise power spectral \ndensity,  No  =  a;/ En.  The performance of the synapse depends on the  SN R  and not on \nthe absolute values of Eh  or an.  In the above model, by regarding synaptic parameters as \nconstants, we  have tacitly ignored history dependent effects like paired-pulse facilitation, \nvesicle  depletion,  calcium buffering.  etc,  which endow the  synapse  with the  nature  of a \nsophisticated nonlinear filter (Markram and Tsodyks,  1997). \n\nb)  N.:\" V..:~ .. \n\nSpike ~ SpIke \n\na) \n\n'\" \nm(t) '~t \",!t) \nn(t) \n\nX=l \n\nY=l \n\nEffective Continuous \nEstimation Channel \n\nl\u00b7P. \n\nEffective Elinm. \nDetection Channel \n\nFigure  2: \n(a)  Effective  chan(cid:173)\nnel  model  for  signal  estima(cid:173)\ntion.  met),  met),  net)  denote \nthe stimulus, the best linear es(cid:173)\ntimate,  and  the  reconstruction \nnoise respectively. (b) Effective \nchannel model for signal detec(cid:173)\ntion.  X  and  Y  denote  the  bi(cid:173)\nnary variables corresponding to \nthe  input  and  the  decision  re(cid:173)\nspectively.  Pi  and  Pm  are  the \neffective error probabilities. \n\n3  Signal Estimation \n\nLet us assume that the  spike train of the presynaptic neuron can be modeled as a doubly \nstochastic Poisson process with a rate A(t)  =  k(t) * met)  given as a convolution between \nthe  stimulus met)  and a filter k(t).  The stimulus is drawn from a probability distribution \nwhich we assume to  be Gaussian.  k(t)  =  exp( -tiT) is  a  low-pass filter  which models \nthe phenomenological relationship between a neuron's firing rate  and its input current.  T \nis  chosen to  correspond to  the  membrane time  constant.  The  exact form  of k(t)  is  not \ncrucial and the above form is assumed primarily for analytical tractability.  The objective is \nto find the optimal estimator ofm(t) from the post-synaptic voltage v(t), where optimality \nis  in a  least-mean square sense.  The optimal  mean-square estimator is,  in general,  non(cid:173)\nlinear  and  reduces  to  a  linear filter  only  when  all  the  signals  and  noises  are  Gaussian. \nHowever,  instead  of making this assumption,  we  restrict ourselves to the  analysis of the \noptimal  linear  estimator,  met)  =  get)  * vet),  i.e. \nthe  filter  get)  which  minimizes  the \nmean-square error E  =  (m(t)  - m(t))2)  where  (.)  denotes an ensemble average.  The \noverall estimation system shown in Fig.  1 can be characterized by an effective continuous \nchannel (Fig.  2a) where net)  = met)  - met)  denotes the  effective reconstruction noise. \nSystem performance can be quantified by E,  the lower E, the better the  synapse at signal \ntransmission. The expression for the optimal filter (Wiener filter) in the frequency domain is \ng(w)  = Smv( -w)/Svv(w) where Smv(w)  is the cross-spectral density (Fourier transform \nof the cross-correlation Rmv) ofm(t) and set)  and Svv(w)  is the power spectral density of \nvet).  The minimum mean-square error is given by, E  =  a~ - Is  I Smv(w)  12  / Svv(w) dw. \nThe set S  =  {w  1 Svv (w)  =J.  O}  is called the support of Svv (w). \n\n\f204 \n\nA. Manwani and C.  Koch \n\nAnother measure of system performance is the mutual information rate I (m; v)  between \nm(t)  and v(t),  defined as the rate  of information transmitted by v(t)  about s(t).  By the \nData Processing inequality (Cover 1991), l(m, v)  ~ l(m, m). A lower bound of l(m, m) \nand thus of l(m; v)  is given by the simple expression lib  = ~ Is log2[~::/w/l dw  (units \nof bits/sec).  The  lower bound is  achieved when n(t)  is  Gaussian  and  is  independent of \nm(t).  Since the spike train s(t)  =  L 6(t - ti) is a POiSSOl!process with rate k(t) * m(t), \nits power spectrum is  given by the  expression, Sss(w)  =  >'+  1 K(w)  12  Smm(w)  where \n).  is the  mean firing rate.  We  assume that the mean (J..Lm)  and variance (CT~) of m(t)  are \nchosen such that the probability that >.(t)  < 0 is  negligible1 The vesicle  release  process \nis the spike train gated by the binary channel and so it is also a Poisson process with rate \n(1  - E1 )>.(t).  Since v(t)  =  L aih(t - ti) +  n(t) is  a filtered  P~isson process, its power \nspectral density is given by Svv (w)  =1  H(w)  12  {(J..L~+CT~)(1-E1)>'+J..L~(1-E1)21 K(w)  12 \nSmm(w)}  +  Snn{w).  The  cross-spectral density  is  given by the  expression Svm(w)  = \n(1  - Et)J..LaSmm(w)H(w)K(w).  This allows us to write the mean-square error as, \n\nThus, the power spectral density ofn(t) is given by Snn  =  >'eff(w)  +  Self(w).  Notice \nthat if K (w)  ---+  00, E  ---+  0  i. e.  perfect reconstruction takes  place in the  limit  of high \nfiring rates.  For the parameter values chosen,  SefJ{w)  \u00ab  >'e//(w),  and can be ignored. \nConsequently, signal estimation is shot noise limited and synaptic variability increases shot \nnoise by a factor N syn  =  (1  +  eVa2 ) /  (1  - E1)'  For eVa  =  0.6 and E1  =  0.6, N syn  =  3.4, \nand for eVa  =  1 and E1  =  0.6, N syn  =  5.  If m(t) is chosen to be white, band-limited to \nBm  Hz,  closed-form expressions for E  and lib can be obtained.  The expression for lib  is \ntedious and provides little insight and so we present only the expression for E  below. \n\nE(r,BT ) =  CTm [1- ~-B tan  (~)l \n\n2 \n\n,1   -1  BT \n1+,  T \n\n+, \n\nE  is a monotonic function of, (decreasing) and BT (increasing).  ,can be considered as \nthe effective number of spikes available per unit signal bandwidth and BT  is the ratio of \nthe signal bandwidth and the neuron bandwidth.  Plots of normalized reconstruction error \nEr  =  E/CT~ and llb versus mean firing rate ().) for different values of signal bandwidth Bm \nare shown in Fig. 3a and Fig. 3b respectively. Observe that lib (bits/sec) is insensitive to Bm \nfor firing rates upto 200Hz because the decrease in quality of estimation (E increases with \nBm) is compensated by an increase in the number of independent samples (2Bm) available \nper second. This phenomenon is characteristic of systems operating in the low SNR regime. \nlib  has the  generic form,  llb  =  B  log(1  +  S/(N B)), where B, S and N  denote  signal \nbandwidth, signal power and noise power respectively.  For low SNR, I  ~ B  S / (N B)  = \nS / N,  is independent of B.  So  one  can argue that,  for  our choice of parameters, a  single \nsynapse is  a low SNR system.  The analysis generalizes very easily to the case of multiple \nsynapses where all are driven by the same signal s (t). (Manwani and Koch, in preparation). \nHowever, instead of presenting the rigorous analysis, we appeal to the intuition gained from \nthe  single  synapse case.  Since a  single synapse can  be  regarded as  a  shot noise  source, \nn parallel synapses can be treated as n parallel noise  sources.  Let us make the plausible \n\nlWe choose pm and O'm  so that X =  30').  (std of ,X)  so that Prob['x(t)  ~ 0]  < 0.01. \n\n\fSynaptic Transmission:  An Information-Theoretic Perspective \n\n205 \n\nassumption that these noises are uncorrelated. If optimal estimation is carried out separately \nfor each synapse  and  the  estimates  are  combined optimally,  the  effective noise  variance \nis  given  by the  harmonic  mean of the  individual variances i.e.  l/u~eff  =  Li l/u~i. \nHowever,  if the  noises are  added first  and optimal estimation is  carried out with respect \nto  the sum, the effective noise variance is given by the arithmetic mean of the individual \n::::  Li u~dn2.  If we  assume that  all  synapses  are  similar so  that \nvariances,  i.e.  u~ef f \nU~i  =  u 2,  u~ef f  =  u 2 In.  Plots of Er  and  Jib  for the  case  of 5  identical synapses are \nshown in Fig.  3c and Fig. 3d respectively.  Notice that  Jib  increases with Bm  suggesting \nthat the system is no longer in the low SNR regime.  Thus, though a single synapse has very \nlow capacity, a small amount of redundancy causes a considerable increase in performance. \nThis is consistent with the fact the in the low S N R regime, J increases linearly with S N R , \nconsequently, linearly with n, the number of synapses. \na) \n\nb) \n\nx  x \n\n0 \n\n0 \nx  x  x  x \n\n0  ~  ~  +  ~  ..  + \n\n00 0   00 0   0 \n\n..  +  +  +  + \n0 \n\n0 \n\n0 \n\no.a \n\nx  x  x \n\nx  x  x \n\nX  X  X \n\nX  X  X \n\nx \no \n\nB  = 10Hz \nm \nBm- 25Hz \n\nBm= 50 Hz \n\n-\n\n-\n\n-\n\nBm=75HZ \nBm: 100Hz \n\n1. \n\n12 \n\nx \no \n\n-\n\n-\n\n-\n\nB  = 10Hz \nm \nB=25Hz \nm \n\nBm=  50Hz \nBm=75Hz \nBm= 100 Hz \n\no.s \n\n~ e \nW 0.7 \n\"0 \nQ)m \n.~ 1U  0.8 \nE ~ o \n\n0.5 \n\nZ \n\no.a \n\n'- 0.8 g \n\nW \n\"0  0.7 \nQ) \n.~ \n(Q  o.s \n\nE o \n\nZ  0.5 \n\n0 .\u2022 \n\n20 \n\n40 \n\nl1li \n\nl1li \n\n100 \n\n120 \n\n140 \n\n1l1li \n\n180 \n\n200 \n\nl1li \n\n80 \n\n100 \n\n120 \n\n140 \n\n180 \n\n180 \n\n200 \n\n~ . \nx  0 \n\n..  +\"-:  ....  __  .. \n\n0 \n\n+  + \n\n.. \n\n-\n\n+  +  +  - - - - - - : (cid:173)\n\n..  . . \n\n..  +  +  + \n\no \n\n0 \n\no  0 \n\no  0 \n\n1. \n\n12 \n\nI \n\nQ) \n\nU IO \n(/) UiS \n.\"t::: \n.0 \n-\nQ) \n\nS \n\n~. \n.E2 \n.5 \n\nFiring Rate (Hz) \n\n20 \n\n40 \n\n80  ~  ~  ~  ~  ~  -\n\nl1li \nFiring Rate (Hz) \n\nFigure 3:  Er and!,b vs.  mean firing rate (X) for n = I [(a) and (b)]  and n = 5 [(c) and (d)]  identical \nsynapses respectively (different values of Em) for signal estimation.  Parameter values are  101  = 0.6, \n100  = 0, eVa = 0.6,  ts = 0.5 msec, T  =  I Omsec,  (7n  = 0.1  mY,  En = 100 Hz. \n\n4  Signal Detection \n\nThe  goal in signal detection is  to  decide which  member from  a finite  set of signals  was \ngenerated by a source, on the basis of measurements related to the output only in a statistical \nsense.  Our example corresponds to its simplest case, that of binary detection. The objective \nis  to  derive an optimal spike  detector based on the  post-synaptic voltage  in a  given time \ninterval.  The criterion of optimality is minimum probability of error (Pe ).  A false  alarm \n\n\f206 \n\nA.  Manwani and C.  Koch \n\n(FA) error occurs when a spike is falsely detected even when no presynaptic spike occurs \nand a miss error (M) occurs when a spike fails to be detected. The probabilities of the errors \nare denoted by P, and Pm respectively. Thus, Pe =  (1- Po)  P, +Po Pm where Po  denotes \nthe a priori probability of a spike occurrence.  Let X  and Y  be binary variables denoting \nspike occurrence and the decision respectively.  Thus, X  = 1 if a spike occurred else X  = \nO.  Similarly, Y  =  1 expresses the decision that a spike occurred.  The posterior likelihood \nratio  is  defined as \u00a3(v)  = Pr(v  I X  = l)/Pr(v  I X  = 0)  and  the  prior likelihood as \n\u00a30  =  (1  - Po)/Po.  The optimal spike detector employs the  well-known likelihood ratio \ntest, \"If\u00a3(v)  ~ \u00a30 Y=lelseY=O\". When X  = 1,v(t) = ah(t)+n(t) elsev(t) = n(t). \nSince a is a random variable, \u00a3(v)  = (f Pr(v I X  = 1; a)  P(a)  da)/ Pr(v I X  = 0).  If \nthe noise n( t)  is Gaussian and white, it can be shown that the optimal decision rule reduces \nto  a matchedfilte?, i.e.  if the correlation, r  between v(t)  and  h(t)  exceeds a particular \nthreshold  (denoted by TJ),  Y  = 1  else  Y  = O.  The  overall  decision system  shown  in \nFig.  1 can be treated as  effective binary channel (Fig.  2b).  The system perfonnance can \nbe quantified either by Pe  or J (X; Y), the mutual infonnation between the binary random \nvariables,  X  and Y.  Note  that even when n(t)  = 0  (SN R  = 00),  Pe  =j:.  0  due to  the \nunreliability of vesicular release. Let Pe*  denote the probability of error when S N R  =  00. \nIf EO  = 0, Pe*  = Po  El  is the minimum possible detection error. Let PJ and P~ denote FA \nand M errors when the release is ideal (El  =  0,  EO  =  0). It can be shown that \n\nPe = Pe*  + P~[Po(1- Ed  -\n\n(1  - Po)EO]  + PJ[(l - Po)(l  - EO)  - PoEl] \n\nP, =  PJ '  Pm  =  P~ + El (1  - P~ + PI) \n\nBoth PJ and P~ depend on TJ.  The optimal value ofT) is chosen such that Pe is minimized. \nIn general, PJ and P~ can not be expressed in closed-fonn and the optimal 'f}  is found using \nthe graphical ROC analysis procedure. Ifwe normalize a such that /-La  =  1, PJ and P~ can \nbe parametrically expressed in tenns ofa nonnalized threshold 'f}*,  PJ  =  0.5[1- Er f('f}*)], \nP~ = 0.5[1+ Iooo  Erf(TJ*  - JSNRa) P(a)  da].  J(X;Y) can be computed using the \nfonnula  for  the  mutual  infonnation for  a  binary channel,  J  =  1i (Po  (1  - Pm)  +  (1  -\nPo)  P,) - Po 1i(Pm )  - (1- Po)1i(P, ) where 1i(x) = -x log2 (x) - (1- x) log2(1- x)  is \nthe binary entropy function.  The analysis can be generalized to the case of n syna!Jses but \nthe  expressions involve n-dimensional integrals which need to be evaluated numerically. \nThe Central Limit Theorem can  be  used  to  simplify the  case  of very large  n.  Plots  of \nPe  and J(X; Y)  versus n for different values of SNR (1,10,00) for the case of identical \nsynapses  are  shown in Fig.  4a and Fig.  4b respectively.  Yet  again,  we  observe the  poor \nperfonnance of a single synapse and the substantial improvement due to redundancy.  The \nlinear increase of J with n is similar to the result obtained for signal estimation. \n\n5  Conclusions \n\nWe  find  that  a  single  synapse  is  rather  ineffective  as  a  communication device  but with \na  little  redundancy neuronal communication can  be  made  much  more  robust.  Infact,  a \nsingle  synapse  can  be  considered as  a  low  SNR  device,  while  5  independent synapses \nin parallel  approach a  high  SNR  system.  This  is  consistently echoed  in the  results  for \nsignal estimation and signal detection.  The values of infonnation rates we obtain are very \nsmall compared to numbers obtained from some peripheral sensory neurons (Rieke et.  ai, \n1996).  This could be due to an over-conservative choice of parameter values on our part \nor could  argue  for the  preponderance of redundancy in neural  systems.  What  we  have \npresented above are preliminary results of work in progress and so the path ahead is much \n\n2 For deterministic a,  the  result is  well-known,  but even  if a is a one-sided random variable,  the \n\nmatched filter can be shown to be optimal. \n\n\fSynaptic Tranrmission:  An lnformation-Theoretic Perspective \n\n207 \n\nb) \n\n-4-- SNR = In!. \n..... SNR=10 \n--SNR= 1 \n\n.\"r;:====~\"'-'-------::::::::~ \n\na) \n\n~ e w 0. \n\n0.' \n\n0 ... \n\n-4-- SNR = In!. \n..... SNR=10 \n--SNR=1 \n\ni' 0.7 \n~ 0 \u2022\u2022 \n~ ... \n:0 \n* ex:  0.4 \nc: \n.~  0.:1 \nE \n\u00a3  .., \n\n00.2 \n\no~,  --~~2~~--~3--~--~'--~~ \n\nNumber of Synapses (n) \n\n~~~--~2~----~3------~'----~ \n\nNumber of Synapses (n) \n\nPe  (a)  and  l,b  (b)  vs.  the  number  of synapses,  n, (different values  of SN R) for  signal  detection. \nSNR =  Inf.  corresponds  to  no  post-synaptic  voltage  noise.  All  the  synapses  are  assumed  to  be \nidentical.  Parameter values are po = 0.5,  101  = 0.6,  100  = 0,  eVa = 0.6, ts = 0.5 msec,  T  = 10  msec, \nan = 0.1  mY,  Bn = 100  Hz. \n\nlonger than the  distance  we  have covered so  far.  To  the best of our knowledge, analysis \nof distinct individual components of a neuron from an communications standpoint has not \nbeen carried out before. \n\nAcknowledgements \n\nThis research was supported by NSF, NIMH and the Sloan Center for Theoretical Neuro(cid:173)\nscience.  We thank Fabrizio Gabbiani for illuminating discussions. \n\nReferences \n\nBekkers, J.M., Richerson, G.B.  and Stevens, C.F.  (1990) \"Origin of variability in quantal \nsize in cultured hippocampal neurons and hippocampal slices,\" Proc.  Natl.  Acad.  Sci.  USA \n87: 5359-5362. \nBialek,  W.  Rieke,  F.  van  Steveninck,  R.D.R.  and Warland,  D.  (1991) \"Reading a  neural \ncode,\" Science 252:  1854-1857. \n\nCover, T.M., and Thomas, lA. (1991) Elements of Information Theory.  New York:  Wiley. \n\nKom,  H.  and  Faber,  D.S.  (1991) \"Quantal  analysis  and  synaptic  efficacy  in  the  CNS,\" \nTrends Neurosci.  14: 439-445. \n\nMarkram, H. and Tsodyks, T. (1996) \"Redistibution of synaptic efficacy between neocorti(cid:173)\ncal pyramidal neurons,\" Nature 382: 807-810. \n\nRieke, F.  Warland, D. van Steveninck, R.D.R. and Bialek, W.  (1996) Spikes:  Exploring the \nNeural Code.  Cambridge: MIT Press. \n\nStevens, C.F.  (1994) \"What form should a cortical theory take,\" In:  Large-Scale Neuronal \nTheories of the Brain, Koch, C. and Davis, J.L., eds., pp. 239-256. Cambridge: MIT Press. \n\n\f", "award": [], "sourceid": 1477, "authors": [{"given_name": "Amit", "family_name": "Manwani", "institution": null}, {"given_name": "Christof", "family_name": "Koch", "institution": null}]}