{"title": "Finding the Key to a Synapse", "book": "Advances in Neural Information Processing Systems", "page_first": 138, "page_last": 144, "abstract": null, "full_text": "Finding the Key to a Synapse \n\nThomas Natschlager &  Wolfgang Maass \nInstitute for Theoretical Computer Science \n\nTechnische Universitat Graz, Austria \n{tnatschl, maass}@igi.tu-graz.ac.at \n\nAbstract \n\nExperimental data have shown that synapses are heterogeneous: different \nsynapses respond with different sequences of amplitudes of postsynaptic \nresponses to the same spike train.  Neither the role of synaptic dynamics \nitself nor the  role  of the  heterogeneity of synaptic  dynamics  for  com(cid:173)\nputations in neural circuits is  well  understood.  We present in this article \nmethods that make it feasible to compute for a given synapse with known \nsynaptic parameters the spike train that is optimally fitted  to the synapse, \nfor example in the  sense that it produces the largest sum of postsynap(cid:173)\ntic responses.  To  our surprise we find  that most of these optimally fitted \nspike  trains  match  common firing  patterns  of specific  types  of neurons \nthat are discussed in the literature. \n\n1  Introduction \n\nA large number of experimental studies have  shown  that biological synapses have an  in(cid:173)\nherent dynamics, which controls how  the pattern of amplitudes of postsynaptic responses \ndepends on the temporal pattern of the incoming spike train.  Various  quantitative models \nhave been proposed involving a small number of characteristic parameters, that allow us to \npredict the response of a given synapse to  a given spike train once proper values for these \ncharacteristic  synaptic  parameters  have  been  found.  The analysis  of this  article  is  based \non  the  model  of [1],  where  three  parameters U,  F,  D  control  the  dynamics  of a  synapse \nand a fourth  parameter A - which corresponds to  the  synaptic \"weight\" in static  synapse \nmodels - scales the absolute sizes of the postsynaptic responses.  The resulting model pre(cid:173)\ndicts  the amplitude  Ak  for  the  kth  spike  in  a  spike  train  with  interspike intervals  (lSI's) \n.60 1 , .60 2 ,  \u2022 .. ,.6ok-l through the equations l \n\nAk = A\u00b7 Uk' Rk \nUk  =  U +Uk-l(1- U)exp(-.6ok-dF) \nRk = 1 + (Rk-l  - Uk-1Rk-l  - 1) exp( -.6ok-d D) \n\n(1) \n\nwhich  involve two  hidden  dynamic  variables  U  E  [0,1]  and  R  E  [0,1]  with  the  initial \nconditions Ul  =  U and Rl  =  1 for the first spike.  These dynamic variables evolve in  de(cid:173)\npendence of the synaptic parameters U,  F,  D  and the interspike intervals of the incoming \n\nITo  be precise:  the term Uk-1Rk-l in Eq.  (1)  was erroneously replaced by  ukRk-l in  the cor(cid:173)\n\nresponding Eq.  (2) of [1].  The model that they actually fitted to their data is the model considered in \nthis article. \n\n\fA \n\n0 .75 \n\nu \n!  0.5 \n\nC \n\n0 .25 \n\n0.75 \n\no  0 \n\nB \n\ninput spike train \n\nfA \n\nI IIIIII \nI I fi I \nF2  I lUI I \n\na \n\nIIIIIII  IIIII \nIII \nII m!  I I ]  I II \nI II \n~ lIn!  Un \n\n2 \n3 \ntime [sec) \n\n4 \n\n5 \n\nFigure  1:  Synaptic heterogeneity.  A The parameters U, D, and F  can be determined for \nbiological synapses.  Shown is the distribution of values for inhibitory synapses investigated \nin  [2]  which  can  be  grouped into  three mayor classes:  facilitating  (Ft), depressing  (F2) \nand recovering  (F3).  B  Synapses produce quite  different outputs  for the  same input for \ndifferent values of the parameters U,  D, and F.  Shown are the amplitudes Uk  \u2022 Rk  (height \nof vertical  bar)  of the  postsynaptic  response  of a  FI-type  and  a  F2-type  synapse  to  an \nirregular input spike train.  The parameters for synapses fA  and F2  are the mean values for \nthe synapse types FI and F2 reported in [2]:  (U, D, F) = (0.16,45 msec, 376 msec) for FI , \nand (0.25,706 msec, 21 msec) for F2 \u2022 \n\nspike train. 2  It is reported in [2]  that the synaptic parameters U,  F,  D  are quite heteroge(cid:173)\nneous, even within a single neural circuit (see Fig.  IA). Note that the time constants D  and \nF  are in the range of a few hundred  msec.  The synapses investigated in [2]  can be grouped \ninto  three  major classes:  facilitating  (FI),  depressing  (F2)  and  recovering  (F3).  Fig.  IB \ncompares  the  output of a  typical FI-type and  a typical F2-type synapse  in  response  to  a \ntypical irregular spike train.  One can see  that the  same input spike train  yields markedly \ndifferent outputs at these two synapses. \n\nIn this article we address the question which temporal pattern of a spike train is optimally \nfitted  to  a given synapse characterized by the three parameters U,  F,  D  in a certain sense. \nOne possible choice is to look for the temporal pattern of a spike train which produces the \nlargest integral of synaptic current.  Note that in the case where the dendritic integration is \napproximately linear the integral  of synaptic current is  proportional to  the sum  'E~=l A . \nUk . Rk of postsynaptic responses. We would like to stress, that the computational methods \nwe will  present are not restricted to  any particular choice of the  optimality criterion.  For \nexample one can use them also to compute the spike train which produces the largest peak \nof the  postsynaptic  membrane  voltage.  However,  in  the  following  we  will  focus  on  the \nquestion which temporal pattern of a spike train produces the largest sum 'E~=l A\u00b7 Uk . Rk \nof postsynaptic responses (or equivalently the largest integral of postsynaptic current). \n\nMore precisely, we fix a time interval T , a minimum value ~min for lSI's, a natural number \nN , and synaptic parameters U, F, D . We then look for that spike train with N  spikes during \nT  and lSI's 2::  ~min that maximizes 'E~=l A\u00b7 Uk' Rk.  Hence we  seek for a solution(cid:173)\nthat is  a sequence ofISI's ~l' ~2' ... ,  ~N-I -\n\nto the optimization problem \n\nN \n\nmaximize LA. Uk  . Rk under  L ~k ~ T  and ~min ~ ~k' 1 ~ k  < N  . \n\nN-I \n\n(2) \n\nk=l \n\nk=l \n\nIn Section 2 of this article we present an algorithmic approach based on dynamic program-\n\n2It should be noted that this deterministic model predicts the cumulative response of a population \n\nof stochastic release sites that make up a synaptic connection. \n\n\fming that is  guaranteed to  find  the  optimal  solution  of this  problem  (up  to  discretization \nerrors),  and exhibit for major types  of synapses temporal patterns  of spike trains  that are \noptimally  fitted  to  these  synapses.  In  Section 3 we  present a  faster heuristic  method for \ncomputing optimally fitted  spike trains, and apply it to  analyze how their temporal pattern \ndepends on the number N  of allowed spikes during time interval T, i.e., on the firing rate \nf  = NIT.  Furthermore we analyze in  Section 3 how changes in the synaptic parameters \nU, F, D  affect the temporal pattern of the optimally fitted  spike train. \n\n2  Computing Optimal Spike Trains for Common Types of Synapses \nDynamic Programming  For T  = 1000 msec and  N  = 10  there are  about 2100  spike \ntrains among which one wants to find the optimally fitted one. We show that a computation(cid:173)\nally  feasible  solution to  this  complex optimization problem can be achieved via dynamic \nprogramming.  We  refer to  [3]  for the  mathematical background of this  technique,  which \nalso underlies the computation of optimal policies in reinforcement learning.  We consider \nthe discrete time dynamic system described by the equation \n\nXl  =  (U, 1, 0)  and  Xk+1  =  g(Xk, ak)  for  k  =  1, ... , N  - 1 \n\n(3) \n\nwhere Xk  describes the state of the system at step k, and ak is the \"control\" or \"action\" taken \nat step k.  In  our case Xk  is  the triple  (Uk, Rk, tk) consisting of the values of the dynamic \nvariables  U  and  R  used  to  calculate  the  amplitude  A  . Uk  .  Rk  of the  kth  postsynaptic \nresponse,  and  the  time  tk  of the  arrival of the  kth  spike at the  synapse.  The \"action\" ak \nis  the  length  Ilk  E  [Ilmin , T  -\ntkJ  of the  kth  lSI  in  the  spike  train  that  we  construct, \nwhere  Ilmin  is  the  smallest possible  size  of an  lSI  (we  have  set  Ilmin  = 5 msec in  our \ncomputations).  As  the function gin Eq. (3) we  take the function which maps  (Uk, Rk, tk) \nand  Ilk  via Eq. (1)  on  (uk+l,Rk+l,tk+1)  for  tk+1  = tk  +  Ilk.  The  \"reward\"  for  the \nkth  spike is  A  . Uk  . Rk, i.e., the amplitude of the postsynaptic response for the kth spike. \nHence maximizing the total reward J(Xl) = 2:~=1 A\u00b7 Uk\u00b7 Rk is equivalent to solving the \nmaximization problem (2). The maximal possible value of J l  (Xl) can be computed exactly \nvia the equations \n\nIN(XN) =  A\u00b7 UN\u00b7 RN \nJk(Xk) = \n\nmax \n\n~E[~min,T-tkl \n\n(A\u00b7 Uk\u00b7 Rk + Jk+1(g(Xk, Il))) \n\n(4) \n\nbackwards  from  k  = N  - 1  to  k  = 1.  Thus  the  optimal  sequence  al, ... , aN-l  of \n\"actions\" is the sequence Ill, .. .  , IlN -1 of lSI's that achieves the maximal possible value \nof 2:~=1 A . Uk  . Rk .  Note that the evaluation of Jk(Xk) for a single value of Xk  requires \nthe evaluation of Jk+1 (Xk+1) for many different values of Xk+1. 3 \n\nThe \"Key\" to a Synapse  We  have applied the dynamic programming approach to three \nmajor types  of synapses reported in  [2].  The results  are  summarized in  Fig. 2 to  Fig.  5. \nWe  refer informally to  the  temporal pattern  of N  spikes  that maximizes  the  response  of \na  particular  synapse  as  the  \"key\"  to  this  synapse.  It is  shown  in  Fig. 3  that  the  \"keys\" \nfor  the  inhibitory synapses  Fi  and  F2  are  rather specific  in  the  sense  that  they  exhibit a \nsubstantially  smaller postsynaptic response on  any  other of the  major types  of inhibitory \nsynapses reported in  [2].  The  specificity  of a \"key\" to  a synapse is  most pronounced for \nspiking frequencies f  below 20 Hz.  One may speculate that due to this feature a neuron can \nactivate -\na particular subpopulation of its  target \nneurons  by  generating  a  series  of action  potentials  with  a  suitable  temporal  pattern,  see \n\neven without changing its  firing rate -\n\n3When one solves Eq. (4) on a computer, one has to replace the continuous state variable Xk  by a \ndiscrete variable Xk , and round Xk+l  := g(Xk'~) to  the nearest value of the corresponding discrete \nvariable Xk+l.  For more details about the discretization of the  model we refer the reader to [4]. \n\n\f0.75 \n\n~ .!!!..  0.5 \no \n\n0.25 \n\nFl \n\nIII  I I  I  I  I  I  I  I  I  I  I \n\nII \n\nI \n\nI \n\nF3  II \ni.~ 0.5 \no \n\na  a \n\n0.25 \n\nu \n\n0.2 \n\n0.4 \n\n0.6 \n\n0.8 \n\ntime [sec] \n\nFigure  2:  Spike  trains  that maximize  the  sum  of postsynaptic  responses  for  three  com(cid:173)\nmon  types  of synapses  (T  =  0.8 sec,  N  =  15  spikes).  The  parameters  for  synapses \nFi,  F2, and  F3  are  the  mean  values  for  the  synapse  types  FI,  F2  and  F3  reported  in \n[2] :  (U, D, F} = (0.16,45 msec, 376 msec}  for Fl ,  (0.25,706 msec, 21 msec}  for F2 ,  and \n(0.32, 144 msec, 62 msec} for F3 . \n\nkey to synapse Fl \nII I I I I  I  I  I  I  I  I  I  I \nkey to synapse F2 \nIIIII \n\nIIII \n\nresponse of a \n\nresponse of a \n\nFl-type synapse  F2-type synapse \n\nI \nII \n\n({/ % \n\n81% I \nI \n\nthe  \"keys\"  to  the  synapses  Fl  and  F2  -\n\nFigure 3:  Specificity of optimal spike trains. The optimal spike trains for synapses Fl and \nobtained for T  = 0.8 sec  and  N  = 15 \nF2  -\nspikes are  tested on the synapses Fl  and F2 .  If the \"key\" to  synapse Fl  (F2 )  is  tested on \nthe synapse Fl (F2 )  this  synapse produces the maximal  (l00 %) postsynaptic response.  If \non the other hand the  \"key\" to  synapse Fl  (F2) is  tested on synapse F2  (Fl) this  synapse \nproduces significantly less postsynaptic response. \n\nFig.  4.  Recent experiments [5,  6]  show  that neuromodulators can control the firing  mode \nof cortical neurons.  In  [5]  it is  shown that bursting  neurons  may  switch  to  regular firing \nif norepinephine is  applied.  Together with the specificity  of synapses to  certain temporal \npatterns these findings point to one possible mechanism how neuromodulators can change \nthe effective connectivity of a neural circuit. \n\nRelation to discharge patterns  A noteworthy aspect of the \"keys\" shown in Fig. 2 (and \nin  Fig.  6  and  Fig.  7)  is  that  they  correspond  to  common  firing  patterns  (\"accommodat(cid:173)\ning\",  \"non-accommodating\", \"stuttering\", \"bursting\"  and  \"regular firing\") of neocortical \ninterneurons reported under controlled conditions in vitro  [2,  5]  and in  vivo  [7].  For ex(cid:173)\nample  the  temporal patterns of the  \"keys\"  to  the  synapses Fl ,  F2, and  F3  are  similar to \nthe discharge patterns of \"accommodating\" [2], \"bursting\" [5, 7], and \"stuttering\" [2] cells \nrespectively. \n\nWhat is the role of the parameter A?  Another interesting effect arises if one compares \nthe optimal values of the sum Ek=l Uk  .  Rk (i.e.  A = 1) for synapses H, F 2 ,  and  F3  (see \nFig.  5A) with  the  maximal values of E~=l A . Uk  \u2022  Rk  (see Fig.  5B), where we have set \n\nN \n\n-\n\n-\n\n-\n\n\fsynaptic response \n\nkey to synapse Fl \n\n11111111  I I I I  I I I \n\nFl lo--\n\nsynaptic response \n\nFl i 0--\n\nkey to synapse F2 \n\n-te--=-III  -----....:11=---------=.11_-=-1 ~< I \n\nFigure 4:  Preferential addressing of postsynaptic targets.  Due to the specificity of a \"key\" to \na synapse a presynaptic neuron may address (i.e. evoke stronger response at) either neuron \nA  or B,  depending  on  the  temporal  pattern  of the  spike  train  (with  the  same  frequency \nf  = NIT) it produces (T = 0.8 sec and N  = 15 in this example). \n\nF2 \n\n(1-\n\nA  4  ,-----\n\n-\n\n-\n\nB  15 \n(3 \n~10 \n\n,-----\n\nr - - - - -\n\n-\n\nA::1 \n\nA::1 \n\nA::1 \n\nA::3.24 \n\nA=7.76 \n\nA:3.44 \n\no \n\nFl \n\nF2 \n\nF3 \n\no \n\nFl \n\nF2 \n\nF3 \n\nFigure 5:  A Absolute values of the sums 2::=1 Uk  .  Rk if the key to  synapse Pi  is  applied \nto  synapse Pi, i  =  1,2,3. B Same as panel  A except that the value of 2::=1 A . Uk  .  Rk is \nplotted.  For A we  used the value of G max  (in nS) reported in  [2].  The quotient max I min \nis  1.3 compared to  2.13 in panel A. \n\nA equal to  the value of Gmax reported in  [2].  Whereas the values of Gmax  vary  strongly \namong different synapse types (see Fig. 5B), the resulting maximal response of a synapse \nto its proper \"key\" is almost the same for each synapse.  Hence, one may speculate that the \nsystem is designed in such a way  that each synapse should have an  equal influence on the \npostsynaptic neuron when it receives its  optimal  spike train.  However,  this effect is  most \nevident for a spiking frequency f  = NIT of 10 Hz and vanishes for higher frequencies. \n\n3  Exploring the Parameter Space \n\nSequential Quadratic Programming  The numerical approach for approximately com(cid:173)\nputing optimal spike trains that was used in  section 2 is sufficiently fast  so that an  average \nPC can carry out any of the computations whose results were reported in Fig. 2 within a few \nhours.  To be able to address computationally more expensive issues we used a a nonlinear \noptimization algorithm known as \"sequential quadratic programming\" (SQP)4 which is the \nstate of the art approach for heuristically solving constrained optimization problems such \nas  (2).  We  refer the reader to  [8]  for the  mathematical background of this  technique and \nto  [4] for more details about the application of SQP for approximately computing optimal \nspike trains. \n\nOptimal Spike  Trains for  Different Firing  Rates  First  we  used  SQP  to  explore  the \neffect of the spike frequency f  = N IT on the temporal pattern of the optimal spike train. \nFor the  synapses  PI,  P2 ,  and Pa  we  computed  the  optimal  spike  trains  for  frequencies \n\n4We used the implementation  (function  constr) which is contained in  the MATLAB Optimiza(cid:173)\n\ntion Toolbox (see http : //www . ma thworks . com/products/ optimiz a tion/). \n\n\fkeys to Fl synapse \n\n111111 11111 11 \n\nh  40 11 111111 111 1111111111111111111111111 \n\n--~ 35 111 11 1111 11 \n!. 30 11 111 11111  11111 111111 \ng 25 111 11111 1  11 111111 1  II I \n~ 20 11 1111 1 1  1 1 111 1 1 \n~ 15 1111 11 1  111 1  I  I \n\nIII \n\nI\n\nI \n\nh  40 ~ \n\n11 11  --~ 35 1 \n\nII \n\"-. 30 I \nC  25 1 \nt:: \ng.  20 1 \nQ) \n~ 15 1 \n\nkeys  to F2  synapse \nI  I  1  I  I  I  I \nI  I  I  1  1  1 I \nI  I  I  1  1  1 \n\nI \n\nI \n\nkeys to F3  synapse \n\nh  40 111111111 11111 1111 111 1111 11111 11111 11 \n\n11 111111 11111 11111 111111 111111 \n\n11 11 1111 111 111111 111 1111 \n\n111 111 111 1111 111 1111 \n\nII \n\nII \n\nII \n\nII \n\nI I \n\nII \n\nI I  I \n\n--~ 35 \n\nII \n\"-. 30 \n\nC  25 \nt:: \ng.  20 \nQ) \n<./::  15 \n\nQ) \n\n0 \n\n0.2 \n\n0.4 \n0.6 \ntime [sec] \n\n0.8 \n\n0.2 \n\n0.4 \n0.6 \ntime [sec] \n\n0.8 \n\n0 \n\n0.2 \n\n0.4 \n0.6 \ntime [sec] \n\n0.8 \n\nFigure 6:  Dependence of the  optimal  spike  train  of the  synapses FL  F2 ,  and  F3  on  the \nspike frequency f  = NIT (T = 1 sec, N  = 15, ... ,40). \n\n0 .60 \n0.50 \n0 .45 \n0.40 \n0 .35 \n::)  0 .30 \n0 .25 \n0.20 \n0.15 \n0 .10 \n\nI \nI \nI \n\nI \n\nI \nI \nI \n\nI \nI \nI \nII \nII \n\nI \nI \nI \n\nI \nI \nI \n\nII \nII \n\n1111 \n\nIII \n\nI \nI \nI \n\nII \nII \n\nI \nI \nI \n\nIII \n\n11111 \n\ni \n0 \n\n11111 \n\ni \n\n0.25 \n\nI \nI \nI \n\nI \nI \nI \n\nII \nII \n\n1111 \n\ni \n\n0 .5 \n\nI \nI \nI \n\nI \nI \nI \n\nI \nI \nI \n\nIII \n\nII \nII \n\n11111 \n\nI \nI \nI \n\nIII \n\nII \nII \n\n1111 \n\ni \n\n0.75 \n\ntime [sec] \n\nI \nI \nI \n\nI \nI \nI \nII \n\nII \n\nI \nI \nI \n\nII \nII \nII \nI  II \nIII \n1111 \n\nFigure 7:  Dependence of the optimal spike train on the synaptic parameter U.  It is  shown \nhow the optimal spike train changes if the parameter U is varied. The other two parameters \nare  set  to  the  value  corresponding  to  synapse  F3:  D  = 144 msec  and  F  = 62 msec. \nThe  black bar to  the  left marks  the  range of values  (mean  \u00b1  std)  reported in  [2]  for  the \nparameter U.  To  the right of each spike train we have plotted the corresponding value of \nJ = Ef=i ukRk  (gray bars). \n\nranging from 15 Hz to 40 Hz.  The results are summarized in Fig.  6.  For synapses Fi  and \nF2  the characteristic spike pattern (Fi  ...  accommodating, F2  ...  stuttering) is  the same for \nall  frequencies.  In  contrast,  the  optimal spike train for  synapse F3  has  a phase  transition \nfrom \"stuttering\" to \"non-accommodating\" at about 20 Hz. \n\nThe Impact of Individual Synaptic Parameters  We will now address the question how \nthe  optimal spike train depends on the individual synaptic parameters U,  F, and  D.  The \nresults for the case of F3-type synapses and the parameter U are summarized in Fig. 7.  For \nresults with regard to  other parameters and synapse types we refer to [4].  We have marked \nin  Fig.  7 with  a black bar the range of U for F3-type synapses reported in  [2].  It can  be \nseen that within this parameter range we find \"regu]ar\" and \"bursting\" spike patterns.  Note \nthat the sum of postsynaptic responses J  (gray horizontal bars in Fig. 7) is not proportional \nto  U.  While U increases from 0.1  to 0.6 (6 fold change) J  only increases by a factor of 2. \nThis  seems to be interesting since the parameter U  is  closely related to  the initial  release \nprobability of a synapse, and it is a common assumption that the \"strength\" of a synapse is \nproportional to its initial release probability. \n\n\f4  Discussion \n\nWe  have  presented  two  complementary  computational  approaches  for  computing  spike \ntrains  that  optimize a  given  response criterion for  a given  synapse.  One  of these  meth(cid:173)\nods is based on dynamic programming (similar as in reinforcement learning), the other one \non  sequential quadratic programming.  These computational methods are not restricted to \nany particular choice of the optimality criterion and the synaptic model.  In [4]  applications \nof these methods to other optimality criteria, e.g. maximizing the specificity, are discussed. \n\nIt turns out that the spike trains that maximize the response of Fl-, F2- and F3-type synapses \n(see Fig.  1) are well known firing patterns like \"accommodating\", \"bursting\" and \"regular \nfiring\"  of specific  neuron  types.  Furthermore for  Fl- and  F3-type synapses  the  optimal \nspike train agrees with the most often found firing pattern of presynaptic neurons reported \nin  [2],  whereas  for F2-type  synapses  there is  no  such agreement;  see  [4].  This  observa(cid:173)\ntion provides the first glimpse at a possible functional role of the specific combinations of \nsynapse types and neuron types that was recently found in [2]. \n\nAnother noteworthy aspect of the optimal spike trains is their specificity for a given synapse \n(see  Fig.  3).:  suitable  temporal firing  patterns  activate  preferentially  specific  types  of \nsynapses.  One  potential functional  role  of such  specificity  to  temporal  firing  patterns  is \nthe possibility of preferential addressing of postsynaptic target neurons (see Fig. 4).  Note \nthat there is experimental evidence that cortical neurons can switch their intrinsic firing be(cid:173)\nhavior from \"bursting\" to  \"regular\" depending on neuromodulator mediated inputs [5,  6]. \nThis findings provide support for the idea of preferential addressing of postsynaptic targets \nimplemented by the interplay of dynamic synapses and the intrinsic firing  behavior of the \npresynaptic neuron. \n\nFurthermore our analysis provides the platform for a deeper understanding of the specific \nrole of different synaptic parameters, because with the help of the computational techniques \nthat we have introduced one can now see directly how the temporal structure of the optimal \nspike train for  a  synapse depends on the individual synaptic parameters.  We  believe that \nthis inverse analysis is essential for understanding the computational role of neural circuits. \n\nReferences \n[1]  H. Markram, Y.  Wang, and M. Tsodyks.  Differential signaling via the same axon  of neocortical \n\npyramidal  neurons.  Proc.  Natl. Acad. Sci. , 95:5323- 5328, 1998. \n\n[2]  A. Gupta, Y.  Wang,  and H.  Markram.  Organizing principles for a diversity  of GABAergic  in(cid:173)\n\nterneurons and synapses in the neocortex.  Science,  287:273- 278, 2000. \n\n[3]  D.  P.  Bertsekas.  Dynamic  Programming  and  Optimal  Control,  Volume  1.  Athena  Scientific, \n\nBelmont, Massachusetts,  1995. \n\n[4]  T. Natschlager and  W.  Maass.  Computing the  optimally  fitted  spike  train  for  a  synapse.  sub(cid:173)\n\nmitted  for  publication,  electronically  available  via  http : //www. igi . TUGr a z .at/igi/ \ntn a tschl/psfiles/synkey-journal. ps . gz, 2000. \n\n[5]  Z. Wang and D. A. McCormick.  Control of firing mode of corticotectal and corticopontine layer \nV burst generating neurons by norepinephrine. Journal of Neuroscience, 13(5):2199-2216, 1993. \n\n[6]  J.  C.  Brumberg,  L.  G.  Nowak,  and  D.  A.  McCormick.  Ionic  mechanisms  underlying  repet(cid:173)\n\nitive  high  frequency  burst  firing  in  supragranular  cortical  neurons.  Journal  of Neuroscience, \n20(1):4829-4843, 2000. \n\n[7]  M. Steriade, I. Timofeev, N.  Diirmiiller, and F.  Grenier.  Dynamic properties of corticothalamic \nneurons  and local cortical interneurons generating fast rhytmic (30-40 hz)  spike bursts.  Journal \nof Neurophysiology, 79:483-490,  1998. \n\n[8]  M.  J.  D.  Powell.  Variable  metric  methods  for  constrained  optimization. \n\nIn  A.  Bachem, \nM  Grotschel ,  and  B.  Korte,  editors,  Mathematical  Programming:  The  State  of the  Art, pages \n288- 311. Springer Verlag, 1983. \n\n\f", "award": [], "sourceid": 1813, "authors": [{"given_name": "Thomas", "family_name": "Natschl\u00e4ger", "institution": null}, {"given_name": "Wolfgang", "family_name": "Maass", "institution": null}]}