{"title": "On the Computational Complexity of Networks of Spiking Neurons", "book": "Advances in Neural Information Processing Systems", "page_first": 183, "page_last": 190, "abstract": null, "full_text": "On the  Computational Complexity of Networks of \n\nSpiking  Neurons \n(Extended Abstract) \n\nWolfgang Maass \n\nInstitute for  Theoretical Computer Science \n\nTechnische  Universitaet  Graz \n\nA-80lO  Graz, Austria \n\ne-mail:  maass@igi.tu-graz.ac.at \n\nAbstract \n\nWe investigate the computational power of a formal model for net(cid:173)\nworks of spiking neurons,  both for  the assumption of an unlimited \ntiming precision, and for  the case of a limited timing precision.  We \nalso prove upper and lower bounds for the number of examples that \nare  needed  to train such networks. \n\n1 \n\nIntroduction and  Basic Definitions \n\nThere  exists substantial evidence  that timing phenomena such  as  temporal  differ(cid:173)\nences  between  spikes  and  frequencies  of oscillating  subsystems  are  integral  parts \nof various  information processing  mechanisms  in  biological  neural  systems  (for  a \nsurvey  and references  see  e.g.  Abeles,  1991; Churchland and Sejnowski,  1992; Aert(cid:173)\nsen,  1993).  Furthermore simulations of a  variety  of specific  mathematical models \nfor  networks of spiking neurons have shown  that temporal coding offers interesting \npossibilities for  solving  classical  benchmark-problems such  as  associative  memory, \nbinding, and pattern segmentation (for an overview see Gerstner et al., 1992).  Some \naspects  of these  models have  also  been studied  analytically, but  almost nothing is \nknown about their computational complexity (see Judd and Aihara, 1993, for some \nfirst  results  in  this  direction).  In  this article  we  introduce  a  simple formal model \nSNN  for  networks  of spiking neurons  that allows us  to model the  most  important \ntiming phenomena of neural nets (including synaptic modulation), and we prove up(cid:173)\nper and lower bounds for its computational power and learning complexity.  Further \n\n\f184 \n\nWolfgang  Maass \n\ndetails to the results reported  in this article  may be found  in Maass,  1994a,1994b, \n1994c. \nDefinition of a  Spiking Neuron Network  (SNN):  An SNN N  consists  of \n- a finite  directed  graph  {V, E}  (we  refer to  the  elements  of V  as  \"neurons\" \n\nand  to  the  elements  of E  as  \"synapses\") \n\n- a subset  Yin  S;  V  of input  neurons \n- a subset  Vout  S;  V  of output  neurons \n- for  each  neuron  v E V  - Yin  a threshold-function  9 v : R+  -+ R  U {oo} \n\n{where  R+ := {x E R  : x  ~ O}) \n\n- for  each  synapse  {u, v}  E  E  a  response-function  \u00a3u,v \n\n:  R+  -+  R  and  a \n\nweight- function  Wu,v  : R+  -+ R \n\n. \n\nWe  assume  that  the firing  of the  input  neurons v  E  Yin  is  determined from  outside \nof N,  i.e.  the  sets  Fv  S;  R+  of firing  times  (\"spike  trains\")  for  the  neurons  v  E \nYin  are  given  as  the  input  of N.  Furthermore  we  assume  that  a  set  T  S;  R+  of \npotential firing  times 7iiiSfjeen  fixed. \nFor  a  neuron  v  E  V  - Yin  one  defines  its  set  Fv  of firing  times  recursively.  The \n,  and for  any  s  E  Fv  the  next \nfirst  element  of Fv  is \nlarger  element  of Fv  is \n,where  the \npotential function  Pv : R+ -+ R  is  defined  by \nL \n\ninf{t  E  T  :  t  >  sand Pv(t)  ~ 0 v(t  - s)} \n\ninf{t  E  T  : Pv(t)  ~ 0 v(O)} \n\nPv(t)  := 0 +  L \n\nu : {u, v}  E EsE Fu  : s < t \n\nwu,v(s)  . \u00a3u,v(t - s) \n\nThe  firing  times  (\"spike  trains\")  Fv  of the  output  neurons  v  E  Vout  that  result  in \nthis  way  are  interpreted  as  the  output  of N. \nRegarding  the  set  T  of potential  firing  times  we  consider  in  this  article  the  case \nT  =  R+  (SNN  with continuous time)  and the  case  T  =  {i\u00b7 JJ  : i  E  N} for  some JJ \nwith 1/ JJ  E  N  (SNN  with discrete  time). \nWe  assume  that for  each  SNN  N  there  exists  a  bound  TN  E  R  with  TN  >  0 such \nthat  0 v(x)  = 00  for  all  x  E  (0, TN)  and  all  v  E  V  - Yin  (TN  may  be  interpreted \nas  the minimum of all  \"refractory periods\"  Tref  of neurons in N).  Furthermore we \nassume  that  all  \"input  spike  trains\"  Fv  with  v  E  Yin  satisfy  IFv  n [0, t]l  <  00  for \nall t  E R+.  On the basis of these  assumptions one  can  also  in the  continuous  case \neasily  show  that the firing  times are  well-defined  for  all  v  E  V  - Yin  (and occur  in \ndistances of at least  TN)' \nInput- and  Output-Conventions:  For  simulations between  SNN's  and  Turing \nmachines  we  assume  that  the  SNN  either  gets  an  input  (or  produces  an  output) \nfrom  {O, 1}*  in the form of a  spike-train (i.e.  one bit per  unit  of time), or encoded \ninto the phase-difference  of just two spikes.  Real-valued input or output for an SNN \nis  always encoded  into the phase-difference  of two spikes. \n\nRemarks \na)  In  models  for  biological  neural  systems one  assumes  that  if x  time-units have \n\n\fOn  the  Computational  Complexity  of Networks  of Spiking  Neurons \n\n/85 \n\npassed  since  its last firing,  the  current  threshold  0 11 (z)  of a  neuron  v  is  \"infinite\" \nfor  z  <  TreJ  (where  TreJ  =  refractory  period  of neuron  v),  and  then  approaches \nquite rapidly from above some constant  value.  A  neuron  v  \"fires\"  (i.e.  it sends  an \n\"action potential\"  or  \"spike\"  along its axon)  when  its current  membrane potential \nPII (t)  at  the  axon  hillock  exceeds  its  current  threshold  0 11 .  PII (t)  is  the  sum  of \nvarious postsynaptic potentials W U ,II(S). t: U ,II(t - s).  Each of these terms describes an \nexcitatory (EPSP) or  inhibitory (IPSP)  postsynaptic  potential at the axon hillock of \nneuron v at time t, as a result of a spike that had been generated by a  \"presynaptic\" \nneuron u at time s, and which has been transmitted through a synapse between both \nneurons.  Recordings of an EPSP typically show a function that has a constant value \nc (c =  resting membrane potential; e.g.  c = -70m V) for  some initial time-interval \n(reflecting  the  axonal and synaptic transmission time),  then  rises  to  a  peak-value, \nand finally  drops  back  to  the  same constant  value  c.  An  IPSP  tends  to  have  the \nnegative shape of an EPSP.  For  the sake of mathematical simplicity we  assume in \nthe  SNN-model  that  the  constant  initial and  final  value  of all  response-functions \nt:U ,1I  is equal to 0 (in other words:  t:U ,1I  models the  difference between a postsynaptic \npotential and the  resting  membrane potential c).  Different  presynaptic  neurons  u \ngenerate  postsynaptic  potentials of different  sizes  at  the  axon  hillock  of a  neuron \nv,  depending  on  the size,  location  and  current  state  of the  synapse  (or  synapses) \nbetween  u  an? v.  This effect  is modelled by  the  weight-factors W U ,II(S). \nThe  precise  shapes  of threshold-,  response-,  and weight-functions  vary  among dif(cid:173)\nferent  biological neural systems,  and even within the same system.  Fortunately one \ncan  prove  significant  upper  bounds for  the  computational complexity  of SNN's N \nwithout any assumptions about the specific shapes of these functions of N.  Instead, \nwe  only  assume that they are of a  reasonably simple  mathematical structure. \nb) In order to prove  lower  bounds for  the  computational complexity of an SNN N \none  is  forced  to  make more specific  assumptions about  these  functions .  All  lower \nbound results  that are reported  in this  article require only  some rather  weak  basic \nassumptions  about  the  response- and  threshold-functions.  They  mainly  require \nthat  EPSP's  have  some  (arbitrarily  short)  segment  where  they  increase  linearly, \nand some  (arbitrarily  short)  segment  where  they  decrease  linearly  (for  details  see \nMaass,  1994a,  1994b). \nc) Although the model SNN  is  apparently more  \"realistic\"  than all  models for  bio(cid:173)\nlogical neural  nets  whose  computational complexity has  previously  been  analyzed, \nit deliberately  sacrifices  a  large number of more  intricate biological  details for  the \nsake  of mathematical tractability.  Our  model  is  closely  related  to  those  of (Buh(cid:173)\nmann and Schulten,  1986),  and (Gerstner,  1991,  1992).  Similarly as  in  (Buhmann \nand Schulten,  1986)  we  consider here  only the deterministic  case. \nd) The model SNN is also suitable for investigating algorithms that involve synaptic \nmodulation at various time-scales.  Hence one can investigate within this framework \nnot only the complexity of algorithms for supervised and unsupervised learning, but \nalso the potential computational power of rapid weight-changes  within the course of \na  computation.  In  the  theorems of this  paper  we  allow  that  the  value  of a  weight \nWU,II(S)  at a firing  time s  E Fu  is  defined  by  an  algebraic  computation  tree (see  van \nLeeuwen,  1990)  in  terms of its value  at previous  firing  times s'  E  Fu  with  s'  < s, \nsome  preceding  firing  times  s < s  of arbitrary  other  neurons,  and  arbitrary  real(cid:173)\nvalued parameters.  In this way WU,II(S)  can be defined by different rational functions \n\n\f/86 \n\nWolfgang  Maass \n\nof the abovementioned arguments, depending on the numerical relationship between \nthese  arguments  (which  can  be  evaluated  by  comparing first  the  relative  size  of \narbitrary rational functions  of these  arguments).  As  a  simple special  case  one  can \nfor  example increase  wu \u2022tI  (perhaps  up  to some specified  saturation-value)  as  long \nas neurons  u  and v  fire  coherently,  and decrease  wu \u2022tI  otherwise. \n\nFor the sake of simplicity in the statements of our results we assume in this extended \nabstract  that  the  algebraic  computation  tree  for  each  weight  w U \u2022tI  involves  only \n0(1) tests  and rational functions of degree  0(1)  that  depend  only  on  0(1) of the \nabovementioned arguments.  Furthermore we  assume  in Theorems 3,  4  and 5  that \neither each weight is an arbitrary time-invariant real, or that each current weight is \nrounded off to bit-length poly(1ogpN') in binary representation, and does not depend \non  the  times  of firings  that  occured  longer  than  time 0(1)  ago.  Furthermore  we \nassume in Theorems 3 and 5 that the parameters in the algebraic computation tree \nare rationals of bit-length O(1ogpN'). \n\ne)  It is  well-known  that the  Vapnik-Chervonenkis  dimension  {\"VC-dimension\"}  of \na neural net N  (and the pseudo-dimension for the  case of a neural net N  with  real(cid:173)\nvalued output,  with some suitable fixed  norm for  measuring the error)  can  be  used \nto bound the number of examples that are needed  to train N  (see  Haussler,  1992). \nObviously  these  notions  have  to  be  defined  differently  for  a  network  with  time(cid:173)\ndependent  weights.  We  propose  to  define  the  VC-dimension  (pseudo-dimension)of \nan SNN N  with time-dependent  weights as  the VC-dimension (pseudo-dimension) \nof the class of all functions that can be computed by N  with different assignments of \nvalues to the real-valued  (or rational-valued) parameters of N  that are  involved in \nthe definitions of the  piecewise  rational response-,  threshold-,  and weight-functions \nof N.  In a  biological neural system N  these  parameters might for  example reflect \nthe  concentrations of certain  chemical substances  that are known  to modulate the \nbehavior of N. \nf) The focus in the investigation of computations in biological neural systems differs \nin  two  essential  aspects  from  that  of classical  computational  complexity  theory. \nFirst, one is not only interested in single computations of a neural net for unrelated \ninputs z, but also in its ability to process an interrelated sequence  \u00ab(z( i), y( i)} )ieN \nof inputs and  outputs,  which  may for  example include an initial training sequence \nfor  learning or associative memory.  Secondly,  exact  timing of computations is  all(cid:173)\nimportant  in  biological  neural  nets,  and  many  tasks  have  to  be  solved  within  a \nspecific number of steps.  Therefore an analysis in terms of the notion of a  real-time \ncomputation  and  real-time  simulation appears  to  be  more  adequate for  models  of \nbiological neural nets than the more traditional analysis via complexity classes. \nOne  says  that  a  sequence  \u00ab(z(i),y(i)})ieN  is  processed  in  real-time  by  a  machine \nM, if for  every  i  E N  the machine M  outputs  y( i)  within a  constant  number c  of \ncomputation steps after having received  input z(i).  One says that M'  simulates M \nin  real-time (with delay  factor  ~), if every  sequence  that is  processed  in real-time \nby  M  (with  some  constant  c),  can  also  be  processed  in  real-time  by  M'  (with  a \nconstant  ~ . c).  For SNN's M  we  count  each spike in M  as a  computation step. \nThese  definitions imply that a  real-time simulation of M  by  M' is a  special case of \na linear-time simulation, and hence that any problem that can be solved by  M  with \na  certain time complexity ten),  can be solved  by  M' with time complexity O(t(n\u00bb \n\n\fOn  the  Computational  Complexity  of Networks  of Spiking  Neurons \n\n187 \n\n(see  Maass,  1994a,  1994b, for  details). \n\n2  Networks of Spiking Neurons with Continuous  Time \n\nTheorem 1:  If the  response- and  threshold-functions  of the  neurons  satisfy  some \nrather weak  basic  assumptions  (see  Maass,  1994a,  1994b),  then  one  can  build from \nsuch  neurons  for  any  given  dEN an  SNN NTM(d)  of finite  size  with  rational \ndelays  that  can  simulate  with  a suitable  assignment  of rational values  from  [0, 1]  to \nits weights  any  Turing  machine  with  at  most d  tapes  in  real-time. \n\nFurthermore  NTM(2)  can  compute  any  function  F  :  {0,1}*  -- {0,1}*  with  a \n\nsuitable  assignment  of real  values  from [0,\"1]  to  its weights. \n\nThe fixed  SNN  NTM(d)  of Theorem  1  can  simulate Turing  machines  whose  tape \ncontent is much larger than the size of NTM (d),  by encoding such tape content into \nthe phase-difference  between two oscillators.  The proof of Theorem 1 transforms ar(cid:173)\nbitrary computations of Turing machines into operations on such phase-differences. \n\nThe  last  part  of Theorem  1  implies  that  the  VC-dimension  of some finite  SNN's \nis  infinite.  In  contrast  to  that  the  following  result  shows  that  one  can  give  finite \nbounds for  the  VC-dimension of those  SNN's that only  use  a  bounded numbers of \nspikes  in  their  computation.  Furthermore the last  part  of the  claim of Theorem 2 \nimplies that  their  VC-dimension  may  in fact  grow  linearly  with  the  number  S  of \nspikes that occur in  a  computation. \n\nTheorem 2:  The  VC-dimension  and  pseudo-dimension  of any  SNN N  with  piece(cid:173)\nwise  linear response- and threshold-functions,  arbitrary  real-valued  parameters  and \ntime-dependent  weights  (as  specified  in  section  1)  can  be  bounded  (even  for  real(cid:173)\nvalued  inputs  and  outputs)  by  D(IEI . WI  . S(log IVI + log S\u00bb \nif N  uses  in  each \ncomputation  at  most S  spikes. \n\nFurthermore  one  can  construct SNN's (with  any  response- and threshold-functions \n\nthat satisfy our basic  assumptions,  with fixed  rational parameters and rational time(cid:173)\ninvariant  weights)  whose  VC-dimension  is  for  computations  with  up  to  S  spikes  as \nlarge  as O(IEI . S). \nWe  refer  to  Maass,  1994a,  1994c, for  upper  bounds  on  the  computational power  of \nSNN's with continuous time. \n\n3  Networks of Spiking Neurons with Discrete Time \n\nIn  this  section  we  consider  the  case  where  all  firing  times  of neurons  in  N  are \nmultiples of some  J.l  with  1/ J.l  EN.  We  restrict  our  attention  to  the  biologically \nplausible case  where  there  exists some tN  ~ 1 such  that for  all z  > tN  all  response \n\nfunctions  \u00a3U,II(Z)  have  the  value \u00b0 and  all  threshold  functions  ell(z)  have  some \narbitrary  constant  value.  If tN  is  chosen  minimal with  this  property,  we  refer  to \nPN  :=  rtN/J.ll  as  the  timing-precision  ofN.  Obviously  for  PN  =  1  the  SNN  is \nequivalent  to  a  \"non-spiking\"  neural  net  that  consists  of linear  threshold  gates, \nwhereas  a  SNN  with continuous time may be viewed  as the opposite extremal case \nfor  PN  -- 00. \n\n\f188 \n\nWolfgang  Maass \n\nThe following result provides a significant upper bound for the computational power \nof an SNN with discrete time, even  in the presence  of arbitrary real-valued parame(cid:173)\nters  and weights.  Its proof is technically rather  involved. \n\nTheorem 3:  Assume that N  is an SNN with timing-precision PJII,  piecewise polyno(cid:173)\nmial response- and piecewise  rational threshold-functions  with  arbitrary real-valued \nparameters,  and  weight-functions  as  specified  in  section  1. \n\nThen  one  can  simulate N  for  boolean  valued  inputs  in  real-time  by  a  Turing  ma(cid:173)\nchine  with  poly(lVl, logpJII,log l/TJII)  states  and poly(lVl, logpJII, tJII/TJII)  tape-cells. \nOn  the  other hand  any  Turing  machine  with  q  states  that  uses  at  most s  tape(cid:173)\ncells  can  be  simulated in  real-time  by  an  SNN N  with  any  response- and threshold(cid:173)\nfunctions  that  satisfy  our  basic  assumptions,  with  rational  parameters  and  time(cid:173)\ninvariant  rational weights,  with O(q)  neurons,  logpJII  = O(s),  and tJII/TJII  = 0(1). \n\nThe  next  result  shows  that  the  VC-dimension  of any  SNN  with  discrete  time  is \nfinite,  and grows proportionally to logpJII.  The proof of its lower bound combines a \nnew  explicit  construction  with that of Maass,  1993. \n\nTheorem 4:  Assume  that  the  SNN N  has  the  same  properties  as  in  Theorem  3. \nThen  the  VC-dimension  and  the  pseudo-dimension  of N  (for  arbitrary  real  valued \ninputs)  can  be  bounded  by O(IEI\u00b7IVI\u00b7logpJII),  independently  of the  number of spikes \nin  its  computations. \n\nFurthermore  one  can  construct  SNN's  N  of this  type  with  any  response- and \nthreshold-functions that satisfy our basic  assumptions,  with rational parameters and \ntime-invariant  rational  weights,  so  that N  has  (already  for  boolean  inputs)  a  VC(cid:173)\ndimension  of at  least  O(IEI(logpJII + log IE!\u00bb. \n\n4  Relationships to other Computational Models \nWe  consider here  the relationship  between  SNN's with discrete  time and recurrent \nanalog  neural nets.  In the latter no  \"spikes\"  or other non-trivial timing-phenomena \noccur,  but  the  output  of a  gate  consists  of the  \"analog\"  value of some squashing(cid:173)\nor  activation function  that  is  applied  to  the  weighted  sum  of its  inputs.  See  e.g. \n(Siegelmann and Sontag, 1992) or (Maass,  1993) for recent results about the compu(cid:173)\ntational power of such models.  We consider in this section a perhaps more \"realistic\" \nversion of such modelsN, where the output of each gate is rounded off to an integer \nmultiple of some  ~ (with  a  EN).  We  refer  to  a  as  the  number  of activation  levels \nof N. \nIt is an interesting open problem whether such analog neural nets (with gate-outputs \ninterpreted  as firing  rates)  or networks  of spiking neurons  provide a  more adequate \ncomputational model for  biological neural systems.  Theorem 5 shows  that in spite \nof their quite different  structure the computational power of these two models is in \nfact  closely  related. \n\nOn  the  side  the  following  theorem  also  exhibits  a  new  subclass  of deterministic \nfinite  automata (DFA's)  which turns out to be of particular interest  in the context \nof neural nets.  We say that a  DFA M  is a  sparse  DFA  of size s if M  can be realized \nby  a  Turing  machine  with  s  states  and  space-bound  s  (such  that  each  step  of M \ncorresponds  to one step of the Turing machine).  Note that a sparse  DFA may have \nexponentially in s many states, but that only poly(s) bits are needed  to describe its \n\n\fOn  the  Computational  Complexity  of Networks  of Spiking  Neurons \n\n189 \n\ntransition function.  Sparse  DFA's  are  relatively  easy  to  construct,  and  hence  are \nvery  useful  for  demonstrating  (via Theorem 5)  that  a  specific  task  can  be  carried \nout on  a  \"spiking\"  neural  net  with a  realistic  timing precision  (respectively  on  an \nanalog neural net  with a  realistic number of activation levels). \n\nTheorem 5:  The  following  classes  of machines  have  closely  related  computational \npower in  the  sense  that  there  is  a  polynomial p  such  that  each  computational model \nfrom  any  of these  classes can  be  simulated in  real-time (with  delay-factor ~ p(s\u00bb)  by \nsome  computational model from  any  other class  (with  the  size-parameter s  replaced \nby  p(s\u00bb): \n\n\u2022  sparse  DFA's  of size  s \n\u2022  SNN's  with 0(1)  neurons  and  timing precision 23 \n\u2022  recurrent  analog  neural  nets  that  consist  of O( 1)  gates  with  piecewise  ra(cid:173)\n\ntional  activation  functions  with  23  activation  levels,  and  parameters  and \nweights  of bit-length $  s \n\n\u2022  neural nets  that  consist  of s  linear threshold  gates  (with  recurrencies)  with \n\narbitrary  real weights. \n\nThe result of Theorem 5 is  remarkably stable since it holds no matter whether  one \nconsiders just SNN's N  with 0(1) neurons that employ very  simple fixed  piecewise \nlinear response- and  threshold-functions  with  parameters of bit-length 0(1)  (with \ntN/TN  =  0(1)  and  time-invariant  weights  of bit-length  $  s),  or  if one  considers \nSNN's N  with  s  neurons  with arbitrary piecewise  polynomial response- and piece(cid:173)\nwise rational threshold-functions with arbitrary real-valued parameters, tN/TN  ~ s, \nand time-dependent  weights (as specified  in section  1). \n\n5  Conclusion \n\nWe  have  introduced  a  simple formal  model  SNN  for  networks  of spiking  neurons, \nand  have  shown  that  significant  bounds  for  its  computational power  and  sample \ncomplexity can  be derived  from  rather  weak  assumptions about  the  mathematical \nstructure  of its  response-,  threshold-,  and  weight-functions.  Furthermore  we  have \nestablished quantitative relationships between  the computational power of a  model \nfor  networks  of spiking  neurons  with  a  limited  timing  precision  (i.e.  SNN's  with \ndiscrete  time) and a  quite  realistic  version  of recurrent  analog neural  nets  (with  a \nbounded number of activation levels).  The simulations which  provide  the proof of \nthis  result  create  an  interesting  link  between  computations  with  spike-coding  (in \nan SNN) and computations with frequency-coding  (in analog neural nets).  We  also \nhave established such relationships for  the case of SNN's with continuous time (see \nMaass  1994a,  1994b,  1994c),  but  space  does  not  permit  to  report  these  results  in \nthis article. \n\nThe Theorems 1 and 5 of this article establish the existence  of mechanisms for  sim(cid:173)\nulating arbitrary  Turing machines  (and  hence  any  common computational model) \non an SNN.  As  a  consequence  one can now  demonstrate that a  concrete  task  (such \nas  binding,  pattern-matching, associative  memory) can  be  carried  out on  an  SNN \nby simply showing that some arbitrary common computational model can carry out \nthat task.  Furthermore one  can bound the required  timing-precision of the SNN  in \nterms of the  space  needed  on a  Turing machine. \n\n\f190 \n\nWolfgang  Maass \n\nSince  we  have  based  our  investigations on  the  rather refined  notion  of a  real-time \nsimulation, our results provide information not only about the possibility to imple(cid:173)\nment computations, but also  adaptive  behavior on networks of spiking neurons. \n\nAcknowledgement \nI  would like to thank Wulfram Gerstner for  helpful  discussions. \nReferences \nM.  Abeles.  (1991)  Corticonics:  Neural  Circuits of the Cerebral Cortex.  Cambridge \nUniversity  Press. \nA.  Aertsen.  ed.  (1993)  Brain Theory:  Spatio-Temporal Aspects  of Brain Function. \nElsevier. \nJ.  Buhmann,  K.  Schulten.  (1986)  Associative  recognition  and storage  in  a  model \nnetwork of physiological neurons.  Bioi.  Cybern.  54:  319-335. \n\nP.  S.  Churchland, T. J. Sejnowski.  (1992)  The Computational Brain.  MIT-Press. \n\nW.  Gerstner. \n(1991)  Associative  memory  in  a  network  of  \"biological\"  neurons. \nAdvances  in  Neural  Information  Processing  Systems,  vol.  3,  Morgan  Kaufmann: \n84-90. \nW.  Gerstner,  R.  Ritz,  J.  L.  van  Hemmen.  (1992)  A  biologically  motivated  and \nanalytically soluble model of collective oscillations in the cortex.  Bioi.  Cybern.  68: \n363-374. \nD.  Haussler.  (1992)  Decision theoretic generalizations of the  PAC model for  neural \nnets and other learning applications.  Inf  and  Comput.  95:  129-161. \nK.  T.  Judd,  K.  Aihara.  (1993)  Pulse  propagation  networks:  A  neural  network \nmodel that uses  temporal coding by action potentials.  Neural Networks 6:  203-215. \n\nJ.  van  Leeuwen,  ed.  (1990)  Handbook  of Theoretical  Computer Science,  vol.  A: \nAlgorithms and Complexity.  MIT-Press. \n\nW.  Maass.  (1993)  Bounds  for  the  computational power  and  learning  complexity \nof analog  neural  nets.  Proc.  25th  Annual  ACM  Symposium  on  the  Theory  of \nComputing,  335-344. \n\nW. Maass.  (1994a) On the computational complexity of networks of spiking neurons \n(extended  abstract).  TR  393  from  May  1994  of the  Institutes  for  Information \nProcessing  Graz (for  a  more detailed  version see  the file  maass.spiking.ps.Z in  the \nneuroprose  archive). \nW.  Maass. \nspiking neurons.  Neural  Computation,  to appear. \n\n(1994b)  Lower  bounds  for  the  computational  power  of  networks  of \n\nW.  Maass.  (1994c)  Analog computations on networks of spiking neurons (extended \nabstract).  Submitted for  publication. \n\nH. T. Siegelmann, E.  D. Sontag.  (1992) On the computational power of neural nets. \nProc.  5th  ACM- Workshop  on  Computational Learning  Theory,  440-449. \n\n\f", "award": [], "sourceid": 926, "authors": [{"given_name": "Wolfgang", "family_name": "Maass", "institution": null}]}