{"title": "A Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation", "book": "Advances in Neural Information Processing Systems", "page_first": 566, "page_last": 573, "abstract": "", "full_text": "A  Hodgkin-Huxley Type Neuron Model \nThat  Learns  Slow Non-Spike  Oscillation \n\nKenji  Doya* \n\nAllen I.  Selverston \nDepartment of Biology \n\nUniversity of California,  San Diego \n\nLa Jolla,  CA  92093-0357,  USA \n\nPeter F.  Rowat \n\nAbstract \n\nA  gradient  descent  algorithm  for  parameter  estimation  which  is \nsimilar to those used for  continuous-time recurrent neural networks \nwas  derived for  Hodgkin-Huxley type  neuron models.  Using mem(cid:173)\nbrane  potential  trajectories  as  targets,  the  parameters  (maximal \nconductances,  thresholds and slopes of activation curves, time con(cid:173)\nstants)  were  successfully estimated.  The  algorithm was  applied  to \nmodeling  slow  non-spike  oscillation  of an  identified  neuron  in  the \nlobster stomatogastric ganglion.  A model with three ionic  currents \nwas  trained  with  experimental  data.  It revealed  a  novel  role  of \nA-current for  slow oscillation below -50  mY. \n\n1 \n\nINTRODUCTION \n\nConductance-based  neuron  models,  first  formulated  by  Hodgkin  and  Huxley  [10], \nare  commonly used for  describing  biophysical mechanisms  underlying neuronal be(cid:173)\nhavior.  Since  the  days  of  Hodgkin  and  Huxley,  tens  of new  ionic  channels  have \nbeen identified [9].  Accordingly, recent H-H  type models have tens  of variables and \nhundreds  of parameters  [1,  2].  Ideally,  parameters  of H-H  type  models  are  deter(cid:173)\nmined  by voltage-clamp  experiments on  individual  ionic  currents.  However,  these \nexperiments are often very difficult or impossible to carry out.  Consequently, many \nparameters must be hand-tuned in computer simulations so that the model behavior \nresembles  that of the real neuron.  However,  a manual search in  a high dimensional \n\n*current address:  The Salk Institute, CNL,  P.O. Box 85800, San Diego, CA 92186-5800. \n\n566 \n\n\fA Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation \n\n567 \n\nI \n\nFigure  1:  A  connectionist's view  of the H-H  neuron model. \n\nparameter  space  is  very  unreliable.  Moreover,  even  if a  good  match  is  found  be(cid:173)\ntween the model and the real neuron,  the validity of the parameters is  questionable \nbecause  there  are,  in  general,  many  possible  settings  that  lead  to  apparently  the \nsame  behavior . \n\nWe  propose  an automatic parameter tuning algorithm for  H-H  type neuron models \n[5].  Since  a  H-H  type  model  is  a  network  of sigmoid  functions,  multipliers,  and \nleaky integrators (Figure 1),  we  can tune its parameters in  a  manner similar to the \ntuning of connection  weights in  continuous-time neural  network models  [6,  12].  By \ntraining  a  model  from  many  initial  parameter  points  to  match  the  experimental \ndata,  we  can systematically estimate a  region  in  the parameter space,  instead  of a \nsingle  point. \n\nWe first  test if the parameters of a spiking neuron model can be identified from the \nmembrane potential trajectories.  Then we  apply the learning algorithm to a  model \nof slow  non-spike  oscillation  of an  identified  neuron  in  the  lobster  stomatogastric \nganglion  [7].  The  resulting  model  suggests  a  new  role  of A-current  [3]  for  slow \noscillation in  the membrane potential range  below -50  m V. \n\n2  STANDARD FORM  OF IONIC  CURRENTS \n\nHistorically,  different forms  of voltage dependency curves  have been used  to repre(cid:173)\nsent the  kinetics  of different  ionic  channels.  However,  in  order  to  derive  a  simple, \nefficient  learning  algorithm,  we  chose  a  unified  form  of voltage  dependency  curves \nwhich is  based on statistical physics of ionic  channels  [11]  for  all  the ionic  currents \nin  the model. \n\nThe dynamics of the membrane potential v  is  given by \n\nGil = I  - LIj, \n\nj \n\n(1) \n\nwhere G  is the membrane capacitance and I  is externally injected current.  The j-th \nionic  current Ij  is  the product of the maximum conductance 9j, activation variable \n\n\f568 \n\nDoya, Selverston, and Rowat \n\naj,  inactivation  variable  bj ,  and  the  difference  of the  membrane  potential  v  from \nthe reversal potential Vrj.  The exponents Pi  and  qj  represent multiplicity of gating \nelements in the ionic channels and are usually an integer between 0 and 4.  Variables \naj  and  bj  are  assumed  to obey the first  order differential equation \n\nTheir steady states  ajoo  and bjoo  are sigmoid functions  of the  membrane potential \n\n(2) \n\nxoo(v) = \n\n1 \n()'  (x=aj,bj ), \n\n1 + e-~'\" v-v\", \n\n(3) \n\nwhere  Vx  and  Sx  represent  the  threshold  and  slope  of the  steady  state  curve,  re(cid:173)\nspectively.  The  rate  coefficients  ka \u00b7 (v)  and  kb \u00b7 (v)  have  the  voltage  dependence \n[11] \n\n] ]  \n\nk  (  ) - 1 \n-\nx  v  -\ntx \nwhere tx  is  the time  constant. \n\nh sx( v  - vx) \n, \n\ncos \n\n2 \n\n3  ERROR GRADIENT  CALCULUS \n\nOur goal  is  to minimize  the average error over a  cycle  with period T: \n\nE  =  ~ iT ~(v(t) - v*(t\u00bb2dt, \n\n(4) \n\n(5) \n\nwhere v*(t)  is  the target  membrane potential trajectory. \nWe first  derive the gradient of E  with respect to the model parameters ( ... , Oi,  ... ) = \n( ... , 9j, va], Saj' taj' ... ).  In  studies of recurrent  neural  networks,  it  has  been  shown \nthat teacher forcing is very important in training autonomous oscillation patterns [4, \n6,  12,  13].  In H-H type models, teacher forcing drives the activation and inactivation \nvariables by  the target membrane potential v*(t)  instead of vet)  as  follows. \n\nx = kx(v*(t\u00bb\u00b7 (-x +xoo(v*(t\u00bb) \n\n(x = aj,bj ). \n\nWe  use  (6)  in  place of (2)  during training. \n\nThe effect  of a small  change in  a parameter Oi  of a  dynamical system \n\nx = F(X; ... , Oi,  ... ), \n\nis  evaluated by the variation equation \n\n.  of \nof \ny  =  oX y  + OOi' \n\n(6) \n\n(7) \n\n(8) \n\nwhich  is  an  n-dimensional  linear  system with  time-varying  coefficients  [6,  12].  In \ngeneral, this variation calculus requires O(n 2 )  arithmetics for each parameter.  How(cid:173)\never, in  the case of H-H  model  with teacher forcing,  (8)  reduces  to a  first  or second \norder  linear  system.  For  example,  the  effect  of  a  small  change  in  the  maximum \nconductance 9j  on  the  membrane potential v  is  estimated by \n\n(9) \n\n\fA Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation \n\n569 \n\nwhere  GCt)  =  l:k 9kak(t)Pkbk(t)Qk  is  the  total  membrane  conductance.  Similarly, \nthe effect  of the activation threshold va]  is  estimated by the equations \n\nGiJ = -G(t)y - 9jpjaj(t)pj- 1bj (t)Qj(v(t)  - Vrj)  Z, \n\nZ = -kaj(t) [z + 8;j {aj(t) + ajoo(t) - 2aj(t)aj oo(t)}]  . \n\n(10) \n\nThe  solution  yet)  represents  the  perturbation  in  v  at  time  t,  namely  8;b~).  The \nerror gradient  is  then given  by \n\n1  fT \n\n.. \n\naE \nOBi  =  T  Jo  (v(t) - v  (t))  OBi  dt. \n\nav(t) \n\n(11) \n\n4  PARAMETER UPDATE \n\nBasically, we can use arbitrary gradient-based optimization algorithms, for example, \nsimple gradient descent or conjugate gradient descent.  The particular algorithm we \nused  was  a  continuous-time version of gradient descent  on normalized  parameters. \n\nBecause the parameters of a H-H  type model have different physical dimensions and \nmagnitudes,  it is  not  appropriate to perform simple  gradient descent on  them.  We \nrepresent each parameter  by the default  value Oi  and  the deviation Bi  as  below. \n\n(12) \n\nThen we  perform gradient descent on the normalized parameters Bi . \nInstead  of updating  the  parameters  in  batches,  i.e.  after  running  the  model  for  T \nand integrating the error gradient by (11),  we  updated the parameters  on-line using \nthe running  average of the  gradient as follows. \n\n. \n\n1 . .   av(t) OBi \nTa.D. o; =  -.D.o, + T(v(t) - v  (t))  OBi  oBi' \n\n(13) \nwhere Ta  is  the  averaging time  and  \u20ac \nis  the learning rate.  This on-line scheme was \nless susceptible to 2T-periodic parameter oscillation than batch update scheme and \ntherefore  we  could  use  larger learning rates. \n\nBi  =  -\u20ac.D. o, , \n\n5  PARAMETER ESTIMATION OF  A  SPIKING MODEL \n\nFirst,  we  tested  if a  model  with  random  initial  parameters  can  estimate  the  pa(cid:173)\nrameters  of  another  moqel  by  training  with  its  membrane  potential  trajectories. \nThe  default  parameters  Bi  of the  model  was  set  to  match  the  original  H-H  model \n[10]  (Table 1).  Its membrane potential trajectories at five  different levels of current \ninjection (I = 0,15,30,45, and 60J..lA/cm2 )  were used alternately as the target v*(t). \nWe ran  100  trials after initializing Bi  randomly in  [-0.5,+0.5].  In 83  cases,  the error \nbecame  less  than  1.3  m V  rms  after  100  cycles  of training.  Figure  2a is  an  exam(cid:173)\nple  of the  oscillation  patterns of the  trained  model.  The  mean  of the  normalized \n\n\f570 \n\nDoya, Selverston, and Rowat \n\nTable  1:  Parameters of the spiking neuron  model.  Subscripts  L,  Na  and  K  speci(cid:173)\nfies  leak,  sodium  and  potassium currents,  respectively.  Constants:  C=1J.lF/cm2 , \nvNa=55mV,  vK=-72mV,  vL=-50mV,  PNa=3,  QNa=l,  PK=4,  QK=PL=qL=O, \nLlv=20mV,  (=0.1, Ta  =  5T. \n\ndefault  value iii  mean \n-0.017 \n-0.002 \n0.006 \n-0.052 \n-0.103 \n0.012 \n-0.010 \n0.093 \n0.050 \n-0.021 \n-0.061 \n-0.073 \n\n0.1 \nl/mV \n0.5  msec \n-62.0  mV \n-0.09 \nl/mV \n12.0  msec \n40.0  mS/cm2 \n-50.0  mV \n0.06 \nl/mV \n5.0  msec \n\n()i  after learning \ns.d. \n0.252 \n0.248 \n0.033 \n0.073 \n0.154 \n0.202 \n0.140 \n0.330 \n0.264 \n0.136 \n0.114 \n0.168 \n\n0.3  mS/cm \n120.0  mS/cm2 \n-36.0  mV \n\ngL \ngNa \nVaNa \nSaNa \ntaNa \nVbNa \nSbNa \ntbNa \ngK \nVaK \nSaK \ntaK \n\nv[ \n\na_No [ \n\nb_Na[ ________ \n\na_K [-------.....-\n\ntaX \n\nsaK \n\nvaK \n\ngK \n\nIbNa \n\nsbNa \n\nvbNa \n\ntaNa \n\nsaNa \n\nvaNa \n\ngNa \n\ngL \n\no \n\n10 \n\ntime (ms) \n\n20 \n\n30 \n\ngL  gNa  vaNa saNa taNa vbNasbNa IbNa  gK  vaK  saK \n\ntaK \n\n(a) \n\n(b) \n\nFigure  2:  (a)  The  trajectory  of the  spiking  neuron  model  at  I  =  30J.lA/cm2 \u2022  v: \nmembrane potential (-80 to +40 mY).  a and b:  activation and inactivation variables \n(0  to  1).  The  dotted  line  in  v  shows  the  target  trajectory  v*(t).  (b)  Covariance \nmatrix of the normalized parameters Oi  after learning.  The black and white squares \nrepresent negative and  positive covariances,  respectively. \n\n\fA Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation \n\n571 \n\nTable  2:  Parameters of the  DG  cell  model.  Constants:  C=1J.lF/cm2 ,  vA=-80mV , \nVH=  -lOmV,  vL=-50mV,  PA=3,  qA=l,  PH=l,  QH=PL=qL=O,  ~v=20mV, (=0.1, \nTa  = 2T. \n\niJ\u00b7 \nt \n0.01 \n50 \n-12 \n0.04 \n7.0 \n-62 \n-0.16 \n300 \n0.1 \n-70 \n-0.14 \n3000 \n\ngL \ngA \nVaA \nSaA \ntaA \nVbA \nSbA \ntllA \ngH \nVaH \nSaH \ntaH \n\ntuned  (}i \n\n0.025  mS  cm \n41.0  mS/cm2 \n-11.1  mV \n0.022  1/mV \n7.0  msec \n-76  mV \n\n-0.19  1/mV \n292  msec \n\nv[ \n\na~[ \n\nb~[ \n\n'_H[ \n\n---\n\n-\n\n0.039  mS/cm2 \n-75.1  mV \n\n-0.11  1/mV  ~ ~ \"\"\"'\" \n\n4400  msec \n\nI_L \n\nI _A \n\n10000 \n\n20000 \n\ntlme(msl \n\n30000 \n\n40000 \n\n50000 \n\nFigure 3:  Oscillation pattern of the DG  cell  model.  v:  membrane potential (-70  to \n-50  mY).  a  and  b:  activation  and inactivation variables  (0  to  1) .  I:  ionic  currents \n(-1  to +1  pAlcm2 ). \n\nparameters iii  were  nearly  zero  (Table  1),  which  implies  that the  original  parame(cid:173)\nter values  were successfully estimated  by  learning.  The standard  deviation of each \nparameter indicates how  critical its setting is  to replicate  the given oscillation  pat(cid:173)\nterns.  From the  covariance matrix of the parameters  (Figure 2b),  we  can estimate \nthe distribution of the solution points in  the  parameter space. \n\n6  MODELING  SLOW NON-SPIKE  OSCILLATION \n\nNext we  applied the algorithm to experimental data from  the  \"DG cell\"  of the lob(cid:173)\nster stomatogastric ganglion  [7].  An  isolated  DG  cell  oscillates  endogenously  with \nthe  acetylcholine  agonist  pilocarpine  and  the  sodium  channel  blocker  TTX.  The \noscillation  period  is  5 to 20  seconds  and  the membrane  potential is  approximately \nbetween -70  and -50  m V.  From voltage-clamp data from  other stomatogastric neu(cid:173)\nrons  [8],  we  assumed  that  A-current  (potassium current  with inactivation)  [3]  and \nH-current (hyperpolarization-activated slow inward current) are the principal active \ncurrents in  this voltage range.  The default  parameters for  these currents were taken \nfrom  [2]  (Table  2). \n\n\f572 \n\nDoya, Selverston, and Rowat \n\nionic  currents \n\n. /  \n\n~ ~ \n\n../ \n\n.. .. .... ~ \n\nV  ..-\n\n~ \n\n2 \n\n,r  .. \n.. \n1 \n, \n~' \no \nW \nIf \" \n\n-2 \n\n-60  -40  -20 \n\n0 \n\n20 \n\n40 \n\nv \n\n(mV) \n\nFigure 4:  Current-voltage curves of the DG cell model.  Outward current is positive. \n\nFigure  3  is  an  example  of the  model  behavior  after  learning  for  700  cycles.  The \nactual  output  v  of the  model,  which  is  shown  in  the  solid  curve,  was  very  close \nto  the  target  output  v*(t),  which is  shown  in  the  dotted  curve.  The  bottom three \ntraces show  the ionic  currents  underlying  this slow  oscillation.  Figure 4 shows  the \nsteady state  I-V  curves  of three  currents.  A-current  has  negative  conductance in \nthe  range  from  -70  to  -40  m V.  The  resulting  positive  feedback  on  the  membrane \npotential destabilizes a  quiescent state.  If we  rotate the I-V diagram 180  degrees,  it \nlooks  similar  to  the I-V  diagram for  the  H-H  model;  the  faster  outward  A-current \nin our model takes the role of the fast inward sodium current in the H-H  model and \nthe slower inward  H-current  takes the role  of the outward potassium current. \n\n7  DISCUSSION \n\nThe  results  indicate that  the gradient descent  algorithm  is  effective for  estimating \nthe  parameters of H-H  type  neuron models from  membrane potential trajectories. \n\nRecently,  an  automatic  parameter  search  algorithm  was  proposed  by  Bhalla  and \nBower  [1].  They chose only the maximal conductances as  free  parameters and used \nconjugate gradient descent .  The error gradient was estimated by slightly changing \neach of the parameters.  In our approach, the error gradient was  more efficiently de(cid:173)\nrived by utilizing the variation equations.  The use of teacher forcing and parameter \nnormalization  was essential for  the  gradient descent  to work. \n\nIn order for  a  neuron  to be  an  endogenous  oscillator,  it is  required  that a  fast  pos(cid:173)\nitive feedback  mechanism is  balanced  with  a  slower  negative feedback  mechanism. \nThe most  popular  example is  the  positive feedback  by the sodium current and  the \nnegative  feedback  by  the  potassium  current  in  the  H-H  model.  Another  common \nexample is  the inward calcium current counteracted by the calcium dependent out(cid:173)\nward  potassium  current.  We  found  another  possible  combination  of positive  and \nnegative feedback  with  the help  of the  algorithm:  the  inactivation of the  outward \nA-current and  the activation of the slow inward H-current. \n\n\fA Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation \n\n573 \n\nAcknowledgements \n\nThe  authors  thank  Rob  Elson  and Thom Cleland  for  providing physiological  data \nfrom stomatogastric cells.  This study was supported in part by ONR grant N00014-\n91-J-1720. \n\nReferences \n[1]  U.  S.  Bhalla and  J.  M.  Bower.  Exploring  parameter space  in  detailed  single \nneuron  models:  Simulations  of the  mitral  and  granule  cells  of  the  olfactory \nbulb.  Journal  of Neurophysiology,  69:1948-1965,  1993. \n\n[2]  F. Buchholtz, J. Golowasch, I. R.  Epstein, and E.  Marder.  Mathematical model \nof an  identified  stomatogastric ganglion  neuron.  Journal  of Neurophysiology, \n67:332-340,  1992. \n\n[3]  J.  A.  Connor,  D.  Walter,  and  R.  McKown.  Neural  repetitive  firing,  modifi(cid:173)\n\ncations  of the  Hodgkin-Huxley  axon  suggested  by  experimental  results  from \ncrustacean  axons.  Biophysical  Journal,  18:81-102,  1977. \n\n[4]  K.  Doya. Bifurcations in the learning of recurrent neural networks.  In  Proceed(cid:173)\nings  of 1992  IEEE  International  Symposium  on  Circuits  and  Systems,  pages \n6:2777-2780,  San Diego,  1992. \n\n[5]  K.  Doya and  A.  I.  Selverston.  A  learning  algorithm for  Hodgkin-Huxley  type \nneuron models.  In  Proceedings  of IJCNN'93,  pages 1108-1111,  Nagoya,  Japan, \n1993. \n\n[6]  K.  Doya and  S.  Yoshizawa.  Adaptive  neural  oscillator  using  continuous-time \n\nback-propagation learning.  Neural Networks,  2:375-386,  1989. \n\n[7]  R.  C.  Elson  and  A.  I.  Selverston.  Mechanisms  of gastric  rhythm generation \nin  the  isolated  stomatogastric ganglion  of spiny lobsters:  Bursting pacemaker \npotential, synaptic interactions,  and  muscarinic modulation.  Journal  of Neu(cid:173)\nrophysiology,  68:890-907,  1992. \n\n[8]  J.  Golowasch  and  E.  Marder.  Ionic  currents of the  lateral  pyloric  neuron  of \nstomatogastric  ganglion  of the  crab.  Journal  of Neurophysiology,  67:318-331, \n1992. \n\n[9]  B.  Hille.  Ionic  Channels  of Excitable  Membranes.  Sinauer,  1992. \n[10]  A.  L.  Hodgkin  and  A.  F.  Huxley.  A  quantitative  description  of  membrane \ncurrents and  its  application to conduction and excitation in  nerve.  Journal  of \nPhysiology,  117:500-544,  1952. \n\n[11]  H.  Lecar,  G.  Ehrenstein,  and  R.  Latorre.  Mechanism  for  channel  gating  in \nexcitable bilayers.  Annals of the  New  York Academy of Sciences,  264:304-313, \n1975. \n\n[12]  P.  F.  Rowat and A.I. Selverston.  Learning algorithms for  oscillatory networks \n\nwith  gap junctions and  membrane currents.  Network,  2:17-41,  1991. \n\n[13]  R. J. Williams and D.  Zipser.  Gradient based learning algorithms for  recurrent \nconnectionist  networks.  Technical Report  NU-CCS-90-9,  College  of Computer \nScience,  Northeastern  University,  1990. \n\n\f", "award": [], "sourceid": 783, "authors": [{"given_name": "Kenji", "family_name": "Doya", "institution": null}, {"given_name": "Allen", "family_name": "Selverston", "institution": null}, {"given_name": "Peter", "family_name": "Rowat", "institution": null}]}