{"title": "Bifurcation Analysis of a Silicon Neuron", "book": "Advances in Neural Information Processing Systems", "page_first": 731, "page_last": 737, "abstract": null, "full_text": "Bifurcation Analysis of a Silicon Neuron \n\nGirish N. Patel] , Gennady s. Cymbalyuk2,3, \n\nRonald L. Calabrese2, and Stephen P. DeWeerth1 \n\nlSchool of Electrical and Computer Engineering \n\nGeorgia Institute of Technology \n\nAtlanta, Ga. 30332-0250 \n\n{girish.patel, steve.deweerth} @ece.gatech.edu \n\n2Department of Biology \n\nEmory University \n\n1510 Clifton Road, Atlanta, GA 30322 \n{gcym, rcalabre}@biology.emory.edu \n\n3Institute of Mathematical Problems in Biology RAS \nPushchino, Moscow Region, Russia 142292 (on leave) \n\nAbstract \n\nWe  have developed a VLSI  silicon  neuron  and  a corresponding  mathe(cid:173)\nmatical  model  that  is  a two  state-variable system. We  describe the  cir(cid:173)\ncuit implementation and compare the  behaviors  observed in  the  silicon \nneuron and the mathematical model. We also perform bifurcation analy(cid:173)\nsis of the mathematical model by  varying the externally applied current \nand show that the behaviors exhibited by the silicon neuron under corre(cid:173)\nsponding  conditions  are  in  good  agreement  to  those  predicted  by  the \nbifurcation analysis. \n\n1  Introduction \n\nThe use of hardware  models  to  understand dynamical  behaviors  in  biological  systems  is \nan  approach  that  has  a long  and fruitful  history  [1 ][2].  The implementation  in  silicon  of \noscillatory  neural  networks  that  model  rhythmic  motor-pattern  generation  in  animals  is \none recent addition to these modeling efforts [3][4]. The oscillatory patterns generated by \nthese  systems  result  from  intrinsic  membrane  properties  of individual  neurons  and  their \nsynaptic  interactions  within  the  network  [5].  As  the  complexity  of these  oscillatory  sili(cid:173)\ncon systems increases, effective mathematical analysis becomes increasingly more impor(cid:173)\ntant  to  our understanding their  behavior.  However,  the  nonlinear dynamical behaviors  of \nthe  model  neurons  and  the  large-scale  interconnectivity  among  these  neurons  makes  it \nvery  difficult  to  analyze  theoretically  the behavior  of the  resulting  very  large-scale  inte(cid:173)\ngrated  (VLSI)  systems.  Thus,  it  is  important  to  first  identify  methods  for  modeling  the \nmodel neurons that underlie these oscillatory systems. \n\nSeveral  simplified  neuronal  models  have  been  used  in  the  mathematical  simulations  of \npattern generating networks  [6][7][8] . In  this  paper,  we describe the implementation of a \n\n\f732 \n\nG.  N  Patel,  G.  S.  Cymbalyuk,  R.  L.  Calabrese and S.  P.  DeWeerth \n\ntwo-state-variable silicon neuron  that has been used effectively to develop oscillatory net(cid:173)\nworks  [9][10].  We  then derive  a mathematical  model  of this  implementation  and analyze \nthe  neuron  and  the  model  using  nonlinear  dynamical  techniques  including  bifurcation \nanalysis  [11].  Finally,  we compare the experimental data derived from  the  silicon neuron \nto that obtained from the mathematical model. \n\n2  The silicon model neuron \n\nThe schematic  for  our silicon  model  neuron  is  shown  in  Figure  1.  This silicon  neuron  is \ninspired  by  the  two-state,  Morris-Lecar  neuron  model  [12][ 13].  Transistor  M I  '  analo(cid:173)\ngous to the voltage-gated calcium channel in the Morris-Lecar model, provides an instan(cid:173)\ntaneous  inward  current  that  raises  the  membrane  potential  towards  V High  when  the \nmembrane is depolarized. Transistor  M2 ' analogous to  the voltage-gated potassium chan(cid:173)\nnel  in  the Morris-Lecar model,  provides  a delayed outward current that lowers  the mem(cid:173)\nbrane  potential  toward  V Low  when  the  membrane  is  depolarized.  V H  and  V L  are \nanalogous  to  the  half-activation  voltages  for  the  inward  and  outward  currents,  respec(cid:173)\ntively. The  voltages  across  C I  and C2  are  the  state  variables  representing  the  membrane \npotential,  V,  and the  slow \"activation\"  variable of the  outward current,  W,  respectively. \nThe  W -nullcline  represents  its  steady-state  activation  curve.  Unlike  the  Morris-Lecar \nmodel, our silicon neuron model does not possess a leak current. \n\nUsing current conservation at node  V, the net current charging CI  is given by \n\nwhere  iH  and  iL  are  the  output currents  of a differential  pair  circuit,  and  a p  and  aN \ndescribe the ohmic effects of transistors M J  and  M2, respectively. The net current into C2 \nis given by \n\n(1) \n\nwhere  ix  is  the output current of the  OTA,  and  ~p and  ~N account for ohmic effects  of \nthe pull-up and the pull-down transistors inside the OTA. \n\n(2) \n\nVHigh \n\nOTA \n\nv \n\nw \n\n'------<>--- V Low \n\nFigure  1: Circuit diagram of the silicon neuron.  The circuit incorporates analog building \nblocks including two differential  pair circuits composed of a bias current,  IB H, and \ntransistors M4-Ms, and a bias current, IBL, and transistors M6-M7' and a single follower(cid:173)\nintegrator circuit composed of an operational transconductance amplifier (OTA), Xl  in the \nconfiguration shown and a load capacitor, C2. The response of the follower-integrator \ncircuit is similar to a first-order low-pass filter. \n\n\fBifurcation Analysis of a Silicon Neuron \n\n733 \n\nThe  output  currents  of the  differential-pair  and  an  OTA  circuits,  derived  by  using  sub(cid:173)\nthreshold  transistor  equations  [2],  are  a  Fenni  function  and  a  hyperbolic-tangent  func(cid:173)\ntion, respectively [2]. Substituting these functions for  iH,  iL, and  ix  in (1) and (2) yields \n\n. \n\nC 1 V  = Iexta p + IBH \n\ne \n\nK(V-YH) / U T \n\nK(W - YL) / UT \n\ne \n\nK(V _ YH) / UT a p -\n\nIBL \n\nK(W _ YL) / U T aN \n\nl+e \n\nl+e \n\nwhere \n\na p  = \n\nI\n\nv - V High / UT \n\n-e \n\naN  = 1 - e \n\nY Low - V I UT \n\n~N = 1- e \n\n- W / UT \n\n(3) \n\n(4) \n\nU T  is  the  thennal  voltage,  V dd  is  the  supply  voltage, and  K  is  a  fabrication  dependent \nparameter. The tenns  a p  and  aN  limit the  range of V  to  within V High  and V Low'  and the \nterms  ~p and  ~N limit the range of W  to within the supply rails (Vdd and Gnd). \n\nIn  order  to  compare  the  model  to  the  experimental  results,  we  needed  to  determine  val(cid:173)\nues  for  all  of the  model  parameters.  V Hi!\\h'  V Low'  V H'  V L ' and  V dd  were directly mea(cid:173)\nsured  in  experiments.  The  parameters  IBH  and  IBL  were  measured  by  voltage-clamp \nexperiments  performed  on  the  silicon  neuron.  At  room  temperature,  UT :::::  0.025  volts. \nThe value  of  K  :::::  0.65  was  estimated  by  measuring  the  slope  of the  steady-state activa(cid:173)\ntion  curve  of inward current.  Because  W  was  implemented as  an  inaccessible  node,  IT \ncould  only  be  estimated. Based  on  the  circuit design,  we  can  assume  that  the  bias  cur(cid:173)\nrents  IT  and  IBH  are  of the  same  order  of magnitude.  We  choose  IT:::::  2.2  nA  to fit  the \nbifurcation diagram (see Figure 3). Cl and C2, which are assumed to be identical accord(cid:173)\ning to the physical design, are time scaling parameters in the model. We choose their val(cid:173)\nues (Cl = C2 = 28 pF) to fit frequency dependence on  lext  (see Figure 4). \n\n3  Bifurcation analysis \n\nThe  silicon  neuron  and  the  mathematical  model!  described  by  (3)  demonstrate  various \ndynamical  behaviors  under  different  parametric  conditions.  In  particular,  stable  oscilla(cid:173)\ntions and steady-state equilibria are observed for different values of the externally applied \ncurrent,  I ext . We focused our analysis on the  influence of I ext  on the neuron behavior for \ntwo  reasons:  (i)  it  provides  insight  about  effects  of synaptic  currents,  and  (ii)  it  allows \ncomparison with  neurophysiological experiments in  which polarizing current is  used as  a \nprimary control parameter. The main results of this work are presented as  the comparison \nbetween  the  mathematical  models  and  the  experimental  data  represented  as  bifurcation \ndiagrams and frequency dependencies. \nThe  null clines  described  by  (3)  and  for  lext  = 32 nA  are  shown  in  Figure  2A.  In  the \nregime  that  we  operate  the  circuit,  the  W -null cline is  an  almost-linear curve  and  the  V(cid:173)\nnullcline  is  an  N-shaped curve. From  (3),  it  can  be  seen  that  when  IBH + lext > IBL  the \nnullclines  cross  at  (V, W):::::  (VHigh, VHigh )  and  the  system  has  high  voltage  (about  5 \nvolts) steady-state equilibrium. Similarly, for  Iext  close to zero, the system has  one stable \nequilibrium point close to  (V, W) :::::  (V Low'  V Low). \n\n!The  parameters  used  throughout  the  analyses  of  the  model  are  V Low  =  0 V , \nV High  = 5 V,  V L = V H = 2.5  V,  I BH  = 6.5  nA ,  I BL  = 42  nA,  IT  = 2.2  nA , \nV dd  = 5  V,  Vt  = 0.025  mV, and  K  = 0.65. \n\n\fG.  N  Patel, G. S.  Cymba/yule,  R.  L.  Calabrese and S.  P.  De Weerth \n\nW-nullcline \n\n/ \n\n/ @ \n\n/ \n\nV-nullcline \n\n734 \n\nA \n\n2.85 \n\n2.8 \n\n2.75 \n\n~ 2.7 \n(5 \n> \n.....,.. \n~ 2.65 \n\n2.6 \n\n2.55 \n\n2.5 \n\n2.45~ ______ ~ ______ ~ ______ ~ ______ ~ ______ ~1 \n\nI \n\no \n\n2 \n\n3 \n\n4 \n\n5 \n\nV (volts) \n\nB \n\n3.2 \n\n3 \n\n2.8 -In -l2.6 \n\n> \n\n2.4 \n\n2.2 \n\n2 \n\n0 \n\n5 \n\n10 \n\n15 \n20 \ntime  (msec) \n\n25 \n\n30 \n\n35 \n\nFigure 2:  Nullclines  and trajectories  in  the  model  of the silicon neuron for lex! = 32 nA. \nThe  system exhibits a stable limit-cycle (filled circles),  an  unstable limit-cycle (unfilled \ncircles),  and  stable equilibrium point.  Unstable limit-cycle separates  the  basins  of \nattraction of the stable limit-cycle and stable equilibrium point. Thus, trajectories initiated \nwithin the area bounded by  the unstable limit-cycle approach the stable equilibrium point \n(solid line in  A's  inset,  and \"x's\" in  B). Trajectories  initiated outside the unstable limit(cid:173)\ncycle  approach  the  stable limit-cycle .  In  A,  the  inset  shows  an  expansion  at  the \nintersection of the  V - and  W -nullclines. \n\n\fBifurcation Analysis of a Silicon Neuron \n\n735 \n\nA \n\n5 \n\n4 \n\nExperimental data \n........................... ~ \n\u2022 \n\u2022 \n\u2022 \n\u2022 \n.. \n\u2022 \n\u2022 \n\u2022 \n\u2022 \n.. \u2022 \nI \n\u2022 \n\n\u2022 \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\n\u2022 \n\u2022 \u2022 \n\u2022 \u2022 \n\nx  xxxx \u2022\u2022 x \n\n\",>0< \n\nx \n\nx \n\n~ \n\n~ \n\n003 \n.-\n0 \n2:-\n>2  x \n\n>t#*  xxxx \n\n1 \n\n0 \n\nx \n\n0 \n\n10 \n\n20 \nlext \n\n30 \n\n(nAmps) \n\n40 \n\n50 \n\nB \n\nModeling data \n\n4 \n\n5 \n\n( \n\u2022 \n\u2022 \n\n....................... -. \n. \n\u2022 \n\u2022 \n\u2022 \n0  rt---------; \n,J \n> ->2 \n\n003 \n.-\n\n\u2022 \n\n. \n'-....................... \n\n\u2022 \n\u2022 \n~ \n\n1 \n\n0 \n\n0 \n\n10 \n\n20 \n30 \nI ext  (nAmps) \n\n40 \n\n50 \n\nFigure  3:  Bifurcation  diagrams  of the  hardware  implementation (A)  and  of the \nmathematical model (B) under variation of the externally applied current. In A, the steady(cid:173)\nstate equilibrium potential of V  is denoted by \"x\"s. The maximum and minimum values \nof V  during stable oscillations are denoted by  the filled circles.  In  B, the  stable and \nunstable equilibrium points  are denoted by  the solid and dashed curve, respectively,  and \nthe minimum and maximum values  of the stable and unstable oscillations are denoted by \nthe  filled  and unfilled circles, respectively.  In  B,  limit-cycle oscillations appear and \ndisappear via sub-critical Andronov-Hopf bifurcations. The bifurcation diagram (B) was \ncomputed with the LOCBIF program [14]. \n\n\fG. N.  Patel,  G. S.  Cymbalyuk,  R.  L.  Calabrese and S.  P.  De Weerth \n\n736 \n\nA \n\nExperimental data \n\n100~--~--~----~~ \n\nModeling data \n\n80  \u2022 \n\u2022 \n\n-\nN \n~ \n>.  60 \ng \n~  40 \nt:T \n~ \nLL  20 \n\n\u2022 \n\u2022  .~~.. \n\u2022  \u2022\u2022\u2022 \n... \n\n\u2022 \n.., \n\n. \n\nB \n\n100 \n\n- 80 \nN ::c ->.  60 \n\n0 c \nQ) \n:l  40 \nt:T \nQ) ... \nLL  20 \n\n\u2022 \n\u2022 \n\n\u2022 \n\n. \n\n\u2022 \u2022\u2022\u2022 \n\n\u2022 \n\u2022 \n\u2022 \n\u2022 \u2022\u2022 \n~ \n\n~. \n\no~--~----~------~ \n\no \n\n20 \n\n10 \nlext  (nAmps) \n\n30 \n\n0 \n\n0 \n\n20 \n\n10 \nlext  (nAmps) \n\n30 \n\nFigure 4:  Frequency dependence of the  silicon neuron (A) and the mathematical model \n(B) on the externally applied current. \n\nFor moderate values of lext  ([1  nA,34 nA)), the stable and unstable equilibrium points are \nclose  to  (V, W) :::::  (V H' V L)  (Figure  3).  In  experiments  in  which  lext  was  varied,  we \nobserved a hard loss  of the  stability  of the  steady-state equilibrium and  a transition into \noscillations  at  lext  = 7.2  nA  (I ext  = 27.5  nA).  In  the  mathematical  model,  at the  criti(cid:173)\ncal  value  of  lext  =  7.7  nA  (lext  =  27.8  nA), an  unstable limit cycle  appears  via  a sub(cid:173)\ncritical Andronov-Hopf bifurcation. This unstable limit cycle merges with the stable limit \ncycle at the fold  bifurcation  at  lext  = 3.4 nA  (lext  = 32.1  nA). Similarly, in the  experi(cid:173)\nlext  =  2.0 nA \nments,  we  observed  hard \n(I ext  =  32.8  nA).  Thus,  the  system  demonstrates  hysteresis.  For  example.  when \nlext  =  20 nA  the  silicon  neuron  has  only  one stable regime,  namely,  stable  oscillations. \nThen  if external  current is  slowly  increased  to  lext  = 32.8  nA. the  form  of oscillations \nchanges.  At this  critical  value  of the  current,  the  oscillations suddenly  lose stability,  and \nonly  steady-state  equilibrium  is  stable.  Now,  when  the  external  current  is  reduced,  the \nsteady-state equilibrium  is  observed  at the  values  of the  current  where  oscillations  were \npreviously  exhibited. Thus, within the ranges  of externally  applied currents  (2.0,7.2)  and \n(27.5,32.8),  oscillations  and  a  steady-state  equilibrium  are  stable  regimes  as  shown  in \nFigure 2. \n\nloss  of  stability  of  oscillations  at \n\n4  Discussion \n\nWe have developed a two-state silicon neuron and a mathematical model that describes the \nbehavior of this neuron. We  have shown experimentally and verified mathematically that \nthis silicon neuron has three regions of operation under the variation of its external current \n(one of its parameters). We also perform bifurcation analysis of the mathematical model \nby varying the externally applied current and show that the behaviors exhibited by the sili(cid:173)\ncon neuron under corresponding conditions are in  good agreement to those predicted by \nthe bifurcation analysis. \n\nThis analysis  and  comparison to experiment is  an important step toward  our understand(cid:173)\ning  of a variety  of oscillatory  hardware networks that we  and  others  are developing.  The \n\n\fBifurcation Analysis of a Silicon Neuron \n\n737 \n\nmodel  facilitates  an  understanding  of the  neurons  that  the  hardware  alone does  not pro(cid:173)\nvide. In  particular for  this  neuron,  the  model  allows  us  to determine  the  location  of the \nunstable fixed  points and the types of bifurcations that are exhibited. In higher-order sys(cid:173)\ntems,  we  expect  that  the  model  will  provide  us  insight  about  observed  behaviors  and \ncomplex bifurcations  in  the  phase space.  The good  matching between  the model  and the \nexperimental  data  described  in  this  paper  gives  us  some  confidence  that  future  analysis \nefforts will prove fruitful. \n\nAcknowledgments \n\nS.  DeWeerth  and  G.  Patel  are  funded  by  NSF  grant  IBN-95 II 721 ,  G.S.  Cymbalyuk  is \nsupported by Russian Foundation of Fundamental Research grant 99-04-49112, R.L. Cal(cid:173)\nabrese and G.S. Cymbalyuk are supported by  NIH grants NS24072 and NS34975. \n\nReferences \n\n[1]  Van  Der Pol,  B (1939) Biological rhythms considered as  relaxation oscillations In H. \nBremmer and c.J. Bouwkamp (eds) Selected Scientific Papers,  Vol  2,  North Holland \nPub. Co.,  1960. \n\n[2]  Mead, C.A. Analog VLSI and Neural Systems. Addison-Wesley, Reading, MA,  1989. \n[3]  Simoni, M.E, Patel, G.N., DeWeerth, S.P., &  Calabrese, RL. Analog VLSI model of \nthe  leech  heartbeat  elemental  oscillator.  Sixth  Annual  Computational  Neuroscience \nMeeting,  1997. in Big Sky, Montana. \n\n[4]  DeWeerth,  S.,  Patel,  G.,  Schimmel,  D.,  Simoni,  M.  and  Calabrese,  R  (1997).  In \nProceedings  of the  Seventeenth  Conference  on  Advanced  Research  in  VLSI,  RB. \nBrown and A.T. Ishii (eds), Los Alamitos, CA: IEEE Computer Society,  182-200. \n\n[5]  Marder, E.  & Calabrese, RL. (1996) Principles of rhythmic motor pattern generation. \n\nPhysiological Reviews 76 (3):  687-717. \n\n[6]  Kopell,  N.  &  Ermentrout,  B.  (1988)  Coupled  oscillators  and  the  design  of central \n\npattern generators. Mathematical biosciences 90:  87-109. \n\n[7]  Skinner, EK., Turrigiano, G.G., &  Marder, E. (1993) Frequency and burst duration in \n\noscillating neurons and two-cell networks. Biological Cybernetics 69:  375-383. \n\n[8]  Skinner, EK., Gramoll, S., Calabrese, R.L., Kopell, N. & Marder, E. (1994) Frequency \ncontrol  in  biological  half-center  oscillators.  In  EH. Eeckman (ed.),  Computation  in \nneurons and neural systems, pp. 223-228, Boston: Kluwer Academic Publishers. \n\n[9]  Patel,  G.  Holleman,  J.,  DeWeerth,  S.  Analog  VLSI  model  of  intersegmental \n\ncoordination with nearest-neighbor coupling. In , 1997. \n\n[10] Patel,  G.  A  neuromorphic  architecture for  modelling  intersegmental  coordination. \n\nPh.D. dissertation, Georgia Institute of Technology,  1999. \n\n[11] J.  Guckenheimer  and  P. Holmes.  Nonlinear  Oscillations,  Dynamical  Systems,  and \nBifurcation  of Vector  Fields.  Applied  Mathematical  Sciences,  42.  Springer-Verlag, \nNew York, New York, Heidelberg, Berlin,  1983. \n\n[12] Morris, C. and Lecar, H. (1981) Voltage oscillations in the barnacle giant muscle fiber. \n\nBiophys. J, 35: 193-213. \n\n[13] Rinzel, J.  & Ermentrout, G.B. (1989) Analysis of Neural Excitability and Oscillations. \nIn  C.  Koch  and  I. Segev  (eds)  Methods  in  Neuronal  Modeling  from  Synapses  to \nNetworks.  MIT press, Cambridge, MA. \n\n[14] Khibnik,  A. I. , Kuznetsov,  Yu.A.,  Levitin,  v.v.,  Nikolaev,  E.V.  (1993)  Continuation \ntechniques  and  interactive  software  for  bifurcation  analysis  of ODEs  and  iterated \nmaps. Physica D 62 (1-4): 360-367. \n\n\f", "award": [], "sourceid": 1690, "authors": [{"given_name": "Girish", "family_name": "Patel", "institution": null}, {"given_name": "Gennady", "family_name": "Cymbalyuk", "institution": null}, {"given_name": "Ronald", "family_name": "Calabrese", "institution": null}, {"given_name": "Stephen", "family_name": "DeWeerth", "institution": null}]}