{"title": "A Recurrent Neural Network Model of Velocity Storage in the Vestibulo-Ocular Reflex", "book": "Advances in Neural Information Processing Systems", "page_first": 32, "page_last": 38, "abstract": null, "full_text": "A Recurrent Neural Network Model of \n\nVelocity Storage in the Vestibulo-Ocular Reflex \n\nThomas J. Anastasio \n\nDepartment of Otolaryngology \nUniversity of Southern California \n\nSchool of Medicine \n\nLos Angeles,  CA 90033 \n\nAbstract \n\nA three-layered neural  network model was used to  explore  the  organization of \nthe  vestibulo-ocular  reflex  (VOR).  The  dynamic  model  was  trained  using \nrecurrent back-propagation to produce compensatory, long duration eye muscle \nmotoneuron  outputs  in  response  to  short  duration  vestibular  afferent  head \nvelocity  inputs.  The  network  learned  to  produce  this  response  prolongation, \nknown  as  velocity  storage,  by  developing  complex,  lateral  inhibitory  interac(cid:173)\ntions among the interneurons.  These had the low baseline, long time constant, \nrectified  and  skewed  responses  that  are  characteristic  of  real  VOR  inter(cid:173)\nneurons.  The  model  suggests  that  all  of these  features  are  interrelated  and \nresult from lateral inhibition. \n\n1 SIGNAL PROCESSING IN THE VOR \nThe  VOR  stabilizes  the  visual  image  by  producing  eye  rotations  that  are  nearly  equal \nand  opposite  to  head  rotations  (Wilson  and  Melvill  Jones  1979).  The  VOR  utilizes \nhead  rotational  velocity  signals,  which  originate  in  the  semicircular canal  receptors of \nthe  inner  ear,  to  control  contractions  of the  extraocular  muscles.  The  reflex  is  coor(cid:173)\ndinated by brainstem interneurons in the vestibular nuclei (VN),  that relay  signals from \ncanal afferent sensory neurons to eye muscle motoneurons. \n\n32 \n\n\fA Recurrent Neural Network Model of Velocity Storage \n\n33 \n\nThe  VN  intemeurons,  however,  do  more  than just relay  signals.  Among  other func(cid:173)\ntions,  the  VN neurons process  the  canal afferent signals,  stretching out their time con(cid:173)\nstants by about four times before transmitting this signal to the motoneurons.  This time \nconstant  prolongation,  which  is  one  of the  clearest  examples  of signal  processing  in \nmotor neurophysiology,  has  been  termed velocity  storage  (Raphan  et  al.  1979).  The \nneural mechanisms underlying velocity storage, however,  remain unidentified. \n\nThe  VOR is bilaterally  symmetric  (Wilson  and  Melvill Jones  1979).  The semicircular \ncanals  operate  in  push-pull  pairs,  and  the  extraocular  muscles  are  arranged  in \nagonist/antagonist  pairs.  The  VN  are  also  arranged  bilaterally  and  interact  via  in(cid:173)\nhibitory commissural connections.  The commissures are necessary  for velocity storage, \nwhich is eliminated by cutting the commissures in monkeys (Blair and Gavin  1981). \n\nWhen  the  overall  V OR  fails  to  compensate  for  head  rotations,  the visual  image is not \nstabilized but moves across the  retina at a velocity  that  is  equal  to  the  amount of VOR \nerror.  This  'retinal slip'  signal is transmitted back to the  VN,  and is known to  modify \nVOR  operation  (Wilson  and  Melvill  Jones  1979).  Thus  the  VOR  can  be  modeled \nbeautifully as  a  three-layered neural network,  complete with  recurrent connections and \nerror signal back-propagation at the VN level.  By  modeling  the  VOR  as  a neural  net(cid:173)\nwork,  insight can be gained into the global organization of this reflex. \n\nFigure  1:  Architecture  of  the \nHorizontal  VOR  Neural  Network \nModel.  lhc and  rhc,  left and  right \nhorizontal  canal  afferents;  Ivn  and \nrvn,  left  and  right VN  neurons;  lr \nand  mr,  lateral  and  medial  rectus \nmotoneurons  of the left eye.  This \nand  all  subsequent  figures  are \nredrawn  from  Anastasio  (1991), \nwith permission. \n\n\f34 \n\nAnastasio \n\n2 ARCHITECTURE OF THE VOR NEURAL NETWORK MODEL \nThe  recurrent  neural  network  model  of the  horizontal  VOR  is diagrammed  in  Fig.  1. \nThe input units represent afferents from the left and  right horizontal semicircular canals \n(thc and rhc).  These are the canals and afferents that respond to yaw head rotations  (as \nin  shaking  the  head  'no').  The  output units  represent  motoneurons  of the  lateral  and \nmedial  rectus  muscles  of the  left  eye  (Ir  and  mr).  These  are  the  motoneurons  and \nmuscles  that  move  the  eye  in  the  yaw plane.  The units in  the  hidden layer correspond \nto interneurons in the VN,  on both the left and right sides of the brainstem (Ivnl, Ivn2, \nrvnl and  rvn2).  All units compute the weighted sum of their inputs and  then pass this \nsum through the sigmoidal squashing function. \n\nTo represent the  VOR relay,  input project to  hidden units and hidden project to  output \nunits.  Commissural connections are modeled as lateral interconnections between hidden \nunits on opposite sides of the brainstem.  The model  is constrained to  allow only those \nconnections  that  have  been experimentally  well  described in mammals.  For example, \ncanal afferents do not project directly to motoneurons in mammals,  and so direct connec(cid:173)\ntions from input to output units are not included in the model. \n\nEvidence  to  date  suggests  that  plastic  modification  of synapses  may  occur  at  the  VN \nlevel but not at the motoneurons.  The weights of synapses  from hidden to  output units \nare  therefore  fixed.  All  fixed  hidden-to-output weights have  the  same absolute value, \nand are arranged in a reciprocal pattern.  Hidden units lvnl and  Ivn2  inhibit lr and ex(cid:173)\ncite mr;  the opposite pattern obtains for  rvnl  and  rvn2.  The connections to  the hidden \nunits,  from  input  or  contralateral  hidden  units,  were  initially  randomized  and  then \nmodified by  the continually  running,  recurrent back-propagation algorithm of Williams \nand Zipser (1989). \n\n3 TRAINING AND ANALYZING mE VOR NETWORK MODEL \nThe  VOR  neural  network  model  was  trained  to  produce  compensatory  motoneuron \nresponses  to  two  impulse head accelerations,  one  to  the  left and the  other  to  the  right, \npresented repeatedly  in  random  order.  The preset  impulse  responses  of the  canal  af(cid:173)\nferents  (input units)  decay  with a time  constant of one network cycle or tick (Fig.  2,  A \nand B,  solid).  The  desired  motoneuron  (output unit)  responses  are equal  and  opposite \nin  amplitude  to  the  afferent  responses,  producing  compensatory  eye  movements,  but \ndecay with a time constant four times longer,  reflecting velocity  storage (Fig.  2,  A and \nB,  dashed).  Because  of the  three-layered architecture of the VOR,  a delay of one net(cid:173)\nwork cycle is introduced between the input and output responses. \n\nAfter about  5000  training  set  presentations,  the  network  learned  to  match  actual  and \ndesired  output  responses  quite  closely  (Fig.  2,  C  and  D, \nsolid  and  dashed, \nrespectively).  The input-to-hidden connections arranged themselves in a reciprocal pat(cid:173)\ntern,  each input unit exciting the ipsilateral hidden units and inhibiting the contralateral \nones.  This arrangement is also observed for the actual VOR (Wilson and Melvill Jones \n1979).  The  hidden-to-hidden  (commissural)  connections  formed  overlapping,  lateral \ninhibitory feedback loops.  These loops mediate velocity storage in the network.  Their \nremoval  results  in a  loss  of velocity  storage  (a  decrease  in output time  constants from \nfour  to  one  tick),  and  also  slightly increases output unit  sensitivity  (Fig.  2,  C  and D, \ndotted). \nThese  effects  on  VOR  are  also  observed  following  commissurotomy  in \nmonkeys (Blair and Gavin  1981). \n\n\fA Recurrent Neural Network Model of Velocity Storage \n\n35 \n\nO~.--------------------~ \n\n8 \n\n~~~ \n\nI '  I' ,  \\ ,  , \n, \n, \n, \n.... -----.... -~-.. --~ \n\n--\n\n' \n\nV \n\n, \n, \nt' \n\" \n\" , \n\n10 \n\n20 \n\n30 \n\n40 \n\n50 \n\n60 \n\no \n\nA \n\nI \n\" I '  , \\ \n,  , , \n, \n\n( \n\n, --\n\n--\n\n\". , \n\nI \n\n, \n, \n, , \n, , \n\" - , \n\" \n\nI \n\n0.85 \n\n(f') \nUJ \n0.8 \n(f') \nZ  0.55 \n0 \nCl. \n(f') \n\n0.5 \n\nUJ a:  0.45 \n~ \nZ \n::::> \n\n0.4 \n\n0.35 \n\no \n\n10 \n\n20 \n\n30 \n\n40 \n\n50 \n\n60 \n\n0.6 \n\n0.85 r-----------.--, c \n\nen \nUJ \nen \nz \no  0.55 \n~ .: ~ V'---'::;:=-=---.J  \\.------~--..... \n\n\\\\ \n~~ \n\n~ \nZ \n::::> \n\n0.4 \n\n0.35 \n\no \n\n20 \n\n10 \n50 \nNETWORK CYCLES \n\n30 \n\n40 \n\n60 \n\n0.8 \n\n. \n\nI \nI' \n\nD.55 \n\nD.5 \n\n0.45 \n\n0.4 \n\n0.35 \n\n0.85 \n\n0 \n\n0.6 \n\n-\n\n0.45 \n\n0.4 \n\n0 \n\n0.35  '-_'----_'----_'----''-----'_-J \n60 \n\n10 \n50 \nNETWORK CYCLES \n\n30 \n\n40 \n\n20 \n\nFigure 2:  Training  the  VOR  Network Model.  A  and  B,  input unit  responses  (solid), \ndesired  output unit responses  (dashed),  and incorrect output responses of initially  ran(cid:173)\ndomized network  (dotted);  Ihc  and lr in A,  rhc  and  mr in B.  C and  D,  desired output \nresponses  (dashed),  actual  output  responses  of trained  network  (solid),  and  output \nresponses following removal of commissural connections (dotted);  lr in C,  mr in D. \n\nAlthough all the hidden units project equally strongly to  the output units,  the inhibitory \nconnections between them,  and their response patterns, are different.  Hidden units lvnl \nand  rvnl have developed strong mutual inhibition.  Thus units Ivnl  and rvnl  exert net \npositive  feedback  on themselves.  Their responses appear as  low-pass  filtered  versions \nof the input unit responses (Fig.  3, A,  solid and dashed).  In contrast, hidden units Ivn2 \nand rvn2 have almost zero mutual inhibition,  and  tend to  pass the sharply peaked input \nresponses  unaltered  (Fig.  3,  B,  solid  and  dashed).  Thus  the  hidden  units  appear  to \nform parallel integrated (lvnl and rvnl) and direct (lvn2 and rvn2) pathways to  the out(cid:173)\nputs. \nThis  parallel  arrangement  for  velocity  storage  was  originally  suggested  by \nRaphan and coworkers (1979).  However, units Ivn2 and rvn2 are coupled to units rvnl \nand  Ivnl,  respectively,  with  moderately  strong  mutual  inhibition.  This  coupling  en(cid:173)\ndows  units  Ivn2  and  rvn2  with  longer  overall  decay  times  than  they  would  have  by \nthemselves.  This  arrangement  resembles  the  mechanism  of feedback  through  a neural \nlow-pass  filter,  suggested by  Robinson  (1981)  to  account  for  velocity  storage.  Thus, \nthe  network  model  gracefully  combines  the  two  mechanisms  that have  been  identified \nfor  velocity  storage,  in  what  may  be  a  more  optimal  configuration  than  either  one \nalone. \n\n\f36 \n\nAnastasio \n\n0.5 'f' \n\n0.4 \n\n, \n'  ' \n\n0.8 \n\n0.3 \n\n0.2 \n\n0.1 \n\n0.8 \n\nen \nUJ \nen z \n0 \na.. \nen \nUJ \na: \n.... \nz \n:::> \n\nCJ) \nUJ \nCJ) \nZ \n0 \na.. \nCJ) \nUJ \na: \n.... \nZ \n:::> \n\nA .. , \n\nA \n\n. . . . . . . . .   I \n\n,. \n, \n\n,  I \n\n, \nI \n> \n\nB \n\n0.8 \n\n0.4 \n\n, , \n0.3  ~  , \n\" \n\nt\"'''''''' \"'\"'' \n0.5  r\"'''''''''' \n, V -------,  ,. ----------\n\n' \n\" \n\" \n\\ \n\n0.2 \n\n0.1 \n\nI \n\n10 \n\n20 \n\n30 \n\n40 \n\n50 \n\n60 \n\n0 \n\n0 \n\n10 \n\n20 \n\n30 \n\n40 \n\n50 \n\n60 \n\n0.7  .: \n\n\" \n\" \n0.8  1, \n' , \n, , \n0.5  .( \n\n0.4 \n\n0.3 \n\n0.2 \n\n0.1 \n\n0 \n\n\\. \n, , \n\" \n, \n\" \n\" \n\" \n\n20 \n\n10 \n50 \nNETWORK CYCLES \n\n30 \n\n40 \n\nC \n\n0.8 \n\n0.7 \n\n0.8 \n\nI \nII \n.11 \n, , \n\" \n, \n\n0.3 \n\n0.2 \n\n0.1 \n\n0 \n\n60 \n\n0 \n\n0.5  (  - --------\n\n0.4 \n\n----------\n\n~ \n\n, , , \n\" \" \n\" \n) \n\n20 \n\n10 \n50 \nNETWORK CYCLES \n\n30 \n\n40 \n\n60 \n\nFigure 3:  Responses of Model VN Intemeurons.  Networks trained with (A and B)  and \nwithout  (C  and  D)  velocity  storage.  A  and  C,  rvnl,  solid;  lvnl,  dashed.  B and D, \nrvn2,  solid; Ivn2, dashed.  rhc, dotted, all plots. \n\nBesides  having  longer  time  constants,  the  hidden  units  also  have lower baseline  firing \nrates and higher sensitivities than the input units (Fig.  3,  A and B).  The lower baseline \nforces  the hidden units to  operate closer to  the bottom of the  squashing  function.  This \nin  tum causes  the hidden units  to  have  asymmetric  responses,  larger in  the  excitatory \nthan in the inhibitory directions.  Actual VN intemeurons also have higher sensitivities, \nlonger  time  constants,  lower  baseline  firing  rates  and  asymmetric  responses  as  com(cid:173)\npared to canal afferents (Fuchs and Kimm  1975; Buettner et a1.  1978). \n\nFor  purposes  of comparison,  the  network  was  retrained  to  produce  a  VOR  without \nvelocity  storage  (inputs  and  desired  outputs  had  the  same  time  constant  of one  tick). \nAll  of the  hidden  units  in  this  network  developed  almost  zero  lateral  inhibition.  Al(cid:173)\nthough  they also had higher sensitivities than  the input units,  their responses  otherwise \nresembled input responses (Fig.  3,  C and D).  This demonstrates that the long time con(cid:173)\nstant,  low baseline and  asymmetric responses of the hidden units  are all  interrelated by \ncommissural  inhibition  in  the  network,  which  may  be  the  case  for  actual  VN  inter(cid:173)\nneurons as well. \n\n\fA Recurrent Neural Network Model of Velocity Storage \n\n37 \n\n4 NONLINEAR BEHAVIOR OF THE VOR NETWORK MODEL \nBecause  hidden  units  have  low  baseline  firing  rates,  larger  inputs  can  produce  in(cid:173)\nhibitory hidden unit responses that are forced into the low-sensitivity region of squash(cid:173)\ning  function  or even  into  cut-off.  Hidden unit  cut-off breaks  the  feedback  loops  that \nsub serve  velocity  storage.  This produces nonlinearities  in  the responses of the  hidden \nand output units. \n\nFor example,  an  impulse  input  at  twice  the  amplitude  of the  training  input  produces \nlarger output unit responses (Fig.  4,  A,  solid),  but these  decay at a faster  rate  than ex(cid:173)\npected  (Fig.  4,  A,  dot-dash).  Faster  decay  results  because  inhibitory  hidden  unit \nresponses are  cutting-off at  the higher amplitude  level  (Fig.  4,  C,  solid).  This cut-off \ndisrupts velocity storage,  decreasing the  integrative properties of the hidden units  (Fig. \n4,  C,  solid) and increasing output unit decay rate. \nNonlinear responses are even more apparent with sinusoidal input.  At low input levels, \nthe output responses are also  sinusoidal and  their phase lag relative to  the input is com(cid:173)\nmensurate with their time constant of four ticks (Fig.  4,  B,  dashed).  As sinusoidal in-\n\n0.75 \n\nen \n~ 0.65  -\nZ \n0  0.55 \na.. en \nw  0.45 \na: \nI-\nZ  D.35  -\n:::> \n\n0.25 \n\n0 \n\n0.8 \n\n10 \n\n20 \n\n30 \n\n40 \n\n50 \n\n0.8  -\n\nen \nw en \nZ \n0 \na.. en \nw a: \nI- 0.2 \nZ \n:::> \n\n0.4 \n\n20 \n\n10 \n50 \nNETWORK CYCLES \n\n40 \n\n30 \n\nA \n\n0.7 \n\n0.8 \n\n0.5 \n\n0.3 \n\n0.8 \n\n0 \n\neo \nC \n\neo \n\n10 \n\n20 \n\n30 \n\n40 \n\n50 \n\neo \n\n20 \n\n10 \n50 \nNETWORK CYCLES \n\n40 \n\n30 \n\neo \n\nFigure 4:  Nonlinear Responses of Model  VOR Neurons.  A and C,  responses of Ir (A) \nand  rvnl  (C)  to  impulse  inputs  at  low  (dashed),  medium  (training,  dotted)  and  high \n(solid)  amplitudes.  A,  expected lr response at high input amplitude with time  constant \nof four ticks (dot-dash).  B and D,  response of lr (B)  and rvnl  (D)  to  sinusoidal inputs \nat low (dashed),  medium (dotted) and high (solid) amplitudes. \n\n\f38 \n\nAnastasio \n\nput  amplitude  increases,  however,  output  response  phase  lag  decreases,  signifying  a \ndecrease  in  time  constant  (Fig.  4,  B,  dotted  and  solid).  Also,  the  output  responses \nskew,  such  that  the  excursions  from  baseline  are  steeper than  the  returns.  Time  con(cid:173)\nstant decrease and skewing with increase in head rotation amplitude are also characteris(cid:173)\ntic  of the  VOR  in  monkeys  (Paige  1983).  Again,  these  nonlinearities  are  associated \nwith hidden unit cut-off (Fig.  4,  D,  dotted and solid),  which disrupts velocity storage, \ndecreasing  time  constant and phase lag.  Skewing results as  the system time constant is \nlowered at peak and raised again midrange throughout each cycle of the responses.  Ac(cid:173)\ntual  VN neurons in monkeys exhibit similar cut-off (rectification) and skew (Fuchs and \nKimm 1975; Buettner et al.  1978). \n\n5 CONCLUSIONS \nThe VOR lends itself well  to  neural network  modeling.  The results  summarized here, \npresented  in  detail  elsewhere  (Anastasio  1991),  illustrate  how  neural  network analysis \ncan be used to  study the organization of the  VOR,  and how its organization determines \nthe response properties of the neurons that subserve this reflex. \n\nAcknowledgments \n\nThis work was  supported by  the Faculty Research and Innovation Fund of the Univer(cid:173)\nsity of Southern California. \n\nReferences \n\nAnastasio,  TJ  (1991)  Neural  network  models  of velocity  storage  in  the  horizontal \nvestibulo-ocular reflex.  BioI Cybern 64:  187-196 \n\nBlair SM,  Gavin M (1981)  Brainstem commissures and control of time constant of ves(cid:173)\ntibular nystagmus.  Acta Otolaryngol 91:  1-8 \n\nBuettner UW,  Buttner U,  Henn  V  (1978)  Transfer  characteristics  of neurons  in  ves(cid:173)\ntibular nuclei of the alert monkey.  I  Neurophysiol 41:  1614-1628 \n\nFuchs  AF,  Kimm  I  (1975)  Unit  activity  in  vestibular  nucleus  of the  alert  monkey \nduring  horizontal  angular  acceleration  and  eye  movement.  I  Neurophysiol  38:  1140-\n1161 \n\nPaige  GC  (1983)  Vestibuloocular  reflex  and  its  interaction  with  visual  following \nmechanisms  in  the  squirrel  monkey.  1.  Response  characteristics  in normal  animals.  I \nNeurophysiol49:  134-151 \n\nRaphan Th,  Matsuo V,  Cohen B (1979)  Velocity  Storage in the vestibulo-ocular reflex \narc (VOR).  Exp Brain Res 35:  229-248 \n\nRobinson DA (1981) The use of control systems analysis in the neurophysiology of eye \nmovements.  Ann Rev Neurosci 4:  463-503 \n\nWilliams RJ,  Zipser D  (1989)  A learning algorithm for continually running fully recur(cid:173)\nrent neural networks.  Neural Comp  1:  270-280 \n\nWilson VI,  Melvill Iones G  (1979)  Mammalian  Vestibular Physiology.  Plenum Press, \nNew York \n\n\f", "award": [], "sourceid": 309, "authors": [{"given_name": "Thomas", "family_name": "Anastasio", "institution": null}]}