{"title": "On Tropistic Processing and Its Applications", "book": "Neural Information Processing Systems", "page_first": 262, "page_last": 269, "abstract": null, "full_text": "262 \n\nON  TROPISTIC  PROCESSING  AND  ITS  APPLICATIONS \n\nManuel  F.  Fernandez \n\nGeneral  Electric  Advanced  Technology  Laboratories \n\nSyracuse,  New  York  13221 \n\nABSTRACT \n\nThe  interaction  of  a  set  of  tropisms  is  sufficient  in  many \n\ncases  to  explain  the  seemingly  complex  behavioral  responses \nexhibited  by  varied  classes  of  biological  systems  to  combinations  of \nstimuli.  It  can  be  shown  that  a  straightforward  generalization  of \nthe  tropism  phenomenon  allows  the  efficient  implementation  of \neffective  algorithms  which  appear  to  respond  \"intelligently\"  to \nchanging  environmental  conditions.  Examples  of  the  utilization  of \ntropistic  processing  techniques  will  be  presented  in  this  paper  in \napplications  entailing  simulated  behavior  synthesis,  path-planning, \npattern  analysis  (clustering),  and  engineering  design  optimization. \n\nINTRODUCTION \n\nThe  goal  of  this  paper  is  to  present  an  intuitive  overview  of \na  general  unsupervised  procedure  for  addressing  a  variety  of  system \ncontrol  and  cost  minimization  problems.  This  procedure  is  hased  on \nthe  idea  of  utilizing  \"stimuli\"  produced  by  the  environment  in  which \nthe  systems  are  designed  to  operate  as  basis  for  dynamically \nproviding  the  necessary  system  parameter  updates. \n\nThis  is  by  no  means  a  new  idea:  countless  examples  of  this \n\napproach  abound  in  nature,  where  innate  reactions  to  specific \nstimuli  (\"tropisms\"  or  \"taxis\"  --not  to  be  confused  with \n\"instincts\")  provide  organisms  with  built-in  first-order  control \nlaws  for  triggering  varied  responses  [8].  (It  is  hypothesized  that \n\"knowledge\"  obtained  through  evolution/adaptation  or  through \nlearning  then  refines  or  suppresses  most  of  these  primal  reactions). \nSeveral  examples  of  the  implicit  utilization  of  this  approach \n\ncan  also  be  found  in  the  literature,  in  applications  ranging  from \nbehavior  modeling  to  pattern  analysis.  Ve  very  briefly  depict  some \nthese  applications,  underlining  a  common  pattern  in  their \nformulation  and  generalizing  it  through  the  use  of  basic  field \ntheory  concepts  and  representations.  A more  rigorous  and  detailed \nexposition  --regarding  both  mathematic  and \napplication/implementation  aspects-- is  presently  under  preparation \nand  should  be  ready  for  publication  sometime  next  year  ([6]). \n\nTROPISMS \n\nTropisms  can  be  defined  in  general  as  class-invariant  systemic \n\nresponses  to  specific  sets  of  stimuli  [6].  All  time-invariant \nsystems  can  thus  be  viewed  as  tropistic  provided  that  we  allow  all \npossible  stimuli  to  form  part  of  our  set  of  inputs.  In  most \ntropistic  systems,  however,  response- (or  time-)  invariance  applies \nonly  to  specific  inputs:  green  plants,  for  example,  twist  and  grow \nin  the  direction  of  light  (phototropism),  some  birds'  flight \npatterns  follow  changes  in  the  Earth's  magnetic  field \n(magnetotropism),  various  organisms  react  to  gravitational  field \n\n\u00a9 American Institute of Physics 1988 \n\n\f263 \n\nvariations  (geotropism),  etc. \n\nTropism/stimuli  interactions  can  be  portrayed  in  term~  of  the \n\nsuperposition  of  scalar  (e.g.,  potential)  or  vector  (e.g.,  force) \nfields  exhibiting  properties  paralleling  those  of  the  suitably \nconstrained  \"reactions\"  we  wish  to  model  [1J,[6J.  The  resulting \nfield  can  then  be  used  as  a  basis  for  assessing  the  intrinsic  cost \nof  pursuing  any  given  path  of  action,  and  standard  techniques  (e.g., \ngradient-following  in  the  case  of  scalar  fields  or  divergence \ncomputation  in  the  case  of  vector  fields)  utilized  in  determining  a \nresponse*.  In  addition,  the  global  view  of  the  situation  provided  by \nfield  representations  suggest  that  a  basic  theory  of  tropistic \nbehavior  can  also  be  formulated  in  terms  of  energy  expenditure \nminimization  (Euler-Lagrange  equations).  This  formulation  would \nyield  integral-based  representations  (Feynman  path  integrals \n[4],[11])  satisfying  the  observation  that  tropistic  processes \ntypically  obey  the  principle  of  least  action. \n\nAlternatively,  fields  may  also  be  collapsed  into  \"attractors\" \n\n(points  of  a  given  \"mass\"  or  \"charge\"  in  cost  space)  through  laws \ndefining  the  relationships  that  are  to  exist  among  these \n\"at tractors\"  and  the  other  particles  traveling  through  the  space. \nThis  provides  the  simplification  that  when  updating  dynamically \nchanging  situations  only  the  effects  caused  by  the  interaction  of \nthe  attractors  with  the  particles  of  interest  --rather  than  the \nwhole  cost  field-- may  have  to  be  recalculated. \n\nFor  example,  appropriately  positioned  point  charges  exerting \non  each  other  an  electrostatic  force  inversely  proportional  to  the \nsquare  of  their  distance  can  be  used  to  represent  the  effects  of  a \ncoulombic-type  cost  potential  field.  A  particle  traveling  through \nthis  field  would  now  be  affected  by  the  combination  of  forces \nensuing  from  the  interaction  of  the  attractors'  charges  with  its \nown.  If  this  particle  were  then  to  passively  follow  the  composite  of \nthe  effects  of  these  forces  it  would  be  following  the  gradient  of \nthe  cost  field  (i.e.,  the  vector  resulting  from  the  superposition  of \nthe  forces  acting  on  the  particle  would  point  in  the  direction  of \nsteepest  change  in  potential). \n\nFinally,  other  representations  of  tropism/stimuli  interactions \n\n(e.g.,  Value-Driven  Decision  Theory  approaches)  entail  associating \n\"profit\"  functions  (usually  sigmoidal)  with  each  tropism,  modeling \nthe  relative  desirability  of  triggering  a  reaction  as  a  function  of \nthe  time  since  it  was  last  activated  [9].  These  representations  are \n\nfield \n\nthrough \n\nthe \n\ninsight \n\ninto \n\n* \n\nIn  order \n\nrepresentations \n\nto  bring  extra \n\ntropism/stimuli \ninteractions  and  simplify  their  formulation,  one  may  exchange  vector \nand  scalar \nof \nappropriately  selected  mappings.  Some  of  the  most  important  of  such \nmappings  are  the  gradient  operator \nthe \nis  proportional  to  a \ngradient  of  a  scalar  --potential--\nthought  of \n\"force\"  --vector-- field),  the  divergence  (which  may  be \nas  performing \nthat \nperformed  in  scalar  fields  by  the  gradient),  and  their  combinations \na  scalar-to-scalar  mapping  which  can  be \n(e.g., \nvisualized  as  performing  on  potential  fields \nthe  equivalent  of  a \nsecond  derivative  operation. \n\n(particularly  so  because \n\nfunction  analogous \n\nthe  Laplacian, \n\nutilization \n\nin  vector \n\nfields  a \n\nfield \n\nto \n\n\f264 \n\n.-\n.' \n/.~ \n\n\u2022  Model fly as a positive geotropislic point of mass M. \n\u2022  Model fence slakes as negalive geotropislic poinls \n\n\u2022 \n\nwith  masses m 1 ,  m z \u2022 \u2022\u2022\u2022\u2022  mit\u00b7 \nAt each  update time compute sum of forces acting on \nfrog: \n\nF  \u2022  k \n\nH \n\nd 2 \n\" \n\n\u2022 \n\nCompute frog's heading and acceleration  based on \nthe ensuing force;  then  update frog's position. \n\nFigure  1:  Attractor-based  representation  of  a  frog-fenee-fly \n\nscenario  (see  [1)  for  a  vector-field  representation).  The  objective \nis  to  model  a  frog's  path-planning  decision-making  process  when \napproaching  a  fly  in  the  presence  of  obstacles.  (The  picket  fence  is \nrepresented  by  the  elliptical  outline  with  an  opening  in  the  back, \nthe  fly  --inside  the  fenced  space-- is  represented  by  a  \"+~  sign, \nand  arrows  are  used  to  indicate  the  direction  of  a  frog's  trajectory \ninto  and  out  of  fenced  area). \n\n\fparticularly  amenable  to  neural-net  implementations  [6J. \n\nTROPISTIC  PROCESSING \n\n265 \n\nTropistic  processing  entails  building  into  systems  tropisms \n\nappropriate  for  the  environment  in  which  these  systems  are  expected \nto  operate.  This  allows  taking  advantage  of  environment-produced \n\"stimuli\"  for  providing  the  required  control  for  the  systems' \nbehavior. \n\nThe  idea  of  tropistic  processing  has  been  utilized  with  good \n\nresults  in  a  variety  of  applications.  Arbib  et.al.,  for  example, \nhave  implicitly  utilized  tropistic  processing  to  describe  a \nbatrachian's  reaction  to  its  environment  in  terms  of  what  may  be \nvisualized  as  magnetic  (vector)  fields'  interactions  [1]. \n\nVatanabe  (12)  devised  for  pattern  analysis  purposes  an \n\ninteraction  of  tropisms  (\"geotropisms\")  in  which  pattern  \"atoms\"  are \nattracted  to  each  other,  and  hence  \"clustered\",  subject  to  a \nsquared-inverse-distance  (\"feature  distance\")  law  similiar  to  that \nfrom  gravitational  mechanics.  It  can  be  seen  that  if  each  pattern \natom  were  considered  an  \"organism\",  its  behavior  would  not  be \nconceptually  different  from  that  exhibited  by  Arbibian  frogs:  in \nboth  cases  organisms  passively  follow  the  force  vectors  resulting \nfrom  the  interaction  of  the  environmental  stimuli  with  the \norganisms'  tropisms.  It  is  interesting,  though,  to  note  that  the \n\"organisms'\"  behavior  will  nonetheless  appear  \"intelligent\"  to  the \ncasual  observer. \n\nThe  ability  of  tropistic  processes  to  emulate  seemingly \n\nrational  behavior  is  now  begining  to  be  explored  and  utilized  in  the \ndevelopment  of  synthetic-psychological  models  and  experiments. \nBraitenberg,  for  example,  has  placed  tropisms  as  the  primal  building \nblock  from  which  his  models  for  cognition,  reason,  and  emotions \nevolve  [3]**;  Barto  [2]  has  suggested  the  possibility  of  combining \ntropisms  and  associative  (reinforced)  learning,  with  aims  at \nenabling  the  automatic  triggering  of  behavioral  responses  by \npreviously  experienced  situations;  and  Fernandez  [6]  has  used \nCROBOTS  [10],  a  virtual  multiprocessor  emulator,  as  laboratory  for \nevaluating  the  effects  of  modifying  tropistic  responses  on  the  basis \nof  their  projected  future  consequences. \n\nOther  applications  of  tropistic  processing  presently  being \n\ninvestigated  include  path-planning  and  engineering  design \noptimization  [6].  For  example,  consider  an  air-reconnaissance \nmission  deep  behind  enemy  lines;  as  the  mission  progresses  and \nunexpected  SAM  sites  are  discovered,  contingency  flight  paths  may  be \ndeveloped  in  real  time  simply  by  modeling  each  SAM  or  interdiction \nsite  as  a  mass  point  towards  which  the  aircraft  exhibits  negative \ngeotropistic  tendencies  (i.e.,  gravitational  forces  repel  it),  and \nmodeling  the  objective  as  a  positive  geotropistic  point.  A  path  to \n\n**  Of  particular  interest  within  the  sole  context  of  Tropistic \nProcessing \nis  Dewdney's  [5]  commented  version  of  the  first  chapters \nof  Braitenberg's  book  [3J,  in  which  the  \"behavior\"  of  mechanically \nvery  simple  cars,  provided  with  \"~yes\"  and  phototropism-supporting \nconnections  (including  Ledley-type  \"neurons\"  [4J),  is  \"analyzed\". \n\n\f266 \n\n\u2022 \n\n\u2022\u2022 . \"\":,~ \n\n\u2022 \n\n\u2022 \u2022 \n\u2022 ,. \n\u2022 \n\u2022\u2022\u2022 -. \n\u2022 \n\nill' \",:\" \n\n\u2022 \u2022  \u2022 \n\u2022 \u2022 \u2022  \" \n, \n\u2022 \n\n\u2022 \n\n\u2022 \u2022  \u2022  \u2022 A \u2022\u2022 \n\n\u2022 \n\n\u2022\u2022 \u2022 \n\u2022 \n\u2022 \n\u2022 \n\u2022 \n\u2022 \u2022 \u2022  \u2022 \u2022 \n\u2022 \n\n\u2022 \n\n\u2022 \u2022 \n\n~!::. \n\n*' \n\n~:::  \u2022 --\n\u2022 \u2022 \u2022 \u2022 \n\u2022 \u2022 .. \n\n\u2022 \n\n-\u2022 \n\n\u2022 \u2022  \u2022 \n\n\u2022 \ne \n\n,,-\n\n\u2022 \ne \n\n.~~ \n\n8 \n\n.. \n\nFigure  ~  (Geotropistic  clustering  ~2]):  The  problem  being  portrayed \nhere  is  that  of  clustering  dots  distributed  in  [x,y]-space  as  shown \nand  uniformly  in  color  ([red,blue,green]).  The  approach  followed  is \nthat  outlined  in  Figure  1,  with  the  differences  that  normalized \n(Hahalanobis)  distances  are  used  and  when  merges  occur,  conservation \nof  momentum  is  observed.  Tags  are  also  kept  --specifying  with  which \ndots  and  in  what  order  merges  occur-- to  allow  drawing  cluster \nboundaries  in  the  original  data  set.  (Efficient  implementation  of \nthis  clustering  technique  entails  using  a  ring  of  processors,  each \nof  which  is  assigned  the  \"features\"  of  one  or  more  \"dots\"  and  the \ntask  of  carrying  out  computations  with  respect  to  these  features.  If \nthe  features  of  each  dot  are  then  transmitted  through  the  ring,  all \nthe  forces  imposed  on  it  by  the  rest  will  have  been  determined  upon \ncompletion  of  the  circuit). \n\n\f267 \n\nthe  target  will  then  be  automatically  drawn  by  the  interaction  of \nthe  tropisms  with  the  gravitational  forces.  (Once  the  mission  has \nbeen  completed,  the  target  and  its  effects  can  be  eliminated, \nleaving  active  only  the  repulsive  forces,  which  will  then  \"guide\" \nthe  airplane  out  of  the  danger  zone). \n\nIn  engineering  design  applications  such  as  lens  modeling  and \n\ndesign,  lenses  (gradient-index  type,  for  example)  can  be  modeled  in \nterms  of  photons  attempting  to  reach  an  objective  plane  through  a \nthree-dimensional  scalar  field  of  refraction  indices;  modeling  the \nprocess  tropistically  (in  a  manner  analogous  to  that  of  the \nair-reconnaissance  example  above)  would  yield  the  least-action  paths \nthat  the  individual  photons  would  follow.  Similarly,  in \n\"surface-of-revolution\"  fuselage  design  (\"Newton's  Problem\"),  the \ncharacteristics  of  the  interaction  of  forces  acting  within  a  sheet \nof  metal  foil  when  external  forces  (collisions  with  a  fluid's \nmolecules)  are  applied  can  be  modeled  in  terms  of  tropistic \nreactions  which  will  tend  to  reconfigure  the  sheet  so  as  to  make  it \npresent  the  least  resistance  to  friction  when  traversing  a  fluid. \n\nAdditional  applications  of  tropistic  processing  include  target \ntracking  and  multisensor  fusion  (both  can  be  considered  instances  of \n\"clustering\")  [6],  resource  allocation  and  game  theory  (both  closely \nrelated  to  path-planning)  [9],  and  an  assortment  of  other \ncost-minimization  functions.  Overall,  however,  one  of  the  most \nimportant  applications  of  tropistic  processing  may  be  in  the \nmodeling  and  understanding  of  analog  processes  [6],  the  imitation  of \nwhich  may  in  turn  lead  to  the  development  of  effective  strategies \n\nPAST EXPERIENCE \n(e.g.  MEMORY MAPS) \n\nM \n\nBASIC \nmOPISM \nFUNCTION \n\nOBSERVATlONS \n\nPREDICTED  (i.e. MODELLED) \nOUTCOUE \n\np \n\nRESPONSE \n\nRESPONSE \nFUNCTION \n\nTROPISM-BASED SYSTEM \n\nFigure  3:  The  combination  of  tropisms  and  associative  (reinforced) \nlearning  can  be  used  to  enable  the  automatic  triggering  of \nbehavioral  responses  by  previously  experienced  situations  [2].  Also, \nthe  modeled  projection  of  the  future  consequences  of  a  tropistic \ndecision  can  be  utilized  in  the  modification  of  such  decision  (6J. \n(Note  analogy  to  filtering  problem  in  which  past  history  and \npredicted  behavior  are  used  to  smooth  present  observations). \n\n\f268 \n\ni \n\n-5000.0 \n\n-33\u00bb.3 \n\n-I \n\n\u2022  \u2022  ''''.7 \n\n.lll3.J \n\n5000.' \n\n-5000.0 \n\n-3l\u00bb.l \n\n-, \n\n\u2022 \n\n\u2022 \n\n''''.7 \n\nlJl3.J \n\n5000.0 \n\ni , \n\ni , \n\noD \n01 \n\n\"' to , \n\ni , \n\ni , \n\n3lJ3.J \n\n5000.0 \n\nFigure  4:  Simplified  representation  of  air-reconnaissance  mission \nexample  (see  text):  objective  is  at  center  of  coordinate  axis,  thick \ndots  represent  SAM  sites,  and  arrows  denote  airplane's  direction  of \nflight  (airplane's  maximum  attainable  speed  and  acceleration  are \nconstrained).  All  portrayed  scenarios  are  identical  except  for \ntropistic  control-law  parameters  (mainly  objective  to  SAM-sites  mass \nratios  in  the  first  three  scenarios).  Varying  the  masses  of  the \nobjective  and  SAM  sites  can  be  interpreted  as  trading  off  the \nrelative  importance  of  the  mission  vs.  the  aircraft's  safety,  and \ncan  produce  dramatically  differing  flight  paths,  induce  chaotic \nbehavior  (bottom-left  scenario),  or  render  the  system  unstable.  The \nbottom-right  scenario  portrays  the  situation  in  which  a  tropistic \ndecision  is  projected  into  the  future  and,  if  not  meeting  some \ncriterion,  modified  (altering  the  direction  of  flight  --e.g., \nfollowing  an  isokline--,  re-evaluating  the  mission's  relative \nimportance  --revising  masses--,  changing  the  update  rate,  etc.). \n\n\f269 \n\nfor  taking  full  advantage  of  parallel  architectures  [11]***.  It  is \nthus  expected  that  the  flexibility  of  tropistic  processes  to  adapt \nto  changing  environmental  conditions  will  prove  highly  valuable  to \nthe  advancement  of  areas  such  as  robotics,  parallel  processing  and \nartificial  intelligence,  where  at  the  very  least  they  will  provide \nsome  decision-making  capabilities  whenever  unforeseen  circumstances \nare  encountered. \n\nACKNOVLEDGEMENTS \n\nSpecial  thanks  to  D.  P.  Bray  for  the  ideas  provided  in  our \nmany  discussions  and  for  the  development  of  the  finely  detailed \nsimulations  that  have  enabled  the  visualization  of  unexpected \naspects  of  our  work. \n\nREFERENCES \n\n[1]  Arbib,  M.A.  and  House,  D.H.:  \"Depth  and  Detours:  Decision  Making \nin  Parallel  Systems\".  IEEE  Vorkshop  on  Languages  for  Automation: \nCognitive  Aspects  in  Information  Processing;  pp.  172-180  (1985). \n[2]  Barto,  A.G.  (Editor):  \"Simulation  Experiments  with  Goal-Seeking \n\nAdaptive  Elements\".  Avionics  Laboratory,  Vright-Patterson  Air \nForce  Base,  OH.  Report  #  AFVAL-TR-84-1022. \n\n(1984). \n\n[3]  Braitenberg,  V.:  Vehicles:  Experiments  in  Synthetic  Psychology. \n\nThe  MIT  Press.  (1984). \n\n[4]  Cheng,  G.C.;  Ledley,  R.S.;  and  Ouyang,  B.:  \"Pattern  Recognition \n\nwith  Time  Interval  Modulation  Information  Coding\".  IEEE \nTransactions  on  Aerospace  and  Electronic  Systems.  AES-6,  No.2; \npp.  221-227  (1970). \n\n[5]  Dewdney,  A.K.:  \"Computer  Recreations\".  Scientific  American. \n\nVol.256,  No.3;  pp.  16-26  (1987). \n\n[6]  Fern6ndez,  M.F.:  \"Tropistic  Processing\".  To  be  published  (1988). \n[7J  Feynman,  R.P.:  Statistical  Mechanics:  A Set  of  Lectures. \n\nFrontiers  in  Physics  Lecture  Note  Series-zI982). \n\n[8]  Hirsch,  J.: \n\n\"Nonadaptive  Tropisms  and  the  Evolution  of \n\nBehavior\".  Annals  of  the  New  York  Academy  of  Sciences.  Vol.223; \npp.  84-88  (1973). \n\n[9]  Lucas,  G.  and  Pugh,  G.: \n\n\"Applications  of  Value-Driven \n\nAutomation  Methodology  for  the  Control  and  Coordination  of \nNetted  Sensors  in  Advanced  C**3\".  Report  #  RADC-TR-80-223. \nRome  Air  Development  Center,  NY.  (1980). \n\n[10]  Poindexter,  T.: \n\n\"CROBOTS\".  Manual,  programs,  and  files  (1985). \n\n2903  Vinchester  Dr.,  Bloomington,  IL.,  61701. \n[11J  Vallqvist,  A.;  Berne,  B.J.;  and  Pangali,  C.: \n\n\"Exploiting \n\nPhysical  Parallelism  Using  Supercomputers:  Two  Examples  from \nChemical  Physics\".  Computer.  Vol.20,  No.5;  pp.  9-21  (1987). \n\n[12]  Vatanabe,  S.:  Pattern  Recognition:  Human  and  Mechanical. \n\nJohn  Viley  & Sons;  pp.  160-168  (1985). \n\n***  Optical  Fourier  transform  operations,  for  instance,  can  be \nin  high-granularity  machines  through  a  procedure  analogous \nlens  simulation  example,  with  processors \n\nmodeled \nto  the  gradient-index \nrepresenting  diffraction-grating  \"atoms\"  [6]. \n\n\f", "award": [], "sourceid": 28, "authors": [{"given_name": "Manuel", "family_name": "Fern\u00e1ndez", "institution": null}]}