{"title": "Neural Network Models of Chemotaxis in the Nematode Caenorhabditis Elegans", "book": "Advances in Neural Information Processing Systems", "page_first": 55, "page_last": 61, "abstract": null, "full_text": "Neural network models of chemotaxis \nthe nematode  Caenorhabditis  elegans \n\n\u2022 In \n\nThomas C. Ferree, Ben A.  Marcotte, Shawn R. Lockery \n\nInstitute of Neuroscience, University of Oregon, Eugene,  Oregon 97403 \n\nAbstract \n\nWe  train recurrent networks  to control chemotaxis in a  computer \nmodel of the nematode  C.  elegans.  The model  presented  is  based \nclosely on the body mechanics, behavioral analyses, neuroanatomy \nand neurophysiology of  C.  elegans,  each imposing constraints rel(cid:173)\nevant  for  information  processing.  Simulated  worms  moving  au(cid:173)\ntonomously in simulated chemical environments display  a  variety \nof chemotaxis strategies similar to those of biological worms. \n\n1 \n\nINTRODUCTION \n\nThe nematode  C.  elegans  provides a unique opportunity to study the neuronal ba(cid:173)\nsis of neural computation in an animal capable of complex goal-oriented behaviors. \nThe adult hermaphrodite is only  1 mm long,  and has  exactly 302  neurons  and 95 \nmuscle cells.  The morphology of every cell and the location of most electrical and \nchemical synapses are known precisely (White et al.,  1986), making C.  elegans espe(cid:173)\ncially attractive for  study.  Whole-cell recordings are now  being made on identified \nneurons in the nerve ring of C.  elegans to determine electrophysiological properties \nwhich  underly  information processing in this animal  (Lockery  and  Goodman,  un(cid:173)\npublished).  However,  the strengths and  polarities of synaptic connections are not \nknown,  so  we  use  neural  network  optimization to find  sets  of synaptic  strengths \nwhich reproduce actual nematode behavior in a simulated worm. \n\nWe  focus  on  chemotaxis,  the ability to move  up  (or  down)  a gradient of chemical \nattractants (or repellants).  In the laboratory, flat  Petri dishes  (radius = 4.25  cm) \nare prepared with  a  Gaussian-shaped field  of attractant at the center,  and worms \nare allowed to move freely  about.  Worms propel themselves forward  by  generating \nan undulatory body wave, which produces sinusoidal movement.  In chemotaxis, the \nnervous  system  generates  motor  commands which  bias  this movement  and  direct \n\n\f56 \n\nT.  C.  Ferree, B. A.  Marcotte and S.  R. Lockery \n\nthe animal toward higher attractant concentration. \n\nAnatomical constraints pose important problems for  C.  elegans during chemotaxis. \nIn particular, the  animal detects the presence of chemicals with  a  pair of sensory \norgans  (amphids)  at the tip of the nose,  each containing the processes of multiple \nchemosensory neurons.  During normal locomotion, however,  the animal moves on \nits side so that the two amphids are perpendicular to the Petri dish.  C.  elegans can(cid:173)\nnot,  therefore,  sense  the gradient  directly.  One  possible strategy for  chemotaxis, \nwhich  has  been suggested previously  (Ward,  1973), is  that the animal computes a \ntemporal derivative of the local concentration during a single head sweep, and com(cid:173)\nbines this with some form  of proprioceptive feedback indicating muscle contraction \nand  the direction of head  sweep,  to compute the spatial gradient for  chemotaxis. \nThe existence of this and other strategies is discussed  later. \nIn Section 2, we derive a simple model of the nematode body which produces realis(cid:173)\ntic sinusoidal trajectories in response to motor commands from the nervous system. \nIn Section  3,  we  give  a  simple model  of the  C.  elegans  nervous  system  based  on \npreliminary physiological data.  In Section 4, we use a stochastic optimization algo(cid:173)\nrithm to determine sets of synaptic weights which control chemotaxis, and discuss \nsolutions. \n\n2  BIOMECHANICS OF  NEMATODE ORIENTATION \n\nNematode locomotion has  been studied in detail  (Niebur and Erdos,  1991;  Niebur \nand Erdos,  1993).  These authors derived Newtonian force  equations for  each mus(cid:173)\ncular  segment  of the  body,  which  can  be  solved  numerically to  generate  forward \nsinusoidal movement.  Unfortunately,  such  a  thorough  treatment is  computation(cid:173)\nally intensive and not practical to use with network optimization.  To  simplify the \nproblem we first recognize that chemotaxis is a behavior more of orientation than of \nlocomotion.  We therefore derive a set of biomechanical equations which  direct the \nhead to generate sinusoidal movement, which can be biased by the network toward \nhigher chemical concentrations. \nWe focus our attention on the point (x,1/)  at the tip of the nose, since that is where \nthe  animal senses  the chemical environment.  As  shown  in  Figure  1 ( a),  we  assign \na  velocity  vector fJ  directed  along the midline of the first  body  segment,  i.e.,  the \nhead.  Assuming that the worm moves forward  at constant speed  v,  we  can write \nthe velocity vector as \n\nfJ(t) = (~;, ~~) = (vcos(}(t),vsin(}(t)) \n\n(1) \n\nwhere  x,  y  and  (}  are  measured  relative  to  fixed  coordinates  in  the  Petri  dish. \nAssuming that the  worm  moves  without  lateral slipping and  that the undulatory \nwave  of muscular  contraction  initiated  in  the  neck  travels  posteriorally  without \nmodification, then each body segment simply follows the one previous (anterior) to \nit.  In this  way,  the  head  directs  the movement  and  the rest  of the body  simply \nfollows. \n\nFigure  l(b)  shows  an  expanded  view  of the  neck  segment.  As  the  worm  moves \nforward,  the posterior boundary of that segment assumes  the position held  by  its \nanterior neighbor at a slightly earlier time.  If L  is  the total body length and N  is \n\n\f57 \nNeural Network Models of Chemotaxis \nthe number of body segments, then this time delay is 6t  ~ L/Nv.  (For L = 1 mm, \nv  = 0.22 mm/s and  N  = 10  we  have  6t  ~ 0.45 s,  roughly  an order of magnitude \nsmaller than the relevant behavioral time scale:  the head-sweep period T  ~ 4.2 s.) \nIf we define the neck angle aCt)  ==  (h(t) - 92 (t),  then the above arguments imply \n\naCt) = 91 (t) - 91 (t - 6t)  ~ dt 6t \n\ndOl \n\n(2) \n\nwhere  the  second  relation  is  essentially  a  backward-Euler  algorithm  for  dOl/dt. \nSince 9 ==  91 ,  we  have reached the intuitive result that the neck angle a determines \nthe  rate of turning dO/dt.  Note  that while  91  and  92  are  defined  relative to  the \nfixed  laboratory coordinates, their difference a  is invariant under rotations of these \ncoordinates,  and  can therefore be  viewed  as  intrinsic to the body.  This  allows us \nto derive an expression for  a  in terms of muscle cell  contraction, or motor neuron \ndepolarization, as follows. \n\n(a) \n\nv \n\n(b) \n\nNeck \n\nT \nIv \n1 \n\nFigure 1:  Nematode body mechanics.  (a) Segmented model of the nematode body, \nshowing the direction of motion v.  (b)  Expanded view of the neck segment, showing \ndorsal  (D)  and ventral (V)  neck  muscles. \n\nNematodes maintain nearly constant volume during movement.  To incorporate this \nconstraint, albeit approximately, we  assume that at all times the geometry of each \nsegment is such that (ID -10) =  -(Iv -10), where 10  ==  L/N is the equilibrium length \nof a relaxed segment.  For small angles a, we  have a  ~ (Iv -ID)/d, where d is  the \nbody diameter.  The dashed lines in Figure 1(b) indicate dorsal and ventral muscles, \nwhich  are believed to  develop  tension  nearly  independent  of length  (Toida  et  al., \n1975).  When  contracting,  these  muscles  must  work  against  the  elasticity  of the \ncuticle, internal fluid  pressure, and elasticity and developed tension of the opposing \nmuscles.  If these elastic forces act linearly, then TD-TV ~ k (Iv-ID), where TD and \nTv are dorsal and ventral muscle tensions, and k is an effective force  constant.  For \nsimplicity, we further assume that each muscle develops tension linearly in response \nto the voltage of its corresponding motor neuron,  i.e.,  TD,v  =  E  VD,V,  where  E  is a \npositive constant, and VD  and Vv  are dorsal and ventral motor neuron voltages. \n\nCombining these results, we have finally \n\n~~ = \"1  (VD(t)  - Vv(t\u00bb) \n\n(3) \n\n\f58 \n\nT.  C.  Ferree, B. A.  Marcotte and S. R.  Lockery \n\nwhere\"Y =  (Nv/L)\u00b7 (E/kd).  With appropriate motor commands, equations (I) and \n(3)  can be integrated numerically to generate sinusoidal worm trajectories like those \nof biological worms.  This model  embodies the  main  anatomical features  that are \nlikely  to be important in  C.  elegans  chemotaxis,  yet is  sufficiently  compact  to  be \nembedded in a  network optimization procedure. \n\n3  CHEMOTAXIS  CONTROL CIRCUIT \n\nC.  elegans  neurons  are  tiny  and  have  very  simple morpologies:  a  typical  neuron \nin  the  head  has  a  spherical  soma  1-2  pm  in  diameter,  and  a  single  cylindrical \nprocess  60-80  pm  in  length  and  0.1-0.2  pm  in  diameter.  Compartmental mod(cid:173)\nels,  based  on  this  morphology  and  preliminary  physiological  recordings,  indicate \nthat  C.  elegans  neurons  are effectively isopotential  (Lockery,  1995).  Furthermore, \nC.  elegans neurons do not fire  classical all-or-none action potentials, but appear to \nrely primarily on graded signal propagation (Lockery and Goodman, unpublished). \nThus,  a  reasonable starting point for  a network model is to represent each neuron \nby a single isopotential compartment, in which voltage is the state variable, and the \nmembrane conductance in purely ohmic. \n\nAnatomical data indicate that the C.  elegans nervous system has both electrical and \nchemical  synapses,  but  the  synaptic  transfer  functions  are  not  known.  However, \nsteady-state synaptic transfer functions for  chemical synapses have been measured \nin Ascaris s'U'Um, a related species of nematode, where it was found that postsynaptic \nvoltage is  a  graded function  of presynaptic voltage,  due to tonic neurotransmitter \nrelease  (Davis  and  Stretton,  1989).  This  voltage  dependence  is  sigmoidal,  i.e., \nVpOlt  \"-J  tanh(Vpre ).  A simple network model which captures all of these features is \n\nT~ =  -Vi + Vmax tanh(f3 t Wij  (Vj  - Vj\u00bb)  + ~Iilm{t) \n\n3=1 \n\n(4) \n\nwhere Vi  is  the voltage of the  ith  neuron.  Here all  voltages are measured relative \nto  a  common  resting  potential,  Vmax  is  an  arbitrary  voltage  scale  which  sets  the \noperational range of the neurons, and f3  sets the voltage sensitivity of the synaptic \ntransfer function.  The weight  Wij  represents  the  net strength  and  polarity of all \nsynaptic connections from neuron j  to neuron i, and the constants Vj  determine the \ncenter of each transfer function.  The membane time constant  T  is  assumed  to be \nthe same for  all cells,  and will be discussed further later.  Note that in  (4),  synap(cid:173)\ntic transmission occurs instantaneously:  the time constant T  arises from capacitive \ncurrent through the cell membrane, and is unrelated to synaptic transmission.  Note \nalso that the way  in which  (4)  sums multiple inputs is not unique,  i.e.,  other sig(cid:173)\nmoidal models which sum inputs differently are equally plausible, since no data on \nsynaptic summation exists for  either C.  elegans or Ascaris 8'U'Uffl. \n\n. \n\nThe  stimulus  term  ~8t1m(t)  is  used  to  introduce  chemosensation  and  sinusoidal \nlocomotion  to  the  network  in  (4).  We  use  i  =  1  to  label  a  single  chemosensory \nneuron at the tip of the nose,  and i = n - 1 ==  D  and i = n ==  V  to label dorsal and \nventral  motor  neurons.  For  simplicity  we  assume  that  the  chemosensory  neuron \nvoltage responds linearly to the local chemical concentration: \n\n(5) \n\n\fNeural Network Models of Chemotaxis \n\n59 \n\nwhere  Vchem  is  a  positive  constant,  and  the local  concentration C(x, y)  is  always \nevaluated at the instantaneous nose position. \n\nIn the previous section, we emphasized that locomotion is effectively independent of \norientation.  We therefore assume the existence of a central pattern generator (CPG) \nwhich  is  outside the chemotaxis control circuit  (4).  Thus,  in addition to synaptic \ninput from other neurons, each motor neuron receives a sinusoidal stimulus \n\n(6) \n\nwhere VCPG  and w = 211\" /T  are positive constants. \n4  RESULTS  AND  DISCUSSION \nEquations  (1),  (3)  and  (4),  together  with  (5)  and  (6),  comprise  a  set  of n + 3 \nfirst-order  nonlinear  differential  equations,  which  can be solved  numerically given \ninitial conditions and a specification of the chemical environment.  We use a fourth(cid:173)\norder  Runge-Kutta algorithm  and  find  favorable  stability  and  convergence.  The \nnecessary body parameters have been measured by observing actual worms (Pierce \nand Lockery,  unpublished):  v = 0.022  cm/s, T  = 4.2 s and '\"Y  =  0.8/(2VcPG).  The \nchemical  environment  is  also  chosen  to  agree  roughly  with  experimental  values: \nC(x,y) = Coexp(-(x2 + y2)/-Xb),  with Co = 0.052 p.mol/cm3  and -Xc  = 2.3 cm. \nTo  optimize networks to control chemotaxis,  we  use  a  simple simulated annealing \nalgorithm which  searches  over the  (n2 + 3)-dimensional space of parameters Wij, \n{3,  Vchem  and  VCPG.  In the  results shown  here,  we  used  n  = 12,  and  set V;  = O. \nEach set of the resulting parameters represents a  different  nervous system for  the \nmodel worm.  At the beginning of each run, the worm is initialized by  choosing an \ninitial position (xo, Yo),  an initial angle 00 ,  and by setting V.  = O.  Upon numerically \nintegrating, simulated worms move autonomously in their environment for  a prede(cid:173)\ntermined amount of time,  typically the real-time equivalent of 10-15 minutes.  We \nquantify the performance, or fitness, of each worm during chemotaxis by computing \nthe average chemical concentation at the tip of its nose over the duration of each \nrun.  To avoid lucky scores, the actual score for each worm is obtained by averaging \nover several initial conditions. \n\nIn Figure  2,  we  show  a  comparison of tracks  produced  by  (a)  biological and  (b) \nsimulated worms  during chemotaxis.  In each  case,  three worms  were  placed in a \ndish  with a radial gradient and allowed to move freely for  the real-time equivalent \nof 15  minutes.  In  (b),  the three worms have the same neural parameters  (Wij,  (3, \nVchem,  VCPG),  but different initial angles 00 \u2022  In both (a)  and  (b),  all three worms \nmake initial movements, then move toward the center of the dish and remain there. \nIn  other  optimizations,  rather  than  orbit  the  center,  the  simulated  worms  may \napproach the center asymptotically from one side,  make simple geometric patterns \nwhich  pass through the  center,  or exhibit a  variety  of other  distinct strategies for \nchemotaxis.  This is similar to the situation with biological worms, which also have \nconsiderable variation in the details of their tracks. \nThe behavior shown in Figure 2 was produced using T  = 500 ms.  However, prelimi(cid:173)\nnary electrophysiological recordings from  C.  elegans neurons suggest that the actual \nvalue may be as much as an order of magnitude smaller, but not bigger (Lockery and \nGoodman,  unpublished).  This presents  a  potential  problem for  chemotaxis  com-\n\n\f60 \n\nT.  C.  Ferree, B. A. Marcotte and S.  R.  Lockery \n\nputation,  since shorter time constants require greater sensitivity to small changes \nin  O(x, y)  in  order  to compute  a  temporal derivative,  which  is  believed  to  be  re(cid:173)\nquired.  During optimization,  we  have  seen  that for  a  fixed  number of neurons n, \nfinding optimal solutions becomes more difficult as T  is decreased.  This observation \nis  very  difficult  to quantify,  however,  due  to the existence of local maxima in the \nfitness  function.  Nevertheless, this suggests that additional mechanisms may need \nto be included  to understand  neural  computation in  C.  elegana.  First,  time- and \nvoltage-dependent conductances will modify the effective membrane time constant, \nand  may  increase the effective  time scale  for  computation by  individual neurons. \nSecond, more neurons and synaptic delays will also move the effective neuronal time \nscale closer to that of the behavior.  Either of these will allow comparisons of O(z, II) \nacross greater distances, thereby requiring less sensitivity to compute the gradient, \nand potentially improving the ability of these networks to control chemotaxis. \n\n2em \n\nFigure  2:  Nematodes  performing chemotaxis:  (a)  biological  (Pierce  and  Lockery, \nunpublished), and  (b)  simulated. \n\nWe  also  note,  based on  a  variety  of other results,  not  shown  here,  that the  head(cid:173)\nsweep strategy, described in the introduction, is by no means the only strategy for \nchemotaxis  in  this  system.  In  particular,  we  have  optimized networks  without  a \nCPG,  i.e.,  with  VCPG  = 0 in  (6),  and found  parameter sets that successfully  con(cid:173)\ntrol chemotaxis.  This presents the possibility that even worms with a CPG do not \nnecessarily compute the gradient based on lateral movement of the head,  but may \ninstead respond only to changes in concentration along their mean trajectory.  Sim(cid:173)\nilar results have  been reported previously, although based on a somewhat different \nbiomechanical model  (Beer and Gallagher, 1992). \nFinally,  we  have  also  optimized discrete-time networks,  obtained by  setting T  = 0 \nin  (4)  and updating all units synchronously.  As  is  well-known,  on relatively short \ntime scales (- T) such  a system tends to \"overshoot\"  at each successive time step, \nleading  to  sporadic  behavior  of the  network  and  the  body.  Knowing  this,  it  is \ninteresting that simulated worms with such a nervous system are capable of reliable \nbehavior over longer time scales,  i.e.,  they successfully perform chemotaxis. \n\n\fNeural Network Models of Chemotaxis \n\n61 \n\n5  CONCLUSIONS  AND FUTURE WORK \n\nThe  main  result  of this  paper is  that  a  small  nervous  system,  based  on  graded(cid:173)\npotential neurons, is capable of controlling chemotaxis in a worm-like physical body \nwith the dimensions of C.  elegans.  The model presented is based closely on the body \nmechanics,  behavioral analyses,  neuroanatomy and neurophysiology of C.  elegans, \nand is a reliable starting point for more realistic models to follow.  Furthermore, we \nhave established the existence of chemotaxis strategies that had not been anticipated \nbased on behavioral experiments with real worms. \n\nFuture work will involve both improvement of the model and analysis of the result(cid:173)\ning solutions.  Improvements will include introducing voltage- and time-dependent \nmembrane conductances, as this data becomes available, and more realistic models \nof synaptic transmission.  Also, laser ablation experiments have been performed that \nsuggest which interneurons and motor neurons in C.  elegans may be important for \nchemotaxis (Bargmann, unpublished), and these data can be used to constrain the \nsynaptic connections during optimization.  Analyses  will  be aimed  at determining \nthe  role  of individual  physiological and  anatomical features,  and  how  they  func(cid:173)\ntion together to govern the collective properties of the network as  a  whole during \nchemotaxis. \n\nAcknowledgements \n\nThe authors would like to thank Miriam Goodman and Jon Pierce for  helpful dis(cid:173)\ncussions.  This  work  has  been  supported  by  NIMH  MH11373,  NIMH  MH51383, \nNSF  IBN  9458102,  ONR N00014-94-1-0642, the Sloan Foundation, and The Searle \nScholars Program. \nReferences \nBeer,  R.  D.  and  J.  C.  Gallagher  (1992).  Evolving dynamical  neural  networks  for \nadaptive behavior,  Adaptive Behavior 1(1):91-122. \n\nDavis,  R.  E.  and  A.  O.  W.  Stretton  {1989}.  Signaling properties  of Ascaris ma(cid:173)\ntorneurons:  Graded active responses, graded synaptic transmission, and tonic trans(cid:173)\nmitter release,  J.  Neurosci.  9:415-425. \nLockery, S.  R.  (1995).  Signal propagation in the nerve ring of C.  elegans,  Soc.  Neu(cid:173)\nrosd.  Abstr.  569.1:1454. \n\nNiebur, E. and P. Erdos (1991).  Theory of the locomotion of nematodes:  Dynamics \nof undulatory progression on a surface,  Biophys.  J.  60:1132-1146. \n\nNiebur,  E.  and P.  Erdos  (1993).  Theory of the locomotion of nematodes:  Control \nof the somatic motor neurons by interneurons, Math.  Biosci.  118:51-82. \n\nToida, N.,  H.  Kuriyama, N.  Tashiro and Y.  Ito {1975}.  Obliquely striated muscle, \nPhysiol.  Rev.  55:700-756. \n\nWard,  S.  (1973).  Chemotaxis  by  the  nematode  Caenorhabditis  elegans:  Iden(cid:173)\ntification  of  attractants  and  analysis  of  the  response  by  use  of  mutants, \nProc.  Nat.  Acad.  Sci.  USA  10:817-821. \nWhite, J. G.,  E.  Southgate, J. N.  Thompson and S.  Brenner (1986).  The structure \nof the nervous system of C.  elegans,  Phil.  Trans.  R.  Soc.  London 314:1-340. \n\n\f", "award": [], "sourceid": 1199, "authors": [{"given_name": "Thomas", "family_name": "Ferr\u00e9e", "institution": null}, {"given_name": "Ben", "family_name": "Marcotte", "institution": null}, {"given_name": "Shawn", "family_name": "Lockery", "institution": null}]}