{"title": "Simulation and Measurement of the Electric Fields Generated by Weakly Electric Fish", "book": "Advances in Neural Information Processing Systems", "page_first": 436, "page_last": 443, "abstract": null, "full_text": "436 \n\nSIMULATION AND MEASUREMENT OF \nTHE ELECTRIC FIELDS GENERATED \n\nBY WEAKLY ELECTRIC FISH \n\nBrian Rasnow1, Christopher Assad2, Mark E. Nelson3 and James M. Bow~ \n\nDivisions of Physics1 ,Elecbical Engineerini, and Biolo~ \n\nCaltech, Pasadena, 91125 \n\nABSTRACT \n\nThe weakly electric fish, Gnathonemus peters;;, explores its environment by gener(cid:173)\nating pulsed elecbic fields  and detecting small pertwbations in the fields  resulting from \nnearby objects.  Accordingly, the fISh  detects and discriminates objects on  the basis of a \nsequence of elecbic \"images\" whose temporal and spatial properties depend on  the  tim(cid:173)\ning of the fish's electric organ discharge and its body position relative to objects in its en(cid:173)\nvironmenl  We are interested in investigating how these fish utilize timing and body-po(cid:173)\nsition during exploration to aid in object discrimination.  We have developed a fmite-ele(cid:173)\nment simulation of the fish's self-generated electric  fields  so as  to  reconstruct the elec(cid:173)\ntrosensory consequences of body position and electric organ discharge timing in the fish. \nThis paper describes this finite-element simulation system and presents preliminary elec(cid:173)\ntric field measurements which are being used to tune the simulation. \n\nINTRODUCTION \n\nThe active positioning of sensory structures (i.e. eyes, ears, whiskers, nostrils, etc.) \nis characteristic  of the information  seeking  behavior of all exploratory animals.  Yet,  in \nmost existing computational models and  in  many  standard experimental paradigms,  the \nactive aspects of sensory processing are either eliminated or controlled (e.g. by stimulat(cid:173)\ning fIXed  groups of receptors  or by stabilizing images).  However, it is clear that the ac(cid:173)\ntive positioning of receptor surfaces directly affects the content and quality of the sensory \ninfonnation  received by  the  nervous  system.  Thus. controlling  the  position  of sensors \nduring  sensory  exploration  constitutes  an  important  feature  of an  animals  strategy  for \nmaking sensory discriminations. Quantitative study of this process could very  well shed \nlight on  the algorithms and  internal  representations  used  by  the  nervous  system  in  dis(cid:173)\ncriminating peripheral objects. \n\nStudies  of the  active  use  of sensory  surfaces generally  can  be expected  to  pose a \nnumber of experimental challenges. This is because, in many animals, the sensory surfac(cid:173)\nes involved are themselves structurally complicated, making it difficult to reconstruct p0-\nsition sequences or the consequences of any repositioning.  For example, while the sen-\n\n\fSimulation and Measurement of the Weakly Electric Fish \n\n437 \n\nsory  systems of rats have been the subjects of a great deal  of behavioral (Welker,  1964) \nand neurophysiological study (Gibson & Welker,  1983), it is extremely difficult to even \nmonitor the movements of the perioral surfaces (lips, snout, whiskers) used by these ani(cid:173)\nmals in their exploration of the world let alone reconstruct the sensory consequences. For \nthese reasons  we  have  sought an  experimental  animal  with  a  sensory  system  in  which \nthese sensory-motor interactions can be more readily quantified. \n\nThe experimental animal which we have selected for studying the control of sensory \nsurface position during exploration is a member of a family  of African  freshwater  fish \n(Monniridae) that use self-generated electric fields  to detect and discriminate objects in \ntheir environment (Bullock &  Heiligenberg,  1986).  The electrosensory system  in  these \nfish relies on an \"electric organ\" in their tails which produces a weak pulsed electric field \nin the surrounding environment (significant within  1-2 body lengths) that is then detected \nwith  an  array  of electrosensors that are extremely  sensitive to  voltage  drops  across  the \nskin.  These \"electroreceptors\" allow the fISh  to respond to  the perturbations in the elec(cid:173)\ntric  field resulting from  objects in the environment which differ in conductivity from  the \nsurrounding water (Fig.  1). \n\nobject \n\n.. conducting \n\u2022 \nIIID  electric organ \n\n\u00a7  electroreceptors \n\nelectric \nfield lines \n\nFigure  1.  The peripheral  electrosensory  system  of Gnathonemus  petersii \nconsists of an \"electric organ\" current source at the base of the tail and sev(cid:173)\neral  thousand  \"electroreceptor\"  cells  distributed  non uniformly  over  the \nfish's body.  A conducting object near the fish causes a local increase in the \ncurrent through the skin. \n\nThese fISh  are nocturnal, and rely  more on  their electric sense than on any  other sensory \nsystem in perfonning a wide range of behaviors (eg. detecting and localizing objects such \nas  food).  It is also known  that these fish execute exploratory movements, changing their \nbody  position  actively  as  they  attempt  an  electrosensory  discrimination  (Toerring  & \nBelbenoit, 1979).  Our objective is to understand how these movements change the distri(cid:173)\nbution of the electric field on the animals skin, and to determine what, if any, relationship \nthis has to the discrimination process. \n\nThere are several clear advantages of this system for our studies.  First, the electrore-\n\n\f438 \n\nRasnow, Assad, Nelson and Bower \n\nceptors are in a  fixed  position  with  respect to  each other on  the surface of the animal. \nTherefore, by knowing the overall body position of the animal it is possible to know the \nexact spatial relationship of electroreceptors with respect to  objects in the environment. \nSecond, the physical equations governing the self-generated electric fIeld in the fish's en(cid:173)\nvironment are well  understood.  As a consequence, it is relatively straightforward to re(cid:173)\nconstruct perturbations in the electric field resulting from  objects of different shape and \nconductance.  Third,  the  electric  potential  can  be readily  measured,  providing  a  direct \nmeasure of the electric field at a distance from  the fish which can be used to reconstruct \nthe potential difference across the animals skin.  And finally, in the particular species of \nfish  we  have chosen  to  work with, Gnathonemus petersii,  individual  animals execute a \nbrief (100 J.1Sec) electric organ discharge (BOD) at intervals of 30 msec to a few seconds. \nModification of the firing pattern is 1cnown to be correlated with changes in the electrical \nenvironment (Lissmann,  1958).  Thus, when the electric organ discharges, it is probable \nthat the animal is interested in \"taking a  look\" at its surroundings.  In few  other sensory \nsystems is there as direct an indication of the attentional state of the subject. \n\nHaving stated the advantages of this system  for  the study we have undertaken, it is \nalso  the  case that considerable effort will  still  be necessary  to  answer the questions  we \nhave posed.  For example, as described in  this paper, in order to use electric field mea(cid:173)\nsurements made at a distance to infer the voltages across the surface of the animal's skin, \nit is necessary to  develop a computer model of the fish  and its environment.  This will \nallow  us  to predict the field on the animal's skin surface given different body  poSitions \nrelative to objects in the environment.  This paper describes our first steps in constructing \nthis simulation system. \n\nExperimental Approach and Methods \n\nSimulations of Fish Electric Fields \n\nThe electric potential, cll(x), generated by the EOD of a weakly electric fish in a fish \n\ntank is a solution ofPoisson's equation: \n\nVe(pVell) = f \n\nwhere  p(x)and f(x)  are the impedance magnitude and source density at each  point x  in(cid:173)\nside and surrounding the fish.  Our goal is to solve this equation for ell  given the current \nsource density, f, generated by the electric organ and the impedances, p, corresponding to \nthe properties of the fish  and external objects (rocks, worms, etc.).  Given  p and f.  this \nequation  can  be  solved  for  the  potential  ell  using  a  variety  of iterative approximation \nschemes.  Iterative methods, in general, first discretize the spatial domain of the problem \ninto a set of \"node\" points, and convert Poisson's equation into a  set of algebraic equa(cid:173)\ntions with the nodal potentials as the unknown parameters.  The node values, in this case, \neach  represent an independent degree of freedom  of the system  and, as a  consequence, \nthere are as many equations as there are nodes.  This very large system of equations can \n\n\fSimulation and Measurement of the Weakly Electric Fish \n\n439 \n\nthen be solved using a variety of standard techniques, including relaxation methods, con(cid:173)\njugate gradient minimization, domain decomposition and multi-grid methods. \n\nTo simulate the electric fields generated by a fish,  we currently use a 2-dimensional \nfmite  element domain discretization (Hughes,  1987) and conjugate gradient solver.  We \nchose  the  finite  element method  because  it allows  us  to  simulate  the  electric  fields  at \nmuch  higher resolution in the area of interest close to the animal's body where the elec(cid:173)\ntric field is largest and where errors due to the discretization would be  most severe.  The \nfmite  element method is based on  minimizing a global  function  that corresponds to  the \npotential energy of the electric field.  To compute this energy, the domain is decomposed \ninto a large number of elements, each  with  uniform impedance (see Fig. 2).  The global \nenergy is expressed as a sum of the contributions from each element, where the potential \nwithin each element is assumed to be a linear interpolation of the potentials at the nodes \nor vertices  of each element The conjugate gradient solver determines  the  values  of the \nnode potentials which minimize the global energy function. \n\n1\\  IVrv'V \n\n1\\  1\\  rv:V  J  J 1\\/1\\ \n\n.J 1\\/  1\\11\\/1\\/1\\11\\/1\\/1\\  1\\11\\/[\\  1\\1 IV \n\nV \n\nv \n\nv \n\nr--.. \n\n7' \n\n[7 \n\nIf\\ If\\  '\\  '\\ V\\ V If\\ J\\ 1'\\  'w  l/\\ V 11'\\  '\\ 1/  :1'\\ \n\n'\\  '\\ '\\ V  '\\ 1'\\\" '\\ 11\\ 1/\\  \\  '\\ V \n\nFigure 2.  The inner region of a fmite element grid constructed for simulat(cid:173)\ning in 2-dimensions the electric field generated by an electric fish. \n\nMeasurement of Fish Electric Fields \n\nAnother aspect of our experimental  approach  involves  the  direct  measurement of \nthe potential generated by a fish's EOD in a fish tank using arrays of small electrodes and \ndifferential amplifiers.  The electrodes and electronics have a high impedance which min(cid:173)\nimizes their influence on the electric fields they are designed to measure.  The electrodes \nare made by pulling a 1mm glass capillary tube across a heated tungsten filament, result(cid:173)\ning in a fine tapered tip through which a 1~ silver wire is run.  The end of this wire is \nmelted in a flame leaving a 200J,un ball below the glass insulation. Several electrodes are \nthen mounted as an array on a microdrive attached to a modified X-Yplotter under com(cid:173)\nputer control and giving better than  1mm positioning accuracy.  Generated potentials are \namplified by a factor of 10 - 100, and digitized at a rate of 100kHz per channel with a 12 \nbit AID  converter using a Masscomp  5700 computer.  An  array processor searches this \n\n\f440 \n\nRasnow, Assad, Nelson and Bower \n\ncontinuous stream of data for EOD wavefonns. which are extracted and saved along with \nthe position of the electrode array. \n\nCalibration of the Simulator \n\nResults \n\nIn order to have confidence in the overall system, it was fD'St necessary to calibrate \nboth the recording and the simulation procedures.  To do this we set up relatively simple \ngeometrical arrangements of sources and conductors in a fish  tank for which the potential \ncould be found analytically.  The calibration source was an electronic \"fake fish\" circuit \nthat generated signals resembling the discharge of Gnathonemus. \n\nPoint current source \n\nA point source in a 2-dimensional box  is  perhaps the  simplest configuration  with \nwhich to initially test our electric field reconstruction system.  The analytic solution for \nthe  potential from  a point current source centered  in  a grounded.  conducting  2-dimen(cid:173)\nsional box is: \n\n00  sm(\"2  sm  L \n4>(x.  y) =  L \nn  =1 \n\n.  (.n7t).  (n7tx).  h (.n7ty ) \nsm  \\L \nri1t \n\nn L  cosh(T) \n\nOur  fmite  element  simulation.  based  on  a  regular  80  x 80  node  grid  differs  from  the \nabove expression by less than  1 %. except in  the elements adjacent to the source.  where \nthe potential change across these elements is large and is not as accurately reconstructed \nby  a linear  interpolation  (Fig.  3).  Smaller elements surrounding  the  source  would  im(cid:173)\nprove the accuracy. however. one should note the analytic solution is infmite at the loca(cid:173)\ntion  of the \"point\" source whereas  the  measured and  simulated  sources  (and real  fish) \nhave finite current densities. \n\nTo measure the real equivalent of a point source in a 2-dimensional box. we used a \nlinear  current  source  (a  wire)  which  ran  the  full  depth  of a  real  3-dimensional  tank. \nMeasurements made in the midplane of the  tank agree with  the simulation and analytic \nsolution to better than 5% (Fig. 3.).  Uncertainty in the positions of the ClUTent source and \nrecording  sites  relative  to  the  position  of the  conducting  walls  probably  accounts  for \nmuch of this difference. \n\n\fSimulation and Measurement of the Weakly Electric Fish \n\n441 \n\n1----~~--~--~----~-------\n\no  - measured \nx  - simulated \n\n00 \n\n2 \n\n4 \n\n6 \n\n8 \n\n10  12  14  16 \n\nFigure 3.  Electric potential of a point current source centered in a grounded \n2-dimensional box. \n\ndislaDce from  source \n\nMeasurements of Fish Fields and 2-Dimensional Simulations \n\nCalibration of our fmite element model  of an electric fish  requires direct measure(cid:173)\nments of the electric potential close to a discharging fish.  Fig. 4 shows a recording of a \nsingle EOD sampled with  5 colinear electrodes near a restrained fish.  The wavefonn is \nbipolar, with the fIrst phase positive if recorded near the animals head and negative if re(cid:173)\ncorded near the tail (relative to a remote reference).  We used the peak amplitude of the \nlarger second phase of the  wavefonn  to  quantify  the  electric potential recorded at each \nlocation.  Note that the potential reverses sign at a point approximately midway along the \ntail.  This location corresponds to the location of the null potential shown in Fig. 5. \n\n1500 \n\n1000 \n\n$' \n\n5500 I  0  -1r-\n\n-soo \n\n-1000 \n\no \n\n200 ~sec \n\nFigure 4.  EOD waveform sampled simultaneously from 5 electrodes. \n\n\f442 \n\nRasnow, Assad, Nelson and Bower \n\nMeasurements of EODs from a restrained fish exhibited an extraordinarily small vari(cid:173)\nance in amplitude and waveform over long periods of time.  In fact, the peak-peak ampli(cid:173)\ntude of the EOD varied by less than 0.4% in a sample of 40 EOD's randomly chosen dur(cid:173)\ning  a 30 minute period.  Thus we are able to directly compare waveforms  sampled se(cid:173)\nquentially without renonnalizing for fluctuations in EOD amplitude. \n\nFigure  5  shows  equipotential  lines  reconstructed  from  a  set of 360  measurements \nmade in the midplane of a restrained Gnathonemus.  Although the observed potential re(cid:173)\nsembles that from a purely dipolar source (Fig. 6), careful inspection reveals an asymme(cid:173)\ntry between the head and tail of the fISh.  This asymmetry can be reproduced in our simu(cid:173)\nlations  by  adjusting  the  electrical  properties  of the  fish.  Qualitatively,  the  measured \nfields can be reproduced by assigning a low impedance to the internal body cavity and a \nhigh impedance to the skin.  However, in order to match the location of the null potential, \nthe skin impedance must vary over the length of the body.  We are currently quantifying \nthese parameters, as described in the next section. \n\n!!!!m 1'!I!fl!IPf!~m II \n\u2022\u2022 1 .. . ...... 1  \u2022\u2022\u2022 !~ ....... . \n\nFigure 5.  Measured  potentials (at peak of second phase of EOD) recorded \nfrom  a  restrained  Gnathonemus  petersii  in  the  midplane  of  the  fish. \nEquipotential  lines  are  20  m V apart.  Inset shows relative  location  of fish \nand sampling points in the fISh  tank. \n\nFigure 6.  Equipotential lines from  a 2-dimensional  finite element simula(cid:173)\ntion of a dipole using the grid of Fig.  2.  The resistivity of the fish  was  set \nequal to that of the sWToundings in this simulation. \n\n\fSimulation and Measurement of the Weakly Electric Fish \n\n443 \n\nFuture Directions \n\nThere  is  still  a  substantial  amount  of work  that  remains  to  be  done  before  we \nachieve our goal of being able  to fully  reconstruct the pattern of electroreceptor activa(cid:173)\ntion  for any arbitrary body position in any particular environment.  First.  it is clear that \nwe require more information about the electrical structure of the fISh  itself.  We need an \naccurate representation of the internal impedance distribution p(x) of the body and skin \nas well as of the source density f(x) of the electric organ.  To some extent this can be ad(cid:173)\ndressed as an inverse problem, namely given the measured potential cl>(x), what choice of \np(x)  and  f(x)  best reproduces  the  data.  Unfortunately,  in  the  absence  of further  con(cid:173)\nstraints, there are many equally valid solution, thus we will need to directly measure the \nskin and body impedance of the fish.  Second, we need to extend our finite-element sim(cid:173)\nulations  of the  fish  to  3-dimensions  which,  although  conceptually  straight forward,  re(cid:173)\nquires  substantial  technical  developments  to  be  able  to  (a)  specify  and  visualize  the \nspace-filling set of 3-dimensional finite-elements (eg. tetrahedrons) for arbitrary configu(cid:173)\nrations, (b) compute the solution to the much larger set of equations (typically a factor of \n100-1(00) in a reasonable time, and (c) visualize and analyze the resulting solutions for \nthe  3-dimensional electrical fields.  As a possible solution to (b), we are developing and \ntesting a parallel processor implementation of the simulator. \n\nReferences \n\nBullock, T. H.  & Heiligenberg, W. (Eds.) (1986). \"Electroreception\", Wiley & Sons, \n\nNew York. \n\nGibson, J. M. & Welker. W. I.  (1983).  Quantitative Studies of Stimulus Coding in First(cid:173)\n\nOrder Vibrissa Afferents of Rats.  1.  Receptive Field Properties and Threshold \nDistributions.  Somatosensory Res. 1:51-67. \n\nHughes, T. J.  (1987).  The Finite Element Method: Linear Static and Dynamic Finite \n\nElement Analysis.  Prentice-Hall, New Jersey. \n\nLissmann. H.W. (1958).  On the function and evolution of electric organs in fish.  J. Exp. \n\nBioi. 35:156-191. \n\nToening, M. J. and Belbenoit. P. (1979).  Motor Programmes and Electroreception in \n\nMonnyrid Fish.  Behav. Ecol. Sociobiol. 4:369-379. \n\nWelker, W. I. (1964).  Analysis of Sniffing of the Albino Rat  Behaviour 22:223-244. \n\n\f", "award": [], "sourceid": 152, "authors": [{"given_name": "Brian", "family_name": "Rasnow", "institution": null}, {"given_name": "Christopher", "family_name": "Assad", "institution": null}, {"given_name": "Mark", "family_name": "Nelson", "institution": null}, {"given_name": "James", "family_name": "Bower", "institution": null}]}