436 
SIMULATION AND MEASUREMENT OF 
THE ELECTRIC FIELDS GENERATED 
BY WEAKLY ELECTRIC FISH 
Brian Rasnow 1, Christopher Assad 2, Mark E. Nelson 3 and James M. Bower 3 
Divisions of Physics 1, Electrical Engineering 2, and Biology 3 
Caltech, Pasadena, 91125 
ABSTRACT 
The weakly electric fish, Gnathonemus petersii, explores its environment by gener- 
ating pulsed electric fields and detecting small perturbations in the fields resulting from 
nearby objects. Accordingly, the fish detects and discriminates objects on the basis of a 
sequence of electric "images" whose temporal and spatial properties depend on the tim- 
ing of the fish's electric organ discharge and its body position relative to objects in its en- 
vironment. We are interested in investigating how these fksh utilize timing and body-po- 
sition during exploration to aid in object discrimination. We have developed a fmite-ele- 
ment simulation of the fish's self-generated electric fields so as to reconstruct the elec- 
trosensory consequences of body position and electric organ discharge timing in the fish. 
This paper describes this finite-element simulation system and presents preliminary elec- 
tric field measurements which are being used to tune the simulation. 
INTRODUCTION 
The active positioning of sensory structures (i.e. eyes, ears, whiskers, nostrils, etc.) 
is characteristic of the information seeking behavior of all exploratory animals. Yet, in 
most existing computational models and in many standard experimental paradigms, the 
active aspects of sensory processing are either eliminated or controlled (e.g. by stimulat- 
ing fixed groups of receptors or by stabilizing images). However, it is clear that the ac- 
tive positioning of receptor surfaces directly affects the content and quality of the sensory 
information received by the nervous system. Thus, controlling the position of sensors 
during sensory exploration constitutes an important feature of an animals strategy for 
making sensory discriminations. Quantitative study of this process could very well shed 
light on the algorithms and internal representations used by the nervous system in dis- 
criminating peripheral objects. 
Studies of the active use of sensory surfaces generally can be expected to pose a 
number of experimental challenges. This is because, in many animals, the sensory surfac- 
es involved are themselves stmcutrally complicated, making it difficult to reconstruct po- 
sition sequences or the consequences of any repositioning. For example, while the sen- 
Simulation and Measurement of the Weakly Electric Fish 437 
sory systems of rats have been the subjects of a great deal of behavioral (Welker, 1964) 
and nenrophysiological study (Gibson & Welker, 1983), it is extremely difficult to even 
monitor the movements of the perioral surfaces (lips, snout, whiskers) used by these ani- 
mals in their exploration of the world let alone reconstruct the sensory consequences. For 
these reasons we have sought an experimental animal with a sensory system in which 
these sensory-motor interactions can be more readily quantified. 
The experimental animal which we have selected for studying the control of sensory 
surface position during exploration is a member of a family of African freshwater fish 
(Mormiridae) that use self-generated electric fields to detect and discriminate objects in 
their environment (Bullock & Heiligenberg, 1986). The electrosensory system in these 
fish relies on an "electric organ" in their tails which produces a weak pulsed electric field 
in the surrounding environment (significant within 1-2 body lengths) that is then detected 
with an array of electrosensors that are extremely sensitive to voltage drops across the 
skin. These "electroreceptors" allow the fish to respond to the perturbations in the elec- 
tric field resulting from objects in the environment which differ in conductivity from the 
surrounding water (Fig. 1). 
conducting 
 object 
Il electric organ 
[g] electroreceptors 
__ electric 
field lines 
Figure 1. The peripheral electrosensory system of Gnathonemus petersii 
consists of an "electric organ" current source at the base of the tail and sev- 
eral thousand "electroreceptor" cells distributed nonuniformly over the 
fish's body. A conducting object near the fish causes a local increase in the 
current through the skin. 
These fish are nocturnal, and rely more on their electric sense than on any other sensory 
system in performing a wide range of behaviors (eg. detecting and localizing objects such 
as food). It is also known that these fish execute exploratory movements, changing their 
body position actively as they attempt an electrosensory discrimination (Toerdng & 
Belbenoit, 1979). Our objective is to understand how these movements change the distri- 
bution of the electric field on the animals skin, and to determine what, if any, relationship 
this has to the discrimination process. 
There are several clear advantages of this system for our studies. First, the electrore- 
438 Rasnow, Assad, Nelson and Bower 
ceptors are in a fixed position with respect to each other on the surface of the animal. 
Therefore, by knowing the overall body position of the animal it is possible to know the 
exact spatial relationship of electroreceptors with respect to objects in the environment. 
Second, the physical equations governing the self-generaxl electric field in the fish's en- 
vironment are well understood. As a consequence, it is relatively straightforward to re- 
construct perturbations in the electric field resulting from objects of different shape and 
conductance. Third, the electric potential can be readily measured, providing a direct 
measure of the electric field at a distance from the fish which can be used to reconstruct 
the poten6al difference across the animals skin. And finally, in the particular species of 
fksh we have chosen to work with, Gnathonemus petersii, individual animals execute a 
brief (100 gsec) electric organ discharge (EOD) at intervals of 30 msec to a few seconds. 
Modification of the firing pattern is known to be correlated with changes in the electrical 
environment (Lissmann, 1958). Thus, when the electric organ discharges, it is probable 
that the animal is interested in "taking a look" at its surroundings. In few other sensory 
systems is there as direct an indication of the attentional state of the subject. 
Having stated the advantages of this system for the study we have undertaken, it is 
also the case that considerable effort will still be necessary to answer the questions we 
have posed. For example, as described in this paper, in order to use electric field mea- 
surements made at a distance to infer the voltages across the surface of the animal's skin, 
it is necessary to develop a computer model of the fish and its environment. This will 
allow us to predict the field on the animal's skin surface given different body positions 
relative to objects in the environment. This paper describes our first steps in constructing 
this simulation system. 
Experimental Approach and Methods 
Simulations of Fish Electric Fields 
The electric potential, l>(x), generated by the EOD of a weakly electric fish in a fish 
tank is a solution of Poisson's equation: 
Vo(pV) = f 
where p(x)and f(x) are the impedance magnitude and source density at each point x in- 
side and surrounding the fish. Our goal is to solve this equation for  given the current 
source density, f, generated by the electric organ and the impedances, p, corresponding to 
the properties of the fish and external objects (rocks, worms, etc.). Given p and f, this 
equation can be solved for the potential  using a variety of iterative approximation 
schemes. Iterative methods, in general, first discretize the spatial domain of the problem 
into a set of "node" points, and convert Poisson's equation into a set of algebraic equa- 
tions with the nodal potentials as the unknown parameters. The node values, in this case, 
each represent an independent degree of freedom of the system and, as a consequence, 
there are as many equations as there are nodes. This very large system of equations can 
Simulation and Measurement of the Weakly Electric Fish 439 
then be solved using a variety of standoral techniques, including relaxation methods, con- 
jugate gradient minimization, domain decomposition and multi-grid methods. 
To simulate the electric fields generated by a fish, we currently use a 2-dimensional 
finite element domain discretization (Hughes, 1987) and conjugate gradient solver. We 
chose the finite element method because it allows us to simulate the electric fields at 
much higher resolution in the area of interest close to the animal's body where the elec- 
tric field is largest and where errors due to the discretization would he most severe. The 
finite element method is based on minimizing a global function that corresponds to the 
potential energy of the electric field. To compute this energy, the domain is decomposed 
into a large number of elements, each with uniform impedance (see Fig. 2). The global 
energy is expressed as a sum of the contributions from each element, where the potential 
within each element is assumed to be a linear interpolation of the potentials at the nodes 
or vertices of each element. The conjugate gradient solver determines the values of the 
node potentials which minimize the global energy function. 
Figure 2. The inner region of a finite element grid constructed for simulat- 
ing in 2-dimensions the electric field generated by an electric fish. 
Measurement of Fish Electric Fields 
Another aspect of our experimental approach involves the direct measurement of 
the potential generated by a fksh's EOD in a fish tank using arrays of small electrodes and 
differential amplifiers. The electrodes and electronics have a high impedance which min- 
imizes their influence on the electric fields they are designed to measure. The electrodes 
are made by pulling a lmm glass capillary tube across a heated tungsten filament, result- 
ing in a fine tapered tip through which a 100gin silver wire is run. The end of this wire is 
melted in a flame leaving a 200pro ball below the glass insulation. Several electrodes are 
then mounted as an array on a microdrive attached to a modified X-Y plouer under com- 
puter control and giving better than lmm positioning accuracy. Generated potentials are 
amplified by a factor of 10 - 100, and digitized at a rate of 100kHz per channel with a 12 
bit A/D converter using a Masscomp 5700 computer. An array processor searches this 
0 Rasnow, Assad, Nelson and Bower 
continuous stream of data for EOD waveforms, which are extracted and saved along with 
the position of the electrode array. 
Results 
Calibration of the Simulator 
In order to have confidence in the overall system, it was first necessary to calibrate 
both the recording and the simulation procedures. To do this we set up relatively simple 
geometrical arrangements of sources and conductors in a fish tank for which the potential 
could be found analytically. The calibration source was an electronic "fake fish" circuit 
that generated signals resembling the discharge of Gnathonemus. 
Point current source 
A point source in a 2-dimensional box is perhaps the simplest configuration with 
which to initially test our electric field reconstruction system. The analytic solution for 
the potential from a point current source centered in a grounded, conducting 2-dimen- 
sional box is: 
 (x, y)=  
,nn, . ,nx, . . ny 
sin ,- ) sn I,'- )stun (--) 
n = 1 n L cosh(-?) 
Our finite element simulation, based on a regular 80 x 80 node grid differs from the 
above expression by less than 1%, except in the elements adjacent to the source, where 
the potential change across these elements is large and is not as accurately reconstructed 
by a linear interpolation (Fig. 3). Smaller elements surrounding the source would im- 
prove the accuracy, however, one should note the analytic solution is infinite at the loca- 
tion of the "point" source whereas the measured and simulated sources (and real fish) 
have finite current densities. 
To measure the real equivalent of a point source in a 2-dimensional box, we used a 
linear current source (a wire) which ran the full depth of a real 3-dimensional tank. 
Measurements made in the midplane of the tank agree with the simulation and analytic 
solution to beuer than 5% (Fig. 3.). Uncertainty in the positions of the current source and 
recording sites relative to the position of the conducting walls probably accounts for 
much of this difference. 
Simulation and Measurement of the Weakly Electric Fish 
4]-  o- measured . 
00 2 4 6 8 10 12- 14 16 
dislance from source 
Figure 3. Electric potential of a point current source centered in a grounded 
2-dimensional box. 
Measurements of Fish Fields and 2-Dimensional Simulations 
Calibration of our finite element model of an eleclric fish requires direct measure- 
ments of the electric potential close to a discharging fish. Fig. 4 shows a recording of a 
single EOD sampled with 5 colinear electrodes near a restrained fish. The waveform is 
bipolar, with the ft phase positive if recorded near the animals head and negative if re- 
corded near the tail (relative to a remote reference). We used the peak amplitude of the 
larger second phase of the waveform to quantify the electric potential recorded at each 
location. Note that the potential reverses sign at a point approximately midway along the 
tail. This location corresponds to the location of the null potential shown in Fig. 5. 
1500 V 
0 200 gsec 
Figure 4. EOD waveform sampled simultaneously from 5 electrodes. 
2 Rasnow, Assad, Nelson and Bower 
Measurements of EODs from a resrained fish exhibited an extraordinarily small vari- 
ance in amplitude and waveform over long periods of time. In fact, the peak-peak ampli- 
tude of the EOD varied by less than 0.4% in a sample of 40 EOD's randomly chosen dur- 
ing a 30 minute period. Thus we are able to directly compare waveforms sampled se- 
quentially without renommlizing for fluctuations in EOD amplitude. 
Figure 5 shows equipotential lines reconstructed from a set of 360 measurements 
made in the midplane of a restrained Gnathonemus. Although the observed potential re- 
sembles that from a purely dipolar source (Fig. 6), careful inspection reveals an asymme- 
try between the head and tail of the fish. This asymmetry can be reproduced in our simu- 
lations by adjusting the electrical properties of the fish. Qualitatively, the measured 
fields can be reproduced by assigning a low impedance to the internal body cavity and a 
high impedance to the skin. However, in order to match the location of the null potential, 
the skin impedance must vary over the length of the body. We are currently quantifying 
these parameters, as described in the next section. 
Figure 5. Measured potentials (at peak of second phase of EOD) recorded 
from a restrained Gnathonemus petersii in the midplane of the fish. 
Equipotential lines are 20 mV apart. Inset shows relative location of fish 
and sampling points in the fish tank. 
Figure 6. Equipotential lines from a 2-dimensional finite element simula- 
tion of a dipole using the grid of Fig. 2. The resistivity of the fish was set 
equal to that of the surroundings in this simulation. 
Simulation and Measurement of the Weakly Electric Fish 443 
Future Directions 
There is still a substantial amount of work that remains to be done before we 
achieve our goal of being able to fully reconstruct the pattern of electroreceptor activa- 
tion for any arbitrary body position in any particular environment. First, it is clear that 
we require more information about the electrical structure of the fish itseft. We need an 
accurate representation of the internal impedance distribution p(x) of the body and skin 
as well as of the source density f(x) of the electric organ. To some extent this can be ad- 
dressed as an inverse problem, namely given the measured potential (x), what choice of 
p(x) and f(x) best reproduces the data. Unfortunately, in the absence of further con- 
straints, there are many equally valid solution, thus we will need to directly measure the 
skin and body impedance of the fish. Second, we need to extend our finite-element sim- 
ulations of the fish to 3-dimensions which, although conceptually straight forward, re- 
quires substantial technical developments to be able to (a) specify and visualize the 
space-filling set of 3-dimensional finite-elements (eg. tetrahedrons) for arbitrary configu- 
rations, Co) compute the solution to the much larger set of equations (typically a factor of 
100-1000) in a reasonable time, and (c) visualize and analyze the resulting solutions for 
the 3-dimensional electrical fields. As a possible solution to Co), we are developing and 
testing a parallel processor implementation of the simulator. 
References 
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Order Vibrissa Afferents of Rats. 1. Receptive Field Properties and Threshold 
Distributions. Somatosensory Res. 1:51-67. 
Hughes, T.J. (1987). The Finite Element Method: Linear Static and Dynamic Finite 
Element Analysis. Prentice-Hall, New Jersey. 
Lissmann, H.W. (1958). On the function and evolution of electric organs in fish. J. Exp. 
Biol. 35:156-191. 
Toerring, M. J. and Belbenoit, P. (1979). Motor Programmes and Electroreception in 
Mormyrid Fish. Behar. Ecol. Sociobiol. 4:369-379. 
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