ON THE NATURE OF ELECTROSENSING IN THE FISH

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• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): 00022031
• Release Decision: RIPPUB
• Original Classification: U
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• Document Creation Date: September 25, 2024
• Document Release Date: December 27, 1977
• Publication Date: August 2, 1971

. . ON THE NATURE OF ELECTROSENSING IN THE FISH
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. August 1971
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ABSTRACT
An evaluative review of the electmsensiug literature was carried out
with the intention of determining the nature of the electrosensing mechanism
and its sensitivity. It was found that the biolnical dsta base was weak. It
was, however, usefUl in the development of a mathematical model and mathematical
analyses of the sense mechanism and its function. In the course of the analyses,
we suggest a working hypothesis on the nature of the sense mechanism. We also
collapse the various sensor coding schemes that have been proposed into one
scheme. The function of the mathematical Model of the sensor that was developed
was explored with the use of a computer. The fishes' function at the system
level was also considered and possible mechanisms defined.
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TABU OF CONTENTS
ABSTRACT
INTAODUCTION 3.
NATURE OF TEE BIOLOGICAL SYSTEM 2
Generator Organ,
Receptor Organ 7
Gymnotid receptors.
Mormyrid receptors.
System Function, Measurement Technique & Sensitivity 11
Electrophysiological & behavioral techniques,
Size of tank required for valid experimental data.
POSSIBLE RECEPTOR MECNANiSM AND NEURAL CODING 17
Mechanism.
MODEL: DEVELOPMENT, FUNCTION, AND SENSIIIvIxf 27
Receptor Level 28
Development.
Function.
Sensitivity.
System Level 54
CONCLUSIONS 55
REFERENCES 58
APPENDIX 61
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1.
IBTRODUCTION
It has only been a short time since certain fish were identified as
having a previously unknown sensing system, an electrosensing system. It
was observed that these fish apparently detect and classify objects that
enter into and perturb a weak electrical field that the fish itself gener-
ates. With further investigation it was found that this sense is more
generally found among fishes than was first thought. Data also appeared
indicating that same fish, such as the shark and goldfish, use a passive
electrosensing system in that the fis).. does not seen to generate its awn
electrical field. Rather, it seems to detect electrical signals, possibly
muscle potentials, generated by objects coming into its area.
Although there is new a fairly substantial data base, we find that
very little can be applied to the development and understanding of sense
mechanism and sensitivity. This is due in part to the fact that pioneering
data in this area, as it is. in most areas, tend to have faults no matter
how ,tompetent the investigators. Further, the data base contains very little
behavioral data. Thus, there is little information available on system sensi-
tivity and function.
In sum, though there are individual investigators contributing quite
useful data to the data base, as a whole the data base is weak. Thus, we
have undertaken several testa which may allow an assessment of the fishes'
electrosensing mechanism and capability, using the data presently available.
First, through limited experimental work with electrical fields, sen-
sors, and objects in various size bodies of water we have gathered data which,
when taken. with the mathematical analyris, allows us to interpret much of the
data now available. This analysis also provides a specification for tank size,
fish location, and attachments, that vill yield valid data in future studies.
Second, we have suggested as a working hypothesis an electrosensor
mechanism. This hypothesis is subject to test and thereby may provide the
means for collapsing the current multiple crude categorizations of the re-
ceptor that is so typical of a new area of investigation. The hypothesis may
also provide a basis for analyzing higher interactions in the fishes.' nervous
system and thereby increase our understanding of the sense.
Third, we indicate in the following the !linkage among the various neural
coding schemes suggested for the fish and show their essential identity.
Fourth, we develop a mathematical model of the fish based upon the use-
able experimental data. A set of equations describing function is developed
on the model. These equations are linked to available experimental data.
The mathematical model is analysed by a computer to ascertain the sensitivity.
requirements of the fish at the receptor and to determine the effects of mani-
pulating a ,L;:lwber of variables. These variables include fish size, object
size, objet.t electrical characteristics, object distance from the fish, direction
and angle of the object from the fishes' axis, etc.
We briefly discuss the fishes' function it the systems level and close
with our conclusions concerning the electric sense.
NATURE OF THE BIOLOGICAL SYSTEM
Both marine and fresh water species of strongly and weakly electric
fish have evolved. :Strongly electric fish are defined as those that dis-
charge their electric generating organs reactively to stun prey or resist
capture; Weakly electric fish are defined as those that detect and classify
objects by the object perturbing the electrical field formed by the electric
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generating organ vhich normally emits a continuous pattern of pulses. The
electric field so set up is not strong enough to stun other fish.
There are numerous species of weakly electric freshwater fish but
most can be classified as either gymnotids which are South American in origin
or mormyrids which are common in Africa. The two groups have many similarities
and some differences in physical structure and in the function of their elec-
trical field generating organs and receptor organs. Other weakly electric
fish include Gymnarchus, an African fish, probably related to the mormyrids,
And sternarchid, a South American fish that is probably related to the
Gymnotids.
Generator 2nem
An understanding of the structure and funutaon of the electrical field
Aenerator organ is of importance in understanding receptor function. Thus,
generator function will be considered first.
Te cells of the generatfmg organ are referred to in the literature
as electroplaques, eleotroplax, electroplates, or electrocstes. We shall
follow Bennett (1970) and use the term electrocytes. The electrocytes are
derived from the mesoderm (Szabo, 1966),the same type of embryonic tissue
as muscle except in the South American family Sternarchidae. The origin
of the electrocytes of the sternarchids is the same embryonic tissue from
which the neural system is derived, the ectoderm (Steinbach, 1970).
Electrocytes of mesodermal origin are typically disc shaped, but may
also be drum shaped or tubular. Electrocytes of ectodermal origin are U
shaped processes from the spinal cord. The electrocytes of the granotid,
Hypopomus, are between 300 -5nn 1 Ir. diameter and about 200 p thick. The
electrocytes of Sternopygus on the other hand are rod-shaped and much longer
4
than those of Hypopomus. They are about 1-2 mm in the anterior posterior
direction and 200 u in diameter. These cells are packed together tightly
with little extracellular space, whereas the electrocytes of Hypopomus -are
separated by a considerable amount of extracellular space.
The electrocytes are "stacked" in columns in the rear portion of the
fish's body to form the electric generating organ. For example, the electric
organ of Gnathonemus, a mormyrid, is located just in front of the tail fin
and extends forward less than 1/5 of the fish's body length. Gymnarchus'
electric generating organ extends from the tail fin to nearly the midpoint.
The generating organs of the gymnotid Gymnotus, and of Sternarchus extend
further from the tail fin almost to the back of the head.
The weakly electric freshwater fish can be categorzed in terms of
patterns of discharge: those with variable frequency and those with con-
stant frequency. Constant frequency fish are defined as those that discharge
their electric generating organs at a virtually constant rate eves when
strongly stimulated by an experimenter. Some of there are Eigenmannia,
Sternopygus, and the sternarchids. These differences are not absolute, how-
ever, and there are species differences in basic rate. The generating organ
of the mormyrid Gnathonemus for example, is reported (Bennett, 1970) to dis-
charge at frequencies of 30-100 pulses per second (pps). Gymnarchus is re-
ported to discharge at a frequency of about 250 pps; Gymnotus has a frequency
rate of 40-60 pps; Eigenmannia emits pulses at a rate of 250-400 pps; Sterno-
pygus firer. at 60-100 pps; Steatogenys emits pulses at 40-60 pps; aLd Hypo-
pomus at 2-20 pps (Hagiwara and Morita, 1963).1 Sternarchids discharge at
1. Each type of fish has a waveform that is specific to itself. Therefore,
although Gymnotus and Steatogenys have the samv.frevencies, their wave-
:orms.are different. These differences in waveform may be functions of
the experimenters' competence in engineering.
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rates of 600-2000 pulses per second (Erskine, Howe & Weed, 1966). Fish that
are reported to emit at variable frequency generally increase their discharge
rate markedly when stimulated. Fish that exhibit this characteristic are
the mormyrids (Mandriota, et al, 1965), Hypopomus, Steatogenys, and Gymnotus
(Larimer and McDonald, 1968). It should be noted that constant frequency
fish do vary their frequency under certain circumstances. These circumstances
include the presence of another signal with frequency close to the fishes'.
For example, Eigenmsnnia which has an organ discharge rate of 4oc pps shifts
its frequency 10 to 20 pps when confronted with a 400 pps signal (Larimer &
McDonald, 1968). In this context, also, is the observation that Gymnarchus
temporarily ceases its discharge entirely when presented with a signal mimick-
ing another Gymnarchus or when startled (Bennett,1970).2
The mechanisms for controlling electric organ output are in the med-
ullary portion of the brain and appear to be similar among weakly electric
fish. A small group of cells in the medulla are aUtoactive and fire syn-
chronously, apparently acting as a pacemaker. Their discharge appears to
trigger another group of cells in the medulla commonly referred to as med-
ullary "relays". .Axons from the medullary relay cells descend as part of
the spinal cord to synapse on spinal relay neurons. These in turn communi-
cate the signal to the electrocytes. The electrocytes of the e.:.'ectric gen-
erating organ fire synchronously because of one or more compensatory mechan-
isms in the relay pathway from the pacemaker cells.' One mechanism is vari-
ation in length of the pathway to the electracytes.i The axons to the more
distant electrocytes extend in the straightest possible line but those to
the less distant electrocytes fcllov a circultous-pattex.t. A second means
of maintaining synchronization involve a delay line mechanism whereby the
pathways to the electrocytes differ in conduction velocities.
2. If a passive electric sense is more common than is thought, this could
be a protective reaction.
A number of investigators have measured the voltage output of the
generating organ. Hypopomus is reported to gsnerate a voltage of 8 volts
peak to peak when electrodes are placed on the head and tail with the fish
more or less out of the water. The same fish in water is reported to generate
a voltage of from 13 to 200 mdllivolts. The in-water measurements were takes
with two stainless steel electrodes, one placed in front of the fish and one
placed behind the fish. The distance between the electrodes was not given
nor was the distance between the electrodes and the fish given. In general,
we find that inadequate information is given in the reports of voltage
measurements of the electric organ output.
Based upon the inadequate information that is reported on voltage
measurements and upon measurements that we have made in water, we would
suggest ignoring the measurements reported in the literature. In measure-
ments in our laboratory simulating the reported data, we found that the
water acts as a very high distributive resistance. When an oscilloscope
is used in the typically reported fashion to measure the fishes' voltage
output the input impedence lf the scope is being placed in parallel with the
resistance of the water. Even when a high input impedence scope is used,
there is a loading effect upon the circuit. Thus, we believe, based upon
our measurements and the reported investigations, that the investigators
have been inadvertently loading down the fish's electric field generator
through the use of their measuring devices.
We can summarize the salient points by saying that these fish generate
a pulsed electrical field in the water. Tlie generator is located in the
posterior portion of the body. The generator components have their outputs
synchronized by a clock. In some species the clock is more or less invarient,
in others it varies, in part, as a function of external events. The reason
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for this difference among species is unknown. The voltage output of the
generator and the effective range of the field are unknown due to inadequate
measurement technique.
Recentor Organ
The weakly electric freshwater fish are reported to have both active
and passive sensory systems. The active syrtem primarily detects disturb-
ances in the fish generated E field. The passive system is primarily sensi-
tive to energy provided by extrinsic sources. We are not so sure that the
data really indicates two such systems in the same fish, but we shall follow
the convention for the time being. There is better evidence that there are
a number of fish, such as sharks and gold fish, that have good passive
electrosensing-systems but no active system. These latter fish and passive
systems ar, not considered, as such, in this paper.
Gymnotid receptors. There are two basic types of electroreceptor
organs reported in the literature. The differences may be more apparent than
real in terns of function.
The ampullary organs are believed to be the passive system sensors.
They consist of cells that maintain a continuous rhythmic background firing
(low rate spontaneous impulses from the receptor to the brain). Thus, they
are referred to as tonic receptors. This background firing appears to be
unrelated to electric organ discharge.
The background firing shifts smoothly to a higher or lower rate in
response to the electrical sources moving into the fish's range. The response
to a brief stimulus, for example, is acceleration followee by deceleration.
The acceleration phase can outlast the stimulus and according to Bennett
(1970) there is accomodation to maintained stimuli. These receptors a:1-e
sensitive to lov frequency electrical ftaids and to changes in a DC field.
Their response to an applied current is a. monotonic increase.
The active system sensors are called tubers:us organs. They are more
rapidly adapting than tonic receptors. They are sensitive to relatively
high frequency stimuli and are insensitive to applied DC. Their firing is
related to electric organ discharge in that they respond with a train of
pulses to each electric organ discharge. Thus, they are referred to as
phasic receptors.
As seen on the skin, the ampullary and tuberous organs differ. They
also differ in appearance from mechanorecsptors, i.e., canal organs and
free neurom: -ts. The tuberous organ appears on the skin surface as a
single small pore, even though it has no opening. The ampullary organs
appear as a group of small pores. As an indication of the number of recept-
ors found on a fish, It can be noted that Lisamanr and MUllinger (1Y68) found
that there Imre P;COO ampullary and tuberous organs on a 6 cm. long Steato-
genys. Most receptors, about 95 percent,, are phasic receptors according to
Lissmann and MUllinger (1968).'
In considering the fine structure of the receptor orams, it (An be
noted that the ampullary organ has the appearance of a flask with a narrow
duct (5-20 0 in diameter) leading from the skin surface to a cavity (30-
40 z in diameter) that is located 100-50C u within the skin. EMbedded in
the cavity wall with only a small surface exposed are the sensing cells of
the organ. These sensing cells are 10-15 i in diameter with each organ
containing two to eight of them. Some microvilli 0.8 u long are irregularly
distributed on the exposed surface of the sensing cells. Filling the duct
and cavity is a jelly-like substance with no known function. All sense cells
in one organ feed their signals to the same myelinated nerve fiber. The nerve
is unmyelinated 'ithin the organ, having lost its myelin sheath and dividing
before entering the orran.
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OVILL
RECEPTOfi CELLS
SUPPORTING CELL
NERVE FIBER
EP!TNICLIAL CELLS
WRING CELLS
ELLULAR USER Of CAPSULE
RECCPTOR cr LI
KW. TERMINAL
UPPORTING CELL
--NERVE FISER
SION SWAGE
ELLY MICK
RECEPTOR CELL
MUCOUS SUBSTANCE
RECEPTOR CELL
f'
BIGEMENT MEMBRANE
SUSSENSORV PLATFORM
NERVE
RECEPTOR CELL
SUBSENSORT PLATFORM
BASEMENT MEMBRANE
NERVE
Fig.1 a) Schematic drawing of the two types of ampulla of gymno-
tids, b) Schematic drawing of the tuberous organ of the gymnotid,
c) Schematic drawing of the mormyromart of the mormyrid, d) Sche-
matic drawind of the tuberous organ of the mormyrid.
Att
' Pe�flmer!����,..
9
There are a great many clusters of five to fifteen ampullary receptor
cells on the head. On the body there are fewer clusters and they tend to
be restricted to 3 bands that extend longitudinally along the fish.
The tuberous organ consists of a bulb shaped invagination of the skin
as shown in Fig. lb. The side of the bulb is composed of 10 to 50 layers
of flattened cells for a. total thickness of 2-5 u. The bottom the bulb
is made up of supporting cells upon which the numerous sensing cells rest.
The sensing cells are 25-30 u long and project somewhat like rods into the
cavity of the bulb. They are ordered such that the gap between adjacent
sensory cells is relatively constant. Each sensory cell is covered on the
cavity end with microvilli 0.7 u long. The cavity is filled with a fluid
or possibly jelly-like substance. Loose epithelial-like cells fill much of
the cavity above the sensory cells and appear to plug the pore to the surface.
The sensory cells feed their signals to a single nerve which, in most Cases,
loses its myelin sheath where it pwises into the tuberous organ. In a small
proportion of the tuberous organs the myelin sheath is retained until the
nerve fiber enters the sensory cell. The tuberous organs are randomly dis-
tributed on the head, where they are most numerous, and on the anterior half
of the body. On the posterior half of the body the tuberous organs are found
in four longitudinal bands.
Mormyrid receptors. In Mormyrids, the electroreceptors are referred
to as mormyromasts and Knollenorons (Szabo, 1961). The mormyromast is a
two level organ that contains at the surface level sensory cells (type A)
similar to the ampullary sensory cells and at the second level sensory cells
(type B) similar to the sensory cells of the tuberous organ of the gymnotids.
Types A and B sensory cells are always separately innervated.
3. The fish being described is Hypopomus artedi, a species of gymnotid.
Details vary slightly from species to species.
10
The type A sensory cells form one or two concentric aureoles at the
base of a "jelly sphere" located near the surface of the skin as shown ..11
Fig. lc. In the center of this aureole, a small duct leads to a more deeply
situated sensory chamber in the skin within which the type B cells are
located. The inner surface of the duct wall bears tiny microvilli. The
duct as well as the lower Lensory chamber is filled with a_mucons.substance.
TVo to five sensory cells occupy the lower sensory chaaber. The type
B cells with their supporting cell platform though similar to the tuberous
organ are smaller. They do not completely fill up the sensory chamber and
their free surfaces bear a large number of micrOvilli.
The type B sensory cells in a mormyromast are innervated by a single
nerve fiber which splits immediately after penetration through the supporting
cells into several branches to serve the sensory cells. Where the nerve joins
the type B sensory cell membrane a rod like projection, 0.5 u in size, occurs
within the sensory cell.
Each type A sensory cell is encircled by several accessory cells. The
sensory cells and their accessory cells are bottle-shaped. The apical or
tip portion of both sensory and accessory cells contact the jelly sphere.
The nerve fibers innervating type A cells lose their myelin sheath before
entering the receptor organ and pass among the accessory cells to contact
the sensory cells. As with type B cells, where the nerve joins the sensory
cell, there is a rod present at the sensory cell membrane.
The mormyrids also have receptor organs, knollenorgans, which are some-
what similar to the tuberous organs of the gymnotids. Derbin and Szabo (1968)
describe them as being cmposed of three or four sensory cell complexes one
of which is shown in Fig. id. Each complex is a single sensory cell attached
to a highly differentiated supporting platform of cells. The organ is inner-
vated by single nerve fiber which is derived from a nerve that appears to
U
serve many sensory cells.
The sensory cell lies in and almost completely fills a cavity in the
skin at the surface. The wall of the cavity is formed by flattened-epithel-
ial cells. The interior epithelial cells have microvilli-like promisees
which densely pack the space about the sensory cell. The cavity has a rela-
tively large opening toward the supporting cells through which the sensory
cell contacts tile nerve endings and supporting cells. The sensory cell itself is
35-40 p in diameter.
In sum then, the weakly electric fishes of South America, the gymnotids
and of Africa, the mormyrids both seem to have receptor organs that are
similar in some respects but differ in other respects. Though there are
structural differences in receptor organs within and between species, the
evidence suggesting that there are differences in function is rather weak.
We shall now consider this matter of the receptor organ and. rYstem function.
System Function, Measurement Technique,and Sensitivity
This section will of necessity be short since therein relatively little
data which is acceptable from both a biological and engineering standpoint.
Thus, we will discuss the three primary techniques that have been used to
obtain data on function, discuss their deficiencies, and estimate from the
data the probaole system function and sensitivity. Two of the techn:ques are
electrophysiological and the third is behavioral.
Electroohysiological and behavioral techniquee. In one electrophysio-
logical technique the fish is anesthesized and fixed to a wooden plate in
the normal swimming position. The wooden plate is then tilted into the water
so that the body is submerged and the head exposed to the air. The regular
respiratory movements and oxygenation are muintained by eprayieg a fine jet
of water into the mouth of the fish. The dorsal branch of the lateral line
12
nerve which lies immediately under the dorsal skin at the head, is then
surgically exposed. After desheathing it, fine nerve strands are separated
by microdisection. Then silver-silver chloride electrodes are applied to a
strand and single nerve fiber responses are recorded under various stimula-
tion conditions.
The other electrophysiological technique involves '..estricting the fish's
movement by placing it in 3 to 5 inches of water in a small glass or plastic
tank. Electric dire:barges are then detected with monitoring equipment connect-
ed to the water via electrodes suspended in the experimental tanks.
The data obtainea by the above provide insight into system operation but
are not very useful in evaklating the function or sensitivity of receptors CT
systems. First, in those cases where anesthesia was used, a question can be
raised on the effect of the reural function. Second, the investi-
gators were looking only at the isolated sensor signal under grossly abnormal
stimulation conditions. Third, the isolated sensor data, even if collected under
reasonably normal stimulation conditions reveal little about system function.
Fourth, the engineering is typically questionable for one reason or another.
This fourth reason is also the prime problem with the yet to be described be-
havioral technique. For example, Agalides (1965) did extensive work on these
fish, much of it being excellent. However, he used a small tank which would
distort the fishes' field, he did not control impedance within normal limits,
& he had extraneous objects in the fishes' field. Clark, Granath, Mincoff &
Sachs (1967) used stainless.ateel electrodes which distorted the fishes' field.
Hagiwara, Szabo, Eager & Suga (1965, 1967) all show waveforms in their reports
which appear to be riding on an increasing DC potential. It appears as though
their electrodes underwent a significant polarization during the experiment.
The experimentors will not offer an explanation for this observation. Mandri -
ota's investigations (1965) are characterized by very poor experimental techniques.
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13
Not only did he use silver electrodes, a small tank, etc, but he used as a
punishment with his behavioral training technique an electrical shock sufficient
to visibly jerk the fish; shock while studying the function of electrosensing
fish.
The foregoing is sufficient indication of the deficiencies encountered.
We shall turn now to the behavioral technique that has been used, the technive
that can most directly answer the question of sensitivity. In this technique,
the free swimming fish is conditioned to respond to a certain stimulus. When
it responds correctly it is rewarded. The stimuli used hay. .. been an applied
voltage gradient across the fish's tank or objects of different conductivity
hidden within clay pots. With this technique, tne sensitivity and function of
the entire system can be tested.-
The limits of sensitivity found can best be summed up by stating that
the fish could detect the presence of a glass rod 2 millimeters in diameter in
s. clay pot but would fail to respond to a glass rod of 0.8 millimeters in dia-
meter in the pot (Lissmann, 1958). This limited statement of sensitivity is as
much as the state of the art provides. And even this statement can be question-
ed since the tank used does not meet the specifications derived below.
Somewhat in to this behavioral technique have been a limited number
ot data gathering expeditions into the fishes' natural environment. The
published results are rather limited. About the only thing that has been
found is that the fishes have about the SKKO pulse repetition rate in natural
conditions as they do in the laboratory. It has also been found that the
weakly electric fish are nocturnal creatures. Other results can not be
accepted due to deficiencies in engineering.
In sum then, we can conclude very little about sensitivity and system
function from the available biological data. About all that can be said is
that the fish is reported to be quite sensitive and qualitative observations
would seem to bear this out. But for reasons indicated above, there is no
adequate quantitative data.
Size of tank required for valid exrerimental data. One of the prima
deficiencies in the reported work-is the use of a tank of inadequate size or
with extraneous objects in the field. These distort the field and seriously
effect the data obtained.
We have experimentally explored the effect of various objects and tank
size on a simulated fish field and found that all objects and even the walls
of small pools distort the .7ield to some extent. A quantification of this
effect is defined in the calculations presented below in which we determine
the specification of the tank needed for acceptable experimental work.
We assume that the fish is locnted centrally within a cylinder. With
this assumption, we study llow the potential varies as a function of cylinder
length assuming an infinite radius for the cylinder. Next, we assume the
cylinder has infinite lengtn and see haw current varies with radius. With
this information, we will be able to determine reasonable lengths, widths,
and depths for experimental containers for electric fish research. We will
disregard all.interfaces in this development because our ultimate intention
is to determine when these interfaces can be disregarded.
The equations which express potential as a function of distance arek
for y > + a
4. The five unnumbered equations used in this sectim, are developed in a later
section. They are numbered in the later section as 27a, 27b, 27c, 48,51,
but appear in this order here. Definitions of symbols can be found in the
appendix.
we
�
for 7 -a>y> - +a V al --a--
c
for -
a >
V
�
15
2y
w c
Plotting for different values of L in Fig. 2a we obtain the required cylinder
length. Only the positive direction is plotted because the negative direction
is identical extzpt the sign is reversed. Bow long tbe cylindrical tank should
be is difficult to determine precisely. As &minimum though we can say that
there should be 5 electric organ lengths of :ister in front and in back of the
fish at all times daring the experiment.
TO determine the cylinder radius required for the tank IN we can modify
the; limits on the integral expressing the current I in equation 48 of our
later development. This equation is
I-
(2
0
QLoR
4 ir c (R +L -
2
3/2
d d R
The only limit which needs to be modified is the infinity symbol. We
replace this with v and solving as before we find the current to be
QLa
2c
-1
1/2
Equation 1 may be expressed in closed form as
(1)
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�������� ��������,=�11�1100/ �������
1.3
1.2
0.4
0.1
100
90
GO
70
110
SO
40
30
20
I0
0
I. 31. SL 71. N. RI. 131.
ORGAN LENGTH
3
3L SL 71. U. III. ix
REGIARED WIDTH AND DEPTH
IN ORGAN LENGTHS
Fig.2 a) This illustrates the potential on the positive side of the
dipole electric generating organ. The potential on the negative side
is the negative reflection of this. b) This indicates the percent
of maximum electric generator current which will flow between the posi-
tive and negative ends a the organ as a function of the width and
depth of the tank, assuming a free field expression for the current
generated.
' The maximum current is expressed in equation 51 of the later development.
It is
I al --9-2-
C
By letting v = +ri and solving equation 2 for different values of n, we
have obtained a plot of current as a function of width or depth. This plot
is shown in Fig. 2b. to obtain 90% of the maximum current requires 5 elec�
tric organ lengths on each side of the fish. To get 95% would require 10
lengths. More than 95% of the maximum current would be almost impossible
to obtain in the laboratory.
Thus for experimental results to be considered valid there should be
at least 5 electric organ lengths of water surrounding the fish. For free
swimming experiments, it would be best to have at least 10 electric organ
lengths about the fish. When the fish must be restrained near the surface,
10 organ lengths of water in all other directions should be the minimum.
It should tlso be noted that to simulate infinity in research, the tank
must be connected to earth ground and made to conduct. This does not seem
to have been done in past research.
17
POSSIBLE RECEPTOR MECHANISM AND NEURAL CODING
The possible mechanism discussed below can not be said to be the mech-
anism that the fish actually uses in its detection and classification of objetw.
It is, however, derived from our review and antolysis of the available data,
from results Of the limited experimentation that we carried out to clarify
some of t!le available data, and from our knowledge of auditory and labyrin-
thine szatem function. The postulating of this mechanism, viewing the electro-
reneptors and auditory receptors as evolutionary derivations of the same primi-
tive receptor, provides a testable hypothesis of receptor function. It also
pruvides a basis that can be of assistance in determining receptor sensitivity.
Mechanism. The inner ear is a fluid filled cavity with a complex membrane
structure. It is notable for the complex electrical fields that are generated
within it by external events and its organized bands of hair-like sensing cells.
Early concepts of pressure waves in the fluid bending the hair cells and thereby
triggering signals to the brain are very much question. Some of the never
concepts implicate an intermediate electrical field sensing mechanism in the
hair cells. The precise nature of this is not clear but elements of O'Leary's
(1970) recent experimentation and theorizing on the inner ear labyrinthine
system appear to be quite applicable to the weakly electric fishes' sensing
system. These elements combined with other information on auditory and balance
sensor function will be discussed below to the extent that they have bearing on
our modeling of the fishes' sensing system.
Since Dahlman (1960) has shown that hair cell membranes are apparently
impermeable to ions, O'Leary assumes that the detection of fields are due
18
to electrostatic forces as opposed to ionic current. In his analysis, he
points up that dissipative energy loss of an electric field in a dielectric
is generally associated with movement of charge Carriers. These movements
in an electric field result in an effect called polarization. Van Beek
(1967) has pointed out that the average molecular dipole moment Paa in a
heterogeneous system is the vector sum of induced (electronic) polarization
resulting from the relative displacement of electrons and nuclei, dipolar
polarization resulting 1"rom the partial alignment in the direction of the
field of molecules with permanent dipole moments. and interfacial (Maxwell-
Wagner) polarization occurring at boundaries between the components of a
heterogeneous system. Jackson (1962) has indicated that Pmoi is related to
the macroscopic polarization P (electric dipole moment per unit volume) and
the macroscopic electric field E by
P ( = N Pmlo1 = xe E
where N is the number of molecules per unit volume
xe is the electric susceptibility.
(3)
From this and his own experimental data, O'Leary (1970) suggests
that a weak electric field in the fluid filled inner ear might be detected by
hair cells by the polarization it induces in long-chain filaments of poly-
atomic molecules in the cilia. Since Van Beer's (1967) studies of dielectric
behavior of colloidal solutions indicate that particles such as polystyrene
spheres are frequently surrounded by electric double layers when they are
dispersed In dilute KC1 solutions, it is conceivable that low frequency
electric fields polarize the molecules by inducing dipole moments in the
OPAIMVW
� 19
double layers. There is also evidence along this line from Eeller, et al
(1960) & Saito, et al (1966). . Polarization can also occur by a mechanism
suggested by Frohlich's observation (1958) that large molecules can have
CH3, CaO, or OH groups that are in themselves dipolar, but have a net dipole
moment of zero due to the vector sum of all the moments. These molecules
then behave like nonpolar molecules in that their polarization are of the
- induced (electronic) type with resonant. frequencies in the optical range.
With these two possible polarization mechanisms, O'Leary develops a
theoretical basis for accepting. an electric field sensing mechanism. This
mechanism encompasses more data than a mechanical model.
Starting with Jackson's (1962) Observation thart. a charge e which is
displaced a distance x is bounded by a restoring force F given by
2
F a - m uo x
where m is the mass of the charge
wo is the radian frequency of harmonic oscillation
(4)
He goes on to consider the effect of a field on a.charge. The action of the
field E causes the charge to be displaced a distance x from its equilibrium
position. From Newton's third law we know
eE a m wo2 x
(5)
The induced dipole moment is then defined for one ellctron da
e2 E
2
m w
o
(6)
If there are Z electrons per molecule with f of them bound by a restoring
force.- mj2 x, then the indUced dipole moment is
e2
1, E �L�
ind m 2
41 wi.
where Z Z f
ii
(7)
Temperature is not a variable in equation 7 so the induced polarization
would not be disrupted by thermal agitation. Thus, the sensitivity of this
effect for the detection of weak E fields would be limited by quantum con-
siderations rather than by the classical limit of kT, These quantum limita-
tions are determined by the magnitude of the allowable shifts in energy levels
of the molecules caused .by the field, considered as a small perturbation,
relative to the energy levels of the molecules in the absence of the field.
O'Leary suggests that this polarization mechanism has great sensitivity.
He estimates it for the inner ear with the following argument. If the behav-
ioral threshold for stimmlueenergy is indeed close to lkT 4 x 10-14 ergs/
molecule as suggested by deVries (1949), the corresponding wave number 1,1
for an energy transition of IkT
7n.
hc/1 ss kT
(8)
would be 1/X = 200 cm:4 if this energy were entirely absorbed by a single
molecule. The threshold eneray would probably be distributed among numerous
molecules. Thus, the polarization of a single Molecule weld occur for
energies much smaller than kT, i.e., for transitions of far less than
-1
200 ma .
The oceurrence of hyperfine splittings in the Stark effect suggests
a transductive mechanism based on. polarization would be sufficiently sensi-
tive for the detection of threshold stimuli. Herzberg (1950) and others
have studie4 spectrosuz.-pically the splitting of energy levels by an electric
field (the Stark effect). Splittings of about 10-3 ma-1 have been observed
from diatomic molecules with field strengths in the range of 102 - 103 Was.
Smaller *hyperfine" splitings were then predicted theoretically and observed
using high resolution spectrometers.
Based upon the foregoing, transduction in the hair cell can be viewed
as a quantum amplification process that is modulated by the average microscop-
ic polarization of an ensemble of long-chain molecules associated with the
cilia.
This development suggests, accepting it for the electric fish sensor,
that the electric sensor would need protection from mechanical stimuli. Both
menhanical and electrical stimuli might be detected by this molecular effect
in the electrical sensor because mechanical forces should also affect the �
microscopic polarization of long-ehain molecules. The bending or shearing
of cilia that contain long-chain molecules could result in a change in the
23
that is essentially independent of generator output, changing as a function
of an environmental stimulus. Qualitative equations can be written for the
output frequency of the phasic and tonic organs. These are:
Fphaslc
NF
generator
= Ftonic resting
f
m
N � 1,2,3
n is 1,2,3
� (stimulus)
(9)
(10)
There is a general relationship between the physical classification
scheme discussed in an earlier section and the tonic-phmac scheme just
described. Receptors fit for the most part into two classes - tonic recept-
ors that are sensitive to low frequency stimuli and are of the ampullary
type, and phasic receptors which are sensitive to high frequencies and are
of the tuberous type.
The other useful classification scheme involves the apparent coding
employed by the electroreceptor for transmission of information to the brain.
One fish, HYpopomus, has electroreceptors which respond to each dis-
charge of the electric organ with a succession of short pulses. Each "pulse
train" contains eight or more pulses (Nagivara, AMMO, & Negishi, 1962).
The electroreceptors of Gymnotus and Staetogenes respond with one to six
pulses to each generator discharge (Nagivara 81 Morita, 1962). Some studies
indicate that the number of pulses in each train can be related to the potential
near the receptor. This has been referred to as "number coding".
10
24
In another weakly electric fish, Eigenmannia, each organ discharge
does not produce a receptor output. If the fishes' electrical field is
distorted though, we find that the receptor output is at most one impulse
per generator organ discharge. If we decrease the distortion we find the
receptor output to be one impulse per every two generator organ discharges
and so on. In other words the chance that a receptor will fire is related
to the stimulus intensity acting on it. This type of coding is called
"probability" coding. (Hagivara & Morita, 1962)�
Another fish, Sternopigus, was throughly studied by Bullock and
Chichibu (1965). They found fibers that carry one impulse per organ dis-
charge. They noted a phase or time relationship with the intensity of the
stimulus. This is referred te, as phase or latency coding. They also found
other nerve fibers that maintained a rhythmical firing out of phase with
tla electric organ discharge. The frequency of these receptors changed as
a function of the intensity of the electric field near the receptor. This
type of coding is frequency coding and is characteristic of ampullary organs.
As a weak generalization, it appea...s that ampullary receptors give
tonic responses with frequency coded information and tuberous receptors
give phasic responses with either number, phase, latency, probability or
frequency coded information. But we can develop a stronger generalization.
Hagivara and Morita (1962) suggest a model ror probability coding
based on an assumed nerve threshold. By making certain assumptions about
the threshold curve, we .have extended their model to encompass all coding
schemes proposed to date. Their model was originally developed for tuberous-
phasic receptors, but we can easily extend it to ampullary -tonic receptors
with a simple modification. We shall develop below this all encompassing
coding scheme since it suggests the nature of the system function.
:11
25
The threshold model as reported by Hagivara and Morita (1962) assumes
that after a receptor fires, the threshold resets to some high value. The
threshold then begins to decay until the intensity at the receptor is este;
to or momentarily exceeds the threshold. At this point, the receptor fires,
and the threshold resets.
This will form the basic model on which we will elaborate in order to
encompass the different oho:ling mechanisms. First, we will discuss some gen-
eral properties of biological threshold curves. Then, we will define the
relationships between the threshold curve and the electric organ output which
are required by the available data. Lastly, we will detail some of the meas-
ures which could be made to quantify the threshold function.
It is established in biology that nerves can not fire during or instant-
aneously after a previous firing. There is also a biological basis for defin-
ing the threshold curve as one describing an exponential decay from some
value; T the maximum threshold, to T the minimum threshold. In reality.
MAX min
the minimum value probably continues to decay with time. But for situations
of repeated sampling, we can approximate it as T , a constant value. Finally,
it is probable that the threshold curve shifts as a function of the needs of
the fish. Such shifts, if understood, could be modeled by changing T ,
T , or the exponential time constant T.
max
With these facts in mind, we can describe the threshold function for
time after each firing as
4.
T [max - Tmir
] Tc +n + S
(a)
Further, is a very good apyroximation we could say
TraT e- Tc +T min + S (t)
'sax
(12),
The general shape of such a function is shown in Pig 3a. To describe
probability coding, we must assume that 4 Tc> > 1/f. In such cases,
the electric organ can discharge several times without firing the nerve.
The number of times it must discharge before activating the nerve is a
function of the stimulus potential relative to the threshold. The higher
the potential the sooner the nerve will fire. This is illustrated in fig. b.
We can also see that phasic coding requires Tain > O. If Tian were
zero, the sensor would reset Independently of the electric organ and one
form of tonic coding would be observed.
To describe number coding we must assume that 4 Ta < < 1/f. In such
bases the electric organ will cause the receptor to fire more than once each
time it discharges. The number of times the receptor will fire is proportion-
al to the intensity at the receptor as illustrated in Ti.. 3c. Again, for
phasic coding Tulin must be greater than zero.
As 14 Tc becomes approximately the same as 1/f, several interesting
possibilities occur: phasic coding, latency coding, and interpulse interval
coding. phase and latency coding are illustrated in Fig.4a. They are one
and the same. Interpulse interval coding is illustrated in Fig.4b. It is
also interesting that when 4 Tc is about two or three times l/f a combinx-.
tion of probability and number coding can be anticipated. Such coding has
been observed for Sternopygus (Hagiwara and Morita, 1962).
It should be noted that this model does not account for the observed
output of phasic receptors while the generator is between pulses. As has
A
I
tea
0
31.
Tot MIEN PRIOR IISCEPTON OUTPUT IN
TINE CONSTANTS
1 /kid&
POUNTIAL
SMARM
3 WON
�3 INTENSITY
1 1
Tess
POICNTIAL
O^ p.
:2
r &
tO 1.
le a
NO
NEMO*
MMCNSITT
Tone
NIG* NOW
111111 III
TINE
LOU
INTENSITY
ITT
Fig.3 a, This is a mathematical description of the threshold of the receptor
to the intensity of the stimulating electrical potential as a function of
time following prior receptor output. b) This illustrates the relationship
between the electric potential at the receptor, the receptor threshold state,
and receptor output when the receptor threshold decay time constant is great-
er than the repetition rate of the electric generating organ. In this case
the apparent coding is commonly referred to as probability or frequency cod-
ing. c) This is comparable to case b, but the decay time constant is much
less than the duration of the electric organ output. In this case the out-
put of the electroreceptor is said to be pulse count coding.
NNW
POTDITIAL AT
It [COTO II
MICOIMM
I.
LOW
TI
II IS N
MC OS UM
III � III I
TIM
MP
LOW
Fig.4 a) This illustrates the relationship between the electric
potential at the receptor, the receptor threshold state and receptor
output when the generator organ rate is just smaller than the recipro�
cal Of the decay time constant. The output of the electroreceptor in
this case is typically referred to as phasic or latency coding. b)
This illustrates the relationship between the electric potential at
the receptor, the receptor threshold state, and receptor output when
the generator organ rate is just larger than the reciprocal of the
decmy time constant. The output of the eleotroreceptor in this case
is typically referred to as interpulse interval coding.
27
been noted previously, there are experimental reports that phasic receptors
have an output apparently unrelated to generator output. We do not know if
this is due to this not being the best fit model, to our having insufficient
data to incorporate those particular observations into the model, the exist -
ance of a buffering capability at the receptor, or if the reports report arti-
facts due to faulty technique.5 But whichever is the case, the utility of
the model is not affected. It can usefully be used as a unifying framework
for studies of threshold, sensitivity, and response time across all weakly
eloctric fish. With such a framework, sensitivity could be well defined
experimentally and the mechanism of the sense better understood.
MODEL: DEVELOPMENT, FUNCTION, AND SENSITIVITY
In outline, the fishes' sensing systea-appears to function as follows:
The generating organ emits an intermittent electric potential or current.
This results in an electric (dipole-like) field in the water surrounding
the fish. Objects within the environment and also the environmental bound-
aries distort the electric field. This distortion causes a change in the
electric field near the fish which we shall refer to as the stimulus. The
receptors measure the electric field or properties of it thereby providing
information that is processed by the fish's nervous system.
This system model generally agrees with data reported by Lissmann
(1963) from conditioned response experiments. In his experiments, Oymnar-
chus was trained to respond to changes in the conductivity of objects placed
in a sealed container. Positive reinforcement was a food reward, and nega-
tive reinforcement or punishment was the insertion of a metallic object into
the fish's tank. This punishment was probably not sufficient to reasonably
test the threshold of the biological system. But by using this method
Lissaann & Machin (l9,8) determined a threshold to potential change of
5. If Tain were below the mechanical noise level, phasic receptors would appear
to be tonic receptors.
�
about 0.15 uviam.
We shall also model the fish as a dipole, but a dipole that differs
from the one suggested by investigators such as Lissaann. He suggested that
the fish is a head to tall dipole. This suggestion, however, neither fits
with the physical location of the electric generating organ as determined
anatomically nor with the function data reported by Bullock and Chichfbu
(1965). As noted in an earlier section, the electric generating organ has
been found to be located near the tail in most if not all weakly electric
fish. Bullock and Chichibu (1965) observed the zero potential plane to be
perpendicular to the fish and found it located approximately one quarter of
the way toward the head measuring from the tall toward the head. Thus, we
use as our model a dipole field as illustrated in Fig. 5a.
We will develop a simplified model of the receptor and then discuss
system function. We will use the dipole concept described above as well as
the hypothesis on sensor function derived from above. Through the develop-
ment of a set of equations and a computer analysis, we will obtain an approxi-
mate solution for the response of the receptors due to perturbations caused
by an object in the field. Through this, ye will determine critical vari-
ables in the sense function and quantify their effect. With the informa-
tion so derived, we will consider the practical implications of the sense.
The symbols used are dnfined in the appendix. The definitions are
critical as is the recognition that coordinate transformations are necessary
and will be used.
Receptor Level
Development. Fig. 5b illustrates a simple dipole. It consists of two
conducting spheres of radius a separated by a distance L. One sphere is
positively, charged to a total charge of +Q. The other sphere is
Fig.5 a) This represents the electric field intensity surrounding
the weakly electric fish as indicated by the data of Bullock &
Chichibu (1965). b) This illustrates the model used to calculate
electric potential, current, and field intensity, due to a fish's
electric generating organ. The model consists of two spherical
conductors of radius a separated by an on center distance L.
negatively charged to a total charge of -Q. We will assume that a is much
less than L. When this is the case, the charge on each sphere can be assumed
to be evenly distributed about the surface. The electric potential (voltage)
is defined to be the increment in work required tc... move an increment of
charge from infinity to a given point in space, or
W
V in (13)
a Q
Recall that the fundamental work equation states that work equals the kinetic
energy minus the gained potential energy or
W =ICE-PE
If we move a very small charge very very slowly along the line which passes
through thirty� charged spheres, the kinetic energy is essentially zero. The
'work is the negative of the potential energy. , Potential energy is defined by
the integral
a
PE
(15)
where f is the force acting on the test charge as it is moved
from x toy
dr is an increment of distance in the path between the
points x and y
The electric force is a conservative force. Thus, if a test charge is acted
on by more than one charge, we can determine the potential energy due to each
charge and find the total potential energy by addition. In other woes if
P En represents the potential energy due to the nth charge, and if there are
. � � 30
a total of N charges the total potential energy is
P Etota E P En
(16)
The potential energy due to a charged sphere can be easily found.
The electric force may be found from Coulomb's Law which states
(17)
where Q is the tots]. charge on the sphere�'
qt is the charge of the test charge
r is the distance between the two charges
c is the dielectric of the media
ir is a vector directed away from the center of the charge
on a straight line
Using Gauss's Law it can be shown that within the charged conducting
sphere there is no electric force. Thus, if the radius of the sphere is a,
the force is
F 1 Iri < lal 0 (18)
If we call the line which passes through the two charges the y axis, we can
find the potential energy at any point on the axis by solving equation 15.
We substitute the force from equation 17 and obtain
1Y 4
Jr
PE
The solution is
.PE
Q qt
Q it
4 w e r2
r
d r (19)
13' +I > lal (20)
If we assign the dummy variable y- for the negative charge, we can solve
for the negatively charged sphere
P E - 4.
cl qt
The potential energy for the posisively charged sphere is
P E +
Q qt
The total potential energy for the dipole system is
P Etotal a
4 w e
Q qt
(21)
(22)
(23)
If we define a coordinate system as shown in Fig.6,we find that the absolute
values of y- and y+ depend on our location on the y axis as follows:
4
CONDUCTOR
RADA'S 4
-Li
/2
4.0
�
Fig.6 Simplified model which will be used in the study ot the electro-
static characteristics of the electric field generated by the fish.
The field generating organ is assumed to be composed of two conducting
spheres of radius a separated by a distance L. For convenience, the
origin is taken to be the midpoint between the two conducting spheres.
[ 1
ty -1 IY +I
11
for y > Ir;
L
for � 7> -
2
1Y+I Y -
1Y-I Y
2
2
L L
for - 7 > y 17+1
2
L
Il-1 ' - --y
4
(2Ita)
(24b)
(24c)
(24d)
(24e)
(24f)
The total potential energy can be found from equation 23 and equations
24a to f in each region. Recalling that the kinetic energy is zero and
applying equation 13 in terms of the test charge we gee:
a
=
a
This yields
V 3i [ 2
4 w e br +1
(25)
(26)
- Substituting the appropriate values of y+ and y- we obtain:
for y > L +a v=
2 4 w c
� for -L-a> y
2
L�) + L
2 2
2 2 . �2
forL-a>y�- L+a V
e
1(it r)
where V is the absolute voltage (V lod a 0)9
Q is the charge on either conductor,
L is the distance between the conductors,
S is a constant known as the dielectric
y is a continuous variable representing an absolute
scale with o being located between the positive and
negative charge as indicated in Fig. 6.
Evation 27b can now be employed to find the relative voltage between
the two spheres. This is the voltage which should be measured in the labor-
atory.
(27a)
V a
4 w
[
2y
Let us define v to be the relative voltage between two spheres. We
note that v is the value of the voltage at the positive sphere minus the
value of the voltage at the negative sphere. By substituting y = L - a
into equation 27b, we find the voltage on the positive sphere to be
If (4.) L - 2a
4 v e a(L-a)
(28)
(27b)
(270)
34
Substituting y = -
�.)
negative sphere to be
+ a into equation 27b we find the voltage on the
- L + 2a
V (-) = (29)
w c a (L - a)
Solving for v we find .
V = V (+) - V (-)
or 2
( 31')
v =
a 1 - a
c
(30)
(31)
Whcn a/L is less than 1/10, equation 31 can be approximated within 5 percent
by the relatio...-.sbip
Q
v
2 ITEM
The capacitance of an object is defined as
C
(32)
(33)
where Q is the charge on one symetric part of the object
V is the voltage across the object.
Determining the capacitance of the dipole from equation 31, it is found
to be
35
4
C
- 11)
L.
(34)
a -...-.21 2wea ,(1
V (1 - 2a
L
)
This may be approximated within 5 percent for a/L less than 1/10 as
C 21 2 w e a
(1 I.
(35)
It is useful to determine the resistance of the dipole we have just
considered. Since we have already evaluated the voltage between the charges,
if we can find an expression for the current that flows between the two points,
we can solve for the resistance from Ohms Law.
(36)
where R is the resistance
v is the voltage
I is the current
Two equal and opposite charges create an electric field. If we can
solve for the magnitude and angle of th. electric field 14 we can determine
the current density from the relationship
(37)
36
where S is the current density
is the conductivity of the media.
Once the current density is known, the current I can be found from the
surface integral
I = .
where the integral is over any closed surface
(38)
de is an element of surface taken to have a unit vector
located normal to the surface.
The electric field is a vector quantity. Thus, the total field is the
vector sum of the field due to the negative charge 1- and the field due to
the positive charge i+. Symbolically we can write
(39)
where I is the total field
Due to symmetry, the most convenient surface to use for cur current
integral is the plane which forms the perpendicular bisector of the line
segment joining the two equal and opposite charges. This plane is illus-
trated in Fig. 7a. The electric. field due to the positive charge at any
point p is defined by the equation
E+
w c D+2
fp+
37
where D+ is the distance between the positive charge and the
iD is a unit vector located in the direction of D+ away
from the positive charge at p.
The electric field due to the negative charge at the same point p is
-Q
2
4weD
IMP
where D is the distance from the negative charge
rD is a unit vector at p directed the D line away
from the negative charge.
The minus sign in equation 41 is due to the fact that unlike charges attract.
So long as the point p is on a plane which forms the perpendicular
bisector of the line segment between the two charges, the distances D* and
D are equal. They can be found from the equation
DD_ {R2 4. 42
D-
_ (s2).
1/2
We have defined the y axis to be the line which passes through the
two charges. We note that at the point p of Fig.Ta, the elebtrical field
due to either charge has both y and radial components. Due to symmetry, how-
ever, the radial components cancel each other and the net field in the plane
is parallel to the y axis. Also due to symmetry,the y magnitude of the y
components are ecual. Thus the total electric field is
-2 Q
Sin 0 i
4 n E D+2 total
(43)
� We note that 0 a a because they are opposite angles .Sin mcan be determined
from Fig.Ta by
Sin a 18 Sin 0 a. 1_22
D+
Substituting equation 44 into equation 43
-Q L
total a 3 iy
4 n e D+
(44)
(45)
Substituting equation 42 into equation 45, and equation 45 into equation 37
yields
a Q L
3 a 046)
4wc R2+L2 3/2
r
[
The surface of interr4tion is considered to be small ring segments in
the x-z plane about the y axis. Fig.Tb illustrates this concept. We note
that the electric field is perpendicular to the x-z plane which makes the
integral defined by equation 38 easy to evaluate. A surface element tor the
ring shown in Fig. Tb is
ds a R d dR
(47)
NEM ����
emb Oft
ed �
0����
11
Fig./ a) This figure illustrates the relationship between E- and E+
due to the electric fields created by the charges Q= and Qt, respec-
tively. �Note that the vector sum of these two electric fields is
parallel to the y axis for any point P in the plane which forms the
set of perpendicular bisectors of the two charges. b) This figure
indicates the quantities that need be defined in order to evaluate the
surface integral required by equation number thirty-eight.
39
wher: ds is the surface element,
d 40 is an increment of angle O which is an angle about
the y axis. .
dR is an increment in the distance vector R.
Substitution of equations 47 and 46 into equation 38, and defining
the surface intearal, we obtain
( 0 4 QLoR
r e (R2 +.L2 3/2
Solving the inner integral yields
riLaR
2 c (112 + L2)
Equation 49 can be solved to yield
I
Q L
2c
In closed form
I.
Q a
3/2
1.
d dR (48)
We have previously solved for the voltage between the charges in
equation 32. Dividing the voltage v by the current I, equation 51 yields
the resistance as defined by equation 36
CI (1 - a/1)
V
R
which reduces to
2 sea
1
R a
2 waa (1- --a )
(52)
(53)
We may now use the derived information. As a first approximation let
us assure that the radius of each charged sphere, a, is 1 cm and that the
length of the generator organ, L, is 10 em. The conductivity of fresh water
is about 2 m . The resistance which loads the generator organ is
about 62,000 ohms6 . The dielectric of fresh water is 0.707 x 10-9 fd/m.
This means that the capacitance which loads the dipole is 40 pfd. This large
resistance and small capacitance indicate that only a small current flows.
Thus, an electrostatic approach to the electric fish problem can be justi-
fied on electrical grounds as well as on the previously discussed theoretical
biological grounds.
Function. Accepting now the electrostatic model, we will consider the
effect of an intruding object on an individual receptor. As a simplification
the intruding object will be assumed to be a sphere. Such an object moving
into the fish's field will modify the potentials along the fish's sur:sce.
To obtain a solution for these modifications, we will first consider the
#.1eetric field produced by the dipole generator organ undisturbed by the
6. In a practical model, the resistance loadinr of the renerator can be
controlled by the choice of generator electrode sine.
41��������������������
41
perturbating object. We will calculate its magnitude at an arbitrary point
p. We will then consider the effect at point p of a perturbating object
located in a uniform field. Then we will transform the perturbation portion
of our solution back to the original coordinate system.
Once we have obtained an appropriate solution, we will assume and fix
certain variables. Then we will study the nature of. the fishes classifica-
tion techniques by plotting our results for given receptors along the fish.
It should be noted that this analysis is three dimensional awl although not
conceptually difficult, it is somewhat complex. Fig. 8a illustrates the
problem.
The electric field is defined as the force that would be exerted per
unit charge on a small test charge at a given point
(p) (p) / qt
(54)
The force on a test charge q+due to another charge Q can be found from
Coulomb's Law as
qt Q
41D2 iD
where E is the dielectric of the media
D is the distance between the charges
iD is a unit vector directed away from each charge at
the charge.
The electric field due to the positive charge in the dipole system is
iR
4 w kw R2
(55)
156)
Fig.8 a) This illustrates the variables involved in our model when
a spherical perturbing object is present in the model's environment.
Note the several coordinate systems that must be considered In the
analysis. b) This illustrates the functions required to express R'
and 0' in terms of R, 0, and L.
�������������1. �������������������RIP.R.101,M������������� ......������������Par.e.r.rdn.YM WiraMwranw" ������-��
The electric field due to the negative charge is
-Q
E - =
4 w zw R
(57)
The electric field is a vector quantity. Thus, we must perform vector
addition in order to determine the total field t. It is easier to determine
the components of the field due to each charge and then to add the components.
Thus. we will concern ourselves first with the y component and then with the
x-z component
The y component of the electric field due to the positive charge is
Ey+ = 1E+1 Cos 0
(58)
Likewise the.y component of the electric field due to the negative charge is
E = li-1 Cos e'
(59)
The x-z components of the electric fields are independent of the angle #
although the actual x and z components are not. The x-z component of the
total electric field will be considered to be the component within the
x-z plane at an angle # from the axis. For the positive charge it is
Ex.z = 1t4.1 Sin 0
and for the negative charge it is
E = It-1 Sin 0'
x-z I
(6o)
(61)
43
In order to treat these variables by standard mathematical techniques,
it is necessary to express R in terms of R and 0, and 0' in terms of R
and G. Fig.8b illustrates the factors which will help us do this. Recog-
nizing that R" is the hypotenuse of a right triangle whose sides are R sin
0 and L + R cos 0, we find
1/2
and
[R2
+ L2 + 2 R L Cos 0]
(62)
-
0' a tau1
R Sin
(63)
L + It Cos
It is useful to note the trigonometric functions for 0'. They are
R Sin e
Sin 0-
(64)
1/2
[1.12 + L2 + 2 R L Cos 1
and
L + R Cos
Cos 0' a
[S2 + L2 + 2 R L Cos e] 1/2
(65)
Combining equations 62, 64, and 65 with equations 56 and 57 and substituting
into equations 58, 59, 60 and 67 yields
Q Cos 0
(66)
4 w cv R2
7+
B
-41 L + R Co. 0
(67)
1. w c
+ L2 + 2 R L Co-s--072-
7-
41,
44
Q Sin 0
E +
x-z 4 v cv R2
-Q R Sin
- m
x-z 312
4 Iv fR2+ L2 + 2 R L Cos 0]
(68)
(69)
We can now determine the total components of the electric field. The
y component of the electric field at any point R, 0 about the dipole is
independent of
f4
and
is
Cos 0
4w
R2
[
R2 + L2 + 2 R L Cos 0 3/2]
L + R Cos
(70)
The component of the electric field in the x-z plane is radial and independent
of 0. It is
E Q Sin 0! 11
x-z
B2
4 w cv
[H2 + L2 + 2 B L cos 0]
To continue our derivation, we must deterthine the magnitude III and
the angle (relative to the y axis) 0 of the electric field at any poirt in
space. The magnitude may be found from the rules of vector addition as
II [ E 2 + E 21 1/2
x-z
(72)
45
The angle found by studying the geometry of the situation is
-1
0 = tan
E5.
E.
(13)
Manipulating the expressions in equations 70 and 71 per equation 72 yields
FPI
0 = tan-1
-s
[1
2 R2 (R + L Cos 0
[R2 + L2 2 R L Cos el 3/2
R
2 R L Cos el
(74)
2
Sine 1[S2 + L2 + 2 R L Cos 8]
3/2
3]-R
[.!2 + L2 + 2 R L Cos 0]
3/2
Cos - R2 + L + R3 Cos
(75)
These rather complicated equations completely describe the electric field
due to the dipole in a continuous media of dielectric eir.
To obtain a first approximation of the perturbation due to a sphere
of radius r at a location R , (0, 0 , relative to the dipole coordinates,
o o
we assume that the sphere is located within a unifosm field of strength En
at an angle 8 relative to a line parallel to the y dipole axis through the
sphere center. We will further assume that the value of this field is the
value of the dipole field at the center of the sphere, disrezarding the effect
or the sphere. If the sphere is small relative to the dipole length or if
it is very small relative to the separation between the dipole and the sphere,
46
the assumption will permit a solution within acceptable limits. The equations
which describe the electric field due to the dipole could be simplified to
a good approximation if the separation between the dipole and the object
sphere is more than ten dipole lengths. Since the biological data is not
adequate, we do not know the range of the electric sense. Consequently, we
will not approximate the .field at this point.
� Assuming a uniform field (without the perturbation) and expressing
the potential in terms of the coordinate system centered at the object
we obtain
Uo -Eo 0 Cos a
(76)
The surface charge on the perturbating sphere is exactly like a dipole. Thus,
the potential is of the form
A COS (2
Ux 2
2 -
P
where:A is a constant to be determined from the boundary conditions.
(77)
Finally the potential inside the sphere is of the same form as the potential
due to the original field or
U = - B p Cos a
(78)
where B is a constant to be determined from the boundary conditions.
The potential outside the sphere is the sum of U0 and Ux or
47
�
= - Eo 0 Cos a +
total
A Cos a
The electrostatic boundary conditions require that when � . r
_ c
-a0
a uI
Utotal
andikthat viten 0 a r
utotal = UI
C
a p
Substituting equations 78 and 79 into equat5or 87 5-1F3C:.
Eo r3 -A 2 B r3
Substituting equations 78 and 79 into equation 60 yields
E + 2 A e = r3 e B
X
O ;,
We can solve, for A on B in equation 82 and 83 to find
and
A =
3le
Cx + 2 elf
x ��� ex
ex + 2 ew
Eo
r3Eo
(79)
(80)
(81)
(82)
(83)
(814)
(85)
148
The potential U0 was assumed to be the potential due to the epole field.
We have an accurate expression for this field. The potential within the sphere .
does not effect the potential near the dipole. The critical term is the potent-
ial Ux which is the approx,mate modification of the dipole field due to the
spherical object. Substituting equation 85 into equation 77 yields
x w r3
- c .
Ux
2 tEl Cosa
�
ex - 2 ew
the perturbation potential.
(86)
We recall the value of Eo to be the value of equation 74 when R =o and
when 0 = Go or
E =
w e R 2
v o 1
+ 1.2
2 R - L Coo Go)
o o
2
+ 2 H2 Cos e, 3!2
0
� (87)
H014
r02 + L2 + 2 Ro L Cos 00]2
and we recall that a is measured relative to an angle 0 which is equation 75
evaluated at Ro, Go or
3 = tan-1
4.4
Sin e� [ito + L2 + 2 Ro L Cos 00] - R03]
3/2
[R02 + L2 + 2 Ro L Cos 00 3/2
Cos ee R02 L + R03 Cos 00
(88)
sawalt.,����mvemr."....rr.7410,,..ekrerwwomprivuyimpt.ir.l.ert"".. 1������ � ...7������� 7
45
To complete our analysis, we must express p and a in terms of R, 0, 41,
Ro, and 00. We find
p [if Cos - Ro Cos
this can be reduded to
00]
2
+[Re Sin Go R Sin 0 Cos (0 - #0)]
R Sin 0 Sin (0 0 00)] 1 1/2
(89)
= R2 + Ro2 - 2 R Ro [Cos Go Cos 0 - Sin 00 Sin' 0 Cos (# - 40)
We can solve for a in terms of R, 0, 00, 0, and p to be
]
R Sin
7. - 00 - 0
si 360� [Sin-1
2
1/2
(90)
We now have a multitude of equations but they provide a basis for a
computer study of the effect of the variables. Thus, we will review the salient
ones and group them in an orderly fashion for computer study.
Sensitivity., The magnitude of the electric field at any point in space
can be found from equation 74 when R + Ro and 0 a Op. Equation 74 becomes
equation 87 which is assumed linear throughout the perturbation. The magnitude
of the electric field is
5*
4 n eV Ro2
[1
2 R2 (Ro + L Cos 00)
o
[ + L2 + Ro L Cos 00 3/2
(92)
J1/2
[ 2
Ro + L" + 2 Ro L Cos
00]
2
5C
The angle of this field relative tz the y axis in the Ro -y axis plane is given
by equation 75 which for R = R0, 0 = Go becomes
Sin 00 [ [ o
R 2 + L2 + 2 Ro L Cos 001 3/2 - R03]
-1
6 m tan
+ L2 + 2 Ro L Cos.00] /2 Cos Go - Ro2 L + R03 Cos 00
3
This field creates a pertmahation field in a remote object of radius r and
dielectric ex. The potential of the perturbation field is given by equation
86 which is
r3
x - v
U = E
X Cos a
02 o
ex + 2 ew
(94)
(93)
Unfortunately, o and a are in terms of a secondary coordinate system. Our
primary coordinate system is Ft, 000. Equation 90 expresses p in terms of
R, Ro, 0, Go, 0, and .0 . This is the first place where the perturbation angle
.0 is important. The expression for o is
[0 m R2 4. R 2 ..
2 R Ro Cos Go Cos 0 - Sin 0 Sin Go COS (40 "" SO)
The angle a can be expressed in terms of R, H, G,0,,$, and .0.
However, it is simpler to express it as
a m 360 - ISin-1
0 R Sin (
L
el.))eo ""
(96)
The undistorted potent!al about the fish is the negative gradient of the sum
of equations 70 and 11 or
51
Uo =
+ L2 + 2 R L Cos 01
[ 1 1
1/2
(97)
Speaking anthropomorphically, the fish knows Uo, R, L, 0, e, Q, and
4 w ew. It must determine Ro, 00 40, r and 4wex. To find these variables,
it makes an analysis of the potential function U0 + U. Exactly how this
analysis is made is unknown.
As an approach to determine how the fish might operate, we shall find
the ratio U /Uo at different points along the fish for different values of
x
ex, r, Ro, 190, and (40 - 40). The equations are rattier complex, and require
a computer analysis. Thus, a fortran computer program was written which
manipulates and evaluates the desired variables.
In the computer program, certain variables have been assigned values for
reasons that are discussed below.
The length lf the generator organ has been set at one meter. In this
way, measures can be referenced in terms of generator organ lengths. Thus,
range, perturbating object size, and electroreceptor locations are all dis-
cussed in terms of generator organ lengths. The dielectric of the perturbat-
'ing object has been expressed in terms of the dielectric of water. In other
words the analysis is in terms of ev/ex rather than ex or ev themselves.
The fish is defined as a cylinder two times as long as the electric gen-
erator organ, with a radius 0:2 times the generator length. Six longitudinal
bands of eleven receptors are assigned ulong the length of the cylinder. Three
bands, each band 15 degrees apart, are located on each side of the fish. The
center band on each side is assumed to be in the same plane as the center of
the perturbing object and the axis of the cylinder.
52
The computer program analysed various combinations of four factors:
1) the ratio of e w /e x or the ratio of the dielectric constant of water to
the dielectric constant of the object, 2) the distance in generator lengths
from the center of the cylinder to the center of the perturbing object,
3) the angle in radians formed by the cylinder axis and the vector from the
cylinder center to the center of the perturbing object, and 4) the radius of
the perturbing object in generator lengths,
With each combination of the above factors, the program had the computer
manipulate and print out values of three variables that describe receptor
position and also the associated ratio of perturbation potential to free field
potential. The three variables were manipulated to show the effect on the
perceived potential ratio. These variables are defined as: (R) the line
segment from the center of the cylinder to the receptor on the cylinder sur-
face,(0) the angle formed by the intersection of line segment R and the cylinder
axis, (e - ejthe angle defined by the intersection of the plane passing
through the center of the object and cylinder axis and the plane passing through
the cylinder axis and a band of eleven receptors on the fish's surface. /n
the actual printout, this angle was taken for each of six receptors defined
by the same radius r and angle e (Ux/ U0) the ratio of the potential due to the
perturbation and that due to the dipole effect in the free field.
The following are the primary conclusions from the computer analysis of
the electrostatic model:
The value of the signal (disturbance/free field) is the same at receptors
15 degrees above and below the receptors on the plane defined by the cylinder
axis, receptor band, and center of the perturbing object.
The magnitude of the signal is largest in most circumstances at the head
end of the simulated fish. (This may explain the high concentration of electro-
receptors on the head of the actual. fish).
The magnitude of the signal is smallest at the cylinder surface closest .
to the two poles of the generator dipole.
Assuming the fish can detect a signal of one part per mil1ioa7(humaneCan
detect sounds 1/1,0001000t1% normal speech loudness) the fish can easily detect
objects of dielectric 0.1 times water whose radius is 0.1 generator organ .
lengths, at distances to the side of 6 generator lengths. It would 'have arri-
T7
culties at 10 Lengths or for dielectrics of 0.5 or 5 times water.
With the assumed sensitivity, the fish could detect objects 0* dial
0.1 water, assuming object radius is 0.1 generator organ lengths at110-eI
tric organ lengths if approached from front or rear.
Objects of dielectric 10 times water could be detected at a considerable
angle from the direct front or rear approaches at 10 lengths.
An indication of receptor sensitivity for objects of dielectric 10 times
water and 0.1 'water with a radius of one length is given in table I.
Table I
Receptor Sensitivity
Detection Distance
1 length
10 lengths
100 lengths
1000 lengths
10,000 lengths
Receptor Discriminatimo
1 part / 100
1 part / 10,000
1 part / 1,000;000'
1 part / 100,000,000
1 part / 1010
The analysis shows that the signal is about twice as large for dielectrics
10 times water as it is for obect:. wiih dielectrics 0.1 times water.
7. The dielectric of water is approximately 81, plastics are about 8, air
is about 1, and metals would be extremely hieh, virtually infinite in
many cases.
1
,
54
System Level
Although the data does not exist for the fish, it is reasonable to
assume that it has a data processing capability similar to that found in other
comparable organisms. Thus, it is likely that the fish can use inhibitory
and facilitating circuits to sharpen the aforementioned data from the receptor,
extract signal from noise and classify multiple incoming signals. In essence,
it would function as a system at the receptor level with interactions among
receptors and at the whole organism level involving the receptors, generator,
and brain. For example, there is evidence that a plot of receptor potential
along a band of receptors would yield, for a single perturbing object, a uni-
modal curve (Hagiwara & Morita, 1963). We might suggest that the configuration
of the curve is a function of the overall impedance of the perturbing object,
defining impedance as the sus, of the resistance and the reactance of the object.
The reactance of an object is given by
�^�
1
X= 2 wFL4f (98)
2rFC
where X is reactance
is frequency
L Is the inductance in henrys of the object under observation.
C is the capacitance in farads of the object under observation.
As may be seen, by operating as a system by coordinating the generator
and receptor function, the fish by changing generator frequency can induce a
lower or higher overall effective impedance in the object. If an object had
an impedance very similar to that of water, the fish could enhance its dis-
crimination and classification ability by varying its frequency; making the
object create a greater or lesser potential gradient at the receptor. Thus, if
,S.LTa. � ����� ����������
55
the fish was trying to locate a certain known object, it could adjust its
frequency to optimize its detection of the object. Using this system approach
it would also be possible for the fish to sense differences in objects-that
have the same exterior physical appearance. This would be done through vary-
ing the frequency and sensing and comparing the changes in the reactance of
the objects. At least some species of fish seem to be using the foregoing
system approach. In the earlier discussion of the available biological data,
it was noted that a generator frequency shifting technique was used.
There also appear to be mechanisms that can be used to optimize detection
and classification of one stationary object among several stationary objects
or a moving object among stationary objects. For example, to detect a moving
object several scans could be carried out, stored, and compared. In this way,
stationary objects in the field would be nulled and only objects of changing
impedance or location would be perceived.
CONCLUSION
Although there is a fairly substantial data base, we find that very
little can be applied to the development of an understanding of sense mechanism
and sensitivity. This is due in part to the fact that pioneering data in this
area, as it is in most areas, tend to have faults no matter how competent the
inves.Idators. One of the prime deficiencies in the reported work is the use
of a tank of inadequate size or with extraneous objects in the field. These
distort the field and seriously effect the data obtained. Further, the data
base contains very little behavioral data.
Thus, we undertook several tasks to provide a basis to assess the fishes'
electrosensing mechanism and capability, using the data presently available.
Through limited experimental work with electrical fields, sensors, and
,lbjects in various size bodies of water we have gathered data which, when
taken with the mathematical analysis, provides a specification for tank size,
fish location, and attachments, that will yield valid data in future studies.
We found that for experimental results to be considered valid there should
be at least 5 electric organ lengths of water surrounding the fish. For free
swimming experiments, it would be best to have at least 10 electric organ
lengths about the fish. When the fish must be restrained near the surface,
10 organ lengths of water in all other directions should be the minimum.
It should also be noted that to simulate infinity in research, the tank must
be connected to earth ground and made to conduct. This does not seem to have
been done in past research.
We have suggested as a working hypothesis an electrosensor mechanism.
This hypothesis is subject to test and thereby may provide the means for
collapsing the current multiple crude categorizations of the receptor that is
so typical of a new area of investigation. The hypothesis may also provide a
basis for analyzing higher interactions in the fishes' nervous system and
thereby increase our understanding of the sense.
We have also, through mathematical analysis, shown the linkage among the
various neural coding schemes suggested for the fish and have shown their
essential identity.
We have also developed a mathematical model of the fish based upon the
useable experimental data. A set of equations describing function was develop�
ed on the model and these equations linked to available experimental data.
The mathematical model was analysed by a computer to ascertain the sensitivity
of the fish at the receptor and to determine the effects of manipulating a
number of variables. These variables included fish size, object size, object
electrical characteristics, object distance from the fish, direction and angle
of the object from the fishes' axis, etc. From the computer anal�sis, we
�.:
51
determined the sensitivity to various perturbing objects under a variety of
conditions and found the fish to be quite sensitive particularly in certain
directions.
�
Erskine, R.T., Howe, D.W., Weed, B.C. The discharge period of the weakly
electric fish Sternarchus aIbifrons. American Zoologist, 1966, 6, 79.
Frohlich, H. Theory of Dielectrics. Oxford University Press, London,1958.
Granath, L.P., Sachs, H.G. Erskine, F.T. III. Electrical sensitivity of
a weakly electric fish. Life Sciences, 1967, 6, 2373-2377.
Hegivera, S., Kusano, K., Negishi, I. Physiological properties of electro-
receptors in some gymnotids. J. General Physiology, 1962, 45, 600-601.
&givers, S. & Aorita, H. Coding mechanisms of electroreceptor fibers in
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Hagivera, S., Szabo, T., and &ger, P.S. Physiological properties of electro-
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Jackson, J.U. Classical E1ectrody2cmics. John Wiley & Sons, Inc. N.Y. 1962.
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Lissmann,H.W. Electric location by fishes. Sci. Amer., 1963, 50-59.
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60
O'Leary, D.P. An electrokinetic model of transduction in the semicircular
canal. Biophysical Journal, 1970, 10, 859-875.
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61
APPENDIX
Symbols
The angle between the Y-axis and the radius vector from the positive
charge to an arbitrary point in space.
O The angle between the Y-axis and the radius vector from the negative
charge to an arbitrary point in space.
00 The angle between the Y-axis and the radius vector from the charge
to the center of the perturbing object.
The radius vector from the positive charge to an arbitrary point in
space.
R' The radius vector from the negative charge to an arbitrary point in
space.
Ro The radius vector from the positive charge to the center of the
perturbing object.
4 The angle between the X-axis and the X-s projection of the radius
vector from the positive charge to an arbitrary point in space
So The angle between the X-axis and the X-z projection of the radius
vector from the positive charge to the center of the perturbing
object
The charge on either side of the dipole.
The length of the dipole.
The dielectric of the water.
The dielectric of the perturbation.
The diameter of the spherical perturbing object.
The electric field at the center of the perturbation as if the
perturbation were not present.
�
Ux The potential due to the interaction between the dipole field and
the perturbation.
w A coordinate centered at Mc perturbation opposed to the vector
Eo and in the Y axis -Ro plane.
� A coordinate perpendicular to the w axis in the Y-Po plane
originating at the center of the perturbation.
0 The angle between Eo ane. a line parallel to the Y-axis at the center
of the perturbation.
A radius vector in the v, m coordinate system to an arbitrary point
in space.
a The angle between the radius vector and the coordinate v.
The electric field due to the positive charge.
Unit vector in the direction YR
The electric field due to the aegative.charge.
Unit vector in the direction of R.
The X-z component of the electric field.
Potential due to uniform approximation of the dipole field.
ir
U1 Potential inside sphere due to uniform approximation of dipole field.
The Y component of the electric field
� End point of charge path.
a Radius of sphere in dipole arrangement.
� Current
� Macroscopic polarizat:on
� The number of molecules per volume.
Related to the macroscopic polarization (electric dipole moment per
m� os
volume).
Xe The electric susceptibility.
� Storing force.
X End point of charge path
rind Induced dipole moment.
� Threshold function.
� Maxim um threshold.
n Minimum threshold.
m� m
Tc Eiponential time constant.
� Charge
y+ Dummy variable (positive charge).
y- Dummy variable (negative charge).
^ Absolute voltage.
� Voltage
� Capacitance
� Current density.
� Distance between charges
A A constant to be determined from the boundary conditions.
� A constant to be determined from the bcuadary conditions.
X Reactance
Inductance
Potential in terms'of the coordinate system.
wo Radian irequency of harmonic oscillation.
� Reactance in polar corrdinates.
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