[PAGE 1]
ON THE NATURE OF ELECTROSENSING IN THE FISH

[PAGE 2]
ABSTRACT

An evaluative review of the electrosensing literature was carried out with the intention of determining the nature of the electrosensing mechanism and its sensitivity. It was found that the biological data 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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[PAGE 3]
TABLE OF CONTENTS

ABSTRACT
INTRODUCTION....
NATURE OF THE BIOLOGICAL SYSTEM............. 2
Generator Organ .. 3
Receptor Orgen........ 7
Gymnotid receptors.
Mormyrid receptors.
System Function, Measurement Technique & Sensitivity..................ll
Electrophysiological & behavioral techniques.
Size of tank required for valid experimental data.
POSSIBLE RECEPTOR MECHANISM AND NEURAL CODING.............. ....17
Mechanism.
Coding.
MODEL: DEVELOPMENT, FUNCTION, AND SENSITIVITY........... ….27
Receptor Level....... …28
Development.
Function.
Sensitivity.
System Level. …54
CONCLUSIONS.......... ....55
REFERENCES......... …58
APPENDIX....... ....61

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[PAGE 4]
INTRODUCTION

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 some fish, such as the shark and goldfish, use a passive

electrosensing system in that the fish does not seem to generate its own

electrical field. Rather, it seems'to detect electrical signals, possibly

muscle potentials, generated by objects coming into its area.

Although there is now 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 competent 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 tasks 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 analysis, allows us to interpret much of the

data now available. This analysis also provides a specification for tank size,

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[PAGE 5]
fish location, and attachments, that will yield valid data in future studies.

Second, we have suggested as a vorking 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

aystem 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 number of variables. These variables include fish size, object

size, object electrical characteristies, object distance from the fish, direction

and angle of the object from the fishes' axis, etc.

We briefly discuss the fishes' function at 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 eleetric 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

[PAGE 6]
generating organ which 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 Organ

An understanding of the structure and function of the electrical field

generator organ is of importance in understanding receptor function. Thus,

generator function will be considered first.

The cells of the generating organ are referred to in the literature

as electroplaques, electroplax, electroplates, or electrocytes. 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, l970).

Electrocytes of mesodermal origin are typically disc shaped, but may

also be drum shaped or tubular. Electrocytes of ectodermai origin are U

shaped processes from the spinal cord. The electrocytes of the gymnotid,

Hypopomus, are between 300-500 μ in diameter and about 200 μ thick. The

electrocytes of Sternopygus on the other hand are rod-shaped and much longer

3

[PAGE 7]
than those of Hypopomus. They are about l-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 forvard 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 categorized 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 even when

strongly stimulated by an experimenter. Some of these 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, l970) to dis-

charge at frequencies of 30-l00 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-

pomus at 2-20 pps (Hagiwara and Morita, l963). Sternarchids discharge at

1. Each type of fish has a waveform that is specific to itself. Therefore, although Gymnotus and Steatogenys have the same frequencies, their waveforms.are different. These differences in waveform may be functions of the experimenters' competence in engineering.

[PAGE 8]
rates of 600-2000 pulses per second (Erskine, Howe & Weed, l966). 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, l965), 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, Eigenmannia which has an organ discharge rate of 400 pps shifts

its frequency 10 to 20 pps when confronted with a 400 pps signal (Larimer &

McDonald, l968). 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 electric 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 electrocytes. The axons to the more

distant electrocytes extend in the straightest possible line but those to

the less distant electrocytes follow s circuitcus pattern. A second means

of maintaining synchronization involve a delay line mechanism whereby the

pathways to the electrocytes differ in conduction velocities.

5

2. If a passive electric sense is more common than is thought, this could be a protective reaction.

[PAGE 9]
A number of investigators have measured the voltage output of the

generating organ. Hypopomus is reported to generate 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 10 to 200 millivolts. The in-water measurements were taken

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 of 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. The 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

[PAGE 10]
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.

Receptor Organ

The weakly electric freshwater fish are reported to have both active

and passive sensory systems. The active system 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 are 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 terms 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 followed by deceleration.

The acceleration phase can outlast the stimulus and according to Bennett

(1970) there is accomodation to maintained stimuli. These receptors are

sensitive to low freçuency electrical fields and to changes in a DC field.

[PAGE 11]
Their response to an applied current is a monotonic increase.

The active system sensors are called tuberous 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 mechanoreceptors, i.e., canal organs and

free neuromasts. 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 Lissmann and Mullinger (1968) found

that there were 2,000 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 organs, it can be noted that the ampullary organ has the appearance of a flask with a narrow duct (5-20 μ in diameter) leading from the skin surface to a cavity (30- 40 μ in diameter) that is located l00-500 μ 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 l0-15 u in diameter with each organ containing two to eight of them. Some microvilli 0.8 μ 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 within the organ, having lost its myelin sheath and dividing before entering the orran.

8

[PAGE 12]
Fig.l a) Schematic drawing of the two types of ampulla of gymnotids, b) Schematic drawing of the tuberous organ of the gymnotid, e) Schematic drawing of the mormyromast of the mormyrid, d) Schematic drawing of the tuberous organ of the mormyrid.

[PAGE 13]
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.3

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 of the bulb

is made up of supporting cells upon which the numerous sensing cells rest.

The sensing cells are 25-30 μ 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 μ 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 passes 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, wherethey 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 Knollenorgans (Szabo, l967). 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.

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3. The fish being described is Hypopomus artedi, a species of gymnotid. Details vary slightly from species to species.

[PAGE 14]
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 in

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 sensory chamber is filled with a\_mucous substance.

Two to five sensory cells occupy the lower sensory chamber. 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 receptcr organs, knollenorgans, which are some-

what similar to the tuberous organs of the gymnotids. Derbin and Szabo (1968)

describe them as being composed of three or four sensory cell complexes one

of which is shown in Fig. ld. 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

[PAGE 15]
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 processes

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 the-nerve endings and supporting cells. The sensory cell itself i

35-40 μ 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 system function.

System Function, Measurement Technique,and Sensitivity

This section will of necessity be short since there'is 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 probable system functior. and sensitivity. Two of the techniques are

electrophysiological and the third is behavioral.

Electrophysiclogical and behavioral techniques. In one electrophysio-

logical techniçue the fish is anesthesized and fixed to a wooden plate in

the normal swimming position. The wocúen plate is then tilted into the water

so that the body is submerged and the head exposed to the air. The regular

respiratory movemerts and oxygenation are maintained by spraying a fine jet

of water into the mouth of the fish. The dorsal branch of the lateral line

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[PAGE 16]
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 restricting the fish's

movement by placing it in 3 to 5 inches of water in a small glass or plastic

tank. Electric discharges are then detected with monitoring equipment connect-

ed to the water via electrodes suspended in the experimental tanks.

The data obtained by the above provide insight into system operation but

are not very useful in evaluating the function or sensitivity of receptors or

systems. First, in those cases where anesthesia was used, a question can be

raised on the effect of the anesthesia on neural 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.

havioral technique. For example, Agalides (1965) did extensive work on these

distort the fishes' field, he did not control impedence within normal limits,

& he had extraneous objects in the fishes' field. Clark, Granath, Mincoff &

Sachs (1967) used stainless·steel electrodes which distorted the fishes' field.

Hagiwara, Szabo, Enger & 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 vill nct offer an explanation for this observation. Mandri-

ota's investigations (1965) are characterized by very poor experimental techniques.

12

[PAGE 17]
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 technique

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 have been an applied

voltage gradient across the fish's tank or objects of different conductivity

hidden within clay pots. With this technique, the 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

a clay pot but would fail to respond to a glass rod of 0.8 millimeters in dia-

meter in the pot (Lissmann, l958). 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 akin to this behavioral technique have been a limited number

of 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 same 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

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[PAGE 18]
would seem to bear this out. But for reasons indicated above, there is no adequate quantitative data.

Size of tank required for valid experimental data. One of the prime deficiencies in the reported workis 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 field 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 located centrally within a cylinder. With this assumption, we study how the potential varies as a function of cylinder length assuming an infinite radius for the cylinder. Next, we assume the cylinder has infinite length and see how 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 cen be disregarded.

The equations which express potential as a function of distance $\sum I _ { 2 } ^ { 3 } x _ { 0 } ^ { 4 }$

$$
\mathrm { f o r } \ y > \frac { I } { 2 } + \tt a
$$

$$
v = \frac { Q } { \frac { q } { q _ { \mathrm { ~ \scriptsize ~ 4 ~ 7 ~ 5 ~ } } } } \quad \quad \left[ \frac { I } { \left( \mathbf { y } \cdot \frac { I } { 2 } \right) \left( \mathbf { y } \cdot \frac { I } { 2 } \right) } \right]
$$

14

4. The five unnumbered equations used in this section 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.

[PAGE 19]
$$
\begin{array} { r l r } { \mathtt { f o r } \frac { \mathtt { L } } { 2 } - \mathtt { a } \simeq \mathtt { y } \ Z - \ \frac { \mathtt { L } } { 2 } \ + \mathtt { a } } & { \qquad \mathtt { V } = \frac { \mathtt { Q } } { \mathtt { k } _ { \mathrm { ~ T ~ } } \mathtt { e } } } & { \left[ \frac { 2 \ \mathtt { y } } { \left( \frac { \mathtt { L } } { 2 } - \mathtt { y } \right) \left( \frac { \mathtt { L } } { 2 } \ + \mathtt { y } \right) } \right] } \end{array}
$$

$$
\cos - \frac { \pi } { 2 } - a > y
$$

$$
v = \frac { Q } { \frac { 4 \pi \epsilon } { 1 + \pi } } \left[ \frac { 1 } { \left( \frac { \frac { 1 } { 2 } - y } { 2 } \right) \left( \frac { \frac { 1 } { 2 } } { 2 } + y \right) } \right]
$$

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 except the sign is reversed. How long the cylindrical tank should be is difficult to determine $\frac { 1 } { \tt p r e c i s e l y }$ . As a minimum though we can say that there should be 5 electric organ lengths of water in front and in back of the fish at all times during the experiment.

To determine the cylinder radius required for the tank w, we can modify the limits on the integral expressing the current I in equation 48 of our later development. This equation is

$$
\texttt { I } = \int \limits _ { 0 } ^ { \infty } \Bigg ( \int \limits _ { 0 } ^ { 2 \pi } \frac { \texttt { e l g } \texttt { R } } { \texttt { i n e } \left( \texttt { R } ^ { 2 } + \texttt { L } ^ { 2 } \right) } \frac { \texttt { 3 } / 2 } { 3 / 2 } \hat { \texttt { 0 } } \texttt { d } \phi \Bigg ) \texttt { d } \texttt { R }
$$

The only limit which needs to be modified is the infinity symbol. We replace this with w and solving as before we find the current to be

$$
I = \frac { \mathrm { ~  ~ \sf ~ Q ~ L ~ o ~ } } { 2 \mathrm { ~  ~ \sf ~ \left[ ~ \left( ~ \frac { ~ \partial ~ \tau ~ } { ~ \partial ~ \tau ~ } ~ + ~ \frac { ~ \partial ~ \tau ~ } { ~ \partial ~ \tau ~ } ~ \right) ~ \tau ~ ^ { \mu } ~ \right] ~ o ~ } }\tag{2}
$$

Equation l may be expressed in closed form as

15

[PAGE 21]
$$
I = \frac { Q \texttt { L g } } { 2 \texttt { \textbf { E } } } \left[ \begin{array} { c c } { \texttt { \small 2 } } & { \texttt { \small 1 } } \\ { \texttt { \small 1 } } & { \texttt { \small - } } \end{array} \right]\tag{2}
$$

The maximum current is expressed in equation 51 of the later development. It is

$$
I = \frac { Q _ { \sigma } } { \sum \limits _ { \varepsilon } }
$$

By letting $\begin{array} { l } { \displaystyle \mathfrak { w } = \frac { \mathrm { L } \mathfrak { n } } { 2 } } \end{array}$ 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 electric 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 lesst 5 electric organ lengths of water surrounding the fish. For free swimming experiments, it would be best to have at least l0 electric organ lengths about the fish. When the fish must be restrained near the surface, l0 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.

16

[PAGE 22]
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 objects.

It is, however, derived from our review and analysis of the available data,

from results of the limited experimentation that we carried out to clarify

some of the available data, and from our knowledge of auditory and labyrin-

thine system function. The postulating of this mechanism, viewing the electro-

receptors and auditory receptors as evolutionary derivations of the same primi-

tive receptor, provides a testable hypothesis of receptor function. It also

provides 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 eellsand thereby

triggering signals to the brain are very much in question. Some of the newer

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 Dohlman (1960) has shown that hair cell membranes are apparently

impermeable to ions, O'Leary assumes that the detection of fields are due

17

[PAGE 23]
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 $\tt P _ { m o l }$ in a heterogeneous system is the vector sum of induced (electronic) polarization resulting from the relative displacement of electrons and nuclei, dipolar polarization resulting from 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 $\mathtt { P _ { m o l } }$ is related to the macroscopic polarization P (electric dipole moment per unit volume) and the macroscopic electric field E by

$$
\frac { p } { s } = N ( p _ { \frac { p } { s } 0 2 } ) = x _ { p } \frac { p } { s }\tag{3}
$$

where N is the number of molecules per unit volume ${ \tt X } _ { \tt S }$ is the electric susceptibility.

From this and his own experimental data, 0'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 polyatomic 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 KCl solutions, it is conceivable that low frequency electric fields polarize the molecules by inducing dipole moments in the

18

[PAGE 24]
double lavers, There is also evidence along this line from Heller, et al

(196o) & Saito, et al (l966).. Polarization can also occur by a mechanism

suggested by Frohlich's observation (1958) that large molecules can have

CHg, C=0, 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 that a charge e which is

displaced a distance x is bounded by a restoring force F given by

$$
\mathfrak { F } : - \mathfrak { m } _ { 0 } ^ { 2 } \textbf { x }\tag{4}
$$

where m is the mass of the charge

is the radian frequency of harmonic oscillation vo

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

$$
e \mathbf { E } = \pi \begin{array} { c } { \begin{array} { r } { \mathbf { \epsilon } _ { \mathbf { w } } } \\ { \mathbf { \epsilon } _ { \mathbf { 0 } } } \end{array} } \end{array}\tag{5}
$$

19

[PAGE 25]
The induced dipole moment is then defined for one electron as

$$
\mathbb { P } _ { \mathtt { i n d } } = \mathtt { e x } \approx \frac { \mathtt { e } ^ { 2 } \mathtt { E } } { \mathtt { m } \ * { \omega _ { 0 } } ^ { 2 } }\tag{6}
$$

If there are Z electrons per molecule with f, of them bound by a restoring $\mathbf { f } _ { \mathbf { \lambda } _ { \mathbf { j } } }$ force- m w, x, then the induced dipole moment is $j ^ { 2 }$

$$
\begin{array} { r } { \mathbb { P } _ { \mathtt { i n d } } = \frac { e ^ { 2 } } { \mathtt { m } } \quad \underbrace { \mathtt { r } _ { \mathtt { i } } } _ { \mathtt { j } } \quad \frac { \mathtt { f } _ { \mathtt { j } } } { \mathtt { m } _ { \mathtt { j } } ^ { 2 } } \quad \mathtt { s } } \end{array}\tag{7}
$$

$$
\mathbf { w i r e r e } \texttt { Z = L f } _ { \mathbf { j } }
$$

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 considerations rather than by the classical limit of kT. These quantum limitations 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 behavioral threshold for stimulusenergy is indeed close to lkT4 $x 1 0 ^ { - 2 h }$ ergs/ molecule as suggested by deVries (1gh9), the corresponding wave number $I _ { \beta _ { h } }$ for an energy transition of lkT

[PAGE 26]
$$
hbar { \mathsf { c } } / { \lambda } ~ = ~ { \tt { k } } { \tt { r } }\tag{8}
$$

would be $I / \lambda \simeq 2 0 0 c m ^ { - 1 }$ if this energy were entirely absorbed by a single molecule. The threshold energy would probably be distributed among numerous molecules. Thus, the polarization of a single molecule would occur for. energies much smaller than kT, i.e., for transitions of far less than 200 cm-1.

The occurrence of hyperfine splittings in the Stark effect suggests a transductive mechanism based on.polarization would be suffieiently sensitive for the detection of threshold stimuli. Herzberg (1950) and others have studied spectroscopically the splitting of energy levels by an electric field (the Stark effect). Splittings of about $1 0 ^ { - 3 } \cos ^ { - 2 }$ have been observed from diatomic molecules with field strengths in the range of $1 0 ^ { 2 } - 2 0 ^ { 3 } v / \cos$ Smaller "hyperfine" splittings 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 microscopic 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 mechanical 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-chain molecules. The bending or shearing of cilia that contain long-chain molecules could result in a change in the

21

[PAGE 27]
net dipole moment and should also be detected by a transduction mechanism

that was sensitive to microscopic polarization. The structure of the fishes

receptor organs suggests that there is such protection. This would be necess-

ary to minimize noise in the system.

If we are correct in suggesting that this is the type of receptor mech-

anism that is used by the fish for electrosensing, then it would provide a

basis for accepting the qualitative statements on the high sensitivity of the

fishes' electrosensors, Lacking well controlled behavioral data on sensitivity,

we would hesitate though to conclude that the fish is as sensitive as this

analysis suggests.

We have now considered the structure and function of the generator organ

and receptor organ. Now we shall look at the next level of the nervous system.

We shall consider the coding of nerve impulses by the receptor organ, its

transmission toward the brain, and the implications of the coding in under-

standing the electrosensing function and sensitivity.

Coding. As has been noted earlier, the literature is encumbered with

multiple classification schemes for electroreceptors. This has the potential

for confusion and does little to increase our understanding of receptor func-

tion. For example, one scheme is based on external physical appearance, e.g-

small, medium and large mormyromasts, another on total configuration, e.g.

ampullary and tuberous, and there are other crude classification schemes.

Two somewhat more useful schemes also exist. One is used by almost

everyone who discusses electroreceptors. In this scheme, the output of the

receptor is related to the output of the generating organ. The electro-

receptors are said to be either phasic or tonic. Phásic receptors respond

(give an output) at some integer division of the generator frequeney with

a string of from one to n pulses. Tonic receptors maintain a steady output

22

[PAGE 28]
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:

$$
\mathbb { F } _ { \mathtt { p h a s i c } } = \frac { \overset { \mathtt { N F } } { \mathtt { F e n e r a t o r } } } { \texttt { n } } \quad \cdot \left\{ \begin{array} { l } { \texttt { N = 1 } , 2 , 3 , \ldots \dots } \\ { \texttt { n = 1 } , 2 , 3 , \ldots \dots } \end{array} \right\}\tag{9}
$$

$$
\mathbb { F } _ { \sf t o r i c } \ \stackrel { = } { \mathbb { F } } _ { \sf r e s t i n g } \ + \ \frac { \Delta \mathrm { ~ \bf ~ f ~ } } { \mathrm { ~ \texttt ~ { ~ m ~ v ~ } ~ } } \quad \cdot \quad ( \sf s t i m i z a ) \mathrm { { u s } } \ )\tag{10}
$$

There is a general relationship between the physical classification scheme discussed in an earlier section and the tonic-phasic scheme Just described. Receptors fit for the most part into two classes - tonic receptors 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 discharge of the electric organ with a succession of short pulses. Each "pulse train" contains eight or more pulses (Hagiwara, Kusano, & Negishi, l962). The electroreceptors of Gymnotus and Staetogenes respond with one to six pulses to each generator discharge (Hagiwara & 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".

23

[PAGE 29]
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. (Hagiwara & Morita, l962),

Another fish, Sternopygus, 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 tó as phase or latency coding. They also found

other nerve fibers that maintained a rhythmical firing out of phase with

the 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 appears 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.

Hagiwara and Morita (1962) suggest a model tor 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.

24

[PAGE 30]
The threshold model as reported by Hagiwara 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 equal 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 coding mechanisms. First, we will discuss some general 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 measures which could be made to quantify the threshold function.

It is established in biology that nerves can not fire during or instantaneously after a previous firing. There is also a biological basis for defining the threshold curve as one describing an exponential decay from some value; $\tt _ { I m a x }$ the maximum threshold, to Tin the minimum threshold. In reality, 1 $\mathtt { P _ { m i n } }$ the minimum value probably continues to decay with time. But for situations of repeated sampling, we can approximate it as $\mathtt { T } _ { \mathtt { m i n } }$ , 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 $\mathfrak { T } _ { \mathtt { m i n } }$ $\Im _ { \mathrm { z a x } } .$ or the exponential time constant $\mathtt { r _ { e } }$

With these facts in mind, we can describe the threshold function for time after each firing as

$$
I ^ { \prime } = \left[ \mathbb { T } _ { \mathtt { m a x } } \ ^ { \mathtt { - } } \ \overset { _ { } } { \mathtt { m i r } } \right] \ - \ \frac { \mathtt { t } } { \mathbb { T } _ { \mathtt { c } } } \ + \ \mathbb { T } _ { \mathtt { m i n } } + \mathtt { s } \ \left( \mathtt { t } \right)\tag{11}
$$

25

[PAGE 31]
Further, as a very good approximation we could say

$$
\mathfrak { T } \approx _ { \mathtt { m a x } } ^ { \mathtt { m a x } } \ \mathtt { e } ^ { - \mathrm { ~ \frac { t } { T } ~ \mathtt { c } ~ + ~ \mathtt { T } _ { \mathtt { m i n } } ~ + ~ s _ \varepsilon ( t ) ~ } }\tag{12}
$$

The general shape of such a function is shown in Fig 3a. To describe probability coding, we must assume that 4 $I _ { c } > > _ { \frac { 1 } { 2 } / \frac { 1 } { 2 } }$ 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.3b.

We can also see that phasic coding requires $\mathbb { T } _ { \mathtt { m i n } } > \mathbb { O } . \quad \mathbb { f } \mathbb { T } _ { \mathtt { m i n } }$ were zero, the sensor would reset independently of the electric organ and one form of tonic coding would be observed.

To deseribe number coding we must assume that 4 $\tt { I } _ { c } < < I / I$ In such cases 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 proportional to the intensity at the receptor as illustrated in Fig. 3c. Again, for phasic coding $\mathtt { T } _ { \mathtt { m i n } }$ must be greater than zero.

As 4 T becomes approximately the same as l/f, several interesting $\mathfrak { T } _ { \mathfrak { c } }$ 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 T. is about two or three times l/f a combina- $\mathfrak { I } _ { \mathfrak { c } }$ 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

[PAGE 34]
been noted previously, there are experimental reports that phasic receptors

have an output apparently unrelated to generator cutput. 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.? 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

electric 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 system 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, Gymnar-

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

Lissmann & Machin (1958) determined a threshold to potential change of

5. If $\mathfrak { T } _ { \mathtt { m } \mathtt { i n } }$ were below the mechanical noise level, phasic receptors would appear

to be tonic receptors.

27

[PAGE 35]
about 0.15 uv/cm.

We shall also model the fish as a dipole, but a dipole that differs

from the one suggested by investigators such as Lissmann. He suggested that

the fish is a head to tail 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 Chichibu

(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 tail 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, we 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 defined 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

28

[PAGE 37]
negatively charged to a total charge of -Q. We will asstre 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 to move an increment of charge from infinity to a given point in space, or

$$
v = \frac { a \pi } { \lambda \theta }\tag{13}
$$

Recall that the fundamental work equation states that work equals the kinetic energy minus the gained potential energy or

$$
\texttt { W } = \texttt { K E - P E }\tag{14}
$$

If we move a very small charge very very slowly along the line which passes through the two charged spheres, the kinetic energy is essentially zero. The work is the negative of the potential energy. Potential energy is defined by the integral

$$
\texttt { P E } \overset { \Delta } { = } \int _ { \texttt { X } } ^ { \texttt { Y } } \frac { * } { \texttt { P } } \cdot \hat { \mathrm { d } } \boldsymbol { \tau }\tag{15}
$$

where F is the force acting on the test charge as it is moved $\ddagger { }$

$$
\mathtt { f r o m \textbf { x } } \mathtt { t o } \ y
$$

$$
\mathtt { p c i n t s \textbf { x } \ e n d \textbf { y } }
$$

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 words if P $\mathtt { E _ { \mathtt { n } } }$ represents the potential energy due to the n" charge, and if there are $n ^ { t h }$

[PAGE 38]
a total of N charges the total potential energy is

$$
\texttt { F } _ { \texttt { t o t a l } } = \sum _ { \texttt { n } = 1 } ^ { \texttt { N } _ { \texttt { N } } } \texttt { \textbf { F } } _ { \texttt { n } }\tag{16}
$$

The potential energy due to a charged sphere can be easily found.

The electric force may be found from Coulomb's Law which states

$$
\frac { a _ { \mathrm { ~ \bf ~ Q ~ } } } { \tilde { \mathrm { ~ \bf ~ r ~ } } } = \mathrm { ~ \frac { ~ \mathsf { ~ Q ~ } ~ \mathsf { ~ \mathsf { ~ Q ~ } ~ } ~ } { ~ \mathsf { ~ \bf ~ l ~ } ~ \mathrm { ~ \bf ~ r ~ } ^ { 2 } ~ } ~ } \hat { \mathrm { ~ \bf ~ i ~ } } _ { \mathrm { ~ \bf ~ r ~ } }\tag{17}
$$

where Q is the total charge on the sphere

$q _ { t }$ is the charge of the test charge

r is the distance between the two charges

e is the dielectric of the media

$^ \dag \mathrm { \Pi _ { \pm } ^ { \mathrm { ~ \tiny ~ { ~ \frac ~ { 1 ~ } ~ } ~ } } }$ 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

$$
F \left\{ \left| x \right| < | z | \right\} = 0\tag{18}
$$

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.

30

[PAGE 39]
We substitute the force from equation 17 and obtain

$$
p _ { \texttt { E } } = \texttt { \textbf { \textit { f } } } \frac { \texttt { \textbf { \textit { Q } } } _ { \texttt { t } } } { \texttt { \textbf { \textit { h r e r } } } ^ { 2 } } \quad \quad \texttt { \textbf { \textit { a r } } }\tag{19}
$$

The solution is

$$
\texttt { P E } = \texttt { \frac { \texttt { \textsf { Q } } \texttt { \texttt { q } } } { \texttt { \textsf { \textsf { q } } \texttt { \textsf { r } } } } } \left[ - \begin{array} { l } { \texttt { \cdot } } \\ { \texttt { \frac { 1 } { r } } } \end{array} \right] ^ { \mid \mathbf { y } \mid \texttt { \textsf { q } } } \left. \begin{array} { l } { \begin{array} { r l } { \vert \mathbf { y } \vert } & { { } } \\ { \vert \mathbf { y } \vert \rule { 0 ex } { 5 ex } } \end{array} } \end{array} \right. \qquad \left. \mathbf { y } + \right. \ge \left. \mathbf { a } \right.\tag{20}
$$

If we assign the dummy variable y- for the negative charge, we can solve for the negatively charged sphere

$$
\texttt { P E } = + \frac { \texttt { \textsf { Q } } _ { \texttt { \textsf { t } } } } { \texttt { \textsf { \textsf { u } } } \texttt { \textsf { t } } | \texttt { \textsf { y } } - 1 }\tag{21}
$$

The potential energy for the posicively charged sphere is

$$
\texttt { P E + = - } \frac { \texttt { Q } _ { \texttt { d } } } { \texttt { * } \texttt { r e l } \texttt { v * l } } .\tag{22}
$$

The total potential energy for the dipole system is

$$
p _ { \texttt { E } _ { \texttt { t o t a l } } } = \frac { \texttt { Q } _ { \texttt { t } } } { \texttt { l a r e } } \left\{ \frac { 1 } { \lvert \mathbf { y } - \mathbf { \sigma } \rvert } \frac { 1 } { \lvert \mathbf { y } \rvert + \mathbf { \beta } \rvert } \right\}\tag{23}
$$

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:

[PAGE 41]
$$
\mathrm { f o r } \textbf { y } > \frac { \texttt { L } } { 2 }
$$

$$
| y + | = y - \frac { 1 } { 2 }
$$

$$
| y - | = y + \frac { 1 } { 2 }\tag{24a}
$$

$$
\cos \frac { \frac { 1 } { 2 } } { 2 } > y > - \frac { 1 } { 2 } \vert y + \vert = \frac { 1 } { 2 } - y\tag{24b}
$$

$$
| y - | = \frac { I } { 2 } + y\tag{24c}
$$

$$
\mathrm { f o r } - \frac { \mathrm { L } } { 2 } > \mathrm { y }\tag{24d}
$$

$$
| y + 1 = \frac { I } { 2 } - y
$$

$$
| y - | = - \frac { I } { 2 } - y\tag{24e}
$$

(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 l3 in terms of the test charge we get:

$$
v = \frac { \partial } { \partial t } \left\{ - \frac { 2 a _ { t } } { 1 + \pi s } \left[ \frac { 2 } { \sqrt { y - 1 } } - \frac { 2 } { \sqrt { y + 1 } } \right] \right\}\tag{25}
$$

This yields

$$
v = \frac { Q } { 4 \pi \epsilon } [ \frac { 1 } { | y + | } - \frac { 1 } { | y - 1 } ]\tag{26}
$$

32

[PAGE 42]
Substituting the appropriate values of y+ and y- we obtain: $\overline { { \mathfrak { n } } } \overline { { \mathfrak { g } } }$ $\ y +$

$$
\mathrm { f o r } \ y > \frac { I } { 2 } + a
$$

$$
\forall = \frac { Q } { \mathrm { ~ i ~ \pi ~ \in ~ } } \left[ \frac { \tilde { \mathbf { z } } } { \left( \mathbf { y } \cdot \frac { \tilde { \mathbf { z } } } { 2 } \right) \left( \mathbf { y } \cdot \frac { \tilde { \mathbf { z } } } { 2 } \right) } \right]\tag{27a}
$$

$$
\cos \frac { I _ { 1 } } { 2 } - a > y > - \frac { I _ { 1 } } { 2 } + a
$$

$$
\gamma = \frac { Q } { \frac { 4 \pi ~ 5 } { 2 } } \left[ \frac { 2 ~ y } { \left( \frac { 1 } { 2 } - y \right) \left( \frac { 1 } { 2 } + ~ y \right) ~ } \right]\tag{27b}
$$

$$
f _ { \overrightarrow { a } } - f _ { \overrightarrow { a } } - a > y
$$

$$
\gamma = \frac { Q } { \frac { 1 + \pi \pi } { 4 \pi \frac { r } { c } } } \left[ \frac { \frac { 1 } { 2 } } { \left( \frac { \frac { 1 } { 2 } } - y \right) \left( \frac { 1 } { 2 } + x ^ { 2 } \right) } \right]\tag{27e}
$$

where V is the absolute voltage $( \because \| \cos _ { i } = 0 )$

Q is the charge on either conductor,

L is the distance between the conductors,

ε is a constant knowm 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 $\mathtt { F i g }$ . 6.

Equation 27b can now be employed to find the relative voltage between the two spheres. This is the voltage which should be measured in the laboratory.

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 $y = \frac { L } { 2 } - a$ into equation 27b, we find the voltage on the positive sphere to be

$$
v ( t ) = \frac { \theta } { 4 \pi \epsilon } \frac { I - 2 a } { a ( L - a ) }\tag{28}
$$

[PAGE 43]
Substituting $y = - \frac { I } { 2 } + a$ into equation 27b we find the voltage on the negative sphere to be

$$
v ( - ) = \frac { Q } { 4 \pi \varepsilon } \frac { - I + 2 a } { a ( L - a ) }\tag{29}
$$

Solving for v we find

$$
\texttt { v } = \texttt { V } \left( + \right) - \texttt { V } \left\{ - \right\}\tag{30}
$$

or

$$
\gamma = \frac { Q } { \frac { \pi } { 4 \pi } \varepsilon } \frac { 2 } { a } \frac { \left( 1 - \frac { 2 \pi } { L } \right) } { \left( \mathrm { ~ i ~ } ^ { - } \frac { a } { L } \right) } .\tag{31}
$$

Whcn $a / I$ is less than 1/10, equation 31 can be approximated within 5 percent by the relatioship

$$
v = \frac { Q } { 2 \pi \in A } ( 1 - \frac { a } { I } )\tag{32}
$$

The capacitance of an object is defined as

$$
c = \frac { Q } { V }\tag{33}
$$

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

34

[PAGE 44]
$$
C = \frac { Q } { v } = 2 \pi = 2 \frac { ( 1 - \frac { c } { 1 } ) } { ( 1 - \frac { 2 a } { 1 } ) }\tag{34}
$$

This may be approximated within 5 percent for $a / I$ less than 1/10 as

$$
c = 2 \pi \in \theta ( 1 + \frac { a } { 1 } )\tag{35}
$$

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.

$$
R = \frac { v } { I }\tag{36}
$$

where R is the resistance

v is the voltage

I is the current

Two equal and opposite charges ereate an electric field. If we can solve for the magnitude and angle of the electric field E, we can determine $\vec { \Xi } ,$ the current density J from the relationship $\vec { \pm }$

$$
⨏ \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } \Sigma \mathbf { \Sigma } \mathbf { \Sigma } \Sigma \Sigma \mathbf { } \Sigma \Sigma \Sigma \Sigma \mathbf { } \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma\tag{37}
$$

35

[PAGE 45]
where $\vec { \mathfrak { J } }$ is the current density

σ is the conductivity of the media.

Once the current density is known, the current I can be found from the surface integral

$$
I = \int \limits _ { 5 } ^ { + } d s \cdot d s\tag{38}
$$

where the integral is over any closed surface

ds 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 E- and the field due to $\sumint \limits _ { z = 0 } ^ { \infty }$ the positive charge $\vec { \tt E } +$ .Symbolically we can write

$$
\frac { \texttt { + } } { \texttt { E } } = \frac { \texttt { + } } { \texttt { E } } + \frac { \texttt { \ * } } { \texttt { \ * } }\tag{39}
$$

where $\scriptstyle { \frac { \pm } { \Xi } }$ is the total field

Due to symmetry, the most convenient surface to use for our current integral is the plane which forms the perpendicular bisector of the line segment joining the two equal and opposite charges. This plane is illustrated in Fig. 7a. The electric field due to the positive charge at any point p is defined by the equation

$$
\mathbf { E } + = \frac { \mathbf { Q } } { \mathbf { \phi } _ { \mathbf { 4 } } \cdot \pi ~ \in ~ \mathbb { D } _ { + } ^ { 2 } } - i \hat { \mathbf { \phi } } _ { \mathbf { D } + }\tag{40}
$$

36

[PAGE 46]
where D. is the distance between the positive charge and the $\mathbb { D } _ { + }$

point p

$i _ { \mathfrak { d } _ { \star } }$ 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

$$
\sum \limits _ { E = \mathrm { ~ \frac ~ { 4 ~ \pi ~ \varepsilon ~ \partial ~ } ~ } _ { = } } ^ { + } i _ { \mathrm { ~ \frac ~ { 5 ~ } ~ { ~ 2 ~ } ~ } } ^ { - 2 } i _ { \mathrm { ~ \frac ~ { 5 ~ } ~ { ~ 2 ~ } ~ } } ^ { 2 }\tag{41}
$$

where D is the distance from the negative charge

$i _ { \mathfrak { d } _ { \overline { { \mathbf { \Pi } } } } }$ is a unit vector at p directed by the D line away from the negative charge.

The minus sign in equation 4l 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 $\mathbb { D } _ { + }$ D are equal. They can be found from the equation

$$
D _ { + } = v _ { - } = \{ \tt R ^ { 2 } + \frac { I ^ { 2 } } { 4 } \} 1 2\tag{42}
$$

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.7a, the electrical field due to either charge has both y and radial components. Due to symnetry,however, 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 equal. Thus the total electric field is

37

[PAGE 47]
$$
\begin{array} { r } { \dot { \bar { \Sigma } } _ { \mathrm { t o t a l } } = \frac { - 2 \mathrm { ~ \mathcal { Q } ~ } } { \mathrm { ~ \mathcal ~ { ~ 4 ~ } ~ } \bar { \pi } \mathrm { ~ \in ~ } \bar { \mathcal { D } } _ { \pm } { } ^ { 2 } } \qquad \mathrm { s i n ~ \mathrm { ~ \mathcal ~ { ~ B } ~ } ~ } \hat { \dot { \mathrm { ~ \mathcal ~ { ~ z } ~ } } } } \end{array}\tag{43}
$$

We note that $B = a$ because they are opposite angles.Sin acan be determined from Fig.7a by

$$
\sin \alpha = \sin \beta = \frac { I _ { 1 } / 2 } { I _ { + } }\tag{44}
$$

Substituting equation 44 into equation $\downarrow 3$

$$
\begin{array} { r } { \dot { \bar { \mathbf { E } } } _ { \mathtt { t o t a l } } = \frac { - \textsc { e f f } } { \mathrm { i } _ { \texttt { \tiny f f f } } \mathrm { \texttt { \leq } \hat { \mathbf { U } } _ { \mathtt { f } } } ^ { 3 } } \cdot \hat { \bar { \mathbf { j } } } _ { \texttt { \textbf { y } } } \quad . } \end{array}\tag{45}
$$

Substituting equation 42 into equation 45, and equation 45 into equation 37 $\yen 2$ $\dot { 4 } 5$ yields

$$
\vec { \mathbb { J } } = \frac { \texttt { C Q L } } { \texttt { \textbf { \downarrow ~ \pi ~ \in ~ } } \left[ \mathbb { R } ^ { 2 } + \frac { \mathbb { L } ^ { 2 } } { \mathbb { L } } \right] } ~ \overline { { 3 / 2 } }\tag{46}
$$

The surface of integrution is considered to be small ring segments in the x-z plane about the y axis. $\mathtt { F i g . 7 b }$ 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 for the ring showm in $\yen 123,45$ , 7b is

$$
\exists \mathbf { s } ~ = ~ \mathbb { R } ~ \hat { \texttt { a } } \vec { \boldsymbol { * } } ~ \partial \mathbb { R }\tag{47}
$$

[PAGE 49]
where ds is the surface element,

d  is an increment of angle φ which is an angle about the y axis..

dR is an increment in the distance vector R.

Substitution of equations ${ \tt i 7 }$ and 46 into equation 38, and defining the surface integral, we obtain

$$
\texttt { I } = \texttt { \smallint } ( \begin{array} { l l } { \int } & { \frac { \texttt { Q L ( F R } } { \texttt { \texttt { q } } \pi \texttt { \texttt { c } } } } \\ { \circ } & { \frac { \texttt { Q L ( S R } } { \texttt { \texttt { q } } \pi \texttt { \texttt { c } } } } \end{array} \frac { \texttt { N } } { \texttt { \texttt { q } } } ) ^ { 3 / 2 } \texttt { \textsf { d } } \phi ) \texttt { \textsf { d } }\tag{48}
$$

Solving the inner integral yields

$$
I = \int \limits _ { 0 } ^ { \infty } \frac { q \texttt { L o R } } { 2 \texttt { \sigma } } \frac { q \texttt { L o R } } { ( \pi ^ { 2 } + \frac { q ^ { 2 } } { 4 } ) } 3 / 2 \texttt { \sigma } d \texttt { R }\tag{49}
$$

Equation 49 can be solved to yield

$$
\texttt { I } = \frac { \texttt { Q I a } } { \texttt { 2 } } ^ { * } \left[ \frac { \texttt { - 1 } } { \left( \stackrel { \gtrsim } { \mathrm { ~ \varepsilon ~ } } ^ { 2 } + \frac { \varepsilon ^ { 2 } } { 4 } \right) ^ { \frac { 2 } { 4 } / 2 } } \right] _ { \circ } ^ { \circ }\tag{50}
$$

In el

$$
I = \frac { Q \sigma } { E }\tag{51}
$$

We have previously solved for the voltage between the charges in equation 32. Dividing the voltage v by the current I, equation 51 yields

39

[PAGE 50]
the resistance as defined by equation 36

$$
\mathbb { R } = \frac { \mathbb { V } } { \ I } = \frac { \frac { \mathbb { Q } \cdot ( 1 - { \mathbf { a } } / 1 ) } { 2 \pi \textup { t e g } } } { \frac { \mathbb { Q } \cdot \mathbb { G } } { \varepsilon } } .\tag{52}
$$

which reduces to

$$
\frac { 1 } { x } = \frac { 1 } { 2 \pi \sigma _ { A } } ( 1 - \frac { a } { I } )\tag{53}
$$

We may now use the derived information. As a first approximation let us assure that the radius of each charged sphere, a, is l cm and that the length of the generator organ, L, is l0 em. The conductivity of fresh water is about $1 0 ^ { - 3 } \approx ^ { - 1 } \mathtt { a } ^ { - 1 }$ . The resistance which loads the generator organ is about $6 2 , 0 0 0 \ \circ n \cos ^ { 6 }$ The dielectric of fresh water is $0 . 7 0 7 \times 1 0 ^ { - 9 } \mathrm { f } \mathrm { d } / \pi .$ 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 justified 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 objeet on an individual receptor. As a simplification the intruding object will bé assumed to be a sphere. Such an object moving into the fish's field will modify the potentials along the fish's surface. To obtain a solution for these modifications, we will first consider the "lectric field produced by the dipole generator organ undisturbed by the

40

h. in a practical model, the resistance loading. of the fonerator can be controlled by the choice of generator elcctrode size.

[PAGE 51]
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 classification techniques by plotting our results for given receptors along the fish. It should be noted that this analysis is three dimensional and 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

$$
\vec { E } ( p ) = \vec { F } ( p ) / q\tag{54}
$$

The force on a test charge q+due to another charge Q can be found from Coulomb's Law as

$$
\vec { F } = \frac { q _ { \theta } } { \frac { \frac { q _ { \theta } } { 4 } } \pi \sum \limits _ { \varepsilon } \frac { \partial } { \partial } } \hat { \mathbf { i } } _ { \mathbb { D } }\tag{55}
$$

where E is the dielectric of the media

D is the distance between the charges

$\sum \limits _ { i _ { \frac { j } { D } } } ^ { s }$ 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

$$
\frac { \partial } { \partial t } + = \frac { 2 } { 4 \pi \varepsilon _ { y } \varepsilon ^ { 2 } } - i _ { z }\tag{56}
$$

41

[PAGE 53]
The electric field due to the negative charge is

$$
\frac { \frac { \pi } { 3 } } { \frac { \pi } { 3 } \pi ^ { 2 } } - \frac { \pi } { \frac { \pi } { 3 } }\tag{57}
$$

The electric field is a vector quantity. Thus, we must perform vector addition in order to determine the total field $\bf { \Xi } _ { \Xi } ^ { \pm } .$ 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

$$
z _ { y + } = | z _ { + } | \cos \theta\tag{58}
$$

Likewise the y component of the electric field due to the negative charge is

$$
z _ { y } = | z - 1 \cos \theta ^ { \prime }\tag{59}
$$

The x-z components of the electric fields are independent of the angle φ 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 φ from the axis. For the positive charge it is

$$
\tt { \tau } _ { \tt { z } _ { x + \tt { z } } } ^ { \tt } = | \vec { \tt { z } } _ { + } | \tt { \tau } _ { \tt { s i n } } \ e\tag{60}
$$

and for the negative charge it is

$$
\tt { E } _ { \tt X - \tt Z } = | \vec { \tt E } - | \tt \nabla \tt S i n \Theta \Theta ^ { - }\tag{61}
$$

42

[PAGE 54]
In order to treat these variables by standard mathematical techniques, it is necessary to express $\Re ^ { \star }$ in terms of R and 0, and $\theta ^ { \star }$ in terms of R and 0. Fig.8b illustrates the factors which will help us do this. Recognizing that $\bar { \Xi } ^ { \star }$ is the hypotenuse of a right triangle whose sides are R sin 0 and $\texttt { \pm } \texttt { \emptyset }$ cos $\mathfrak { G } _ { \mathfrak { s } }$ we find

$$
\tt R _ { - } ^ { - } = \ \left[ \tt R ^ { 2 } + I _ { 2 } ^ { 2 } + 2 \tt R \tt I \ c o s \ \Theta \Theta \right] ^ { 1 / 2 }\tag{62}
$$

and

$$
\theta ^ { - } = \tan ^ { - 1 } \frac { \frac { A \sin \theta } { \cos \theta } } { I + A \cos \theta }\tag{63}
$$

It is useful to note the trigonometric functions for $\Theta ^ { \star }$ . They are

$$
\sin \theta ^ { - } = \frac { \tt { R } \sin \theta } { \left[ \tt { R } ^ { 2 } + I ^ { 2 } + 2 \tt { R } \tt { I } \cos \theta \right] } \frac { 1 / 2 } { 1 / 2 }\tag{64}
$$

and

$$
\cos \theta ^ { - } = \frac { I + R \cos \theta } { \sqrt { R ^ { 2 } + I ^ { 2 } + 2 R \tan \theta } } ] ^ { 1 / 2 }\tag{65}
$$

Combining equations $\mathtt { 6 2 }$ $6 4$ , and $6 5$ with equations $5 6$ and $^ { 5 7 }$ and substituting into equations 58, 59, 60 and $6 7$ yields

$$
\mathrm {  ~ { \cal ~ E } ~ } _ { \mathrm { y } + } = \frac { \mathrm {  ~ { \cal ~ Q } ~ } \mathrm {  ~ { \bar { ~ } \cos ~ } ~ } \Theta } { \mathrm {  ~ { \bar { ~ } \mathrm { 4 } ~ } ~ } \mathrm {  ~ { \pi ~ } ~ } \varepsilon _ { _ { M } } \mathrm {  ~ { \cal ~ R } ~ } ^ { 2 } } \quad \mathrm {  ~ \cdot ~ }\tag{66}
$$

$$
\frac { z _ { \mathrm { ~ \textsc ~ { ~ k ~ } ~ } } } { \mathrm { ~ \textsc ~ { ~ k ~ } ~ } } = \frac { - \textsc { Q } \textsc { i } + \textsc { R } \cos \textsc { \theta } } { \mathrm { ~ \textsc ~ { ~ k ~ } ~ } \kappa _ { \mathrm { ~ \textsc ~ { ~ w ~ } ~ } } \left[ \boldsymbol { \mathrm { { R } } } ^ { 2 } + \mathrm { ~ \textsc ~ { ~ L } } ^ { 2 } + 2 \textsc { r } \textsc { i } \textsc { c } \cos \textsc { \theta } \right] ^ { 3 / 2 } }\tag{67}
$$

43

[PAGE 55]
$$
E _ { x = z } + \frac { Q \sin \theta } { 4 \pi \varepsilon _ { y } R ^ { 2 } }\tag{68}
$$

$$
\begin{array} { r l } { \tt \sum _ { x = z } - \mathrm { ~ \tt ~ = ~ } } & { = \tt \frac { - \tt { Q } \cdot \tt { R } \leq \tt { i n } \ \theta } { \tt \frac { 1 } { \tt { d } \cdot \tt { r } \cdot \tt { e _ { w } } \cdot \tt { e _ { w } } \cdot \left[ \tt { R } ^ { 2 } + \tt { L } ^ { 2 } + \tt { Q } \cdot \tt { R } \ { \tt { L } } \cos \theta \right] } \ ^ { 3 / 2 } } } \end{array}\tag{69}
$$

We can now determine the total components of the electric field. The y component of the electric field at any point R, θ about the dipole is independent of φ and is

$$
\tt \frac { \sigma _ { E _ { y } } } { \tt \sigma } = \frac { \psi _ { d } } { \tt \sigma _ { U } \varepsilon _ { w } } \left[ \frac { \cos \theta } { \tt R } ^ { 2 } - \frac { I + \tt { R } \cos \theta } { \tt R ^ { 2 } + I ^ { 2 } + 2 \tt { R } I \cos \theta } \frac { 3 / 2 } { 3 / 2 } \right]\tag{70}
$$

The component of the electric field in the x-z plane is radial and independent of 0. It is

$$
\begin{array} { r l } { \mathbb { E } _ { _ { X = 2 } } = \frac { \texttt { Q S i n } \oplus } { \texttt { i f f e } _ { \texttt { w } } } } & { \left[ \frac { 1 } { \texttt { R } ^ { 2 } } - \frac { \texttt { R } } { \left[ \texttt { R } ^ { 2 } + \texttt { L } ^ { 2 } + \texttt { Z R L C o s } \ominus \right] } \right] ^ { 3 / 2 } } \end{array}\tag{71}
$$

To continue our derivation, we must determine the magnitude e $| \vec { \pmb { \mathrm { \bf z } } } |$ and the angle (relative to the y axis) β of the electric field at any point in space. The magnitude may be found from the rules of vector addition as

$$
\begin{array} { r l } { | \vec { \mathbf { E } } | } & { { } = \left[ \begin{array} { c } { \vec { \mathbf { E } } _ { \mathbf { y } } ^ { \ 2 } + \vec { \mathbf { E } } _ { \mathbf { x } - \mathbf { z } } ^ { \ 2 } } \end{array} \right] ^ { 1 / 2 } } \end{array}\tag{72}
$$

44

[PAGE 56]
The angle found by studying the geometry of the situation is

$$
\beta = \tan ^ { - 1 } \frac { E _ { x - 2 } } { E _ { y } } .\tag{73}
$$

Manipulating the expressions in equations 70 and 71 per equation 72 yields

$$
\begin{array} { r l } { | \overrightarrow { \mathbf { E } } | = \frac { \mathrm { ~  ~ e ~ } } { \mathrm { ~  ~ i ~ } \pi ~ \epsilon _ { _ { \mathrm { { W } } } } ~ \mathrm { ~  ~ { ~ R } ~ } ^ { 2 } } } & { { } [ 1 - \frac { 2 ~ \mathrm { ~  ~ R ~ } ^ { 2 } ~ \mathrm { ~ \scriptstyle ( ~ R ~ + ~ L ~ c o s ~ } ~ \Theta ) } { [ \mathrm { ~  ~ R ~ } ^ { 2 } ~ + ~ \mathrm { ~ \bf ~ L ~ } ^ { 2 } ~ 2 ~ \mathrm { ~  ~ R ~ L ~ c o s ~ } ~ \Theta ] ~ ^ { 3 / 2 } } + \frac { \mathrm { ~  ~ \scriptstyle ~ ( ~ 7 4 ) ~ } } { \mathrm { ~  ~ \scriptstyle ~ ( ~ 7 4 ) ~ } }  } \\ { \mathrm { ~ \quad ~ } } & { { }  \frac { \mathrm { ~  ~ R ~ } ^ { 4 } } { \mathrm { ~  ~ \scriptstyle ~ \cdot ~ } } \frac { \mathrm { ~  ~ R ~ } ^ { 4 } } { \mathrm { ~  ~ \scriptstyle ~ \cdot ~ } } \frac { 1 } { 2 }  } \\ { \mathrm { ~ \quad ~ } } & { { }  \frac { \mathrm { ~  ~ R ~ } ^ { 2 } ~ + ~ \mathrm { ~ \bf ~ L ~ } ^ { 2 } ~ + ~ 2 ~ \mathrm { ~  ~ R ~ L ~ c o s ~ } ~ \Theta ] ~ ^ { 2 } } { [ \mathrm { ~  ~ R ~ } ^ { 2 } ~ + ~ \mathrm { ~ \bf ~ L ~ } ^ { 2 } ~ + ~ 2 ~ \mathrm { ~  ~ R ~ L ~ c o s ~ } ~ \Theta ] ~ ^ { 2 } } ~ ] ^ { 1 / 2 } } \end{array}
$$

$$
\begin{array}{c} \begin{array} { r l } { 8 = \tan ^ { - 2 } } & { \left[ \begin{array} { l l l } { \tan { \Theta } } & { \left[ \tan ^ { 2 } \Theta + \tt L ^ { 2 } + 2 \tt R \tt L \cos \Theta } & { 3 / \tt 2 } & { \tt \tt \tt - R ^ { 3 } \right] } \\ { \tt S i n \Theta } & { \left[ \tt R ^ { 2 } + \tt L ^ { 2 } + 2 \tt R \tt L \cos \Theta \right]} \\ { \tt L ^ { 2 } + \tt L ^ { 2 } + 2 \tt R \tt L \cos \Theta } \end{array}  ^ { 3 / 2 } \right]} & { \tt c o s \Theta - \tt R ^ { 2 } + \tt L + \tt R ^ { 3 } \cos \Theta } \end{array}   \\ & { \quad \tt \left[ \tt R ^ { 2 } + \tt L ^ { 2 } + 2 \tt R \tt L \cos \Theta \right] ^ { 3 / 2 } \quad \tt C o s \Theta \Theta - \tt R ^ { 2 } + \tt L + \tt R ^ { 3 } \cos \Theta } \\ & { \quad \tt \qquad \quad \ldots , \quad \tt c . . , } \end{array}\tag{75}
$$

These rather complicated equations completely describe the electric field due to the dipole in a continuous media of dielectric €. $\varepsilon _ { \mathbf { w } } .$

To obtain a first approximation of the perturbation due to a sphere of radius r at a location $\mathtt { R } _ { \odot } , \mathtt { P } _ { \mathtt { e } } \odot _ { \mathtt { o } }$ , relative to the dipole coordinates, we assume that the sphere is located within a uniform field of strength $\curvearrowright$ $\tt E _ { o }$ at an angle β 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, disregarding the effect of 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,

45

[PAGE 57]
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

$$
u _ { 0 } = - E _ { 0 } \texttt { o c o s a }\tag{76}
$$

The surface charge on the perturbating sphere is exactly like a dipole. Thus, the potential is of the form

$$
v _ { x } = \frac { r _ { 0 } s _ { a } } { p ^ { 2 } }\tag{77}
$$

where A is a constant to be determined from the boundary conditions.

Finally the potential inside the sphere is of the same form as the potential due to the original field or

$$
u _ { \tt L } = - \texttt { b p } \cos \texttt { a }\tag{78}
$$

where B is a constant to be determined from the boundary conditions.

The potential outside the sphere is the sum of U and ${ \tt u } _ { \tt o }$ ${ \mathfrak { v } } _ { \mathbf { x } } \ { \mathfrak { o r } }$

46

[PAGE 58]
$$
u _ { \pm \mathsf { o t a l } } = - \mathsf { E _ { \circ } } \circ \mathsf { C o s } \circ \mathsf { \ Q } \cdot \frac { \mathsf { A _ { \theta } } \mathsf { C o s } \circ \mathsf { a } } { \mathsf { e } ^ { 2 } }\tag{79}
$$

The electrostatic boundary conditions require that when $\beta ~ \equiv ~ \mathfrak { T }$

$$
- \ \varepsilon _ { _ { \texttt { W } } } \frac { \texttt { \texttt { U } } _ { \texttt { t o t a l } } } { \texttt { \texttt { 3 } } _ { \texttt { f } } } = - \ \varepsilon _ { _ { \texttt { X } } } \ - \frac { \texttt { \texttt { u } } _ { \texttt { I } } } { \texttt { \texttt { s } } _ { \texttt { 0 } } }\tag{80}
$$

and that when $\rho \ r = \tt { r }$

$$
\mathtt { U } _ { \mathtt { t o t a l } } = \mathtt { U } _ { \mathtt { I } }\tag{81}
$$

Substituting equations 78 and 79 into equatior 83 $\tilde { s } \tilde { \mathcal { \Sigma } } ^ { \star }$

$$
z _ { 0 } = r ^ { 3 } - A = B r ^ { 3 }\tag{82}
$$

Substituting equations 78 and 79 into equation 60 yields

$$
\varepsilon _ { y } \varepsilon _ { 0 } + 2 A \varepsilon _ { y } = r ^ { 3 } \varepsilon _ { z } B\tag{83}
$$

We can solve for A on B in equation 82 and 83 to find

$$
\tt { \tt \tt _ { B } = \frac { 3 \varepsilon _ { w } } { \varepsilon _ { x } + 2 \varepsilon _ { w } } \varepsilon _ { z } }\tag{84}
$$

and

$$
A = \frac { \epsilon _ { x } - \epsilon _ { y } } { \epsilon _ { x } + 2 \epsilon _ { y } } - r _ { 0 } ^ { 3 } r _ { 0 }\tag{85}
$$

47

[PAGE 59]
The potential ${ \tt u } _ { \circ }$ was assumed to be the potential due to the dipole 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 potential U which is the approximate modification of the dipole field due to the $\mathfrak { v } _ { \mathbf { x } }$ spherical object. Substituting equation 85 into equation 77 yields

$$
\frac { \varepsilon _ { x } = \varepsilon _ { x } - \varepsilon _ { y } } { \varepsilon _ { x } - 2 \varepsilon _ { y } } \frac { r ^ { 3 } } { p ^ { 2 } } \mathbb { E } _ { 0 } ( \cos \alpha\tag{86}
$$

the perturbation potential.

We recall the value of $\mathtt { z } _ { \circ }$ to be the value of equation 74 when $R = \frac { R } { 0 }$ and when $\theta = \theta _ { 0 }$ or

$$
E _ { _ 0 } = \frac { Q } { 1 + \pi ~ { \epsilon _ { _ { \mathsf { W } } } } ~ \mathsf { R } _ { _ 0 } } ^ { 2 }  \\ { \mathbb { E } _ { _ { Q } } = \frac { Q } { 1 + \pi ~ { \epsilon _ { _ { \mathsf { W } } } } ~ \mathsf { R } _ { _ 0 } } ^ { 2 } } [ \mathbb { 1 } - \frac { 2 ~ \mathsf { R } _ { _ 0 } ^ { \mathrm { ~ 2 ~ } } ~ ( \mathbb { R } _ { _ 0 } - \mathbb { L } ~ \mathbb { C } \circ \mathbb { s } ~ \mathsf { \Gamma } \circ _ { _ 0 } ) } { [ \mathbb { R } _ { _ 0 } ^ { \mathrm { ~ 2 ~ } } + \mathbb { L } ^ { 2 } + 2 ~ \mathsf { R } _ { _ 0 } ^ { \mathrm { ~ 2 ~ } } ~ \mathbb { C } \circ \mathsf { s } ~ \mathsf { \Gamma } \mathsf { } \mathsf { \Gamma } \mathsf { } \mathsf { ) } _ { _ 0 } ] ~ { ^ { 3 / } } _ { 2 } } ~ +  \mathbb { E } ( \mathbb { R } _ { _ 0 } + \mathbb { L } ^ { \mathrm { ~ 3 ~ 2 ~ } } \circ \mathbb { R } _ { _ 0 } - \mathbb { L } ) \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 2 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 2 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 2 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 2 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 2 } \mathbb { 1 } \mathbb { 1 } \mathbb { 1 } \mathbb { 2 } \mathbb { 1 } \mathbb { 1 } \mathbb { 2 } \mathbb { 1 } \mathbb { 1 } \mathbb { 2 } \mathbb { 1 } \mathbb { 1 } \mathbb { 2 } \mathbb { 1 } \mathbb { 1 } \mathbb { 2 } \mathbb { 1 } \mathbb { 1 } \mathbb { 2 } \mathbb { 1 } \mathbb { 1 } \mathbb\tag{87}
$$

$$
\frac { \texttt { \texttt { R } } ^ { 4 } } { [ \texttt { \^ { R } } ^ { 2 } + \texttt { L } ^ { 2 } + \texttt { 2 } \texttt { \^ { R } } \texttt { \^ { L } } \texttt { \^ { C o s } } \theta _ { 0 } ] ^ { 2 } } ] ^ { 1 / 2 }
$$

and we recall that a is measured relative to an angle β which is equation 75 evaluated at $\pi _ { o }$ $\theta _ { \circ }$ or

$$
\begin{array}{c} 3 = \tan ^ { - 1 } [ \frac { \sin \theta _ { 0 } } { \tan ^ { 2 } \theta _ { 0 } } [ [ \begin{array} { c c } { \mathrm { ~ , ~ } } \\ { \mathrm { ~ } } \\ { \sin ^ { 2 } \theta _ { 0 } } \\ { \mathrm { ~ } } \\ { \mathrm { ~ } } \\ { \sqrt { \mathrm { ~ \frac { 8 ~ } { ~ } { ~ } ^ { 2 ~ } + ~ L ^ { 2 ~ } + ~ 2 ~ \mathcal ~ { ~ R } _ 0 ~ L ~ c o s ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } ^ { 3 / 2 } } } & { 3 / 2 } \\ { \mathrm { ~ } } \\ { \mathrm { ~ } } \\ { \mathrm { ~ } \tan ^ { 2 } \theta _ { 0 } } \end{array} ] ^ { 3 / 2 } \mathrm { ~ c o s ~ } \mathrm { ~ } \tan ^ { 2 } \mathrm { ~ } \mathrm { ~ } _ { 0 } ^ { 2 } \mathrm { ~ }  \end{array} ] ,
$$

48

[PAGE 60]
To complete our analysis, we must express p and α in terms of R, θ, φ, $R _ { a }$ , and $G _ { \odot } .$ We find

$$
\begin{array} { r l } { \mathsf { \Lambda } _ { \odot } ^ { \mathsf { ^ { * } } } = } & { \left[ \left[ \mathsf { R } \mathsf { \Lambda } _ { \odot \texttt { \Theta } \sim \textsc { \tiny R } _ { \odot } \texttt { \mathsf { C o s } } \ h _ { \circ } } \right] ^ { \mathrm { ~ \texttt ~ { ~ 2 ~ } ~ } } + \left[ \mathsf { R } _ { c } \texttt { \tiny a i n } \Theta _ { \circ } - \texttt { \tiny R S i n } \Theta _ { \mathsf { C o s } } \left( \phi - \phi _ { \circ } \right) \right] ^ { \mathrm { ~ \texttt ~ { ~ 2 ~ } ~ } } + \right. } \\ & { \left. \left[ \mathsf { R } \texttt { \textbf { S i n } } \Theta \mathrm { \texttt { S i n } } \left( \phi = \phi _ { \circ } \right) \right] ^ { \mathrm { ~ \texttt ~ { ~ 2 ~ } ~ } } \right] ^ { \mathrm { ~ \texttt ~ { ~ 1 / 2 ~ } ~ } } \quad \quad ( \ P _ { \sigma } \operatorname { ^ { \prime } } \left( \phi _ { \circ } \phi _ { \circ } \right) \phi _ { \circ } ) ^ { \mathrm { ~ \texttt { ~ ~ 2 ~ } ~ } } } \\ & { \quad \quad \quad \left. \left( \mathbb { R } \lambda \mathrm { ~ i n ~ } \theta \mathrm { ~ \mathbb { ~ S i n } ~ } \left( \phi = \phi _ { \circ } \right) \right] ^ { \mathrm { ~ \texttt ~ { ~ 2 ~ } ~ } } \right] ^ { \mathrm { ~ \texttt { ~ ~ 1 / 2 ~ } ~ } } } \end{array}\tag{89}
$$

this can bé reduced to

$$
\begin{array} { r l } { \rho = } & { { } \tt R ^ { 2 } + \tt R _ { 0 } ^ { 2 } - 2 \tt ~ R ~ \tt R _ { 0 } ~ \left[ \tt C o s ~ \tt \otimes _ { \tt O } \tt ~ \tt C o s ~ \tt ~ \otimes - \tt ~ \tt S i n ~ \tt \otimes _ { \tt O } ~ \tt S i n ~ \tt \otimes ~ \tt C o s ~ \tt ~ ( \tt \otimes - \tt ~ \tt \otimes _ { \tt O } ) \right] ^ { ~ \mathrm { 1 / 2 } } } \end{array}\tag{90}
$$

We can solve for a in terms of R, θ, $\Game$ , B, and $\circ$ to be

$$
a = 3 6 0 ^ { \circ } - [ \sin ^ { - 1 } \frac { 8 \sin ( \theta - \theta _ { 2 } ) } { \theta } ] - \theta _ { 0 } - \theta\tag{91}
$$

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 $\Im \{ 4$ when $R ^ { \prime } + R _ { 0 }$ and $\Theta \equiv \Theta _ { \pmb { \alpha } }$ .Equation $7 4$ becomes equation $8 7$ which is assumed linear throughout the perturbation. The magnitude of the electric field is

$$
\begin{array} { r } { \bar { \bar { \mathbf { E } } } _ { 0 } = \frac { \mathsf {  ~ Q ~ } } { \bar { \mathrm { ~ I ~ } } _ { 0 } \bar { \textbf { \tiny { \wedge } } } \bar { \mathbf { e } } _ { \breve { { \mathbf { { Q } } } } } ^ { 2 } } \left[ 1 - \frac { 2 \mathrm {  ~ ~ R ~ } _ { 0 } ^ { 2 } \bar { \mathrm { ~ \tiny ~ \left( R ~ _ 0 ~ + ~ L ~ C o s ~ \right)} ~ \bar { \mathsf { \Theta } } _ { 0 }  } } { \left[ \bar { \mathrm {  ~ R ~ } } _ { 0 } ^ { 2 } + \mathrm {  ~ L ~ } _ { 0 } ^ { 2 } + \mathrm {  ~ R ~ _ 0 ~ L ~ C o s ~ } ~ \bar { \mathsf { \Theta } } _ { 0 } \right] ^ { 3 / 2 } 2 } + \frac { \bar { \mathrm {  ~ \Lambda ~ } } _ { 0 } ^ { 2 } } { \bar { \mathrm { ~ I ~ } } _ { 2 } ^ { 2 } } \right. } \\ { \left. \qquad \frac { 2 \mathrm {  ~ R ~ } _ { 0 } ^ { 4 } } { \bar { \mathrm { ~ I ~ } } _ { 0 } ^ { 2 } + \mathrm {  ~ L ~ } _ { 0 } ^ { 2 } + \mathrm {  ~ R ~ _ 0 ~ L ~ C o s ~ } ~ \bar { \mathsf { \Theta } } _ { 0 } } \right] ^ { 1 / 2 } } \end{array}
$$

49

[PAGE 61]
The angle of this field relative to the y axis in the $\sum \limits _ { 0 } - y$ axis plane is given by equation 75 which for $R = R _ { 0 } , \ \Theta = \ \Theta _ { \circ }$ becomes

$$
\theta = \tan ^ { - 1 } \left[ \frac { \sin \theta _ { 0 } } { \displaystyle \left[ \hat { \lambda } _ { 0 } ^ { \mathrm { ~ 2 ~ } } + \mathrm { ~ L ~ } ^ { 2 } + \mathrm { ~ 2 ~ } \hat { \kappa } _ { 0 } ^ { \mathrm { ~ ~ 2 ~ } } \cos \theta _ { 0 } \right] ^ { 3 / 2 } \mathrm { ~ c o s ~ } \Theta _ { 0 } } \right] ^ { 3 / 2 } - \mathrm { ~ \frac { \kappa _ { 0 } \kappa _ { 0 } ^ { ~ 3 } } { \kappa } ~ } \Bigg ] \qquad \mathrm { ~ , ~ }\tag{93}
$$

This field creates a pertu:bation field in a remote $\tt o b j$ ect of radius r and dielectric $\varepsilon _ { \mathbf { x } } ,$ The potential of the perturbation field is given by equation 86 which is

$$
u _ { x } = \frac { e _ { x } - e _ { y } } { e _ { x } + 2 e _ { y } } \frac { x ^ { 3 } } { p ^ { 2 } } s _ { 0 } \cos a\tag{94}
$$

Unfortunately, p and α are in terms of a secondary coordinate system. Our primary coordinate system is R, θ,φ. Equation 90 expresses ρ in terms of $R , R _ { 0 } , \ldots \varnothing , \varnothing _ { 0 } , \ldots \varnothing$ ,and $\$ 0$ . This is the first place where the perturbation angle $\phi _ { 0 }$ is important. The expression for ρ is

$$
{ \boldsymbol { \mathfrak { p } } } = \left[ { \texttt { R } } ^ { 2 } * { \texttt { R } } _ { 9 } ^ { 2 } - 2 { \texttt { R } } { \texttt { R } } _ { 9 } \quad { \texttt { C o s } } \otimes _ { 9 } { \texttt { C o s } } \otimes - { \texttt { S i n } } \otimes { \texttt { S i n } } \Theta _ { 0 } { \texttt { C o s } } ( \phi - \phi _ { 0 } ) \right] \left[ { \begin{array} { c } { 2 / 2 } \\ { 3 / 2 } \end{array} } \right] ,\tag{}
$$

The angle a can be expressed in terms of $\ R , \ R _ { 0 } , \ L _ { 0 } , \ L _ { 0 } , \phi$ , and $\$ 0$

However, it is simpler to express it as

$$
a = 3 6 0 ^ { \circ } - [ \sin ^ { - 1 } ( \frac { 8 \sin ( \theta - \theta _ { 0 } ) } { \theta } ) ] - \theta _ { 0 } - \theta\tag{96}
$$

The undistorted potential about the fish is the negative gradient of the sum of equations 70 and 71 or

50

[PAGE 62]
$$
v _ { 0 } = \frac { Q } { \frac { 1 } { 4 } \pi \varepsilon _ { \texttt { w } } } \left[ \frac { 2 } { \texttt { R } } - \frac { 2 } { \left[ \texttt { R } ^ { 2 } + \texttt { L } ^ { 2 } + 2 \texttt { R L } \cos \theta \right] ^ { 1 / 2 } } \right]\tag{97}
$$

Speaking anthropomorphically, the fish knows ${ \mathfrak { U } } _ { \circ } , \ { \mathfrak { R } } , \ { \mathfrak { L } } , \ { \mathfrak { Q } } , \ { \ \Phi } , \ { \ Q } .$ and 4  E\_. It must determine $\varepsilon _ { \mathrm { w } }$ $\mathbb { R } _ { \circ } , \ \in _ { 0 } \ \oplus _ { 0 } , \ \pmb { \tau }$ and $4 \pi \varepsilon _ { _ { \mathbf { X } } } .$ To find these variables, it makes an analysis of. the potential function $\mathbb { I } _ { \circ } + \mathbb { I } _ { \mathbf { x } }$ . Exactly how this analysis is made is unknown.

As an approach to determine how the fish might operate, we shall find the ratio $\mathbb { U } _ { \mathbf { x } } / \mathbb { U } _ { 0 }$ at different points along the fish for different values of $\varepsilon _ { \mathbf { x } } , \mathrm { ~ } \mathbf { r } , \mathrm { ~ } \mathcal { R } _ { \mathbf { \Phi } _ { \mathbf { Q } } } , \mathrm { ~ } \mathcal { Q } _ { \mathbf { \Phi } _ { \mathbf { Q } } }$ , and $( \phi - \phi _ { \circ } )$ . The equations are rather 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 of 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 discussed in terms of generator organ lengths. The dielectric of the perturbating object has been expressed in terms of the dielectric of water. In other words the analysis is in terms of $E _ { w } / E _ { \mathbf { x } }$ rather than $\varepsilon _ { _ { \textrm { X } } } \circ \tau { \mathrm { ~ \tt ~ E ~ } } _ { _ { \textrm { Y } } }$ themselves.

The fish is defined as a cylinder two times as long as the electric generator organ, with a radius 0.2 times the generator length, Six longitudinal bands of eleven receptors are assigned along 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.

51

[PAGE 63]
The computer program analysed various combinations of four factors:

1) the ratio of $\varepsilon _ { \mathbf { w } } / \varepsilon _ { \mathbf { x } }$ 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, (φ - φ)the 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. In

the actual printout, this angle was taken for each of six receptors defined

by the same radius r and angle 0 $\{ \mathbb { U } _ { \mathbb { X } } / \mathbb { U } _ { \mathbb { O } } \}$ 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).

52

[PAGE 64]
l0 times water as it is for objects with dielectrics 0.l times water.

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 million (humans can objects of dielectric 0.l times water whose radius is 0.l generator organ lengths, at distances to the side of 6 generator lengths. It would have difficulties at 10 lengths or for dielectrics of 0.5 or 5 times water.7

With the assumed sensitivity, the fish could detect objects of dielectric 0.1 water, assuming object radius is 0.1 generator organ lengths at 10 electric organ lengths if approached from front or rear.

Objects of dielectric l0 times water could be detected at a considerable angle from the direct front or rear approaches at l0 lengths.

An indication of receptor sensitivity for objects of dielectric l0 times water and O.l water with a radius of one length is given in table I.

The analysis shows that the signal is about twice as large for dielectrics

7. The diclectric of water is approximately 81, plastics are about 8, air is about l, and metals would be extremely high, virtually infinite in many cases.

[PAGE 65]
Systen 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 elassify 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 unimodal curve (Hagiwara & Morita, 1963). We might suggest that the configuration of the curve is a function of the overall impedence of the perturbing object, defining impedance as the sum of the resistance and the reactance of the object. The reactance of an object is given by

$$
x = 2 \pi \operatorname { F } \frac { 1 } { 2 } \frac { 1 } { \pi \operatorname { F } \operatorname { c } }\tag{98}
$$

where X is reactance

F 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 discrimination and classification ability by varying its frequency; making the object create a greater or lesser potential gradient at the receptor. Thus, if

54

[PAGE 66]
the fish was trying to locate a certain known object, it could adjust its

frequeney 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 frequeney 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

investigators. 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

objects in various size bodies of water we have gathered data which, when

55

[PAGE 67]
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 l0 electric organ

lengths about the fish. When the fish must be restrained near the surface,

l0 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 deseribing 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 analysis, we

56

[PAGE 68]
determined the sensitivity to various perturbing objects under a variety of conditions and found the fish to be quite sensitive particularly in certain directions.

57

[PAGE 69]
REFERENCES

58

[PAGE 70]
59

[PAGE 71]
60

[PAGE 72]
APPENDIX

Symbols

The angle between the Y-axis and the radius vector from the positive charge to an arbitrary point in space.

0- The angle between the Y-axis and the radius vector from the negative charge to an arbitrary point in space.

$\Theta _ { \alpha }$ The angle between the Y-axis and the radius vector from the charge to the center of the perturbing object.

R 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.

$\mathtt { a } _ { \mathtt { o } }$ The radius vector from the positive charge to the center of the perturbing object.

中 The angle between the X-axis and the X-z projection of the radius vector from the positive charge to an arbitrary point in space

The angle between the X-äxis and the X-z projection of the radius vector from the positive charge to the center of the perturbing object

Q The charge on either side of the dipole.

L The length of the dipole.

$\varepsilon _ { _ { \omega } }$ The dielectric of the water.

$\varepsilon _ { \mathbf { x } }$ The dielectric of the perturbation.

r The diameter of the spherical perturbing object.

$\vec { \mathtt { E } } _ { \circ }$ The electric field at the center of the perturbation as if the perturbation were not present.

61

[PAGE 73]
$\mathtt { u } _ { \mathbf { x } }$ The potential due to the interaction between the dipole field and the perturbation.
W A coordinate centered at Mc perturbation opposed to the vector $\mathtt { E _ { o } }$ and in the Y axis $- R _ { 0 }$ plane.
m A coordinate perpendicular to the w axis in the $Y - P _ { 0 }$ plane originating at the center of the perturbation.
B The angle between $\mathtt { E _ { \circ } }$ and a line parallel to the Y-axis at the center of the perturbation.
p A radius vector in the w, m coordinate system to an arbitrary point in space.
a The angle between the radius vector and the coordinate v.
$\vec { \tt { E } } +$ The electric field due to the positive charge.
$\hat { \mathfrak { i } } _ { \mathfrak { x } }$ Unit vector in the direction YR
$\sumint \limits _ { E } ^ { \infty } d x$ The electric fieid due to the negative charge.
$i _ { \frac { j } { \sum _ { j } } }$ Unit vector in the direction of $\Re ^ { \ast }$
$\Xi _ { \mathbf { x } }$ The X-z component of the"electric field. - z
$u _ { 0 }$ Potential due to uniform approximation of the dipole field.
${ \mathfrak { u } } _ { 1 }$ Potential inside sphere due to uniform approximation of dipole field.
$\mathbf { E } _ { \mathbf { y } }$ The Y component of the electric field
$\mathbf { y }$ End point of charge path. Radius of sphere in dipole arrangement.
I Current
P Macrcscopic polarization
N The number of molecules per volume.
$\mathfrak { P } _ { \mathtt { m o s } }$ Related to the macroscopic polarization (electric dipole moment per volume).

62

[PAGE 74]
$x _ { p }$ The electric susceptibility.

E Storing force.

X End point of charge path

$\bar { \bf P } _ { \mathrm { i n d } }$ Induced dipole moment.

T Threshold function.

$\mathbb { T } _ { \mathtt { m a x } }$ Maximum threshold.

$\mathbb { T } _ { \mathtt { m i n } }$ Minimum threshold.

$\ I _ { c }$ Exponential time constant.

q Charge

y+ Dummy variable (positive charge).

y- Dummy variable (negative charge).

y Absolute voltage.

V Voltage

C Capacitance

J Current density.

D Distance between charges

A A constant to be determined from the boundary conditions.

B A constant to be determined from the boundary conditions.

X Reactance

L Inductance

U Potential in terms'of the coordinate system.

$w _ { 0 }$ Radian frequency of harmonic oscillation.

J Reactance in polar corrdinates.

63