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. . ON THE NATURE OF ELECTROSENSING IN THE FISH

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August 1971

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ABSTRACT

An evaluative review of the electrosensing literature was carried out vith the intention of determining the nature of the electrosensing mechanism ard its sensitivity. It was found that the biological data base was wenk. It vas, hovever, 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 vorking bypothesis on the nature of the sense mechaniam. We also collapse the various sensor coding schemes that have been proposed into one scheme. The fumnction of the mathematical model of the sensor that vas developed vas expiored vith the use of a computer. The fishes' function at the syster level was also considered and possibls mechanisms defined.

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TABLE OF CONTENTS

ABSTRACT
TRODUCTIO
NATURE OF THE BIOLOGICAL SYSTEM.... 2
Generator Orga
Receptor Organ
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.
Syste Lvel..54
CONCISIO5
REFERENCES58
APDI1

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INTRODUCTION

It has only been a short time since certain fish were identified as having a previously unknon sensing system, an electrosensing system. It vas observed that these fish apparently detect and classiry objects that enter into and perturb a weak electrical field that the fish itself generates. 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 owm electrical field. Rather, it seems'to detect electrical signals, possibly muscle potentials, generated by objects coming into its area.

Although there is ncv a fairly substantial data base, ve 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 :ompetent the investigators. Further, the data base contains very little behavioral data. Thus, there is little information available on system sensitivity and fumction.

In sum, though there are individual investigators contributing quits useful data to the data base, as a vhole the data base is weak. Thus, ve 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 vork vith electrical fields, sensors, and objects in various size bodies of vater ve 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 specirication for tank size,

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fish location, and attachments, that will yield valid data in future studies.

Second, we havt suggested as a vorking bypothesis 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 nev area of investigation. The hypothesis mey also provide a basis for analyzing higher interactions in the fishes' nervous system and thereby increase our understanding of the sense.

Third, we indicate in the folloving 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 useable experimental data. A set of equations describing fumction is developed on the model. These equations are linked to available experimental dats. The mathemstical model is analysed by a computer to ascertain the sensitivity requirements of the fish at the receptor and to determine the effests of manipulating a Laber of variables. These variables include fish size, object size, objest electrical characteristics, object distance from the fish, directics and angle of. the object from the fishes' axis, etc.

We briefly discuss the fishes' function at the systems level and close with our concluaions concerning the electric sense.

NATURE OF THE BIOLOGICAL SYSTEM

Both marine and fresh water species of strongly and veakly electric fish have evolved. Strongly electric fish are defined as those that discharge their electric generating organs reactively to stun prey or resist capture. Weakly electric fish are defined as those that detect and classiry objects by the object perturbing the electrical field formed by the electric

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generating organ which normally emits a continuous pattern of pulses. The electric fieli 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 vhich 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 electrical rield generating organs and receptor organs. Other veakly 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 funcuion of the electrical field generator organ is of importance in understanding receptor function. Thus, generator fumction vill 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, l966) ,the same type of embryonic tissue as muscle except in the South American family Sternarchidae. The origin of the electrocytes of the sternarchids is the same embryonic tissue from which the neural system is derived, the ectoderm(Steinbach, 1970).

Electrocytes of mesodermal origin are typically disc shapei, but may also be drum shaped cr tubular. Electrocytes of ectolermal origin are U shapeù processes from the spinal cord. The electrocytes of the gymnotid, Hypopomus, are between 300-50n  diameter and about 200 μ thick. The electrocytes of Sternopygus on the other hand are: rod-shaped and much longer

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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 tear portion of the fish's body to form the electric generating organ. For example, the electric organ of Gnathonemus, a mormyrid, is located just in front of the tail fin and extends forward less than 1/5 of the fish's body length. Gymmarchus' 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 freshvater fish can be categoi'zed in terms of patterns of discharge: those with variable frequency and those with constant frequency. Constant frequency fish are defined as those that discharge their electric generating organs at a virtually constant rate evem when strongly stimulated by an experimenter. Some of these are Eigenmannia, Sternopygus, and the sternarchids. These differences are not absolute, hovever, and there are species differences in basic rate. The generating organ of the mormyrid Gnathonemus for example, is reported (Bennett, l970) to discharge at frequencies of 30-l00 pulses per second (pps). Gymnarchus is reported to discharge at a frequency of about 250 pps; Gymmotus has a frequency rate of 40-60 pps; Eigenmannia emits pulses at a rate of 250-k00 pps; Sternopygus fires at 60-l00 pps; Steatogenys emits:pulses at 40-60 pps; ard Hypopomus at 2-20 pps (Hagiwera and Morita, 1963).1 Sternarchids discharge at

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1. Each type of fish has a waveform that is specific to itself. Therefore, although Gymnotus and Steaiogenys have the sam: frequenciss, their vavelorms are different. These differences in vaveform may be functions of the experimenters' compctence in engineering.

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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 Gymmotus (Larimer and McDonald, l968). 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, 1968). In this context, also, is the observation that Gymnarchus temporarily ceases its discharge entirely when presented vith a signal mimicking another Gymnarchus or when startled (Bennett,1970).2

The mechanisms for controlling electric organ output are in the medullary portion of the brain and appear to be similar among veakly electric fish. A small group of cells in the medulla are autoactive and fire syuchronously, apparently acting as a pacemaker. Their discharge appears to trigger another group of cells in the medulla commonly referred to as medullary "relays". Axons from the medullary relay cells descend as part of the spinal cord to synapse on spinal relay neurons.: These in turn cormumicate the signal to the electrocytes. The electrocytes of the eiectric generating organ fire synchronously because of one or more compensatory mechan. isms in the relay patkway from the pacemaker cells.' One mechaniam is variation in length of the pathway to the electrucytes.i The axons to the more distant electrocytes -extend in the straightest possible line but those to the less distant electrocytes fcllov a circuitous patter.l. A second means of maintaining synckronization involve a delay line mechanism whereby the pathvays to the electrocytes differ in conduction velocities.

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

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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 vater is reported to generate a voltage of from l0 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 betveen the electrodes was not given nor was the distance betveen the electrodes and the fish given. In general, we find that inadequate information is given in the reports of voltage measw'ements of the electric organ output.

Based upon the inadequate iuformation that is reported on voltage measurements and upon measurements that ve have made in water, ve would suggest ignoring the measurements reported in the literature. In measurements in our laboratory simulating the reported data, ve found that the vater acts as a very high distributive resistance. When an oscilloscope is usei 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 sater. Even vhen a high input impedence scope is used, there is a loading effect upon the circuit. Thus, we believe, based upon cur measurements and the reported investigations, that the investigators have been inadvertently loading dowm the fish's electric field generator through the use of their measuring devices.

We can summarize thè saiient points by saying that these fish generate a pulsed electrical field in the vater. The generacor is located in the posterior portion of the body. The generator components have their outputs synchronized by a clock. In some species the clock is more or less invarient, in others it varies, in part, as a function of external events. The reason

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for this difference among species is unknowm. The voltage output of the generator and the efrective range of the field are unknowm due to inadequate measurement technique.

Receptor Urgan

The veakly electric freshwater fish are reported to have both active and passive sensory systems. The active syetem primarily detects disturbances in the fish generated E field. The passive system is primarily sensitive to energy provided by extrinsic sources. We are not so sure that the data really indicates two such systems in the same fish, but ve shall follov the convention for the time being. There is better evidence that there are a mumber 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.

Gymmotid 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 lover rate in response to the electrical sources moving into the fish's range. The response to a brief stimulus, for exauple, is acceleration folloved by deceleratior. The acceleration phase can outlast the stimulus and according to Bennett (1970) therr is accomodation to maintained stimuli. These receptors are sensitive to lov frequency electrical fisids and to changes in a DC field.

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Their response to ar applied current is a monotonic increase.

The active system sensors are called tubercus organs. They are more rapidly adapting than tonic receptors. They are Eensitive to relatively high frequency stimuli and are insensitive to applied DC. Their firirg 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 tubercus organs di?fer. They 1 also differ in appearance from mechanorec:ptors, i.e., canal orgens and free neurom: ts. The tuberous organ appears on the skia surface as a single small pore, even though it has no opening. The ampullary organs appear as a group of small pores. As an indicaticn of the number of receptors found on a fish, it can be noted that Lissmanr and Mullinger (1s68) found that there were ?.c00 ampullary and tuberous organs on a 6 cr. long Steatogenys. Most receptors, about 95 percent, are phasic receptors according to Lissmann and Mullinger (1968).

In considering the fine structure or the receptor organs, it cn be noted that the ampullary organ has the appearance of a flask with a narrow duct (5-20  in diameter) leaiing from the skin surface to a cavity (30- 40 u in 'diameter) that is located l00-50C u within the skin. Embedded in the cavity wall with only a small surface exposed are the sensing cells of the organ. These sensing cells are l0-15 u in diameter vith each organ containing twc to eight of them.. Some microvilli 0.8 u long are irregularly distributed on the exposed surface of the sensing cells. Filling the duct and cavity is a jelly-like substance vith no known function. All sense cells in one organ feed their signals to the same myelinated nerve fiber. The nerve is unmyelinated vithin the organ, having lost its myelin sheath and dividing before entering the orran.

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C LOOSE EP!THELIAL CELLS COVERING CELLS CELLULAR LAER OF CAPSULE B RECEPTOR CRLL MERVE TERMINAL SUPPORTING CELL NERVE FIBER

Fig.l a) Schematic drawing of the two types of ampulla of gymnotids, b) Schematic drawing of the tuberous organ of the gyunotid, c) Schematic drawing of the zormyromast of the morwyrid, d) Schematic drawing of the tuberous organ of the mormyrid.

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There are a great many clusters of five to fifteen ampullary receptor cells on the head. On the body there are fever clusters and they tend to be restricted to 3 bands that extend longitudinally along the fish.?

The tuberous organ consists of a bulb shaped invagination of the skin as shown in Fig. lb. The side of the bulb is composed of 10 to 50 layers of flattened cells for a.total thickness of 2-5 u. The bottom ur the bulb is made up of supporting cells upon which the numerous sensing cells rest. The sensing cells are 25-30 u long and project somevhat like rods into the cavity of the bulb. They are ordered such that the gap between adjacent sensory cells is relatively constant. Each sensory cell is covered on the cavity end with microvilli 0.7 u long. The cavity is filled with a fluid or possibly jelly-like substance. Loose epithelial-like cells fill much of the cavity above the sensory cells and appear to plug the pore to the surface The sensory cells feed their signals to a single nerve which, in most cases, loses its myelin shesth where it pusses 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 distributed on the head, where they are most numerous, and on the anterior halr 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 alvays separately innervated.

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

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The type A sensory cells form one or tvo concentric aureoles at the base of a "jelly sphere" located near the surface of the skin as showa 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 aro located. The inner surface of the duct wall bears tiny microvilli. The duct as well as the lower censory chamber is filled with amucous-aubstance.

Tvo to five sensory cells occupy the lover sensory chamber. The type E cells with their supporting cell platform though similar to the tuberous crgan 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 vhich 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 vithin 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 spical 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 celis to contact the sensory cells. As with type B cells, where the nerve joins the sensory cell, there is a rod present at the sensory cell membrane.

The mormyrids also have receptor organs, knollenorgans, vhich are somewhat similar to the tuberous organs of the gymnotids. Derbin and Szabo (1968) describe them as being cmposed of three or four sensory cell complexes one of which is shown in Fig. ld. Each complex is a single sensory cell attached to a highly differentiated supporting platform of cells. The organ is innervated by single nerve fiber which is derived from a nerve that appears to

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serve many sensory cells.

The sensory cell lies in and almost coupletely fills a cavity in the skin at the surface. The wall of the cavity is formed by flattened.epithelial cells. The interior epithelial cells have microvilli-like procr:eses which densely pack the space about the sensory cell. The cavity has a relatively large opening toward the supporting cells through which the sensory cell contacts the·nerve endings and supporting cells. The sensory cell itself is 35-40 μ in diameter.

In sum then, the weakly electric fishes of South Americs, 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 vithin and between species, the evidence suggesting that there are differences in function is rather veak. We shall now consider this matter of the receptor organ and eystem function. Systex Function, Measurement Technique,and Sensitivity

This section vill of necessity be short since there'is relatively little dats which is acceptable from both a biological and engineering standpoint. Thus, ve 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 techn:ques are electrophysiological and the third is behavioral.

Electrophysiological and behavioral techniques. In one electrophysiological techniçue the fish is anesthesized and fixed to a wooden plate in the zormal swimming position. The vocden plate is then tilted into the water so that the body is submerged and the head exposed to the air. The regular respiratory movements and oxygenation are muintained by eprayirg a fine jet of water into the mouth of the fish. The dorsal branch of the lateral line

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nerve which lies immediately under the dorsal skin at the head, is then. surgicaily exposed. After desheathing it, fine nerve strands are separated by microdisection. Then silver-silver chloride electrodes are applied to a strand and single uerve fiber responses are recorded under various stimulation conditions.

The other electrophysiological technique involves zestricting the fiah's movement by placing it in 3 to 5 inches of vater in a small glass or plastic tank. Electric discharges are then detected vith monitoring equiument connected to the vater via electrodes suspended in the:experimental tanks.

The data obtainea by the above provide insight into system operation but are not very useful in evaluating the function or sensitivity of receptors c systems. First, in those cases vhere anesthesia vas used, a question can be raised on the erfect of the anesthesia-en reural function. Second, the investigators were looking only at the isolated sensor :signal under grossly abnormal stimulation conditions. Third, the isolated sensor data, even if collectei undur reasonably normal stimulation conditions reveal little about system function. Fourth, the engineering is typically questionable for one reason or another. This fourth reason is also the prime problem vith the yet to be described behavioral technique. For example, Agalides (1965) did extensive vork on these fish, much of it being excellent. However, he used a small tank which would distort the fishes' field, he did not control impedence vithin normal limits, & he had extraneous objects in the fishes' field. Clark, Granath, Mincoff & Sachs (1967) used stainless·steel electrodes vhich distorted the fishes' field. Hagivara, Szabo, Enger & Suga (1965, 1967) all show vaveforms in their reports which appear to be riding on an increasing DC potential. It appears as though their electrodes undervent a significant polarization during the experiment. The experimentors will not offer an explanation for this observation. Mandriota's investigatinns (1965) are characterized by very poor experimental technique

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Not only did he use silver electrodes, a small tank, etc, but he used as a pwiishment with his behavioral training technique an electrical shock sufficient to visibly Jerk the fish; shock vhile 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 ansver the question of sensitivity. In this technique, the free swimming fish is conditioned to respond to s certain stimlus. When it responds correctly it is rewarded. The stimuli used hav. 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 foundi can best be sumed up by stating that the fish couid detect the presence of a glass rod 2 millimeters in diameter in a clay pot but vould fail to respond to a glass rod of 0.8 millimeters in diameter 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 questioned since the tank used does not meet the specifications derived below.

Somevhat ukin to this behavioral technique have been a limited mumber ot data gathering expeditions into the fishes! natural environment. The published results are rather limited. About the only thing that has been found is that the fishes have about the 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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would seem tobear 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 valls of small pools distort the ield to some extent. A quantification of this effect is defined in the calculations presented below in vhich ve dətermine the specification of the tank needed for acceptable experimental vork.

We assume that the fish is locsted centrally within a cylinder. With this assumption, ve study kow the potential varies as a function of cylinder length assuming an infinite radius for the cylinder. Next, we assume the cylinder has infinite lengtn and see how current varies with radius. With this information, ve vill be able to determine reasonable lengths, widths, and depths for experimental containers for electric fish research. We vill disregard all .interfaces in this development bscause our ultimate intention is to determine vhen these interfaces can be disregarded.

The equations which express potential as a function of distance $\cos e ^ { \frac { \alpha } { 4 } }$

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

$$
v = \frac { 0 } { \frac { 1 } { 4 \pi } c } \left[ \frac { 2 } { \left( y - \frac { 2 } { 2 } \right) \left( y + \frac { 2 } { 2 } \right) } \right]
$$

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4. The five unnumbered equations used in this sectioi 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.

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$$
\begin{array} { r l } { \tt c o r \frac { 1 } { 2 } - \tt a > y > - \frac { 1 } { 2 } + \tt a } & { { } \tt v = \frac { 0 } { \frac { 1 } { 2 } + \tt a } } \\ { \tt c o r } & { { } \tt c o s \left[ \tt c o s \tt a > \tt a > \tt a \tt a \right] } \end{array}
$$

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

Pictting for different values of L in $\overline { 8 } 4 _ { \bigotimes } .$ 2s ve obtain the required cylinder length. Only the positive direction is plotted because the negative direction is identical except the sign is reversed. Hov long the cylindrical tank should be is difficult to determine precisely. As a minimum though ve can say that there should be 5 electric organ lengths of sster in front and in back of the fish at all times Caring the experiment.

To determine the cylinder radius required for the tank ${ \pmb v } _ { \pmb { \imath } }$ we can modify the limits on the integral expressing the current I in equation 48 of our later development. This equation is

$$
I = \int \limits _ { 0 } ^ { \infty } \left( \int \limits _ { 0 } ^ { 2 \pi } \frac { \mathrm { ~ e ~ } z \mathrm { ~ d ~ } \alpha } { \mathrm { ~ i ~ r ~ e ~ } \left( \pi ^ { 2 } + \frac { z ^ { 2 } } { R } \right) ^ { 3 / 2 } } - \mathrm { ~ a ~ } \phi \right) \mathrm { ~ d ~ } \alpha
$$

The only limit vhich needs to be modified is the infinity symbol. We replace this vith v and solving as before ve find the current to be

$$
I = \frac { \mathrm { ~  ~ \ l ~ } _ { 0 } } { 2 \mathrm { ~  ~ c ~ } } \left[ \frac { - 2 }  \left( \frac { \mathrm { ~  ~ n ~ } ^ { 2 } + \frac { \mathrm { ~  ~ r ~ } ^ { 2 } } { \mathrm { ~  ~ r ~ } } ^ { 2 } \right) \mathrm { ~  ~ \ l ~ } ^ { 2 } } { \displaystyle \int _ { 0 } ^ { \mathrm { ~  ~ \ l ~ } ^ { 2 } + \frac { \mathrm { ~  ~ r ~ } ^ { 2 } } { \mathrm { ~  ~ r ~ } } ^ { 2 } } } \right] ^ { \mathrm { ~  ~ v ~ } } _ { 0 }\tag{1}
$$

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Equation l may be expressed in closed form as

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$$
I = \frac { Q _ { \mathrm { ~ L ~ O ~ } } } { 2 \mathrm { ~ e ~ } } \left[ \frac { 2 } { L } - \frac { 1 } { \left( v ^ { 2 } + \frac { 1 } { L } ^ { 2 } \right) ^ { 1 / 2 } } \right]\tag{2}
$$

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

$$
I = \frac { 9 0 } { \frac { 1 } { 2 } }
$$

By letting $v = \frac { I \pi } { 2 }$ and solving equation 2 for different values of a, ve have obtained a plot of current as a function of width or depth. This plot is shown in Fig. 2b. To obtain 90% of the maximum current requires 5 elec- $\pmb { \mathcal { F } } \pmb { \mathcal { L } } _ { \pmb { \mathcal { B } } } .$ tric organ lengths on each side of the fish. To get 95% would require 10 lengths. More than 95% of the maximum current would be almost impossible to obtain in the laboratory.

Thus for experimental results to be considered valid there should be at lesst 5 electric organ lengths of water surrounding the fish. For free swimming experiments, it would be best to have at least 10 electric organ lengths about the fish. When the fish must be restrained near the surface, 10 orgen lengths of vater in all other directions should be the minimum. It should slso 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.

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POSSIBLE RECEPTOR MECHANISM AND NEURAL CODING

The possible mechanism discussed below can not be said to be the mechanism that the fish actually uses in its detection and classification of objects. It is, hovever, derived from our revier and anslysis of the available data, from results of the limited experimentation that ve carried out to clarify some of the available data, and from our knovledge of auditory and labyrinthine sstem function. The postulating of this mechanism, viewing the electroreceptors and auditory receptors as evolutionary derivations of the same primitive receptor, provides a testable, hypothesis of receptor fumction. It also provides a basis that can be of assistance in determining receptor sensitivity.

Mechanism. The inner ear is a fluid filled cavity vith a complex membrane strurture. It is notable for the complex electrical fields that are generated vithin it by external events and its organized bands of hair-like sensing cells. Early concepts of pressure vaves in the fluid bending the hair eellsand thereby triggering signals to the brain are very much i: question. Some of the never concepts implicate an intermediate electrical field sensing mechanism in the hair cells. The precise nature of this is not clear but elements of O'Leary's (1970) recent experimentation and theorizing on the inner ear labyrinthine system appear to be quite applicable to the veakly electric fishes' sensing system. These elements combined vith other information on auditory and balance sensor function vill be discussed below to the extent that they have bearing on our modeling of the fishes' sensing system.

Since Dohlman (1960) has showm thet hair cell membranes are apparently impermeable to ions, O'Leary assumes that the detection of fields are due

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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. Thase movements in an electric field result in an effect called polarization. Van Beek $( 2 9 6 7 )$ has pointed out that the average molecular dipole moment ${ \pmb { \mathrm { \tilde { \varepsilon } } } } _ { \mathbf { \tilde { \varepsilon } } }$ ina heterogeneous system is the vector sum of induced (electronic) polarization resulting frou the relative displacement of electrons and nuclei, dipolar polerization resulting rom the partial aligmment in the direction of the field of molecules with permanent dipole moments. and interfacial (Maxvell-Wagner) polarization occurring at boundaries betveen the components of a heterogeneous system. Jackson (1962) has indicated that $\mathfrak { P } _ { \mathtt { m o l } }$ is related to the macroscopic polarization P (electric dipole moment per unit volume) and the macroscopic electric field E by

$$
p = y ( p _ { \mathrm { m o l } } ) = x _ { 0 } , z\tag{3}
$$

where N is the mumber of molecules per unit volume $\pmb { x _ { \alpha } }$ is the electric susceptibility.

From this and his own experimental data, O'Leary (1970) suggests that a weak electric field in the fluid filled inner ear might be detected by hair cells by the polarization it induces in long-chain filaments of polyatomic molecules in the cilia. Since Van Beer's (1967) studies of dielectric behavior of colloidal solutionsindicate 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 lov frequency electric fields polarize the molecules by inducing dipole moments in the

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[PAGE 25]
double layers. There is also evidence along this line from Feller, et al (1060) & Saito, et al (1966). . Polarization can also occur by a mechazism suggested by Frohlich's observation (1958) that large molecules can have $C _ { 3 } \cdot$ C=0, or OH groups that are in themselves dipolar, but have a net dipole moment of zero üue 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 vith 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 thar a charge e which is displaced a distance x is bounded by s restoring force F given by

$$
\mathtt { P } = - \mathtt { m } _ { 0 } ^ { 2 } \mathtt { x }\tag{4}
$$

where m is the mass of the charge

$^ { \infty } _ { \circ }$ is the radian frequency of harmonic oscillation

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

$$
\cos = \frac { 2 } { 0 } \times\tag{5}
$$

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[PAGE 26]
The induced dipole moment is then defined for one eirctron ds

$$
P _ { 1 23 d } = \cos = \frac { e ^ { 2 } E } { 2 4 0 ^ { 2 } }\tag{6}
$$

If there are Z electrons per molecule with f. of them bound by a restoring $\pmb { \varepsilon _ { g } }$ $\cos \cos \cdot - \sin ^ { 2 } \ x _ { 2 }$ then the induced dipole moment is

$$
E _ { \mathrm { 4 n d } } = \frac { e ^ { 2 } } { n } = \frac { f _ { 1 } } { n ! }\tag{7}
$$

$$
\operatorname { s h e r e } 2 = \sum _ { j } \mathbb { E } _ { j }
$$

Temperature is pot a variable in equation 7 su the induced pclarization world 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 allovable 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 lkT≈ k $\yen 10$ ergs/ molecule as suggested by deVries (1949), the corresponding vave number 1/λ for an energy transition of LkT

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

The ocuurrence 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 spectrosuopically the splitting of energy levels by an electric field (the Stark effect). Splittings of about $2 0 ^ { - 3 } \cos ^ { - 2 }$ have been observed from diatomic molecules with field strengths in the range of $2 0 ^ { 2 } - 2 0 ^ { 3 } = 1 0$ Smalier "hyperfine" splitings vere then predicted theoretically and observed using high resulution 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 menhanical and electrical stimuli might be detected by this molecular effect in the electrical sensor because mechanical forces should also affect the microscopic polarization of long-chain molecules. The bending or shearing of cilia that contain long-chain molecules could result in a change in the

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hc/λ = kT

(8)

[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:

$$
{ \mathbb { P } } _ { \mathrm { p h a s i c e } } = \frac {  { \mathbb { P } } _ { \mathrm { g e n e r a t o r } } } {  { \mathbb { n } } } \quad \cdot \ \left\{ \begin{array} { l } {  { \mathbb { N } } = 1 , 2 , 3 , \cdots \cdots } \\ {  { \mathbb { n } } = 1 , 2 , 3 , \cdots \cdots } \end{array} \right\}\tag{9}
$$

$$
P _ { \mathrm { t o n d e } } = F _ { \mathrm { r e s t i n g } } + \frac { \Delta \textsf { f } } { \texttt { n v } } \cdot ( \mathrm { s t } \mathrm { m } 1 1 0 8 )\tag{10}
$$

There is & general relationship between the physical classification scheme discussed in an earlier section and the tonic-phssic 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 ccding employed by the electroreceptor for transmission of information to the brain.

One fish, Hypopomus, has electroreceptors vhich respond to each discharge of the electric organ vith a succession of short pulses. Each "pulse train" contains eight or more pulses (Hagivara, Kusano, & Negishi, l962). The electroreceptors of Gymnotus and Staetogenes respond vith one to six pulses to each generator discharge (Hagiwara & Morita, l962). Some studies indicate that the mumber of pulses in each train can be related to the potential near the receptor. This has been referred to as "number coding".

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In another weakly electric fish, Eigenmannia, each organ discharge does not produce a receptor output. If the fishes' electrical field is distorted though, ve find that the receptor output is at most one impulse per generator organ discharge. If we decrease the distortion ve 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 vill fire is related to the stimulus intensity acting on it. This type of coding is called "probability" coding. (Hagivara & Morita, 1962),

Another fish, Sternopygus, was throughly studied by Bullock and Chichibu (1965). They found fibers that carry one impulse per organ discharge. They noted a phase or time relationship vith the intensity of the stimulus. This is referred tò as phase or latency coding. They also found other nerve fibers that maintained a rhythmical firing out of phase with ti a 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 requency coding and is characteristic of ampullary organs.

As a weak generalization, it appea's that ampullary receptors give tonic responses with frequency coded information and tuberous receptors give phasic responses with either mumber, phase, lateney, probability or frequency coded information. But we can develop a stronger generalization.

Hagivara and Morita (1962) suggest a model ror probability coding based on an assumed nerve threshold. By making certain assumptions about the threshold curve, we.have extended their model to encompass all coding schemes proposed to date. Their model was originally developed for tuberousphasic receptors, but we can easily extend it to ampullary-tonic receptors vith a simple modification. We shall develop belov this all encompassing coding scheme since it suggests the nature of the system function.

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[PAGE 30]
Thc threshold model as reported by Hagivara and Morita (1962) assumes that after a receptor fires, the threshold resets to some high value. The threshold then begins to decay until the intensity at the receptor is equal to or momentarily exceeds the threshold. At this point, the receptor fires, and the threshold resets.

This will form the basic model on vhich we will elaborate in order to encompass the different coding mechanisms. First, we vill discuss some general properties of biological threshold curves. Then, ve will define the relationships betveen the threshold curve and the electric organ output which are required by the available data. Lastly, we vill detail some of the measures vhich 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; $\pmb { \tau } _ { \mathrm { m a x } }$ the maximum threshold, to $\pmb { x } _ { \mathrm { s t a } }$ the minimum threshold. In reality, the minimum value probably continues to decay with time. But for situations of repeated sampling, ve can approximate it as ${ \pmb { \tau } } _ { \mathbf { z } \mathbf { z } , \mathbf { \dot { z } } }$ 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 $\pmb { \tau _ { \mathrm { s t a } } }$ $\pmb { \tau } _ { \mathbf { z } \mathbf { z } } .$ or the exponèntial time constant $\pmb { \tau _ { e } }$

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

$$
I = \left[ \mathbb { T } _ { \operatorname* { m a x } } - \mathbb { T } _ { \operatorname* { m i r } } \right] - \frac { t } { \mathbb { T } _ { \mathrm { e } } } + \mathbb { T } _ { \operatorname* { m i n } } + \mathrm { ~ S ~ } \left( t \right)\tag{21}
$$

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[PAGE 31]
$$
\mathfrak { T } \approx _ { \mathrm { a x } } ^ { } \mathrm { e } ^ { - \frac { t } { \mathfrak { T } _ { \mathrm { c } } } } + \mathfrak { T } _ { \mathrm { m i n } } + \mathrm { s } _ { \mathrm { \Gamma } } ( \mathfrak { t } )\tag{12}
$$

The general shape of such a function is shown in Fig 3a. To describe probability coding, we must assume that 4 $\frac { \pi } { c } > > \frac { 1 } { 2 } / \frac { 1 } { 5 }$ In such cases, the electric organ can discharge several times without firing the nerve. The mumber of tines 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 $\mathbb { T } _ { \mathtt { m } 4 \mathtt { n } } > 0 , \quad \mathbb { T } ^ { \mathtt { m } } \mathbb { T } _ { \mathtt { m } 4 \mathtt { n } }$ were zero, the sensor vould reset independently of the electric organ and one forn of tonic coding would be observed.

To describe mumber coding we must assume that 4 $\frac { \pi } { c } < < 1 1 2$ 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 vill fire is proportional to the intensity at the receptor as illustrated in Pis. 3c. Again, for phasic coding $\pmb { \underline { { \tau } } } _ { \mathbf { s } 4 \mathbf { s } }$ must be greater than zero.

As4 $\pmb { \tau _ { \pmb { c } } }$ becomes approximately the same as $\mathbf { 1 } / \mathbf { \hat { r } _ { 3 } }$ several interesting possibilities occur: phasic coding, latency coding, and interpulse interval coding. Phase and latency coding are illustrated in Fig.4a. They are one and the same. Interpulse interval coding is illustrated in Fig.4b. It is also interesting that when 4 $\mathfrak { r } _ { \mathfrak { e } }$ is about two or three times $\pmb { \mathscr { n } }$ a combination 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

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Further, as a very good approximation ve could say

[PAGE 32]
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been noted previously, there are experimental reports that phasic receptors have an output apparently unrelated to generator output. We do not knov 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 existance of a buffering capability at the receptor, or if the reports report artifacts due to faulty technique.5 But whichever is the case, the utility of the model is not affected. It can usefully be used as a unirying framework for studies of threshold, sensitivity, and response time across all veakly electric fish. With such a framework, sensitivity could be vell 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 vater surrounding the fish. Objects within the environment and also the envirommental boundaries distort the electric field. This distortion causes s change in the electric field near the fish vhich ve 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, @ymnar
chus was trained to respond to changes in the conductivity of objects placed
in a sealed container. Pusitive reinforcement was a food revard, and nega
tive reinforcement or punishment was the insertion of & 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 & threshold 'to potential change of
5. If Tmin vere below the mechanical noise level, phasic receptors would sppear to be tonic receptors.

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[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 vith the fumction 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 vill develop a simplified model of the receptor and then discuss system function. We will use the dipole concept described above as vell as the hypothesis on sensor fumction derived from above. Through the development 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 trís, we will determine critical variables in the sense fumction and quantify their effect. With the information so derived, we vill 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

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[PAGE 36]
Fig.5 a) This represents'the electric field intensity surroumding the weakly electric fish as indicated by the data of Bullock & Chichibu (1965). b) This illustrates the model used to calculate electric potential, current, and field intensity, due to a fish's electric generating organ. The model consists of two spherical conductors of radius a separated by an on center distance L.

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negatively charged to a total charge of -Q. We will assume that a is much less than $\pmb { \tau _ { * } }$ 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 vork required to wove an increment of charge from infinity to a given point in space, or

$$
\nabla = \frac { \partial \textbf { w } } { \partial \textbf { \textsf { Q } } }\tag{13}
$$

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

$$
\mathbf { w } = \mathbf { x } \textbf { z } - \textbf { p z }\tag{14}
$$

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 vork is the negative of the potential energy., Potential energy is defined by the integral

$$
P E \triangleq \int \limits _ { X } ^ { Y } \frac { d } { d t } \cdot d z\tag{15}
$$

where $\pmb { \widehat { \widetilde { \tau } } }$ is the force acting on the test charge as it is moved from x to y dr is an increment of distance in the path between the points x and y

The electric force is a conservative force. Thus, if a test charge is acted on by more than one charge, we can determine the potential energy ¿ue to each charge and find the total potential energy by addition. In other wors if P $\Xi _ { \mathfrak { n } }$ represents the potential energy due to the $n ^ { t h }$ charge, and if there are

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a total of N charges the total potential energy is

$$
\texttt { P E } _ { \mathrm { t o t a l } } = \sum \limits _ { \texttt { m } } ^ { \texttt { N } } \texttt { P E } _ { \texttt { n } }\tag{16}
$$

The poteptial energy due to a charged sphere can be easily found. The electric force may be found from Coulomb's Law which states

$$
\overrightarrow { F } = \frac { \theta _ { 4 } } { 4 \pi \epsilon z ^ { 2 } } - \widehat { i } _ { z }\tag{17}
$$

where Q is the total charge on the sphere

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

r is the distance between the two charges

e is the dielectric of the media

$\widehat { \mathbf { \downarrow } } _ { \mathbf { \ 5 } }$ is a vector directed avay from the center of the charge on a straight line

Using Gauss's Lav 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

$$
z \{ | z | < | | z | \} = 0\tag{18}
$$

If we call the line vhich passes through the two charges the y axis, we can find the potential energy at any point on the axis by solving equation 15.

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

$$
p _ { \textrm { \tiny \textrm { E } = } } \frac { | { \textrm { y } } \cdot | } { \int _ { \texttt { \tiny { 4 } + } } ^ { } \frac { Q _ { \textrm { \tiny { q } } } } { 2 } }\tag{19}
$$

The solution is

$$
\cdot \texttt { P E } : \frac { \texttt { e q } } { 4 \pi \epsilon } [ - \frac { 1 } { 2 } ] { } ^ { \frac { \mid x + 1 } { 2 } } .\tag{20}
$$

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

$$
\geq E = \frac { a _ { t } } { 4 \geq e | y - 1 }\tag{21}
$$

The potential energy for the posicively charged sphere is

$$
\textcircled { 2 } z + = - \frac { 8 a _ { 4 } } { 4 \pi e ! \pi 1 }\tag{22}
$$

The total potential energy for the dipole system is

$$
p _ { \mathrm { \ t o t a l } } = \frac { q _ { \mathrm { \ d } } } { \mathrm { \ d } \mathrm { \ d } \mathrm { \ x } } \left\{ \frac { 1 } { \vert \mathbf { y } - \mathbf { \nabla } \vert } \frac { 1 } { \vert \mathbf { y } + \vert } \right\}\tag{23}
$$

If we define a coordinate system as showm in $P _ { 2 } ( 3 . 6 , 2 0 0$ find that the absolute values of y- and y+ depend on our location on the y axis as follows:

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$$
\cos y > \frac { I } { 2 }
$$

$$
\vert y + \vert = y - \frac { texttt { L } } { 2 }
$$

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

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

$$
\vert y - 1 = \frac { 2 } { 2 } + y\tag{24e}
$$

$$
\cos - \frac { I } { 2 } > y\tag{24d}
$$

$$
\vert y + \vert = \frac { I } { 2 } - y
$$

$$
\vert y - 1 = - \frac { I } { 2 } - y\tag{24e}
$$

(24f)

The total potential energy can be found from equation 23 and equations 2ha to f in each region. Recalling that the kinetic energy is zero and applying equation l3 in terms of the test charge ve gec:

$$
v = \frac { \partial } { \partial t } \left\{ - \frac { \theta _ { q } } { 1 + \pi \epsilon } \left[ \frac { 1 } { \vert y - 1 \vert } - \frac { 2 } { \vert y + 1 } \right] \right\}\tag{25}
$$

This yields

$$
v = \frac { Q } { 1 + \pi \epsilon } [ \frac { 2 } { | y + | } - \frac { 2 } { | y - 1 } ]\tag{26}
$$

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[PAGE 42]
Substituting the appropriate values of y+ and $y - z \geq$ obtain:

$$
\cos y > \frac { z } { z } + a
$$

$$
v = \frac { e } { \frac { 3 \pi \pi } { 4 \pi \epsilon } } \left[ \frac { 2 } { ( y - \frac { 2 } { 2 } ) ( y + \frac { 2 } { 2 } ) } \right]\tag{27a}
$$

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

$$
\eta = \frac { Q } { \frac { 4 \pi ~ 2 } { 2 } } \int \frac { 2 y } { \left( \frac { 5 } { 2 } - y \right) \left( \frac { 5 } { 2 } + y \right) }\tag{27b}
$$

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

$$
v = \frac { a } { \frac { b } { \frac { b } { c } \pi \epsilon } } [ \frac { \frac { c } { \sqrt { \frac { c } { 2 } } - y } \sqrt { ( \frac { c } { 2 } + z ) } } { \left( \frac { c } { 2 } - y \right) \left( \frac { c } { 2 } + z \right) } ]\tag{27e}
$$

where V is the absolute voltage $( \nabla 1 - 1 = 0 )$

Q is the charge on either conductor,

L is. the distance between the conductors,

ε is a constant known as the dielectric

y is a continuous variable representing an absolute scale with o being located between the positive and negative charge as indicated in Fig. 6.

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

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

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Substituting $y = - \frac { 2 } { 3 } + 8$ into equation 27b ve find the voltage on the negative sphere to be

$$
v ( - ) = \frac { q } { 4 \pi \epsilon } \frac { - 1 + 2 a } { a ( 1 - a ) }\tag{29}
$$

Solving for v ve find.

$$
\nabla = \nabla ( + ) - \nabla ( - )\tag{30}
$$

or

$$
v = \frac { q } { q \pi \epsilon } \frac { 2 } { a } \frac { ( 1 - \frac { 2 \epsilon } { c } ) } { ( 1 - \frac { a } { c } ) }\tag{31}
$$

Whcn $a / \tau$ is less than 1/10, equation 31 can be approximated vithin 5 percent by the reletioship

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

The capacitance of an object is defined as

$$
c = \frac { a } { v }\tag{33}
$$

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 3l, it is found to be

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35

$$
c = \frac { Q } { v } = 2 \pi = 0 \frac { ( 1 - \frac { a } { v } ) } { ( 1 - \frac { 2 a } { v } ) }\tag{34}
$$

This may be approximated vithin 5 percent for a/L less than 1/10 as

$$
c = 2 \pi c _ { 2 } \frac { 3 } { 2 } )\tag{35}
$$

It is useful to determine the resistance of the dipole ve have Just considered. Since ve have already evaluated the voltage betveen the charges, if we can find an expression for the current that flows between the two points, ve can solve for the resistance from Ohms Lav.

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

where R is the resistance

v is the voltage

I is the current

Two equal and opposite charges creste an electric field. If ve can solve for the magnitude and angle of the electric field E, we can determine $\widehat { \mathbf { z } } _ { \bullet }$ the current density $\mathfrak { F }$ from the relationship

$$
\gamma = \sigma \frac { \ast } { \mathbf { z } }\tag{37}
$$

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36

where $\pmb { \mathfrak { J } }$ is the current density

o is the conductivity of the media.

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

$$
I = \int \limits _ { 3 } ^ { + } \frac { \partial } { \partial s } d z\tag{38}
$$

vhere 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 $z _ { - }$ and the field due to the positive charge E+. Symbolically we can write

$$
\widehat { \Xi } = \widehat { \Xi } + \widehat { \Xi } -\tag{39}
$$

where $\pmb { \vec { \mathrm { z } } }$ is the total field

Due to symmetry. the most convenient surface to use for cur current integral is the plane vhich 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

$$
E + = \frac { Q } { 4 \pi \in D _ { + } ^ { 2 } } - i \sqrt { D + }\tag{40}
$$

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where D. is the distance between the positive charge and the $\mathfrak { d } _ { \ast }$

point p

$^ { 1 } v _ { \phi }$ 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

$$
E = \frac { - a } { 4 \pi e ^ { 2 } } = i \frac { i } { 2 }\tag{41}
$$

where D is the distance from the negative charge

$\widehat { \widehat { \mathbf { \phi } } } _ { \mathbb { D } }$ is a unit vector at p directed by the D line avay $0 、$ 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 betveen the two charges, the distances D. and $\mathfrak { d } _ { * }$ D are equal. They can be found from the equation

$$
D _ { + } = 0 _ { - } = [ R ^ { 2 } + \frac { I ^ { 2 } } { 4 } ] ^ { 2 / 2 }\tag{42}
$$

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.Ta, the elettrical field due to either charge has both y and radial components. Due to symmetry,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

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[PAGE 47]
$$
\begin{array} { r } { \overrightarrow { \mathbf { E } } _ { \mathrm { t o t a l } } = \frac { - 2 \mathrm { ~  ~ Q ~ } } { { \mathrm { ~  ~ \psi ~ } } _ { \ast } \mathrm { ~  ~ \pi ~ } _ { E } \mathrm { ~  ~ D ~ } _ { \ast } ^ { 2 } } \qquad \mathrm { s i n ~  ~ B ~ } _ { 1 } ^ { \prime } , } \end{array}\tag{43}
$$

We note that β s a because they are opposite angles.Sin acan be determined from Fig.7a by

$$
\sin a = \sin B = \frac { I _ { 1 } / 2 } { I _ { 2 } }\tag{44}
$$

Substituting equation 44 into equation 43

$$
\frac { \sum _ { \mathrm { ~ t o t a l ~ } } - \mathrm { ~ e ~ f ~ f ~ } } { 4 \mathrm { ~ r ~ e ~ p ~ } _ { \mathrm { + } } ^ { 3 } } : \frac { \hat { z } } { y }\tag{45}
$$

Substituting equation 42 into equation 45, and equation 45 into equation 37 yields

$$
\frac { x } { 3 } = \frac { 0 . 8 [ \lambda } { 4 \pi \in [ \frac { \pi ^ { 2 } } { 4 } + \frac { \pi ^ { 2 } } { 4 } ] ^ { 3 / 2 } }\tag{46}
$$

The surface of integrution is considered to be small ring segments in the x-z plane about the y axis. Fig.Tb illustrates this concept. We note that the electric field is perpendicular to the x-z plane vhich makes the integral defined by equation 38 easy to evaluate. A surface element for the ring shown in Fig. 7o is

$$
d \mathbf { z } = \mathcal { R } \textbf { d } \phi \ \mathrm { d } \mathbf { R }\tag{47}
$$

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Fig.7 a) This figure illustrates the relationship betveen E- and E+ due to the electric fields created by the charges Q- and $a > .$ ,respectively. Note that the vector sum of these two electric fields is parallel to the y axis for any point P in the plane which forms the set of perpendicular bisectors of the two charges. b) This figure indicates the quantities that need be defined in order to evaluate the surface integral required by equation number thirty-eight.

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wher: ds is the surface element, d  is an increment of angle  which is an angle about the y axis.. dR is an increment in the distance vector R.

Substitution of equations 47 and 46 into equation 38, and defining the surface integral, ve obtain

$$
\texttt { I } = \int \limits _ { 0 } ^ { \infty } \Bigg ( \int \limits _ { 0 } ^ { 2 \pi } \frac { \texttt { Q L o R } } { \texttt { \frac { L r c } { 6 } } \left( \texttt { R } ^ { 2 } + \underbrace { \texttt { L } ^ { 2 } } _ { \texttt { 4 } } \right) ^ { 3 / 2 } } \texttt { d } \phi \Bigg ) \texttt { d }\tag{48}
$$

Solving the inner integral yields

$$
I = \int \limits _ { 0 } ^ { \infty } \frac { q \texttt { L o n } } { 2 \texttt { c } \left( \pi ^ { 2 } + \frac { v ^ { 2 } } { h } \right) } 3 / 2 d R\tag{49}
$$

Equation k9 can be solved to yield

$$
I = \frac { \mathrm { ~ \bf ~ Q ~ L ~ O ~ } } { 2 \mathrm { ~ \bf ~ e ~ } } [ \frac { - 1 } { \left( \mathrm { ~ \bf ~ R ~ } ^ { 2 } + \frac { L ^ { 2 } } { L } \right) ^ { \frac { 1 } { 2 / 2 } } } ] _ { 0 } ^ { \frac { 1 } { 2 } }\tag{50}
$$

In cl

$$
I = \frac { 8 0 } { 6 }\tag{52}
$$

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

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[PAGE 50]
$$
R = \frac { V } { I } = \frac { \frac { 9 . 6 ( 1 - 0 . 1 2 ) } { 2 \pi \textrm { c a } } } { \frac { 9 . 6 } { \frac { 9 } { 5 } } }\tag{52}
$$

which reduces to

$$
R = \frac { 2 } { 2 \pi 0 0 } ( 2 - \frac { a } { 2 } )\tag{53}
$$

We may now use the derived information. As a first approximation let us assure that the radius of each charged sphere, a, is 1 cm and that the length of the generator organ, L, is 10 cm. The conductivity of fresh water is about $1 0 ^ { - 3 } 8 ^ { - 1 } = 2 ^ { - 1 }$ . The resistance which loads the generator organ is about $6 2 . 0 0 0 0 m ^ { 6 }$ The dielectric of fresh vater is $0 . 7 0 7 \times 1 0 ^ { - 9 } \neq 0 . 1 2$ This means that the capacitance which loads the dipole is 40 pfd. This large resistance and small capacitance indicate that oniy 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 object 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 modiry the potentials along the fish's surcace. To obtain a solution for. these modifications, vs will first consider the "leetric field produced by the dipole generator. organ undisturbed by the

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40

the resistance as defined by equation 36

In a practical modcl, the resistance loading. of the pruorator can be controlled by the choice of penerator clectrode "izo.

[PAGE 51]
41

perturbating object. We will calculate its magnitude at an arbitrary point p. We vill then consider the effect at point p of a perturbating object located in a uniform field. Then ve will transform the perturbation portion of our solution back to the original coordinate system.

Once ve 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 ad although not conceptually difficult, it is somewhat complex. $\tt R t B .$ 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

$$
E ( p ) = \frac { 1 } { 5 } ( p ) / \rho _ { q }\tag{54}
$$

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

$$
\beta = \frac { q _ { \mathrm { ~ Q ~ } } } { \mathrm { ~ i ~ m ~ c ~ n ~ } ^ { 2 } } = i\tag{55}
$$

where E is the dielectric of the media

D is the distance between the charges

$\widehat { \frac { 1 } { 2 } }$ 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

$$
z _ { 1 } = \frac { 0 } { z _ { \mathrm { ~ \bf ~ r ~ } } z ^ { 2 } } - i \frac { z _ { \mathrm { ~ \bf ~ r ~ } } } { z _ { \mathrm { ~ \bf ~ r ~ } } z ^ { 2 } }\tag{56}
$$

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Fig.8 a) This illustrates the variables involved in our model vhen a spherical perturbing object is present in the model's environment. Note the several coordinate systoms that must be considered in the analysis. b) This illustrates the functions required to express $\pmb { \mathbb { R } } ^ { \star }$ and $\flat ^ { \bullet }$ in terms of R, 0, and L.

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The electric field due to the negative charge is

$$
\frac { \frac { 1 } { x } } { 5 } = \frac { - 8 } { 4 \times e _ { y } x ^ { 2 } } i _ { z }\tag{57}
$$

The electric field is a vector quantity. Thus, we must perform vector addition in order to determine the total field È. It is essier to determine the components of the field due to each charge and then to add the components. Thus. we vill 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

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

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

$$
z = \vert \overrightarrow { z } - 1 \cos \theta \vert\tag{59}
$$

The x-z components of the electric fields are independent of the angle although the actual x and z components are not. The x-z component of the total electric field vill be considered to be the component within the ,x-z plane at an angle  from the axis. For the positive charge it is

$$
z _ { x = z } = | \sum \limits _ { z = 1 } ^ { \infty } | \sin \theta\tag{60}
$$

and for the negative charge it is

$$
z _ { x = z } = | \frac { \sqrt { z } - 1 } { z } | \sin \theta ^ { \circ }\tag{61}
$$

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[PAGE 54]
In order to treat these variables by standard mathematical techriiques, it is necessary to express $\pmb { \mathtt { R } } ^ { \star }$ in terms of R and θ, and $\theta ^ { \ast }$ in terms of R and 0. Fig.8b illustrates the factors vhich vill help us do this. Recognizing that R' is the hypotemuse of a right triangle whose sides are R sin $\pmb { \lambda } ^ { * }$ 0 and L + R cos θ, we find

$$
z ^ { - } = [ z ^ { 2 } + z ^ { 2 } + 2 z \cos \theta ] ^ { 2 / 2 }\tag{62}
$$

and

$$
\theta ^ { \prime } = \tan ^ { - 2 } \frac { R \sin \theta } { I + R \cos \theta }\tag{63}
$$

It is useful to note the trigonometric functions for 9". They are $9 ^ { \circ }$

$$
\sin \theta ^ { - } = \frac { \pi \sin \theta } { \sqrt { R ^ { 2 } + I ^ { 2 } + 2 \pi \times \cos \theta } } ] ^ { 1 / 2 }\tag{64}
$$

and

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

Combining equations 62, 64, and 65 with equations 56 and 57 and substituting $6 5$ into equations 58, 59, 60 and 67 yields

$$
\mathbb { E } _ { y + } = \frac { \mathrm { ~ \textsf ~ { ~ Q ~ o ~ s ~ } ~ } \theta } { \mathrm { ~ \textsf ~ { ~ 4 ~ r ~ e ~ } ~ } \theta ^ { 2 } }\tag{66}
$$

$$
z _ { y = } = \frac { - 0 . \dot { z } + \mathrm { ~ R ~ } \cos \theta } { \frac { 1 } { 4 - z _ { y } } \left[ \mathrm { R } ^ { 2 } + z _ { z } ^ { 2 } + 2 \mathrm { ~ R ~ L ~ c o s ~ } \theta \right] ^ { 3 / 2 } }\tag{67}
$$

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[PAGE 55]
44

$$
\frac { \theta _ { x = z } } { 4 \pi c _ { y } \pi ^ { 2 } }\tag{68}
$$

$$
\varepsilon _ { x = z } - \frac { - 0 . 8 \sin \theta } { \mathrm { ~ \texttt ~ { ~ i ~ r ~ } ~ c ~ } _ { \mathbf { v } } \left[ \ R ^ { 2 } + \ L ^ { 2 } + 2 \mathrm { ~ \texttt ~ { ~ R ~ L ~ C o s ~ } ~ } \theta \right] ^ { 3 / 2 } }\tag{69}
$$

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  and is

$$
z _ { y } = \frac { q } { 1 + \pi \epsilon _ { y } } \left[ \frac { \cos \theta } { \sin ^ { 2 } \theta } - \frac { I + { \tt R C o s } \theta } { { \tt R } ^ { 2 } + { \tt L } ^ { 2 } + 2 \mathrm { ~ R ~ L ~ C o s } \theta } \frac { 3 / 2 } { 3 / 2 } \right]\tag{70}
$$

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

$$
z _ { x = z } = \frac { \mathrm { ~ Q ~ S i n ~ } \theta } { \mathrm { ~ 4 ~ } \pi \mathrm { ~ c _ { w } ~ } } \left[ \frac { 1 } { \mathrm { ~ \pi } ^ { 2 } } - \frac { \mathrm { ~ R ~ } } { \left[ \mathrm { ~ \pi } ^ { 2 } + \mathrm { ~ L ~ } ^ { 2 } + 2 \mathrm { ~ R ~ L ~ C o s ~ } \theta \right] } \mathrm { ~ 3 / 2 ~ } \right]\tag{71}
$$

To continue our derivation, we must deterhine the magnitude [E| and the angle (relative to the y axis) β of the electric field at any poirt in space. The magnitude may be found from the rules of vector addition as

$$
| \overrightarrow { z } | = [ e _ { y } ^ { z } + e _ { x - z } ^ { - 2 } ] ^ { 2 / 2 }\tag{72}
$$

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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

$$
\begin{array} { r l } { | \overrightarrow { \mathbf { g } } | = \frac { \textsf { q } } { \textsf { l a r c } _ { \mathbf { w } } \mathbf { g } ^ { 2 } } } & { \left[ 2 - \frac { 2 \mathrm { ~ \& ~ } ^ { 2 } \mathrm { ~ \tiny ~ ( ~ R ~ + ~ L ~ C o s ~ } \boldsymbol { \Theta } ) } { \left[ \mathrm { ~ \& ~ } ^ { 2 } + \mathrm { ~ L ~ } ^ { 2 } \mathrm { ~ 2 ~ \& ~ L ~ C o s ~ } \boldsymbol { \Theta } \right] ^ { 3 / 2 } } + \frac { ( T \mathsf { 4 } ) } { \sqrt { 2 } } \right. } \\ & { \qquad \cdot \qquad \left. \frac { \mathrm { ~ \& ~ } ^ { \mathsf { h } } } { \left[ \mathrm { ~ \& ~ } ^ { 2 } + \mathrm { ~ L ~ } ^ { 2 } + 2 \mathrm { ~ \& ~ L ~ C o s ~ } \boldsymbol { \Theta } \right] ^ { 2 } } \right] ^ { 1 / 2 } } \end{array}
$$

$$
\beta = \tan ^ { - 2 } { \left[ \begin{array} { l } { \displaystyle \sum _ { \Delta = \texttt { c m } } \left[ \left[ \Re ^ { 2 } + z ^ { 2 } + z ^ { 2 } + z \texttt { R L C o s } \Theta \right] ^ { 3 / 2 } \quad \quad \quad \quad \quad \_ { \Delta } ^ { 3 } \right] } \\ { \displaystyle \left[ \overline { { \texttt { R } } } ^ { 2 } + z ^ { 2 } + z \texttt { R L C o s } \Theta \right] ^ { 3 / 2 } \quad \scriptstyle \cos \Theta - \ R ^ { 2 } + z + \texttt { R } ^ { 3 } \cos \Theta } \\ { \displaystyle \left[ \overline { { \texttt { C } } } ^ { 2 } + z ^ { 2 } + z \texttt { R L C o s } \Theta \right] \quad \quad \quad \quad \hfill , \ldots , } \end{array} \right] }\tag{75}
$$

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

To obtain a first approximation of the perturbation due to a sphere of radius r at a location $z _ { 0 } = \phi _ { 0 } = 0 _ { 0 }$ relative to the dipole coordinates, we assume that the sphere is located within a unifosm field of strength $\Xi _ { \circ }$ 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,

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[PAGE 57]
the assumption will permit a solution vithin 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 knov the range of the electric sense. Consequently, ve vill 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

$$
u _ { 0 } = - E _ { 0 } \circ \cos \alpha\tag{76}
$$

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

$$
v _ { z } = \frac { A \cos \alpha } { 2 }\tag{T7}
$$

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

$$
u _ { z } = - \infty , \cos \alpha\tag{78}
$$

where B is a constant to be determined from the boundary conditions. The potential outside the sphere is the sum of ${ \mathfrak { v } } _ { \circ }$ and ${ \pmb v } _ { \pmb x } ^ { \prime } \pmb { \alpha }$

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[PAGE 58]
$$
v _ { \tt t o t a l } = - e _ { 0 } \cdot \cos \alpha + \frac { A \cos \alpha } { \theta ^ { 2 } }\tag{79}
$$

The electrostatic boundary conditions require that when . r

$$
\frac { \partial \psi } { \partial \psi } = \frac { \partial \psi _ { \pm } } { \partial \psi } - \epsilon _ { x } \frac { \partial \psi _ { \pm _ { \pm } \varrho \ t a l } } { \partial \theta } = - \epsilon _ { x } \frac { \partial \psi _ { \pm } } { \partial \theta }\tag{80}
$$

ande that when p s r

$$
v _ { \tt t o t a l } = v _ { \tt I }\tag{81}
$$

Substituting equations 78 and 79 into equatior 8 J.e.

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

Substituting equations 78 and 79 into equation 60 yields

$$
2 \varepsilon _ { _ { \mathrm { ~ v ~ } } } \varepsilon _ { _ { 0 } , ^ { + } } 2 \mathrm { ~ A ~ } \varepsilon _ { _ { \mathrm { ~ v ~ } } } = { \tau } ^ { 3 } \mathrm { ~ e ~ } _ { _ { \tt \tt \tt \tt _ { X } } } B\tag{83}
$$

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

$$
\ B = \frac { 3 \epsilon _ { y } } { \epsilon _ { z } + 2 \varepsilon _ { y } } = \sum \limits _ { 0 } ^ { \cdot }\tag{84}
$$

and

$$
A = \frac { C _ { x } - C _ { y } } { C _ { x } + 2 _ { z _ { y } } } = 3 E _ { 0 }\tag{85}
$$

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

$$
\cdot \mathrm { ~  ~ v ~ } _ { x } = \frac { c _ { x } - c _ { y } } { c _ { x } - 2 c _ { y } } \frac { x ^ { 3 } } { p ^ { 2 } } | z _ { 0 } | \cos \alpha\tag{86}
$$

the perturbation potential.

We recall the value of E to be the value of equation 7k when $\Xi _ { \circ }$ $a = 8 。$ and vhen $\theta = \theta _ { 0 }$ or

$$
\varepsilon _ { _ 0 } = \frac { \mathrm { ~  ~ \rho ~ } } { \mathrm { ~  ~ \frac ~ { ~ 4 ~ } ~ { ~ 4 ~ } ~ r ~ } { \mathrm { ~  ~ \sigma ~ } _ { \mathrm { w } } } ^ { 2 } } \left[ \begin{array} { c } { { \mathrm { ~  ~ \rho ~ } } } \\ { { \mathrm { ~ \overline { { ~ } } - \mathrm { ~  ~ \sigma ~ } ~ } \frac { 2 \mathrm { ~  ~ \cal ~ R ~ } _ { 0 } ^ { 2 } \mathrm { ~  ~ \langle ~ \mathcal ~ { ~ R } _ { 0 } ~ - ~ \mathcal ~ { ~ L } ~ c o s ~ \mathcal ~ { ~ O } _ { 0 } ~ } \rangle ~ } { \mathrm { ~  ~ \sigma ~ } \mathrm { ~ \overline { { ~ } } \sigma ~ } \mathrm { ~  ~ \rho ~ } } } } \\ { { \mathrm { ~  ~ \rho ~ } } } \end{array} \right] ^ { - 1 } - \frac { 2 \mathrm { ~  ~ \sigma ~ } _ { 0 } ^ { 2 } \mathrm { ~  ~ \langle ~ \mathcal ~ { ~ R } _ { 0 } ~ - ~ \mathcal ~ { ~ L } ~ c o s ~ \mathcal ~ { ~ O } _ { 0 } ~ \rangle ~ } } { \left[ \begin{array} { c } { { \mathrm { ~  ~ \sigma ~ } _ { 0 } } ^ { 2 } + { \mathrm { ~  ~ \sigma ~ } _ { \mathrm { ~ \tiny ~ { ~ \widehat ~ { ~ } } } } } ^ { 2 } + 2 \mathrm { ~  ~ { ~ \sigma ~ } _ { 0 } } ^ { 2 } \mathrm { ~  ~ \ c o s ~ \mathcal ~ { ~ \theta } _ { 0 } ~ } } \\ { { \mathrm { ~  ~ \rho ~ } } } \end{array} \right] ^ { - 3 } \mathrm { ~  ~ \sigma ~ } _ { 1 2 } ^ { 2 } } + \frac { \mathrm { ~  ~ \sigma ~ } _ { 0 } ^ { 2 } \mathrm { ~  ~ \langle ~ \mathcal ~ { ~ O } _ { 0 } ~ - ~ \mathcal ~ { ~ O } _ { 0 } ~ \rangle ~ } } { \mathrm { ~  ~ \sigma ~ } _ { 0 } ^ { 2 } } .\tag{87}
$$

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

and we recall that a is measured relative to an angle β which is equation 75 evaluated at $a _ { 0 } \cdot 0 _ { 0 }$ or

$$
\bf \Pi _ { 3 } = \tan ^ { - 1 } \left[ \frac { \sin \theta _ { 0 } } { \left[ \frac { \left[ R _ { 0 } + z ^ { 2 } + 2 \textrm { R } _ { 0 } \textrm { L C o s } \Theta _ { 0 } \right] } { \left[ R _ { 0 } ^ { 2 } + z ^ { 2 } + 2 \textrm { R } _ { 0 } \textrm { L C o s } \Theta _ { 0 } \right] } \lambda ^ { 2 } \left[ 2 \textrm { \texttt { B } } _ { 0 } ^ { 3 } \right] \right]}   { \left[ \frac { \left[ R _ { 0 } \textrm { L } ^ { 2 } + z ^ { 2 } + 2 \textrm { R } _ { 0 } \textrm { L C o s } \Theta _ { 0 } \right] } { \left[ \frac { R _ { 0 } \textrm { L } ^ { 2 } } { 8 } + z ^ { 2 } + 2 \textrm { R } _ { 0 } \textrm { L C o s } \Theta _ { 0 } \right] } \lambda ^ { 3 } I _ { 2 } \right. \left. \cos \theta _ { 0 } \textrm { e } - \textrm { R } _ { 0 } ^ { 2 } \textrm { L } + \textrm { R } _ { 0 } ^ { 3 } \cos \theta _ { 0 } \right]} { \left. \frac { \left[ R _ { 0 } \textrm { L } ^ { 2 } + \textrm { R } _ { 0 } \textrm { L } ^ { 3 } \right] } { \left[ \frac { R _ { 0 } \textrm { L } ^ { 2 } } { 8 } + z ^ { 2 } + 2 \textrm { R } _ { 0 } \textrm { L C o s } \Theta _ { 0 } \right] } \lambda ^ { 3 } I _ { 2 } \right. }   \right]\tag{88}
$$

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[PAGE 60]
To complete our analysis, we must express p and a in terms of R, θ, \$, $\bar { \mathfrak { a } } _ { \mathfrak { o } ^ { \bullet } }$ and $\mathfrak { e } _ { \bullet } .$ We find

$$
\begin{array} { r l } & { \rho \bullet \left[ \begin{array} { l l l } { \left[ \mathtt { R } \odot \mathtt { e a s \ \Theta } \odot \mathtt { R _ { 0 } \ \subset o s \ \ \Theta _ { 0 } } \right] } & { ^ { 2 } } & { \mathtt { \Gamma _ { + } } \left[ \mathtt { R _ { e } \ \otimes \mathtt { i n \ \Theta } \Theta _ { 0 \circ } \ - \mathtt { R \ S i n \ \Theta _ { 0 } \ \subset o s \ \ \left( \phi - \phi _ { 0 } \right) } } \right] ^ { 2 } \ + } \\ & { \left[ \mathtt { R _ { \alpha } \otimes \mathtt { i n \ \Theta } \otimes \mathtt { i n \ ( \phi _ { 0 } \otimes \varphi _ { 0 } ) } } \right] ^ { 2 } } \end{array} \right] ^ { 2 } } \\ & { \quad \mathrm { ~ t h ~ a ~ m ~ \mathtt { ( \mathtt { N \otimes \mathtt { i n \ \Gamma _ { 0 } \otimes \mathtt { i n \ \Gamma _ { 0 } \otimes \mathtt { i n \ \Gamma _ { 0 } } } } ) } ~ } } 2 \left[ \begin{array} { l l l } { \ \mathtt { 1 / 2 } } & & { \mathtt { ( \mathtt { S \otimes \mathtt { i n \ \ ( \phi _ { 0 } \otimes \varphi _ { 0 } ) } } ) } } \\ & { \mathtt { ( \mathtt { S \otimes \mathtt { i n \ \ ( \phi _ { 0 } \otimes \mathtt { i n \ ( \phi _ { 0 } \otimes \varphi _ { 0 } ) } ) } } ) } } \end{array} \right] ^ { 2 } } \\ & { \quad \mathrm { ~ t h ~ a ~ m ~ \mathtt { ( \mathtt { N \otimes \mathtt { i n \ ( \phi _ { 0 } \otimes \mathtt { i n \ ( \phi _ { 0 } \otimes \varphi _ { 0 } ) } ) } } ~ } } } \end{array}\tag{9}
$$

$$
\rho = \texttt { R } ^ { 2 } + \texttt { R } _ { 0 } ^ { 2 } = 2 \texttt { R } \texttt { R } _ { 0 } \left[ \cos \theta _ { 0 } \texttt { c o q } \theta - \mathtt { S i n } \theta _ { 0 } \texttt { S i n } \theta \texttt { C o u } ( \phi - \phi _ { 0 } ) \right] ^ { 1 / 2 }\tag{90}
$$

We can solve for a in terms of R, 0, Oo, B, and p to be ${ \mathfrak { o } } _ { \mathfrak { o } } ,$

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

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 fashior for computer study.

Sensitivity. The magnitude of the electric field at any point in space can be found from equation 74 when $R + R _ { o }$ and $\theta = \theta _ { 0 }$ . Equation 74 becomes equation 87 which is assumed linear throughout the perturbation. The magnitude of the electric field is

$$
\begin{array} { r l r } { \mathbb { E } _ { 0 } = \frac { \displaystyle \mathsf { q } } { \displaystyle \mathsf { i } _ { \mathrm { ~ f ~ r ~ } \in \mathbb { N } _ { 0 } } \mathbb { R } _ { 0 } ^ { 2 } } } & { { } \left[ \mathbb { 1 } - \frac { \displaystyle 2 \mathbb { R } _ { 0 } ^ { 2 } \left( \mathbb { R } _ { 0 } + \mathbb { L } _ { \mathrm { ~ C ~ O ~ E ~ } _ { 0 } } \right) } { \displaystyle \left[ \mathbb { R } _ { 0 } ^ { 2 } + \mathbb { L } ^ { 2 } + \mathbb { R } _ { 0 } ^ { 2 } \mathbb { ~ L ~ C ~ o ~ n ~ } \mathsf { e } _ { 0 } \right] ^ { 3 / 2 } } + \mathbb { 1 } _ { \mathrm { ~ f ~ r ~ f ~ } \in \mathbb { Z } } \mathbb { 1 } _ { \mathrm { ~ f ~ r ~ f ~ } _ { 0 } } ^ { 2 / 2 } \right] } & { } \\ { \mathbb { E } _ { 0 } = \frac { \displaystyle \mathsf { R } _ { 0 } ^ { 4 } } { \displaystyle \mathsf { i } _ { \mathrm { ~ f ~ r ~ } \in \mathbb { N } _ { 0 } } \mathbb { R } _ { 0 } ^ { 2 } } } & { { } \left[ \mathbb { 1 } _ { 0 } ^ { 2 } + \mathbb { L } ^ { 2 } + \mathbb { 2 } \mathbb { \ R } _ { 0 } ^ { 2 } \mathbb { \Lambda } _ { 0 } \mathbb { \Lambda } _ { \mathrm { ~ C ~ O ~ a ~ } _ { 0 } } \right] ^ { 2 / 2 } } & { } \end{array}
$$

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5C

The angle of this field relaive to the y axis in the $\mathtt { R } _ { \mathtt { O } } \mathtt { \Gamma } \mathtt { - } \mathtt { J }$ axis plane is given by equation 75 which for $R = R _ { 0 } , \theta = \theta _ { 0 }$ becomes

$$
s = \tan ^ { - 1 } \left[ \frac { \sin \theta _ { 0 } } { \displaystyle \sum _ { 0 } s ^ { 2 } + \texttt { L } _ { 0 } ^ { 2 } + 2 \texttt { R } _ { 0 } \texttt { L } \cos \theta _ { 0 } } \right] ^ { 3 / 2 } ~ \frac { \sin ^ { 3 } 2 } { \texttt { C o n } _ { 0 } } \left[ \begin{array} { l l } { \hat { \log } _ { 0 } } & { \displaystyle \sum _ { 1 } s ^ { 2 } + \texttt { L } _ { 0 } ^ { 2 } + 2 \texttt { R } _ { 0 } } \\ { \hat { \log } _ { 0 } ^ { 2 } + \texttt { L } _ { 0 } ^ { 2 } + 2 \texttt { R } _ { 0 } \texttt { L } \cos \theta _ { 0 } } \end{array} \right] ^ { - 3 } / 2 ~ \hat { \log } _ { 0 } ~ \texttt { - R } _ { 0 } ^ { 2 } ~ \texttt { L + R } _ { 0 } ^ { 3 } ~ \cos ~ \theta _ { 0 }  ~ .\tag{93}
$$

This field crestes a pertubation field in a remote object of radius r and dielectric e. The potential of the perturbation field is given by equation $\varepsilon _ { \mathbf { x } } \cdot$ 86 which is

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

Unfortunately, p and a are in terms of a secondary coordinate system. Our primary coordinate system is R, 0,\$. Equation 90 expresses p in terms of $R _ { 0 } , R _ { 0 } , \theta _ { 0 } , \theta _ { 0 } ,$ \$, and $\spadesuit _ { \bullet }$ . This is the first place where the perturbation angle $\spadesuit _ { \bullet }$ is important. The expression for p is

$$
0 = [ \ R ^ { 2 } + \ R _ { 0 } ^ { 2 } - 2 \textrm { R R } _ { 0 } \cos \theta _ { 0 } \cos \theta - \sin \theta \sin \theta _ { 0 } \cos \theta ( \phi - \phi _ { 0 } ) ] ]\tag{95}
$$

The angle a can be expressed in terms of $\mathbb { R } , \mathbb { R } _ { 0 } , \mathbb { \Theta } _ { \mathrm { p } } \mathbb { \Theta } _ { \mathrm { o b } } \mathbb { \phi } ;$ and $\spadesuit _ { \bullet }$

Hovever, it is simpler to express it as

$$
a = 3 6 0 ^ { \circ } - [ \sin ^ { - 2 } ( \frac { 8 \sin ( 0 - \theta _ { 0 } ) } { 0 } ) ] - \theta _ { 0 } = 8\tag{'96}
$$

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

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$$
v _ { 0 } = \frac { 0 } { 1 + \pi e _ { v } } \left[ \frac { 1 } { r _ { 0 } } - \frac { 1 } { \sqrt { R ^ { 2 } + I ^ { 2 } + 2 \pi \times \cos \theta \sqrt { 2 / R ^ { 2 } } } } \right]\tag{97}
$$

Speaking anthropomrphically, the fish knows ${ \mathfrak { v } } _ { \circ } , { \mathfrak { R } } _ { \bullet } , { \mathfrak { L } } _ { \bullet } \otimes _ { \bullet } \oplus _ { \bullet } { \mathfrak { a } } _ { \bullet }$ and 4 π E. It must determine R, 00 o, r and $\mathfrak { a } _ { \mathfrak { o } ^ { \prime } }$ $4 \pi \mathfrak { c } _ { \mathfrak { x } } .$ To find these variables, it makes an analysis of. the potential function $\theta _ { \circ } + \theta _ { \mathbf { z } }$ . Exactly how this analysis is made is unknown.

As an approach to determine hov the fish might operate, we shall find the ratio $\pmb { \sigma } _ { \pmb { \nu } } / \pmb { \sigma } _ { \bullet }$ at different points along the fish for different values of $\varepsilon _ { x ^ { 3 } } , \varepsilon _ { \mathrm { { p } } } , \mathcal { R } _ { \mathrm { { o } } } , \theta _ { 0 } ,$ and $( \phi - \phi _ { \infty } )$ . The equations are rether coaplex, and require a computer analysis. Thus, a fortran computer program was written which manipulates and evaluates +he desired variables.

In the computer program, certain variables have been assigned values for reasons that are discussed below.

The length f 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 cbject 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 vater. In other words the analysis is in terms of $\rho _ { y }$ rather than $\overline { x } _ { \mathrm { ~ } } \circ \overline { x } _ { \mathrm { ~ } } \in _ { \mathfrak { v } }$ themselves.

The fish is defined as a cylinder two times as long as the electxic 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 bunds, 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 samc plane as the center of :he pertirbing object and the axis of the cylinder.

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The computer program analysed various combinations of four factors: 1) the ratio of $\epsilon _ { y } / \epsilon _ { x }$ 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 surface,(0) the angle formed by the intersection of line segment R and the eylinder axis, (φ - o)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 θ $( 0 _ { x } / \ y _ { o } )$ 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 electroreceptors on the head of the actual fish).

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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 2 detect sounds 1/l,000,000th normal speech Loudness) the fish can easily detect objects of dielectric 0.l times vater whose radius is 0.l generator organ lengths, at distances to the side of 6 generator lengths. It would have difriculties at 10 lengths or 1or dielectrics of 0.5 or 5 times water.?

With the assumed sensitivity, the fish could detect objects odielectaic 0.1 vater, assuming object radius is 0.1 generator organ lengths at:10 eleetric organ lengths if approached from front or rear.

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

An indication of receptor sensitivity for objects of dielectric l0 times vater 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 dieleetrics 10 times water as it is for objects with dielectrics O.l times water.

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

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System Level

Although the data does not exist for the fish, it is reasonable to assume that it has a data processing capability similar to that found in other comparable organisms. Thus, it is likely thmt the fish can use irhibitory and facilitating circuits to sharpen the aforementioned data from the receptor, extract signal from noise and classify multiple incoming signals. In essence, it would function as a system at the receptor level vith interactions among receptors and at the whole organism level involving the receptors, generator, und brain. For example, there is evidence that & plot of receptor potential along a band of receptors would yield, for a single perturbing object, a unimodal curve (Hagiwara & Morita, l963). We might suggest that the configuration of the curve is.a function of the overall impedence of the perturbing object, defining impedance as the su of the resistance and the reactance of the object. The reactance of an object is given by

$$
x = 2 \pi r \frac { 1 } { 2 } \frac { 1 } { 2 \pi r }\tag{98}
$$

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 cooráinating the generator and receptor function, the fish by changing generator frequency can induce a lover or higher ovorall effective impedance in the object. If an object had an impedance very similar to that of water, the fish couid 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

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the fish was trying to locate a certain known object, it could adjust its frequency to optimize its detection of the object. Using this system approach it would also be possible for the fish to sense dirferences in objects.-that have the same exterior physical appearance. This would be done through varying the frequency and sensing and comparing the changes in the reactance of the objects. At least some species of fish seem to be using the foregoing system approach. In the earlier discussion of the available biological data, it was noted that s generator frequency shifting technique vas 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 exemple, to detect a moving object several scans could be carried out, stored, and compared. In this vay, stationary objects in the field would be nulled and only objects of changing impedance or location vould be perceived.

CONCLUSION

Although there is a fairly substantial data base, ve 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 invescigators. One of the prime deficiencies in the reported vork 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, ve undertook several tasks to provide a basis to assess the fishes' electrosensing mechanism and capability, using the data presently available.

Through limited experimental work vith electrical fields, sensors, and objects in various size bodies of water we have gathered data which, when

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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 éxperiments, 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 mechanisn. This bypothesis 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 bypothesis may also provide a basis for analyzing higher interactions in the fishes' nervous system and thereby increase our understanding of the sense.

We have also, through mathematical analysis, shown the linkage among the various neural coding schemes suggested for the fish and have shown their essential identity.

We have also developed a mathematical model of the fish based upon the useable experimental data. A set of equations describing function was developed 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 mumber 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 analsis, ve

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

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APPENDIX

Symbols

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

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

00 The angle between the Y-axis and the radius vector from the charge to the center of the perturbing object.

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.

RO The radius vector from the positive charge to the center of the perturbing object.

申 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-axis 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.

The dielectric of the vater.

cx The dielectric of the perturbation.

r The diameter of the spherical perturbing object.

The electric field at the center of the perturbation as if the perturbation were not present.

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The potential due to the interaction betveen the dipole field and the perturbation.

W A coordinate centered at Mc perturbation opposed to the vector $\tt _ { o }$ and in the Y axis -R, plane. $\mathbf { - R } _ { \odot }$

A coordinate perpendicular to the v axis in the Y-P, plane $\pmb { \mathrm { \check { z } } } - \pmb { \mathrm { \check { z } } } _ { \pmb { \mathrm { \check { \diamond } } } }$ originating at the center of the perturbation.

B The angle between E and a line parallel to the Y-axis at the center $\mathtt { e } _ { \circ }$ of the perturbation.

p A radius vector in the w, m coordinate system to' an arbitrary point in space.

The angle betveen the radius vector and the coordinate v.

É+ The electric field due to the positive charge.

$\widehat { \sf \sf { \sf \sf { \sf { \sf { \sf { \sf { \sf { \Lambda } } } } } } } } _ { \overline { { \sf { \sf { \sf { \sf { \Lambda } } } } } } }$ Unit vector in the direction YR

E The electric field due to the segative charge.

$i _ { \overline { { \overline { { \mathbf { r } } } } } } .$ Unit vector in the direction of R".

$\pmb { \mathrm { g } } _ { \pmb { x } }$ \_z The X-z component of the electric field.

${ \mathfrak { v } } _ { \circ }$ Potential due to uniform approximation of the dipole field.

${ \mathfrak { v } } _ { \mathbf { 1 } }$ Potential inside sphere due to uniform approximation of dipole field.

$\pmb { \ 5 }$ The Y component of the electric field

y End point of charge path.

Radius of sphere in dipole arrangement.

I Current

P Macroscopic polarization

N The number of molecules per volume.

Pmos Related to the macroscopic polarization (electric dipole moment per volume).

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${ \pmb x } _ { \pmb { \varrho } }$ The electric susceptibility.

Storing force.

x End point of charge path

$\mathfrak { P } _ { 1 \mathbf { a d } }$ Induced dipole moment.

T Threshold function.

$\pmb { \tau } _ { \mathrm { m a x } }$ Maximum threshold.

$\pmb { \underline { { \tau } } } _ { \mathbf { m } 4 \mathbf { a } }$ Minimum threshold.

$\clubsuit _ { \mathfrak { c } }$ Exponential time constant.

q Charge

y+ . Dummy variable (positive charge).

y-Dummy variable (negative charge).

V Absolute voltage.

4 Voltage

C Capacitance

J Current density.

D Distance betveen charges

A A constant to be determined from the boundary conditions.

B A constant to be determined from the bcuidary conditions.

x Reactance

L Inductance

Potential in terms'of the coordinate system.

$^ { \omega } \circ$ Radian irequency of harmonic oscillation.

J Reactance in polar corrdinates.

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