ON THE INATURE OF ELECTROSENSING IN THE FISH August 1971 ABSTRACT An evaluative review of the electrosensing literature was carried out with the intention of determining the nature o-l' the electrosensing mechanism and its sensitivity. It was found that the biological data base was weak. It was, however, useful in the development of a mathematical model and mathematical analyses of the sense mechanism and its function. 5 In the course of the analyses, we suggest a working hypothesis on the nature of the sense mechanism. We also collapse the various sensor coding schemes that have been proposed into one scheme. The function of the mathematical model of the sensor that was developed was explored with the use of a cnmputer. The fishes' function at the system level was also considered and possible..-mechanisms defined. TABLE OF CONTENTS ABSTRACT INTRODUCTION .................................................. NATURE OF THE BIOLOGICAL SYSTEM ...................................... 2 Generator Organ ..................................................... 3 Receptor Organ ................................................. 0 ...... T 2 Gymnotid receptors. Mormyrid receptors. System Function, Measurement Technique & Sensitivity .................. 11 Electrophysiological & behavioral techniques Size of tank requir d for valid experimental data. POSSIBLE RECEPTOR MECHANISM AND NEURAL CODING ......................... 1T Mechanism. 2Coding. MODEL: DEVELOPMENT, FUNCTION, AND SENSITIVITY ........................ 2T Receptor Level ....................................................... 28 Development. Function. Sensitivi System Leve'L ........................................................... 54 CONCLUSIONS .................................5.......................... 55 REFERENCES ............................................................ 58 AppENDix .............................................................. 61 INTRODUCTION It has only been a short time since certain fish were identified as having a previously unknown sensing system, an electrosensing system. It was observed that these fish apparently detect and classify objects that enter into and perturb a weak electrical field that the fish itself gener- ates. With further investigation it was found that this sen2se is more generally found among fishes than was first thought. Data also appeared indicating that some fish, such as the shark and goldfish, use a passive electrosensing system in that the fish does not seem to generate its own electrical field. Rather, it seems'to detect electrical signals, possibly muscle potentials, generated by objects coming into its area. Although there is now a fairly substantial d2ata base, we find that very little can be applied to the development and understanding of sense mechanism and sensitivity. This is due in part to the fact that pioneering data in this area, as it is in most areas, tend to have faults no matter how competent the investigators. Further, the data base contains very little behavioral data. Thus, there is little information available on system sensi- tivity and function2. In sum, though there are individual investigators contributing quite useful data to the data base, as a whole the data base is weak. Thus, we have undertaken several tasks which may allow an assessment of the fishes' electrosensing mechanism and capability, using the data presently available. First, through limited experimental work with electrical fields, sen- sors, and objects in various size 2bodies of water we have gathered data which, when taken with the mathematical analysis, allows us to interpret much of the data now available. This analysis also provides a specification for tank size, fish location, and attacnments, that will yield valid data in future studies. Second, we have suggested as a working hypothesis an electrosensor mechanism. This hypothesis is subject to test and ',hereby may provide the means for collapsing the current multiple crude categorizations of the re- ceptor that is so typical of a new area of investigation. The hypothesis may also2 provide a basis for analyzing higher interactions in the fishes' nervous system and thereby increase our understanding of the sense. Third, we indicate in the following the linkage among -,%--,he various neural coding schemes suggested for the fish and show their essential identity. Fourth, we develop a mathematical model of the fish based upon the use- able experimentpl data. A set of equations describing2 function is developed on the model. These equations are linked to available experimental data. The mathematical model is analysed by a computer to ascertain the sensitivity requirements of the fish at the receptor and to determine the effects of mani- pulating a number of variables. These variables include fish size, object size, object 'electrical characteristics, object distance from the fish, direction 2 and angle of the object from the fishes' axis, etc. We briefly discuss the fishes' function at the systems level and close with our conclusions concerning the electric sense. NATURE OF THE BIOLOGICAL SYSTEM Both marine and fresh water species of strongly and weakly electric fish have evolved. Strongly electric fish are defined as those that dis- charge their electri5c generating organs reactively to stun prey or resist capture. Weakly electric fish are defined as those that detect and classify objects by the object perturbing the electrical field formed by the electric 3 generating organ which normally emits a continuous pattern of pulses. The- electric field so set up is not strong enough to stun other fish. .There are numerous species of weakly electric freshwater fish but most can be classified as either gymnotids which are South American in origin or mormyrids which are common 2in Africa. The two groups have many similarities and some differences in physical structure and in the function of their elec- trical field generating-organs and receptor organs. Other weakly electric fish include Gymnarchus, an African fish, probably related to the mormyrids, and sternarchid, a South American fish that is probably related to the Gymnotids. Generator Organ An 2understanding of the structure and function of the electrical field generator organ is of importance in understanding receptor function. Thus, generator function will be considered first. The cells of the generating organ are referred to in the literature as electroplaques, electroplax, electroplates, or electrocytes. We shall follow Bennett(1970) and use the term electrocytes. The electrocyt2es are derived from the mesoderm (Szabo,,- 1966),the same type of embryonic tissue as muscle except in the South American family Sternarchida.e. 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 mesodermql origin are typically disc shaped, but may also be dr2um shaped or tubular. Electrocytes of ectodermal origin are U shaped processes from the spinpl cord. The electrocytes of the gymnotid, Hypopomus, are between 300-500 p in diameter and about e'00 p thick. The electrocytes of Sternopygus on the other hand are rod-shaped and much longp-r 4 than those of HypoDomus. They are about 1-2 mm in the anterior poste rior direction and 200 p in diameter. These cells are packed together tightly with little extracellular space, whereas the electrocytes of Hypopomus.-are 2 separated by a considerable amount of extracellular space. The electrocytes are "stacked" in columns in the rear portion of the fish's body to form the electric generating organ. For example, the electric organ of Gnathonemus, a mormyrid, is located just in front of the tail fin and extends forward less than 1/5 of the fish's body length. Gymnarchus' electric generating organ ext2ends from the tail fin to nearly the midpoint. The generating organs of the gymnotid Gymnotus, and of Sternarchus extend further from the tail fin almost to the back of the head. The weakly electric freshwater fish can be categorized in terms of patterns of discha--ge: those with variable frequency and those with con- stant frequency. Constant frequency fish are defined as those that discharge 2 their electric generating organs at a virtually constant rate even when strongly stimulated by an experimenter. Some of these ar e Eigenmannia, Sternopygus, and the sternarchids. These differences are not absolute, how- ever, and there are species di-fferences in basic rate. The generating organ of the mormyrid Gnathonemus for example, is reported (Bennett, 1970) to dis- 2charge at frequencies of 30-100 pulses per second (pps). Gymnarchus is re- ported to discharge at a frequency of about 250 pps; Gymnotus has a frequency rate of 40-60 pps; Eigenmannia emits pulses at a rate of 250-400 pps; Sterno- pygus fires at 60-100 pps; Steatogenys emits pulses at 40-60 pps; and Hypo- pomus at 2-20 pps (Hagiwara and Morita, 1963). 1 Sternarchi6ds discharge at 1. Each type of fish has a waveform that is specific to itself. Therefore, although Gymnotus and Steatogenys have the same.frequencies, their wave- ,orms.are different. These differences in waveform may be functions of the experimeritf-rs' comt)etence in engineering. 5 rates of 600-2000 pulses per second (Erskine, Howe & Weed, 1966). Fish that are reported to emit at variable frequency generally increase their discharge rate markedly when stimulated. Fish that exhibit this characteristic are the mormyrids (Mandriota, et al, 1965), Hypopomus, Steatogenys, and Gymnotus (Larimer and McDonald, 1968). It should be noted that constant frequency fish do vary their frequency under certain circumstances. These circumstances include the presence of another signal with frequency close to the fishes'. For example, Eige=annia 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 co2ntext, also, is the observation that Gymnarchus temporarily ceases its discharge entirely when presented with a signal mimick- ing another Gymnarchus or when startled (Bennett,1970).2 The mechanisms for controlling electric organ output are in the med- ullary portion of the brain and appear to be similar among weakly electric fish. A small group of cells in the medulla are autoactive and fire syn- c2hronously, apparently acting as a pacemaker. Their discharge appears to trigger another group of cells in the medulla commonly referred to as med- ullary firelays". Axons from the medullary relay cells descend as part of the spinal cord to synapse on spinal relay neurons. These in turn communi- cate the signal to the electrocytes. The electrocytes of the electric gen- erating organ fire synchronously because 2of one or more compensatory mechan- isms in the relay pathway from the pacemaker cells. One mechanism is vari- ation in length of the pathway to the electrocytes. The axons to the more distant electrocy-tes extend in the straightest possible line but those to the less distant electrocytes follow a circuitous pattern. A second means of maintaining synchronization involve a delay line mechanism whereby the pathways 4to the electrocytes differ in conduction velocities. 2. If a passive electric sense is more common than is thought, this could be a protective reaction. A number of investigators have measured the voltage output of the generating organ. Hypopomus is reported to generate a voltage of 8 volts peak to peak when electrodes are placed on the head and tail with the fish more or less out of the water. The same fish in water is reported to generate a voltage of from 10 to 200 millivolts. The in-water measurements were taken with two stai2nless steel electrodes, one placed in front of the fish and one placed behind the fish. The distance between the electrodes was not given nor was the distance between the electrodes and the fish given. In general, we find that inadequate information is given in the reports of voltage measurements of the electric organ output. Based upon the inadequate information that is reported on voltage 2 measurements and upon measurements that we have made in water, we would ,suggest ignoring the measurements reported in the literature. In measure- ments in our laboratory simulating the reported data, we found the" the water acts as a very high distributive resistance. When an oscilloscope is used in the typically reported fashion to measure the fishes' voltage output the input impedence of the scope is bein2g placed in parallel with the resistance of the water. Even when a high input iinpedence scope is used, there is a loading effect upon the circuit. Thus, we believe, based upon our measurements and the reported investigations, that the investigators have been inadvertently loading down the fish's electric field generator through the use of their measuring devices. We can summarize th9e salient points by saying that these fish generate a pulsed electrical field in the water. The generator is located in the posterior portion of the body. The generator components have their outputs synchronized by a clock. In some species the clock is more or less invarient, in others it varies, in part, as a function of external events. The reason 7 for this difference among species is unknown. The voltage output of the generator and the effective range of the field are unknown due to inadequate measurement technique. Receptor Organ The weakly electric freshwater fish are reported to have both active and passive sensory systems2. The active system primarily detects disturb- ances in the fish generated E field. The passive system is primarily sensi- tive to energy provided by extrinsic sources. We are not so sure that the data replly indicates two such systems in the same fish, but we shall follow the convention for the time being. There is better evidence that there are a number of fish, such as sharks and gold fish, that have good pass2ive electrosensing systems but no active system. These latter fish and passive systems are not considered, as such, in this paper. Gymnotid receptors. There are two basic types of electroreceptor organs reported in the literature. The differences may be more apparent than real in terms of function. The ampullary organs are believed to be the passive system sensors. They consist of2 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 ba@kground firing shifts smoothly to a higher or lower rate in response to the electricsl sources moving into the fish's range. The response to a brief7 stimulus, for example, is acceleration followed by deceleration. The acceleration phase can outlast the stimulus and according to Bennett (1970) @here is accomodation to maintained stimuli. These receptors are sensitive to low freauency electrical fields and to changes in a DC field. 8 Their response to an applied current is a monotonic increase. The active system sensors are called tuberous organs. They are more rapidly adapting than tonic receptors. They are sensitive to relatively high frequency stimuli and are insensitive to applied DC. Their firing is related to electric orga2n discharge in that they respond with a train of pulses to each electric organ discharge. Thus, they are referred to as phasic receptors@ As seen on the skin, the ampullary and tuberous organs differ. They also differ in appearance from mechanoreceptors, i.e., canql organs and free neuromasts. The tuberous organ appears on the skin surface as a single small2 pore, even though it has no opening. The ampullary organs appear as a group of small pores. As an indication of the number of recept- ors found on a fish, it can be noted that Lissmann and Mullinger (1968) found that there were 2,000 ampullary and tuberous organs on a 6 cm. long Steato- genys. Most receptors, about 95 percent, are phasic receptors according to Lissm2ann and Mullinger (1968). In considering the fine structure of the receptor organs, it can be noted that the ampullary organ has the appearance of a flask with a narrow duct (5-20 p in diameter) leading from the skin surface to a cavity (30- 40 p in diameter) that is located 100-500 v within the skin. Embedded in the cavity wall with only a small 2surface exposed are the sensing cells of the organ. These sensing cells are 10-15 p in diameter with each organ containing two to eight of them. Some microvilli 0.8 p long are irregularly distributed on the exposed surface of the sensing cells. Filling the duct and cavity is a jelly-like substance with no known function. All sense cells in one organ feed their s3ignals to the same myelinated nerve fiber. The nerve is unmyelinated within the organ, having lost its myelin sheath and dividing before entering t@if, orr,,iri. DUCT WALLS MICROVILLI RECEPTOR C L SUPPORTING CEL -NERVE FIBER LOOSE EPITHELIAL CELLS OVERING CELLS 2 ELLULAR LAYER OF CAPSULE RECT:PTOR crLL. NFR,VL TERMINAL UPPORTING CELL NERVE FIBER -SVJN SURFACE 2 ELLY SPHERE RECEPTOR CELL MUCOUS SUBSTANCE FACEPTOR CELL ASEMENT MEMBRANE SUBSENSORY PLATFORM NERVE 2 -RECEPTOR CELL SUSSENSORY PLATFORM BASEMETJT MEMBRANE -NERVE Fig.1 a) Schematic drawing of the two types of ampulla of gy=o- tids, b) Schematic drawing of the tuberous organ of the gymnotid, 7 c) Schematic drawing of the mormyromast of the mormyrid, d) Sche- matic drawing of the tuberous organ of the mormyrid. 9 There are a great many clusters of five to fifteen ampullary receptor cells on the head. On the body there are fewer clusters and they tend to be restricted to 3 bands that extend longitudinally along the fish.3 The tuberous organ consists of a bulb shaped invagination of the skin as sh2own 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 -p. The bottom of the bulb is made up of supporting cells upon which the numerous sensing cells rest. The sensing cells are 25-30 P long and project somewhat like rods into the cavity of the bulb. They are ordered such that the gap between adjacent sen2sory cells is relatively constant. Each sensory cell is covered on the cavity end with microvilli 0.7 P 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 eel-Is feed their signals to a single nerve which, in most cases, loses its myelin2 sheath where it passes into the tuberous organ. In a smpll proportion of the tuberous organs the myelin sheath is'retained until the nerve fiber enters the sensory cell. The tuberous organs are randomly dis- tributed on the head, where,,they are most numerous, and on the anterior half of the body. On the posterior half of the body the tuberous organs are found- in four lo2ngitudinal bands. Mormyrid recedtors. In Mormyrids, the electroreceptors are referred to as mormyromasts and Knollenorgans (Szabo, 1967). The mormyromast is a two level organ that c6ntains 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 cell8s of the tuberous organ of the gymnotids. Types A and B sensory cells are always separately innervated. 3. The fish being described is Hypopomus artedi, a species of gymnotid. Details vary slightly from species to species. 10 The type A sensory cells forn one or two concentric aureoles at the base of a "Jelly sphere" located near the surface of the skin as shown in Fig. lc. In the center of this aureole, a small duct leads to a more deeply situated sensory chamber in the skin within which the type B cells are located. The inner surface2 of the duct wall bears tiny microvilii. The duct as well as the lower sensory chamber is filled with a-.=-,cous substance. Two to five sensory cells occupy the lower sensory chamber. The type B cells with their supporting cell platform though similar to the tuberous organ are smaller. They do not completely fill up the sensory chamber and their free surfaces bear a large numb2er of microvilli. The type B sensory cells in a mormyromast are innervated by a single nerve fiber which splits immediately after penetration through the supporting cel-Is into several branches to serve the sensory cells. Where the nerve joins the type B sensory cell membrane a rod like projection, 0.5 p in size, occurs within the sensory cell. Each type A sensory cel2l is encircled by several accessory cells. The sensory cells and their accessory cells are bottle-shaped. The apical or tip portion of both sensory and accessory cells contact the jelly sphere. The nerve fibers innervating type A cells lose their myelin sheath before entering the receptor organ and pass among the accessory cells to contact the sensory cells. As with type B cells, where the nerve joins the sensory 2 cell, there is a rod present at the sensory cell membrane. The mormyrids also have receptor organs, knollenorgans, which are some- what similar to the tuberous organs of the gymnotids. Derbin and Szabo (1968) describe them as being composed of three or four sensory cell complexes one of which is shown in Fig. ld. Each complex is a single sensory cell attached to a highly differentiated supporti7ng platform of cells. The organ is inner- vated by single nerve fiber which is derived from a nerve that appears to serve many sensory cells. The sensory cell lies in and almost completely fills a cavity in the skin at the surface. The wall of the cavity is formed by flattened-epithel- ial cells. The interior epithelial cells have microvilli-like processes which densely pack the space about the sensory cell. The cavity has a rela- tively large opening toward the support2ing cells through which the sensory cell cont-acts the-nerve endings and supporting cells. The sensory cell itself 35-40 ii in diameter. In sum then, the weakly electric fishes of South America, the gymnotids and of Africa, the mormyrids both seem to have receptor organs that are similar in some respects but differ in other respects. Though there are structu2ral differences in receptor organs within and between species, the evidence suggesting that there are differences in function is rather weak. We shall now consider this matter of the receptor organ and system funct-@-(-)Li. System Function, Measurement Technique,and Sensitivity. This section will of necessity be short since the-ze,is relatively little da2ta which is acceptable from bo'.h a biological and engineering standpoint. I'hus, we will discuss the three primary techniques %ha+. have been used to obtain data on function, discuss their deficiencies, and estimate from the data the probable system functior- and sensitivity. Two of the techniques are electrophysiological ard the third is behavioral. Electrophysi2ological and behavioral techniques, In one electrophysio- logical techniciie the fish is anesthesized and fixed to a wooden plate in the normal swimming position. The wocden plate is then tilted into the water so that the body is submerged and the head exposed to the air. The regular respirat(=y movemerits and oxygenat'@on are maintained by spi-aying a fine jet of wat0er into the mouth of the fish. The dorsal branch of the lateral line 12 nerve which lies i=nediately under the dorsal skin at the head, is then - surgically exposed. After desheathing it, fine nerve strands are separated by microdisection. Then silver-silver chloride electrodes are applied to a strand and single nerve fiber responses are recorded under various stimula- tion conditions2. The other electrophysiological technique involves restricting the fish's movement by placing it in 3 to 5 inches ofwater in a small glass or plastic tank. Electric discharges are then detected with monitoring equipment connect- ed to the water via electrodes suspended in the experimental tanks. The data obtained by the above provide insight into system operation but are not very useful in eval2uating the function or sensitivity of receptors or systems. First, in those cases where anesthesia was used, a question can be raised on the effect of the anesthesia on neural function. Second) the investi- gators were looking only at the isolated sensor signal under grossly abnormal stimulation conditions. Third, the isolated sensor data, even if collected under reasonably normal stimulation conditions reveal little abo2ut system function. Fourth, the engineering is typically questionable for one reason or another. This fourth reason is also the prime problem with the yet to be described be- havioral technique. For example, Agalides (1965) did extensive work on these fish, much of it being excellent. However, he used a small tank which would distort the fishes' field, he did not control impedence within normal limits, 2 & he had extraneous objects in the fishes' field. Clark, Granath, Mincoff & Sachs (1967) used stainless-@teel electrodes which distorted the fishes' field. Hagiwara, Szabo, Enger & Suga (1965, 1967) all show waveforms in their reports which appear to be riding on an increasing DC potential. It appears as though their electrodes underwent a significaiit polarization during the experimen1t. The experimentors will not offer an explanation for *,his observation. Mandri- otals investigations (1965) are characterized by verf poor experimental techniques. Not only did he use silver electrodes, a small tank, etc, but he used as a punishment with his behavioral training technique an electrical shock sufficient to visibly jerk the fish; shock while studying the function of ele--trosensing fish. The foregoing is sufficient indication of the deficiencies encountered. We shall turn now to the behavioral technique that has been used, the te2chnique that can most directly answer the question of sensitivity. In this technique, the free swimming fish is conditioned to respond to a certain stimulus. When it responds correctly it is rewarded. The stimuli used have been an applied voltage gradient across the fish's tank or objects of different conductivity hidden within clay pots. With this technique, the sensitivity and function of the entire 2system can be tested. The lim-,Lts of sensitivity found can best be simtned up by stating that the fish could detect the presence of a glass rod 2 millimeters in diameter in a clay pot but would fail to respond to a glass rod of 0.8 millimeters in dia- meter in the pot (Lissmann, 1958). This limited statement of sensitivity is as mu2ch as the state of the art provides. And even this statement can be question- ed since the tank used does not meet the specifications derived below. Somewhat akin to this behavioral technique have been a limited number of data gathering expeditions into the fishes' natural environment. The published results are rather limited. About the only thing that has been found is that the fishes have about the same 2pulse 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 repor1ted to be quite sensitive and qualitative observations 14 would seem to bear this out. But for reasons indicated above, there is no adequate quantitative data. Size of tank required for valid experimental data. One of the prime deficiencies in the reported work-is the use of a tank of inadequate size or with extraneous objects in the field. These distort the field and seriously 2 effect the data obtained. We have experimentally explored the effect of various objects and tank size on a simulated fish field and found that all objects and even the walls of small pools distort the field to some extent. A quantification of this effect is defined in the calculations presented below in which we determine the specification of the tank needed for acceptable experimental work. 2 We assume that the fish is located centrally within a cylinder. With this assumption, we study how the potential varies as a function of cylinder length assuming an infinite radius for the cylinder. Next, we assume the cylinder has infinite length and see how current varies with radius. With this information, we will be able to determine reasonable lengths, widths, and depths for experimental containers2 for electric,fish research. We will disregard all interfaces in this'-development because our ultimate intention is to determine when these interfaces can be disregarded. The equations which express potential as a function of distance areh L L for y > -:@ + a v L 2 4 ir c y 2 (y + L2 -4. The five unnumbered equations used in this section are developed in a later section. They axe numbered in the later section as 27a, 27b, 27c, 48,51, but a-opear in this order here. Definitions of symbols can be found in the appendix. 5 for L a > y > L + a V 4 7r c + y 2 2 L L 2 L for a > y V 4 n e L y (2L + y 2 Plotting for different values of L in Fig. 2a we obtain the required cylinder length. Only the positive direction is plotted because the negative direction is identical except the 2sign is reversed. How long the cylindrical tank should I be is difficult to determine precisely. As a minimii?n though we can say that there should be 5 electric organ lengths of water in front and in back of the fish at all times during the experiment. To determine the cylinder radius required for the tank w, we can modify the limi2ts on the integral expressing the current I in equation 48 of our later development. This equation is co 2 7r Q L R 2 2 3/2 d d R 4 1T E R + L 0 0 The only limit which needs to be modified is the infinity symbol. We replace2 this with w and solving as before we find the current to be w Q L a I 2 2 1/2 2 c R + L 0 Equation 1 may be expressed in closed form as L/a or %@,'Z" V12 i.2 "I 0.9 0.8 2 0.7 0.0 0.5 0.4 0.3 Q 2 L 3 L 5 L 7 L 9L [2IL 13L ORGAN LENGTH 100 to z i-I so cc 70 2 60 50 x c 40 30 0 20 I-- z 2 10 de 0 L 3L 5L 7L 9L liL 13L REOTARED WIDTH AND DEPTH IN ORGAN LENGTHS 2 Fig.2 a) This illustrates the potential on the positive side of the dipole electric generating organ. The potential on the negative side is the negative reflection of this. b) This indicates the percent of maximum electric generator current which will flow between the posi- tive and negative ends of the organ as a fanction of the width and 0 depth of the tank, assuming a free field expression for the current generated. Q L a 2 (2) 2 L w2 + L 2 1/2 L The maximum current is expressed in equation 51 of the later development. It is I c By letting Ln d 2solving equation 2 for different values of n, we w 2- Em have obtained a plot of current as a function of width or depth. This plot is shown in Fig. 2b. to obtain 90% of the maximum current requires 5 elec- tric organ lengths on each side of the fish. To get 95% would require 10 lengths. More than 95% of the maximum current would be almost impossible to obtain in the laboratory. Thus for experimental tesults to be considered valid there should be at least 5 electric organ lengths of water surrounding the fish. For free sw-itninirig 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,1 10 organ lengths of water in all other directions should be the minimum. It should also be noted that to simulate infinity in research, the tank must be connected to earth ground and made to conduct. This does not seem to have been done in past research. 17 POSSIBLE RECEPTOR MECHANISM AND NEURAL CODING The possible mechanism discussed below can not be said to be the mech- anism that the fish actually uses in its detection and classification of objects. It is, however, derived from our review and analysis of the available data, from results of the limited expe2rimentation that we carried out to clarify some of the available data, and from our knowledge of auditory and labyrin- thine system function. The postulating of this mechanism, viewing the electro- receptors and auditory receptors as evolutionary derivations of the same primi- tive receptor, provides a testable. hypothesis of receptor function. It also provides a basis that can be of assistance in determining receptor 2 sensitivity. Mechanism. The inner ear is a fluid filled cavity with a complex membrane structure. It is notable for the complex electrical fields that are generated within it by external events and its organized bands of hair-like sensing cells. Ea.rly concepts of pressure waves in the fluid bending the hair cellsand thereby triggering signals to the brain are very much in question. Some of the newer2 concepts implicate an intermediate electrical field sensing mechanism in the hair cells. The precise nature of this is not clear but elements of O'Leary's (1970) recent experimentation and theorizing on the inner ear labyrinthine system appear to be quite applicable to the weakly electric fishes' sensing system. These elements combined with other information on auditory and balance sensor function8 will be discussed below to the extent that they have bearing on our modeling of the fishes' sensing system. Since Dohlman (1960) has shown that hair cell membranes are apparently impermeable to ions, O'Leary assumes that the detection of fields are due to electrostatic forces as opposed to ionic current. In his analysis, he points up that dissipative energy loss of an electric field in a dielectric is generally associated with movement of charge carriers. These movements in an electric field result in an effect called polarization. Van Beek (1967) has pointed out that the average molecular dipole moment P in a 2 moi heterogeneous system is the vector sum of induced (electronic) polarization resulting from the relative displacement of electrons and nuclei, dipolar polarization resulting from the partial alignment in the direction of the field of molecules with permanent dipole moments. and interfacial (Maxwell- 2Wagner) polarization occurring at boundaries between the components of a heterogeneous system. Jackson (1962) has indicated that P is related to moi the macroscopic polarization P (electric dipole moment per unit volume) and the macroscopic electric field E by P = N X E 2 (3) e where N is the number of molecules per unit volume x is the electric susceptibility. e 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 2 hair cells by the polarization it induces in long-chain filaments of poly- atomic molecules in the cilia. Since Van Beer's (1967) studies of dielectric behavior of colloidal solutiom indicate that particles such as polystyrene spheres are frequently surrounded by electric double layers when they are dispersed 'in dilute KC1 solutions, it is conceivable that low frequency5 electric fields polarize the molecules by inducing dipole moments in the 19 double layers. There is also evidence along this line from Heller, et al (1960) & Saito, et al (1966).. Polarization can also occur by a mechanism suggested by Frohlich's observation (1958) that large molecules can have CH 3) C=O, or OH groups that are in themselves dipolar, but have a net dipole moment of2 zero due to the vector sum of all the moments. These molecules then behave like nonpolar molecules in that their polarization are of the induced (electronic) type' with resonant-frequencies in the optical range. With these two possible polarization mechanisms, O'Leary develods a theoretical basis for accepting,an electric field sensing mechanism. This mechanism encompasses more data than a mechanical model. 2 Starting with Ja:ckson's (1962) 6bservation that a charge e which is disdlaced a distance x is bounded by a restoring force F given by F m w 2 x (4) 0 where m is the mass of the charge w0 is the radian frequency of harmonic oscillation He1 goes on to consider the effect of a field on a charge. The action of the field E causes the charge to be displaced a distance x from its equilibrium position. From Newton's third law we know eE = m w 2 x (5) 0 The induced dipole moment is then defined for one electron as e2 E p ex ind 2 m w 0 If there are Z electrons per molecule with f of them bound by a restoring 2 2 force - m w X, then the induced dipole moment is e2 E fA .- E (T) pind m 2 wJ. where Z = E fJ J Temperature2 is not a variable in equation T so the induced polarization would not be disrupted by thermal agitation, Thus, the sensitivity of this effect for the detection of weak E fields would be limited by quantum con- siderations rather than by_the classical limit of kT. These quantum limita- tions are determined by the magnitude of the allowable shifts in energy levels of the molecules caused 2by the field, considered as a small perturbation, relative to the energy levels of the molecules in the absence of the field. O'Leary suggests that this polarization mechanism has great sensitivity. He estimates it for the inner ear with the following argument. If the behav- ioral threshold for stimul-usenergy is indeed close to lkT'-- 4 x 10-14 ergs/ molecule as suggested7 by devries (1949), the corresponding wave number 1/1 for an energy transition of lkT 21 hc/I km (8) would be 1/;k 200 cm71 if this energy were entirely absorbed by a single molecule. The threshold energy would probably be distributed among numerous molecules. Thus, the polarization of a single molecule would occur for 2 energies much smaller than kT, i.e., for transitions of far less than 200 CM-1. The occurrence of hyperfine splittings in the Stark effect suggests a transductive mechanism based on-polarization would be sufficiently sensi- tive for the detection of threshold stimuli. Herzberg (1950) and others have studied spectroscopically the splitting of energy levels by an electric 2 Splittings of about -3 1 field (the Stark effect). 10 cm- have been observed 2 .3 from diatomic molecules with field strengths in the range of 10 - 10 v/cm. Smaller "hyperfine" splittings were then predicted theoretically and observed 2 using high resolution spectrometers. Based upon the foregoing, transduction in the hair cell can be viewed as a quantum amplification process that is modulated by the average microscop- ic polarization of an ensemble o--F' long-chain molecules associated with the cilia. This development suggests, accepting it for the electric fish sensor, that the electric sensor would 3need protection from mechanical stimuli. Both mechanical and electrical stimuli might be detected by this molecular effect in the electrical sensor because mechanical forces should also affect the microscopic polarization of long-chain molecules. The bending or shearing of cilia that contain long-chain molecules could result in a change in the 22 net dipole moment and should also be detected by a transduction mechanism that was sensitive to microscopic polarization. The structure of the fishes' receptor organs suggests that there is such protection. This would be necess- ary to minimize noise in the system. If we are correct in suggesting 2that this is the type of receptor mech- anism that is used by the fish for electrosensing, then it would provide a basis for accepting the qua7-itative statements on the high sensitivity of the fishes' electrosensors. Lacking well controlled behavioral data on sensitivity, we would hesitate though to conclude that the fish is as sensitive as this analysis suggests. We hav2e now considered the structure and function of the generator organ and receptor organ. Now we shall look at the next level of the nervous system. We shall consider the coding of nerve impulses by the receptor Organ, its transmission toward the brain, and the implications of the coding in under- standing the electrosens--'ng function and sensitivity. Coding. As has been noted earlier, the literature2-is encumbered with multiple classification schemes for electroreceptors. This has the potential for confusion and does little to increase our understanding of receptor func- tion. For example, one scheme is based on external physical appearance, e.g. small, medium and large mormyr(--)masts, another on total configuration, e.g. ampullary and tuberous, and there are other crude clas2sification schemes. Two somewhat more useful schemes also exist. One is used by almost everyone who discusses electroreceptors. In this scheme, the output of the receptor is related to the output of the generating organ. The electro- receptors are said to be either phasic or tonic. Phdsic receptors respond (give an output) at some integer division of the generator frequency with 2 a string of from one to ri pul,-,es. Tonic -eceptors maintain a steady output 23 that is essentially independent of generator output, changing as a function of an environmental stimulus. Qualitative equations can be written for the output frequency of the phasic and tonic organs. These are: NF generator N = 1,2,3 ...... F m 2 (9) phasic n n = 1,2,3 ...... F F +. A f (stimulus) (10) tonic resting m v There is a general relationship between the physical classification scheme discussed in an earlier section and the tonic-phasic scheme just 2 described. Receptors fit for the most part into two classes - tonic recept- ors that are sensitive to low frequency stimuli and are of the ampullary type, and phasic receptors which are sensitive to high frequencies and are of the tuberous type. The other useful classification scheme involves the apparent coding employed by the electroreceptor for transmission of information to the brain. 2 One fish, Hypopomus, has electroreceptors which respond to each dis- charge of the electric organ with a succession of short pulses. Each "pulse train" contains eight or more pulses (Hagiwara, Kusano, & Negishi, 1962). The electroreceptors of Gymnotus and Staetogenes respond with one to six pulses to each generator discharge (Hagiwara & Morita, 1962). Some studies 8 indicate that the number of pulses in each train can be related to the potential near the receptor. This has been referred to as "number coding". 24 In another weakly electric fish, Eigermannia, each organ discharge does not produce a receptor output. If the fishes' electrical field is distorted though, we find *.hat the receptor output is at most one impulse per generator organ discharge. If we decrease the distortion we find the receptor o2utput to be one impulse per every two generator organ discharges and so on. In other w(:,rds the chance that a receptor will fire is related to the stimulus intensity acting on it. This type of coding is called liprobability" coding. (Hagivara & Morita, 1962), Another fish, Sternopygus, was throughly studied by Bullock and Chichibu (i965). They found fi2bers that carry one impulse per organ dis- charge. They noted a phase or time relationship with the intensity of the stimulus. This is referred t6 as phase or latency coding. They also found other nerve fibers that maintained a rhythmical firing out of phase with the electric organ discharge. The frequency of these receptors changed as a function of the intensity of the electric2 field near the receptor. This type of coding is frequency coding and is characteristic of ampullary organs. As a weak generalization, it appears that ampullary receptors give tonic responses with frequency coded information and tuberous receptors give phasic responses with either number, phase, latency, probability or frequency coded information. But we can develop a stronger generalization. 2 Hag2Lwara and Morita (1962) suggest a model t'or probability coding based on an assumed nerve threshold. By making certain assumptions about the threshold curve, we-have extended their model to encompass all coding schemes proposed to date. Their model was originally developed for tuberous- phasic receptors, but we can easily extend it to ampullary-tonic receptors4 with a si-mdle modification. We shall develop below this all encompassing coding scheme since it suggests the nature of the system function. 25 The threshold model as reported by Hagiwara and Morita (1962) assumes that after a receptor fires, the 'threshold resets to some high value. The threshold then begins to decay until the intensity at the receptor is equa4 to or momentarily exceeds the threshold. At this point, the receptor fires, and the threshold resets. 2 This will form the basic model on which we will elaborate in order to encom-Dass the different coding mechanisms. First, we will discuss some gen- eral properties of biological threshold curves. Then, we will define the relationships between the threshold curve and the electric organ output which are required by the available data. Lastly, we will detail some of the meas- ures which could be made to quantify the threshol2d function. It is established in biology that nerves can not fire during or instant- aneously after a previous firing. There is also a biological basis for defin- ing the threshold curve as one describing an exponential decay from some value; Tmax the maximum threshold, to Tmin the minimum threshold. In reality, the minimum value probably continues to decay with time. But for situations o2f repeated sampling, we can approximate it as Tmin , 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 Tmin' Tmax or the exponential time constant Tc With these facts in mind, we can describe the threshold function for time after each firing as t 4 T =[TM. T T + T . + S (t) m c min Further, as a very good approximation we could say t e T + T . + S (t) (12) ma.x c min The general shape of such a function is shown in Fig 3a. To describe probability coding, we must assume that 4 Tc llf. In such cases, 2 the electric organ can discharge several times without firing the nerve. The number of times it must discharge before activating the nerve is a function of the stimillus 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 Tmin > 0. If Tmin were zero, the 2sensor would reset independently of the electric organ and one form of tonic coding would be observed. To describe number coding we must assume that 4 Tc < < llf. In such cases the electric organ will cause the receptor to fire more than once each time it discharges. The number of times the receptor will fire is proportion- Al to the intensity at the receptor as illustrated in Fig. 3c. Again, for phasic coding T . must be greater than zero. min As 4 T becomes approximately the same as llf, several interesting c possibilities occur: phasic coding, latency coding, and interpulse interval coding. Phase and latency coding are illustrated in Fig.4a. They are one and the same. Interpulse interval co2ding is illustrated in Fig.4b. It is also interesting that when 4 Tc is about two or three times 1/f a combina- tion of probability and number coding can be anticipated. Such coding has been observed for Sternopygus (Hagi-wara and Morita, 1962). It should be noted that this model does not account for the observed output of phasic receptors while the generator is be2tween pulses. As has T A 0 To,* 0 T, aTe 3Tc 4Tc t TOIK ARTER PRIOR RECEPT2OR OUTPUT IN TI PAC CO#4$TA UTS Too% THITESHOL.D p T p 2 Twa L L i u HIGH MEDIUM INTENSITY LOW INTENSITY INTENSITY 4L TIMC 2 THRESHOLD Toes i c TM T 't TIAL AT Ce'l ER 0 2 MEDIUM LOW lum HIGH L TIME Fig.3 a) This is a mathematical description of the threshold of the receptor to the intensity of the stimulating elec2trical potential as a function of time following prior receptor output. b) This illustrates the relationship between the electric potential at the receptor, the receptor threshold state, and receptor output when the receptor threshold decay time constant is great- er than the rep etition rate of the electric generating organ. In this case the apparent coding i6s commonly referred to as probability or frequency cod- ing. c) This is comparable to case b, but the decay time constant is much less than the duration of the electric organ output. In this case the out- put of the electroreceptot is said to be pulse count coding. TMRCSHOL POTENTIAL AT Too& RECEPTOR A I man NIGH MIEDIUM LOW u 0 2 TIME POTENTIA& AT ITCC PTO HRESHOL To" B 0 lot* HIGH MEDIUM L(NW CO,2 ,nmE Fig.4 a) This illustrates the relationship between the electric potential at the receptor, the receptor threshold state and receptor output when the generator organ rate is just smaller than the recipro- cal of the decay time constant. The output of the electroreceptor in this case2 is typically referred to as phasic or latency coding. b) This illustrates the relationship between the electric potential at the receptor, the receptor threshold state, and receptor output when the generator organ rate is just larger than the reciprocal of the decay time constant. The output of the electroreceptor in this case is typically referred to a0s interpulse interval coding. 27 been noted previously, there are experimental reports that phasic -eceptors have an output apparently unrelated to generator output. We do not know if this is due to this not being the best fit model, to our having ihsufficient data to incorporate those Darticular observations into the model, the exist- ance of a buffering 2capability at the receptor, or if the reports report arti- facts due to faulty technique.5 But whichever is the case, the utility of the model is not affected. It can usefully be used as a unifying framework for studies of threshold, sensitivity, and response time across all weakly electric fish. With such a framework, sensitivity could be well defined experimentally and the mechanism of the sense better understood. 2 MODEL: DEVELOPMENT, FUNCTION, AND SENSITIVITY in outline, the fishes' sensing system appears to function as follows: The generating organ emits an intermittent electric potential or current. This results in an electric (dipole-like) field in the water surrounding the fish. Objects within the environment and also the environmental bound- aries distort the electric field. This distort2ion causes a change in the electric field near the fish which we shall refer to as the stimillus. 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, g7=ar- ch2us was trained to respond to changes in the conductivity of objects placed in a sealed container. Positive reinforcement was a food reward, and nega- tive reinforcement or punishment was the insertion of a metallic object into the fish's tank. This punishment was probably not sufficient to reasonably test the threshold of the biological system. But by using this method Lissmann & Machin (1958) determined a threshold7 to potential charge of 5. If Tmin were below the mechanical noise level, phasic receptors would appear to be tonic receptors. 28 about 0.15 uvl'cm. We sh%;Il 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 lo2cation of the electric generating organ as determined anatomically nor with the function data reported by T@illock 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. 1:@illock and Chichibu (1965) observed the zero potential plane to be perpend2icular to the fish and found it located approximately one quarter of the way toward the head measuring from the tail toward the head. Thus, we use as our model a dipole field as illustrated in Fig. 5a. We will develop a simplified model of the receptor and then discuss system function. We will use the dipole concept described above as well as the hypothesis on sensor function derived 2from above. Through the develop- ment of a set of equations and a computer analysis, we will obtain an approxi- mate solution for the response of the receptors due to perturbations caused by an object in the field. Through this, we will determine critical vari- ables in the sense function and quantify their effect. With the informa- tion so derived, we will consider the practical implicati2ons 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 positivel6y charged to a total charge of +Q. The other sphere is 0 Fi F-i :y :3, til 0 m m @-4 m 04 P, 0 0 0 c+ c+ :y %-n Ili pi P. C+ P. 2 0 0 ul 04 Id (D 0 Fi 0 :1 c+ @O (D m m aN P #i tj M to P3 C+ 0 P2O C+ H.- & P. P. (D r, 04 0 0 En P. (D 04 @-i @r C+ (D 2 ct- En CI- q O P. P F3 P C+ Fi (1) r_ P, m r P, P, ED H- FJ El C+ 0 m 2 cl, 0 " " 0 0 C-4 P, P. P3 C-1- C+ Fi Fim Pi H. P,m 0 0 ol 0 0 C+ C-4 2 o 0 :y P. En c+M c+ M 0 P. M ::$,P m m 00m 13, ti C+ tn0 c+ En P. pi P, Fi- m C+ r2p ti 01.t0 94 Fi C+ P, P. C+ P, En 0 U) En 4 z m " P. C+ 0 m Pi ct 94 Ell C+ C+ (I 'd 0 0 rA m @y P N m 0 0 0 Fi w 0 0 t-J. In 9 Fi P, m 04 negativ'ely charged to a total charge of -Q. We will asstmre that a is much less than L. When this is the case, the charge on each sphere can be-assumed to be evenly distributed about the surface. The electric potential (voltage) is defined to be the increment in work required to move an increment-of charge from infinity to a given point in space, or a 2w V (13) Recall that the fundamental work equation states that work equals the kinetic energy minus the gained potential energy or W = K E - P E (14) If we move a very small charge very very slowly along the line which passes through the two cha2rged spheres, the kinetic energy is essentially zero. The work is the negative of the potential energy. Potential energy is defined by the integral y A f P E F - dr x whereF is the force acting on the test charge as it is moved from x to y 2 dr is an increment of distance in the path between the points x and y The electric force is a conservative force. Thus, if a test charge is acted on by more than one charge, we can determine the potential energy due to each charge and find the total potential energy by addition. In other words if P E represents the potential energy due to the nth charge, and if there are 6 n 30 a total of N charges the total potential energy is N P E P E (16) total n n=l The potential energy due to a charged sphere can be easily found. Th2e electric force may be found from Coulomb's Law which states q_t F r 4 ir c r (lT) where Q is the total charge on the sphere cit is the charge of the test charge r is the distance between the two charges 2 c is the dielectric of the media ir is a vector directed away from the center of the charge on a straight line Using Gauss's Law it can be shown that within the charged conducting sphere there is no electric force. Thus, if the radius of the sphere is a, the force is F Irl < lal 0 (18) 1 If we call the line which passes through the two charges the y axis, we can find the potential energy at any point on the axis by solving equation 15. We substitute the force from equation 1T and obtain ly Q qt p E d r 4 7i c r The solution is ly + I P E '4t 2 ly +1 > lal 4 w c r Go If we assign the dummy variable y- for the negative charge, we can solve for the negatively charged sphere q (21) P E + 4 Tr el y -21 The potential energy for the posi-.;ively charged sphere is Q cit (22) P E + 4 n The total potential energy for the dipole system is P E Q qt 1 1 5(23) total 4 ir c . ly- Fy-+ If we define a coordinate system as shown in Fig.6,we find that the abso,.ute values of y- and y+ depend on our location on the y axis as follows: CONDUCTOR RADIUS G ab y L + 12 /2 Fig.6 Simplified model2 which will be used in the study of the electro- static characteristics of the electric field generated by the fish. The field generating organ is assumed to be composed of two conducting spheres of radius a separated by a distance L. For convenience, the origin is taken to be the midpoint between the two conducting spheres. 32 for y > L L 2 ly+l = y - "24a@ 2 L ly-I = y + 2 (24b) for L > y > L I Y+ I = L 2 2 2 -'y (24c) ly-I = L +y (24d) 2 for L > y 2 2 ly+l = L -Y (24e) 2 ly_l = L 2 'y (24f The total potential energy can be found from equation 23 and equations 24a to f in each reg2ion. Recalling that the kinetic energy is zero and applying equation 13 in terms of the test charge we get: Q qt 1 (25) V 4 7r E ly -1 ly +1 This yields 1 9 1 V Q 4 Tr E ly +1 I Y -1 Substituting the appropriate values of y+ and y- we obtain: Q L for y > L + a V = (27a) 4 y - L)(y + L) 2 2 2 r 2 y for L a > y > L + a V Q (27b) 4 L Y) (L + y (2 2 L for L a > y v 2 (27c) 2 4 L y L + (2 (2 where V is the absolute voltage (V o), is the charge on either conductor, L is the distance between the conductors, 2 c 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 labor- atory. 2 Tet us define v to be the relative voltage between two spheres. We note that v is the value of the voltage at the positive sphere minus the value of the voltage at the negative sphere. By substituting y L a 7 into equation 27b, we find the voltage on the positive sphere to be v I L - 2a 4 7r c a (L-a) 8 34 Substituting y L + a into equation 27b we find the voltage on the 1) negative sphere to be Q L + 2a v (-) = (29) 4 7T E a (L - a) 2 Solving for v we find v v v (30) or Q 2 1 L V (31) 4 1T c Whcn a/L is less than 1/10, equation 31 can be approximated within 5 percent by the relatio----ship 2 V a (32) 2 n c 6 The capacitance of an object is defined as Q C = --q- (33) where Q is the charge on one symetric part of the object V is the voltage across the object,. Determin8ing the capacitance of the dipole from equation 31, it is fourd to be 35 a Q L) (34) c - = 2 n ea v 2a This may be approximated within 5 percent for a/L less than 1/10 as C 2 c a 1 + a (35) L It is useful to determine the resistance of the dipole we have just consider Since we have already evaluated the ed voltage between the charges, if we can find an expression for the current that flows between the two points, we can solve.for the resistance from Ohms Law. v R 4 (36) where R is the resistance v is the voltage I is the current Two equal and opposite charges create an electric field. If we can solve for the magnitude and angle of the electric field E, we can determine the current density i from the relationship 36 where J is the current density a is the conductivity of the media. Once the current density is known, the current I can be found from the surface integral I J . ds (38) f where the integral is over any closed s2urface 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 @- and the field due to the positive charge E+. Symbolically we can write E = E+ + E- (39) 2 where E is the total field Due to symmetry, the most convenient surface to use for our current integral is the plane which forms the perpendicular bisector of the line segment joining the two equal and opposite charges. This plane is illus- trated in Fig. 7a. The electric field due to the positive charge at any point p is defined by the equation E+ 4 -a c D2 6 D+ (40) 37 where D+ is the distance between the positive charge and the point p i is a unit vector located in the direction of D+ away D -from the positive charge at p. The electric field due to the negative charge at the same point p is -Q 2 1 E- D (41) 2 4 ir cD- where D- is the distance from the negative charge i is a unit vector at p directed by the D line away D- from the negative charge. The minus sign in equation 41 is du'e to th2e fact that unlike charges attract. So long as the point p is on a plane which forms the perpendicular bisector of the line segment between the two charges, the distances D+ and D are equal. They can be found from the equation 1/2 2 L2 D D R + (42) We have defined the y axis to be the line which p4asses through the two charges. We note that at the point p of Fig-7a, the electrical field due to either charge has both y and radial components. Due to symite'.ry,how- ever, the radial components cancel each other and the net field in the plane is narallel to the y axis. @lso due to symetry,the y magnitude of the y COMDonents are ec.ual. Thus the total electric field is -2 Q Etotal 4 Tr c D2 sin 0 iy (43) We note that a because they are opposite angles.Sin acan be determined from Fig-7a by L/2 Sin a Sin 8 (44) 2 D Substituting equation 44 into equation 43 -Q L E (45) total 4 ir c D 3 y Substituting equation 42 into equation 45, and equation 45 into equation 37 yields L 2 j 2 (46) 4 7r c + L2] 3/2 'i@ The surface of integr,.;tion is considered to be small ring segments in the x-z plane about the y axis. Fig.Th illustrates this concept. We note that the electric field is perpendicular to 5the x-z plane which makes the integral defined by equation 38 easy to evaluate. A surface element 'for the ring shown in Fig. 7b is ds R d dR (47) En P- EA 10 C+ P, 'lj :1 (DPH- r- H- rL ci' @l -4 (D aq P. P) (D Po0Fl C+ :A 0PPt H 1.40 (D C,+ (D-P) CD 'd FJ C'+ H. El) (D :31 :i '-I C-+ :4 (D C+ C,+ Id00 (D ::r (D C+ ID -,Q (DnC+ (D 11 PI @r (D P(D C+0 :3- C+ C+ 04 (D C.+ C+ (D 2 P. Fl- En (D H- (D (D LI) El) (D0(D P, Fi C+0"0m0 1 1 @:r C,+ C,+ (1) 2P0P0C+ (D EA '-4 (DP :@ U)Pcil M0'd9C.+ (D P(D"0MEl) clt P, P. P, H. C+ @j0C+ 0a' 2::r rt :1 (D (D 110 C+ C.+ 1-Y C.+ (D (D :3' (D 0:3mmH P ::s0C+ (D 2 rn 0C.+ 9L C+ Fl- 3@o ct0 @3'F014 En 2 MM:3- En CD C4 P - rn C,+0Id C+ Fi :3, C,+ P .0 2 r) P, 0:31" :+ P- En m + 0@-j P, Fl- tn En 1 ct C+ ta rD0tzj C++ 39 where ds is the surface element, d 0 is an increment of angle 0 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, we obtain co 2Tr 2 f f L a R 4 7r E R2+.L2 d 0 dR (48) Solving the inner integral yields co (4 L a R f c:.(R2 2 3/2 dR (49) 2 + L 0 Equation 49 can be solved to yield co 2 Q L -1 (50) I 2 2 1/2 2 c R + L In closed form -0 Q a (51) c We have previously solved for the voltage between the charges in equation 32. Dividing the voltage v by the current I, equation7 51 yields .4o the resistance as defined by equation 36 (1 - a/1) v 2 c a R (52) which reduces to 1 a 2 R 1- (53) 2 it a a L 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 -3 -i -1 2 is about 10 0 m . The resistance which loads the generator organ is 6 -9 about 62,000 ohms . The dielectric of fresh water is 0.707 x 10 fd/m. This means that the capacitance which loads the dipole is 40 pfd. This large resistance and small capacitance indicate that only a small2 current flows. Thus, an electrostatic approach to the electric fish problem can be justi- fied on electrical grounds as well as on the previously discussed theoretical b4.ological 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 bLi assum2ed to be a sphere. Such an object moving into the fish's field will modify the potentials along the fish's surface. To obtain a solution for these modifications, we will first consider the --lectric field produced by the dipole i-,cnerator or[,ari uridistiirbod by the Iii a practicil model, the resistance loadiiir of the f-,t,,,i(5,rator call be coritrolled I)y the choice of (,en#-rat.or t-lecti-ode 41 perturbating object. We will calculate its magnitude at an arbitrary point p. We will then consider the effect at point p of a perturbating object located in a uniform field. Then we will transform the perturbation portion of our solution back to the original coordinate system. Once we have obtained an appro2priate solution, we will assume and fix certain variables. Then we will study the nature of the fishes classifica- tion techniques by plotting our results for given receptors along the fish. It should be noted that this analysis is three dimensional and although not conceptually difficult, it is somewhat complex. Fig. 8a illustrates the problem. The electric field is defined a2s the force that would be exerted per unit charge on a small test charge at a given point (54) E (p) = F (p) / q-t The force on a test charge q+due to another charge Q can be found from Coulomb's Law as qt 2 (55) 4 D2 D where E is the dielectric of the media D is the distance between the charges ,A@ iDis a unit vector directed away from each charge at the charge. The electric field due to the positive charge in the dipole system is 8 E+ 156) 4 ir c R2 R w @3 0 P. P, PClt- U) 04 P (D Id U) C't (D (D ct (D (D C), -4 (D Id (D En P " It LI)0 CT, 2 0 H- Ct P, @rq 11 P. P3 (4 0 C+ & (D ct rt. c@. En PL P C+ clt L* clt :31 (D En 2 C+ In .4 C+ (D P (D U) I H- (D P c+ EA 0, ::J' (D 0 C+ C+ En (+ 0 r :3 @:s C+ C+ 0 11 Cy, (D m eD m 0 P. 0 0 P. r_ C+ (D I 0 1 m (D tl E3 m P, .4 0 x 1 :3 0 (D En c+ En C+ 42 The electric field due to the negative charge is -.).- -Q E - = i 4 -n c R'2 R' (57) w The electric field is a vector quantity. Thus, we must2 perform vector addition in order to determine the total field E. It is easier to determine the components of the-field due to each charge and then to add the components. Thus, we will concern ourselves first with the y component and then with the x-z component. The y component of the electric field due to the positive charge is EY+ = IE,l COS 0 2 (58) L4@kewise the y component of the electric field due to the negative charge is E = I'E-1 COS 0- (59) y 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 3 total electric field will be considered to be the component within the x-z plane at an angle 0 from the axis. For the positive charge it is EX-Z = JE.1 Sin 0 (6o) and for the negative charge it is E I'E-1 Sin 0- (61) X-Z 43 In order to treat these variables by standard mathematical techni-ques, it is necessary to express R' in terms of R and 0, and OA in terms of R and 0. Fig.8b illustrates the factors which will help us do this. Recog- nizing that R' is th2e hypotenuse of a right triangle whose sides are R sin 0 and L + R cos 0, we find 1/2 2 2 R.' R + L + 2 R L Cos 01 (62) and R Sin 0 OA = tan 2 (63) L + R Cos- 0 It is useful to note the trigonometric functions for OA. They are R Sin 0 Sin OA =[R 2 + L2 + 2 R L Cos 0 1/2 (64) and L + R Cos-0 2 Cos OA = 2 2 1/2 (65) [R + L + 2 R L Cos 0 1 Combining equations 62, 64, and 65 with equations 56 and 57 and substituting into equations 58, 59, 60 and 67 yields Q -'Os 0 E (66) Y+ 4 7r c R2 v -Q L + R Cos 0 Ey- 2 2 3/2 (67) 4 R + L + 2 R L Cos 0 w 44 Q Sin 0 E + (68) X-Z 4 R2 w -Q R Sin 0 (69) EX-Z 2 2 3/2 2.4 it ew [R + L + 2 R L Cos 0 We can now determine the total components of the electric field. The y component of the electric field at any point R, 0 about the di-pole is independent of 0 and is Cos 0 L + R Cos 0 Ey 2 2 2 3/2 (70) 4 c R R + L + 2 R L C2os E) w The component of the electric field in the x-z plane is radial and independent of 0. It is R E Q Sin 0 1 X-Z 4 -ff c R5' - [R 2 + L2 + 2 R L Cos 0 3/2 (71) w To continue our derivation, we mu4st deterfaine the magnitade l@l and the angle (relative to the y axis) 0 of the electric field at any point in space. The magnitl-tde may be found from the rules of vector addition as 2 2 1/2 Ey + EX-Z (72) 45 The angle found by studying the geometry of the situation is E X-Z a = tan (73) Ey Manipulating the expressions in equations 70 and 721 per equation 72 yields 2 2 R (R + L Cos 0) (74) E + 4 c R2 R2 + L 22 R L Cos 0 3/2 w 2 1/2 R 4 [R2 + L2 + 2 R L Cos 0 2 3/ 2 tan -1 Sin E) [R2 + L2+ 2 R L Cos 01 -R 3 3/ 2 [R2+ L 2 + 2 R L Cos 01 2 Cos 0 - R 2 + L + R 3 Cos 0 (75) 7'hese rather complicated equations completely describe the electric field due to the dipole in a continuous media of dielectric c.. To obtain a first approximation of the perturbation due to 2 a sphere of radius r at a location R 0 , relative to the dipole coordinates, 0 0 we assume that the sphere i.s located within a uniform field of strength E0 at an angle a 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 v0alue of the dipole field at the center of the sphere, disre@.-arLI.ing 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, 46 *,he assumption will permit a solution within acceptable limits. The equations which describe the electric field due to the dipole could be simplified to a good approximation if the separation between the dipole and the object sphere is more than ten dipole lengths. Since the biological data is not adequate, we do not know the range of the electric s2ense. Consequently, we will not approximate the field at this point. Assuming a uniform field (without the perturbation) and expressing the potential in terms of the coordinate system centered at the object we obtain U -E0 P Cos a (76) The surface charge on the perturbating sphere is exactly like a dipole. Thus, the potential is of the form 2 A Cos a U = (77) x 2 p 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 uL B p Cos a 6 (78) where B is a constant to be determined from the boundary conditions, The potential outside the sphere is the sum of U0 and Ux or 47 Utota., E p Cos a A Cos a (79) 0 p The electrostatic boundary conditions require that when p r a u a ui c tot2al c (8o) w p x p and that when p = r utotal u I (8i) Substituting equations 78 and 79 into equatic-l- 3 3 E r A = B r 2 (82) 0 Substituting equations 78 and 79 into equation 60 yields c E + 2 A c = r3c B (83) w 0 w x We can solve for A on B in equation 82 and 83 to find 3 c 2 w B (84) c + 2 E 0 x w -ind c x w 3 A r E (85) c + 2 E 0 x w 48 The potential U 0 was assumed to be the potential due to the dipole field. We have an accurate expression for this field. The potential within the sphere does not effect the potential near the dipole. The critical term is the potent- ial U xwhich is the approximate modification of the dipole field due to the spherical object. Substituting equ2ation 85 into equation 77 yields C C .3 x w r U - JE I Cos a (86) x 2 0 cx 2 cw p the perturbation potential. We recall the value of E0to be the value of equation 74 when R Ro and 2when 0 = 00 or 2 R 2 (R - L Cos 00) 0 0 + (87) E0 2- 1 2 2 2 3/ 4 c R R + L +2 R Cos Oj 2 v 0 0 0 2 1/2 4 R 0 2+ L2 + 2 R L Cos 0 2 i ro 0 0 and we recall that a is measured relative to an angle $ whic2h is equation 75 evaluated at Rolp 00 or 3/2 Sin 00 R + L2 + 2 R L Cos 0 - R 3 0 0 0 3 tan-1 L - - (88) 3/2 7 R 2 + L2 + 2 R L Cos 00 Cos 00 - R 2 L + R 3 Cos 00 - 0 0 0 0 45 To complete our analysis, we must express p and a in terms of R, 0, 0, Ro-i and 00. We find 2 2 [R Cos 0 - R Cos 0 +[R, Sin 00 R Sin 0 Cos (o - oo)] + 2 0 01 1/2 2 IR Sin 0 Sin Oo)] (89) this can be reduded to 1/2 p = R2+ R 02 _ 2 R R0 Cos 0. Cog 0 - Sin 00 Sin2 0 Cos (90) We can solve for a in terms of R, 0, 00, and p to be O- R Sin (O a = 360 Sin- p (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 revie2w the salient ones and group them in an orderly fashion for computer study. Sensitivity. The magnitude of the electric field at any point in space can be found from equation 74 when R,+ R 0 and 0 = 00. Equation 74 becomes equation 87 which is assumed linear throughout the perturbation. The magnitude of the electric field is 2 R 2 (R + L Cos Oo) 2 Q 0 0 F, + (92) 0 4 7 c R 2 [R 2 + L2 + R L Cos 0,] '12 w 0 0 0 L - 1/2 R 4 0 0 2 2 IR0 + L + 2 R0L Cos 001 5 The angle of this field relative tD the y axis in the R0 -y axis plane is given by equation 75 which for R = Ro, 0 00 becomes 3/ 2 2 2 3 Sin (Do + L + 2 R L Cos 00 R 2 0 0 0 tan 3/ (93) 2 + L 2 + 2 R L Cos 00 2 Cos 0, - R 2 3 10 0 1 0 L + R0 Cos 00 This field creates a pertii!bation field in a remote object of radius r and 2 dielectric c . The potential of the perturbation field is given by equation x 86 which is c c r3 u x - w E Cos a (94) x c + 2 E p2 0 x v Unfortunately, p and a are in terms of a secondary coordinate s2ystem. Our primary coordinate system is R, 0,0. Equation 90 expresses p in terms of R, R0 0, 00, 0, and 00 . This is the first place where the perturbation angle is important. The expression for p is 1/2 p 2+ R 2 _ 2 R R Cos 00 Cos"O - Sin 0 Sin 0, Cos (o - 200) 0 0 (95) The angle a can be expressed in terms of R, R 0 0,00.@, and @o However, it is simpler to express it as R Sin 0 - 00) a = 3600 Sin-1 -.00 - a (96) 1 p The undistorted potential about the fish is the negative gradient of the sum of equations 70 and 71 or 51 U0 (97) 4 Tr cw L R [R2 + L2 + 2 R L Cos 01 1/2 Speaking anthropomorphically, the fish knowsU0 R, L, 0, and 4 -ff cw. It must determine Rol 00 00, r and 47E2 X' To find these variables, it makes an analysis of-the potential function U + U . Exactly how this 0 x analysis is made is unknown. As an approach to determine how the fish might operate, we shall find the ratio Ux/u0 at different points along the fish for different values of Ex1, r, R0, 00, and @DO). The equations are rather complex, and require a computer analysis. Thus, a fortran computer program was written which -manipulates and evaluates the desired variables. In the computer program, certain variables have been assigned values for reasons that are discussed below. The length of the generator organ has been set at one meter. In this way, measures can be referenced in terms of ge2nerator organ lengths. Thus, range, perturbating cbject size, and electroreceptor locations are all dis- cussed in terms of generator organ lengths. The dielectric of the perturbat- ing object has been expressed in terms of the dielectric of water. In other words the analysis is in terms of cw/Ex rather than cx or cw themselves. The fish is defined as a cylinder two times as long as the elect0-k-ic gen- erator organ, with a radius O.'2 times the generator length, Six longitudinal bands of eleven receptors are assigned along the length of the cylinder. Three bands, each band 15 degrees apart, are located on each side of the fish. The center band on each side is assumed to be in the same plane as the center of !,he pertlirbing object and the axis of the cylinder. 52 The computer program analysed various combinations of four factors: 1) the ratio of e /E 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 cylinder2 axis and -',he 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 wer2e manipulated to show the effect on the perceived potential ratio. These variables are defined as: (R) the line segment from the center of the cylinder to the receptor on the cylinder sur- face,(O) the angle formed by the intersection of line segment R and the cylinder axis, (o - @,)the angle defined by the intersection of the plane passing through the center of the object and cylinder axis and the plane pa2ssing through the cylinder axis and a band of eleven receptors on the fish's surface. In the actual printout, this angle was taken for each of six receptors defined by the same radius r and angle 0 (Ux/ u0 ) the ratio of the potential due to the perturbation and that due to the dipole effect in the free field. The following are the primary conclusions from the computer analysis of the electrostatic m2odel: 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 I-Ln most circumstances at the ',,lead e,-,d of the simulated fish. (This may explain the high concentration of electro- 3receptors on #"he head of the actual.fish). The magnitude of the signal is smallest at the cylinder surface closest to the two poles of the generator dipole. Assuming the fish can detect a signal of one part per million (humans can detect sounds 1/1,000,000 th normal speech loudness) the fish can easily detect objects of dielectric 0.1 times water whose radius is 0.1 generator organ 2 lengths, at distances to the side of 6 generator lengths. It would have diffi- culties at 10 lengths or for dielectrics of 0.5 or 5 times water. 7 With the assumed sensitivity, the fish could detect objects of dielectric 0.1 water, assuming object radius is 0.1 generator organ lengths at 10 elec- tric organ lengths if approached from front or rear. Objects of diele2ctric 10 times water could be detected at a considerab7-e angle from the direct front or rear approaches at 10 lengths. An indication of receptor sensitivity for objects of dielectric 10 times water and 0.1 water with a radius of one length is given in table I. Table I Receptor Sensitivity Detection Distanc2e Receptor Discrimination 1 length 1 part 100 10 lengths I part 10,000 100 lengths 1 part 1,000,000 1000 lengths 1 part 1002,000,000 10 10,000 lengths 1 par'. 10 The analysis shows that the signal is about twice as large for dielectrics 10 times water as it is for objects with dielectrics 0.1 times water. f 'i'tie (iielectric of water is3 approximately 81, plastics ire about 8, air is about 1, and metals would be extremely hi,ch, virtually infiiiite in m,ljiy cases. 54 System Level Although the data does not exist for the fish, it is reasonable to assume that it has a data processing capability similar to that found in other comparable organisms. Thus, it is likely that the fish can use inhibitory and facilitating circuits to sharpen the aforementioned data from the receptor, extract signal from noise2 and classify multiple incoming signals. In essence, it would function as a system at the receptor level with interactions among receptors and at the whole organism level involving the receptors, generator, and brain. For example, there is evidence that a plot of receptor potential along a band of receptors would yield, for a single perturbing object, a uni- moda.1 curve (Hagiwara & Morita, 1963). We might suggest that th2e configuration of the curve is a function of the overall impedence of the perturbing object, defining impedance as the sum of the resistance and the reactance of the object. The reactance of an object is given by 1 X 2 7 F L ( :@# (98) 2 7 F C where X is reactance 2 F is frequency L is the inductance in henrys of the object under observation. C is the capacitance in farads of the object under observation. As may be seen, by operating as a system by coordinating the generator and receptor function, the fish by changing generator frequency can induce a lower or higher overall effective impedance in the object, If an object had an impedance very5 similar to that of water, the fish could enhance its dis- crimination and classification ability by varying its frequency; making the object create a greater or lesser potential gradient at the receptor. Thus, if 55 the fish was trying to locate a certain known object, it could adjust its frequency to optimize its detection of the object. Using this system approach it would also be possible for the fish to sense differences in objects that have the same exterior physical appearance. This would be done through vary- ing the fr2equency and sensing and comparing the changes in the reactance of the objects. At least some species of fish seem to be using the foregoing system approach. In.the earlier discussion of the available biological data, it was noted that a generator frequency shifting technique was used. There also appear to be mechanisms that can be used to optimize detection and classification of one stationary object among2 several stationary objects or a moving object among stationary objects. For example, to detect a moving object several scans could be carried out, stored, and compared. In this way, stationary objects in the field -would be nulled and only objects of changing impedance or location would be perceived. CONCLUSION Although there is a fairly substantial data base, 2we find that very little can be applied to the development of an understanding of sense mechanism and sensitivity. This is due in part to the fact that pioneering data in this area, as it is in most areas, tend to have faults no matter how competent the investigators. One of the prime deficiencies in the reported work is the use of a tank of inadequate size or with extraneous objects in the field. These 2 distort the field and seriously effect the data obtained. Further, the data base contains very little behavioral data. Thus, we undertook several tasks to provide a basis to assess the fishes' electrosensing mechanism and capability, using the data presently available. Through limited experimental work with electrical fields, sensors, and ,ob in various size bo9dies of water we have gathered data which, when .jt2cts 56 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 e2xperiments, it would be best to have at least 10 electric organ lengths about the fish. When the fish must be restrained near the surface, 10 organ lengths of water in all other directions should be the minimum. It should also be noted that to simulate infinity in research, the tank must be connected to earth ground and made to conduct. This does not seem to have been done in past research. 2 We have suggested as a working hypothesis an electrosensor mechanism. This hypothesis is subject to test and thereby may provide the means for collapsing the current multiple crude categorizations of the receptor that is so typical of a new area of investigation. The hypothesis may also provide a basis for analyzing higher interactions in the fishes' nervous system and thereby increase our understanding of t2he sense. We have also, through mathematical analysis, shown the linkage among the various neural coding schemes suggested for the fish and have shown their essential identity. We have also developed a mathematical model of the fish based upon the useable experimental data. A set of equations describing function was develop- ed on the model and these equations linked to available experime7ntal data. The mathematical model was analysed by a computer to ascertain the sensitivity of the fish at the receptor and to determine the effects of manipulating a number of variables. These variables included fish size, object size, object electrical characteristics, object distance from the fish, direction and angle of the object from the fishes' axis, etc. From the computer analysis, we deter.mined the sensitivity to various perturbing objects under a variety 6f conditions and found the fish to be auite sensitive particular--L'y in certain directions. 58 REFERENCES .galides, E. Investigation of electric and magnetic sensitive f4-shes.-- Final report, Office of Naval Research, 1965. Lljure, E.F. Neuronal control system of electric organ discharges in Mormyridae. University Microfilms, Inc., An2n Arbor, 1964. 3ennett, M.V.L. Comparative physiology of electric organs. Ann. Review of physiology, 1970, 32; 471-528. Bennett, M.V.L. Electroreceptors in Mormyrids. Cold Spring Harbor Symposia on Quantitative Biology, 1965, XXX, 245-262. Bennett, M.V.L. Electrosensory systems in electric fish. Finpl report, Air Force Of2fice of Scientific Research, 1969. Bennett, M.V.L. Mechanisms of Electroreception. Lateral line detectors. P. Cahn, ed., Indiana University University Press, 1967, 313-393. Bennett, M.V.L. Neural control of electric organs. The central nervous system and fish behavior. David Ingle, ed., University of Chicago, 1968, 147-167. Bennett, 2M.V.L., Pappas, G., Aljure, E., Nakajima, Y. Physiology and ultra- structure of electrotonic junctions. II. Spinal and medullary electromotor nuclei in mormyrid fish. J. Neurophysiology, 1967, 30; 180-208. Bennett, M.V.L. Pappas, G., Gimenez M., Nakajima, Y. Physiology and ultra- structure of electrotonic Junctions.IV. Medullary electromotor nuclei in 2 gymnotid fish. J. Neurophysiology, 1967, 30; 236-300. Bullock, T.H. Biological Sensory Vistas in Science. University of New Mexico Press, 1968, 176-2o6. Bullock, T.H., Chichibu, S. Further analysis of sensory coding in electro- receptors of electric fish. Proceedings of the National Academy of Sciences, 1965, 54, 422-429. 2Clark, L., Granath, L.P. A measure of the threshold sensitivity of Gymnotus carapo to electric fields. American Zoologist, 1967, 7, 130. Derbin, C., Szabo, T. Ultrastructure of an electroreceptor (Knollenorgan) in the mormyrid fish Gnathonemus petersii. 1, J. Ultrastructure Res., 1968, 22, 269-484. Dohlmaii, G. Electro-physiologie et physio patho6logie de llappareil vestibul- aire. (Some aspects of the mechanisms of vestibula hair cell stimulation). Confin. Neurol. 1960, 20, 169. Enger, P.S. and Szabo, T. Effect of temperature on the discharge rates of the electric organ of some gymotids. Comp.Biochem. Physiol, 1968, 27, 625-627. 59 Erskine, R.T., Howe, D.W., Weed, B.C. The discharge period of the weakly electric fish Sternarchus albifrons. American Zoologist, 1966, 6, 79. Frohlich, H. Theory of Dielectrics. Oxford University Press, London,1958. Granath, L.P., Sachs, H.G. Erskine, F.T. III. Electrical 2 sensitivity of a weakly electric fish. Life Sciences, 1967, 6, 2373-2377. Hagiwara, S., Kusano, K., Negishi, I. Physiological properties of electro- receptors in sdme gy=otids. J. General Physiology, 1962, 45, 6oo-6ol. Hagiwara, S. & 14orita, H. Coding mechanisms of electroreceptor fibers in some electric fish. J. Neurophysiology, 19263, 26, 551-67. Hagiwara, S., Szabo, T., and Enger, P.S. Physiological properties of electro- receptors in the electric eel, Electrophorus electricus. J. Neurophysiol., 1965, 28, 775-83. Heller, J., Teixeira-Pinto, A., Nejelski, L. & Cutler, J. Experimental Cell Research, 1960, 20, 548 Herzberg, G. Molecular Spectra.and Molec2ular Structure. D. Van Nostrand Co., Inc., Princeton, 1950. Jackson, J.U. Classical Electrodynamics. John Wiley & Sons, Inc. N.Y. 1962. Larimer, J.L. & MacDonald, J.S. Sensory feedback from electroreceptors to electromotor pacemaker centers, in gymnotids. American J. of Physiology, 1968, 215, 1253-1261. Lissnann,H.W.2 Electric location by fishes. Sci. Amer., 1963, 50-59. Lissmann,H.W" Machin, K.E. The mechanism of object location in Gymnarchus niloticus and similar fish. Journal Exptl. Biology, 1958, 35, 451-86. Lissmann,F.R.S. & '4ulli-nger Ann M. Organization of ampullary electric receptors in Gymnotidae. Proceeding of the Royal Society, 1968, 2 169, 335-378. Lissmann,H.W. & Schwassman, H.O. Activity rhythm of an electric fish Gym- norhamphiethys hypostomus, ellis, Zei'@schrift fut vergleichende Physio- logie, 1965, 51, 153-171. MacDonald, J.A. & Larimer, J.L. Phase-sensitivity of Gymnotus carapo to low-amplitude electric'stimuli. Z. Vergl. Phy2siologie, 1970, 70, 322-334. '.1andriota, F.J., Thompson, L., Bennett, M.V.L. Classical conditioning of electric organ discharge rate in mormyrids. Sci. 150, 1965, 1740-1742. i-linkoff, L.A., Clark, W.L., Sachs, H.G. Interspike interval analysis of the discharge of a weakly electric Mormyrid fish. American Zoologist 1967, 5 7, 131. 6o O'Leary, D.P. An electrokinetic model of transduction in the semicircular canal. Biophysical Journal, 1970, 10, 859-875. Saito, M.,Schwan, H. & Sehwarz, G. Response of non spherical biological particles to alternating electric fields. Biophysical J. 1966,6,313-27. Steinbach, A.B. Diurna2l movements and discharge characteristics of electric gymnotid fishes in the Rio Negro, Brazil. Biol. Bul,1,1970,138,200-210. Suga, N. Coding in tuberous and ampullary organs of a gy=otid electric fish J. Comp. Neurology, 1967, 131, 437-452. Suga, N. Electrosensitivity of canal and free neuromast organs in a gymnotid electric fish. J. Comparative Neurology, 1967, 1312, 453-458. Suga, N. Electrosensitivity of specialized and ordinary lateral line organs of the electric fish, gymnotus carapo. Lateral Line Detectors. P.Cahn, ed, Indiana University Press, 1967, 394-4og. Szabo, T. & Hagiwara, S. A latency-change mechanism involved in sensory coding of electric fish. (Mormyrids) Physiology & Behavior, 1967, 2 331-335. 2 Szabo, T. The origin of electric organs of Electrophorus electricus. The Anatomical Record, 1966, 155, 103-110. de Vries, H.L. The-minimum perceptible angular acceleration under various conditions. Acta Oto-Laryngeal, 1949, 37, 218. van Beek, L.K.H. The dielectric behavior of heterogeneous systems. Prog. Dielectric, 1967, 7, 69. Wa5chtel, A.W. The ultrastructure relationships of electric organs and muscle. J. Morph, 1964, 114, 325-360. 61 APPENDIX Symbols 0 The angle between the Y-axis and the radius vector from the positive charge to an arbitrary point in space. (DI The angle between the Y-axis and the radius vector from the negative charge to an arbitra2ry point in space. E), 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. R0 The radius vector from 2the positive charge to the center of the perturbing object. 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-dxis and the X-z projection of the radius vector from the positive charge to the center of the perturbing object 6 The charge on either side of the dipole. L The length of the dipole. Ew The dielectric of the water. cx The dielectric of the perturbation. r The diameter of the spherical perturbing object. 0 The electric field at the center of the perturbation as if the perturbation were not present. 62 ux The potential due to the interaction between the dipole field and the perturbation. v A coordinate centered at Mc perturbation opposed to the vector E0 and in the Y axis -R0 plane. m A coordinate perpendicular to the w axis in the Y-P 0 plane 2 originating at the center of the perturbation. The angle between E0 an'" a line parallel to the Y-axis at the center of the perturbation. p A radius vector in the w, m coordinate system to an arbitrary point in space. a The angle between the radius vector and the coordinate w. E+ The electric field due to the positive charge. 2 i Unit vector in the direction YR r E- The electric field due to the negative charge. i Unit vector in the direction of R'. r EX -Z The X-z component of the'-electric field. u0 Potential due to uniform approximation of the dipole field. uI Potential inside sphere due to uniform approximation2 of dipole field. E The Y component of the electric field y y End point of charge path. a Radius of sphere in dipoi-e arrangement. I Current p Macrczcopic polarization N The number of molecules per volume. pmos Related to the macroscopic polarization (electric dipole moment per 9 vol=e) 63 Xe -'he electric susceptibility. E Storing force. x End point of charge path pind Induced dipole moment. T Threshold function. T Maximl,Tn threshold. max T . Minimum threshold. m3.n 2 Tc Exponential time constant. Charge Y+ Dummy variable (positive charge). y- Dummy variable (negative charge). v Absolute voltage. v Voltage c Capacitance i Current density. D Distance between charges A A constant to be determined from the boundary conditions. 9 B A constant to be determined from the boundary conditions. x Reactance L Inductance u Potential in terms-of the coordinate system. w0 Radian frequency of harmonic oscillation. Reactance in polar corrdinates.