Brain Research, 88 (1975) 325-332 © Elsevier ScientificPublishing Company, Amsterdam - Printed in The Netherlands

Voltage attenuation within

Aplysia neurons:

325

the effect of branching pattern

KATHERINE GRAUBARD* Department of Physiology and Biophysics, University of Washington School of Medicine, Seattle, Wash. 98195 (U.S.A.)** (Accepted January 27th, 1975)

Steady-state voltage attenuation for common shapes of Aplysia neurons has been computed, based on anatomical and physiological measurements. Voltages are transmitted effectively between different regions of a cell within the ganglion, except that voltages applied to small diameter processes (where synapses are typically located) are severely attenuated in transmission to those of large diameter. These results are important because the 'electrical distance' from one part of a neuron to other locations within the same cell is a basic determinant of how that neuron functions. Problems such as whether some synapses are more effective than others in influencing cell outputl,a, is or whether a cell can have two or more partially independent output sites2S, 29 are determined in part by the electrical distances within a neuron. In most intracellular studies of neurons, recordings are made only from the cell body. From these recordings inferences are made about events which occur at locations physically distant from the recording site. A major practical problem for neurophysiologists has been to determine what limitations the shape of the cell and its cable properties put on the interpretation of recordings made in the cell body2, 3. Interpretation becomes particularly difficult when input and output sites are intermixed at a distance from the recording site as in cells with axo-axonic or dendrodendritic synapses 2e. Although Aplysia neurons have proven to be popular and useful cells for the study of both synaptic function and of simple neural circuits, little attention has been paid to the functional organization of the individual neurons. Aplysia neurons are structured so that their large cell body, from which most recordings are made, is physically removed from both synaptic sites and from the region of action potential initiation 5,27. Some evidence indicates that a portion of the physically distant neuropile region of these cells is electrically close to the cell soma27,30,31; however, other evidence indicates that some parts of these neurons are electrically distant from the cell bodyla, 20. Thus, in order to understand the behavior of these cells more completely, a study was undertaken to compute electrical distances between different regions of Aplysia abdominal ganglion neurons. Since many of the desired voltage decrements * Present address: Biology Department, University of California at San Diego, La Jolla, Calif. 92037, U.S.A. ** Address to which reprint requests should be sent.

326 c a n n o t be measured directly, the results were computed, using the shape of a b d o m i n a l g a n g l i o n neurons, the m e m b r a n e properties of cells L 1 2 - k l 3 r', a n d the steady-state cable equation. N e u r o n shape was f o u n d by reconstructing cells filled by pressure injection with the dye P r o c i o n Yellow M4RS. The tissue was fixed for a b o u t 1 m o n t h in 10)~ form a l i n in seawater, then was sunk in sucrose a n d frozen sections were cut, m o u n t e d on glass slides, dehydrated, cleared, a n d coverslipped using a non-fluorescent m o u n t ing medium. N o detectable, systematic shrinkage or d i s t o r t i o n was f o u n d with this procedure. The cells which were examined were large (soma diameter 200 # m or more) a n d were usually identified neurons, i n c l u d i n g examples of L2-L6, L I2-LI3, R2 a n d R15. All n e u r o n s studied have at least one large-diameter process or p r i m a r y neurite which originates from the soma a n d continues t h r o u g h the neuropile to enter a nerve trunk. I n addition, all cells also have small-diameter, short processes

.

F

C

~

96

lOmV

t

D

I,

G

~.91

10 o•

==

500~ eooe

1

•. ' "

•

......

2 axon

8,, ,

10 diameter

,

. , ....

i 100/~

H

_d-

Fig. 1. A: a partial reconstruction of an example of cell LI3. Only the largest diameter and longest processes are shown. B: a partial reconstruction of an unidentified left upper quadrant neuron. All processes which entered nerve trunks are shown. This cell also had about 45 small-diameter secondary processes which ended in the neuropile, similar to those shown for cell L13 in A; these have been omitted. C: a tracing of the membrane folds of an R2 axon in the right connective nerve trunk. D: a plot of log Fc v s . log de, where de is the diameter of the circle which encloses the same area as is contained in a cross-section of the process. Fe is the infolding factor, the number by which ztd~ must be multiplied in order to compute the true perimeter of the process cross-section from the diameter, de. The points on the graph for which dc > 50 #m are examples of the R2 axon. E: the voltage response of an example of cell L13 to a current step. Both the current and voltage electrodes were in the cell soma. Note the long time course of the transient, implying a long time constant. No steady-state voltage level is reached in this record; using longer steps, this neuron had a steady-state voltage-current curve which was linear and gave a slope resistance of 8.3 MQ. F: an idealization of the soma and primary processes for a large abdominal ganglion neuron. Membrane folds are not shown. The primary process, which normally enters a nerve trunk at about 1 ram, is assumed to be equivalent to an infinite cable. Soma diameter, ds = 280/~m. Primary process diameter, de = 30 ~m. G: identical to F except that the primary process bifurcates at 500/~m from the soma. All process diameters are 30 #m and both processes past the bifurcation are assumed equivalent to infinite cables. H: identical to F except that a fine process, 2 #m in diameter and 100/~m long, branches from the primary process at a point 1 mm from the soma.

327 which branch from the primary neurite and which terminate within the neuropile (Fig. 1A). In some neurons, the primary neurite branches to form two or more largediameter processes which continue through the neuropile and enter nerve trunks (Fig. 1B). These large-diameter secondary processes can also have small-diameter, short processes which branch from them. The cell shapes described above are idealized in Fig. 1F,G,H and will be used in computing voltage decrements. The lengths and diameters are typical of the large abdominal ganglion neurons studied. Extensive infolding of the plasma membrane occurs in Aplysia neuron cell bodies and in all but the smallest diameter neurites. A quantitative measure of infolding is necessary for calculations of specific membrane resistance, Rm, and of voltage decrements. Since membrane infoldings cannot be seen easily in thick sections of Procion Yellow dyed cells, measurements were made from tissue fixed by immersion in phosphate-buffered OsO4 and embedded in Epon. Sections were cut at 2 #m and were stained with Richardson's stain. Infolding was measured with a map measurer from enlargements of tracings made with the light microscope at × 1000. This method gave little or no evidence of tissue shrinkage or distortion under either the light or electron microscope. Some fine membrane folds were missed by measuring from light microscopic drawings rather than from electron micrographs. However, in the one axon examined with both methods, the difference between the two values was within the range of error for the light microscope measurements. Infolding of soma membrane was very difficult to resolve; the minimum true soma perimeter was 6 × the perimeter estimated from soma diameter alone for cell bodies of about 250 #m diameter. Infolding of neurites was estimated from measurements of axonal infolding in the right connective nerve trunk (Fig. 1C). Axonal infolding was found to increase as a power function of axon diameter (Fig. 1D). These estimates of axonal infolding are used below for processes in the neuropile as well as for those in nerve trunks. To compute voltage decrements using the cable equation, the values of the specific membrane resistance, Rm, and the specific axoplasmic resistivity, R~, must be obtained. For a cell which can be idealized as an isopotential sphere connected to a uniform, infinite cable, it is possible to calculate Rm, Ri, and the specific membrane capacitance, Cm, from the voltage transient response of the cell to applied current steps, where the current is applied and the voltage is measured in the cell soma22, 2~. (The entire cell membrane must have a uniform and constant Rm over the voltage range and time course used.) The shape of the voltage transient (Fig. 1E) can be analyzed to yield values for the membrane time constant (Zm ---- RmCm), the ratio of the input conductance of the cable (ge) to the input conductance of the soma (gs), and the total membrane capacity of the soma (Csoma)z2,23. The steady-state, voltage-current curve gives the input conductance of the cell, gn = g8 + ge, where gs = zrFsds2/Rm and gc = 1/2z~[Fcdea/RmRt]l/~; F is the infolding factor and d is the diameter. Thus when gn, ge/gs, Fs and Fe are known, Rm and Ri can be found directly; Cm can be found from Csoma or Zm15,16,22,2a. Since, in practice, none of the measured parameters is known precisely, the equations were solved by computer, incrementing the parameters over their widest

328 TABLE I Parameter

set

Cm (l~F/sq. cm)

Rm ( ( 2 . sq. cm)

R+ ( g2 • era)

1 2 3

1.3

450,000

1.0 0.8

550,000 690,000

150 90 50

2e (ram)

5.3 8.7 16

possible range of variation. The result is a range of possible combinations of values of Rm, Ri, Cm, and infolding. The results for an example of cell L13 are given in Table I. The two extremes are shown as sets 1 and 3 of Table I. Set 1 gives the combination of parameter values which would produce the greatest amount of voltage attenuation with length (shortest length constant) and which will still fit the measured values of ge/gs, "rm and gn for the L13 cell. Set 3 lists the parameter values which yield the smallest voltage loss with distance. Set 2 indicates the midrange parameter values: it is the set for which the unit membrane capacity is 1.0 #f/sq. cm. Note that the length constant for a 30/zm diameter process is long for all sets, ranging from 5 to 16 mm. A typical distance over which a primary neurite would course from the neuron soma until entering a nerve trunk is 1 mm. The unit membrane capacity for all sets is within the range of values computed for other biological membranes. Previous measurements of Cm for Aplysia neurons failed to consider membrane infolding and the cable effect of the primary neurite: they gave much higher values for Cm 10,11,17. For more generally accepted values of Cm see Cole 6. The value of Rm is higher by several orders of magnitude than that for most biological membranes, but it is lower than the value found by Gorman and Mirolli 15 for the G cell of Anisodoris gastroesophageal ganglion. The Rl values found in this study are typical for axons, but are much lower than the value found by Carpenter et al. 4 in Aplysia neuron cell bodies. Infolding values are in general agreement with those of Mirolli and Talbott 21. These results now provide all the information which is needed to compute voltage decrements. Calculations were made for each of the three idealized neuron shapes shown in Fig. 1F,G,H by means of successive applications of the steadystate cable equation for a finite length cylinder 16. Processes which enter nerve trunks are treated as infinite cables; processes which terminate in the neuropile are assumed to have sealed ends. Fig. 1F diagrams what happens when a steady-state voltage is applied to the isopotential soma and the voltage attenuation is computed for a point in the primary neurite 1 mm from the soma. For the midrange parameter set (set 2), 89 ~ of the applied voltage is transmitted to the 1 mm point (only 11 ~ is lost). For the extreme sets 1 and 3, transmission values are 83 ~ and 94 ~ respectively, When the voltage is applied at the 1 mm point on the cable and one calculates the amount of voltage

329 that reaches the sphere (Fig. 1F), then 96 70 of the applied voltage is seen at the idealized soma for the midrange set (the extreme values are 92 ~ and 98 ~). Thus, for a sphere-cable model, steady-state voltages are transmitted with only 5-157o losses for a length equal to the distance over which many primary neurites would travel within the neuropile before entering nerve trunks. These voltage decrement values are not changed a detectable amount by the addition of forty small-diameter, short length processes of the type shown in Fig. 1A. When the primary neurite (cable) bifurcates (Fig. 1G), then there is more voltage attenuation than occurs in an undivided cable; however, the losses are still small. Steady-state voltage transmission from the soma to a point 1 mm from the soma, and 500 #m past the point of bifurcation, is 84~ (with 74~ and 917o as extremes). In the other direction, the transmission is 9170 (85-95~). Thus, the presence of a single, large-diameter branch does not cause much more voltage loss than occurs in an unbranched cable; however, a series of such branches, as is seen in the unidentified cell of Fig. 1B, would cause large voltage losses in either direction. When the steady-state voltage is applied to the soma and the voltage transmission to the termination of a small-diameter process is calculated (Fig. 1H), the results are the same as those of the simple sphere-cable combination; the voltage loss from the sphere to the 1 mm point in the infinite cable is the same (to two significant figures) as the voltage loss from the sphere to the termination of the smalldiameter branch. This result is due to the very high input resistance of the smalldiameter branch and to its long length constant. Doubling the length of the branch has little effect on the results. The results are dramatically different when the voltage is applied to the termination of the small-diameter branch and the voltage is computed at the spherical soma (Fig. 1H). Only 2 2 ~ (range 14-34~)of the applied voltage reaches the soma. Almost all of this voltage loss occurs at the branch point between the two cables: for set 2 only 23 70 of the voltage applied at the tip of the small-diameter cable arrives in the large cable. (However, over 95 70 of the current is transmitted into the large cable.) This severe voltage attenuation is produced by the resistance mismatch between the small-diameter cable and the rest of the idealized cell. Voltage transmission is very sensitive to the relative diameters of the large and small cable. Thus, for the typical neuron shapes described above, steady-state voltages will be transmitted within the ganglion with only small losses except when the number of large-diameter branches is high or when voltage is being transmitted from a small-diameter process into a large-diameter one. This result is dependent on constant membrane resistivities of the types found for LI3. A neuron with a very non-linear voltage-current curve, with large transient conductance changes for small changes in voltage, or with a very different Rm from the one found for L13 will have different properties. In interpreting these results functionally, it is necessary to know where synapses are located on these neurons. Therefore, an electron microscopic survey of synaptic profiles in the abdominal ganglion was undertaken 16. Synapses were found between small-diameter profiles in the neuropile (Fig. 2). In agreement with other studiesS,la,

330

Fig. 2. An electron micrograph of Aplysia abdominal ganglion neuropile. There are lwo probable synapses and another possible one. P, presynaptic profile; *, postsynaptic site. Note that the diamete~ of all profiles corresponds in size to that of the small secondary processes shown in Fig. I A and idealized in Fig. 1H.

14,z5, cell bodies and axons in nerve trunks were not observed to synapse and no clearly defined synapses onto large-diameter processes were seen. Although Gillette 14 has described seeing a small number of synapses onto the primary process of El0, most of such contacts lacked membrane specializations. The size range of the synaptic profiles seen in the present survey matches the size range of small-diameter processes such as those shown on the L13 cell in Fig. 1A and idealized in Fig. 1H. Thus, it is likely that, for large cells in the abdominal ganglion, many synapses occur on smalldiameter processes. Since synaptic potentials are transients, they will be attenuated more than is indicated by the steady-state calculations described above 9. Thus, while the few synaptic potentials which originate in large-diameter processes might suffer little attenuation in transmission throughout the neuropile or into the soma, those synaptic potentials which occur in small-diameter processes will be severely attenuated in transmission to the rest of the neuron. (However, approximation of reversal potentials by passing current and recording with electrodes in the cell soma, a steady-state measurement, will be accurate to within 10-15 ~ even for PSPs which are severely attenuated by virtue of originating on a small branch; only when there are many large branches will the reversal potential measurements for synapses on far branches be inaccurate.) However, a synaptic input onto a small-diameter process could produce a large voltage change in the process itself, due to the high input resistance

331 there t,t6,24. Thus, input synapses could have a substantial effect in p r o d u c i n g or modifying transmitter release by output synapses located on the same small-diameter process. Such serial synaptic arrays between small-diameter processes do occur in the Aplysia a b d o m i n a l ganglion 16. Thus, it is possible that large synaptic inputs could impinge on an Aplysia neuron without the PSPs appearing in the soma as large, unitary events; yet those synaptic inputs could still act to modulate other inputs t h r o u g h shunting and driving potential interactions 7 and might also modulate or even produce synaptic output f r o m nearby synapses. The evidence that such electrically remote synapses exist and their m o d e o f function must now be sought. I thank C. F. Stevens, W. E. Crill, R. D. L u n d and W. H. Calvin for advice and equipment; the Friday H a r b o r Laboratories and the D e p a r t m e n t o f Ophthalm o l o g y for the use o f their facilities; and A. O. D. Willows, Daniel G a r d n e r and Esther G a r d n e r for their c o m m e n t s on an earlier version o f the manuscript. This w o r k was supported by U S P H S Training G r a n t G M 00260 f r o m the National Institutes o f Health. 1 BARRETT,J. N., AND CRILL, W. E., Influence of dendritic location and membrane properties on the effectiveness of synapses on cat motoneurones, J. Physiol. (Lond.), 239 (1974) 325-345. 2 BURKE, W., AND GINSBORG,B. L., The action of the neuromuscular transmitter on the slow fibre membrane, J. Physiol. (Lond.), 132 (1956) 599-610. 3 CALVIN,W. H., Dendritic synapses and reversal potentials: theoretical implications of the view from the soma, Exp. Neurol., 24 (1969) 248-264. 4 CARPENTER,D. O., HOVEY, M. M., AND BAK, A. F., Measurements of intracellular conductivity in Aplysia neurons: evidence for organization of water and ions, Ann. N. Y. Acad. Sci., 204 (1973) 502-533. 5 COC~ESHALL,R. E., A light and electron microscope study of the abdominal ganglion of Aplysia californica, J. Neurophysiol., 30 (1967) 1263-1287. 6 COLE, K. S., Membranes, Ions, and Impulses, University of California Press, Berkeley, Calif., 1968, 569 pp. 7 DIAMOND,J., The activation and distribution of GABA and L-glutamate receptors on goldfish Mauthner neurones: an analysis of dendritic remote inhibition, J. Physiol. (Lond.), 194 (1968) 669-723. 8 ECCLES,J. C., The Physiology of Nerve Cells, The Johns Hopkins Press, Baltimore, Md., 1957, 270 pp. 9 FALK, G., AND FATT, P., Linear electrical properties of striated muscle fibres observed with intracellular electrodes, Proc. roy. Soc. B, 160 (1964) 69-123. I0 FESSARD,A., ET TAUC, L., Capacit6, r6sistance, et variations actives d'imp6dance d'un soma neuronique, J. PhysioL (Paris), 48 (1956) 541-544. 11 FRANK,K., AND TAUC, L., Voltage-clamp studies of molluscan neuronal membrane properties. In J. HOFFMAN(Ed.), The Cellular Function of Membrane Transport, Prentice-Hall, Englewood Cliffs, N.J., 1964, pp. 113-135. 12 FRAZIER,W. T., KANDEL,E. R., KUPFERMANN,I., WAZIRI,R., AND COGGESHALL,R. E., Morphological and functional properties of identified neurons in the abdominal ganglion of Aplysia ealifornlea, J. Neurophysiol., 30 (1967) 1288-1351. 13 GERSC.ENFELD,H. M., Submicroscopic bases of synaptic organization in gastropod nervous system, Proc. 5th int. Congr. Electron Microsc., 1962. 14 GILLETTE,R., Microstructural and Ultrastructural Studies on Identified Neurons of the Abdominal Ganglion ofAplysia californica, Ph. D. Thesis, University of Toronto, Toronto, 1974. 15 GORMAN,A. L. F., AND MIROLLI,M., The passive electrical properties of the membrane of a molluscan neurone, J. Physiol. (Lond.), 227 (1972) 35-50.

332 16 GRAUBARD, K., Morphological and Eleetrotonic Properties 0! Identified Neurons ~ff the Mo[/u.s~. Aplysia calffbrnica, P h . D . Thesis, University of Washington, Seattle, Wash,, 1973. 17 HUGHES, G. M., Further studies on the electrophysiological anatomy of the left and right gianl cells in Aplysia, J. exp. Biol., 46 (1967) 169-193. 18 JACOBSON, S., AND POLLEN, D. A., Electrotonic spread of dendritic potentials in feline pyramidal cells, Science, 161 (1968) 1351 1353. 19 KEHOE, J. S., The physiological role of three acetylcholine receptors in synaptic transmission in Aplysia, J. Physiol. (Lond.), 225 (1972) 147 172. 20 KEHOE,J. S., AND ASCHER, P., Re-evaluation of the synaptic activation of an electrogenic sodium pump, Nature (Lond.), 225 (1970) 820--823. 21 MIROLU, M., AND TALBOTT, S., The geometrical factors determining the electrotonic properties of a molluscan neurone, J. Physiol. (Lond.), 227 (1972) 19-34. 22 RALL, W., Branching dendritic trees and motoneuron membrane resistivity, Exp. Neurol., I (1959) 491-527. 23 RALL, W., Membrane potential transients and membrane time constant of motoneurons, Exp. Neurol., 2 (1960) 503-532. 24 RALL, W., AND RINZEL, J., Branch input resistance and steady attenuation for input to one branch of a dendritic neuron model, Biophys. J., 13 (1973) 648-688. 25 ROSENBLUT,, J., The visceral ganglion of Aplysia californica, Z. Zellforsch., 60 (1963) 213-236. 26 SHEPHERD, G. M., The Synaptic Organization of the Brain, Oxford University Press, New York, 1974. 27 TAUt, L., Site of origin and propagation of spike in the giant neuron of Aplysia, J. gen. Physiol., 45 (1962) 1077-1097. 28 TAUC, L., AND HUGHES, G. M., Modes of initiation and propagation of spikes in the branching axons of molluscan central neurons, J. gen. Physiol., 46 (1963) 533-549. 29 VAN ESSEN, D. C., The contribution of membrane hyperpolarization to adaptation and conduction block in sensory neurones of the leech, J. Physiol. (Lond.), 230 (1973) 509-534. 30 WACHTEL, H., Isopotential coupling between synaptic regions and cell bodies of Aplysia neurons, Fed. Proc., 29 (1970) 953 (abstract). 31 WAZml, R., KANDEL, E. R., AND FRAZlER, W. T., Organization of inhibition in abdominal ganglion of Aplysia. 11. Posttetanic potentiation, heterosynaptic depression, and increments in frequency of inhibitory postsynaptic potentials, J. Neurophysiol., 32 (1969) 509-519.