The Mechanism of Bursting Pacemaker

Activity in Neurons of the Mollusc

Tritonia diomedia

by

Stephen J. Smith

A dissertation submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

University of Washington

1978

Approved by

(Chairperson of Supervisory Committee)

Program Authorizedto Offer Degree

Date

UNIVERSITY OF WASHINGTON

Date: June 2, 1977

We have carefully read the dissertation entitled The Mechanism of Bursting PacemakerActivity in Neurons of the Mollusc Tritonia Diomedia

submitted byStephen J. Smith

in partial fulfillment of the requirements of the degree of Doctor of Philosophy and recommend its acceptance. In support of this recommendation we present the followingjoint statement of evaluation to be filed with the dissertation.

This work provides an experimental and theoretical analysis of themembrane mechanism underlying the generation of recurrent impulse "bursts"in certain re-identifiable molluscan neurons. The voltage-clamp techniquewas used to characterize the three ionic channels of major importance ininitiating and terminating a burst of impulses. Among these channels, oneis, on pharmacological grounds, a Ca-permeable channel. The two othersare, on the basis of ion substitution experiments, permeable to sodium/calciumand to potassium, respectively. On the basis of previous work by Thompson,it is suggested that this potassium channel is regulated not directly bythe membrane potential but instead by the intracellular Ca++-concentration.In a theoretical section, Ca++-inward currents, the intracellular Ca++-concentration and the properties of the Ca++-sensitive potassium channelare combined with voltage-clamp data of Thompson and of Connor and Stevensin order to reproduce the electrical activity of bursting neurons.

This work is of significance in that it explains how a single nervecell can, without synaptic input, generate an impulse pattern of greatcomplexity.

4)DISSERTATION READING COMMITTEE.

Doctoral Dissertation

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tation may be referred to University Microfilms, 300 North Zeeb Road, Ann

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University of Washington

Abstract

THE MECHANISM OF BURSTING PACEMAKER ACTIVITY

IN NEURONS OF THE MOLLUSC TRITONIA DIOMEDIA

By Stephen J. Smith

Chairperson of the Supervisory Committee: Professor Wolfhard AlmersDepartment of Physiology and

Biophysics

Bursting pacemaker activity was studied in pleural ganglion cell bodies

of the nudibranch mollusc Tritonia diomedia. The work described here is

concerned with the mechanism of the endogenous slow alternation of episodes

of repetitive firing and silence characteristic of bursting pacemaker neurons.

Membrane electrical properties were studied using microelectrode voltage

clamp techniques. Implications of the voltage clamp observations were

investigated by mathematically modelling the observed membrane currents and

then reconstructing membrane potential trajectories. This work is presented

here in three parts.

Part I is an analysis of the prolonged tails of membrane current observed

following depolarizations of voltage clamped bursting pacemaker neurons. The

dependence of these current tails on membrane potential, time, and external

ions indicates that they consist of two distinct slowly decaying components.ti

A normally inward component designated I B activates maximally in less than

one second at positive potentials and decays exponentially with a time con-

stant of 2 - 4 seconds upon return to a negative holding potential. I B is

apparently carried by both sodium and calcium ions. A normally outward cur-

rent component designated I C activates progressively during depolarizations

lasting up to several seconds. The decay of I C upon repolarization has a

very prolonged, non-exponential time course, remaining at detectable levels

for over a minute following long activating pulses. I C evidently reflects a

calcium-activated potassium permeability mechanism: the current is carried

mainly by potassium ions, but activation upon depolarization appears to be

mediated by a voltage-dependent influx of calcium ions. Inward calcium

currents were therefore studied to allow development of a model for the

linkage of calcium influx to I activation. Though the slow current IB

should contribute to calcium influx, a much larger and more rapidly gated

calcium current, designated I D , has also been identified.

Part II develops a set of equations describing the total ionic current

observed in Tritonia bursting pacemaker neurons. The ionic current is repre-

sented as the sum of seven components, distinct in their activation kinetics

and their instantaneous current-voltage relationships. In addition to the

three ionic currents discussed in part I, the seven components include a fast,

inactivating sodium current, II' two voltage-dependent potassium currents, IK

and IA, and a linear leakage current, IL . The descriptions of each component

are based quantitatively on voltage clamp measurements described in part I or

in previously published work. Five of the currents are described using vol-

tage-dependent conductance equations of the form introduced by Hodgkin and

Huxley's analysis of the axonal action potential. The activation kinetics

for I are described using a different formalism which explicitly represents

an involvement of calcium ions. Calculations based on published estimates of

parameters relating to neuronal intracellular calcium metabolism indicate

that the slow kinetics of I activation may directly reflect the kinetics of

calcium ion accumulation near the inner membrane surface.

In part III, the membrane current equations are combined and solved

numerically to predict membrane potential under conditions corresponding to

an unclamped, spatially isopotential membrane region. The resulting solution

is a spontaneous oscillation closely resembling naturally occurring bursting

pacemaker activity: both action potential firing and the slow alternation of

bursts and silent episodes are reproduced. The two slow currents, IB and IC ,

are both necessary to reconstruct bursting activity. The slow variation of

the intracellular calcium ion concentration governing I c activation determines

the long period of the reconstructed pacemaker oscillation. The mathematical

reconstruction constitutes a detailed hypothesis regarding the mechanism of

bursting pacemaker activity and shows that the current components identified

by the voltage clamp analysis are quantitatively sufficient to account for

the phenomenon.

TABLE OF CONTENTS

List of Tables iv

List of Figures

TITLE PAGE

INTRODUCTION 1

Molluscan bursting pacemaker neurons 1

Previous studies of the mechanism of the burst oscillation 4

Rationale for the present study 8

METHODS 11

Voltage clamp studies 11

Description of membrane current 14

Reconstruction of membrane potential 15

RESULTS I: SLOW RELAXATIONS OF VOLTAGE CLAMP CURRENT 17

Characteristics of slow current tails 17

The slow inward current, IB 20

The slow outward current, I C 21

The time course of slow potassium permeability changes 27

The role of calcium entry in the activation of I C 32

The requirement for calcium entry 32

The dependence of calcium entry on time and voltage 34

DISCUSSION OF SLOW CURRENT TAIL ANALYSIS 40

The slow inward current I B 40

The calcium-activated potassium current I C 41

TITLE PAGE

RESULTS II: A MATHEMATICAL MODEL OF THE MEMBRANE CURRENT 45

Multiple component description of membrane current 45

Capacity current 49

The fast sodium current, I I 50

The fast calcium current, ID 54

The slow inward current, I B 57

The delayed potassium current, IK 60

The transient potassium current, IA 63

The calcium-activated potassium current, I C 66

A calcium-coupled model for the activation of I C 69

(1)calcium influx 70

(2)calcium efflux 70

(3) diffusion and binding in cytoplasm 71

Solving for the dependence of Cai(r,t) on time and voltage 73

Comparisons of Cai (a,t) with gc 77

The linear leakage current, IL 80

RESULTS III: THE RECONSTRUCTION OF BURSTING PACEMAKER ACTIVITY 82

Conditions for the reconstruction of membrane potential 82

Features of the reconstructed potential waveform 84

The time course of ionic currents and intra-cellular calcium86during the reconstructed burst cycle

DISCUSSION OF MODEL RESULTS 90

The mechanism of bursting pacemaker-like activity in the model 90

(1)"background" excitability 90

(2)rate-limiting conductance charges 91

Bibliography 95

iii

LIST OF TABLES

TABLE TITLE PAGE

I Seven components of ionic current observed in 46bursting pacemakers.

II Outline of model for intracellular calcium ion 74metabolism.

LIST OF FIGURES

FIGURE TITLE PAGE

1 Intracellular recordings of bursting pacemaker activity 2

2 Procedure for measurement of slow tail currents 18

3 The effect of external potassium concentration on slowtail currents

23

4 The effect of holding potential on the slow currenttail waveform.

25

5 Comparison of tail current and slope conductance timecourses.

26

6 Procedure for estimating the time course of membranepotassium permeability.

28

7 Time course of potassium-dependent current differencesafter pulses of varying durations.

31

8 The effect of Cow and lowered Cam on the slow tailcurrent.

33

9 Inward currents in a bathing solution containing mad . 37

10 Gating parameters: gI . 52

11 Gating parameters: gd. 55

12 Gating parameters: gm. 59

13 Gating parameters: gm. 62

14 Gating parameters: gm. 65

15 Predictions and recordings of decays at -40 mV. 68

16 Prediction of intracellular free calcium concentrationresponse to a depolarizing pulse

76

17 The voltage dependence of and predictions ofcalcium accumulation.

79

18 Reconstructed and recorded membrane potential waveforms. 83

19 Time courses of selected variables during the recon-structed burst cycle.

87

20 Time courses of ionic currents during reconstructedaction potential firing.

89

mCmNOWLEDGMENTS

This dissertation represents a part of a larger effort undertaken

in collaboration with my good friend, Dr. Stuart H. Thompson. Stuart's

1976 University of Washington doctoral dissertation documents another

part of this same effort. moth of our thesis projects grew out of

exploratory experiments we performed together at the Friday Harbor

Laboratories in the fall of 1974. Stuart and I recognized that the

insights gained from our early joint experiments could be pursued most

effectively if we continued to coordinate our efforts. mccordingly,

we embarked on the closely related projects described in our two disser-

tations. Each document reflects the individual author's contributions

to the collaboration, but includes extensive reference to the work of

the other partner. While the experiments, mathematical formulations,

and computations presented in detail here were designed and executed

by myself, I wish to emphasize that the broad concept of the work is

fundamentally a product of the collaboration between myself and Stuart

Thompson.

This project was begun under the supervision of Dr. Charles F.

Stevens. When Dr. Stevens moved from Washington to Yale University,

Dr. Wolfhard mlmers accepted the difficult task of supervising my

efforts at bringing the ongoing project to completion. I am deeply

grateful to both men for their multifaceted advice and support, and

for the examples they each set for me. I would also like to thank Dr.

Arthur C. mrown, Dr. mertil Hille, Dr. L. Donald Partridge, and Dr.

A.O.D. Willows for their advice and support of this project. Finally,

I am grateful for especially helpful discussions to Dr. John Connor,

Phillip Lloyd, Robert Snow and Paul Taghert.

INTRODUCTION

Ever since the introduction of techniques for recording the activity of

individual nerve cells, it has been obvious that the temporal patterning of

action potentials into rhythmic sequences is of central importance to the

coding and transmission of information in nervous systems. Temporal pat-

terns of unit activity clearly related to sensory or motor functions are

observed at various levels in the nervous systems of many different species.

The study of rhythmicity in nervous systems can readily be pursued to the

level of individual cells: it has been shown that certain neurons fire in

endogenously rhythmic patterns even after complete isolation from all other

neurons and any external input. This thesis is a detailed analysis of the

mechanism underlying one such endogenous rhythmic firing pattern in some

identified giant neurons of the nudibranch mollusc, or sea slug, Tritonia

diomedia.

Molluscan bursting pacemaker neurons

The giant ganglion cells of gastropod molluscs have proven to be espe-

cially suitable material for studies of the physiology of individual central

nervous system neurons. Their large size (some are over 500 pm in diameter)

facilitates many microtechniques, including multiple micro-pipette impale-

ment, which are generally much more difficult or impossible to apply to the

smaller neurons found in most other species. mnother important advantage to

working with gastropod ganglion cells is that many identifiable neurons with

constant physiological properties can be recognized reliably from one indi-

vidual preparation to the next. The subjects of the present study are the

three identified bursting pacemaker cells, LP12, LP13 and RP12 (Willows,

Dorsett and Hoyle, 1973) found in the pleural ganglia of Tritonia. Fig. 1

Fig. 1: Intracellular recordings from six different Tritonia

bursting pacemaker cells. The top four traces represent fully

developed bursting pacemaker activity. The two lower traces show

less common firing patterns which are transitional between bursting

and beating (regularly firing) pacemaker activity.

a

2

3

MINNS intracellular microelectrode recordings of the electrical activity in

several of these cells.

The characteristic bursting pacemaker firing pattern is distinguished

by the spontaneous occurrence of bursts of action potentials alternating

with electrically silent intervals. mctivity patterns like those exempli-

fied by Fig. 1 can persist continuously for hours or days under appropriate

experimental conditions. mursting pacemaker neurons similar to those

observed in Tritonia have been observed in many other species of gastropods.

Perhaps the most widely studied is a large, white cell in the parieto-

visceral ganglion of the sea hare, mplysia californica. This cell has been

designated variously "mr" (Arvanitaki and Chalazonitis, 1958), "the para-

bolic burster" (Strumwasser, 1965), and "R15" (Frazier et al., 1967).

Bursting pacemaker activity has also been observed in cells of terrestrial

molluscs (see Gola, 1974), although in such species the appearance of the

bursting pattern sometimes depends on the season and environmental factors

(Gainer, 1972a,b).

The endogenous nature of the bursting pacemaker rhythm was first

directly demonstrated by mlving's (1968) recordings of bursts in R15 somata

isolated by axonal ligation from all possible synaptic input. The same

conclusion was suggested earlier by Strumwasser's (1965) observation that

the phase of the burst rhythm in R15 could be changed by the injection of

current into the soma, ruling out the possibility that the cells merely fol-

low an oscillatory synaptic input. Observations confirming the endogenous

origin of bursting pacemaker rhythms in many other species, including

Tritonia, have since been reported Mater and Kaneko, 1972; Gainer, 1972b;

Thompson, 1576).

4

The physiological function of the bursting cells in Tritonia has not

Pet been determined. In mplysia and in the land snail, Otala lactea, ana-

tomical evidence strongly suggests that the cells serve a neurosecretory

function (Frazier et al., 1967; Gainer, 1972b). Several reports have indi-

cated that secretion by R15 in mplysia may be involved in body water or ion

regulation (Stinnakre and Tauc, 1966, 1969; Kupfermann and Weiss, 1976). m

bursting cell in the land-..slug, Limax, on the other hand, has been shown to

'observe a motor function, driving contractions of the salivary gland muscu-

lature (Prior and Gelperin, 1977). Burst firing patterns have been shown to

be subject to both short and long term modulation by synaptic inputs (mryan-

ltski and Chalazonitis, 1961; Pinsker and Kandel, 1967; Parnas, mrmstrong

and Strumwasser, 1974) and can also be modified by the introduction of cer-

tain specific peptides at concentrations on the order of 109 M (Barker,

Ifshin and Gainer, 1975).

Previous studies of the mechanism of the burst oscillation

The mechanism of the slow burst oscillation has been investigated

intensively in recent years, but has remained imperfectly understood. It is

clear that the observed slow variation in membrane potential is a necessary

step in the cycle of events that maintain the oscillation. It has already

been noted that the rhythm can be modified by injected current, which pre-

sumably acts only to change the membrane potential. With sufficient tonic

hyperpolarizing current the oscillation can be suppressed completely (mryan-

itaki and Chalazonitis, 1961). Similarly, no slow oscillations of membrane

current are observed when membrane potential is held constant by voltage

clamping (Carnevale, 1974; Gola, 1974; Smith, marker and Gainer, 1975).

5

These observations rule out the possibility that the rhythmic variations in

membrane potential merely reflect some underlying, autonomous (e.g., meta-

bolic) oscillation not directly involving a membrane potential-sensitive

mechanism (but see Gradmann and Slayman, 1975, for an example of such a

mechanism in a mutant strain of Neurospora crassa).

Junge and Stephens (1973) showed evidence from current injection exper-.

iments indicating that a slow variation in membrane potassium conductance

occurs during the burst cycle. Subsequent voltage clamp experiments by

Cola (1974) and T. Smith, marker and Gainer (1975) have supported this con-

clusion, and demonstrated that the potassium conductance involved reflects a

time-and voltage-dependent mechanism with intrinsically slow kinetics of

activation and decay. The slow potassium conductance is activated at depol-

arized potentials and decays after bursts on a time scale comparable to the

normal interburst silent intervals. These authors have suggested, therefore,

that this potassium conductance is fundamental to the mechanism of bursting,

its slow gating kinetics acting as the rate limiting step to determine the

low frequency of the burst oscillation.

R. W. Meech and coworkers have developed several lines of evidence sug-

gesting that membrane potassium conductance in molluscan neurons is con-

trolled partially by intracellular calcium ions. Since depolarization of

molluscan neural somata is known to produce a significant influx of calcium

ions (Geduldig and Junge, 1968; Geduldig and Gruener, 1970), Meech and

coworkers have suggested that a component of the potassium activation

observed during depolarization is activated by the accumulation of inward

current carrying calcium ions near the inner membrane surface (Meech and

Strumwasser, 1970; Meech, 1972, 1974a; Meech and Standen, 1975). Meech

6

(1974b) and Eckert and Lux (1976) have suggested that the activation kinetics

of the slow potassium conductance may directly reflect the kinetics of the

accumulation and dissipation of calcium ions inside the cell. mccording to

this view, the slow oscillation of membrane potential characteristic of

bursting activity would occur in intimate connection with an oscillation in

the level of free calcium inside the cell.

The existence of an intracellular calcium oscillation phase locked to

bursts in cell R15 has now been ascertained using both the dye mrsenazo III

(Gorman and Thomas, 1977) and the photoprotein aequorin (Zucker, personal

communication) as optical calcium indicators. In both cases, the optical

signals were of the type expected according to the postulate of a calcium-

mediated potassium conductance. In addition, Gorman and Thomas (1977)

showed that the microinjection of calcium ions into the soma of R15 produced

the expected parallel waveforms of calcium concentration, measured optically,

and potassium conductance, measured as voltage clamp current. Another test

of the calcium oscillation hypothesis, however, has been more equivocal. If

calcium entry is necessary for the activation of a potassium conductance

essential to the slow oscillation, one would expect bursting to require

external calcium ions. mursting pacemaker activity does disappear when

Tritonia neurons are bathed in media of nominally zero calcium concentration

(Smith, unpublished). In neurons of mplysia californica and Otala lactea,

on the other hand, burst-like oscillations in nominally calcium-free media

have been reported (Junge and Stephens, 1973; marker and Gainer, 1975).

Barker and Gainer (1975) showed that such oscillations disappeared when the

divalent ion chelating agent EDTA (ethylene diamine tetraacetic acid) was added

to the bathing medium, so the persistence of slow oscillations in the

7

nominal zero-calcium media might be explained as an effect of residual cal-

cium.

The slow, calcium-mediated potassium conductance may well determine the

duration of the burst cycle, but even so cannot entirely explain the

phenomenon of bursting pacemaker activity. Potassium conductances with similar

properties have been observed in many neurons that do not fire in bursts

(Connor and Stevens, 1971c; mrodwick and Junge, 1972; Meech, 1975; Gola,

1974; Partridge and Stevens, 1976), so some other factor must cooperate with

slow potassium currents to produce bursting. Several authors (Wilson and

Wachtel, 1974; Gola, 1974; Eckert and Lux, 1976) have suggested that a

region of negative or near-zero slope in the steady-state voltage clamp

current-voltage relationship may be a prerequisite for bursting. Partridge

(1975), however, has shown negative slopes, as in bursters, in neurons that

also have a slow potassium conductance similar to that of bursters (Smith,

unpublished), but themselves do not normally fire in bursts. This finding

indicates that some other factor still may be necessary to differentiate

bursters and to fully explain the bursting pattern.

my activating a voltage clamp circuit at different times during bursts

of action potentials in cell R15, Gola (1974) discovered a slowly decaying

inward current which is activated preferentially to the slow potassium cur-

rent early in the burst. Gola did not identify the ionic basis of this

inward current, but suggested that it might cooperate with the slow potas-

sium current to generate the slow oscillation characteristic of bursting

pacemaker activity. S. H. Thompson and I (Thompson and Smith, 1976) have

shown that depolarizing afterpotentials (DmPs) several seconds in duration

follow driven individual spikes in bursting pacemakers. The slow time-

8

course of DmPs suggests that they very probably reflect the same mechanism

as the slow inward current studied by Gola (1974); our analysis indicates

that this current is carried partly by sodium ions and partly by calcium

ions. DAPs were observed in the bursting pacemaker neurons of all five

gastropod species we studied, but were not observed in any non-bursting

neurons, suggesting a possible causal role in bursting pacemaker activity.

Rationale for the present study

The work described in this thesis has been aimed at advancing a com-

plete and quantitative description of bursting pacemaker activity in terms

of underlying membrane properties. The rationale for the present effort is

borrowed from Hodgkin and Huxley's (1952) analysis of the nerve action

potential. Hodgkin and Huxley showed that the main features of the action

potential could be reconstructed from a mathematical model based strictly on

the analysis of voltage clamp data. The voltage clamp technique allowed

Hodgkin and Huxley to analyse the complex ionic permeability properties of

the squid axon membrane into three relatively simple ionic current compo-

vents, with well defined ionic selectivities and activation or gating

kinetics. From voltage clamp data, they developed aset of equations to

describe the time and voltage dependencies of each ionic current component.

The equations for the three current components were then combined and solved

for the condition of constant total membrane current, corresponding to the

natural, unclamped condition of a small, isolated patch of membrane. The

resulting solutions for membrane potential closely resembled recorded mem-

brane action potentials. This mathematical reconstruction established that

Hodgkin and Huxley's voltage clamp analysis had in fact encompassed the

membrane properties essential to action potential firing, and provided a

9

detailed hypothesis as to exactly how the individual current components

cooperate to produce the action potential. The rationale for Hodgkin and

Huxley's analysis revolves around the central role played by membrane poten-

tial in linking the various processes underlying axonal excitation. Evi-

dence discussed above indicates that membrane potential plays a similar role

in the burst oscillation, so I have adopted a strategy similar to Hodgkin

and Huxley's to approach questions concerning which membrane electrical

properties are essential to bursting pacemaker activity, and how current

components identified by voltage clamp analysis could produce slow oscilla-

tions of membrane potential.

The results presented here are divided into three parts. Part I is an

analysis of the slow relaxations of voltage clamp current observed in burst-

ing pacemaker neurons. In part II, a descriptive mathematical model of the

total membrane current is developed on the basis of voltage clamp data pre-

sented in part I in addition to some previously published results. In part

III, this descriptive model is used to reconstruct membrane potential wave-

forms which reproduce many of the main features of bursting pacemaker activ-

ity. The model presented in parts II and III is similar in many respects to

that developed by Hodgkin and Huxley, particularly in that it is developed

as an empirical description of voltage clamp data. The model presented here

differs from Hodgkin and Huxley's in that it has been necessary to represent

a larger number of distinct components of ionic currents, as might be

expected considering the greater complexity of bursting pacemaker activity

compared to axonal action potential firing. It has also been necessary here

to consider the calcium mediation of potassium activation, whereas in the

10

squid axon it is apparently appropriate to describe each current component

as depending directly on membrane potential and time.

Other mathematical models reproducing bursting pacemaker activity have

now appeared in the literature (Both, Finger and Chaplain, 1976; Plant and

Kim, 1976; Gulijarani, Roberge and Matthieu, 1977). The model developed in

this thesis differs in many respects from each of these other efforts, but

the most important distinction is that only the model presented here was

developed strictly from voltage clamp measurements. The model proposed by

Both et al. (1976) does not address the question of the origin of the slow

oscillation at all, it merely assumes a slow cyclic driving function. The

models proposed by Plant and Kim (1976), and Gulijarani et al. (1977), pro-

pose membrane mechanisms which could account for slow oscillations, but the

implications of these models are limited by the fact that important kinetic

parameters were simply chosen to insure oscillation, without experimental

constraints. The major strength of the model presented in parts II and III

of this thesis is that it leads to predictions of burst-like oscillations

from parameters based entirely on voltage clamp measurements.

METHODS

Voltage clamp studies

Three identified giant cells in the pleural ganglia of the sea slug

Tritonia diomedia reliably exhibit bursting pacemaker activity; these

cells are designated RP12, LP12 and LP13 (Willows, Dorsett and Hoyle, 1973).

The three cells are similar to each other in appearance, normal firing

patterns, and all membrane properties that have been studied. The experi-

ments described below have been performed using a two microelectrode voltage

clamp applied to one of the three cells specified in an isolated ganglion

preparation.

Specimens of Tritonia were obtained by trawling in waters near the

San Juan Islands of Washington State. Animals were maintained in a

recirculating natural sea water system at 10 C prior to use. The cerebral-

pedal-pleural ganglion complex was excised with nerve root stumps 1 to 5 mm

in length and pinned to the waxed bottom of a recording and perfusion cham-

ber. The chamber was maintained at a temperature between 10 °C and 11°C.To facilitate microelectrode penetration, the epineureal sheath was exposed

for 15 min to a 0.5%, by weight, solution of collagenase (Worthington

Biochemicals) in the normal bathing solution described below. Preliminary

investigations showed no effect of this enzyme treatment on the cell's

firing patterns or on behavior under voltage clamp conditions. Cells fail-

ing to recover normal bursting pacemaker activity within 30 min after pene-

tration by the two microelectrodes required for voltage clamping were

rejected without further study.

The voltage clamp electronics employed were similar to those described

by Connor and Stevens (1971a), except for two modifications designed to

12

improve control of membrane potential during long depolarizing pulses:

(1) membrane potential was recorded differentially between an intracellular

microelectrode and a voltage reference electrode in the bathing solution,

and (2) the final output amplifier was modified to provide increased gain

at low frequencies, as described by Dionne and Stevens (1975). These

modifications were necessary because the current electrodes tend to polarize

and increase in resistance while passing large and sustained clamp currents.

The internal current and voltage electrode micropipettes were filled with

3 M KC1 and had tip resistances between 2 and 4 N2. All electrodes were

Ag-AgCl in either 3 M KC1 (micropipettes) or sea water (bath electrodes).

Membrane potential and clamp current recordings were stored on FM

magnetic tape. Clamp current was recorded on two separate tracks of the

tape: one track at low amplification to record the large currents flowing

during depolarizing pulses, the other track at high amplification to record

small tail relaxations near zero current with a minimum of noise and drift.

Tape recorded data was played back onto a pen recorder or digitized by a

computer system for display and analysis. A first order low pass filter

(time constant = 24 msec) was used to reduce noise in records of slow

current relaxations.

Solution changes were effected by introducing 10 ml of the new solution

at one end of the recording chamber, while the bath volume was maintained at

0.9 ml by a suction tube located at the opposite end of the chamber. The

normal bathing solution is similar to sea water, with the following composi-

tion: 420 mM-NaC1, 10 mM-KC1, 23.5 mM-MgC1 2 , 25.5 mM-MgSO 4 , 10 mM-CaC12 ,

2 mM-NaHCO. The various altered bathing solutions were the same except3.

13

for the following specified changes: High K+

solution, 390 mM-NaC1,

40 mM-KC1; Cobalt solution, 2.5 mM-MgC1 1 , 1 mM-CaC1 2, 30 mM-CoC1 2 ; marium

solution, OmM-NaC1, 320 mM-Tris C1,100mM-TEm Cl, 49 mM-MgC1 2, OmM-MgSO4 ,

OmM-CaC12' 10mM-maC1

2.

Tritonia bursting pacemaker neurons consist of a roughly spherical

cell body or soma, approximately 300 pm in diameter, and a slender axonal-

dendritic process which ramifies in the neuropil region of the ganglion and

also sends axons out one or more nerve roots. In the present studies,

voltage-sensing and current-passing electrodes were inserted into the soma.

Such electrode placement results in good control of membrane potential in

the soma region of the cell (see Geduldig & Gruener, 1970; Connor & Stevens,

1971a), but not in the axonal-dendritic process.

Two effects of the poor clamping of distant axonal membrane are evident

in whole 'cell voltage clamp current records: (1) capacity transients are

complex and prolonged (up to 10 msec for 90% complete settling after small

voltage steps), presumably reflecting the axoplasmic resistance in series

with part of the membrane capacitance, and (2) at clamping potentials in

the range of spike threshold, small oscillations of membrane current,

presumably due to repetitive action potential firing in electrically remote

axonal regions of the cell, can often be observed. Various schemes for

measuring membrane current across the well-clamped somatic membrane in

isolation from the axonal cable current have been described (see Connor and

Stevens, 1971a; Neher, 1971; Kado, 1973; Kostyuk, Krishtal and Pidoplichko,

1975; Connor, 1977; Lee, mkaike and mrown, 1977), but do not appear to be

necessary in the context of the present investigation. Preliminary in-

vestigations have shown that the slow current tails described below have

14

much the same appearance in intact cells and in microsurgically isolated

somata, indicating that the slow relaxations reflect mainly the behavior

of the soma membrane, which is clamped well in both the intact cell and

the isolated soma preparations. The only obvious manifestations of poor

clamping in the axon region of intact cells, the long capacity transients

and the small current oscillations at moderate depolarizations, do not

interfere with observations of slow current relaxations in the pacemaker

potential range following depolarizing activation pulses. mn intact cell

preparation was used in the present study because such cells display stable

bursting pacemaker activity more reliably than isolated somata.

Description of membrane current

The mathematical description of membrane current developed in part II

of the results below is based on the analysis and measurement of voltage

clamp data presented in part I of the results and in several previously

published studies (Thompson 1976, 1977; Connor and Stevens, 1971a,c). Two

different methods were used to determine the equations describing particular

ionic current components. In some cases, experimental data and theoretical

curves were displayed simultaneously on a computer terminal display

screen. The expressions generating the theoretical curves were then

varied to achieve a good correspondence, judged by eye, between experimental

and theoretical curves. In other cases, where the choice of an appropriate

descriptive equation (e.g. an exponential or a power of an exponential)

Lad already been made, measured data points were transformed mathematically

so that exponential parameters could be determined from a linear slope

and intercept.

Most of the ionic current mechanisms described in this paper are

15

characterized by equations for time constants and steady state activation

values as functions of membrane potential. The equations used are of the

general form:

1 K5-f(V) = K1 { + K2K3I exp , V-

K4(1)

where Kl-K5 are the parameters to fit experimental data points. Solutions

to this equation are sigmoidal in form, as is evident in Figs. 10-14. In

some cases, the time constants might be better described by an equation

giving a bell-shaped curve. Nevertheless, the equations used appear

adequate for the range of membrane potentials relevant to the reconstruction

of bursting pacemaker activity. The general expression indicated by eqn. 1

was chosen purely for convenience in fitting experimental data points,

rather than for any possible physical significance.

Reconstruction of membrane potential

The mathematical reconstruction of membrane potential described in

part III of the results was accomplished using numerical techniques,

implemented on a digital computer, to solve finite difference analogues of

the differential equations developed in part II to describe the membrane

current. The first-order differential equations defining the behavior of

the postulated voltage-dependent gating variables m and h were integrated

using the point slope formula. Approximate solutions to the partial

differential equation representing the radial diffusion of intracellular

calcium ions were found by a method equivalent to resolving the spherical

intracellular space into a series of thin, concentric shells. m first-

order transfer of ions between adjacent shells in such a system

16

approximates the diffusion process. The effects of widely varying finite

variable step sizes for the time and radius were investigated in order to

discover and avoid values which might give rise to errors in the solutions

being sought. Time increments were automatically adjusted during each

computing run, since longer increments were acceptable during quiescent

intervals than during action potentials.

RESULTS I: SLOW RELAXATIONS OF VOLTmGE CLAMP CURRENT

Previous studies have characterized the major ionic currents flowing

during voltage clamped depolarizations of Tritonia bursting pacemaker

neurons (Thompson, 1976, 1977). The results presented here are concerned

primarily with some long-lasting aftereffects of depolarization, observed

as slowly relaxing current tails following the return of the voltage

clamped membrane to negative holding potentials. These slow current

relaxations are of interest because they occur on a time scale similar to

the slow burst oscillation, and because they are observed in the pacemaker

potential range between spike threshold (typically -35 mV to -25 mV) and

the trough of the interburst hyperpolarizing wave (-55 mV to -40 mV). m

rationale for quantitatively separating the slow, subthreshold current into

two distinct components is developed here. m third component of ionic

current, believed to represent an inward calcium current, will also be

described. This putative calcium current has relatively rapid kinetics

of activation and decay, and therefore makes no direct contribution to

slow current tails, but does appear to play an important role in mediating

the activation of one of the two slower current components.

Characteristics of slow current tails

Fig. 2m illustrates the procedure we have used to record tail current

relaxations. The soma membrane is voltage clamped and held at a specified

holding potential (-40 mV in this case) for several minutes to allow the

membrane current to stabilize. The membrane then is stepped to a specified

pulse potential (+2 mV) for a specified duration (3.0 sec) and returned

to the original holding potential. The two lower traces in Fig. 2A display

the resulting membrane current signal at two different amplifications. In

the low gain record, only the large net ionic current flowing during the

Fig. 2: m. Procedure for measurement of slow tail currents. Top

trace: membrane potential controlled by voltage clamp circuitry.

Middle trace: clamp current at low gain showing larger currents

flowing during activation pulse. Bottom trace: simultaneous high-

gain record showing the smaller slow tail current. Current during

the pulse is off scale at this amplification. The flat baseline

represents the level of the steady holding current at this hold

potential. It was drawn to emphasize the waveform of the relaxa-

tion back toward this level. Similar tail current baselines are

drawn in subsequent figures. m. Slow tail currents recorded after

pulses of five different durations (indicated to the left of each

trace) by the method indicated in m. This data was recorded from

the same cell as that shown in m, using the same holding and pulse

potentials.

A

18

mV+21 MEMmRANE POTENTImL

-40 J I 3 sec

100 -

nm 0

-100 -

CLAMP CURRENT: LOW mMPLIFICmTION

2 ]

nm 0 --2

CLAMP CURRENT: HIGH mMPLIFICmTION

BPULSE DURmTIONS (msec) INDICATED NEmR TRmCES

100 - ••••

,w.00..:01.0.101PFeriirir.r...11.~0.44/.111.11.11.~..1ftrall.•■■•■■•■400 •••• •

2 nA

10 sec

HOLDING POTENTImL: -40 mVPULSE POTENTImL: +2 mV

19

pulse and the impulse-like capacity current transients are evident. The

main properties of the ionic currents flowing during such pulses have

been described by Thompson (1976, 1977). The higher gain trace shows the

slow tail current following the pulse: the current during the pulse is far

off scale at this amplification. The flat baseline represents the steady

level of holding current and has been drawn to emphasize the waveform of

the relaxation back to this steady level. It is with such slow current

relaxations that the present analysis is primarily concerned.

Fig. 2m shows the typical effect of varying the duration of the

activating pulse. mfter the shorter pulses, the slow current tail is

dominated by an initially inward transient. As pulse duration is increased,

the waveform undergoes a complex transition to an initially outward form.

There is reason to believe that this transition from inward to outward

current is intimately related to the mechanism of bursting pacemaker

activity. It has been possible to demonstrate such transition behavior

in virtually every active bursting pacemaker cell studied, but never in

non-bursting ganglion cells. Only slow tails of the initially outward

form are normally observed when the procedure of Fig. 2m is applied to

ganglion cells other than identified bursting pacemakers.

The ionic sensitivities and current-voltage relationships of the

inward tails after short pulses are very different from those of the out-

ward tails after long pulses, suggesting that the complex tails may be

a composite of two distinct slowly decaying ionic currents. Several

procedures for studying each of the two slow processes in isolation have

been developed. The two slow currents, designated I B and Ic , are discussed

separately below.

20

The slow inward current IB

The experimental analysis of I B is described in detail in S. H.

Thompson's doctoral dissertation (Thompson, 1976). Only a brief summary

of that analysis will be presented here.

IB decay tails can be studied with relatively little interference from

other slow transients by using only short (<50 msec) activation pulses. The

resulting relaxations are of the inward form shown by the top trace in

Fig. 2B. Such tails behave as if they resulted from the slow decay of

an increased membrane permeability to sodium and calcium ions following

the pulse: the amplitudes of inward tails are reduced by substituting

Tris for sodium or magnesium for calcium in the bathing salines, and the

tails are eliminated by making both substitutions simultaneously. The

tail currents after short pulses are unaffected by changes in the external

potassium concentration, unaffected by addition of the potassium current

blocking agents TEm (tetraethylammonium) and 4-AP (4-aminopyridine), and

unaffected by minor variations in hold potential near the reversal potential

for other components identified as potassium currents. These observations

rule out the possibilities that I B tails could reflect either changes in

potassium permeability or an extracellular accumulation of potassium

ions.

It has not been possible to observe I B activation directly at depolar-

ized potentials. This is because the currents associated with the slow

kinetics defining IB are much too small in comparison to the other currents

activated during membrane depolarization (see Fig. 2). The conductance g B

activated during various depolarizing pulses has instead been estimated by

extrapolating the slow current tails at subthreshold potentials back in

21

time to the ends of pulses and dividing by an appropriate estimate of the

driving force. It is assumed that slow tails measured near the potassium

equilibrium potential reflect I B alone, regardless of activating pulse

duration. This assumption is made in order to reject another slow current,

associated with long pulses, which appears to reflect mainly a variation in

potassium conductance (see the following section). From measurements of

IB tails, it has been possible to estimate the time course and saturating

level of gm activation at potentials ranging from -40 mV to +30 mV during

pulses lasting up to 2 sec. The activation of g m appears to increase

progressively with depolarization over the entire range of potentials

studied, reaching steady-state values in the range 0.02 to 0.04 limhos at

0 mV. The kinetics of activation and decay appear to be approximately

exponential, with the time constants ranging from 2 seconds or more near

-40 mV to 300 msec or less at +20 mV (see Fig. 12, Results II).

The slow outward current I

Observations described in this section suggest that the slow outward

current tails after long activation pulses reflect mainly a relaxation of

membrane permeability to potassium ions. The current governed by this

slowly varying potassium permeability is designated I C„ Transitions from

inward to outward slow tails like that illustrated by Fig. 2B appear to

reflect an increasing activation of the outward IC' relative to the

inward IB' as depolarizing pulse length is increased. Like potassium

currents described in neurons of mplysia and Helix (Meech, 1974b; Meech

and Standen, 1975), I c appears to be activated by the influx of calcium ions

during depolarization, rather than by depolarization per se.

22

Fig. 3 shows evidence that a variable potassium permeability contributes

increasingly to slow tail waveforms as activation pulses are lengthened.

The tail currents in the left column were recorded after pulses of four

different durations in a bathing saline of normal ionic composition. The

slow tails in the right column were recorded under conditions identical

except that the bath potassium concentration was increased from 10 mM to

40 mM by replacing a small fraction of the sodium ions with potassium. m

major effect of the solution change is evident only after the longer activa-

tion pulses. The finding that the tails after short pulses are relatively

unaffected by a change in the electrochemical gradient for potassium ions

is consistent with our previous conclusion that such tails reflect mainly

sodium and calcium currents. The outward tails after the longer pulses, on

the other hand, are completely eliminated by high external potassium. This

observation suggests that such outward tails probably reflect a slow varia-

tion of potassium permeability. The four-fold elevation of external potas-

sium should nearly eliminate the electrochemical gradient for membrane

potassium current at the -40 mV holding potential used in this experiment.

mssuming an intracellular potassium activity of 137.9 mM (equal to the

mean values measured in cell R-15 by Kunze, Walker and Brown, 1971) and

assuming an activity coefficient of 0.69 for K + in the external medium (see

Brown, 1976) the Nernst potentials for potassium are -73.2 mV in the 10 mM

K medium, and -39.3 mV in the 40 mM K+

medium, at 11°C. The slow inward

tails evident after long pulses in high potassium probably reflect mainly

the slow inward current, IB' and possibly a very small potassium current

now flowing inward.

The waveform of slow tail current after a long pulse depends on the

Fig. 3: The effect of external potassium concentration on slow tail

currents. The currents in the left-hand column were recorded in

the normal (10 mM K+) bathing medium. Each tail was recorded at a

holding potential of -40 mil after a pulse to +2 mV. The activation

pulse duration was varied as indicated near the traces (in msec).

The currents in the right-hand column were recorded under condi-

tions identical except that the bathing medium contained 40 mM K+

.

Note that the tails after the shortest pulses are nearly identical

in the two different bathing media. ms pulse length is increased,

tails in the normal medium develop initially outward components,

while the forms of the tails in high potassium remain relatively

constant. The digitized records in this and subsequent figures

are sampled at 200 msec intervals. The short vertical tick-marks

at the left of the holding current baselines in this and subsequent

figures showing digitized current tail records represent the times

of the trailing edges of the activation pulses, that is, the times

at which membrane potential was returned to the holding level.

23

25

•

FV

100 F--. 711°61611

•;

•

400 . • • •, .

•

•

• • 1 nA

10 sec••

• • .1' Al w a , ,if#104441.....,..4repimeivnarivoftentAvi.

t 2000 sibrAemie.rworanfru...).......0-1- 71mb,

HIGH (40 mM) IC+NORMmL (10 mM) +

24

holding potential at which it is measured. Fig. 4 depicts the slow

tails following identical 1 sec pulses to +1 mV at several different holding

potentials from -30 to -65 mV. mt -30 mV and -35 mV, the slow tails are

entirely outward. ms the holding potential is made more negative, the

tails undergo a reversal, but current decays are not always monotonic and

the form, as well as amplitude and sign, of the decay varies with potential.

The data shown in Fig. 4 are consistent with the identification of the

slow outward tails with a variable potassium permeability: the outward tail

current becomes smaller as the holding potential is made more negative and

thus closer to the potassium equilibrium potential. m more precise inter-

pretation of the hold potential dependence of slow tails is difficult

because these tails presumably represent a composite of several processes

with similar relaxation times. The slow tails at the more negative holding

potentials may include effects of a slow, time-dependent inward-going

rectification (see Marmor, 1971), as well as I B and IC' and the earlier

phases of the decay may reflect the recovery from an extracellular accumu-

lation of potassium during the activation pulse (see Eaton, 1972).

If the slow outward tails do in fact reflect a decline in membrane

ionic permeability, it should be possible to observe a corresponding

decrease in membrane conductance during such tails. Fig. 5 illustrates an

experiment designed to test this prediction. The upper trace shows the

slow outward current tail observed at a hold potential of -35 mV after a 4

sec pulse to -1 mV. The lower trace was recorded under conditions identical

except that a 5 Hz square wave, 3 mV peak to peak in amplitude,was superim-

posed on the -35 mV hold potential during the slow decay of outward tail

current. Two aspects of the membrane current response to the imposed voltage

Fig. 4: The effect of holding potential on the slow current tail

waveform. mll tails were recorded after 1 sec pulses to +1 mV,

in the normal bathing solution.- Holding potential was varied as

indicated near each trace. Slow tails were not recorded at more

negative or more positive potentials because the holding currents

required became excessively large and generally failed to stabilize

at steady values.

25

HOLDING POTENTIAL (mV) INDICATED NEAR TRmCES

—35 —

--40

—45 --I

—50 —I

—55 --J

—65 ---I

PULSE POTENTIAL: +1 mV PULSE DURATION: 1 sec

Fig. 5: Comparison of tail current and slope conductance time-

courses after a 4 sec pulse to -1 mV. The tail current at a con-

stant holding potential of -35 mV is shown in the top trace. The

current waveform shown in the lower trace was recorded under

conditions identical except that a voltage square wave (3 mV, p-p)

was added to the -35 mV holding potential beginning 1 sec after the

pulse. m "late" membrane slope conductance is proportional to the

height of the inner envelope of the current response to the square

wave (see text). Note that the inner envelope becomes narrower

along a time-course similar to the decay of net outward current

shown in the upper trace.

26

PULSE

HOLD POTENTIAL

4 sec -35 my-1 my 2 sec

-35 my square wave (3 my p-p)

it

27

square wave are evident in Fig. 5: capacity current spikes coincident

with the square wave voltage steps, and an inner envelope of the quasi-

steady currents approached during each half-cycle of the square wave.

The difference between the quasi-steady currents flowing during any two

successive square-wave half cycles can be divided by the amplitude of the

intervening voltage step to obtain a slope conductance giving a relative

indication of slow variations in membrane conductance. (Note, however,

that the absolute magnitude of the slope conductance so calculated has no

significance, since rapid changes in ionic permeability probably occur

between the time of each voltage step and the subsequent approach to a

quasi-steady current). Fig. 5 shows that the width of the inner current

envelope, which is proportional to the slope conductance defined above,

decreases along a time course similar to that of the slow outward current

tail. The most probable explanation of this finding is that both the

decline in outward current and the decline in slope conductance reflect

a slow decline in membrane permeability to an ion with a reversal potential

negative to -35 mV. Measurements of slope conductance therefore also

support the identification of the slow outward tail with a variable per-

meability to potassium ions, which should have an equilibrium potential

near -70 mV.

The time course of slow potassium permeability changes

The evidence presented above indicates that the slow current tails

observed in the pacemaker potential range reflect at least the two

distinct ionic currents IB and I

C . Fig. 6 illustrates a procedure to

separate the time course of the potassium permeability change underlying I c

from the simultaneous relaxation of I B and any other ionic current not

Fig. 6: Procedure for estimating the time course of membrane potas-

sium permeability. Tails are recorded in media of two different

potassium concentrations, and the differences in current are calcu-

lated point by point in time from digitized records. A. tail

current after a 700 msec pulse to +2 mV at a holding potential of

-40 mV. The normal bathing medium used has 10 mM le. B. tail

current in 40 mM K+

medium. Pulse dimensions and holding potential

the same as in A. C. tail current recorded after return to normal

10 mM le medium. Pulse dimensions and holding potential the same

as in A and B. D. Waveform assumed to represent the time-course

of potassium permeability (see text). Calculated by subtracting

points in B from the mean of points in A and C.

28

A NORMmL e B HIGH le

• o •

1 -: •.21P

.1%

•P•

1 nm

10 sec

C RETURN TO NORMmL le D NORMmL - HIGH le• DIFFERENCE

Nog,ANIONSA,40,

Illiodmmu siuglarimial.maumu,gmt. 411444/444441,WeiWoluslor..

29

involving potassium ions. The procedure is based on the observation,

illustrated in Fig. 3, that slow tail waveforms depend on the concentration

of potassium ions in the bathing solution. If this is due to a slowly

varying potassium permeability, the differences between current tails in

different potassium concentrations should directly reflect the time-course

of that permeability change. Fig. 6m-C shows slow current tails following

a 700 msec pulse to +2 mV, at a hold potential of -40 mV. The tails were

recorded in a normal bathing solution (10 mM Fe, Fig. 6A), then,in the

40 mM le solution (Fig. 6B), and then again in the normal solution(Fig. 6C). Fig. 6D shows the difference between the tail current in high

potassium and the mean of the two tails recorded in normal potassium,

calculated point by point in time during the tail relaxations. Tails

recorded in the normal medium both before and after the high potassium

run were averaged in the hopes of controlling for the gradual rundown of

tail current amplitudes generally evident during this rather lengthy

experimental procedure (20 min was required for each solution change to

equilibrate and 20 min to record tails after ten different pulses in

each solution).

The current difference shown in Fig. 6D should be directly propor-

tional to time-dependent variations in membrane potassium permeability

if one assumes: (1) that only potassium fluxes are affected by the

change in [K+Jbath from 10 mM to 40 mM, , and (2) that [K+

and [1(+

] 0, the

potassium concentrations immediately adjacent to the inner and outer

membrane surfaces, respectively, are constant for a given [le]bath'

If

one further assumes: (3) that [K+

]o

is equal to [K+]bath'

(4) that

membrane potassium permeability is independent of the external potassium

30

concentration, and (5) that [K+] is independent of [m

+]bath' it is

possible to calculate the proportionality constant relating the observed

current differences to variations in a potassium permeability coefficient

of the type expressed in the Goldman-Hodgkin-Katz flux equation (Goldman,

1943; Hodgkin & matz, 1949). If AP represents a deviation of the potassium

permeability coefficient from its steady-state level at a holding potential,

V, and Idiff represents the difference between tail current recorded in

+baths of two different potassium concentrations, [1(+]bath and bath

(expressed here in moles/cm3), we can write, from the Goldman-Hodgkin-

Katz equation,

1 eVF/RT-

Idiff F

2 V yK [K+ ]bath - y K

+] bath

(2)

where R, T, and F have their usual meanings, and yK is an activity

coefficient. It is assumed that yK has a value of 0.69 in the seawater-

like media used (see Brown, 1976). For the data shown in Fig. 6, the

constant APC/Idiff has a value of 8.7 x 10

-9cm sec

-1 . The data in Fig.

6D therefore represents a slow decay of AP c from a value of approximately

3 x 10-8 cm-sec-1 at a time 200 msec after the pulse. The applicability of

assumptions (1) -(5) is considered in the discussion section.

Fig. 7 shows current differences after activating pulses of ten

different durations, ranging from 25 msec to 3 sec. These differences

were obtained by the procedure illustrated in Fig. 6, using the same cell,

and the same holding and pulse potentials. The figure shows that the

potassium-sensitive component of slow current is activated progressively

during the depolarizing pulse up to the longest duration studied.

C RT

25

31

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10" sec

501...V44.11.potkablivArtahricsv

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32

Though noise inherent in the preparation makes exact comparisons difficult,

the decay tails in Fig. 7 appear similar in form over a wide range of pulse

durations and relaxation amplitudes. This situation is in marked contrast

to the complex effect of pulse duration on the total current in normal

bathing media (cf. Fig. 2b), suggesting that the potassium change

separation procedure has successfully isolated an elementary component

of the slow current tails.

The role of calcium entry in the activation of I

m number of studies have found evidence that part of the potassium

current in bursting pacemaker neurons is activated by an influx of calcium

ions during membrane depolarization (Meech, 1974 b, c; Meech and Standen,

1975; Heyer and Lux, 1976; Thompson, 1977). Meech (1974 a, c) found that

recovery from this calcium-dependent potassium activation includes a slow

component, which appears comparable in its time-course to the decay of

IC in Tritonia neurons. mn investigation of the relationship of calcium

entry to the activation of I C has therefore been undertaken.

The requirement for calcium entry. One argument for the proposed mediating

role of calcium ions (Meech and Standen, 1975) cites the suppression of

potassium activation by operations presumed to reduce or eliminate voltage-

dependent calcium currents. Such operations include the removal of calcium

ions from the bathing solution and the addition of certain other divalent

metal cations, such as Cow and Mn++ (see Hagiwara, 1973). Fig. 8 shows

the effect of removing most of the bath calcium, and adding cobalt (Co++

),

on slow current tails in a voltage clamped Tritonia bursting pacemaker. The

upper trace shows the tail current, in a normal bathing solution, at a

holding potential of -30 mV, following a 1 sec depolarization to +1 mV.

Fig. 8: The effect of Co and lowered Ca++ on the slow tail

current after a long pulse. The two current tails were both

recorded at a holding potential of -30 mV, after a 1 sec pulse to

+1 mV. The upper trace was recorded in the normal bathing saline.

The lower trace was recorded in a solution where part of the Mgt

and all but 1 MM of the Ca++ were replaced by Co++.

33

NORMAL (10 mM Ca)

COBALT, LOW CmLCIUM (30 mM Co, imM Ca)

34

The tail is of the initially outward form presumed to reflect mainly the

slow decay of IC. The lower trace shows the very different slow tail

current recorded in the cobalt bathing solution. In this solution, the

tail is initially inward, resembling the slow tails normally observed

at this holding potential only following pulses of much shorter duration

(see Fig. 2). The suppressing effect of cobalt and low calcium on I C

shown in Fig. 8 is representative of observations over a wide range of

pulse dimensions and holding potentials at which outward tails are

normally observed. Other experiments (see Thompson, 1977) show that the

slow outward tails can also be suppressed by the addition of either

Co++

or Mn++ to solutions containing the normal (10 mM) concentration of

calcium, or by simply removing calcium in the absence of Co++

or Mn++

.

mdditional voltage clamp experiments suggesting a role of calcium

entry in the activation of potassium currents in Tritonia neurons have

been described by Thompson (1977). Thompson's study includes a description

of a faster calcium-sensitive component of the outward current flowing

during depolarizing pulses. On the basis of a common requirement for

calcium entry, it seems likely that the calcium-dependent potassium

current observed during pulses and that observed as slow tails after

pulses reflect a single mechanism of potassium transport, with complex

gating kinetics which may depend on both calcium influx and voltage.

The dependence of calcium entry on time and voltage. In considering possible

roles of calcium entry in potassium activation, it should be helpful to

characterize the dependence of calcium entry on time and membrane potential.

One question of particular interest concerns the pattern of calcium entry

during sustained depolarizations. Many published studies have suggested

35

that calcium currents are subject to a voltage dependent inactivation,

possibly similar to that governing sodium currents (Geduldig and Gruener,

1970; Connor and Stevens, 1971; Kostyuk, Krishtal and Doroshenko, 1974; Stan-

den, 1975). The rate and extent of such inactivation bears on the inter-

pretation of the continued, progressive activation of I C observed during

long depolarizing pulses (see Fig. 7). If there is a sustained influx of

calcium during long depolarizations, the progressive activation of I C

may simply reflect the cumulative entry of calcium ions. If, on the other

hand, calcium currents inactivate rapidly and completely, an alternative

explanation for the progressive activation of I C is required. Previous

voltage clamp studies of the calcium current in molluscan somata have

suggested divergent views of the kinetics of calcium inactivation, based

on different types of observations in many different types of cells (see

calcium current references above, but also Kostyuk,Krishtal and Pidoplichko,

1975; mdams and Gage, 1976; Eckert and Lux, 1976; Connor, 1977). I have

attempted, therefore, to analyze the calcium current in Tritonia neurons

for conditions of time and voltage comparable to those employed in

characterizing IC activation.

One possible source of ambiguity in a voltage clamp study of calcium

inactivation arises from the presence of a calcium-dependent potassium

activation mechanism. Calcium activation followed by a potassium activa-

tion dependent on calcium entry may sometimes be difficult to distinguish

from calcium activation followed by calcium inactivation. Both sequences

would appear as an inward current which declines or becomes outward

during sustained depolarizations, and in either case both the initial

inward current and the later outward going component would be reduced

by reductions in external calcium. Connor (1977) has shown evidence that

36

an apparent calcium inactivation in mrchidoris neurons can be attributed

mainly to a calcium dependent potassium activation. He has also shown

that barium ions appear to carry inward currents in a manner very similar

to calcium, but either block or fail to activate the calcium-dependent

potassium current. This finding is important because no other agents

which effectively block this potassium current without blocking calcium

channels have yet been described. (The calcium activated potassium current

in Tritonia neurons appears quite resistant to the effects of external

TEm, even in high doses, Thompson, 1977.)

Fig 9 illustrates an experiment designed to assess the time-course

and voltage dependence of membrane calcium permeability during long

depolarizations of a voltage clamped Tritonia bursting pacemaker cell. The

cell was bathed in a medium in which all sodium was replaced by Tris and

TEm to eliminate inward sodium currents and block the TEA-senstive

fraction of outward potassium current. Fig. 9B shows a representative

record of the membrane current when the cell is depolarized in a medium

where calcium has been replaced by barium. Fig. 9m is the current during

an identical pulse in a solution where magnesium replaces calcium. Since

barium, but not magnesium (Hagiwara, 1973), substitutes for calcium in

carrying inward current, and since calcium-activated potassium current

should be absent in both media, the difference between the currents, as

represented graphically in Fig. 9C, should give an indication of the cal-

cium current that would normally occur during such a pulse. Fig. 9C

indicates that a calcium current of about 25 nm would activate very

rapidly upon depolarization to +22 mV. The current declines only to about

one half by the end of the 5 sec depolarizing pulse, but then turns off

very rapidly on return to the holding potential. The turning on and off

Fig. 9: Inward currents in a Na+-free, Cam-free solution contain-

ing Ba++

. m. Upper trace: voltage clamped membrane potential.

Lower trace: membrane current in Na+-free, Ca

++ -free solution

before adding Ba++

. B. Same as m, except 10 mM Bad (replacing

Mgt)) was added to the bathing solution. Note development of a net

inward current during pulse. C. The current records from m and

B are superimposed. The difference between these currents is

assumed to represent an inward Bad current. D. The dependence

of inward Bad current on membrane potential during pulse. Filled

circles represent peak inward current early in pulse. Open circles

represent inward current near the end of the 3 sec pulse.

•D

•

0 Current- Voltage Relationship

Barium-dependent +22 mV -

-- 3 sec —

-38 mV -

0 nA

-25 nA -

Pulse Potential, mV'10 20 30-20 -10

0-30

A

0 0

o Late0

0

Na, Ca free

50 mM Mg

100 mM TEm

10 mM Ba Traces

added Super-

Imposed ••

Voltage Clamp Currents • • • Early

• -10 nm

-20

•

38

of the inward current evident in Fig. 9B is too rapid to be accurately

resolved in intact Tritonia neurons (because of the long capacity transients

in this preparation, see methods), but an upper limit of 20 msec for 90%

activation and decay can be safely assumed. The decline of inward current

by about one-half evident in Fig. 9C is typical of our observations in

this voltage range.

If inward barium currents accurately represent normal inward calcium

currents, as Connor's data suggests (Connor, 1977), a major influx of

calcium must persist for the duration of the longest pulses used in

investigating IC activation. It is therefore possible that the progessive

activation of I with long activation pulses may simply represent an effect

of cumulative calcium influx. The decline of inward current during the

5 sec pulse shown in Fig. 9B may indicate a partial inactivation of

calcium current, but the inactivation is far from complete. The decline of

inward current is usually even less prominent at lower pulse potentials;

currents often decline by as little as 10-20% during a 5 sec pulse to

0 mV, as evident in Fig. 9D. Fig. 9D shows current-voltage relationships

derived by subtracting currents in magnesium from currents in barium, as

indicated by Fig. 9C.

The calcium current component characterized by findings like those

illustrated in Fig. 9 will be designated I D . We have noted that the slow

inward current IB also appears to be carried partly by calcium ions but

we shall tentatively regard I B and ID as distinct transport processes.

The two currents have completely distinct dependerres on time and mem-

brane potential: ID turns on and off far more rapidly than I B and IB

39

begins to activate at more negative potentials than ID . The maximum

apparent inward calcium fluxes associated with I D are 10 to 20 times

larger than largest calcium fluxes associated with the slower I B kinetics.

It is therefore to be expected that I B will be of little importance to

the activation of calcium-dependent potassium currents whenever depolariz-

ations are large enough to activate ID.

DISCUSSION OF SLOW CURRENT TAIL ANALYSIS

The slow inward current IB

A slowly activating component of inward current in bursting pacemaker

neurons was first reported, nearly simultaneously, by Lux and Eckert (1974)

and by Gola (1974). Though less detailed, the observations reported by Gola .

are very similar to the observations of IB described by Thompson (1976).

Gola did not identify the ionic species carrying the slow inward current,

but he did describe slow tails in the pacemaker potential range'which

closely resemble the I B tails evident after short pulses in Tritonia

neurons. Eckert and Lux (1976) have now described an inward current

(Iin slow) with some properties similar to those of IB' The characteriza-

tions of Iin slow and of IB differ in that Iin slow is regarded as being

highly selective for calcium, while IB appears to be carried by sodium

as well as calcium. The properties of Iin slow' as described by Eckert and

Lux (1976), are actually intermediate between the properties of IB , the

slow inward current, and ID, the fast calcium current, as described

in this thesis and by Thompson (1976). IB and ID are distinguished here

mainly on the basis of their differing activation and decay rates. Were

IB and ID to be regarded as a single mechanism, with very complex

activation kinetics, that mechanism would coincide closely to Eckert and

Lux's Iin slow. More work on the selectivity and activation properties

of the inward current will be necessary to decide which type of characteri-

zation is the more appropriate.

IB is almost surely the ionic current which accounts for the long

depolarizing afterpotentials (DAPs) observed after single action potentials

or bursts in bursting pacemaker neurons (Thompson and Smith, 1976). Like

41

IB' DAPs are observed only in bursting pacemaker neurons. The time-course

of IB and DAPs are similar, and I B and DmPs show a similar dependence on

both sodium and calcium in the external medium.

Since the slow inward current tails persist (with reduced amplitude)

in the absence of either sodium or calcium, but not in the absence of both,

we tentatively describe I B as a single conductance component, imperfectly

selective between sodium and calcium ions. Nevertheless, we have not ruled

out the possibility that the sodium and calcium components might reflect

completely separate transport mechanisms, perhaps even depending somewhat

differently on time and membrane potential.

The'calciuMactivated - potassium current, IC

The activation of long-lasting increases in potassium conductance by

depOlarizing pulses in molluscan neural somata was first noted by Connor

-and Stevens (1971c). Subsequent studies have linked such conductance

increases to post-tetanic hyperpolarization, spike frequency adaptation,

and bursting pacemaker activity (Brodwick and Junge, 1972; Partridge and

Stevens, 1976; Junge and Stephens, 1973; Gola, 1974; T. Smith, Barker and

Gainer, 1975).

Meech and Strumwasser (1970) were the first to show that long-lasting

increases in potassium conductance could be produced by the intracellular

injection of calcium ions. Meech (1974) later showed that calcium entry

during depolarization was probably responsible for the potassium conduct-

ance increases underlying post-tetanic hyperpolarization. In this thesis

and in a preceding paper (Thompson, 1977), it has been suggested that I C

in Tritonia neurons reflects a mechanism similar to that described by

42

Meech. I have also described a voltage-dependent component of calcium

entry, ID, with properties appropriate to account for the activation of

I during membrane depolarization. In the following section (Results II),

I discuss the relationships among ID , the intracellular accumulation of

calcium ions, and the activation of I c in more detail.

The procedure used to measure the time course of the potassium

permeability change underlying I c involves five assumptions, which are

liated in the text above. Only assumptions (1) and (2) are required for the

inference that current differences in different external potassium concen-

trations are directly proportional to potassium permeability. mssumptions

(3), (4), and (5) are necessary only to determine the exact magnitude of the

related permeability change.

mssumption (1) requires that only potassium fluxes be affected by

changes in external potassium concentrations. m deviation from the

required condition would occur, for instance, if external potassium ions

block an ionic channel selective for some other species, or if an exchange

mechanism couples potassium influx to the movements of some other ion. m

deviation from assumption (1) would be misleading in the present context

only if the blocked or coupled ion flux is electrogealc and varies with

a time and voltage dependency similar to that of the potassium permeability

change. The only mechanism known at present that can be considered as a

possible source of such an error is the ouabain-sensitive sodium-potassium

exchange pump. The best available evidence indicates that this transport

mechanism is, indeed,electrogenic in molluscan neurons (Gorman and Marmor,

1970; Thomas, 1972a), and that it has a slow time and voltage dependence

due to its dependence on the internal sodium concentration [Na]..

43

Experiments by Thomas (1972b) have suggested, however, that the pump

rate is affected by [m] o only at concentrations below the physiological

level (i.e., less than 1 mM in snail neurons where the normal [K] o is

4 mM), . Thomas has also shown that sodium loads are cleared with a time

constant on the order of 10 minutes, and that the electrogenic pump

current has a similar time course. These relaxations are much slower

than the decays of I c. For both of these reasons it seems unlikely that

a [K]o - dependent variation in sodium-potassium exchange distorts

the estimates of slow potassium permeability changes presented in this

thesis.

mssumption-(2) requires that potassium concentrations near the

membrane do not vary during slow tail decays, and (3) requires further

that external potassium near the membrane be equal to bath potassium.

The lower baseline level of extracellular potassium implies that external

concentration effects are the most likely to be significant. Eaton

(1972) has presented evidence that a considerable accumulation of

potassium occurs during large depolarizing pulses. Such an effect may

be significant during the first few hundred milliseconds after large

depolarizations, but probably does not distort the estimates of the

slower conductance tails considered above. Eaton's studies and similar

preliminary investigations of my own indicate that the external potassium

levels equilibrate in times on the order of a few hundred milliseconds.

This conclusion is also substantiated by direct measurements of external

potassium near the membrane (vieher and Lux, 1973), in which relaxations

after depolarization were observed to be essentially complete within 1 sec.

Furthermore, I have purposely chosen to record slow tails after activation

pulses of relatively low amplitude to minimize the efflux of potassium

44

during the depolarization. (Peak outward currents during the pulses em-

ployed for the experiment illustrated by Figs. 6 and 7 were only about one-

third of the outward currents during the pulses employed by Eaton, 1972,-

to demonstrate accumulation).

mssumption (4) requires that membrane potassium permeability remain

constant as external potassium is varied. On this point, one can merely

note that a major dependence of potassium permeability on external

potassium concentration has been described only in connection with potas-

sium currents: showing strong inward rectification (Hodgkin and Horowicz,

1959; Noble amiTsien, 1968,1969). The calcium-activated potassium

current, on the other hand, shows an outward rectification characteristic

(Meech and Standen, 1975; Smith, unpublished).

mssumption (5) requires that [if i remain at the same value during

the tail measurements in normal and high potassium media. Though some

increase of [le] i in the high potassium medium is to be expected,

experiments using ion-selective microelectrodes to measure intracellular

potassium activities indicate that the effect would be small for the

conditions of the present study. Russell and Brown (1972) found that

changes of [e] i in mplysia neurons produced by reducing [e] bath tozero occurred with a time constant on the order of hours, and sometimes

began only after a delay comparable to the duration of each run in the

present study. It is to be expected that the smaller fractional changes

in [e]bath imposed in my experiments would produce even more gradual

effects. Miura, Hoffman and Rosen (1977) found that no significant changes

in [K]i occurred in canine cardiac purkinje fibers exposed to changes

in [e]bath comparable to those employed here.

RESULTS II: m MATHEMmTICmL MODEL OF

THE MEMBRmNE CURRENT

Multiple component description of membrane current

This section describes a mathematical model intended to summarize

the electrical behavior of the Tritonia bursting pacemaker cell soma

membrane, as observed under voltage clamp conditions. Voltage clamp

experiments described by Thompson (1976, 1977), along with the results

described in the previous section (Results I), indicate that the'membrane

current in the-physiological voltage range can be regarded as the sum of

a capacity current and seven distinguishable components of ionic current.

Some properties of these ionic current components are summarized in Table

I. The designations II , IK , - Im and IL were introduced by Connor & Stevens's

(1971 a,b) analysis of the ionic current in neurons of the dorid nudibranchs

mrchidoris montereyensis and mnisidoris nobilis. Since these species are

closely related to Tritonia, and since similar ionic currents can be

identified in each species, we adopt the same terminology for the four

analgous components. New designations are introduced for three ionic

current components, IB , IC, and ID, which were not described in the studies

by Connor & Stevens.

Each ionic current component appears to reflect a time-dependent

membrane permeability to sodium, calcium or potassium ions (except that

the ions carrying I L have not been determined) as indicated in Table I.

In order to model the electrical behavior of the ionic current components,

each will be represented as a variable conductance in series with an ionic

battery, as in the work of Connor and Stevens (1971c). mt a membrane

Table I: Summary of the properties of seven components of ionic

current observed in bursting pacemaker neurons. The information

presented in this table was derived as indicated in the more

detailed descriptions of each individual current component in the

text following this section. The column headed "Gating Kinetics"

specifies appropriate descriptive formalisms, most often in terms

of first order, voltage-dependent gating variables like the m and

h in Hodgkin and Huxley's (1952) description of the squid axonal

sodium current. The specific coefficients governing the behavior

of these m- and h-like variables are described by eqns. 6-10 below.

The column headed "T(-40 my)" indicates exponential time constants

approximating changes in activation gating observed at a membrane

potential of -40 mV. The column headed g specifies conductance

scale factors, determined for a cell with an estimated membrane

capacitance of 19.8 nF (see following text).

46

Table I: Seven components of ionic current observed in bursting pacemakers

Current Component MajorPermeantIons

Vrev

(mV)

GatingKinetics

t(-40 mV)

(msec)

g

(umhos)

II Fast sodium Na+

+45 m3h 4 29.7note

IB Fast calcium Cam +130 m 7 0.259

IB Slow inward++

Na+ ,Cay +85 1800 0.026

note

IK Delayed outward -52

notem2

24 2.55

IA Transient outward K+

-63 m4h 18 11.7note

Slow outward K+

-73 Cai depnote

1 to 50seconds

6.7x10-5

note

IL Leakage +7 none none 0.039

notes:

I • subject to complete or nearly complete inactivation at potentialspositive to 0 mV.

IB: gm = gB (0 mV,40). See p. 60.

figure is V axis intercept of instantaneous I K(V), extrapolatedIK: Erevfrom the interval -40 mV < V < -20 mV. See p. 63.

IA' subject to complete or nearly complete inactivation at potentials•

IC : membrane potential dependence assumed to be mediated by changes inintracellular concentration of ionized calcium. The figure for grexpresses the sensitivity to intracellular calcium ions in umhos/riM.

positive to -30 mV.

47

potential V, any component current, I, will be given by

I = g (7-vrev)

(3)

where g is a variable with the dimensions of a conductance and Vrev

is a

reversal, or zero-current, potential. Variations in g represent the

gating characteristics of the particular ionic current component. The

gating process may depend on membrane potential, but it is assumed that

at least on the macroscopic level gating variations occur in a fashion

that is graded and continuous in time. This assumption and eqn. 3 imply

that each compOnent will have a linear instantaneous current-voltage

relationship. While this condition has not yet been verified in a

strict way for each component observed in Tritonia neurons, neither has

there been any evidence of pronounced rectification within the limited

potential range (-50 mV to -30 mV) over which the slow pacemaker

oscillation occurs. In the case of each current component active in this

pacemaker potential range, the quantitative models presented below have

been based mainly on tail, currents measured in this same potential

range. The linear conductance defined by eqn. 3 should therefore be

adequate to approximate the electrical behavior of:each ionic current

component during the slow pacemaker oscillation of pribery interest here.

The ionic transport mechanisms associated with each current component

will sometimes be referred to as the ionic conductances g I , gm , gm, g

gc , g , or gL .

Formalisms of two different types will be used to describe the gating

of the various ionic current components. For all the conductances

except g,,, gating is presumed to depend directly on membrane potential and

48

will be described by equations of the type introduced by Hodgkin & Huxley's

(1952) analysis of the axonal ionic currents. m general form for this

type of equation is

g = imx h (4),

where g is a constant with the dimensions of a conductance and m and h

are dimensionless variables governed by first order differential equations

whose coefficients depend on membrane potential. The m term is used to

represent the activation of conductance by depolarization, and may be

raised to the power x (greater than 1) where necessary to describe a

delay in the increase of conductance. The h term is used to represent

the inactivation of a conductance by depolarization, if such a process

must be described. Both m and h vary between 0 and 1, so the conductance

scale factor g corresponds to the maximum possible, or fully-activated,

conductance associated with a given current component (in one case these

terms have a different significance, see section on IB below). Since the

gating of the conductance g appears to depend on calcium entry, rather

than depending directly and exclusively on membrane potential, the

formalism of eqn. 4 is inappropriate for this component. m new formalism,

explicitly representing the involvement of calcium ions is developed in

a later section.

Measured conductance scale factors and reversal potentials for each

ionic current are specified in Table I, along with some information on

the gating kinetics governing each conductance. The ideal situation of

characterizing all current components completely in a single exemplar

cell has not been achieved. First, the experimental lifetimes of

49

individual neurons are too short for the large number of measurement

operations that would be required. Second, for the fast sodium current

II' Connor and Stevens's(1971) data has been used in preference to the less

reliable measurements available for Tritonia neurons. (The voltage

clamp technique they used is superior to that employed in the present

studies for the resolution of fast current changes, see Methods). In

order to minimize the errors introduced by pooling data on individual

currents from different cells, 1) g I was scaled in direct proportion to

membrane capacitance from Connor and Stevens's measurements_ and 2) gm and gK

were estimated from representative pulses in the same cell used for the

final determinations of kinetics and gc . Estimates of gL , gd, and gm ,

and all reversal potentials remain subject to errors due to individual

variations among bursting pacemaker neurons.

Capacity current

If the somatic membrane is assumed to behave as an ideal capacitor,

and to be spatially isopotential, the somatic capacity current, Icap' will

be given by

I = C dVcap s dt (5)

where Cs is the soma capacitance.

Thompson (1976) used a voltage clamp procedure to estimate Cs in

Tritonia bursting pacemakers, and found values ranging from 9 nF to 46 nF.

When referred to the apparent spherical surface area of the somata, these

values imply a specific membrane capacitance, C m, upwards of 5 pf/cm2

.

This figure greatly exceeds the accepted value for nerve membranes of

50

1 pf/cm2 . mnatomical studies (Mirolli andTalbott, 1972; Graubard, 1975) of

molluscan somata provide an obvious basis for this discrepancy: the sur-

face membrane appears to be highly infolded, forming numerous narrow

clefts ranging up to 15 micrometers in depth.

Electrical measurements of Cs in intact cells are subject to errors

due to poor clamping of the axonal cable. An estimate of Cs based on

anatomical findings is employed here to scale the capacity current.

Anatomical studies of molluscan nerve cells similar in size to the

Tritonia bursters have found that surface infolding results in a 7.5-

fold (4imlliandTalbott,1972) or a 6-fold (Graubard, 1975) increase in

membrane area over that expected from the overall dimensions of the

soma. The somata of the Tritonia bursting cells from which data in the

model have been selected appeared under the dissecting microscope to be

spheres approximately 300 micrometers in diameter. The surface area of

such a sphere is 0.00283 cm2. Assuming an infolding factor of 7, and a

true membrane capacitance of 1.0 pF/cm 2, a value of 19.8 nF has been

adopted for C s .

The fast sodium current, •.„

The bursting pacemaker neurons of Tritonia possess a sodium current

mechanism resembling that first described in the squid giant axon by

Hodgkin and Huxley (1952). Connor and Stevens (1971a,c) found that equations

retaining Hodgkin and Huxley's m3 h formalism could adequately describe

the inward currents in Dorid neurons, though in certain other details

these currents differed from the axonal sodium currents. The sodium

current in Tritonia neurons appears to be similar to the inward current

51

measured by Connor and Stevens (Thompson, 1976). Connor and Stevens's

(1971c) descriptive equations, in their exact form, have therefore been

adopted to represent the sodium current in Tritonia bursting pacemakers.

The sodium conductance g will be described by

gI = gI mI3(V't) hI (V , t)

(6a)

where gI is the fully-activated sodium conductance, and m I (V,t) is an

activation variable and hI(V,t) an inactivation variable analagous to

Hodgkin and Huxley's m and h, respectively. The value of gI in Table I

was obtained by scaling the value measured by Connor and Stevens (1971c) to

maintain the same conductance per unit membrane capacitance. The value

of EI in Table I is equal to that measured by Connor and Stevens (1971a).

The variables mi (V,t) and hI (V,t) are governed by the first order

differential equations:

dmI (V , t)T (V) MIdt

MI(V,t) = mi (V,03)

and

dhI'(V t)

ThI(V) dt+ hI (V ' = hI (V co)

where TmI (V), 1111,(V, 00), ThI (V), and h (V,01) are estimated from the

experimental data in figure 10. The data points were obtained from

voltage clamp measurements by Connor and Stevens (1971c); the smooth curves

in Fig. 10 were fit by eye and are given by

11.5TmI (V) = V + 16 + 0.5 cosec (6d)

1 + exp( 5.75 )

(6b)

(6c)

Fig. 10: The dependence of g i gating parameters on membrane poten-

tial. The symbols represent values estimated from voltage clamp

data by Connor and Stevens (1971a,c). The solid curves were fit by

eye (but see text, p. 84). A. mctivation time constant, TmI

(V).

Eqn. 6d specifies the solid curve. B. Steady-state activation,

m1 3 (V,00). Eqn. 6e specifies the solid curve. C. Inactivation

time constant,ThI(V) . Eqn. 6f specifies the solid curve.

D. Steady-state inactivation, h 1 (V,00). Eqn. 6g specifies the

solid curve.

ml3(V ' m))

53

1 30

V - 11 + exp( )

(6e)

V' = 7 ln(214 - 1) - 21.5 = -30.1 mV

= 2.7

85thI(V) = V + 24 + 4.5 msec (6f)

1 = exp( 3.75 )

1 6hI(V,°°

) 1 + exp(V + 19.5 )

}

4.5

(6g)

The parameter 0 / is incorporated to adjust the steepness of the exponential

foot of m3i (V,00), with only minor effects on the rest of the curve. The

estimation.of this parameter is subject to rather large errors: the data

points at the foot of the curve are unreliable, as they necessarily

represent very small and potentially contaminated inward currents.

One apparent discrepancy in adopting Connor and Stevens's description

of II must be discussed. Noting that some inward current persisted in the

absence of external sodium ions, provided calcium was present, Connor and

Stevens described II as a mixed sodium and calcium current. It has since

been reported that the activation of the sodium and calcium inward currents

can be temporally separated (Kostyuk, Krishtal andftdoplichko,1975; mdams

& Gage, 1976; Connor, 1977). These recent findings indicate that the

sodium current has rapidly activating kinetics like those described by

Connor and Stevens for I I' while the calcium current either does not

inactivate or does so only very slowly. Connor and Stevens's method

of measuring I I neglects any inward current which does not inactivate

rapidly. so their data should correctly represent the sodium component

54

of inward current.

The fast calcium current, ID

Observations discussed earlier in this thesis (see Results I) indi-

cate that Tritonia bursting pacemaker neurons-have a time- and voltage-

dependent inward calcium current. The largest and fastest component of

this calcium current is designated ID: its properties are assumed to

correspond to the prominent inward currents observed in barium-substitUted,

sodium-free bathing salines, as discussed previously. Inward calcium

currents similar to ID have been reported in neurons of several other

gastropod species (Kostyuk, et al., 1975; Eckert and Lux, 1976; mdams

and Gage, 1976; Connor, 1977).

The data points in Fig. 11 represent estimates of the peak calcium

conductance activated by various step depolarizations of a voltage

clamped Tritonia neuron. ms noted previously, the activation and decay

of ID

occur too rapidly to be resolved accurately in the intact cell prep-

aration used in obtaining the data shown in Fig. 11. Connor (1977, per-

sonal communication) has used an isolated cell voltage clamp technique,

however, and achieved a more precise temporal resolution of the calcium

currents in mrchidoris neurons. His preliminary observations indicate that

both the activations and decays are approximately exponential, with time

constants of 10 msec or less. Since the activation and decay of I D

appears to be exponential at all potentials, the gating kinetics can be

approximated by a single potential-dependent first-order variable:

= gD MD, t)

(7a)

Fig. 11: The dependence of steady-state activationon membrane

potential, mD (V,c0). The symbols represent the relative activation

of calcium conductance at various membrane potentials, estimated

from the data shown in Fig. 9. Conductance values were calculated

from the peak inward currents specified in that figure, assuming

ED = +130 mV. The solid curve (specified by eqn. 7C) was fit by

eye.

55

1.00

0.75

0. 50

0.25

0.00-30.-25. -20. -15. -10. -S. 0. S. 10. 15. 20. 25.

MILLIVOLTS

56

where gD is the fully-activated calcium conductance. The value for iD

listed in Table I was determined from the data shown in Fig. 9, assuming

a reversal potential, ED, of +130 my. This value for ED is approximately

the Nernst potential for calcium ions, if it is assumed that the intra-

cellular calcium concentration near the membrane approaches 5 x 10 -7

during pulses such as those imposed in Fig. 9 (see section on intra-

cellular calcium concentrations below). The activation variable is

governed by the differential equation

dri1D(V,t)TmD (V)

MD(V't) = MD (V 'm)dt(7b)

The function mD(V,07) has been determined from the experimental data

points shown in Fig. 11. The curve in that figure corresponds to

mD (V,0*) -1 11.7V + 3.5

)1 + exp(-7.25

(7c)

m fixed value of 7 msec will be adopted for the time constant T D . This

value is in the range suggested by Connor's data, and is just below the

upper limit of 10 to 20 msec. estimated from voltage clamp studies of

Tritonia neurons (see Results I, the dependence of calcium entry on time

and voltage).

The possibility that mayy be subject to some degree of potential-

dependent inactivation has already beet discussed (Results I, calcium

entry). There has also been a suggestion (Heyer and Lux, 1976) that

somatic calcium currents may be subject to a frequency-dependent

facilitation when activated repetitively. Neither inactivation nor

57

facilitation is represented in our description of I D since these phenomena

are probably of secondary quantitative importance in the present context,

and because there remains considerable ambiguity in the experimental

observations of these processes.

The slow inward current IB

m component of inward current characterized by very slow decays

after depolarizing pulses was originally observed in bursting pacemaker

neurons by Gola (1974) and by Eckert and Lux (1975). m voltage clamp

analysis of the analogous inward current in Tritonia neurons, desig-

nated IB' has been described by Thompson (1976). Of all the ionic

current components distinguished in voltage clamp experiments on

Tritonia neurons, IB is the one component which is observed exclusively

in identified bursting pacemaker neurons. Ion substitution experiments

have indicated that IB is carried by both sodium and calcium ions. The

fraction carried by calcium appears to be approximately one-third, and that

by sodium approximately two-thirds (Thompson, 1976).

IB is the smallest of the ionic currents described here. In a given

cell, the largest observed values of IB are less than one hundredth the

maximal values of the sodium current II and less than one tenth the

calcium current ID' IB is, nonetheless, readily distinguished from the

other inward currents by its slow kinetics of decay at negative hold

potentials: II and ID relax to steady values within a few milliseconds

following a depolarizing pulse, whereas IB persists for seconds as an

inward tail. Because of the simultaneous activation of other much

larger ionic currents, I B cannot be observed directly in the positive

potential range. The kinetics of activation at positive potentials

58

must be inferred from observations of the slow tails at negative holding

potentials following depolarizing pulses of various dimensions. The

lack of more direct information on the gating of I B at depolarized

potentials is probably not important for the present purposes: I B

constitutes a significant fraction of the total ionic current only at

the subthreshold potentials where the larger ionic conductances are

gated off.

The gating kinetics of gB , as inferred from tail current measurements,

appear to be approximately exponential in time at all potentials, and I B

does not appear to inactivate markedly with prolonged depolarization

(Thompson, 1976). The gating of gm can, therefore, be adequately described

by an equation with a single voltage-dependent, first order activation

variable, which need not be raised to a power:

gm =

B (V, t)

(8a)

where gm is the estimated steady-state conductance at 0 mV. The value

for "B listed in Table I was determined using a method described by Thompson

(1976), assuming a value of 85 mV for E B . The activation variable m B(V,t)

is governed by

dmB (V,t)T (V) mB + mB (V,t) = mB (V,00)

dt(8b)

The functions TB (V) and mB (V,00) have been estimated from the experimental

data in Fig. 12. The smooth curve and straight line in that figure were

fit by eye and are given by

Fig. 12: The dependence of g m gating parameters on membrane poten-

tial. The symbols represent values estimated from voltage clamp

data as described by Thompson (1976). The solid lines were fit by

eye. m. mctivation time constant, TmB (V). Eqn. 8c specifies the

solid curve. B. Steady-state activation, m B (V, 00). Eqn. 8d

specifies the straight line. (See text, p. 60.)

0. 10. 20.0.00 I1 I

-60.. -SO. -40. -30. -20. -10.MILLIVOLTS

B

e0. a CI

4000.

2000.

6

00. L I I

-60. -SO. -40. -30. -20. -10.MILLIVOLTS

0 .

A

60

3000TmB (V) = V + 40 + 300 msec

1 + exp( 8.5(8c)

mB (7,03) = 1 + V/60 (8d)

A straight line has been used to represent the function mB (V,00) because

the available data did not seem adequate to determine a curve of the

more familiar sigmoidal form. It is difficult to imagine a physical

interpretation for the linear relationship indicated. While it•is not

likely that this function could be strictly correct, it should be

adequate for the present descriptive purposes, over the physiological

voltage range from -50 mV to +30 mV.

The delayed potassium current, IK

The potassium current in Tritonia neurons has been resolved into three

kinetically and pharmacologically distinct components (Thompson, 1977).

mlthough none of these components have properties identical to the squid

axon potassium current, the component we designate I m is the most similar

to the axonal current. IK

activates along a sigmoidal time course follow-

ing step depolarizations, and it decays exponentially after depolarization.

Like the axonal current (EhrensteinandGilbert, 1966), I K undergoes some

inactivation during prolonged depolarization, but the process is slow and

incomplete. IK differs from the axonal potassium current in its response

to certain pharmacological agents, and the activation times of I m are

slower than those of the axonal current by approximately a factor of

ten (Thompson, 1977).

If we neglect the slow inactivation process, the activation of g K

under voltage clamp conditions can be approximated by an equation with a

Fig. 13: The dependence of g K gating parameters on membrane

potential. The symbols represent values estimated from voltage

clamp data, as described by Thompson (1977). The solid curves

were fit by eye (but see text, p. 84). m. mctivation time

constant, Tmm (V). Eqn. 9c specifies the solid curve. B. Steady-

state activation, mm2 (V,0*). Eqn. 9d specifies the solid curve.

60. 1.00

0.75

40.

6

❑

0.50

U❑ 0.25

U. I I I I—60. —40.. —20. 0. 20.

MILLIVOLTS

0.00-40. 20.0 .—20. —10.

MILLIVOLTS10.

❑

63

This descriptidn of gk is quantitatively quite similar to Connor and

Stevens's (1971a,c) description of a potassium current in Archidoris neurons.

It differs in that Connor and Stevens included a secondary activation term

describing a slower component of potassium activation. In Tritonia neurons,

these faster and slower components of potassium conductance can be

pharmacologically separated, so they are described here as two separate

components, andd gc (Thompson, 1977; see also Meech and Standen, 1975).

The value of -52 mV for EK specified in Table I is not the reversal

potential for Ik. II( actually reverses at a potential closer to -60 mV.

The value of -52 mV is used because it provides a much more accurate

description of Ik at the potentials where this component would be expected

to be active under physiological conditions (-40 to +30 mV). The

discrdpancy between this figure and the true reversal potential is due

to a marked outward rectification in the instantaneous current-voltage

relationship for this component (Smith and Thompson, unpublished). The

measured reversal potential of -60 mV is still considerably more positive

than the estimated potassium equilibrium potential of -73.2 mV (see

Results I, the slow outward current, ThisThis difference is presumed to

reflect- an imperfect selectivity of the transport mechanism for potassium

ions, an effect of extracellular potassium accumulation, or both.

The transient potassium current, IA

m current component in Tritonia bursting pacemakers has properties

almost identical to those of the transient potassium current I m described

in Dorid neurons by Connor and Stevens (1969, 1971b,c;see Thompson, 1977).

m is evident in voltage clamp experiments only when holding potentials

or prepulses negative to -40 mV are employed: This component is almost

64

completely inactivated in the range of normal resting potentials. On

depolarization from a sufficiently negative potential, the conductance

gm

activates rapidly, but then declines to zero during maintained depolari-

zations (Thompson, 1977). The sequence of activation and inactivation

observed is analogous to the behavior of the axonal sodium current, except,

of course, that Im is normally outward, rather than inward, in the

physiological potential range.

Connor and Stevens (1971b,c) found that a formalism similar to the m3h

description of sodium current (Hodgkin and Huxley; 1952) could adequately

describe the potential-dependent activation and inactivation of g m. The

rising phase of gm, however, appears to be best fit by the fourth power,

rather than the third power, of a first order activation variable. Like

the conductance studied by Connor and Stevens, gm in Tritonia neurons can

be described by the equation

..._gm = g

m mm4 (V'

hm(V '(10a

wherem is the fully-activated conductance. The value for gm listed in

Table I was determined assuming a reversal potential, E m , of -63 mV, accord-

ing to the method described by Thompson (1977). The activation variable

mm (V,t) and the inactivation variable hm (V,t) are governed by

dmA(V,t)Tmm (V) mm(V,t) = - mm (V,c0) (10b)

dt

dhm(V

't)

Thm (V) hA(V , t) = h

m'CV 00)

dt(10c)

(10e)

and

1 + exp(V 4.'473.9

)

1 10.5

66

The steady-state activation and inactivation functions, m m (V,00) and hm (V,00),

have been determined from the experimental data represented in Fig. 14.

The smooth curves in that figure are given by:

mm4(V ' 1 V + 59 41 + exp( -10.5)

(10d)

The activation and inactivation time constants governing g m appear to have

only a minor dependence on membrane potential (Thompson, 1977; see also

Connor and Stevens, 1971b). For modelling purposes, rmm will be assigned

a fixed value of 18 msec, and Thm will be assigned a fixed value of 260

msec. These values are near the middle of the range of values measured

by Thompson (1977).

The calcium-activated potassium current, I C

m calcium-dependent component of potassium conductance in molluscan

neurons was first described by Meech and coworkers (Meech and Strumwasser,

1970; Meech, 1974a,b,c; Meech and Standen:, 1975). They showed that part of

the potassium activation normally occurring during depolarization is

actually mediated by an influx of calcium through voltage-sensitive

calcium channels. The evidence for such a mechanism includes the

susceptibility of a kinetically distinct component of potassium current to

supression by a variety of different operations known to prevent calcium

entry. In addition, Meech (1974a) has shown that microinjection of calcium

into cells can mimic certain effects of depolarization on the membrane

67

potassium conductance.

m component of potassium current observed in Tritonia neurons has

properties similar in many respects to the currents described by Meech

and his associates. This component, designated I C, is described in detail

in previous sections of this thesis (see Results I), and in Thompson

(1977). The chief characteristic distinguishing I from other components

of potassium conductance is its dependence on calcium entry. I C accounts for

the slowest variations of potassium current in voltage clamped Tritonia

neurons: it activates progressively for several seconds during step

depolarizations and the decays on return to a negative holding potential

require tens of seconds.

Waveforms representing the slow, decay of the conductance g after

various depolarizing voltage clamp pulses are shown in Fig. 15. The data

points in this figure are chord conductances derived from estimates of slow

potassium permeability changes presented above (see Results I, the time-

course of slow potassium permeability changes). Increments in I over

steady-statevalues (Ay, were calculated from increments in permeability

(APC' see eqn. 2) using an appropriate form of the Golman-Hodgkin-Katz

current equation:

F v y •

+ o_y[e ] eVF /RTK

SIC = APC RT 1 - eVF/RT

As in eqn. 2, yK is an activity coefficient with the value 0.69 and the

concentrations are in moles/cm3. m value of 1.379 x 10

-4 moles/cm

3 for

the intracellular potassium acitivty (y[e] i) was determined from the

direct measurements by Kunze,WalkerandBrown(1971). m value of 10_5 moles/

cm3 (10mM) was used for [K1 so the calculated values for AI_ should

Fig. 15: Slow decays of the potassium conductance g c at -40 mV

after pulses to +2 mV for the durations indicated. The times

designated 0 sec correspond to the trailing edges of the specified

activation pulses. The data points are chord conductances derived

from the same data as the points shown in Fig. 7 (see text, p. 67,

for method). The solid curves were calculated from a theory

represented by equs. 7, 8, 12, 13, 16-18. These curves are pro-

portional (see eqn. 18, p. 77) to predictions of the free calcium

concentration near the inner membrane surface after pulses of the

same dimensions used in obtaining the corresponding experimental

data. Both points and curves represent transient conductance

changes only, i.e., the pulse dependent increases over the steady-

state conductance at the holding potential of -40 mV.

60. 3000 msec

4010 20 30

SECONDS

SO.

0.310 30

SECONDS

700 msec

40.

40

e.

100 msec200 msec

40 SO

SIMONDS SECCVDS0 10 30 40 SO 10 20

30

e.

68

69

approximate the increments in I c following the specified pulses in a normal

potassium bathing medium at the holding potential V = -40 mV. The con-

ductance values plotted in Fig. 15 were obtained by dividing the values

for AI by the net driving force on potassium ions at -40 mV (assuming

ER = -73.2 mV, see Results I, the slow outward current, I c). The chord

conductances so calculated are used below to estimate the conductance

scaling factor "gc necessary to quantitatively model I c .

m calcium-coupled model for the activation of I c . Meech and coworkers

(see Meech and Thomas, 1977) have described evidence suggesting that the

calcium-activated potassium conductance may be controlled directly by

the concentration of free calcium at the inner surface of the cell mem-

brane. This hypothesis provides the basis for the description of I

activation kinetics developed in this section. m model incorporating

experimental measurements of membrane calcium transport and cytoplasmic

calcium diffusion and binding is shown to predict kinetics of intracellular

accumulation with a close resemblance to the observed kinetics of I c .

Factors capable of influencing intracellular calcium levels may

conveniently be divided into three categories: (1) the influx across

the .cell surface membrane, (2) the efflux across the surface membrane,

and (3) the intracellular diffusion and binding of calcium. It will be

assumed that the voltage clamp data on inward calcium currents accurately

characterize the calcium influx. Experimental data on efflux, diffusion

and binding in gastropod somata, on the other hand, are scant, but a great

deal of information on these processes has recently become available from

studies on cephalopod giant axons. The model developed here therefore

makes use of data from the squid axon, where necessary, by scaling

70

intensive properties to a geometry representing the approximately spherical

gastropod neural soma.

(1) Calcium influx. The influx of calcium as a function of time and

membrane potential is assumed to occur as implied by eqns. 7 and 8, which

describe the calcium inward currents I D and IB , respectively. The molar

influx of calcium, mnf1ux is then given by:

(ID + 0.33 133)

minf lux z F(12)

where F is Faraday's constant and z = +2 for the divalent calcium ion. The

factor 0.33 represents the observation that only about one third of I B

appears to be carried by calcium ions (see Thompson, 1976).

(2) Calcium efflux. Since the electrochemical gradient for calcium ions

across nerve membranes is large and inwardly directed, the efflux respons-

ible for maintenance of the steady state must be an active process. The

active efflux of calcium has been best characterized by experiments on

internally dialysed squid axons (see Brinley, 1976). Such experiments

indicate that efflux is in approximately direct proportion to the con-

centration of calcium at the inner membrane surface, over most of the range

of possible physiological calcium concentrations (when other factors, such

as internal mTP and external Na are held constant at their normal levels,

see DiPolo, 1973, 1977; Brinley, Spangler andMullins, 1975). It is assumed

that active efflux across the soma membrane is similar to,the process

studied in axons. mt the low internal concentrations expected in neuronal

cytoplasm, the molar efflux of calcium, me fflux' is then given approximately

by

=mefflux

K e 1

whereKe isaconstantx.Aditheunitsofm/secar 1

t) is a variable

representing the free calcium concentration near the membrane (see below).

Some numerical values for Ke determined from published experimental

data on squid axons are indicated in Table II below.

(3) Diffusion and binding in cytoplasm. It will be assumed that calcium

ions entering across the surface membrane may diffuse into the cytoplasm.

It is known, however, that calcium ions in neuronal cytoplasm are subject

to binding and sequestration reactions and that those reactions occur on

a scale sufficient to effect a high degree of buffering to imposed calcium

loads and to restrict diffusion drastically compared to that which would

otherwise be expected in aqueous solution (Luxoro andYaiiez, 1968; Blaustein

amdHodgkia,1969, Baker and Crawford, 1972; see also Rose and Loewenstein,

1975). If it is assumed that free calcium ions diffuse as in dilute

solution, but that bound calcium is immobilized, the differential equation

governing diffusion is:

8[Ca++]2 , ++, 9[CaX] - D V icaat 9t (3)

where [Cam] denotes the concentration of free calcium in some small

volume element, and [CaX] the concentration of the bound form (or forms) of

calcium in that same volume element. D is ,a diffusion coefficient with

the value 6.4x10-6

cm2/sec (see BlausteinandHodgkin, 1969). To use eqn.

14, it is necessary to characterize the kinetics of CaX formation, and to

define specific geometrical and boundary conditions.

71

(13)

72

It is assumed that calcium binding in somatic cytopolasm can be

approximated by a single class of reactive sites, uniformly distributed

throughout the volume of the cell, and that these sites bind calcium in

the manner described by Baker and Schlaepfer's (1975) studies on isolated

squid axoplasm. Those authors described a component of calcium binding,

probably to axoplasmic proteins, which could be approximated by first

order Michaelis-Menten kinetics with a dissociation constant between 300

and 500 nM (the value 400 nM is adopted here) and a total concentration

of sites between 20 and 40 1114/4 axoplasm (30 414 is adopted here). Since

calcium binding'to other proteins with similar estimated affinities (e.g.,

troponin C) is known to be extremely rapid compared to the slow kinetics

of IC' it will be assumed that the binding reaction equilibrates

instanteneously. For the formation of CaX, we can now write:

[X][CaX] - (15)

1 + m /[Ca++

]

where [X] = 30 4M, the total concentration of binding sites, and m =

400 nM, the dissociation constant. It is probable that the uptake of

calcium by mitochondria (Meechandlhomas, 1977; Tiffert, BrinleyaniScarpa,

1977) or other organelles (Baur et al., 1977) may account for the majority ,

of calcium binding under conditions of large calcium loads or high local

concentrations (e.g., microinjection), but such effects will be neglected

here. (There are indications that mitochondrial binding may be less sig-

nificant at low, physiological calcium concentrations than a simple extra-

polation of high dose behavior to low calcium levels would imply, see

Carafoli and Crompton, 1976. m similar conclusion emerges from consideration

of the effects of mitochondrialinhibitorsin axoplasm reported by Baker and

73

Schlaepfer, 1975).

The somata of Tritonia bursting pacemaker neurons are approximately

spherical in overall shape and about 300 p in diameter. There are folds

several microns deep in the surface membrane which may restrict diffusion

under some conditions, but for purposes of calculating slow concentration

relaxations after impulse-like calcium influxes, it will be assumed that

soma geometry can be approximated as a smooth and symmetrical sphere. This

also neglects any effects of the relatively small axonal process'from the

soma. By assuming spherical symmetry, eqns. 14 and 15 can be combined and

rewritten in the form:

3Cai(r,t) D [Cel (r,t)+mx ] 2 a 2 aCa(r,t)fr

at X r2 9r- 3r (16)

where Cai (r,t) represents the concentration of free calcium ions at a

radius r within the cell. The value of r varies from 0 at the center of

the cell to a (150 pm) at the surface membrane.

Solving for the dependence of Cai (r,t) on time and voltage The main features

of the intracellular calcium model developed above are summarized in Table

II. The intracellular calcium distribution will depend in general on

time and membrane potential, due to the time- and voltage-dependence of the

calcium inward currents I D and IB . Depolarizing pulses will result in

pulse-like influxes of calcium ions. After activating pulses, accumulations

of calcium near the membrane will dissipate due to the diffusion of calcium

into the internal volume of cytoplasm, due to the intracellular binding of

calcium ions, and due to active extrusion at the cell surface membrane.

Table II: Summary of factors considered in predictions of intra-

cellular free calcium concentration transients. References and

numerical values of parameters taken from studies on squid axonal

cytoplasm or membrane are included, as the presently available data

on calcium metabolism in gastropod somata are very limited.

74

Table II: Outline of model for intracellular calcium ion metabolism

1. Calcium Entry

Observations: Voltage clamp of calcium inward currents in Tritonia neurons.

Representation: Eqns. 7 and 8; Table I. Time- and voltage-dependent gatingof two conductances, gm and gd .

2. Extrusion of Calcium from Cell

Observations: Tracer calcium efflux from internally dialysed squid axons(work cited immediately below).

Representation: Eqn. 13. Efflux approximated by a direct proportionalityto the instantaneous concentration of calcium ions near the innermembrane surface. Pump sensitivity constants derived from the datapoints at low calcium concentrations presented in several publishedstudies are listed below.

Ke (cm/sec) Reference

1 x 10-4

3 x 10-4

1 x 10 3

3.2 x 103

Blaustein and Russell (1975)

Brinley, Spangler and Mullins (1975)

Di?olo (1973)

Blaustein (1976)

3a. Calcium Binding in Cytoplasm

Observations: Equilibrium dialysis of extruded squid axoplasm with tracercalcium (Baker and Schlaepfer, 1975).

Representation: Eqn. 15. Michaelis-Menten binding to a single class ofsaturable sites (dissociation constant 0.4 I1M) distributed throughoutthe intracellular volume (total concentration 30 4M). Must be evalu-ated simultaneously with 3b below.

3b. Diffusion of Free Calcium in Cytoplasm

Observations: Binding as described in 3a above. Fick's law diffusion ofunbound calcium ions is assumed (see Hodgkin and Keynes, 1957, andBlaustein and Hodgkin, 1969).

Representation: Eqn. 16. Partial differe9ti4 equation for simultaneousdiffusion (diffusion constant 6 x 10 -° cm (sec) and bingli-tigNsitl-■ a_sphere (radius 0.015 cm) approximating the neuronal soma geometry.

Fig. 16: Prediction of intracellular free calcium concentrations

when membrane potential of a spherical cell is pulsed for one sec

(indicated by heavy bar below traces) to +4 mV from a holding

potential of -50 mV. Ke was assigned a value of 4 x 10

4 cm/sec,

which is within the range of values indicated in Table II. mll

other parameters are as indicated in connection with equs. 7, 8,

12, 13, 16 and 17, and were obtained either from voltage clamp

measurements or from the studies on squid axon indicated in Table

2000.Lti

1000.

0. e.

82a--1--.65a

8. 10.2. 4. 6.

5000.

FREE CALCIUM CONCENTRmTIONS

mT SPHERICAL RmDII INDICmTED

RmDIUS OF CELL. = a = 150 um

4000.

3000.

TIME (SECONDS)

76

77

cannot be followed to completion on the time scale used in Fig. 16.

Comparisons of Cai (a,t) with gc . The variable designated Ca i (a,t)

corresponds in the model to the free calcium concentration near the inner

membrane surface and therefore represents the physiological parameter

hypothesized to control or gate g c in some way. The solid curves in

Fig. 15 were drawn from the equation

g = Ca (a 0C C i '

(18)

where Cai (a,t) was calculated in each panel for conditions of time and

membrane potential identical to the corresponding experimental measurements

of g , and g is a constant with the value 0.067 nmho/nM calcium. The

solid curves in Fig. 15 thus represent the simple case where the conductance

g is assumed to be in direct proportion to the calculated calcium con-

centration at the inner membrane surface. This might be approximately the

case, for instance, if calcium ions bind in a one-to-one fashion with

potassium channels, and only a small fraction of the number of possible

calcium binding sites are occupied at the highest concentrations of calcium

considered.

A very close similarity between the measured time-course of g and the

calculated time-course of Ca i(a,t) is evident in Fig. 15. This agreement

is striking since all but one of the parameters used in calculating Ca i(a,t)

where fixed, in advance, from independent experimental data, as described

above. Ke was treated as a free parameter because the available experiment-

al literature indicates a wide range of values under slightly varying condi-

tions. The final value for Ke (4 x 10-4 cm/sec) is within the range of

78

experimentally determined values (see Table II).

It should be emphasized that eqn. 18 and the Ca i (a,t) model predict

not only the time course of individual g c relaxations, but also the progres-

sive activation of larger tails by longer pulses, over the entire 30-fold

range of pulse durations shown in Fig. 15. The constanti c was arbitrarily

chosen to bring about a vertical scaling between Ca i(a,t) and the measure-

ments of gc, but once chosen, the same value of "gc was used in each panel

of Fig. 15.

Fig. 17 shows that the model also predicts a good agreement between

Cai (a,t) and gC when the activating pulse amplitude, rather than duration,

is varied. The data shown in Fig. 17 was recorded from a different cell

than that shown in Fig. 15, so a different value of ; was used for vertical

scaling, but all parameters in the Ca i (a,t) calculation had values the same

as those used in calculating the curves shown in Fig. 15.

The activation and decay kinetics of aree obviously complex. The

fact that these kinetics are naturally predicted by the model developed

here definitely seems to support the hypothesis that g c is controlled by

intracellular calcium ions. Regardless of the ultimate fate of this

mechanistic hypothesis, however, the Ca i (a,t) model and eqn. 18 are

demonstrably adequate to reproduce the observed time- and voltage-dependence

of gc tails. Though the adequacy of the Ca i(a,t) model in predicting other

aspects of activation has not been tested here, the suitability of the

model for tails in the pacemaker potential range implies that it should be

useful in reconstructing the role of g c in the bursting pacemaker oscillation.

3.0 100

2.5

75

3.5

7/

25

0-30. -20. -10. 0. 10. 20.

0.5

1.0

0.0 -40.

50

79

Pulse Potential (mV)

80

The linear leakage current, IL

The voltage clamp analysis upon which the ionic current descriptions

above are based has relied heavily on the measurement of the current

relaxations following step changes in membrane potential. m linear leak-

age current, by definition, undergoes no such relaxation and is therefore

not subject to analysis by the methods employed for the other current

components. The direct experimental measurement of such a linear current

is in fact very difficult in bursting pacemakers, due to the fact that

larger non-linear currents appear to be active in every voltage range.

(A non-linear, inward-going rectification, not described above, becomes

prominent when the membrane is hyperpolarized below the physiological

potential range, see Marmor, 1971). These difficulties notwithstanding,

the effects of a linear leakage component must be considered. It is

known, for instance, that molluscan neural somata have a permeability

to chloride (Barker and Levitan, 1975) which is not represented by any

of the conductance components described above.

Parameters appropriate to represent the linear leakage current in

Tritonia bursting pacemakers have been deduced from voltage clamp data

by a process of elimination. It has been assumed that any membrane con-

ductance evident in the immediate subthreshold potential range -45 mV to

-35 mV, not already accounted for by the variable conductances described

above, is linear leakage described by

IL = gL (V -VL ) (19)

where gL and VL are constants. To estimate the values of gL and VL , the

equations representing the other six ionic currents were solved for

81

the steady state at subthreshold potentials and subtracted from observation

of the steady holding currents measured at these same potentials in an

actual cell. m value of 0.039 pmho was estimated for gL and +7 mV for Vt .

The value of 0.039 imho for gL is close to the value of 0.049 pmho

estimated by Connor and Stevens (1971c). The positive value for VL suggests

that a steady inward sodium or calcium current may contribute to IL.

RESULTS III: THE RECONSTRUCTION OF BURSTING

PACEMmKER ACTIVITY

Conditions for the reconstruction of membrane potential

For the purposes of reconstructing membrane potential it is assumed

that the soma membrane is isopotential and that the effects of cable

currents from the axon region are negligible. The total membrane

current, IM, can now be expressed as the sum of all the membrane current

components described above:

dVIm = Cs + II + IK + Im + IB + Ic + IO + IL (20)

The normal unclamped condition of such a cell corresponds to I m=0. The

reconstruction of membrane potential can thus be accomplished by finding

solutions for voltage as a function of time which satisfy this condition

in eqn. 20 and the subsidiary eqns. 3, 5-1Q, 12, 13, and 16-19.

Fig. 18m shows such a solution which was found using numerical

approximation techniques, implemented on a digital computer. mn intra-

cellular recording of the electrical activity from a typical Tritonia

bursting pacemaker neuron is included for comparison in Fig. 18B. The

computation represented in Fig. 18m was initialized by setting all

variables to values corresponding to the steady state at a clamped holding

potential of -38 mV. The oscillatory pattern evident in the figures was

attained within 150 sec after the simulated release from voltage clamp.

The oscillation appears to persist indefinitely in the form represented

here.

1 1 1 1 1 1 1 10

ta

Membrane Potential (mV)

. . . . . . . . . . .

Membrane Potential (mV)

I I I I I I I 1 I I

H

cm8

roC)ca

ID

r.0

CV

84

The many parameters necessary for the calculation of membrane potential

were as indicated in Tables I dild II, and in connection with the individual

membrane current equations. Values for all parameters were estimated

initially, in advance of any reconstruction calculations, strictly from the

analysis of voltage clamp data. The earliest reconstruction attempts

showed that burst-like solutions were a very characteristic feature of the

model developed here: bursting persists over a wide range of particular

parameter values. It was also discovered, however, that the form of

the predicted burst was strongly dependent on the exact values of

certain parameters which could not be estimated with commensurate accuracy

from voltage clamp data. In order to show that the model could reasonably

reproduce a burst topography close to that observed most typically in

actual cells, minor adjustments to four parameters were made after the

fact of preliminary reconstructions. The principal parameters adjusted

were (I)I and q)K' which describe the steepness at the exponential feet of

the steady-state activation curves governing g I and gk. Compensatory

adjustments to gL and EL , the leakage conductance and reversal potential,

were then required (see section on IL above). mll four parameters remained

within the range of experimental error after adjusting them to achieve the

solution shown in Fig. 18m.

Features of the reconstructed potential waveform

There are striking similarities between the reconstructed potential

waveform and the accompanying recording of actual bursting pacemaker

activity. The firing during the active phase of the burst cycle begins

in both cases with a slight acceleration, the second interspike interval

being somewhat shorter than the first, followed by a gradual deceleration

85

lasting the rest of the burst. The forms of the interburst voltage

trajectories are also quite similar. There is a small depolarized after-

potential lasting a few seconds after the final action potential (see

Thompson and Smith, 1977), then a gradual hyperpolarization, followed by

a long slow depolarization leading up to the next burst. These features

of the interburst interval vary quantitatively from cell to cell, but

their presence in absolutely characteristic of bursting pacemaker activity

in Tritonia neurons. These features are always reproduced by the model

under any conditions which allow slow oscillations.

One important discrepancy between the calculated and naturally

occurring burst waveforns is that the model does not reproduce the

observed tendency for action potential undershoots to reach less negative

values after each successive spike in a burst. This tendency, evident

in the intracellular recording shown in Fig. 18B, is a quite typical

feature observed in the bursting pacemaker neurons of many gastropod

species. mnother very likely related discrepancy, not evident in Fig. 18,

is the failure of the model to reproduce the broadening of successive

action potentials commonly observed in actual bursting cells (Stinnakre

and Tauc, 1973; Eckert and Lux, 1977). These shortcomings may be

explained by the fact that the formal description of I K used in the

model disregards a slow inactivation process evident in voltage clamp

experiments (see Connor and Stevens, 1971a).

Other evident differences between the calculated and recorded bursts

in Fig. 18 are of less concern, since the differing dimensions are highly

variable from one actual cell to another, rather than constant features

of bursting pacemaker activity. Thus, though the waveforms in Fig. 18

86

differ in the durations of the firing bursts, the frequencies of action

potential firing during bursts, the forms of the action potential under-

shoots, and the depths of the interburst hyperpolarizing waves, the

dimensions of the computed oscillation are still well within the range

observed in actual cells (see Fig. 1). On the whole, the similarity

between the two waveforms shown in Fig. 18m and 18B suggests that the

features of the membrane current essential to bursting pacemaker activity

must have been described by the model developed above.

The time-course of ionic currents and intracellular calcium during the

reconstructed burst cycle.

The time-courses of some of the major variables computed in recon-

structing bursting pacemaker activity are shown in Fig. 19. The top

panel shows the same membrane potential waveform as in Fig. 18m, but on

an expanded vertical scale. The middle panel shows the variations of

individual ionic current components, displayed to emphasize variations

on the time scale of the whole burst cycle. The bottom panel in Fig.

19 represents intracellular calcium concentrations calculated during

the burst cycle. One trace is labelled to indicate that it represents

the calcium concentration adjacent to the inner membrane surface, i.e.,

Cai (a,t). It is this concentration to which the conductance g is

assumed to be in direct proportion, as expressed by eqn. 18. The other

trace in the bottom panel represents the mean calcium concentration over

theentireintracellularvolume.ThelargedifferencesbetweenCa.(a,t)

and the mean concentration are indicative of large radial gradients of

calcium concentration within the cell. Because of diffusional delays,

which are greatly augmented by the binding of calcium to fixed sites, the

FIg. 19: Time courses of selected variables calculated in recon-

structing the bur .gt cycle. Time axes in each panel are identical

and simultaneous. Top panel shows same membrane potential waveform

as Fig. 18m. Expanded vertical scale truncates spikes but empha-

sizes subthreshold potential variations. Middle panel shows

variations of individual ionic current components, identified by

capital letters near arrows. For clarity, all traces except I B and

I have been suppressed during the firing burst (but see Fig. 20).

ID is not shown in this figure since it is within the trace width

of zero at all times, except during action potentials. Bottom

panel shows variations of intracellular calcium concentrations

near the inner membrane surface and as a volume average throughout

the cell, as indicated.

87

IONIC CURRENT (NA)3. r---.

C

/B

2.

1.

0 .

-2. -

jK

TIME (sec)

- 34.

-36.

- 38.

-40.

-42.

-44.

MEMBRANE POTENTIAL (MV)

1

Fig. 20: Time courses of ionic currents during reconstructed action

potential firing. Time axes in each panel are identical and simultaneous.

Values shown are from the same computing run as those in Fig. 19, but

with the time scale expanded as indicated on the abscissae. The top

panel shows the reconstructed membrane potential waveform; the lower

two panels show membrane currents plotted with two different vertical

scale factors to accommodate the differing magnitudes of the seven

components. The behavior of ionic currents is essentially the same

during each action potential of the burst, except for the changes in

IB and I evident in Fig. 19.

6.

4.

2

0.

-2.

7.1 6.17.87.0 7.2 7.9 8.07.3 7.4I 7.77.5 7.6

TIME (sEc)

89

MEMBRANE POTENTIAL (MV)40.

30.

0.

10.0.

-ie.

-20.

-30.

-40.

-50. 1

DISCUSSION OF MODEL RESULTS

The mechanism of bursting pacemaker-like activity in the model

The model developed in this thesis has been shown to reproduce the

main features of bursting pacemaker activity. An attempt will be made

here to succinctly summarize the mechanism of the slow pacemaker

oscillation, as it is manifested by the model. The explanation offered

below is based on the assumption that the slow oscillation can be under-

stood in terms of two sets of factors: (1) a "background" of membrane

excitability, including the ability to fire action potentials, an

action potential firing threshold, and the stationary sub-threshold

current-voltage relationship, and (2) conductance changes with slow

time-dependence, which act as rate-limiting steps governing slow variations

in membrane potential. Though such a dichotomy cannot be made in an

entirely rigorous fashion, it is nevertheless proposed here for its

possible heuristic value.

(1) "Background" excitability. ms noted above, the model appears

to fire action potentials by a mechanism much like that described by

Connor and Stevens's (1971c) model of repetitive firing in non-bursting

neurons. It is only in long-term, subthreshold behaviot that the

distinctive behavior of the present model becomes evident. Several

authors have proposed that bursting pacemaker neurons are distinguished

by the form of the subthreshold stationary, or steady-state, current-

voltage relationship (see Introduction, previous studies of the mechanism

of the burst oscillation). While there is reason to doubt that this

current-voltage relationship alone is sufficient to differentiate bursting

and non-bursting neurons, the form of this current-voltage relationship

91

is. indeed important to the behavior of the model developed here. Spontan-

eous bursting is observed only when the parameters g/ and VL are adjusted

to values such that the calculated steady-state values of the total ionic

current are between zero and a fraction of a nanoamp negative over the

entire sub-threshold, pacemaker voltage range. In other words, the steady-

state current-voltage relationship must be nearly flat and just below the

zero-current axis. The effects of I L parameters have been mentioned here

because they provide a simple means of manipulating the steady-state current-

voltage relationship in modelling experiments. mctually, all the ionic

current components but I D contribute to the total subthreshold current.

The currents IK, Ic , and IL add positive conductance in the steady-state,

while the currents II' IB' and Im

contribute negative conductance in the

steady-state (negative because of the voltage-dependence of activation

for the inward currents and because of inactivation for I A). m condition

for bursting in the model appears to be that the positive and negative

conductance contributions are nearly equal in the true steady state.

(2) Rate-limiting conductance changes. The relaxation times of the slow

inward current, IB'

and of the calcium-activated potassium current, IC' are

longer by approximately an order of magnitude (or more), than the relaxation

times of all other current components, when observed under voltage clamped

conditions. It will therefore be assumed that the relaxations of these

two currents are rate-limiting for the slow burst oscillation. The follow- ,

ing simplified account of burst formation kinetics is proposed on the

basis of this assumption.

The time-courses of IB and I during the complete calculated burst

cycle are shown in Fig. 19. Just before the first action potential in a

92

burst, both IB and I are relatively small. The positive membrane

potentials associated with the first spike produce an increment in the

inward IB . The calcium influx associated with that spike leads to an

accumulation of calcium inside the membrane and a consequent increment

in the outward I c . The increment in IB is larger: that current

activates more rapidly than Ic. The net effect of the first spike is

thus a lasting inward current. The membrane quickly fires another

action potential, and even more quickly a third as successive increments

of IB are added. This tendency is soon checked as g m reaches a saturation

value, and reversed as I c activates progressively with successive action

potentials and the accumulation of more and more calcium inside the cell.

Though IB

activates more rapidly, I can attain considerably larger values.

mction potentials occur at a decreasing rate until I c becomes large enough

to prevent the membrane potential from reaching spike threshold. In the

absence of spiking and the associated large influx of calcium, both g m

and g now begin to decay, but gm

decays the more rapidly. The small

depolarizing afterpotential following the final action potential reflects

the residual activation of IB. ms IB decays, the membrane reaches the

trough of the interburst hyperpolarizing wave. The decay of g c proceeds

as calcium which had accumulated near the inner membrane surface diffuses

away and is extruded. mfter sufficient decay of gc, the membrane begins

to depolarize. Depolarization results in a gradual reactivation of I B ,

and the small calcium influx associated with IB

leads to a gradual

reactivation of Ic. The slow interburst depolarization appears to reflect

a delicate balance between a regenerative effect of I B activation and a

degenerative effect of I activation. Eventually, the membrane potential

approaches spike threshold, depolarization becomes rapid, and the burst

93

cycle repeats.

This interpretation of bursting as an interplay of the two currents

IB and I is supported by the comparative experimental observations

discussed previously (see Results I and Thompsom and Smith, 1976). Over

a wide range of cell types, species and experimental conditions, it

appears that bursting is observed only in cells where processes like I B

and IC can both be observed. I

B has not been observed in cells that are

not bursting pacemakers, though I often is. Since IB

appears to be a

unique and distinguishing feature of bursting pacemaker neurons, it is

reasonable to suppose that it may play a pivotal role in burst production.

Though IB may be the unique attribute of bursting pacemaker neurons,

it is clear that the calcium-activated potassium current I c is also

necessary for the reconstruction of bursting pacemaker activity from

voltage clamp data like ours. Meech and Standen (1975) were the first to

suggest that the coupling of calcium and potassium activation may be

fundamental to bursting pacemaker oscillation. Eckert and Lux (1976)

studied the properties of calcium current activation in bursting pace-

makers and suggested the possibility that a slow accumulation and dissi-

pation of intracellular calcium is the rate-limiting process in the

slow oscillation. The calcium-dependent model for gc activation developed

in this thesis shows that these suggestions, in conjunction with estimates

based on known properties of neuronal calcium metabolism, can lead to a

simple and self-consistent explanation of bursting, provided that I B is

also taken into consideration. More direct studies of calcium metabolism

in bursting pacemaker neurons will be necessary to directly test the

calcium hypothesis for slow potassium gating. One recent report is

94

encouraging. In a study of the mplysia bursting pacemaker cell R15,

Thomas and Gorman (1977) used the calcium indicator dye mrsenazo III

to record a spatial average of the intracellular calcium concentration.

They observed an oscillation closely resembling that predicted by the

model developed here.

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VITA

mUTHOR: Stephen J. Smith

DATE OF BIRTH: March 22, 1947

PLmCE OF BIRTH: Santa Monica, California

MOTHER: Lois Marie Gairich

FATHER: Donald J. Smith

SECONDmRY EDUCATION: Mercer Island High SchoolMercer Island, Washington

DEGREES: Reed CollegePortland, OregonB.m. Psychology, 1970

University of WashingtonSeattle, WashingtonPh.D. Physiology-Psychology, 1978