A Model of Graded Synaptic Transmission for Use in Dynamic … · 2014-02-10 · brane currents and synaptic transmission that promote oscillation in these networks are subtle, multifarious,

JOURNALOFNEUROPHYSIOLOGY Vol. 69, No. 4, April 1993. Printed in U.S.A.

A Model of Graded Synaptic Transmission for Use in Dynamic Network Simulations

ERIK DE SCHUTTER, JAMES D. ANGSTADT, AND RONALD L. CALABRESE Department of Biology, Emory University, Atlanta, Georgia, 30322; and The Born Bunge Foundation, University of Antwerp, B2610 Antwerp, Belgium

SUMMARY AND CONCLUSIONS

1. The heartbeat central pattern-generating network of the medicinal leech contains elemental neural oscillators, comprising reciprocally inhibitory pairs of segmental heart interneurons, that use graded as well as spike-mediated synaptic transmission. We are in the process of developing a general computer model of this pattern generator. Our modeling goal is to explore the interaction of membrane currents and synaptic transmission that promote oscillation in heart interneurons. As a first step toward this goal, we have developed a computer model of graded synaptic transmission between reciprocally inhibitory heart interneurons. Previ- ously gathered voltage-clamp data of presynaptic Ca2+ currents and simultaneous postsynaptic currents and potentials (5 mM external [ Ca”‘] ) were used as the bases of the model. .

2. We assumed that presynaptic Ca2+ current was composed of distinct fast (I& and slow (I& components because there are two distinct time courses of inactivation for this current. We fitted standard Hodgkin-Huxley equations (Eq. 1 and 2, APPENDIX) to these components using first-order activation and inactivation kinetics.

3. Graded synaptic transfer in the model is based on calculation of a dimensionless variable [PI. A portion of both JCaF and Itis determined by a factor A contributes to [PI, and a removal factor B decreases [P] (Eq. 4, APPENDIX). [P] can be roughly equated to the [ Ca2+] in an unspecified volume that is effective in causing transmitter release. Transmitter release, and thus postsynaptic conductance, is related to [P] 3 (Eq. 3, APPENDIX).

4. We adapted our model to voltage-clamp data gathered at physiological external [ Ca2+] (2.0 mM) and tested it for shorter presynaptic voltage steps. Presynaptic Ca2+ currents and synaptic transfer were well simulated under all conditions.

5. The graded synaptic transfer model could be used in a network simulation to reproduce the oscillatory activity of a reciprocally inhibitory pair of heart interneurons. Because synaptic transmission in the model is an explicit function of presynaptic Ca2+ current, the model should prove useful to explore the interaction between membrane currents and synaptic transmission that promote and modulate oscillation in reciprocally inhibitory heart interneurons.

INTRODUCTION

Reciprocal inhibitory interactions between neurons are thought to be critical in promoting oscillation in several well-characterized pattern generating networks that control rhythmic behaviors in invertebrates (Benjamin and Elliot 1989; Friesen 1989; Getting 1988, 1989; Satterlie 1989; Sel- verston 1989; Selverston and Moulins 1987) and verte-

brates (Grillner et al. 1989; Roberts et al. 1983). Graded release of transmitter can play an important role in such networks ( DiCaprio 1989; Granzow et al. 1985; Graubard et al. 1983; Miller and Selverston 1985; Raper 1979; Selver- ston 1989; Spencer 1988). The interactions between membrane currents and synaptic transmission that promote oscillation in these networks are subtle, multifarious, and subject to neuromodulation (Harris-War-rick and Marder 199 1); realistic computer models will be necessary in understanding these interactions (Mulloney and Perkel 1988).

Wang and Rinzel ( 1992) have provided a general model for understanding how reciprocally inhibitory neurons oscillate. Their model neurons are minimal, containing a synaptic conductance that is a sigmoidal function of presynaptic membrane potential with a set threshold and in- stantaneous kinetics, a constant leak conductance, and a voltage-gated postinhibitory rebound conductance. They recognize two fundamentally different modes of oscillation, “release” and “escape,” but the applicability of their find- ings to real oscillatory networks is limited, in part by the lumping of independent voltage-gated conductances into one postinhibitory rebound conductance and in part by a simplistic synaptic transfer function. Ideally, a simulation of an oscillatory neural network should be able to accommodate the changes in synaptic transmission that are brought about by changes in membrane potential or by competing currents and chemical modulators that influence Ca2’ entry.

Heartbeat in the leech, Hirudo medicinal& is governed by an oscillatory network of eight heart interneurons (Cala- brese et al. 1989) (Fig. 1 A). Pairs of heart (HN) interneurons in the third and fourth segmental ganglia of the ventral nerve cord form reciprocal inhibitory synapses across the ganglionic midline. Each of these heart interneuron pairs can function autonomously as a neural oscillator, with a period of - 10 s, when their respective segmental ganglion is isolated from the animal (Fig. 1 B) . Normally these independent oscillators are coordinated in their activity owing to inhibitory connections with heart interneurons of the first and second segmental ganglia. The intrinsic membrane properties that promote oscillation and their underlying voltage-gated membrane currents have been well characterized in the interneurons that compose the elemental oscillators (Angstadt and Calabrese 1989, 199 1; Arbas and Cala- brese 1987a,b; Simon et al. 1992). Graded release of trans-

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1226 E. DE SCHUTTER, J. D. ANGSTADT, AND R. L. CALABRESE

B

FIG. 1. Synaptic connectivity in the heartbeat central oscillator. A: schematic showing the synaptic connections of the heart (HN) interneurons in the heartbeat oscillator. Hollow circles: HN cell bodies numbered by their ganglion of origin (G3 or G4). Smaller filled circles: inhibitory synaptic connections. The heart interneurons of ganglia 1 and 2, which have identical connections and electrical properties, are lumped together. Their primary input region and spike initiation sites are located in G4 ( q ) and have been drawn differently from their somata. The contralateral reciprocal inhibitory interactions of the HN ( 3 ) and HN (4) pairs sustain oscillation in the system, whereas the ipsilateral synaptic interactions between the HN ( 1) and HN ( 2 ) neurons and the HN ( 3 ) and HN ( 4) neurons coor- dinate activity in the network. B: simultaneous intracellular recordings from 2 reciprocally inhibitory HN( 4) neurons (R and L) of the heartbeat oscillator in an isolated ganglion. Spontaneous activity of these reciprocally inhibitory neurons consists of alternate bursting in normal physiological saline. Dashed line: membrane potential of -50 mV. After Angstadt and Calabrese ( 199 1) .

mitter is an important, perhaps dominant, component of the reciprocal inhibitory interaction of the interneurons composing an elemental oscillator ( Angstadt and Calabrese 199 1; Arbas and Calabrese 1987a,b). Sensory and other neuromodulatory inputs regulate the period and pattern of the oscillatory network (Arbas and Calabrese 1984, 1990), in some cases by influencing the membrane and synaptic properties of the heart interneurons of the elemental oscillators directly (Simon et al. 1992).

Thus we are in a strong position to proceed in modeling one of these elemental oscillators, which should contribute to our understandings of the mechanisms that promote and modulate oscillation in reciprocally inhibitory neurons. As a first step toward such a model, here we develop an effi- cient ad hoc model of graded synaptic transfer based on a Hodgkin-Huxley type of model of presynaptic calcium currents. Because synaptic transmission in the model is an explicit function of presynaptic Ca2+ current, the model accommodates changes in synaptic transmission brought about by both direct and indirect influences on Ca2+. This model is then incorporated into a simple network simulation to demonstrate that it is compatible with dynamic net-

work simulations and sufficient to sustain oscillation. Pre- liminary reports have appeared in abstract form (De Schut- ter and Calabrese 1990; De Schutter et al. 1989).

METHODS

All simulations were done with the Nodus software (De Schut- ter 1989) on a Macintosh II computer. Simulations were integrated by a forward Euler method with variable time step. Neu- rons were simulated by single compartments with active membrane. Experimental data illustrated are from Angstadt and Calabrese ( 199 1) and were acquired using switching single-elec- trode voltage-clamp and discontinuous current-clamp techniques. The experimentally measured cell capacitance was used. All ionic currents are described by standard Hodgkin-Huxley (H-H) equations (Hodgkin and Huxley 1952).

The synapses between the heart interneurons of the third and fourth segmental ganglia have been extensively studied both physi- ologically and morphologically. Both spike-mediated and graded synaptic transmission occurs between these neurons (Arbas and Calabrese 1987a,b; Nicholls and Wallace 1978b; Thompson and Stent 1976), and the postsynaptic potentials appear to be mediated by Cl- (Calabrese 1979; Nicholls and Wallace 1978a,b) and have a reversal potential near -60 mV (Angstadt and Cala- brese 199 1). The synaptic contacts between reciprocally inhibitory heart interneurons have been shown at the ultrastructural level to occur at the ends of fine neurites where they intermingle near the ganglionic midline (Tolbert and Calabrese 1985). Al- though the morphology would seem to indicate otherwise, it is possible both to record unitary synaptic events from the heart interneuron soma that can be attributed to single quanta released presynaptically and to evoke transmitter release with subthreshold currents injected into the heart interneuron soma (Nicholls and Wallace 1978a,b). These observations indicate that soma and neuritic regions of the heart interneurons are electrically compact and motivated voltage-clamp analysis of graded synaptic transmission in heart interneurons (Angstadt and Calabrese 199 1). This work relates presynaptic Ca2+ currents to postsynaptic current and potential over a range of presynaptic potential ( -60 to -35 mV), that encompasses the slow wave of the heart interneuron’s normal oscillation (-60 to -45 mV) (Fig. 1 B). We con- structed our model to simulate the different voltage-clamp experiments performed in that analysis. Possible errors due to incom- plete space-clamping of the cells (which would result in slower apparent time constants) were ignored in the model.

A major assumption behind the synaptic transfer model is that the Ca2+ currents, measured presynaptically, correspond to the Ca2+ currents at the transmitter release site and so can be modeled by the same equations. This assumption is supported by the close correspondence between the amplitude and waveform of presynaptic Ca 2+ currents and the amplitude and waveform of the postsynaptic current in our experimental results (Angstadt and Cala- brese 199 1). Moreover, we assume that these same Ca2+ currents underlie both graded and spike-mediated transmission. Because no direct experimental data are available about the size of the change in calcium concentrations brought about by Ca2+ influx during the simulated voltage clamps or about the spatial distribution of these changes, we do not simulate calcium concentration explicitly in our synaptic transfer model. Instead, we simulate an empirically defined “effective” concentration of Ca2+ ( [P] ) in an undetermined volume. A portion of the calcium influx in the heart interneuron, determined by a voltage-dependent factor, A, adds linearly to [PI, whereas a voltage- and concentration-dependent removal process, B, decreases [PI, the equilibrium concentration of [P] being zero (Eq. 3). Transmitter release is determined by the third power of [ P], and postsynaptic conductance is

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a linear function of transmitter release. Synaptic delays are not modeled explicitly .

The graded synaptic transfer model contains the calcium currents, with a reversal potential of 135.2 mV corresponding to the Nernst potential for 5 mM Ca2+ outside and 0.1 mM inside. The change in Nernst potential due to calcium accumulation was neglected. Fast sodium currents ( JNa) that underlie action potentials and the hyperpolarization-activated inward current (1iJ ( Angstadt and Calabrese 1989) were presumed to be completely blocked in voltage-clamp experiments and simulations. Outward currents, delayed rectifier current (1k) and the A-current (I*), were probably only partially blocked in voltage-clamp experiments (Angstadt and Calabrese 199 1)) but were assumed to be fully blocked in voltage-clamp simulations. In voltage-clamp simulations, no leak current was used, and the results are compared with leak-subtracted experimental data.

The size of moo and h, and their time constants (7) for the slow Ca2+ current (&) were estimated from the voltage-clamp data. Rate equations were fitted to the estimates manually, using the Excel (Microsoft) program. Simulations of voltage-clamp experiments with the Icas equations were run and the simulated currents subtracted from the real voltage-clamp data to isolate the fast Ca2+ current ( JCaF). The size of ma, h,, and r for the fast Ca2+ current were also estimated, and rate equations were fitted. Simulations of the voltage-clamp experiments with these equations were run, and discrepancies with the experimental data were used to adjust the estimated values for m,, h,, and r for both currents. In addition, different powers for m were tested. The new values were fitted again and the results checked by simulation. This cycle was re- peated several times until the results were satisfactory. The maximum conductance, S, for the Ca2+ currents was determined by trial and error.

Several schemes were tried to simulate [PI, the internal concentration of Ca2+ effective in causing transmitter release, and Eq. 4 (APPENDIX) provided the best results. The optimal values for A and B were determined for each of seven simulated voltage-clamp steps. Equations were then fitted to these values for A and B.

In network simulations of an oscillatory pair of reciprocally inhibitory heart interneurons, several voltage-dependent ionic currents were added to a reciprocal synaptic transfer model (equations for network model, APPENDIX). The equation for & was derived from voltage-clamp data ( Angstadt and Calabrese, 1989) using the methods described above. The equations for &, 1k, and 1* were derived from the literature (Connor and Stevens 197 la,b,c; De Schutter 1986). The equation for JNa was modified so that JNa supported continuous spiking at threshold (-45 mV) and during Ca2+ plateaus (-30 to -35 mV).

RESULTS

Experimentally determined presynaptic calcium current consists of a fast and a slow component

The experimental data on which our graded synaptic transfer model is based consist predominantly of a series of voltage-clamp records where the presynaptic heart interneuron was held at -60 mV and the postsynaptic cell was held at -35 or -40 mV. The presynaptic neuron was then stimulated with a series of 1.5-s voltage steps in 2.5mV increments to -30 mV, and presynaptic and postsynaptic currents were measured. In many experiments, external [ Ca2+] was elevated to 5 mM to enhance Ca2+ currents and transmitter release. Normal external Ca2+ is 1.8 mM, and a corresponding, but smaller data set, was collected at anprox-

imately this concentration (i.e., 2 mM). The detailed kinetic structure of the presynaptic Ca2+ current varied from measurement to measurement even within the same preparation (cf. Fig. 2, B and C). Nevertheless, our analysis (Angstadt and Calabrese 199 1) indicated that the presynaptic Ca 2+ current is composed of two distinct kinetic components, one fast (quickly inactivating) and one slow (slowly inactivating). These two kinetic components were assumed to be distinct currents in the model, so that separate H-H equations describe each. We chose one set of records, where the same cells were used to determine the rela- tions between presynaptic Ca2+ current and both postsynaptic current and postsynaptic potential, to generate the H-H equations and synaptic transfer model, but compared the results against other record sets. At step potentials of -35 mV and higher, inward Ca2’ currents in the presynaptic cell were contaminated by partially blocked outward currents. Beyond -30 mV, contamination from outward currents prevented any meaningful measurements of Ca2+ currents.

Our data allow a reasonably accurate description of presynaptic Ca 2+ currents only over a range of presynaptic voltages from -60 to -35 mV. This range encompasses the voltage excursion of the heart interneuron’s slow wave during normal oscillation, although spikes in the heart interneuron reach - 10 to 0 mV. Assuming initially the same Ca2+ currents underlie both graded and spike mediated transmission, we adapted our Ca2+ current equations so that they would be useful in simulating normal heart interneuron activity, i.e., we wrote equations that would pro- duce realistic behavior up to 0 mV. In our H-H equations, the values for m, and h, for the range -30 to 0 mV were chosen as 1 and 0, respectively, and the values for the 7s approximated. We assumed that 7s of the Ca2’ currents do not change much above -30 mV and that the peak amplitude of the synaptic current remains approximately constant.

The two components of the presynaptic Ca2+ current used in the model (Eqs. 1 and 2, APPENDIX) are illustrated separately in Fig. 2A. IcaF could be fitted with a single closed-state model for m (Eq. 1). The lack of a clear delay in the activation of &, after a voltage step supports m’ kinetics (Hodgkin and Huxley 1952). Because we have not been able to isolate Icas from &, we have no records of the start of activation of &. Thus we cannot rule out that there is a delay in its activation corresponding to multiple closed states. Because lcaF was fitted by a single closed-state model, we assumed this simplest model to be valid for Icas.

The presynaptic Ca2+ currents, both in the chosen set and in other sets, could be simulated with sufficient accuracy to make them useful in a network model (Fig. 2, B and C). Two minor discrepancies were noted. The fast component inactivated too slowly at the lower step potentials. This problem was not considered important because its size is small at these voltages. The amplitude of the fast component is too large at -35 .O mV (and above) and its inactivation does not match the experimental data completely. We think, however, that the shape of the fast Ca2+ current might be deformed by partially blocked outward currents, particularly IA, which were not simulated. The discrepancy


B C A

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FIG. 2. Simulation of presynaptic Ca2+ currents in 5 mM Ca2+. The presynaptic cell was voltage clamped: holding potential at -60 mV with 1,500-ms steps to the potential indicated ( I$,). A : simulated fast (I& and slow (I& calcium currents for each step potential are shown separately. B: comparison of simulated presynaptic Ca2+ currents with experimental data ( 5 mM Ca” ) . Noisy lines: leak-subtracted experimental data. Smooth lines: simulation. C: comparison of simulated presynaptic Ca 2+ currents with experimental data (5 mM Ca2+) for the same preparation as B, but from a different experimental run. Note differences in measured currents between B and C. The simulated heart interneuron had an input capacitance of 500 pF. The conductances (a used in the simulations were 32.5 nS for IcaF and 5.9 nS for Icas .

I

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in peak size might also be caused by calcium-mediated inactivation, which we did not include in the model.

Synaptic transfer is proportional to presynaptic Ca2+ current *

Synaptic transfer in our model works through calculation of a variable, [P] 3, that determines the postsynaptic conductance (Eq. 3, APPENDIX, and Fig. 3 ) . [P] is dimensionless but can be roughly equated with the internal [ Ca2+] (in an unspecified volume) that is effective in causing transmitter release. The increase of [ P] is determined by the presynaptic Ca 2+ currents and a voltage dependent factor A. The decrease in [P] depends on a factor B, which is voltage and concentration dependent. The factors A and B complement each other, and they show an opposite relation to voltage: A increases with depolarization, whereas B decreases (Fig. 4). There is also a peak in the B-versus-voltage relation, which is situated just below the threshold for synaptic activation. A corresponds to a portion of the total Ca2’ influx that cannot effect synaptic release (e.g., such as current entering the cell body), whereas B corresponds to pumps and buffering mechanisms that reduce internal free Ca2+.

In Fig. 3, the change in [P] in the presynaptic cell during the voltage steps is compared with the postsynaptic conductance. This conductance is a linear function of [ PI3 (the scale for [P] has been amplified 200 times). Note that the peak in Ca2+ current is damped so that [P] remains high during the complete voltage step. The power function causes both a significant increase in synaptic delay and in the threshold for activation (note the difference between [P] and gs at -47.5 mV).

The synaptic transfer model simulates postsynaptic

currents produced by long ( 1,500 ms) presynaptic voltage steps quite well (Fig. 5 ). The threshold for synaptic activation and the decrease in synaptic delay with steps to increasing voltage are also well reproduced.

The simulated postsynaptic potentials produced by long ( 1,500 ms) presynaptic voltage steps are shown in Fig. 6. These were simulated with the same model, but a leak current was added to the model with a conductance of 11 nS and a reversal potential of -45 mV (measured experimentally). The postsynaptic potentials are simulated less well than the postsynaptic currents. Several explanations can be offered. Variability in the presynaptic Ca2’ currents (cf. Fig. 2, B and C) and/or postsynaptic currents (Ang- stadt and Calabrese 199 1) between experimental runs on the same neuron could account for at least part of the ob- served discrepancies. Additionally, the fact that we were using a one-compartment model, and thus neglecting the effect of the electrotonic distance from the synaptic sites and the electrotonic structure of the dendritic tree, might also contribute to the inaccuracy of the simulated postsynaptic potentials.

Adaptations and tests of the graded synaptic transfer model

We adapted the Ca2+ current and synaptic transfer models to simulate the data from another preparation, which was bathed in 2 mM Ca2+ ( 1.8 mM Ca2+ is considered to be the normal value for leech physiological saline) (Figs. 7 and 8 ) . Any Ca2+ current variability limits the likelihood that the model we developed can be transferred without some modification. Moreover, in the experimental data, presynaptic and postsynaptic voltage was only accurate only within a 5-mV range (Angstadt and Calabrese 199 1). Thus


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500 ms FIG. 3. Simulation of postsynaptic conductance in 5 mM Ca2’. Same

voltage clamp protocol and presynaptic simulation as in Fig. 2. For all potentials (V,,,), [P] (top, --) and postsynaptic conductance ([PI “) (bottom, - - -) are shown. [P] is dimensionless; it was plotted at a scale of 200 times larger than the conductance.

comparisons between experiments and between different cells must allow for discrepancies of such magnitude. The presynaptic Ca 2+ currents in 2 mM Ca2+ saline could only be simulated reasonably well if the kinetics were voltage shifted to the depolarized side (thus increasing the threshold of activation) : 5 mV shift for &F and 2 mV shift for lcas (Fig. 7). The gS were also changed, and the reversal potential adjusted to accommodate the lower external [Ca2+]. Again the peak currents for voltage steps to -35 mV (and above) were simulated less accurately (Fig. 7).

The simulation of synaptic transfer in the 2 mM Ca2’ experiment (Fig. 8) worked approximately when the standard Eq. 3 (APPENDIX) was used with gs equal to 58 nS (Fig. 8, broken line). The synaptic threshold and delay were simulated well, but the size of the peak current and the slope of the persistent current were less accurate. The simulation improved when the slope of the B curve was increased (Fig. 8, continuous line, Fig. 4 broken line). The peak current was still too large for steps to - 35 mV (and above). This discrepancy, however, is at least partially caused by the corresponding inaccuracy in the simulation of the presynaptic Ca 2+ current.

We also tested the models by attempting to simulate pre- and postsynaptic currents generated by short presynaptic voltage steps (70-100 ms) in both 5 mM and 2 mM Ca2+ (Fig. 9 ), reasoning that our model must behave well under such conditions, if it is to be used in simulations of natural activity. The equations we developed (suitably adjusted for

the external [ Ca2’]) predicted both the presynaptic Ca2+ currents and the postsynaptic currents including the tall currents well. The ability of the models to predict the postsynaptic tail currents, for short pulses, indicates the robust- ness of the B removal factor.

Use of the graded synaptic network simulation

transfir model in a

We developed the model of graded synaptic transmission between reciprocally inhibitory heart interneurons to be used in network simulations of these oscillatory interneurons. To test the applicability of our model to network simulation, we have made a preliminary simulation of a reciprocally inhibitory pair of heart interneurons comprising an elemental oscillator. Each interneuron was modeled as a single compartment and contained I*, I,, &, INa, I&F, and Icas voltage-gated currents as well as the graded synaptic transfer model (See METHODS and equations for network model in APPENDIX). The model (Fig. 10) produces oscillatory activity that approximates the activity of an oscillatory pair of heart interneurons, particularly the envelope of bursting and inhibition (cf. Fig. 1 B). Individual inhibitory synaptic potentials are not well simulated, and the transition between inhibition and spiking is too abrupt (data not shown). Some of the inaccuracies undoubtedly arise because it contains equations for voltage-gated currents adapted from the literature rather than determined from biophysical data on heart interneurons, and because the model contains no explicit formulation for spike-mediated

A 0.15 0.10 0.05 I 1 0.00 J I I / I , L 1 I , I i

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FIG. 4. Decrease factors for effective concentration of Ca2+ [P] vs. membrane potential. A : factor A vs. membrane potential. B: factor B vs. membrane potential. Solid line: standard equation used for most simulations (in 5 mM Ca2’). Broken line: equation adapted for the 2 mM Ca2’ experiment: B = -0.0076 - 0.0003 potential indicated ( VP,,) + 0.011 e-(vpre+49)2/ lo (cf. Fig. 7).

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500 ms FIG. 5. Comparison of simulated postsynaptic currents with experi-

mental data (5 mM Ca2’>. The postsynaptic cell was voltage clamped at -35 mV to record the postsynaptic current (&,,,) . Synaptic reversal potential was -60 mV. The presynaptic cell was voltage clamped: holding potential at -60 mV with 1,500-ms steps to the potential indicated (V,,) . The experimental data (noisy lines) are from the same experimental run and voltage clamp protocol as Fig. 2 B; the smooth lines are the simulation. The simulated presynaptic cell is the same as in Fig. 2, and the simulated postsynaptic cell had a maximum synaptic conductance, @, of 44 nS and a synaptic reversal potential, I?,, of -60 mV.

synaptic transmission. Moreover, there may be other important currents in heart interneurons of which we are not yet aware, e.g., we have discovered a persistent sodium current in heart interneurons (Opdyke and Calabrese un- published results) that we are currently characterizing. Still, the simulation demonstrates that our synaptic transfer model will work in a dynamic network simulation and sup- port oscillation, even in the absence of any model for spike mediated transmission.

Figure 11 illustrates the role of IcaF and Icas in mediating graded transmitter release during oscillation in heart interneurons. & acts only transiently at the beginning of a burst, where it determines the initial [P] and begins transmitter release. &as activates slowly, reaching its maximum value well after 1 CaF has fully inactivated, but the buildup of [P] and transmitter release continue for approximately 1 s. Eventually, as Icas slowly inactivates, [P] starts to diminish and transmitter release falls off rapidly, so that by the end of the burst transmitter release is at -5% of its maximum level.

DISCUSSION

We have developed a model to describe graded synaptic transmission between reciprocally inhibitory heart interneurons that can be applied to simulations of the dynamic behavior of these oscillatory interneurons. Our model is based on the pre- and postsynaptic currents in one prepara-

tion under particular voltage-clamp conditions; it is not a model based on the average behavior of a large sample of neurons. There is some variability among different heart interneurons and even between experimental runs on the same interneuron (cf. Fig. 2, B and C), but the data selected are representative (Angstadt and Calabrese 199 1). We chose this single-sample approach because voltage-clamp data averaged across preparations homogenized the traces so that rapid time constants of activation and inactivation were obscured, i.e., kinetic definition was lost. Moreover, because we could not separate the two components of the Ca2+ current experimentally, it was not possible to determine mGcj and h, accurately for the two components, thus precluding fitting to averages of these values. The fact that experiments from successive runs on the same interneuron (Fig. 2, B and C) or from other heart interneurons (Figs. 7-9) with the same or different duration presynaptic voltage steps could also be simulated (though with less quantitative accuracy) suggests that the presented model can be generalized. Most importantly, the synaptic transfer model is stable over the physiological voltage range (Fig. 5 ). It is adaptable to changes in Ca2+ currents (Figs. 7-9) and it also performs well in dynamic simulations of oscillating heart neurons (Fig. 10).

We have attempted with our model to simulate our physiological data as it was taken. We have ignored the morphological complexity of the heart interneurons, treating them as single isopotential compartments. We have assumed that

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FIG. 6. Comparison of simulated postsynaptic potentials with experimental data (5 mM Ca2’). The postsynaptic cell was held at about -37 mV in current clamp to record the postsynaptic potential ( V,,,,). Synaptic reversal potential was -60 mV. The presynaptic cell was voltage clamped, holding potential at -60 mV and 1,500-ms steps to the potential indicated (V,,,). The experimental data (noisy lines) are from the same experimental run and voltage clamp protocol as Fig. 2C; the smooth lines are the simulation. Postsynaptic potentials were simulated with the same model cell as Fig. 5, but a leak current was added with a conductance of 11 nS and a reversal potential of -45 mV (measured experimentally).

MODEL OF GRADED

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500 ms FIG. 7. Comparison of simulated presynaptic Ca2+ currents with exper-

imental data (2 mM Ca2’). Voltage clamps of presynaptic cell: holding potential at -60 mV with 1,500-ms steps to the potential indicated ( V,,). Noisy lines: leak-subtracted experimental data (I&. Smooth lines: simulation. Simulation included fast ( lcaF) and slow (&-& calcium currents with slightly modified equations compared with Fig. 2 (see text). The gS were also changed: 26.5 nS for fast Ca2+ ( - 18% ) and 3.0 nS for slow Ca2+ (-34%). A reversal potential of 123.8 mV was used corresponding to the lower external [ Ca2+ ] .

our voltage clamps accurately represent global currents in these neurons (i.e., that we have perfect frequency response and perfect space clamp), and we have assumed that the intracellular [ Ca2+ ] important for synaptic release, [ P 1, builds up as a direct result of presynaptic Ca2+ currents in an unspecified volume and then declines by concentration- dependent mechanisms.

We have taken this simplified approach mainly because our ability to gather more fine-grained physiological data is limited and because we believe that complete network simulations must ultimately be based on such simplified neuronal models for computational tractability. Although we have adequate morphological data to construct a geometri- cally accurate model, we can only make guesses about such important physiological parameters as the distribution of ion channels and the nature of intracellular Ca2+ handling systems. We have used standard forms (H-H equations) to describe ionic currents, and whereas we have resorted to ad hoc methods to describe intracellular [ Ca2+], the approach has been straightforward and uses Ca 2+ -handling equations that are related to presynaptic potential, which is a readily measurable quantity. In other systems where the geometri- cal considerations are simpler, such as the neuromuscular junction or the squid giant synapse, workers have developed elaborate synaptic transfer models based on Ca2+ diffusion in a constrained volume and its eventual interaction

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with release factors (Llinas et al. 198 1 b; Yamada and Zucker 1992; Zucker and Fogelson 1986). Such detailed models would be cumbersome to apply to dynamic network simulations.

Presynaptic Ca2+ currents

Our voltage-clamp data indicate that there are two distinct time constants for inactivation of the presynaptic Ca2+ current. Thus we have assumed that presynaptic Ca2’ current is composed of two independent components and have modeled each with an independent kinetic equation. Both of these equations use m’ kinetics rather than the m5 or ~1~ kinetics seen in other formulations (e.g., Llinas et al. 198 1 a). We also assume that, for both components of Ca2’ current, inactivation is solely voltage dependent (cf. Car- bone and Lux 1984). Although we have no direct evidence that internal Ca2+ contributes to the inactivation of either component, we also cannot rule out the possibility. Indeed, the observation that the transient component inactivates more quickly in the presence of 5 mM Ca2+ than in the presence of 2 mM Ca2+ ( cf. Figs. 4 and 10) is consistent with Ca2+ -mediated inactivation. Such Ca2’ -mediated inactivation is common (Eckert and Chad 1984) and has been successfully modeled in both vertebrate (Yamada et

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-35.0

200 pA

FIG. 8. Comparison of simulated postsynaptic currents with experimental data ( 2 mM Ca2’). The postsynaptic cell was voltage clamped at -35 mV to record the postsynaptic current (I,,,). Synaptic reversal potential was -60 mV. The presynaptic cell was also voltage clamped: holding potential at -60 mV with 1,500-ms steps to the potential indicated ( I$,). Noisy lines: experimental data. Data are from the same experimental run and preparation as Fig. 7. Solid lines: simulation with an adapted equation for B (see legend to Fig. 4 B) . Broken lines: simulation with the standard equation for B. Simulated postsynaptic cell was the same as in Fig. 5, except @s was eaual to 58 nS.

500 ms


70 ms

2 mM Ca*+

R 100 ms I Pre

500 pA 5 mM Ca2+

200 ms FIG. 9. Testing the synaptic transfer model with short presynaptic voltage steps. Comparison of simulated presynaptic

Ca2+ currents (I& (top) and postsynaptic currents ( Ipost) (bottom) with experimental data [ 2 mM Ca2+ (A) and 5 mM Ca2+ (I?)]. Voltage clamp of the presynaptic cell with a 70-ms (A ) or 1 00-ms (B) step from -60 to -35 mV. The postsynaptic cell was voltage clamped at -35 mV to record the postsynaptic current. Noisy lines: experimental data. Smooth lines: simulation. A and B are from different experiments. Simulated presynaptic cells were similar to Figs. 2 and 7 for 2 (A ) and 5 mM Ca2+ (B) , respectively. Simulated postsynaptic cells were similar to Figs. 5 and 8 for 2 (A ) and 5 mM Ca2+ ( B) , respectively, but with adjusted gS.

al. 1989) and leech neurons (Johansen et al. 1987). To do. not appreciably affect E,, . Ca2+ accumulation is un- introduce Ca2’ -mediated inactivation into our Ca2+ likely to affect Ica significantly; by the constant-field equa- current model, however, would have introduced a circular tion (Hodgkin and Katz 1949), an increase in [ Ca2+li from

0.1 to 100 PM causes an estimated decrease in &a of < 1% at a membrane potential of -30 mV.

process during the development of the synaptic transfer model, which we wished to avoid. Enhanced outward current ( possibly Ca2+ -mediated) could also account for the apparent change in the inactivation time constant of the transient Ca2+ current in the presence of 5 mM Ca2+. Our voltage-clamp records may be partially contaminated by outward current (primarily 1*, see Fig. 2), and evidence from our lab (Simon et al. 1992) indicates the presence of a Ca2+-sensitive A-like current in heart interneurons.

Internal Ca2+ concentration, [Ca2+]i, in heart interneurons is not known, but was assumed in the model to be 10v7 M as in Johansen et al. ( 1987 ) . Moreover, we assumed that changes in Ca2+ concentration due to inward flow of Ca2+

Synaptic transfer model

Our synaptic transfer model is based on the calculation of [PI, the [ Ca2+] in an unspecified volume that is effective in causing transmitter release. [P] builds up as the integral of the presynaptic Ca2+ current, minus an ineffective portion determined by a voltage-dependent factor A. [P] is reduced

j 20mV

\ lsec

0.5 nA

f

I

kaF

[P] A; J

FIG. 11. One cycle of oscillation (membrane potential for 1 of the heart interneurons is shown in the top 2 traces) from the same simulation

1

as Fig. 10, at expanded time base. Fast Ca2+ current ( ItiF), slow Ca2’ 10mV current (I&, effective concentration of Ca2+ [P] (dimensionless) and 4 set transmitter release ( [ P] 3, scale 660 times larger than [ P] ) are plotted for

FIG. 10. Simulation of normal oscillation of a reciprocally inhibitory the same interneuron. Note that the major contribution to [P] and thus to heart interneuron pair using the synaptic transfer model and the voltage- transmitter release ( [P] 3, during the plateau/ burst phase of the inter- gated currents described in APPENDIX. Membrane potential for the 2 interneuron is Ias, because IcaF inactivates very quickly after the transition to neurons over 4 cycles of oscillation is shown. the plateau/ burst phase.


by a voltage- and concentration-dependent factor B. We have chosen this approach because we wanted synaptic transfer to be a function of known variables, presynaptic potential and presynaptic Ca2’ current. We have no infor- mation on the volume in which Ca2+ is effective in evoking release, which may be highly restricted to synaptic terminals or more global ( e.g., Tank et al. 19 8 8 ) , or what percent- age of the presynaptic Ca2+ current flows into this effective volume. Others have emphasized the difficulties in calcu- lating effective [ Ca2+li in simulations and have pointed out the need for detailed knowledge about Ca2+ channels, their kinetics, density and distribution, and about internal Ca2+ handling by buffers and transporters (Carnevale 1989; Sherman et al. 1989).

Our best results in the synaptic transfer model were ob- tained when we assumed that transmitter release was proportional to the third power of [P] . Similarly, at the squid giant synapse the rising phase of the postsynaptic current is proportional to the third power of the integrated presynaptic Ca2+ current (Augustine et al, 198 5 ) .

The concentration-dependent factor B that decreases [P] probably lumps together diffusion, buffering, and maybe a pump. The non-concentration-dependent factor A is a portion of the Ca2+ that is ineffective in causing release. Per- haps this is Ca2+ that enters regions of the neuron where it cannot effect transmitter release (e.g., the cell body and primary neurite). Because all our measurements of Ca2+ currents were from the cell body, it is reasonable to assume that we measure some Ca2+ current flowing into nonsynap- tic regions. Alternatively, A could represent a voltage-dependent Ca2+ threshold for release.

The voltage dependence of decrease factors A and B is to a certain degree an artifact of the available data; presynaptic potential is accurately known in our voltage-clamp experiments, whereas hard data on Ca2+ handling in invertebrates is limited. Other models of synaptic transfer have also included presynaptic voltage-dependent processes, not in the control of Ca 2+ buildup, but in the activation of Ca 2+ -bind- ing proteins that mediate release (Parnas et al. 1986). More- over, a more “physiological” model similar to that of Ya- mada et al. ( 1989), with one shell, a buffer, and a Ca2+ pump, did not work as well in simulating our data.

Comparisons of our model to other models of synaptic transfer are difficult, because most other models involve transient transmitter release at synapses, such as the squid giant synapse (Llinas et al. 198 1 b), neuromuscular junc- tions (Parnas et al. 1989; Zucker and Fogelson I986), and Aplysia sensory neurons (Gingrich and Byrne 1985, 1987), where release is spike mediated. These models have defined a releasable pool of transmitter that is quickly depleted and must be replenished. Our model, on the other hand, was developed to simulate long-lasting graded release of transmitter. We have assumed that transmitter cannot be depleted. Transmission wanes during presynaptic voltage steps because Ca2+ currents eventually inactivate and because the Ca2+ removal system gradually overtakes Ca 2+ influx at depolarized potentials.

Although the mechanistic significance of our synaptic transfer model is still unclear, it does work very well, both to replicate voltage-clamp experiments and in network sim-

ulations that reproduce the oscillatory activity of heart interneurons.

Uses ofthe synaptic transfer model .

An immediate use of our synaptic transfer model will be to explore, with appropriate parameter searches, the interaction between voltage-gated membrane currents and synaptic transmission that leads to oscillation in reciprocally inhibitory heart interneurons (De Schutter and Calabrese 1990; De Schutter et al. 1989 ) . With our synaptic transfer model, any change in membrane potential, any competing current, or any neuromodulatory influence that alters Ca2+ current in a network simulation will influence synaptic transmission in the network, because synaptic transfer is an explicit function of presynaptic Ca2+ current. Similar synaptic transfer models should be useful in simulating other central pattern generators [e.g., the pyloric and gastric mill networks of the crustacean stomatogastric ganglion (Grau- bard et al. 1983; Miller and Selverston 1985; Raper 1979; Selverston 1989)] or other neural networks [e.g. leg reflex circuits in locusts ( Burrows 1985 )] where graded release of transmitter plays an important role and where neuromodulatory influences can affect presynaptic Ca2+ entry.

APPENDIX

Fast Ca*+ current

I CaF = gmhcvm - ECa)

e = 32.5 nS EC, = 135.2 mV

0.04 m: CY = 1 + c-Wm+W/2.~

P -0.13(Vm + 57)

= 1-e +( V,+57)/2.0

61: a = 0.0050

c+( I’,+61 )/5.6

P 0.028 Z.X

1 + e-(bn+45)/3.0

Slow Ca*+ current

I CaS = iTrnht vm -- ECa)

K= 5.9 nS E Ca = 135 2 mV .

0.0072 in: N =

1 + ,-wn+5w3.5

P 0.19

=I+e +( V,+58)/3.5

h: a = 0.0027

1 + e+(V,n+58)/2.0

P

0.00072(&,+ 67) = l+e +( 1/,+67)/6.5

(1)

(2)


Postsynaptic current Is = iw&J3wpost - Es)

g=44.0nS Es=-600mV .

WI - = Ica - B[P] dt

(Iin nA)

ka = -kaF - kas - A Ica < O.OO:I, = 0.00

A = 0.66 + 0.012Vp,, A < O.OO:A = 0.00 A > 0.29:A = 0.29

B = -0.000101 Vpre + 0.01 1e-(vpre+49)2/10

Equations for halfoscillator

Fast Na+ current

INa = gm3h( Vm - ENa)

8=250nS E,,=45mV

m: a = O.O3(V, + 8.5) 1 _ e-(vm+8*5)/7

P 0.20

=l+e +(v,+16)/6

h: a = 0.0050

1 + e+(vrn+20)12

P 1.0 =

1 + e-(vm+O)ll

Delayed rectljier current IK = gm2h(Vm = EK)

g=325nS E,=-8OmV

m: CY = 0.002(Vm - 9.5)

1 - ~-(&n-9.5)/10

P 0.0026 =

e+( 1/,+40)/ 190

h: a = 0.194

1 + ,-(v,-17.5)/10

P 0.0234 =

e+( v/,+40)/84

A current IA = gm4h( Vm - EA)

g=200nS E,=-80mV

m: CY = O.O0124(V, + 70)

1 _ e-(vrn+70)/ lo

B 0.04 18 =

,+<Y,+40)/41

h: a = 0.0039

1 + e+<Ym+56)/4

P o.oooll(v, + 79) =

1 - e-(vm+79)/3

Hyperpolarization activated current Ih = gm2h(I/m - Eh)

E= 15 nS Eh = -21 mV

(3) P

0.00 15 =1+e +( V,+86)/23 (7)

m: CY = -o.ooools(v, + 43.5)

1 + e+(Y,+43.5)/ 10

Fast Ca2+ current

See Eq. 1. g = 90 nS; EC, = 122.5 mV.

Slow Ca2+ current (4)

See Eq. 2. g = 12.5 nS; EC, = 122.5 mV.

Postsynaptic current

See Eq. 3. g = 44.0 nS; Es = -65 mV.

Leak current

g= 4.5 nS; Es = -45 mV .

We thank Clifford Opdyke, Michael Nusbaum, and Ted Simon for help- ful discussions.

This work was supported by National Institute of Neurological Dis- orders and Stroke Grant NS-24072 to R. L. Calabrese and NRSA Grant NS-08089 to J. D. Angstadt.

Present address: E. De Schutter, Div. of Biology, California Institute of Technology, Pasadena, CA 9 1125; J. D. Angstadt, Dept. of Biology, Siena College, Loudonville, NY 122 11.

(4) Address for reprint requests: R. L. Calabrese, Dept. of Biology, Emory University, 15 10 Clifton Road, Atlanta, GA 30322.

Received 2 1 January 1992; accepted in final form 13 November 1992.

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