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Journal of Physiology (1992), 445, pp. 157-167 With 7 figures Printed in Great Britain

COMPUTED POTENTIAL RESPONSES OF SMALL CULTURED RAT HIPPOCAMPAL NEURONS

BY STAFFAN JOHANSSON* AND PETER ARHEM From the Nobel Institute for Neurophysiology, Karolinska Institutet, S-104 01 Stockholm, Sweden

(Received 15 January 1991) SUMMARY

1. The potential responses of small hippocampal neurons were computed on the basis of a previous mathematical description of the currents recorded under voltageclamp conditions. 2. The computed action potentials were graded with respect to stimulus strength, in accordance with previous experimental findings. 3. The time course of the membrane currents and of the permeabilities and permeability variables during the impulse was computed for different stimulus intensities. 4, The effect of the membrane time constant on the impulse amplitude was investigated. It was concluded that the value of the time constant used was not per se sufficient to explain the amplitude variation of the impulse. 5. The effect of the magnitudes of the different potential-dependent permeabilities on the impulse amplitude was investigated. A Na+ permeability within a certain range caused impulses of variable amplitude, and this variability was affected by the K+ permeability. INTRODUCTION

In two previous papers we described the electrophysiological properties of small cultured hippocampal neurons from rat embryos. The first paper showed that these neurons are capable of generating action potentials with amplitudes that depend on the stimulus strength, and thus deviate from the 'all-or-nothing' principle (Johansson, Friedman & Arhem, 1992). The second paper described a voltage-clamp analysis of the currents underlying these potential responses (Johansson & Arhem, 1992). The currents did not differ largely from those of other neuron types. The initial transient (Na+) current component and the delayed sustained (K+) component were described quantitatively on the basis of earlier membrane descriptions. In the present paper, we use this quantitative description to compute the expected potential responses to different current stimuli. It is shown that graded action potentials are predicted by the equations. Possible explanations of this are suggested, and the effects of varying some permeability parameters analysed. Similar computations of potential responses were first made for the squid giant * To whom correspondence should be addressed. MS 9079

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S. JOHANSSON AND P. ARHEM axon (Hodgkin & Huxley, 1952) and later for the nodal membrane in myelinated nerve fibres (Frankenhaeuser & Huxley, 1964). In extended or modified versions they accurately describe a number of membrane properties (see also Frankenhaeuser, 1965; Frankenhaeuser & Vallbo, 1965; Gestrelius & Grampp, 1983). The model used by Frankenhaeuser & Huxley (1964) served as a basis for the present description (see Johansson & Arhem, 1992) and is here used also for comparisons, since the action potential in this model is essentially an 'all-or-nothing' phenomenon. Some preliminary results have been presented previously (Johansson & Arhem, 1990). METHODS

As stated in the Introduction, the present computations are based on the empirical description of membrane currents in the preceding paper (Johansson & Arhem, 1992). For the computations, we assumed a uniformly activated membrane region with no current spread to neighbouring regions. These conditions were not valid in the present experimental situation. Most neurons studied had neurites (Johansson et al. 1992), to which current could spread. However, since electrophysiological data concerning the neurites are not available, the present treatment serves as an approximation. The computed potential responses were not expected to differ much from a situation in which current spread is possible (cf. membrane action potential and propagated action potential described by Hodgkin & Huxley, 1952). List of symbols U: membrane potential; UR: resting or holding potential; membrane resistance; Rm: membrane capacitance; Cm: initial transient current; INa IK: delayed sustained current; leak current; IL: capacitative current; IC: I s: stimulus current; PNa, PK: permeabilities associated with INa and IK; PNa, P': permeability constants (see Johansson & Arhem, 1991); time. t: Equations The equations used are: dU/dt = Ic/CM, (1) where Ic =Is-(Na+IK+IL), (2) and

IL=(U UR)/Rm. (3) The ion currents INa and IK were calculated according to the empirical current description (eqns (1)-(13)) in the preceding paper (Johansson & Arhem, 1992), with standard values for constants PNa and P' of 1-3 x 10-4 and 2-4 x 10-5 cm s-1 respectively. The standard values for Rm and Cm were 4-3 Gil and 7 0 pF respectively. (The values were derived from the same cell as the one used for the description of the currents. This cell had an estimated membrane area of 100 /m2, giving an estimated resistivity and capacitance per unit area of 4-3 kQ cm2 and 7 0 ,uF cm-2 respectively.) Equation (3) above was justified by the finding of a linear relation between the leak and capacitative currents and the potential (Johansson & Arhem, 1992). Computation of the potential responses of the nodal membrane in the myelinated nerve fibre was based on the equations and constants given by Frankenhaeuser & Huxley (1964). The equations for the marginal p-current were not included. Numerical integration followed the Euler method (Wilson & Bower, 1989), using an integration step of 10-50 ps for the hippocampal model and 5 0 ,us for the faster processes when data from myelinated nerve fibres were used. In some cases an explicit exponential method (Wilson & Bower, 1989) was used, with similar results. All computations were performed on an IBM-compatible personal computer (80286 CPU). The software was written in BASIC.

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The electrophysiological data presented in Fig. 1B were recorded as described in Johansson et al. (1992). RESULTS

Computation of potential responses Our first aim was to compute the action potentials associated with rectangular current pulses of varying amplitude. The results of a series of such computations are A

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Fig. 1. Computed and recorded action potentials. A, computed responses of hippocampal neuron to 10 ms current steps of 30, 35, 40, 45, 50 and 80 pA. B, experimentally recorded hippocampal action potentials (as described by Johansson et al. 1992). Stimulating current steps 35, 40 and 45 pA. Temperature 22 'C. (From the cell used for the standard empirical description of the currents; Johansson & Arhem, 1992.) C, computed action potentials for the nodal model. Parameters from Frankenhaeuser & Huxley (1964; see Methods). Stimulating current steps 0 350, 0 355, 0-360, 0-400, 0 500 and 0 800 mA cm-2. Step duration 1-0 ms to account for the faster time courses compared with those of the hippocampal model.

shown in Fig. 1A, together with experimentally recorded action potentials (Fig. 1 B) from the same cell as that from which the empirical description of the currents was derived (Johansson & Arhem, 1992). Action potential amplitude increased with stimulus current in the computed case as well as in the experimental case. The computed peak-versus-stimulus curve was S-shaped in the suprathreshold stimu-

S. JOHANSSON AND P. ARHEM lation range (cf. curve A in Fig. 4), as typically seen experimentally (Johansson et al. 1992). The general agreement between computed and recorded action potentials was satisfactory, considering the uncertainties in the description of the currents (see Methods), and the individual variability of the experimentally recorded currents (Johansson & Arhem, 1992). Thus, it was concluded that the currents recorded under voltage-clamp conditions were sufficient to generate graded action potentials. Variation of stimulus current duration affected the computed responses, especially the threshold values, but yielded qualitatively similar results. In most computations concerning the hippocampal neurons, a duration of 10 ms was used since this is of the same order of magnitude as the duration of the presumably synaptically induced potential events observed experimentally (Johansson et al. 1992). The recorded currents on which the computations were based were not qualitatively different from those of other excitable membranes not reported to generate graded action potentials. For comparison, action potentials with standard parameters from the nodal membrane of amphibian myelinated nerve fibres (Frankenhaeuser & Huxley, 1964; see Methods) were computed. Figure 1 C shows that for stimulus strengths ranging from just suprathreshold to more than twice this value, the 'nodal' membrane model generates action potentials that are essentially of the 'all-or-nothing' type. Furthermore, the amplitude does not depend much on whether the peak occurs during or after the eliciting current pulse. This agrees with experimental findings (e.g. Frankenhaeuser, 1957). 160

Membrane parameters during the impulse To investigate the mechanisms underlying the amplitude variations in the hippocampal neuron model, we computed the permeabilities during the potential responses (Fig. 2A and B) to a slightly suprathreshold and to a relatively strong current stimulus. Both the Na+ permeability and the K+ permeability rose much faster and to higher values in response to the strong than to the weak stimulus (Fig. 2B). Figure 2C and D shows the changes in the corresponding variables m, h and n during the two different potential responses. Also, the computed currents during the action potential reflect the difference in time course. INa and IK (and also the leak current) followed single-peaked changes of larger amplitude in response to the stronger stimulus as compared with the weaker (Fig. 2E and F).

Effects of modifying the membrane time constant and the permeability constants The time courses of the model parameters shown change with stimulus intensity (Fig. 2). It is not, however, evident which membrane parameters are most critical for the generation of graded action potentials. We therefore investigated the effect of changing some parameters (i.e. permeability constants and membrane time constant) that differed considerably from those of the nodal (essentially 'all-or-nothing', see Fig. 1 C) model. Time constant The membrane time constant of the hippocampal model is 30 ms (resistivity 4.3 kQ cm2, capacitance per unit area 7 0 ,uF cm-2), while in the nodal model it is only 66 /is (33 Q cm2 and 2 ,uF cm-2). Computations were made with the standard hippocampal parameters except for resistance and capacitance, which were set to the

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Fig. 2. Computed membrane parameters during two impulses of different amplitude. A, potential responses to 10 ms current steps of 35 (right) and 50 pA (left). B, Na+ and K+ permeabilities during the impulses in A (left curves associated with the impulse elicited by 50 pA). C and D, variables m, h and n during the impulses elicited by 35 pA (C) and 50 pA (D). E and F, Na+, K+ and leak currents during the impulses (A) elicited by 35 pA (E) and 50 pA (F). 6

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Fig. 3. Effects of the membrane time constant on computed potential responses. A, hippocampal model. Membrane resistance and capacitance were set to the standard values of the nodal model (33 Q cm2 and 2-0 ,F cm-2 respectively). Stimulating rectangular, 10 ms current steps 1 2, 1 3, 1-4, 1 5, 1-7 and 2-0 nA. Note that the lowest curve is almost indistinguishable from an electrotonic response. B, nodal model. Membrane resistance and capacitance were set to the standard values of the hippocampal model (4 3 kQ cm2 and 70 ,uF cm-2 respectively). Stimulating 1 0 ms current steps 60, 70, 80, 100, 200 and 500 ,tA cm2.

nodal model values. Only when strong stimuli were used did the response peak clearly exceed the electrotonic response resulting from IL and Ic (Fig. 3A). A small hump, corresponding to a regenerative response, was distinguished. This component and the total response were, however, graded in amplitude.

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Conversely, computations were made with the standard nodal membrane parameters (Frankenhaeuser & Huxley, 1964; see above), except for resistance and capacitance, which were set to the hippocampal model values. Although a very small amplitude variation was detected, the regenerative potential responses were still E D

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Fig. 4. Computed peak-potential-current relation at various PNa values. Stimulating rectangular current pulses of 10 ms duration were used. A, standard hippocampal data. B-E, PNa increased 2 (B), 5 (C), 20 (D) and 50 (E) times. F, PNa reduced 5 times. C, PNa and PK increased 50 times.

essentially 'all-or-nothing' (Fig. 3B). It was concluded that the amplitude variation of the hippocampal potential responses was not exclusively an effect of the time constant.

Permeability constants The ionic permeabilities of the hippocampal neurons were much smaller than those of the node of the myelinated nerve fibre. In the hippocampal model the constant PNa was 1-3 x 10-4 cm s-1 and P'K was 2-4 x 10-5 cm S-1, while in the nodal model the corresponding constants were 8-0 x 10-3 and 1-2 x 10-3 cm s-' respectively. We made computations with changed permeability constants in the hippocampal model. Effects of PNa variations on peak membrane potential are shown in Fig. 4. When PNa was reduced 5 times (curve F), almost no regenerative potential response was 6-2

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obtained in addition to the response resulting from IL and Ic. In fact, responses smaller than those expected from IL and Ic were obtained at strong stimulus currents, due to the activation of outward currents. On the other hand, when PNa was increased, the potential responses grew larger; but the amplitude variation gradually

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decreased (curves B-E in Fig. 4), giving an essentially 'all-or-nothing' response at high P Na values. Also when the constant P' was raised in proportion to the rise of P Na higher values resulted in less graded potential responses (Fig. 4, curve G). Thus, the effect of raising P Na was not exclusively due to a relative increase of the ratio PNa/PK (see also below). It was concluded that the magnitude of P Na is of importance for the generation of graded impulses. It appears that PNa has to be within a critical range. Values above this range result in an essentially 'all-or-nothing' response, while values below it result in no action potential at all. An unexpected observation was that when PNa was raised 17 times or more relative to the standard value (1-3 x 10-4 cm s-1), the potential following an impulse did not return to the original resting potential, but settled after damped oscillations at a more positive value (Fig. 5). This effect, however, disappeared when P' was raised in proportion to the P Na rise, indicating the necessity of an outward repolarizing current system for normal membrane function. Computations were made with a reduced P', since many of the hippocampal neurons studied (Johansson & Arhem, 1992) showed a lower permeability for the delayed sustained current than that of the mathematical model presented. When P'K was reduced, the amplitude variation clearly increased, with strong stimuli causing responses of higher amplitude than with the standard P' value (Fig. 6). Figure 7

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Fig. 6. Computed peak-potential-current relation at various PK values. Stimulating rectangular current pulses of 10 ms duration were used. A, standard hippocampal data. reduced 5 times. C, PK = 0. B, P'KK

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Fig. 7. Computed potential response at P'K = 24 x 101 cm sl (standard data; lower curve) and PK = 0 (upper curve). Rectangular stimulating current pulses of 40 pA and 10 ms duration were used.

S. JOHANSSON AND P. ARHEM shows an impulse computed with PK set to zero. The impulse was drastically changed compared with the control case, and showed a prolonged repolarization phase. 166

DISCUSSION

The main conclusion of the present investigation is that the graded action potentials observed in small cultured neurons from embryonic rat hippocampi (Johansson et al. 1992) can be predicted theoretically from the currents recorded under voltage-clamp conditions (Johansson & Arhem, 1992). Thus the initial transient current and the delayed sustained current are sufficient to account for the phenomenon: no other potential-activated currents are required. We further conclude that the existence of graded action potentials in this model critically depends on PN, but not on the membrane time constant. The degree of amplitude variation depends on PK when standard hippocampal values are used for the other parameters. The PNa dependence suggests that the amplitude variation is mainly determined by the density of Na+ channels in the cell membrane. Thus the variation is not caused by modified Na+ channel gating or persistent K+ channel activation, as has been suggested for other systems (Bush, 1981). It has been shown theoretically that the squid giant axon also generates impulses of different amplitudes when near-threshold stimuli are given (Fitzhugh & Antosiewicz, 1959). However, the stimulus range for this is extremely narrow, and graded impulses are expected neither to be seen experimentally nor to play any physiological role (Fitzhugh & Antosiewicz, 1959). Furthermore, the computations by Cooley & Dodge (1966) relate to the dependence of the amplitude variation on PNa, Investigating the effects of anaesthetics on the Hodgkin-Huxley model of the squid giant axon, they found reduced Na+ and K+ conductances to be compatible with decrementally propagated responses. This propagation could occur over distances of several centimetres. Experimental results supporting this view have been presented by Lorente de N6 & Condouris (1959). Our computations showed some further differences between the present and earlier models. In the squid giant axon model (Hodgkin & Huxley, 1952) and in that of the amphibian myelinated axon (Frankenhaeuser & Huxley, 1964) the computed Na+ current during the action potential showed a characteristic double-peaked appearance. In the present model the corresponding Na+ current was single-peaked. In the nodal model, the constant PK has only marginal effects on the action potential (Frankenhaeuser & Huxley, 1964), and the function of the K+ current remains obscure. In the present model PK was essential for the time course of the action potential. This was even more pronounced when PNa was increased 17-fold: without IK' the potential did not return after the impulse to the original resting potential. Lastly, other neurons also show membrane properties similar to those of the hippocampal neurons (e.g. Cull-Candy, Marshall & Ogden, 1989); the permeabilities and membrane time constants are probably often closer to the hippocampal than to the nodal values. Graded impulses may therefore be more common in the nervous system than usually assumed. This may be of some importance for our knowledge of information processing in the central nervous system (Johansson et al. 1992).

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We thank Professor B. Frankenhaeuser and Professor S. Grillner for valuable discussions. This work was supported by grants from the Swedish Medical Research Council (project No. 6552) and Karolinska Institutets Fonder. REFERENCES

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