Measurements of dendritic conductance changes to GABA in granule cells of the rat dentate gyrus

0306-4522/88 $3.00+0.00

Neuroscience Vol.24,No. 3,pp. 821-827,1988

Printed

Pergamon Press plc

in Great Britain

0

1988 IBRO

MEASUREMENTS OF DENDRITIC CONDUCTANCE CHANGES TO GABA IN GRANULE CELLS OF THE RAT DENTATE GYRUS T. J. BLAXTER*~ and P. L. CARLEN$$ *Playfair Neuroscience Unit, Departments of Physiology and Medicine, University of Toronto; and JAddiction Research Foundation, 33 Russell Street, Toronto, Ontario, Canada MSS 2Sl magnitude of dendritic conductance changes occurring distantly from the somatic site of recording can be diicult to measure. We have used measurements of the neuronal time constant, T,,, instead of the neuronal input resistance, R,.,, to estimate the resistance decrease that accompanies the depolarizing response of the dendrites of granule cells when GABA is applied. Ahatrae-The

From the changes in T,, we estimated the reversal potential of the response and found that the conductance change accompanying a given GABA-mediated voltage response as measured at the cell body was the same regardless of where in the dendritic tree the drug was applied. On the other hand, R, changes underestimated the increase in conductance of the GABA responses in the distal dendrites and were not

accurate for determining the reversal potential.

The purpose of this paper is to investigate whether it is possible to estimate conductance changes associated with the actions of neurotransmitters and drugs, in this case GABA, even when the response is electrotonically distant from the site of cell impalement and, therefore, current injection. Electrotonic potentials are often used to measure the input resistance (RN). Small hyperpolarizing currents (usually 0. l-l .OnA) are injected into the cell for long enough to reach a steady-state maximum. This period is about four times the neuronal time constant (to). We will refer to these as R, estimations. It is preferable to use the least current consistent with accurate measurement in order to avoid any voltage-dependent conductance changes associated with the current pulse itself; anomalous rectification may be large in some cells, CA1 pyramidal neurons for example.15 Carlen and Durand,6 using an analogue neuronal model, showed that an increase in the input conductance (G,, which equals l/R,) on or close to the cell body was accurately measured by long pulses (i.e. R,

changes), but that the apparent conductance increase fell off rapidly as the equivalent shunt was moved further down the modelled equivalent lumped dendritic cable, and thus became more remote electronictPresent address: Department of Pharmacology, School of Pharmacy, 29-39 Brunswick Square, London, WClN IAX, U.K. §To whom correspondence should be addressed. Abbreuiutions: ACSF, artificial cerebrospinal fluid; C,,,, specific membrane capacitance; E,,, reversal potential of the response to GABA; G,, input conductance; Gh, input conductance in the presence of GABA; L, electrotonic length; R,,,, specific membrane resistance; R,, input resistance; RE, input resistance in the presence of GABA; p. dendritic-somatic conductance ratio; rO, final neuron time constant; rh, time constant in the presence of GABA; r,,,, membrane time constant; r, , first peeled exponential.

ally from the somatic current injection and recording site. In contrast to the R, results, they found that the reduction in T,,, the neuronal time constant, produced by a specified shunt, was almost the same wherever the shunt was situated in the analogue dendritic tree.6 These results held for a compartmental cable model with an electronic length (15) of 1.5 and a soma at one end, and a model with the soma having two cables, each with an L of 1.O. Further detailed discussions of the utility of measurements of changes in z. compared to R, are in Refs 6 and 7. The purpose of this study was to use the R, and 7,, estimations for measuring conductance changes and for estimating the reversal potential of the GABA response. We have chosen granule cells because they behave more passively around the resting membrane potential” than, for example CA1 neurons. Granule cells have been shown to approximate an equivalent cylinder model’ with their dendritic branching satisfying the 312 power law of the theoretical models of dendritic trees developed by Rall.“~” Also the granule cell is relatively electronically compact.9 GABA was used to produce the dendritic conductance changes since reproducible responses lasting several seconds can be obtained from the dendrites with a large conductance increase and large amplitude. We assumed that the resting membrane potential and the reversal potential of the response to GABA are uniform throughout the dendrites. We compared the conductance changes produced by applying GABA to different sites on the dendrites of the impaled neuron using both 7. and R, estimations. EXPERIMENTAL PROCEDURES Transverse slices 300 pm thick were cut using a vibratome from the hippocampus of male Wistar rats (14&16Og). Slices were stored for an hour before being transferred to the

821

822

T.J.

HLAXTER

and P. L.

recording chamber of a modified Haas bath. Intracellular recordings were made at 32..35”C from granule cell somata in the buried blade region of the dentate gyrus (Fig. IA). Artificial cerebrospinal fluid (ACSF) had the following composition (mMk Na+, 154; Kc, 3.25; Ca2+, 2; MgZ+, 2: Cl*,, 131.5; HCO;. 26: H,PO:. 1.25: SO?. 2: and dextrose, 10. The A&F waskquihbrated with &d/, Or, 5% CO, giving a pH of 7.4. Microelectrodes (60-l 50 Ma) were filled with 3 M potassium acetate. Recordings were accepted if the spike height was more than 80mV, the input resistance was greater than 40MR and the resting potential was more negative than -65 mV, RN was determined in the standard way from hyperpolarizing current pulses of SO--lOOms duration with an amplitude of 0.05-0.2 nA. The current was adjusted to give a voltage response from the cell of about 10mV. r,, was estimated from short hyperpolariting current pulses of 0.5ms duration with an amplitude of lO.OnA given at frequencies of up to 5 Hz. Conductance increases to GABA were measured either as the reduction in size of the long pulse (RN), or as the reduction in to as estimated from the short-pulse decay. Analyses of the short pulse was performed on averages of up to 40 sweeps at a steady membrane potential. R, can also be estimated from the short-pulse decay.6 At least three responses to GABA were obtained at each of the three sites in the dendritic tree of the recorded ganule cell. The short-pulse method has advantages over estimating r,, from the long-pulse response as shown in dentate granule cells by Durand et ai.’ The time constant is measured as the decay approaches the resting potential such that errors introduced by non-linearities in the current-voltange relationships of the membrane are minimized. Also, because the pulse is so short, there is less likelihood of turning on at least the slowly activating voltage-de~ndent conductames including anomalous rectification in the hyperpolarizing direction.’ The slowest time constant (1~“) corresponds to the membrane time constant (r,),‘4 assuming that the specific membrane resistance (4) and the specific membrane capacitance (C,,,) are constant through the cell. r,=&C,.

(1)

Since the somatic & is probably lower than the dendritic R,: the absolute value of r, is probably an uude~timate of r,. The present study, however, was concerned with changes in z,, to conductance increase, and not the absolute values. In hippocampal slices, the dendrites of granule cells respond to GARA with depolarization2 This study found the reversal potentird of the response to be about 20 mV less negative than the resting potential. The reversal potential refers to the membrane potential measured by the somatically placed recording electrode as opposed to the equilibrium potential, which is the true subsynaptic potential at which the GABA-mediated voltage response changes polarity.’ When there is an electrotonic separation between the (somatic) site of current injeotion and the (dendritic) subsynaptic site, one expects that the measured reversal potential of the depolarizing GABA response will be more positive to the resting membrane potential than the “true” equi~b~~ potential. The method used was from eqn (5) derived by Ginsborgr2 which is:

(2) where r is the resistance of the electrical pathway activated by the action of a neurotransmitter, which equals l/g, and e is the size of the response to the applied neurotransmittcr assuming no decay due to cable properties. E, is the reversal potential measured relative to the resting membrane potential. G;,==g+G, (3)

CARLEN

where GE,is the input conductance activated in the presence of GABA (l/R&) and C;, is the resting input conductance (l/R,). Substituting eqn (3) into eqn (2) gives: (4) Substituting for g,

and therefore

GARA (1 mM, pH 7) was applied by pressure ejection (150 kPa, 4-50 ms) to the dendrites of the imn&d ceil at intervals’of at least 1 min. Three sites of ap&ation were chosen (Fig. IA); about 1OOpm (one-third of the distance between the ceil body layer and the &sure), about 200 pm ~tw~~~s), and as close to the fissure as possible (about 3OO~m). Steady, flat-topped responses were obtained by rapidly applying short pulses of GABA (Fig. 3). No -desensitization was seen. We aimed for responses of 3, 5, and IO mV. The dendrites of the granule celhs spread out in a fan from the cell body. Therefore. if the number of GABA receptors is constant per unit surface area of the neuron, GABA applied to the proximal region of the dend&ic tree should activate more dendritic branches than GABA applied near the fissure where the density of branches is lower. ft dii appear more blurt to find responses to GABA near the fissure, p&haps for this reason. Long pulses were applied throughout each response and short p&es just before each response and then during the Rat portion of the response. Signals were recorded on tape [Racai FM 5 kHz bandwidth) and were subsequently sampled and analysed by computer. The r0 was measured from the stope of the final exponential decay of the short pulse (Fig. 4). Usually three time constants could be extracted. Changes in conductance were calculated simiiarly From RN and ‘t;, measurements:

RN--Rk

AR,% = -Y

x lo&

JfN

Assuming no change in C,,,, then for a sphere (i.e. no cable),

Y

where 7; is the time constant measured in the presence of GABA. From eqns (7) and (8) if one assumes no decremental decay along a cable, it can be inferred that R,]R, can be written as r&;1 and therefore eqn (6) becomes

The eleetrotonic length, L, is equal to the ratio l/i, where 1 is the length of the equivalent cable, and i is the length constant. Provided the dendritic branching of theseneurons follows the 3/2 power law as has been demons@twi,s L can be calculated from the following equation developed by RaR’9

Regression analysis for Fig. 6 was carried out using the SPSS-X statistical package on a VAX 11/750 and in&&d linear regression, calculation of the multiple correlation coetlicient and analysis of the scatter between the linear equation and the data points. XEsUl‘TS

Results were taken from 19 rmurcms from which GABA responses with a flat top could be obtained.

823

membr~e potential

cell

proximal

distal

tayer

-81 RMP

-3%

-et -96

.

@ABA

applied to dendrites

rewoftse a&e CmV)

a GABA applied to dendrltaa Fig.

1. (A) Diagram of the hippocampal slice and enlarged view of the area of the dentate gyrus where granule cells were impaled. The iRtra~ilu~r electrode approached from the axonal side of the cell layer, and the GABA pipette approached at right angles to the axis of the dendrites. The enlarged view shows the layers from which the responses were obtained. The sites were separated by about 100 pm. (B) Responses obtained by applying GABA in increasing doses to the mid-dendrites. The amount of GABA applied is expressed in multiples of the shortest duration of GABA pulse to produce a response (5ms in this case). Y, = -86mV. resting potential was -?9* 7mV The action ~tentials always overshot zero and ranged from 80 to 120 mV in size. The mean RN was 65 f 10 MQ. The mean 2, was 12.5 &-3.2 ms. The mean electrotonic length, 15,of these 19 neurons was 1.22 It: 0.17 as calculated from eqn (lo), suggesting that these neurons are el~tr~ni~ally “compact”. GABA (I mM) applied to any part of dendrites produced a depolarizing response. Greater amounts of GABA gave larger responses and dose-response relationships could be examined (Fig. IB) from responses obtained in any part of the dend~tic tree. Unlike CA1 pyramidal neurons,1*3*4 no hyperpolarizing responses of the dendrites to GABA were seen in any cell. We tried to reverse the GABA responses by direct injection of d.c. currents. This was SU~S~UI for depolarizing responses from the most proximal denThe

mean

(CS.D.).

-98

-8%

-78

-86

-58 mambrsne potential
Pig. 2. Reversal potential by direct currant injection. Peak depolarizing GABA responses from one cell obtained from the distal and the proximal dendrites. Current was injected intracellularIy to change the membrane potential in an attempt to obtain the reversal potential The holding potentiais are shown at the side. Note that the long-pulse duration for RN dete~nation was 10Oms for the distal responses and 75 ms for the proximal responses. Below, the site of the response is piot~ against the rn~br~e ~t~tial and the reversal potential obtained by extrapolation. A, Proximal responses; A, distal responses. k;, = - 86 mV.

drites which were near enough to the impalement site to be affected by the changing membrane potential. However, as we did not know the membrane potential at the site of the response, the “real” reversal OT eq~j~~b~urn potential is expected to be more negative than the one we observed. Figure 2 shows an attempt to reverse the GABA responses of the distal and proximal dendrites in one cell. The size of the response in the distal dendrites was relatively unaffected by the current injection. The proximal response appeared to partly invert at a holding

x74

T. J. BLAXER

A. DISTAL

and P. L. CARLEN

5. PROXIMAL

w:,

short pulse

long pulse

tong pulse

l

5 mV I 10s l

OABA applied to the dendrites

Fig. 3. Comparison of conductance changes in distal and proximal sites. GABA responses from one cell, Three responses of different sizes arc shown. These responses were produced by varying the duration of the GABA puke. (A) GABA responses from the distal dendrites. Short pulses were applied before and during flat-~ responses to GABA. Note that the chart recorder attemtated the size of the short-pulse response. Long &mating current pulses were applied during responses of similar sixe and show little apparent increase in conductance. (E) GABA responses of similar size obtained in the proximal dendrites are accompanied by a large apparent increase in conductance. V, = -86mV.

of -66 mV. The size of the response is plotted in Fig. 2B. The lines through the points are extrapolated to obtain the apparent reversal potential, which is clearly different for the two sites (see also Fig. 6B). The use of long pulses to measure changes in R, is illustrated in Fig. 3. Three pairs of GABA responses of different sizes were obtained from the distal and proximal dendrites. Long pulses were applied throughout. The decrease in RN accompanying the proximal responses are greater and are more pronounced with larger responses. There was a conductance increase of 50% with the 2 mV response (Fig. 3B), 75% with the 5 mV response and 90% with the 7.5 mV response. Also shown are GABA responses from the distal dendrites with the short-pulse measurements. Control short pulses were taken before the GABA was applied, and then again during the application. These are typical of the eat-top~d responses obtained for the use of short-pulse measurements. The effect of GABA on the short-pulse decay is shown in Fig. 4. The decays are averages of six sweeps. During analysis, each sweep was started a few milliseconds before the current pulse was applied in order to establish the baseline. In Fig. 4A, a current pulse of 10 nA, 0.5 ms was used. The decays became faster as the size of the GABA response increased. The same averages are shown on a semifog scale in Fig. 4B. The slope of the final part of the decay, which is a straight line, represents the slowest time constant, rO.The time constant became faster as the size of the GABA response was increased. of the We compared the r,, and R, measurements potential

a.

~~r*tutgl*wwv

fed

tuna.flW

end 0s L)l!an (ma)

Fig. 4. (A) The voltage response of a cell to a short current pulse of 1OnA lasting O.Sms. Each of the four traces is the average of 10 w-. ollc trace shows the control response and other three are the responses during diRerent sixed GABA responses (2,5 and 10 mV) in the mid-dendritess. (B) The decays of the voltage responses are shown on a semi-log scale. The slope of the final straight-line portion of the decay gives so. V, = -87 mV.

825

Dendritic conductance changes

% decrease

surements are shown in Table 1 and Fig. 6. Using Rf, measurements, the reversal potentials are more positive for the distal responses than for the proximal responses. For the proximal responses, RE,measurements gave reversal potentials close to the values obtained with 7;. The 7; method yields values close together as would be predicted from the results shown in Fig. 5, where responses of apparently similar size from distal, medial and proximal dendrites, measured at the soma, were accompanied by similar increases in conductances.

in resistance

DISCUSSION I

A distal

A

A

mid

proximal

Fig. 5. Resistance changes at three sites as measured by BE, and r; estimations. Decreases in resistance measured as per eqns (7) and (8) from all responses to GABA of 4-6 mV in all cells. 0, Short pulse; 0, long pulse. The differences between the measured decreases in resistance using r; and &., are significant (Wilcoxon signed rank test, paired data) at P < 0.005 for the distal responses (n = 14) and at P -c 0.05 for the mid-dendritic responses ‘(n = 14). The difference for the proximal responses is not significant (n = 7).

increases in conductance accompanying all 4-6 mV responses at each site. RE, measurements gave lower apparent increases in conductance than the r; measurements, especially at the distal dendritic site. The difference between R, and 7; estimations of GABAmediated resistance changes was smaller at the middendritic site, and at the most proximal site the percent decreases in resistance were the same. The resistance decreases measured by RE,are significantly different from each other (Mann-Whitney U-test, P < 0.05and from the values given by Wilcoxon signed rank test; see legend of Fig. 5 for values). The resistance decreases measured with R,, however, were not significantly different between sites (Mann-Whitney U-test). Using the data from the 7; method, we have estimated the reversal potentials of the proximal GABA response from the slope of eqn (6). The value found was 15 mV positive to resting potential. The results at the different sites with RE,and 7; mea-

According to cable theory, the membrane potential of the more remote parts of the cell will be affected less by the current and so the reversal potentials of the GABA responses appear to invert at potentials more positive than those where the observed membrane potential more closely reflects the actual membrane potential at the site of the response.5J6 RE, reflects perisomatic resistance changes near the electrode, and not of the whole system.6 If R(,estimations are used to measure the conductance to a given dendritic shunt, the apparent conductance increase becomes less as the shunt is made further from the site of current injection. The 7; method of estimating conductance changes reflects not only the local change of conductance, but the change throughout the cell, with little regard to the site of generation of this conductance. Combining RE, and 7; measurements, one can infer the localization of a conductance increase.6 Fox and Chan’O have recently used sine-wave impedance analysis to differentiate distal from somatic GABA responses in cultured chick spinal neurons. According to cable theory, a voltage response to GABA (accompanied by a local increase in conductance) in the distal dendrites decrements as it passes to the cell body. If the conductance change of the response was accurately measured by the 7; method, then responses of a given size measured at the soma should be accompanied by a greater increase in conductance if they were evoked at distal sites as compared to proximal sites. However, in these neurons, for the 7; method, the differences in the mea-

Table 1. Apparent reversal potential of the GABA responses at different sites for RE,and r; estimations Method

Site

Reversal potential (mV)

n

SE.

Correlation

RE, K-i BE,

distal mid proximal

23.1 19.9 16.8

41 35 22

1.3 1.0 1.0

0.893 0.913 0.933

ro’

distal mid proximal

14.6 15.3 14.8

41 35 22

0.7 0.8 1.3

0.916 0.916 0.866

70’

ro’

The slope of the relationship between response size and conductance increase equals the apparent reversal potential [eqns (6) and (9). and Fig. 61. All correlations are significant at P < 0.001.

826

T. J. BLAXTERand P. L. CARLEN

sured resistance decrease (or conductance increase) resulting from GABA application between sites were not significant. A large response at the distal end of the dendrite would have only a relatively small effect on the conductance of the whole cell despite having a large effect on the local conductance. Our data show that the same voltage response in the soma is associated with the same increase in conductance, as measured by ri, generated by focal GABA application anywhere in the dendritic tree (Fig. 5). The justification for using ~6 estimations of local dendritic conductance changes is strengthened by the fact that the estimated reversal potentials for GABA application at all three dendritic sites were very similar using T; estimations and eqn (9) (Table 1, Fig. 6). It is assumed that the E,, for GABA is identical anywhere in the dendritic tree. However, when RE, estimations were used for calculating E,,. E,, was more positive than those calculated from T; measurements, even for the most proximal dendritic GABA applications, and the more distal GABA applications had a more positive E,, than the proximal applications (Table 1, Fig. 6), showing an effect of the cable properties when using Rf, measurements. In eqn (4), the conductance change term, g, reflects a local dendritic response to the applied GABA. Since the input conductance of the proximal dendrites is higher than that of the distal dendrites, a given g will produce a larger response distally than proximally. According to previous modellingY6 the input conductance change of the whole system (GN), as measured from the soma, will be almost constant when estimated by ~6 from the same local shunt (g) regardless of its location on the dendritic cable model. The proximal dendritic input conductance is high relative to that of the distal dendrites, tending to shunt the proximally generated voltage response. A.

rwponre 16

f

aira

(mV)

6.

&a

re*ponas

(mV)

. . . \ . .

. 09 .

.

.

.

. \

.

. .

. .

. . . .

. . . .

. . ?

0.0

. ..

1

1

1

1

0.2

0.4

0.6

0.8

time

constant

ratio.

l \

t

1.0

_TO’ G.

C. 30

rs*pon*asize(mv)

1

I

I

r~*lstonce

1.0

0.6

0.0

of tlmo

conItu1t

ratto,

k,E ‘N

To

Fig. 6. Apparent reversal potential of responses &a&d fro& the three sites. (A) Time cons&& ratios pMcd.rygrrinst the size of GABA reqmma obtained at the t&e dcadritic sites in a sin#e cell: distal (0). mid (&) and praxkl (r). Regcessioa lines are drawn for each set of three p&s. The apparent reversal potentials relative to the resting potential (V,= - 73 mV) were distal, + 14 mV; mid, + 15.5 mV, and proximal, + 16.5mV. The reversal potcntia4 of all the responses (mean) was + 14.3 mV. (9) The revereel potential estimated from r; measurements during the GABA responses in distal dendrites. Each point reprcaentsone observation of response size and the accom$aqinI~ resistance decrease r,& . The lower the resistance ratio, the iaqer tbe decrease in resistance (or imxease in conductanoo). The regression line was cakulated using least-squares linear regression. See Table 1 for details of CorreIation Theslopeofthelin~whichisthenvmal is 14.6 f 0.7 mV (S.E.). (C) Regmssion lines for Rb/Jt, measured from long pulses (LP) and r;/rs measured from short pukes (SP) at the three dendcitic response sites. Individual data points arc not shown. Solid lines vt the regression equation (Table 1) within the scatter of the data points. The dashed line is the e&rapdation ba& to the ordinate. Tbe slopes and intercqts are the apparent reversal potentials of the GABA responses.

However, the large voltage response of the distal dendrites will be attenuated electronically before being measured in the cell body. In these neurons, the GABA responses from distal and proximal dendrites

Dendritic conductance changes

of the same size appeared to be accompanied by the same increase in conductance as measured by 7;. This may have important implications for neuronal integration since it implies that distal responses are just as effective at changing the somatic membrane potential for a given overall neuronal conductance change as proximal responses. The compensating mechanism giving rise to equivalent voltage and conductance changes of distal compared to proximal responses is dependent on the transmission ratio, which is defined as the ratio of the measured voltage at the end of the cable to the voltage measured at the soma from a constant cur-

rent injection at one or other end of the equivalent cable (discussed in Ref. 7). The transmission ratio is a function of L and the dendritic-somatic conductance ratio, p.’ The smaller the values of p, the larger the electrical load placed by the soma onto the dendrites. The mean L of these neurons was 1.22, less than that of motoneurons (1.50),’ thereby improving the transmission ratio. The local response size is also dependent on g and G, [eqn(4)]. However, both G, and the transmission ratio are ultimately determined

821

by the same factors: R, and the loading of one part of the cell by the rest of the cell. Graubard and Calvin” have modelled postsynaptic responses for various types of cell, from the Aplysiu L12 neuron (where the loading of the cell by the processes determines the transmission of a response to the soma) to a horizontal cell in the superior colliculus (where the long dendritic cables determine the transmission). In the present study, loading seems to be an important factor in determining the transmission of a response along the dendrites. This is especially true since our value of the electronic length of the whole cell, L, is probably an overestimate because the time constants we measured were affected by the presumably low somatic R,.8-9 Thus we suspect that responses are attenuated less in intact cells than in cells impaled by a microelectrode.

Acknowledgemen&--We thank the Medical Research Council and the Ontario Mental Health Foundation for support, Dr Dominique Durand and Aldo D’Aguanno for helpful comments, Rheinhart Schuller for help with statistics and Mary Cairoli for manuscript preparation.

REFERENCES

Alger B. E. and Nicoll R. A. (1982) Pharmacological evidence for two kinds of GABA receptors on rat hippocampal pyramidal cells studied in oiuo. J. Physiol., Lend. 328, 125141. Assaf S. Y., Crunelli V. and Kelly J. S. (1981) Electrophysiology of the rat dentate gyrus in oirro. In Electrophysiology of Isolated Mammalian CNS Preparations (eds Kerkut G. A. and Wheal H. V.), pp. 153-187. Academic Press, London. Blaxter T. J. and Cottrell G. A. (1985) Actions of GABA and ethylenediamine on CA1 pyramidal neurones of the rat hippocampus. Q. Jl exp. Physiol. 70, 75-93. Blaxter T. J., Carlen P. L., Davies M. F. and Kujtan P. W. (1986) Hyperpolarizing response to y-aminobutyric acid in rat hippocampal pyramidal cells is mediated by a calcium-dependent potassium conductance. J. Physiol., Lond. 373, 181-194. 5. Calvin W. H. (1969) Dendritic synapses and reversal potentials; theoretical implications of the view for the soma. Expl Neural. 24, 248-264. 6. Carlen P. L. and Durand D. (1981) Modelling the postsynaptic location and magnitude of tonic conductance changes resulting from neurotransmitters or drugs. Neuroscience 6, 839-846. I. Carlen P. L., McCrea D. A. and Durand D. (1984) Dendrites and motoneuronal integration. In Handbook of the Spinal Cord (ed. Davidoff R. A.), Vol. 2, pp. 243-267. Marcel Dekker, New York. 8. Durand D. (1984) The somatic shunt cable model for neurons. Biophys. J. 46, 645-653. 9. Durand D., Carlen P. L, Gurevich N., Ho A. and Kunov H. (1983) Electrotonic parameters of rat dentate granule cells measured using short current pulses and HRP staining. J. Neurophysiol. 50, 1080-1097.

10. Fox S. E. and Chan C. V. (1985) Location of membrane conductance changes by analysis of the input impedance of neurons. II Implementations. J. Neurophysiol. 54, 15941606. 11. Fricke R. A. and Prince D. A. (1984) Electrophysiology of dentate gyrus granule cells. J. Neurophysiol. 51, 195-209. 12. Ginsborg B. L. (1967) Ion movements in junctional transmission. Pharmac. Rev. 19, 289-316. 13. Graubard K. and Calvin W. H. (1979) Presynaptic dendrites: implications of spikeless synaptic transmission and dendritic geometry. In The Neurosciences Fourth Study Program (eds Schmitt F. 0. and Worden F. G.), Chapter 18, pp. 317-331. M.I.T. Press. 14. Hodgkin A. L. and Rushton W. A. H. (1946) The electrical constants of crustacean nerve fibre. Proc. R. Sot. B. 133, 444-479. 15. Hotson J. R., Prince D. A. and Schwartzkroin P. A. (1979) Anomalous inward rectification in hippocampal neurons. J. Neurophysiol. 42, 889-995.

16. Llinas R. and Nicholson C. (1976) Reversal properties of climbing fibre potential in cat Purkinje cells; an example of a distributed synapse. J. Neurophysiol. 39, 31 l-323. 17. Rall W. (1962) Theory of physiological properties of dendrites. Ann. N.Y. Acad. Sci. 96, 1071-1092. 18. Rail W. (1967) Distinguishing theoretical synaptic potentials computed for different soma-dendritic distributions of synaptic input. J. Neurophysiol. 30, 1138-l 168. 19. Rall W. (1969) Time constants and electrotonic length constants of membrane cylinders and neurons. Biophys. J. 9, 14831508.

20. Rall W. (1977) Core conductor theory and cable properties of neuron%. In Handbook of Physiology. The Nervous System, Section 1, Part 1, Chapter 3, pp. 39-97. American Physiological Society, Bethesda. (Accepted 13 July 1987)