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Electrotonic coupling between two CA3 hippocampal pyramidal neurons: a distributed cable model with somatic GAP-junction
Bulletin of Mathematical Biology, Vol.57, No. 6, pp. 865 881, 1995
ElsevierScienceInc. © 1995Societyfor MathematicalBiology Printedin GreatBritain.All rightsreserved 009~8240/95$9.50+0.00
Pergamon
0092-8240(95)00328-N
E L E C T R O T O N I C C O U P L I N G B E T W E E N T W O CA3 HIPPOCAMPAL PYRAMIDAL NEURONS: A DISTRIBUTED CABLE MODEL WITH SOMATIC GAP-JUNCTION* R. R. POZNANSKI, W. G. GIBSON and M. R. BENNETT The Neurobiology Laboratory, Department of Physiology and The School of Mathematics and Statistics, University of Sydney, NSW, 2006, Australia A model of a pair of electrotonically coupled CA3 h i p p o c a m p a l p y r a m i d a l n e u r o n s is presented. Each n e u r o n is represented by a tapered equivalent cable attached to a n isopotential soma. The synaptic potential in a n e u r o n s o m a is determined as a consequence of electrical coupling to a n o t h e r soma t h a t receives a synaptic input o n its dendritic tree. Estimates of the coupling resistances, soma i n p u t resistances a n d soma-to-dendritic tree c o n d u c t a n c e ratio show t h a t a substantial current m a y arise in a n e u r o n as a consequence of synaptic activity in a n e u r o n coupled to it. The small increase in decay time due to coupling in the model indicates that actual coupling is between m o r e t h a n just pairs of neurons.
1. Introduction. Dye-coupling between some CA3 pyramidal neurons in slices of the hippocampus obtained from adult mammals is indicated by staining of up to four adjacent neurons in a cluster after the injection of the dye Lucifer Yellow into a CA3 pyramidal neuron (MacVicar and Dudek, 1980; MacVicar et al., 1982). The proportion of neuronal coupling in the hippocampal subfields CA3, CA1 and dentate gyrus is in the range of 30% to 70% (Schuster, 1992). Simultaneous intracellular recording from some pairs of CA3 pyramidal neurons when synaptic transmission is blocked confirms electrical coupling (MacVicar and Dudek, 1981). This electrical coupling is probably mediated by the gap-junctions that have been observed between CA3 pyramidal neurons (Schmalbruch and Jahnsen, 1981). It has been hypothesized that this coupling mediates fast stimulation and graded synchronous activation of the neurons in a cluster, whereas the more plastic and adaptable chemical transmission mediates that between the clusters (Schuster, 1992). The electrical transients arising from current injection experiments as well as * Support under ARC Grant AC9031997 is acknowledged. Correspondence to: Professor Max Bennnett, Neurobiology Laboratory, University of Sydney, NSW 2006, Australia. 865
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voltage-clamp recordings in CA3 hippocampal pyramidal neurons have been developed by Brown et al. (1981), Johnston (1981), Johnston and Brown (1983), Turner and Schwartzkroin (1983), Durand (1984), Turner et al. (1992), Spruston and Johnston (1992) and Spruston et al. (1993) in order to estimate the passive membrane constants of these cells. However, in none of these models was electrical coupling between neurons taken into account. This is important, since theoretical results have shown that the effects of electrotonic coupling on the evaluation of electrotonic properties (with the dendrites not given a distributed representation) may produce erroneous conclusions as to the correct passive membrane properties (Getting, 1974; Rall, 1981; Skrzypek, 1984; Publicover, 1989). In the present work the coupling between two CA3 pyramidal neurons, each with dendritic trees, will be analysed in order to determine the effects of such coupling on the interpretation of synaptic potentials generated at different sites in the dendritic trees. In order to begin a cable analysis of this problem we need to have an appropriate model for the electrotgnic structure of a single CA3 pyramidal neuron. Models that lump an extensive dendritic tree into a reduced equivalent cable (Poznanski, 1988) provide a picture of this electrotonic structure that is not too complex and thus ideal for large neural network simulations. This is especially useful in gaining an understanding of synaptic inputs located close to the soma (as in the case of mossy fibre inputs onto CA3 pyramidal neurons). It will be shown that synaptic transmission in the CA3 pyramidal neurons at the quantal level is complicated when electrical coupling is present, as potentials recorded in one neuron may actually arise due to transmission in an adjacent neuron. This implies that electrotonic coupling may have an important role in synaptic integration at the cellular level within the CA3 hippocampus (Berry and Pentreath, 1977). 2. Mathematical Formulation.
Let
V(Z, T) and U(Z, T) be the membrane
(electrotonic) potentials along the first and second neurons, respectively. The voltage response V(Z, T) to synaptic c u r r e n t Isyn(Z , T ) generated at an arbitrary point along an initially quiescent neuron 1 satisfies the modified cable equation (Poznanski, 1988, equation 3.1a) Vzz--K 1Vz -
V-~l'i,taper/~taperIsyn(T)(~(Z-~
-
~)~--- VT,
--L 1
(1)
where Z and T are, respectively, dimensionless position and time variables, )ttaper = [(Rm/4Ri)Dtape,] 1/2 is the generalized characteristic length parameter, 2 ri, t,per=4Ri/rcDt,per is the core resistance per unit length of the tapering equivalent cable, R m is the membrane resistance, Ri is the resistivity of the intracellular material, Dt,per = D exp(2KxlZ]/3) is the diameter of the tapering equivalent cable, D is the diameter ofa nontapering equivalent cable, Isyn is the
ELECTROTONICALLY COUPLED CA3 HIPPOCAMPAL PYRAMIDAL NEURONS
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synaptic current, K 1 is a negative constant which determines the amount of exponential taper and 6 ( Z + 4) is the Dirac delta-function concentrated at the point Z = - ~ (where ~ is a positive number). The voltage response along an initially quiescent neuron 2 satisfies U z z + K z U z - U= UT,
0
(2)
where K 2 is a negative constant which determines the amount of exponential taper present in neuron 2. At the site of the junction ( Z = 0) two boundary conditions are needed. By applying Kirchhoff's law of conservation of current to the circuit of Fig. 1, we find
(3) and (4)
Ij = It(Z=O + )+ Ia(Z=O +),
where I a is the axial current, I t is the transverse current and Ij is the junctional current. The axial current is given by Ohm's law (with the assumption that the axial current is positive in the increasing Z direction): 1 ri~ Vz(0, T)
I,(Z=0-)=
(5)
U8
la Ri
I Neuron 1
I
I
Z=O"
Z=L 2 Neuron 2
Figure 1. A schematic illustration of a pair of tapered equivalent cables representing the dendritic arborizations of CA3 hippocampal pyramidal neurons coupled at the soma (Z = 0) by a junctional resistance (Rj). (Z is the electrotonic distance variable.) The presence of a gap-junction between the somata of both neurons requires us to distinguish to the left Z = 0 - and to the right Z = 0 + of the point Z = 0. It is assumed that synaptic input is activated at an arbitrary point along neuron 1 and the potential response is recorded at the cell body of neuron 2 (see text for meaning of symbols).
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and 1
I n ( Z = 0 + ) = - ,¢r~ ~ Uz(0, T),
(6)
where ri=4Ri/~zD 2 is the core resistance per unit length of a nontapering equivalent cable and 2 = (RmD/4Ri) ~/2 is the characteristic length parameter of a nontapering equivalent cable. The transverse current or the total outward somatic m e m b r a n e current is the sum of the resistive and capacitative currents given by
It(Z = 0 - ) = V(O, T)/R~ + (CJzm) VT(0, T)
(7)
I t ( Z = 0 + ) = U(0, T)/Rs+(CJ'Vm)UT(O, r ) ,
(8)
and
where R~ is the resistance of the soma, Cs is the capacitance of the soma and Zm= RmCm, where Cm is the m e m b r a n e capacitance, is the m e m b r a n e time constant. The voltage at the gap-junction is discontinuous by the a m o u n t IjRj (representing the drop in voltage across the junctional resistance), where Ij is the axial current through Rj given by Ij = [ V(O, T ) - U(O, T)]/Rj.
(9)
Substitution of (5) to (9) into (3) and (4) yields the required set of coupled b o u n d a r y conditions:
[I+(RJRj)]V(O, r)+yVz(O , T)+aVT(O , T)=(RJRj)U(O, T)
(10)
[I+(RJRj)]U(O, T)--TUz(0 , T)+trUT(0 , T)=(Rs/Rj)V(O , T),
(11)
and
where 7 = Rs/ri). is the dendritic to somatic conductance ratio for a semi-infinite cable (the notation p o~is also used; Jack et al., 1975) and o-= -cs/-cm is the somatic shunt parameter where zs = RsCs is the somatic time constant. If we assume no longitudinal (axial) current flow at either end of neurons 1 and 2 (i.e. sealed-end terminations) then the b o u n d a r y conditions at Z = - L 1 and Z = L 2 a r e
V z ( Z = - L 1, T ) = U z ( Z = L 2 , T ) = 0 .
(12)
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3. Solution for the Potentials. In this section we use Green's function methods to determine a partial analytic solution to the initial-boundary value problems derived in the preceding section, with particular emphasis on obtaining the coupling potential (i.e. the potential at the soma of neuron 2) denoted by Us. The full solution of (2) (at the soma) is the inverse Laplace transform of
uL(0, s)= (Rs/gj)VL(0, s)6 (0, s + 1 + Ji q )2 ,
(13)
where s is the transform variable, UL is the Laplace transform of U, VL is the Laplace transform of V and G~ is the Laplace transform of Green's function given by G*(0, s)= x/~ cosh~/sL2-(K2/Z)sinhx/~L2 /{[(~ - (K2/2 )a)s-- (KE/2 )e2]sinhx/~L 2 + x/~[e2 + as--(K2/2)7]coshx/~L2},
(14)
with ea -- [1 + (Rs/Rj) + 7(Ka/2 ) -- a(1 + K22/4)]. Green's function (14) is derived in Appendix 1. The Laplace transform of the potential in neuron 1 at the stoma can be expressed as VL(0, S)= IL(S)H*(O, -- 4, S+ 1 + ¼K~),
(15)
where IL(S)= Qocca/zm(S + c¢)2, Qo is the charge applied instantaneously, a is a constant parameter and HL*(0, --4, s) is Green's function HL*(0, --4, s)=ri27 exp(--1Kl~){[x/~ cosh~/~L1-(K~/2)sinhx/~sL1]coshw/ss~ --[x/~ sinhx/~L1-(K1/2)cosh~sL1]sinhx/Tss~} /{Tx/~[x/~ sinhx/~L1-(K1/2)coshx/~sL ~] +[x/~ coshxSssLa -- (K1/2 )sinhx/FsL1] 1 2 1 2 • [e x + as-- (RJRj)ZG*(O, s + xK~ ---~K[ )]}, (16)
where ex = 1 + Rs/R j + 7(K1/2 ) - a ( 1 +K~/4). Green's function defined by (16) is derived in Appendix 2. We check (16) against the previously known limiting case of R j ~ and Ka = 0 (Poznanski, 1987). We first note that R j ~ corresponds to a single neuron problem with the potential in the inactive neuron 2 being equal to zero, as shown by letting R j ~ ~ in (13). The case of K~ = 0 yields the well-known
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somatic-shunt cable model (Durand, 1984). By letting R j ~ o o and Kt = 0 in (16) and noting that el = 1 - a , we arrive at H * (0, - 4, s, Rj ~ ~ , K 1 = 0)/ri2
=7 coshx/Fss(~-L1)/{Tx/rss sinhx/~L1 + ( 1 - ~ + ~s)coshx//sLx}.
(17)
Apart for notational differences (L 1 for L, and Z for X), (17) corresponds to (4) with I"= 0 in Poznanski (1987). 4. Parameter Values. The parameter values used in this paper are chosen from two independent studies of CA3 pyramidal cells, one performed by Brown et al. (1981) and the other by Johnston (1981). A table summarizing these electrical parameters is given in Turner and Schwartzkroin (1984). An additional parameter appearing in that table is p, the dendritic-to-somatic conductance ratio, which is related to ~ by Jack et al. (1975): 7 = p coth(L),
(18)
where p is in the range 0 . 8 < p < 2 . 7 and the electrotonic length (L) is in the range 0.79 < L < 1.03. Note that the electrotonic length parameter derived by these authors was based on a formula for a uniform equivalent cylinder (Rall, 1969) and therefore it is expected to be an underestimate of the electrotonic length parameter for a tapering equivalent cable as shown by Poznanski (1988) and Turner and Deupree (1991). In fact, a recent study based on the tapered model indicates that L = 1.65 may be a better estimate (Poznanski, 1994). The p measured under the assumption of equal somatic and dendritic resistivity (i.e. a -- 1) shall be denoted by the subscript "ns" in order to differentiate the value of p when a ¢ 1. Thus p = apn ~ (Durand, 1984) and (18) becomes ~ = p,~a coth(L), leaving 7 in the range 1.0<~<4.1 by choosing Pns=3, and a = 1 and 0.25. A numerical estimate of the resistance of the soma (R~) can be obtained from (Durand, 1984): R s = R N ( I +p),
(19)
where R~ is the input resistance. From MacVicar and Dudek (1981) (see also Dudek et al., 1983) it follows that the coupling ratio CR is given by CR = RN/(R i + Rj).
(20)
Then the ratio R = R s / R j is given by R = (1 + P ) C R / ( 1 - CR). Clusters of CA3 pyramidal cells show electrotonic coupling ratios less than 0.3 (MacViear and Dudek, 1981); thus an estimate of the maximum value of R is about 1.5 (p ~ 2.7, CRY0.3 ). For graphical illustrations of the results, we have used the values R=0.5, 1 and 1.5.
E L E C T R O T O N I C A L L Y C O U P L E D CA3 H I P P O C A M P A L P Y R A M I D A L N E U R O N S
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Most CA3 hippocampal pyramidal neurons show a combined dendritic trunk parameter that is well approximated by a exponential taper (Turner and Schwartzkroin, 1983, 1984; Poznanski, 1988, 1994):
r£
3/2 j=l
d./2
-~ exp(KZ),
(21)
Lj = 1
where K is a parameter which determines the hmount of taper (K< 0) present, dj is the diameter of thejth branch element at a given electrotonic distance (Z) from the soma ( Z = 0), n o is the number of primary trunks emanating from the soma at Z = 0 and n i is the number of branch elements at any given electrotonic distance (Z) from the soma (Z = 0). It is emphasized that (21) is most frequently illustrated in the experimental literature (see Poznanski, 1994, Fig. 1). 5. Results.
F r o m (15) the potential at the soma of neuron 1 is
gs=(Qoa2/~m)~- l {H~(O, - ~ , s + l + ¼K2)/(s + a)2},
(22)
where ~ - 1 represents the inverse Laplace transform operation. Substituting (15) into (13) gives the coupling potential at the soma of neuron 2: 1 2 Us= (Rs/Rj) (Qoa2/.Cm). ~o- l'¢H*t0t L, , -- 4, S + 1 + zK, )
G*(O, s+l+¼K~)/(s+a)2}.
(23)
The inverse transforms in (22) and (23) can be evaluated numerically using a computer software package. The time course of the normalized potential V~ at the soma of the active neuron (1) (i.e. at Z = 0 - ) , in response to a fast synaptic input is shown in Fig. 2. The parameter values are given in the caption. The synaptic input is governed by the a-function a2Te-'T which for a = 50 peaks at T = 0.02 and is essentially zero for T>0.2. The somatic potential has a much slower time course, dependent on the membrane properties. The largest response occurs in (a) where the activation is at a proximal distance (~ = 0.1) and the somatic shunt resistance is largest (o-= 1); the smallest occurs in (d) where the activation is at a distal distance (¢=0.8) and the somatic shunt resistance is smallest (o-= 0.25). The peak amplitude of Vs decreases irrespective of the input site or junctional resistance value if the somatic shunt parameter a is made smaller (by increasing the somatic shunt), and also the voltage decay becomes more rapid. These observations appear to be compatible with the results obtained by Durand (1984) and Poznanski (1987) for the single neuron problem. Figure 3 shows the time course of the normalized potential U~at the soma of the passive neuron (2) (i.e. at Z = 0 +) for the same four cases as the corresponding graphs in Fig. 2. As expected, these potentials peak at later
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FAST SYNAPTIC INPUTS (~=50)
[ R=O.O
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COUPLING STRENGTHS | R=0.5 - ' t R=1.5 ---
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(b) proximal activation 2.0
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SCALED TIME, T
Figure 2. Time course of the normalized potential Vs at the soma of neuron 1 in response to a fast synaptic input (alpha function with c~= 50) activated at distance from its soma. Neuron 1 is coupled to neuron 2 by a junctional resistance Rj. Parameter values for both neurons are electrotonic length L~ =L 2 =0.9, taper parameter K~=K2=--4.75, dendritic-to-somatic conductance ratio y=3~r coth(L) = 4.188a, where a is the ratio of the somatic and membrane time constants. In each graph, the response is given for three values of the coupling strength R=Rs/Rj, where Rs is the somatic shunt resistance for each neuron: R=0, corresponding to neurons 1 and 2 being uncoupled (solid line), R=0.5 (dot~lash line) and R = 1.5 (dashed line). (a), ~ = 0.1 (proximal activation), a = 1; (b), ~ = 0.1, cr=0.25; (c), 4=0.8 (distal activation), a = 1.0; (d), 4=0.8, a=0.25. times than those in the active n e u r o n , with delay being m o s t p r o n o u n c e d for large o- (graphs (a) and (c)). Figures 4 and 5 give the c o r r e s p o n d i n g results for a slower synaptic input (c~= 10) which peaks at T - - 0 . 1 and is effectively zero for T > 0 . 8 . The overall effect of this slower input is to increase the delay in the response in b o t h the active and the passive neurons. Further characterization of the responses for all of the a b o v e cases is given in Table 1; a n o t h e r parameter n o t listed there is the time constant for decay w h i c h is 1.0 w h e n o--- 1 and 0.39 w h e n o-= 0.25 (for b o t h V~ and Us). Figure 6 s h o w s the d e p e n d e n c e of the times to peak and the peak potentials o n o- and R. As R increases, the time to peak for Vs s h o w s little change, but that
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FAST SYNAPTIC INPUTS (~=50) [ R=1.5 - - COUPLING STRENGTHS | R=I.0 - -
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Figure 3. Time course of the normalized potential Us at the soma of neuron 2 in response to a fast synaptic input (alpha function with c~--50) activated at distance from the soma of neuron 1. All the parameter values are the same as for the corresponding graphs in Fig. 2, except that the three values for R = Rs/R j are now R=0.5 (do%dash line), R= t.0 (dotted line) and R= 1.5 (dashed line). of Us increases markedly (Fig. 6(a)). Similarly, the peak of V~ shows only a small decrease (a similar conclusion was reached by Hall (1981), Fig. 7a, for a "lumped parameter" model) whereas the peak of Us undergoes a significant increase (Fig. 6(b)). Figure 6(c) shows that increasing the somatic shunt parameter o- has only a small effect on the time to peak of Vs, but causes a big increase in that of Us. Conversely, increasing o- from very low values causes a large increase in the peak value of Vs (Fig. 6(d)), but very little change in the peak of Us. (Though note that the behaviour is not monotonic; the peak value of Us increases for small o-, peaks at about o-= 0.28 and then decreases steadily.) Since the value of the electrotonic length may be greater than L = 0.9 (Poznanski, 1994) some calculations were repeated for larger L values. This caused little change in the rise and fall times of the potentials, but did lead to some reduction in peak values. F o r example, for L~ = L 2 = 1.8, R = 1.0, o-= 1.0 and the remaining parameters as for Fig. 2, there was a 23% reduction in the peak amplitude of V~ and a 12% reduction in the peak amplitude of Us.
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Figure 4. Time course of the normalized potential V~at the soma of neuron 1 in response to a slow synaptic input (alpha function with c~= 10) activated at distance from its soma. The remaining parameter values are the same as for the corresponding graphs in Fig. 2. 6. Discussion. As has been emphasized by Schuster (1992), the quantitative analysis of gap junctions is an essential component of future brain research. The present work has dealt with a small subset of this problem, in giving a theoretical analysis of the effect of electrotonic coupling of two cells in the context of a model appropriate for the CA3 pyramidal cell. Although the precise values of the coupling strengths and other parameters are not yet known, varying these over reasonable ranges indicates that the peak response in the passive cell could be as high as 40% of that in the activated cell (Table 1). However, this would be an extreme case. Since evidence from dye coupling and morphological studies indicates that neurons are coupled in clusters it would be expected that the potential would spread over several passive cells, with a consequent lowering of the response in each cell. Evidence for more extensive coupling also comes from a consideration of decay times. Fu et al. (1992) have shown that uncoupling hippocampal neurons with ethanol leads to about a 30% increase in the final time constant from the voltage transient, but not much change in the other potential parameters in the active
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Figure 5. Time course of the normalized potential Vs at the soma of neuron 2 in response to a slow synaptic input (alpha function with c~= 10) activated at distance from the soma of neuron 1. The remaining parameter values are the same as for the corresponding graphs in Fig. 3. cell. The results of the present study indicate that as the coupling is reduced to zero there is almost no change in the time to peak (Fig. 6(a); Table 1), a small increase in the peak (Fig. 6(b); Table 1) and a larger increase in the decay time (Fig. 6(c); Table 1). However, this latter increase is still well below 30% (maximum of a b o u t 15%), suggesting that hippocampal neurons may be coupled in a two- or three-dimensional array in order for uncoupling to change the final time constant by such a large amount. (For an analogous problem of current flow in a three-dimensional syncytium, see Bennett et al. (1993); there, it is found that the time course of the membrane potential following simulated transmitter release is much faster than that expected from the value of the membrane time constant.) Electrotonic coupling will have an effect on passive cable parameter estimates of single CA3 hippocampal neurons and this will eventually have to be taken into account. The most recent cable analysis of CA3 pyramidal neurons (Major et al., 1994) does not take this into consideration. Given that for the 20-day old rats used in the study there is evidence of high coupling
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Table 1. Shape indices for the potential responses in neurons 1 and 2. The values used for the parameters L1, L2, K1, K 2 and y are the same as for Fig. 2. T~,°"k is the time to peak, Vpe"k is the peak potential value, Hv is the half width (i.e. the time that the potential is above half its peak value) and Tvh"lf is the half decay time (i.e. the time taken for the potential to decay from its peak value to half that value). These are for the potential V~at the soma of neuron 1; the final four columns are the corresponding quantities for the potential Us at the soma of neuron 2
0~
~
0"
R
50
0.1
0.25
0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5
1.0 0.8
0.25 1.0
10
0.1
0.25 1.0
0.8
0.25 1.0
~vTpeak Vpeak 0.09 0.09 0.08 0.11 0.11 0.11 0.23 0.22 0.22 0.28 0.27 0.27 0.28 0.27 0.27 0.38 0.36 0.35 0.41 0.40 0.40 0.51 0.50 0.49
1.71 1.62 1.54 2.01 1.97 1.94 1.18 1.09 1.02 1.56 1.50 1.46 1.04 0.95 0.88 1.53 1.45 1.39 0.87 0.79 0.73 1.32 t.25 1.19
By 0.28 0.25 0.23 0.65 0.57 0.52 0.40 0.37 0.36 0.79 0.71 0.66 0.54 0.52 0.52 0.95 0.88 0.83 0.57 0.56 0.56 0.98 0.91 0.87
Tvhalf T~¢ak UPe,k 0.22 0.19 0.18 0.57 0.49 0.44 0.28 0.26 0.25 0.63 0.56 0.51 0.36 0.35 0.35 0.70 0.64 0.60 0.37 0.36 0.36 0.71 0.64 0.61
0.31 0.24 0 . 2 6 0.37 0.22 0 . 4 6 0.83 0.18 0.71 0.29 0.62 0.38 0.44 0.20 0.39 0.31 0.37 0 . 3 8 0.95 0.16 0.83 0.26 0.75 0.34 0.51 0.21 0.46 0.31 0.43 0.38 1.01 0.17 0 . 9 0 0.29 0 . 8 2 0.37 0.64 0.18 0.59 0.27 0.56 0.32 1.14 0.15 1.02 0.25 0.95 0.33
Hu
T~alf
0.72 0.61 0.55 1.99 1.74 1.58 0.75 0.65 0.60 1.99 1.74 1.58 0.82 0.74 0.70 2.01 1.77 1.63 0.84 0.77 0.73 2.03 1.79 1.63
0.50 0.43 0.40 1.37 1.21 1.12 0.5t 0.45 0.41 1.37 1.21 1.11 0.53 0.48 0.45 1.37 1.21 1.12 0.53 0.48 0.46 1.37 1.22 1.11
(Reece a n d S c h w a r t z k r o i n , 1991; Schuster, 1992) it w o u l d be advisable to c h e c k if the n e u r o n s used in a cable analysis are indeed c o u p l e d (by, for example, dye injection). T h e present c a l c u l a t i o n c o u l d be e x t e n d e d in a n u m b e r of ways. A n o b v i o u s one is the extension to d e n d r o - d e n d r i t i c c o u p l i n g , w h i c h has been o b s e r v e d e x p e r i m e n t a l l y a n d in fact m a y be the principal site of g a p j u n c t i o n s b e t w e e n CA3 p y r a m i d a l cells ( M c V i c a r a n d D u d e k , 1980; Schuster, 1992). Such c o n n e c t i o n s b e t w e e n the ends of each dendritic cable c o u l d be i n c o r p o r a t e d into the present m o d e l b y replacing (12) b y the following set of c o u p l e d b o u n d a r y conditions:
Vz(--L1, T)= (ri, taper/~taper/Rc)[ V ( - L1, T ) - U(L2, T)3
(24)
ELECTROTONICALLY COUPLED CA3 HIPPOCAMPAL PYRAMIDAL NEURONS (a)
~=1
(b)
877
(y=l
1.0 MS ,z
0.8
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LU O-
o
I-W
zLU 1.0
Us
0.6
}--
on
0.4
Vs
<
~a
F-
LU
0.5
Us
0.2
0.0
0.0 015
0.0
110
115
o.o
210
,'.5 COUPUNG STRENGTH, R
COUPLING STRENGTH, R (c)
G=I
R=I
(d)
1.5
1.0
W
>. 1.0
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O W O
Vs
......
< uJ
0.8
w
0.4
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0.0
015
110
115
o.o
210
11o
(e)
(f)
R=I
R=I
1.5
1,5 W
J
_<
I--
1.0
0.5
0.0
I
IJ
COUPLING STRENGTH, R
< u.I o_
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/
0.2 0.0
I.-Z LU
Vs
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b o.5
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Us
>< O W D
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Figure 6. Dependence of the time to peak, the peak potential and the half decay time on R and a. The values for the parameters LI, Lz, K 1 , K 2 and y are the same as for Fig. 2. A fast synaptic input ( , = 50) at a distal distance ( 4 = 0 . 8 ) is used. In each graph the quantity pertaining to V~ (the potential at the soma of neuron 1) is given by the sold line and that pertaining to Us (the potential at the soma of neuron 2) is given by the dashed line. (a) The times to peak as functions of R for a = !.0; (b) the peak potentials as functions of R for a = 1; (c) the half decay times as functions of R for ~r= 1 ; (d) the times to peak as functions of a for R = 1.0; (e) the peak potentials as functions of ~ for R = 1; (f) the half decay times as functions of ~r for R = 1.
878
R.R. POZNANSKIet al.
and U z ( L 2 , T ) = (ri, taper~taper/Re) [-V(--L1, T ) -
U ( L 2 , T)],
(25)
where R c represents the coupling resistance between the terminal ends of both dentritic cables. This case m a y still be amenable to a semi-analytic treatment. Further extensions would require a numerical approach from the outset. These include the use of action potentials which would allow, for example, the investigation of bursting behaviour (Traub, 1982; T r a u b et al., 1991) and also the extension of the model to networks of cells, coupled both chemically and electrotonically (Traub and Wong, 1983; T r a u b et al., 1985; D u d e k and Traub, 1989). The implications of such coupling for current theories of autoassociative m e m o r y storage in CA3 (Treves and Rolls, 1994; Bennett et al., 1994) needs to be studied in the context of a network model.
APPENDIX
1
The solution of (2) can be written in terms of the Green's function G(Z, - ~, T) by the convolution integral Laplace Transform of the Green's Function for the Inactive Neuron 2.
U(Z, T ) = ( R J R j )
V(O, t7)G(Z, - ~ , T-tT) dtT,
(A1)
where the Green's function G*=G exp[-½KzZ+(I+¼K~)T] is the solution of the initialboundary value problem G T*-- Gzz*
(00)
(A2)
[1 + (RJRj) + 7(K2/2) -- a -- a(K2/4)]G* (0, T) -- 7G* (0, T) + aG *(0, T) = 6(T)exp[(1 + ¼K22)T] (A3) G*(L2, T ) - (K2/2)G*(L2, T)=0
(A4)
G*(Z, 0)=0.
(A5)
Define G*(Z, s) to be the Laplace transform of G*(Z, T) and therefore the subsidiary equation corresponding to (A2) and (A5) is -- d2Gt/dZ 2 + sG t = 0
(A6)
(e2+ ~rs)G*(O, s ) - y dG*(0, s)/dZ= 1
(A7)
dG*(L2, s)/dZ-(K2/2)G*(L2, s)= 0,
(A8)
to be solved with
and
where ~2= [1 + (Rs/Rj)+Y(K2/2)-a(1 +/£22/4)] and s is the transform variable. The solution in
ELECTROTONICALLY COUPLED CA3 HIPPOCAMPAL PYRAMIDAL NEURONS
879
the Laplace transform space of these equations corresponds to the required Green's function defined by (14) at Z = 0 .
APPENDIX
2
Laplace Transform of the Green's Function for the Active Neuron 1. The solution of (1) can be written in terms of the Green's function H(Z, - 4 , T): V(Z, T ) =
I~y.(q)g(z, --4; T-rl) dq,
(B1)
do where the time course of the synaptic current is given by the standard alpha function (Jack et al., 1975) Isy. (T) = (Qo/zm)Ctz T e x p ( - aT) and the Green's function H * = H e x p [ - ½ K 1 Z + ( I + I K ~ ) T ] boundary value problem
(B2) is the solution of the initial-
H x*--Hzz* (--L1 < Z < 0 , - L 1 < ~<0, T > 0 )
elH*(0, T)+TH*(0, T)+aH*(O, T ) =
H*(O, q)e<~+K?/4)~r-n)G(O,T--q) dq
(B3)
(B4)
H~(--L1, T)+(K1/2)H*(--LI, T ) = 0
(B5)
H*(Z, 0) = ri. ,..or'~,..or exp(-½K, a)a(Z+ ¢),
(B6)
where e~ = 1 + (Rs/Rj) + y ( K , / 2 ) - a(1 + K~/4). Define HL* to be the Laplace transform of H*, and therefore the subsidiary equation corresponding to (B3) and (B6) is
-- d i H * / d Z 2 + sH* = ri2 e x p ( - ½K1{)6(Z + 4)
(B7)
[e1 + as -- (Rs/Rj)2G*]H* (0, s) + ~ dH* (0, s)/dZ = 0
(~8)
dH* ( - L 1 , s)/dZ + (K1/2)H* ( - L 1 , s) = 0.
(B9)
to be solved with
and
The solution in the Laplace transform space of these equations corresponds to the required Green's function defined by (16) at Z = 0.
LITERATURE Bennett, M. R., W. G. Gibson and R. R. Poznanski. 1993. Extracellular current flow and potential during quantal transmission from varicosities in a smooth muscle syncytium. Phil. Trans. Roy. Soc. Lond. B342, 89-99.
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Bennett, M. R., W. G. Gibson and J. Robinson. 1994. Dynamics of the CA3 pyramidal neuron autoassociative memory network of the hippocampus. Phil. Trans. Roy. Soc. Lond. B343, 167-187. Berry, M. S. and V. W. Pentreath. 1977. The integrative properties of electrotonic synapses. Comp. Biochem. Physiol. 57A, 289-295. Brown, T. H., R. A. Fricke and D. H. Perkel. 1981. Passive electrical constants in three classes of hippocampal neurons. J. Neurophysiol. 46, 812-827. Dudek, F. E. and R. D. Traub. 1989. Local synaptic and electrical interactions in hippocampus: experimental data and computer simulations. In Neural Models of Plasticity, J. H. Byrne and W. O. Berry (Eds), pp. 378-402. San Diego: Academic Press. Dudek, F. E., R. D. Andrew, B. A. MacVicar, R. W. Snow and C. P. Taylor. 1983. Recent evidence for and possible significance of gap junctions and electrotonic synapses in the mammalian brain. In Basic Mechanisms of Neuronal Hyperexcitability, H. H. Jasper and N. M. van Gelder (Eds), pp. 31-73. Liss: New York. Durand, D. 1984. The somatic shunt cable model for neurons. Biophys. J. 46, 645-653. Fu, P., B. L. Bardakjian and P. L. Carlen. 1992. Ethanol uncouples dentate granule neurons by increasing junctional resistance: a multineuronal system model approach. Neuroscience 51, 47-54. Getting, P. A. 1974. Modification of neuron properties by electrotonic synapses. I. Input resistance time constant, and integration. J. Neurophysiol. 37, 846-857. Jack, J. J. B., D. Noble and R. W. Tsien. 1975. Electric Current Flow in Excitable Cells. Oxford: Clarendon Press. Johnston, D. 1981. Passive cable properties of hippocampal CA3 pyramidal neurons. Cell. Mol. Neurobiol. 1, 41-55. Johnston, D. and T. H. Brown. 1983. Interpretation of voltage clamp measurements in hippocampal neurons. J. Neurophysiol. 50, 464-486. MacVicar, B. A. and F. E. Dudek. 1980. Local synaptic circuits in rat hippocampus: interactions between pyramidal cells. Brain Res. 184, 220-223. MacVicar, B. A. and F. E. Dudek. 1980. Dye-coupling between CA3 pyramidal cells in slices of rat hippocampus. Brain Res. 196, 494-497. MacVicar, B. A. and F. E. Dudek. 1981. Electrotonic coupling between pyramidal cells: a direct demonstration in rat hippocampal slices. Science 213, 782-785. MacVicar, B. A., N. Ropert and K. Krnjevic. 1982. Dye-coupling between pyramidal cells of rat hippocampus in vivo. Brain Res. 238, 239-244. Major, G., A. U. Larkman, P. Jonas, B. Sakman and J. J. B. Jack. 1994. Detailed passive cable models of whole-cell recorded CA3 pyramidal neurons in rat hippocampal slices. J. Neurosci. 14, 4613-4638. Poznanski, R. R. 1987. Techniques for obtaining analytical solutions for the somatic shunt cable model. Math. Biosci. 85, 13-35. Poznanski, R. R. 1988. Membrane voltage changes in passive dendritic trees: a tapering equivalent cylinder model. I M A J. Math. Appl. M e d . & Biol. 5, 113-145. Poznanski, R. R. 1994. Electrotonic length estimates of CA3 hippocampal pyramidal neurons. Neurosci. Res. Commun. 14, 93-100. Publicover, N. G. 1989. Mathematical models of intercellular communication. In Cell Interactions and Gap Junctions, N. Sperelakis and W. C. Cole (Eds), Vol. 1, pp. 183-201. Boca Raton: CRC Press. Rail, W. 1969. Time constants and electrotonic length of membrane cylinders and neurons. Biophys. J. 9, 1483-1508. Rail, W. 1981. Functional aspects of neuronal geometry. In Neurones Without Impulses, A. Roberts and B. M. Bush (Eds). Cambridge: Cambridge University Press. Reece, L. J. and P. A. Schwartzkroin. 1991. Effect oS cholinergic agonists on immature rat hippocampal neurons. Develop. Brain Res. 60, 29-42. Schmalbruch, H. and H. Jahnsen. 1981. Gap-junctions on CA3 pyramidal cells of guinea pig hippocampus shown by freeze-fracture. Brain Res. 217, 175-178.
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Schuster, T. H. 1992. Gap junctions and electrotonic coupling between hippocampal neurons: recent evidence and possible significance--a review. Concepts Neurosci. 3, 135-155. Skrzypek, J. 1984. Electrical coupling between horizontal cell bodies in the tiger salamander retina. Vis. Research 24, 701-711. Spruston, N. and D. Johnston. 1992. Perforated patch-clamp analysis of the passive membrane properties of three classes of hippocampal neurons. J. Neurophysiol. 67, 508-529. Spruston, N., D. B. Jaffe, S. H. Williams and D. Johnston. 1993. Voltage- and space-clamp errors associated with the measurement of electrotonically remote synaptic events. J. Neurophysiol. 70, 781-802. Teranishi, T., K. Negishi and S. Kato. 1983. Dopamine modulates S-potential amplitude and dye-coupling between external horizontal cells in carp retina. Nature 301,243-246. Traub, R. D. 1982. Simulation of intrinsic bursting in CA3 hippocampal neurons. Neuroscience 7, 1233-1242. Traub, R. D., F. E. Dudek. C. P. Taylor and W. D. Knowles. 1985. Simulation of hippocampal afterdischarges synchronized by electrical interactions. Neuroscience 14, 1033-1038. Traub, R. D. and R. Miles. 1991. Neuronal Networks of the Hippocampus. New York: Cambridge University Press. Traub, R. D. and R. K. S. Wong. 1983. Synaptic mechanisms underlying interictal spike initiation in a hippocampal network. Neurology 33, 257-266. Traub, R. D., R. K. S. Wong, R. Miles and H. Michelson. 1991. A model of a CA3 hippocampal pyramidal neuron incorporating voltage-clamp data on instrinsic conductances. J. Neurophysiol. 66, 635-650. Treves, A. and E. T. Rolls. 1994. Computational analysis of the role of the hippocampus in memory. Hippocampus 4, 374-391. Turner, D. A. 1984. Segmental cable evaluation of somatic transients in hippocampal neurons (CA1, CA3, and dentate). Biophys. J. 46, 73-84. Turner, D. A. and P. A. Schwartzkroin. 1983. Electrical characteristics of dendrites and dendritic spines in intracellularly stained CA3 and dentate hippocampal neurons. J. Neurosei. 3, 2381-2394. Turner, D. A. and P. A. Schwartzkroin. 1984. Passive electrotonic structure and dendritic properties of hippocampal neurons. In Brain Slices, R. Dingledine (Ed), pp. 25-50. New York: Plenum Press. Turner, D. A. and D. L. Deupree. 1991. Functional elongation of CA1 hippocampal neurons with aging in Fischer 344 rats. Neurobiol. Agin9 12, 201-210. Turner, D. A., H. V. Wheal, E. Stockley and H. Cole. 1991. Three dimensional reconstruction and analysis of the cable properties of neurons. In Cellular Neurobiology, J. Chad and H. Wheal (Eds), pp. 225-246. Oxford: IRL Press.
R e c e i v e d 18 O c t o b e r 1993 R e v i s e d v e r s i o n a c c e p t e d 3 A p r i l 1995
ElsevierScienceInc. © 1995Societyfor MathematicalBiology Printedin GreatBritain.All rightsreserved 009~8240/95$9.50+0.00
Pergamon
0092-8240(95)00328-N
E L E C T R O T O N I C C O U P L I N G B E T W E E N T W O CA3 HIPPOCAMPAL PYRAMIDAL NEURONS: A DISTRIBUTED CABLE MODEL WITH SOMATIC GAP-JUNCTION* R. R. POZNANSKI, W. G. GIBSON and M. R. BENNETT The Neurobiology Laboratory, Department of Physiology and The School of Mathematics and Statistics, University of Sydney, NSW, 2006, Australia A model of a pair of electrotonically coupled CA3 h i p p o c a m p a l p y r a m i d a l n e u r o n s is presented. Each n e u r o n is represented by a tapered equivalent cable attached to a n isopotential soma. The synaptic potential in a n e u r o n s o m a is determined as a consequence of electrical coupling to a n o t h e r soma t h a t receives a synaptic input o n its dendritic tree. Estimates of the coupling resistances, soma i n p u t resistances a n d soma-to-dendritic tree c o n d u c t a n c e ratio show t h a t a substantial current m a y arise in a n e u r o n as a consequence of synaptic activity in a n e u r o n coupled to it. The small increase in decay time due to coupling in the model indicates that actual coupling is between m o r e t h a n just pairs of neurons.
1. Introduction. Dye-coupling between some CA3 pyramidal neurons in slices of the hippocampus obtained from adult mammals is indicated by staining of up to four adjacent neurons in a cluster after the injection of the dye Lucifer Yellow into a CA3 pyramidal neuron (MacVicar and Dudek, 1980; MacVicar et al., 1982). The proportion of neuronal coupling in the hippocampal subfields CA3, CA1 and dentate gyrus is in the range of 30% to 70% (Schuster, 1992). Simultaneous intracellular recording from some pairs of CA3 pyramidal neurons when synaptic transmission is blocked confirms electrical coupling (MacVicar and Dudek, 1981). This electrical coupling is probably mediated by the gap-junctions that have been observed between CA3 pyramidal neurons (Schmalbruch and Jahnsen, 1981). It has been hypothesized that this coupling mediates fast stimulation and graded synchronous activation of the neurons in a cluster, whereas the more plastic and adaptable chemical transmission mediates that between the clusters (Schuster, 1992). The electrical transients arising from current injection experiments as well as * Support under ARC Grant AC9031997 is acknowledged. Correspondence to: Professor Max Bennnett, Neurobiology Laboratory, University of Sydney, NSW 2006, Australia. 865
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R.R. POZNANSKIet
al.
voltage-clamp recordings in CA3 hippocampal pyramidal neurons have been developed by Brown et al. (1981), Johnston (1981), Johnston and Brown (1983), Turner and Schwartzkroin (1983), Durand (1984), Turner et al. (1992), Spruston and Johnston (1992) and Spruston et al. (1993) in order to estimate the passive membrane constants of these cells. However, in none of these models was electrical coupling between neurons taken into account. This is important, since theoretical results have shown that the effects of electrotonic coupling on the evaluation of electrotonic properties (with the dendrites not given a distributed representation) may produce erroneous conclusions as to the correct passive membrane properties (Getting, 1974; Rall, 1981; Skrzypek, 1984; Publicover, 1989). In the present work the coupling between two CA3 pyramidal neurons, each with dendritic trees, will be analysed in order to determine the effects of such coupling on the interpretation of synaptic potentials generated at different sites in the dendritic trees. In order to begin a cable analysis of this problem we need to have an appropriate model for the electrotgnic structure of a single CA3 pyramidal neuron. Models that lump an extensive dendritic tree into a reduced equivalent cable (Poznanski, 1988) provide a picture of this electrotonic structure that is not too complex and thus ideal for large neural network simulations. This is especially useful in gaining an understanding of synaptic inputs located close to the soma (as in the case of mossy fibre inputs onto CA3 pyramidal neurons). It will be shown that synaptic transmission in the CA3 pyramidal neurons at the quantal level is complicated when electrical coupling is present, as potentials recorded in one neuron may actually arise due to transmission in an adjacent neuron. This implies that electrotonic coupling may have an important role in synaptic integration at the cellular level within the CA3 hippocampus (Berry and Pentreath, 1977). 2. Mathematical Formulation.
Let
V(Z, T) and U(Z, T) be the membrane
(electrotonic) potentials along the first and second neurons, respectively. The voltage response V(Z, T) to synaptic c u r r e n t Isyn(Z , T ) generated at an arbitrary point along an initially quiescent neuron 1 satisfies the modified cable equation (Poznanski, 1988, equation 3.1a) Vzz--K 1Vz -
V-~l'i,taper/~taperIsyn(T)(~(Z-~
-
~)~--- VT,
--L 1
(1)
where Z and T are, respectively, dimensionless position and time variables, )ttaper = [(Rm/4Ri)Dtape,] 1/2 is the generalized characteristic length parameter, 2 ri, t,per=4Ri/rcDt,per is the core resistance per unit length of the tapering equivalent cable, R m is the membrane resistance, Ri is the resistivity of the intracellular material, Dt,per = D exp(2KxlZ]/3) is the diameter of the tapering equivalent cable, D is the diameter ofa nontapering equivalent cable, Isyn is the
ELECTROTONICALLY COUPLED CA3 HIPPOCAMPAL PYRAMIDAL NEURONS
867
synaptic current, K 1 is a negative constant which determines the amount of exponential taper and 6 ( Z + 4) is the Dirac delta-function concentrated at the point Z = - ~ (where ~ is a positive number). The voltage response along an initially quiescent neuron 2 satisfies U z z + K z U z - U= UT,
0
(2)
where K 2 is a negative constant which determines the amount of exponential taper present in neuron 2. At the site of the junction ( Z = 0) two boundary conditions are needed. By applying Kirchhoff's law of conservation of current to the circuit of Fig. 1, we find
(3) and (4)
Ij = It(Z=O + )+ Ia(Z=O +),
where I a is the axial current, I t is the transverse current and Ij is the junctional current. The axial current is given by Ohm's law (with the assumption that the axial current is positive in the increasing Z direction): 1 ri~ Vz(0, T)
I,(Z=0-)=
(5)
U8
la Ri
I Neuron 1
I
I
Z=O"
Z=L 2 Neuron 2
Figure 1. A schematic illustration of a pair of tapered equivalent cables representing the dendritic arborizations of CA3 hippocampal pyramidal neurons coupled at the soma (Z = 0) by a junctional resistance (Rj). (Z is the electrotonic distance variable.) The presence of a gap-junction between the somata of both neurons requires us to distinguish to the left Z = 0 - and to the right Z = 0 + of the point Z = 0. It is assumed that synaptic input is activated at an arbitrary point along neuron 1 and the potential response is recorded at the cell body of neuron 2 (see text for meaning of symbols).
868
R.R. POZNANSKIet al.
and 1
I n ( Z = 0 + ) = - ,¢r~ ~ Uz(0, T),
(6)
where ri=4Ri/~zD 2 is the core resistance per unit length of a nontapering equivalent cable and 2 = (RmD/4Ri) ~/2 is the characteristic length parameter of a nontapering equivalent cable. The transverse current or the total outward somatic m e m b r a n e current is the sum of the resistive and capacitative currents given by
It(Z = 0 - ) = V(O, T)/R~ + (CJzm) VT(0, T)
(7)
I t ( Z = 0 + ) = U(0, T)/Rs+(CJ'Vm)UT(O, r ) ,
(8)
and
where R~ is the resistance of the soma, Cs is the capacitance of the soma and Zm= RmCm, where Cm is the m e m b r a n e capacitance, is the m e m b r a n e time constant. The voltage at the gap-junction is discontinuous by the a m o u n t IjRj (representing the drop in voltage across the junctional resistance), where Ij is the axial current through Rj given by Ij = [ V(O, T ) - U(O, T)]/Rj.
(9)
Substitution of (5) to (9) into (3) and (4) yields the required set of coupled b o u n d a r y conditions:
[I+(RJRj)]V(O, r)+yVz(O , T)+aVT(O , T)=(RJRj)U(O, T)
(10)
[I+(RJRj)]U(O, T)--TUz(0 , T)+trUT(0 , T)=(Rs/Rj)V(O , T),
(11)
and
where 7 = Rs/ri). is the dendritic to somatic conductance ratio for a semi-infinite cable (the notation p o~is also used; Jack et al., 1975) and o-= -cs/-cm is the somatic shunt parameter where zs = RsCs is the somatic time constant. If we assume no longitudinal (axial) current flow at either end of neurons 1 and 2 (i.e. sealed-end terminations) then the b o u n d a r y conditions at Z = - L 1 and Z = L 2 a r e
V z ( Z = - L 1, T ) = U z ( Z = L 2 , T ) = 0 .
(12)
ELECTROTONICALLY
COUPLED
CA3 HIPPOCAMPAL
PYRAMIDAL
NEURONS
869
3. Solution for the Potentials. In this section we use Green's function methods to determine a partial analytic solution to the initial-boundary value problems derived in the preceding section, with particular emphasis on obtaining the coupling potential (i.e. the potential at the soma of neuron 2) denoted by Us. The full solution of (2) (at the soma) is the inverse Laplace transform of
uL(0, s)= (Rs/gj)VL(0, s)6 (0, s + 1 + Ji q )2 ,
(13)
where s is the transform variable, UL is the Laplace transform of U, VL is the Laplace transform of V and G~ is the Laplace transform of Green's function given by G*(0, s)= x/~ cosh~/sL2-(K2/Z)sinhx/~L2 /{[(~ - (K2/2 )a)s-- (KE/2 )e2]sinhx/~L 2 + x/~[e2 + as--(K2/2)7]coshx/~L2},
(14)
with ea -- [1 + (Rs/Rj) + 7(Ka/2 ) -- a(1 + K22/4)]. Green's function (14) is derived in Appendix 1. The Laplace transform of the potential in neuron 1 at the stoma can be expressed as VL(0, S)= IL(S)H*(O, -- 4, S+ 1 + ¼K~),
(15)
where IL(S)= Qocca/zm(S + c¢)2, Qo is the charge applied instantaneously, a is a constant parameter and HL*(0, --4, s) is Green's function HL*(0, --4, s)=ri27 exp(--1Kl~){[x/~ cosh~/~L1-(K~/2)sinhx/~sL1]coshw/ss~ --[x/~ sinhx/~L1-(K1/2)cosh~sL1]sinhx/Tss~} /{Tx/~[x/~ sinhx/~L1-(K1/2)coshx/~sL ~] +[x/~ coshxSssLa -- (K1/2 )sinhx/FsL1] 1 2 1 2 • [e x + as-- (RJRj)ZG*(O, s + xK~ ---~K[ )]}, (16)
where ex = 1 + Rs/R j + 7(K1/2 ) - a ( 1 +K~/4). Green's function defined by (16) is derived in Appendix 2. We check (16) against the previously known limiting case of R j ~ and Ka = 0 (Poznanski, 1987). We first note that R j ~ corresponds to a single neuron problem with the potential in the inactive neuron 2 being equal to zero, as shown by letting R j ~ ~ in (13). The case of K~ = 0 yields the well-known
870
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somatic-shunt cable model (Durand, 1984). By letting R j ~ o o and Kt = 0 in (16) and noting that el = 1 - a , we arrive at H * (0, - 4, s, Rj ~ ~ , K 1 = 0)/ri2
=7 coshx/Fss(~-L1)/{Tx/rss sinhx/~L1 + ( 1 - ~ + ~s)coshx//sLx}.
(17)
Apart for notational differences (L 1 for L, and Z for X), (17) corresponds to (4) with I"= 0 in Poznanski (1987). 4. Parameter Values. The parameter values used in this paper are chosen from two independent studies of CA3 pyramidal cells, one performed by Brown et al. (1981) and the other by Johnston (1981). A table summarizing these electrical parameters is given in Turner and Schwartzkroin (1984). An additional parameter appearing in that table is p, the dendritic-to-somatic conductance ratio, which is related to ~ by Jack et al. (1975): 7 = p coth(L),
(18)
where p is in the range 0 . 8 < p < 2 . 7 and the electrotonic length (L) is in the range 0.79 < L < 1.03. Note that the electrotonic length parameter derived by these authors was based on a formula for a uniform equivalent cylinder (Rall, 1969) and therefore it is expected to be an underestimate of the electrotonic length parameter for a tapering equivalent cable as shown by Poznanski (1988) and Turner and Deupree (1991). In fact, a recent study based on the tapered model indicates that L = 1.65 may be a better estimate (Poznanski, 1994). The p measured under the assumption of equal somatic and dendritic resistivity (i.e. a -- 1) shall be denoted by the subscript "ns" in order to differentiate the value of p when a ¢ 1. Thus p = apn ~ (Durand, 1984) and (18) becomes ~ = p,~a coth(L), leaving 7 in the range 1.0<~<4.1 by choosing Pns=3, and a = 1 and 0.25. A numerical estimate of the resistance of the soma (R~) can be obtained from (Durand, 1984): R s = R N ( I +p),
(19)
where R~ is the input resistance. From MacVicar and Dudek (1981) (see also Dudek et al., 1983) it follows that the coupling ratio CR is given by CR = RN/(R i + Rj).
(20)
Then the ratio R = R s / R j is given by R = (1 + P ) C R / ( 1 - CR). Clusters of CA3 pyramidal cells show electrotonic coupling ratios less than 0.3 (MacViear and Dudek, 1981); thus an estimate of the maximum value of R is about 1.5 (p ~ 2.7, CRY0.3 ). For graphical illustrations of the results, we have used the values R=0.5, 1 and 1.5.
E L E C T R O T O N I C A L L Y C O U P L E D CA3 H I P P O C A M P A L P Y R A M I D A L N E U R O N S
871
Most CA3 hippocampal pyramidal neurons show a combined dendritic trunk parameter that is well approximated by a exponential taper (Turner and Schwartzkroin, 1983, 1984; Poznanski, 1988, 1994):
r£
3/2 j=l
d./2
-~ exp(KZ),
(21)
Lj = 1
where K is a parameter which determines the hmount of taper (K< 0) present, dj is the diameter of thejth branch element at a given electrotonic distance (Z) from the soma ( Z = 0), n o is the number of primary trunks emanating from the soma at Z = 0 and n i is the number of branch elements at any given electrotonic distance (Z) from the soma (Z = 0). It is emphasized that (21) is most frequently illustrated in the experimental literature (see Poznanski, 1994, Fig. 1). 5. Results.
F r o m (15) the potential at the soma of neuron 1 is
gs=(Qoa2/~m)~- l {H~(O, - ~ , s + l + ¼K2)/(s + a)2},
(22)
where ~ - 1 represents the inverse Laplace transform operation. Substituting (15) into (13) gives the coupling potential at the soma of neuron 2: 1 2 Us= (Rs/Rj) (Qoa2/.Cm). ~o- l'¢H*t0t L, , -- 4, S + 1 + zK, )
G*(O, s+l+¼K~)/(s+a)2}.
(23)
The inverse transforms in (22) and (23) can be evaluated numerically using a computer software package. The time course of the normalized potential V~ at the soma of the active neuron (1) (i.e. at Z = 0 - ) , in response to a fast synaptic input is shown in Fig. 2. The parameter values are given in the caption. The synaptic input is governed by the a-function a2Te-'T which for a = 50 peaks at T = 0.02 and is essentially zero for T>0.2. The somatic potential has a much slower time course, dependent on the membrane properties. The largest response occurs in (a) where the activation is at a proximal distance (~ = 0.1) and the somatic shunt resistance is largest (o-= 1); the smallest occurs in (d) where the activation is at a distal distance (¢=0.8) and the somatic shunt resistance is smallest (o-= 0.25). The peak amplitude of Vs decreases irrespective of the input site or junctional resistance value if the somatic shunt parameter a is made smaller (by increasing the somatic shunt), and also the voltage decay becomes more rapid. These observations appear to be compatible with the results obtained by Durand (1984) and Poznanski (1987) for the single neuron problem. Figure 3 shows the time course of the normalized potential U~at the soma of the passive neuron (2) (i.e. at Z = 0 +) for the same four cases as the corresponding graphs in Fig. 2. As expected, these potentials peak at later
872
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et al.
FAST SYNAPTIC INPUTS (~=50)
[ R=O.O
--
COUPLING STRENGTHS | R=0.5 - ' t R=1.5 ---
(a) proximal activation
(b) proximal activation 2.0
z O
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.
.
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SCALED TIME, T
Figure 2. Time course of the normalized potential Vs at the soma of neuron 1 in response to a fast synaptic input (alpha function with c~= 50) activated at distance from its soma. Neuron 1 is coupled to neuron 2 by a junctional resistance Rj. Parameter values for both neurons are electrotonic length L~ =L 2 =0.9, taper parameter K~=K2=--4.75, dendritic-to-somatic conductance ratio y=3~r coth(L) = 4.188a, where a is the ratio of the somatic and membrane time constants. In each graph, the response is given for three values of the coupling strength R=Rs/Rj, where Rs is the somatic shunt resistance for each neuron: R=0, corresponding to neurons 1 and 2 being uncoupled (solid line), R=0.5 (dot~lash line) and R = 1.5 (dashed line). (a), ~ = 0.1 (proximal activation), a = 1; (b), ~ = 0.1, cr=0.25; (c), 4=0.8 (distal activation), a = 1.0; (d), 4=0.8, a=0.25. times than those in the active n e u r o n , with delay being m o s t p r o n o u n c e d for large o- (graphs (a) and (c)). Figures 4 and 5 give the c o r r e s p o n d i n g results for a slower synaptic input (c~= 10) which peaks at T - - 0 . 1 and is effectively zero for T > 0 . 8 . The overall effect of this slower input is to increase the delay in the response in b o t h the active and the passive neurons. Further characterization of the responses for all of the a b o v e cases is given in Table 1; a n o t h e r parameter n o t listed there is the time constant for decay w h i c h is 1.0 w h e n o--- 1 and 0.39 w h e n o-= 0.25 (for b o t h V~ and Us). Figure 6 s h o w s the d e p e n d e n c e of the times to peak and the peak potentials o n o- and R. As R increases, the time to peak for Vs s h o w s little change, but that
ELECTROTONICALLY
COUPLED
CA3 H I P P O C A M P A L
PYRAMIDAL
NEURONS
873
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Figure 3. Time course of the normalized potential Us at the soma of neuron 2 in response to a fast synaptic input (alpha function with c~--50) activated at distance from the soma of neuron 1. All the parameter values are the same as for the corresponding graphs in Fig. 2, except that the three values for R = Rs/R j are now R=0.5 (do%dash line), R= t.0 (dotted line) and R= 1.5 (dashed line). of Us increases markedly (Fig. 6(a)). Similarly, the peak of V~ shows only a small decrease (a similar conclusion was reached by Hall (1981), Fig. 7a, for a "lumped parameter" model) whereas the peak of Us undergoes a significant increase (Fig. 6(b)). Figure 6(c) shows that increasing the somatic shunt parameter o- has only a small effect on the time to peak of Vs, but causes a big increase in that of Us. Conversely, increasing o- from very low values causes a large increase in the peak value of Vs (Fig. 6(d)), but very little change in the peak of Us. (Though note that the behaviour is not monotonic; the peak value of Us increases for small o-, peaks at about o-= 0.28 and then decreases steadily.) Since the value of the electrotonic length may be greater than L = 0.9 (Poznanski, 1994) some calculations were repeated for larger L values. This caused little change in the rise and fall times of the potentials, but did lead to some reduction in peak values. F o r example, for L~ = L 2 = 1.8, R = 1.0, o-= 1.0 and the remaining parameters as for Fig. 2, there was a 23% reduction in the peak amplitude of V~ and a 12% reduction in the peak amplitude of Us.
874
R . R . P O Z N A N S K I et al.
SLOW SYNAPTIC INPUTS (c¢=10) R=0.0 - R=0.5 ---R=1.5 - - -
COUPLING STRENGTHS
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SCALED TIME, T
Figure 4. Time course of the normalized potential V~at the soma of neuron 1 in response to a slow synaptic input (alpha function with c~= 10) activated at distance from its soma. The remaining parameter values are the same as for the corresponding graphs in Fig. 2. 6. Discussion. As has been emphasized by Schuster (1992), the quantitative analysis of gap junctions is an essential component of future brain research. The present work has dealt with a small subset of this problem, in giving a theoretical analysis of the effect of electrotonic coupling of two cells in the context of a model appropriate for the CA3 pyramidal cell. Although the precise values of the coupling strengths and other parameters are not yet known, varying these over reasonable ranges indicates that the peak response in the passive cell could be as high as 40% of that in the activated cell (Table 1). However, this would be an extreme case. Since evidence from dye coupling and morphological studies indicates that neurons are coupled in clusters it would be expected that the potential would spread over several passive cells, with a consequent lowering of the response in each cell. Evidence for more extensive coupling also comes from a consideration of decay times. Fu et al. (1992) have shown that uncoupling hippocampal neurons with ethanol leads to about a 30% increase in the final time constant from the voltage transient, but not much change in the other potential parameters in the active
ELECTROTONICALLY
COUPLED
CA3 HIPPOCAMPAL
PYRAMIDAL
NEURONS
875
SLOW SYNAPTIC INPUTS (~ =10) R=1.5 COUPLING
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Figure 5. Time course of the normalized potential Vs at the soma of neuron 2 in response to a slow synaptic input (alpha function with c~= 10) activated at distance from the soma of neuron 1. The remaining parameter values are the same as for the corresponding graphs in Fig. 3. cell. The results of the present study indicate that as the coupling is reduced to zero there is almost no change in the time to peak (Fig. 6(a); Table 1), a small increase in the peak (Fig. 6(b); Table 1) and a larger increase in the decay time (Fig. 6(c); Table 1). However, this latter increase is still well below 30% (maximum of a b o u t 15%), suggesting that hippocampal neurons may be coupled in a two- or three-dimensional array in order for uncoupling to change the final time constant by such a large amount. (For an analogous problem of current flow in a three-dimensional syncytium, see Bennett et al. (1993); there, it is found that the time course of the membrane potential following simulated transmitter release is much faster than that expected from the value of the membrane time constant.) Electrotonic coupling will have an effect on passive cable parameter estimates of single CA3 hippocampal neurons and this will eventually have to be taken into account. The most recent cable analysis of CA3 pyramidal neurons (Major et al., 1994) does not take this into consideration. Given that for the 20-day old rats used in the study there is evidence of high coupling
876
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al.
Table 1. Shape indices for the potential responses in neurons 1 and 2. The values used for the parameters L1, L2, K1, K 2 and y are the same as for Fig. 2. T~,°"k is the time to peak, Vpe"k is the peak potential value, Hv is the half width (i.e. the time that the potential is above half its peak value) and Tvh"lf is the half decay time (i.e. the time taken for the potential to decay from its peak value to half that value). These are for the potential V~at the soma of neuron 1; the final four columns are the corresponding quantities for the potential Us at the soma of neuron 2
0~
~
0"
R
50
0.1
0.25
0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5
1.0 0.8
0.25 1.0
10
0.1
0.25 1.0
0.8
0.25 1.0
~vTpeak Vpeak 0.09 0.09 0.08 0.11 0.11 0.11 0.23 0.22 0.22 0.28 0.27 0.27 0.28 0.27 0.27 0.38 0.36 0.35 0.41 0.40 0.40 0.51 0.50 0.49
1.71 1.62 1.54 2.01 1.97 1.94 1.18 1.09 1.02 1.56 1.50 1.46 1.04 0.95 0.88 1.53 1.45 1.39 0.87 0.79 0.73 1.32 t.25 1.19
By 0.28 0.25 0.23 0.65 0.57 0.52 0.40 0.37 0.36 0.79 0.71 0.66 0.54 0.52 0.52 0.95 0.88 0.83 0.57 0.56 0.56 0.98 0.91 0.87
Tvhalf T~¢ak UPe,k 0.22 0.19 0.18 0.57 0.49 0.44 0.28 0.26 0.25 0.63 0.56 0.51 0.36 0.35 0.35 0.70 0.64 0.60 0.37 0.36 0.36 0.71 0.64 0.61
0.31 0.24 0 . 2 6 0.37 0.22 0 . 4 6 0.83 0.18 0.71 0.29 0.62 0.38 0.44 0.20 0.39 0.31 0.37 0 . 3 8 0.95 0.16 0.83 0.26 0.75 0.34 0.51 0.21 0.46 0.31 0.43 0.38 1.01 0.17 0 . 9 0 0.29 0 . 8 2 0.37 0.64 0.18 0.59 0.27 0.56 0.32 1.14 0.15 1.02 0.25 0.95 0.33
Hu
T~alf
0.72 0.61 0.55 1.99 1.74 1.58 0.75 0.65 0.60 1.99 1.74 1.58 0.82 0.74 0.70 2.01 1.77 1.63 0.84 0.77 0.73 2.03 1.79 1.63
0.50 0.43 0.40 1.37 1.21 1.12 0.5t 0.45 0.41 1.37 1.21 1.11 0.53 0.48 0.45 1.37 1.21 1.12 0.53 0.48 0.46 1.37 1.22 1.11
(Reece a n d S c h w a r t z k r o i n , 1991; Schuster, 1992) it w o u l d be advisable to c h e c k if the n e u r o n s used in a cable analysis are indeed c o u p l e d (by, for example, dye injection). T h e present c a l c u l a t i o n c o u l d be e x t e n d e d in a n u m b e r of ways. A n o b v i o u s one is the extension to d e n d r o - d e n d r i t i c c o u p l i n g , w h i c h has been o b s e r v e d e x p e r i m e n t a l l y a n d in fact m a y be the principal site of g a p j u n c t i o n s b e t w e e n CA3 p y r a m i d a l cells ( M c V i c a r a n d D u d e k , 1980; Schuster, 1992). Such c o n n e c t i o n s b e t w e e n the ends of each dendritic cable c o u l d be i n c o r p o r a t e d into the present m o d e l b y replacing (12) b y the following set of c o u p l e d b o u n d a r y conditions:
Vz(--L1, T)= (ri, taper/~taper/Rc)[ V ( - L1, T ) - U(L2, T)3
(24)
ELECTROTONICALLY COUPLED CA3 HIPPOCAMPAL PYRAMIDAL NEURONS (a)
~=1
(b)
877
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COUPLING STRENGTH, R (c)
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Figure 6. Dependence of the time to peak, the peak potential and the half decay time on R and a. The values for the parameters LI, Lz, K 1 , K 2 and y are the same as for Fig. 2. A fast synaptic input ( , = 50) at a distal distance ( 4 = 0 . 8 ) is used. In each graph the quantity pertaining to V~ (the potential at the soma of neuron 1) is given by the sold line and that pertaining to Us (the potential at the soma of neuron 2) is given by the dashed line. (a) The times to peak as functions of R for a = !.0; (b) the peak potentials as functions of R for a = 1; (c) the half decay times as functions of R for ~r= 1 ; (d) the times to peak as functions of a for R = 1.0; (e) the peak potentials as functions of ~ for R = 1; (f) the half decay times as functions of ~r for R = 1.
878
R.R. POZNANSKIet al.
and U z ( L 2 , T ) = (ri, taper~taper/Re) [-V(--L1, T ) -
U ( L 2 , T)],
(25)
where R c represents the coupling resistance between the terminal ends of both dentritic cables. This case m a y still be amenable to a semi-analytic treatment. Further extensions would require a numerical approach from the outset. These include the use of action potentials which would allow, for example, the investigation of bursting behaviour (Traub, 1982; T r a u b et al., 1991) and also the extension of the model to networks of cells, coupled both chemically and electrotonically (Traub and Wong, 1983; T r a u b et al., 1985; D u d e k and Traub, 1989). The implications of such coupling for current theories of autoassociative m e m o r y storage in CA3 (Treves and Rolls, 1994; Bennett et al., 1994) needs to be studied in the context of a network model.
APPENDIX
1
The solution of (2) can be written in terms of the Green's function G(Z, - ~, T) by the convolution integral Laplace Transform of the Green's Function for the Inactive Neuron 2.
U(Z, T ) = ( R J R j )
V(O, t7)G(Z, - ~ , T-tT) dtT,
(A1)
where the Green's function G*=G exp[-½KzZ+(I+¼K~)T] is the solution of the initialboundary value problem G T*-- Gzz*
(0
(A2)
[1 + (RJRj) + 7(K2/2) -- a -- a(K2/4)]G* (0, T) -- 7G* (0, T) + aG *(0, T) = 6(T)exp[(1 + ¼K22)T] (A3) G*(L2, T ) - (K2/2)G*(L2, T)=0
(A4)
G*(Z, 0)=0.
(A5)
Define G*(Z, s) to be the Laplace transform of G*(Z, T) and therefore the subsidiary equation corresponding to (A2) and (A5) is -- d2Gt/dZ 2 + sG t = 0
(A6)
(e2+ ~rs)G*(O, s ) - y dG*(0, s)/dZ= 1
(A7)
dG*(L2, s)/dZ-(K2/2)G*(L2, s)= 0,
(A8)
to be solved with
and
where ~2= [1 + (Rs/Rj)+Y(K2/2)-a(1 +/£22/4)] and s is the transform variable. The solution in
ELECTROTONICALLY COUPLED CA3 HIPPOCAMPAL PYRAMIDAL NEURONS
879
the Laplace transform space of these equations corresponds to the required Green's function defined by (14) at Z = 0 .
APPENDIX
2
Laplace Transform of the Green's Function for the Active Neuron 1. The solution of (1) can be written in terms of the Green's function H(Z, - 4 , T): V(Z, T ) =
I~y.(q)g(z, --4; T-rl) dq,
(B1)
do where the time course of the synaptic current is given by the standard alpha function (Jack et al., 1975) Isy. (T) = (Qo/zm)Ctz T e x p ( - aT) and the Green's function H * = H e x p [ - ½ K 1 Z + ( I + I K ~ ) T ] boundary value problem
(B2) is the solution of the initial-
H x*--Hzz* (--L1 < Z < 0 , - L 1 < ~<0, T > 0 )
elH*(0, T)+TH*(0, T)+aH*(O, T ) =
H*(O, q)e<~+K?/4)~r-n)G(O,T--q) dq
(B3)
(B4)
H~(--L1, T)+(K1/2)H*(--LI, T ) = 0
(B5)
H*(Z, 0) = ri. ,..or'~,..or exp(-½K, a)a(Z+ ¢),
(B6)
where e~ = 1 + (Rs/Rj) + y ( K , / 2 ) - a(1 + K~/4). Define HL* to be the Laplace transform of H*, and therefore the subsidiary equation corresponding to (B3) and (B6) is
-- d i H * / d Z 2 + sH* = ri2 e x p ( - ½K1{)6(Z + 4)
(B7)
[e1 + as -- (Rs/Rj)2G*]H* (0, s) + ~ dH* (0, s)/dZ = 0
(~8)
dH* ( - L 1 , s)/dZ + (K1/2)H* ( - L 1 , s) = 0.
(B9)
to be solved with
and
The solution in the Laplace transform space of these equations corresponds to the required Green's function defined by (16) at Z = 0.
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R e c e i v e d 18 O c t o b e r 1993 R e v i s e d v e r s i o n a c c e p t e d 3 A p r i l 1995