A dendritic cable model for the ampli®cation of synaptic
potentials by an ensemble average of persistent sodium
channels
R.R. Poznanski
a,*, J. Bell
b,1 aAdvanced Research Laboratory, Hitachi, Ltd., Hatoyama, Saitama 350-0395, Japan
bDepartment of Mathematics, State University of New York, Bualo, NY 14214-3093, USA
Received 26 October 1999; received in revised form 27 May 2000; accepted 1 June 2000
Abstract
The persistent sodium current density (INaP) at the soma measured with the `whole-cell' patch-clamp
recording method is linearized about the resting state and used as a current source along the dendritic cable (depicting the spatial distribution of voltage-dependent persistent sodium ionic channels). This procedure allows time-dependent analytical solutions to be obtained for the membrane depolarization. Computer simulated response to a dendritic current injection in the form of synaptically-induced voltage change located at a distance from the recording site in a cable with unequally distributed persistent sodium ion channel densities per unit length of cable (the so-called `hot-spots') is used to obtain conclusions on the density and distribution of persistent sodium ion channels. It is shown that the excitatory postsynaptic potentials (EPSPs) are ampli®ed if hot-spots of persistent sodium ion channels are spatially distributed along the dendritic cable, with the local density of INaP with respect to the recording site shown to
spe-ci®cally increase the peak amplitude of the EPSP for a proximally placed synaptic input, while the spatial distribution ofINaP serves to broaden the time course of the ampli®ed EPSP. However, in the case of a
distally positioned synaptic input, both local and nonlocal densities yield an approximately identical en-hancement of EPSPs in contradiction to the computer simulations performed by Lipowsky et al. [J. Neurophysiol. 76 (1996) 2181]. The results indicate that persistent sodium channels produce EPSP am-pli®cation even when their distribution is relatively sparse (i.e., approximately 1±2% of the transient sodium channels are found in dendrites of CA1 hippocampal pyramidal neurons). This gives a strong impetus for the use of the theory as a novel approach in the investigation of synaptic integration of signals in active dendrites represented as ionic cables. Ó 2000 Elsevier Science Inc. All rights reserved.
Keywords: Persistent sodium channels; Dendrites; Synaptic integration; Ionic cable theory; Neuronal modeling; Analytical solutions; GreenÕs function
*Corresponding author. Tel.: +81-492 96 6111, ext. 330; fax: +81-492 96 6006.
E-mail address:[email protected] (R.R. Poznanski).
1
Present address: Department of Mathematics and Statistics, UMBC, Baltimore, MD, USA.
1. Introduction
The classical Sherrington±Eccles doctrine of how a neuron functions, namely the passive conduction of small graded potentials towards the soma for action potential initiation at the axosomatic region, needs to be revised (see Refs. [1,2]) in view of patch-clamp recordings and imaging studies revealing voltage-dependent Na channels distributed predominantly on the somata and primary trunks of some neurons [3], as well as backpropagation of Na action po-tentials [4±6] and initiation of Naaction potentials in distal dendrites [7±11]. In particular, it has been shown in Refs. [12±17] that voltage-activated channels in neocortical neurons are involved in shaping the time course of subthreshold depolarizations. Similar ®ndings have been observed in spinal motoneurons [18], hippocampal pyramidal neurons [19], and retinal neurons [20,21].
The persistent or sustained-type current (INaP) activated in the subthreshold voltage range of
the membrane potential [12,22] was shown to be responsible for the voltage-dependent ampli®-cation of EPSPs. Stuart and Sakmann [17] suggested a persistent sodium current in the axoso-matic region per se to be the ampli®cation mechanism, while Schwindt and Crill [23] predicted enhancement EPSPs by both dendritic INaP and synaptically-activated NMDA receptors. Indeed
the persistent Na current appears to be the major mechanism (see Ref. [24]), but not the only mechanism for the voltage-dependent ampli®cation of EPSPs in dendrites (see Refs. [25,26] for reviews); other possibilities, such as the low-threshold Ca2 current, have also been associated with the amplifying mechanism (see Refs. [27,28]).
Cable theory originally applied to neurons at the turn of this century from the earlier work on trans-Atlantic telegraph cables by Lord Kelvin around 1855 is not ionic, but electrotonic. Al-though Rall [29] pioneered the application of electronic cable theory to dendrites, no attempt to further ionic cable theory by incorporating discrete loci of voltage-dependent ionic channels into cable models with particular reference to the persistent sodium ion channels has been undertaken to the best of our knowledge. Earlier theoretical work dealing with graded potential ampli®cation in dendrites was mostly numerical [30,31] or based on compartmental models [32±39], which approximates the distributed, continuous membrane of the neuron by a set of discrete inter-connected compartments.
Mozrzymas and Bartoszkiewicz [40] presented an alternative approach by discretely juxta-posing values of membrane resistivity to allow for the inclusion of ionic channels. However, they considered only the discrete nature of the membrane resistivity in the absence of voltage-de-pendent ion channels (see also Ref. [41]). Furthermore, they assumed an in®nite membrane re-sistance that ignored the `leaky' nature of the cable at rest, or small voltage excursion from rest. They did show that the assumption of leaky cable holds since the dierence between a segmented cable (cf. Ref. [42]) and a continuous leaky cable was negligibly small. Baer and Tier [43] modeled a cable with a non-linear boundary condition, representing voltage-dependent ionic channels at one end point along the cable. This is mathematically equivalent to a single loci of ionic channels, but did not take into consideration a multiple loci of ionic channels.
paper by Bell and Holmes [45] considered a passive cylindrical cable with active ®lopodia con-tinuously distributed along the dendrite and characterized by a density (i.e., the number of ®lo-podia per unit length). Based on the Baer±Rinzel theory, they assumed the distribution of ®lopodia and channel densities on a Pacinian corpuscle to be a continuum with about one or two active channels per ®lopodia. Toth and Crunelli [46] considered a continuously inhomogeneous ion channel distribution with exponentially decaying densities of voltage-activated channels.
Although such models are satisfactory for large densities of channel distributions, for less dense distributions it is better to consider the cable as a `leaky' (i.e., an RC cable) with densities of voltage-dependent ionic channels discretely imposed at speci®c locations along the cable and referred to as hot-spots (see Fig. 1). The non-linearities occur only at discrete points along the linear cable rather than continuously along the entire cable structure, this is valid provided the cable at rest is mostly passive (i.e., voltage-independent channels are opened along the entire cable).
An a priori assumption of the theory is that persistent sodium ion channels along the soma-dendritic axis occur in discrete patches resembling hot-spots, but this assumption can be made as general as we like by reducing the distance between hot-spots in order to approximate a con-tinuous layer of ionic channels. As yet there is no experimental veri®cation of the validity of this assumption because determination of the spatial distribution of persistent sodium channels in dendrites is a dicult if not an impossible task, and in most cases an average estimate is found based on recording from only a few dendritic patches. However, single-channel recordings from rat hippocampal neurons show a possible homogeneous distribution throughout the neuron [47]. It is stressed, however, that a homogenous distribution may not imply a continuous distribution since recordings using patch-pipettes are always measured discretely along the dendrite, and hence the advantage of the present model over the classical Sabah and Leibovic [48] model is that the latter assumes voltage-dependent conductances are distributed continuously along the entire length of the axonal cable, leaving the problem dicult to solve analytically.
The notion that active propagation in dendrites might occur by saltatory conduction between patches of excitable membrane or hot-spots is not new (see Ref. [49]). The cable model for sal-tatory conduction in myelinated axons as described by FitzHugh [50] consists of segments of passive (RC-leaky) cable interrupted by isopotential patches of excitable membrane described by Hodgkin and Huxley [51]. However, the FitzHugh cable model is segmented because the nodes are intrinsically imbedded in the membrane (cf. Ref. [42]), and therefore it should not be seen as identical to the present cable model where patches of voltage-dependent ion channels or hot-spots constitute active point sources of current on a homogeneous (nonsegmented) leaky cable struc-ture. This is justi®ed if each hot-spot is assumed to occupy an in®nitesimal region represented explicitly by a Dirac-delta function.
Recently, in order to explain saltatory-like wave propagation in cardiac myocytes and skeletal muscle, a model has been formulated based on a non-linear diusion equation with regularly spaced sources called `sparks' comparable to the instantaneous release of Ca2 [55]. This
nu-merical approach is similar to an earlier model [56] implementing the sources at a single grid point. However, only limited analytical solutions were obtained in [55] consisting of a simpler `®re-diuse-®re' model or a linear diusion equation (i.e., a passive RC cable) with an in®nite number of source terms represented by Dirac-delta functions, and referred to as `Dirac-delta function spikes'. Their work can also be related to that found in [52].
Fig. 1. A schematic illustration showing a cable of diameterd(cm) and lengthL(cm): (a) The arrow above the hot-spot re¯ects the notion ofINaP representing a point source of current applied at an in®nitesimal area on the cable. The
symbolNdenotes the number of hot-spots andNdenotes the number of persistent sodium channels in each hot-spot
per unit membrane surface of cable, represented schematically as black dots. (b) An electrical equivalent circuit rep-resentation of the same cable. Note: r
The present theory deals with ampli®cation of non-regenerative potentials (i.e., EPSPs) as a result of linearized ensemble average or macroscopicINaP current sources, and, in a companion
paper, a whole range of synaptic potential ampli®cations produced by persistent sodium channels is investigated in order to include the non-linear properties of clusters of small numbers of ionic channels (see Ref. [57]).
2. Formulation of the model
2.1. The cable equation for discretely imposed persistent sodium channels
LetVbe the depolarization (i.e., membrane potential less the resting potential) in mV, and let INaPbe the transmembrane sodium current density per unit membrane surface of cable in (A/cm).
The voltage response or depolarization in a leaky cable representation of a cylindrical passive dendritic segment withINaP occurring at discrete points along the cable (see Fig. 1) satis®es the
following cable equation:
CmVt d=4RiVxxÿV=Rm
XN
i1
INaP x;t;Vd xÿxi I td xÿx0
XN
z1
Pd xÿxz; t>0; 1
whereI tis the time course of the applied current density per unit membrane surface of cable in A/cm,xthe distance in cm,tthe time in seconds,dthe diameter of the cable in cm,Cm cm=pd
the membrane capacitance (F/cm2),R
m rmpdthe membrane resistivity (Xcm2 ),Ri ripd2=4
the cytoplasmic resistivity (Xcm),Nthe number of hot-spots (dimensionless),Pthe sodium-pump current density which contains ion ¯uxes due to active transport per unit membrane surface of cable in A/cm, anddis the Dirac-delta function re¯ecting the axial position along the cable where the ionic current is positioned in cmÿ1 (see Ref. [58]) with the synaptic (current) injection at
lo-cationxx0 , the pump location atxxz, and the hot-spot locations atxxi. Note subscriptsx
and t indicate partial derivatives with respect to these variables.
Eq. (1) describes the membrane depolarization in a single cable or dendrite, but a possible extension to multiple cables or M dendrites emanating from a common soma can be shown to satisfy
CmVjt dj=4RjiVjxxÿVj=Rjm
X
Nj
i1
IjNaP xj;t;Vjd xjÿxji Ik td xkÿxk0
X
Nj
z1
Pjd xjÿxjz;
t>0 2
forj1;2;. . .M with all other parameters being modi®ed accordingly. Eqs. (1) and (2) can be
further modi®ed to include tapering geometries as discussed in [59,60], respectively.
2.2. Linearization of the persistent sodium current
It is still widely accepted that near the resting potential theI±V relationship is linear and the
linearizing the voltage-dependent ionic currents. This approach was ®rst advocated in [51] and was further developed in [48] who obtained solutions for graded responses in an in®nite cable by applying linearization theory to the voltage-dependent ionic currents (see also Refs. [61,62]).
The application of linearization theory is particularly suited to the analysis of EPSPs not ex-ceeding a few millivolts from rest, but this depends on the particular neuron in question. A better criterion, therefore, should be the voltage range around the resting state before the non-linearity comes into play, which in some mammalian CA1 hippocampal neurons could reach values of up to 30 mV from the resting state (see Fig. 2). However, in Refs. [48,62] linearization theory was applied to the ionic current densities expressed in the Hodgkin±Huxley equations on the as-sumption that the membrane potential did not exceed a few millivolts from the resting potential. A recent paper [63] provides experimental evidence for localized dendritic spike-like potentials not exceeding about 30 mV and for active dendritic conductances linearizing synaptic potentials. Thus dendritic spike-like potentials can be modeled based on the linearization procedure.
The persistent Na transmembrane current density per membrane surface of cable (lA/cm) described in [35] based on the experimental data of French et al. [64] can be represented by
INaP x;t;V gNaPm VV ÿVNaP; t>0 3
withVNaPENaPÿEr,Er ÿ70 mV is the resting membrane potential, andENaP65 mV is the
persistent sodium equilibrium potential (mV). The strength (conductance) of persistent sodium ion channel densities is given by (cf. Ref. [65])
gNaPgNaP N; 4
whereN=pd is the number of persistent sodium channels per unit membrane surface of cable in cmÿ1, andg
NaP 18pS is the maximum attainable conductance of a single sodium channel [66,67].
The dimensionless activation variablem V is governed by the ®rst-order reaction equations [51]:
dm=dtam 1ÿm ÿbmm; 5
where the rate-constants (sec)ÿ1 are taken from [64]
am ÿ1:74 V Erÿ11=fexpÿ V Erÿ11=12:94 ÿ1g; 6
bm0:06 V Erÿ5:9=fexp V Erÿ5:9=4:47 ÿ1g: 7
If the rates of change of (m) with respect to time are altered by a dimensionless factoru[68], then Eq. (5) becomes
udm=dtam 1ÿm ÿbmm; 8
which takes into consideration the slow activation time of persistent sodium current (see [69]). Consequently, it can be shown by linearizing the INaP current in the vicinity of the resting
membrane potential along the same lines as in [48] that a linearized version of Eq. (3) becomes
INaPgNaPmr V ÿVNaP
smr=ugNaPVNaPexp ÿt=smrfmr dam=dVr
dbm=dVr ÿ dam=dVrgV; t>0; 9
wheremramr= amrbmr, smr1= amrbmr, andamr and bmr are Eqs. (6) and (7) with
volt-ages set equal to zero, respectively. It should be noted that Eq. (9) is strictly applicable to small excursions from a resting state (i.e.,V0) of a few millivolts and not greater than about 5 mV (see Fig. 2), but it remains a relatively good approximation before the I±V relationship bends
upwards. Thus, linearization is appropriate for membrane potentials typically under 30 mV, as depicted in Fig. 2.
The inclusion of an outward current would eectively remove thegNaPmr VNaPterm in Eq. (9)
by conservation of current that corresponds to the non-decaying (persistent) level, thus causing the membrane potential to decay over time to its resting state. One interesting candidate for the outward current is a slowly inactivating (calcium-independent) A-type K current (I
AS ) recently
found to have a critical role in the subthreshold depolarization of neostriatal neurons [39,70]. Another is the hyperpolarization-activated current (Ih) known to decrease EPSP amplitude and
duration as well as the time window over which temporal summation of EPSPs takes place [71]. In fact, bothIAS andIh may act in concert to reduce the excitability in the dendrites caused byINaP.
In the present theory the maintenance of zero ionic current at rest (i.e., when I 0) is ac-complished with a sodium pump, which counters the resting ionic ¯uxes in order to maintain equilibrium so that ionic conductances are at their resting values and the membrane potential returns to the resting state. Consequently, the current balance equation at the site of the hot-spot reads:INaPÿP 0, wherePis the outward sodium pump current density.
2.3. Analytical time-dependent solutions
If the outward sodium pump current density is positive (by convention) then P gNaPmrVNaP.
V x;t
is the synaptic current per unit surface (lA/cm). Hereaandb are taken to be constants;bis the charge carried by the synaptic current through the unit membrane surface per unit time. Later we useb0(in dimension ofbper unit time) to be the input charge corresponding to the area found by integrating I t. G is the GreenÕs function corresponding to the solution of the standard linear cable equation (see Appendix A). Here the hot-spot points i and sodium pump points z are juxtaposed on the cable. Techniques for obtaining analytical solutions to such linear problems have been derived for single cables [73,74], for multiple equivalent cylinders [75,76], and for arbitrarily branching cables [77].
The system of Volterra integral equations (10) can be solved numerically or analytically by the method of successive substitutions [78, pp. 182,183]. Both methods have advantages and disad-vantages, but we feel the analytic approach in contrast to a numerical solution provides a better theoretical insight into the problem, especially when faced with network modeling. Adopting the latter approach, Eq. (10) is re-written in the form corresponding to the voltage response at lo-cationxxp to a synaptic (current) injection at locationxx0, and hot-spot locations atxxi:
Vp t fp t gNaP
We formally present the solution to Eq. (11), but readers interested in a more mathematically meticulous treatment should consult [78] for the existence and uniqueness of solutions, the radius of convergence and the speed of convergence.
The Liouville±Neumann series or Neumann series solution of the system of Volterra integral equations (11) is given as a series of integrals:
Vp t fp t gNaP
Rpi t;s;gNaP Mpi t;s X
and thej-fold iterated kernels Mpi j are de®ned inductively by the relations
Mpi 1 t;s
By writing Eq. (12) in terms of the GreenÕs functions explicitly, it can be shown that
Vp t
whereRj, which is the remainder of the series, is
Rjg
It can be shown from the theory of linear integral equations (e.g., Ref. [78]) that the Neumann series solution given by Eq. (13) converges uniformly provided kgNaPGijqk<1 is satis®ed, and
that the remainder (14) tends to zero as j increases (see Ref. [78, pp. 10,11]). To prove this it suces to show that shqGij is bounded ons2(0,t] for 0<h<1, which can be shown by
calcu-lating the remainder of the GreenÕs function (see Ref. [79]).
3. Results
3.1. How hot-spot location with identical number and strength of persistent sodium ion channels aects the membrane potential
Lipowsky et al. [35] used a reduced compartmental model to simulate the eect of various densities and distributions of persistent Na channels on EPSP enhancement. They showed that somatic placement of INaP alone did not signi®cantly elevate the amplitude of EPSPs. They
showed dendritic INaP produced a large `plateau' potential with extreme slowing of the somatic
membrane potential decay. They introduced a transient potassium currentIA [80] in the dendrites
so that the simulated wave form matched the experimental data. They concluded that the most important parameter determining the role of dendritic INaP in synaptic integration is its
distri-butionrather than itslocal density.
However, since they did not consider unequal distributions of dendritic persistent Nachannels in their simulations, we investigated this problem by considering a cable of lengthL1.6 mm that has a diameter of 4 lm, and two variations in the spatial distribution of hot-spots. The ®rst variation is an equal distribution of hot-spots fromx0 tox0:1Llocated at length intervals of 0:1jL=N, where j1;2;. . .N, on the path between the soma x0. The second variation is a
uniform distribution of hot-spots along the entire length of cable at length intervals of jL=N, where j1;2;. . .N, for an identical number of hot-spots (N5) and an identical number of
persistent sodium channels per hot-spot (N 10=pd).
In Fig. 3, we show EPSPs aected by the presence of an outward current or sodium pump for two dierent sites of the synaptic input located at: (i)x0 0:1L(i.e., 160lm from the endx0),
and (ii)x0 0:9L(i.e., 1440lm from the endx0). It is possible to compare a discrete or sparse
distribution of persistent sodium channels as shown in Fig. 3(c) and (d), and a more dense or continuous distribution of persistent sodium channels as shown in Fig. 3(a) and (b) by keeping the total number of persistent sodium channels ®xed in both cases (i.e., NN 50).
It is evident from Fig. 3 that a continuous distribution appears to produce a greater ampli®-cation of EPSPs in comparison with the discrete distribution. Indeed, this dierence can be es-pecially apparent when the synaptic input is located close to the recording point and is in close vicinity to the hot-spots. Even for a very distal synaptic input (i.e., at 0.9L) it can be seen from Fig. 3 that the continuous distribution yields a slightly greater ampli®cation in comparison with the discrete distribution of hot-spots along the cable. These results are compatible with the concept that a higher density of persistent sodium channels per hot-spot, should produce a greater ampli®cation in the voltage response in comparison to a sparser density distribution of persistent sodium channels.
importance of the local density ofINaP for distally placed synapses from the soma or recording
site.
3.2. How the number of hot-spots (N) with identical location and strength of persistent sodium ion channel densities aects the EPSP
The next problem was to investigate how the hot-spot number (N) aects the EPSP, assuming identical location and strength of persistent sodium ion channel densities along the dendritic cable. This was done by considering a cable of lengthL1:6 mm that has a diameter of 4lm, and a uniform distribution of hot-spots fromx0 toxLlocated at length intervals ofjL=N, where
j1;2;. . .N. The current input is assumed to be located at: (i)x0 0:1L(i.e., 160 lm from the
endx0) and (ii) x0 0:9L(i.e., 1440 lm from the endx0) for two Nvalues: (a), (b)N10, Fig. 3. The ampli®cation of electrotonic potentials mediated by persistent sodium hot-spots distributed: (a) and (b) near the x0 region and (c) and (d) uniformly. In each case the location of synaptic input is at: x00:1L and x00:9=L, respectively. The number of hot-spots is assumed to beN5 in each case, together with the following
parameters as used in [35]:Ri200Xcm,Cm1lF/cm2andRm50,000Xcm2,d4lm,k0.158 cm,b00.438
lA/(cm ms),a0.25/ms,gNaP0.143lS/cm andu0.2073. The values ofbandawere selected arbitrarily to yield a
response atx0 of approximately 1 mV (in the absence of hot-spots) for a synaptic input current located atx00:6L,
and the value ofgNaPwas based on the number of persistent sodium channels per hot-spot ofN10/pd. The total
membrane capacitance per unit area Cmcm/pd, where cmc0Ncm (cf. Ref. [65]) is the total capacitance of
membrane per unit length (lF/cm),c
m810ÿ18F is the capacitance associated with a single sodium channel [65],
andc01lF/cm2 is the capacitance of membrane per unit length without any channels. Hence, in theory the total
and (c), (d) N20. In Fig. 4, all EPSPs are measured at the pointx0 in the presence of an outward current or sodium pump. The results in Fig. 4 clearly show that a greater number of hot-spots results in greater ampli®cation of the distal synaptic signal. What is particularly interesting from Fig. 4 is that for a greater number of persistent sodium channels distributed along the cable, the peak amplitude of the EPSP is not signi®cantly increased, but rather the time-course of the EPSP is broadened. Furthermore, the spatial distribution of INaP need not be along the entire
length of the cable to produce signi®cant broadening of the response.
The ampli®cation of small graded potentials like EPSPs produces a much slower time course of decay, which was initially believed to be caused by NMDA receptor activation in dendrites [14,81], but the broadening was not as great as seen here. Furthermore, prolonged broadening of the EPSP time-course is not observed in some neurons (see Ref. [35]), which may indicate the presence of other rapidly gating channels in the nerve cell membrane. However, we believe that the process of linearization of the somatic INaP introduces an inductive element that, if in large
numbers, could be the major cause behind such intense broadening of the EPSP time-course. Thus the linearization ofINaPappears insucient to explain most results related to EPSP modulation.
Only full treatment of the non-linear eects of theI±V relationship could better explain our full understanding of these phenomena (see Ref. [57]).
The question of concern is whether there is a saturation point where further increases to the number of hot-spots distributed in the proximal dendrites results in no ampli®cation to the EPSP. It was found that there is no such optimal number for greatest ampli®cation to EPSP; i.e., a larger
number of hot-spots results in a more enhanced ampli®cation of EPSP (see Table 1). This is not surprising as it simply re¯ects a consequence of linearity in modeling the persistent sodiumI±V
relation. However, it is thought that an optimal density could be found if theINaP relation is left
non-linear (see Ref. [57]). It is clear from Table 1 that only 10 hot-spots distributed uniformly along the cable produce a 43% ampli®cation of the signal, while 20 hot-spots uniformly distrib-uted result in a 100% ampli®cation of the signal. An ampli®cation of 300% results if 50 hot-spots are uniformly distributed along the cable. These results are dependent on both the location of the synaptic input and the density distribution of hot-spots. The data in Table 1 only consider a single location of the synaptic input.
Alzheimer et al. [82] estimated about 17,000±23,000 transient sodium channels on isolated pyramidal neurons (excluding the axon). Given that only a small fraction of the transient channels are persistent sodium channels (1±2%), a rough estimate yields between 170 and 460 persistent sodium channels, which is in the ballpark observed here for moderate ampli®cations of the synaptic potentials (see Table 1). Poznanski and Bell [57], using a dierent approach, have shown similar estimates of persistent sodium channel densities represented as non-linear current sources.
3.3. How the strength (conductance) of persistent sodium ion channel densities with identical location and number of hot-spots aects the EPSP
It is also important to know the exact number of persistent sodium channels per hot-spot (assuming each hot-spot has an equal number of such channels). One way to proceed is to ex-amine the strength (i.e., conductance) of a single hot-spot (i.e., N1) or density of persistent sodium channels. In the case of a somatically placedINaP it has been shown that a 12%
ampli-®cation of distal synaptic input at the soma is observed from the experimental data [35]. In the GreenÕs function representation of the neuron, the soma is assumed to be space-clamped, so the distribution of hot-spots is only permissible at a single point (i.e., xp 0), which theoretically impliesN1. This value is used in the simulation with varyingNvalues to see what conductance value would bring an ampli®cation in the EPSPs to 12%.
The simulations based on the space-clamped soma approximation used in the present model indicate that a maximum attainable conductance ofgNap0:2860lS/cm orN20/pdis required Table 1
Ampli®cation of peak membrane potentiala
N (mV)
2 0.06
5 0.20
10 0.43
20 1.00
30 1.63
40 2.35
50 3.00
a
to produce a 12% ampli®cation in the peak of an EPSP (see Fig. 5). These results also indicate that a single hot-spot, regardless of the density, does not produce the broadening eect seen in Figs. 3 and 4, although greater density indicates a slightly higher peak amplitude. If the density is given by (20/pd)(Dÿ1), then this is equivalent to approximately 16/Dchannels per 10lm2 whereDis the
uniform spacing between successive hot-spots in microns. For example, if there is one persistent sodium channel per 10 lm2, then the spacing between hot-spots will be 16lm or approximately
0.01L (see Fig. 6).
Consequently, the maximum attainable speci®c conductancegmaxgNaP=D1:7875 pS/lm2 is
very close to the average speci®c conductance of 1.7 0.6 pS/lm2 for the persistent sodium
current, but it is only a small fraction (1.5%) of the speci®c conductance of 11311:6 pS/lm2 for
the transient sodium current [64]. Indeed, the estimate of one persistent sodium channel per 10lm2is also a small fraction (1.25%) of the density distribution of transient Nachannels to be about 80 channels per 10 lm2 in dendrites of CA1 hippocampal neurons [47]. These results
suggest that INaP may ¯ow through noninactivating Na channels that are distinct from the
transient Nachannels, or through the same channels with dierent gating mechanisms [82], with the density distribution of the activated channels being very sparse, in the vicinity of 1±2% in comparison to the transient sodium channels.
Fig. 5. The ampli®cation of EPSPs mediated by persistent sodium hot-spots distributed near thex0 region. The number of persistent sodium channels per hot-spot is assumed to be equal with: (a)N6/pd(i.e.,g
NaP0.0858lS/
cm), (b)N12/pd(i.e.,g
NaP0.1716lS/cm), (c)N16/pd(i.e.,gNaP0.2288lS/cm), and (d)N20/pd(i.e.,gNaP
0.2860lS/cm). The same parameters as those in Fig. 3 are used but the number of hot-spots is kept atN1 and a single synaptic location of x00:6Lis chosen. The dashed curve corresponds to the ampli®cation in the membrane
4. Discussion and conclusions
Elements of an ionic cable theory were developed by incorporating the density distribution of ion channels into cable models of neurons with time and voltage-dependent ionic current imposed discretely rather than continuously as assumed in classical non-linear cable theory. This was accomplished by superimposing a ®nite number of sodium current sources at discrete loci along the dendritic cable. The protocol diers from the segmented cable modeling used to investigate the saltatory conduction in axons (see Ref. [50]).
The rationale behind the modeling is the belief that a continuous distribution of voltage-dependent ion-channels in dendrites may be inappropriate because such channels occur more sparsely in dendrites as compared with axons, and so the continuum approximation used in cable theory needs to be modi®ed to include the discreteness of ion-channel density distribution. Experimental evidence for a relatively sparse density distribution of voltage-dependent sodium channels comes indirectly from the observed decrement in the amplitude of back-propagating sodium spikes [5,6].
Fig. 6. The density of persistent sodium channels per 10lm2 of cable is shown for the case ofN20/pd(i.e., 20
In this paper, we introduced various discrete density distributions of dendritic persistent so-dium channels (hot-spots) to study the eects of varying the distributions of persistent soso-dium channels on the enhancement of EPSPs, and to look at how dierent strengths of persistent so-dium densities aect processing of synaptic signals. Given that the contribution of such voltage-dependent ionic channels to dendritic information processing is at present not well understood, our results can elucidate the role of ampli®ed EPSPs in the initiation of sodium spike-like potentials in the dendrites.
The following conclusions emerged from the analysis: (i) Local density de®ned as the number of persistent sodium channels in close vicinity to the recording site, and not the spatial distribution of NaP channels, produces a greater increase in the peak amplitude of the EPSP. That is, peak EPSPs will be ampli®ed the most if a high density of NaP channels are in close vicinity to the activating synapses. (ii) Local density together with the spatial distribution of NaP channels produces broadening in the time-course of the ampli®ed EPSP. That is, the distribution of NaP channels along the entire ionic cable with a higher density near the synaptic region may result in a greater probability of action potential initiation. (iii) An increase in the number of NaP channels without any local density or spatial distribution will only slightly increase the peak amplitude of the ampli®ed EPSP. That is, the notion of NaP channels concentrated at high densities at the axosomatic region of the neuron will have a minimum eect on the dendritic integration of synaptic signals.
Acknowledgements
One of the authors (R.R.P.) is indebted to the late Professor Kohyu Fukunishi, and to Dr Masayoshi Naito for support and administrative assistance. J.B. was partially supported by National Science Foundation grant DMS-9706307.
Appendix A. GreenÕs function for some linear cable structures
The GreenÕs function G x;x0;t corresponds to the response at time t at location x to a unit
impulse at location x0 at time t0, and is given by the solution of the following initial-value
problem:
CmGt x;x0;t d=4RiGxx x;x0;t ÿG x;x0;t=Rm;
G x;x0;0 d xÿx0:
For a multiple cable, the GreenÕs function is de®ned in notation Gk
j xj;xk0;t to represent the
response at a pointxxjin thejth cable to a unit impulse input at a locationxx0along thekth
cable with the above initial-value problem changed to:
CmoGkj xj;xk0;t=ot dj=4Rjmo2Gjk xj;xk0;t=ox2ÿGkj xj;xk0;t=Rjm;
A.1. Single cable with sealed ends
This case represents a single ®nite cable of length (L) in cm with both ends sealed (i.e.,
Gx 0;x0;t Gx L;x0;t 0). A representation which converges fast for larget is [72]
An alternative representation which converges more rapidly at small times is given by [59]
G x;x0;t Rm=kexp ÿt=sm
whereH : represents the Heaviside step function.
A.2. Single cable with lumped-soma
conduc-An x0 2ccoshn=sinwnL=kf c=wnÿwnL=k c=k x0ÿLtanhng coswnL=kf 2cL=k
This case represents a multiple number (P) of ®nite cables of length (Lj ) with a lumped-soma
attached to the end xj0 (i.e., G 0;xj;t Gt 0;xj;t ÿcGx 0;xj;t 0, where c
and the coecientsEkn are de®ned as
Ekn2ck 2
The coecients Ek0 need particular care since the ®rst root is x00 with the corresponding
coecient
[1] S.J. Redman, A quantitative approach to integrative function of dendrites, in: R. Porter (Ed.), International Review of Physiology: Neurophysiology II, vol. 10, University Park Press, Baltimore, MD, 1976.
[2] N.N. Urban, G. Barrionuevo, Active summation of excitatory postsynaptic potentials in hippocampal CA3 pyramidal neurons, Proc. Nat. Acad. Sci. USA 95 (1998) 11450.
[4] G. Stuart, N. Spruston, B. Sakmann, M. Hausser, Action potential initiation and backpropagation in neurons of the mammalian CNS, Trends Neurosci. 20 (1997) 125.
[5] H.-Y. Jung, T. Mickus, N. Spruston, Prolonged sodium channel inactivation contributes to dendritic action potential attenuation in hippocampal pyramidal neurons, J. Neurosci. 17 (1997) 6639.
[6] C.M. Colbert, J.C. Magee, D.A. Homan, D. Johnston, Slow recovery from inactivation of Na channels
underlies the activity-dependent attenuation of dendritic action potentials in hippocampal CA1 pyramidal neurons, J. Neurophysiol. 17 (1997) 6512.
[7] W. Regehr, J. Kehoe, P. Ascher, C. Armstrong, Synaptically triggered action potentials in dendrites, Neuron 11 (1993) 145.
[8] S.B. Colling, H.V. Wheal, Fast sodium action potentials are generated in the distal apical dendrites of rat hippocampal CA1 pyramidal cells, Neurosci. Lett. 172 (1994) 73.
[9] M. Andreasen, J.D.C. Lambert, Regenerative properties of pyramidal cell dendrites in area CA1 of rat hippocampus, J. Physiol. (Lond.) 483 (1995) 421.
[10] G. Stuart, J. Schiller, B. Sakmann, Action potential initiation and propagation in rat neocortical pyramidal neurons, J. Physiol. (Lond.) 505 (1997) 617.
[11] N.L. Golding, N. Spruston, Dendritic sodium spikes are variable triggers of axonal action potentials in hippocampal CA1 pyramidal neurons, Neuron 21 (1998) 1189.
[12] C.E. Stafstrom, P.C. Schwindt, M.C. Chubb, W.E. Crill, Properties of persistent sodium conductance and calcium conductance of layer V neurons from cat sensorimotor cortex in vitro, J. Neurophysiol. 53 (1985) 153.
[13] A.M. Thomson, D. Girdlestone, D.C. West, Voltage-dependent currents prolong single-axon postsynaptic potentials in layer III pyramidal neurons in rat neocortical slices, J. Neurophysiol. 60 (1988) 1896.
[14] B. Sutor, J.J. Hablitz, EPSPs in rat neocortical neurons in vitro. II Involvement of NMDA receptors in the generation of EPSPs, J. Neurophysiol. 61 (1989) 621.
[15] R.A. Deisz, G. Fortin, W. Zieglgansberger, Voltage dependence of excitatory postsynaptic potentials of rat neocortical neurons, J. Neurophysiol. 65 (1991) 371.
[16] A. Nicoll, A. Larkman, C. Blakemore, Modulation of EPSP shape and ecacy by intrinsic membrane conductances in rat neocortical pyramidal neurons in vitro, J. Physiol. (Lond.) 468 (1993) 693.
[17] G. Stuart, B. Sakmann, Ampli®cation of EPSPs by axosomatic sodium channels in neocortical pyramidal neurons., Neuron 15 (1995) 1065.
[18] P.C. Schwindt, W.E. Crill, Membrane properties of cat spinal motoneurons, in: R.A. Davido (Ed.), Handbook of the Spinal Cord: Anatomy and Physiology, vols. 2 & 3, Marcel Dekker, New York, 1984.
[19] J.C. Magee, D. Johnston, Synaptic activation of voltage-gated channels in the dendrites of hippocampal pyramidal neurons, Science 268 (1995) 301.
[20] A.M. Vallet, J.A. Coles, J.C. Eilbeck, A.C. Scott, Membrane conductances involved in ampli®cation of small signals by sodium channels in photoreceptors of drone honey bee, J. Physiol. (Lond.) 456 (1992) 303.
[21] S. Hidaka, A.T. Ishida, Voltage-gated Na current availability after step- and spike-shaped conditioning
depolarizations of retinal ganglion cells, P¯ugers Arch. 436 (1998) 497.
[22] C.E. Stafstrom, P.C. Schwindt, W.E. Crill, Negative slope conductance due to a persistent subthreshold sodium current in cat neocortical neurons in vitro, Brain Res. 236 (1982) 221.
[23] P.C. Schwindt, W.E. Crill, Ampli®cation of synaptic current by persistent sodium conductance in apical dendrite of neocortical neurons, J. Neurophysiol. 74 (1995) 2220.
[24] B.A. MacVicar, Depolarizing prepotentials are Nadependent in CA1 pyramidal neurons, Brain Res. 333 (1985)
378.
[25] C.P. Taylor, Na currents that fail to inactiviate, Trends Neurosci. 16 (1993) 455.
[26] W.E. Crill, Persistent sodium current in mammalian central neurons, Annu. Rev. Physiol. 58 (1996) 349. [27] T. Gillessen, C. Alzheimer, Ampli®cation of EPSPs by low Ni2 and amiloride-sensitive Ca2channels in apical
dendrites of rat CA1 pyramidal neurons, J. Neurophysiol. 77 (1997) 1639.
[28] N.N. Urban, D.A. Henze, G. Barrionuevo, Amplifcation of perforant-path EPSPs in CA3 pyramidal cells by LVA calcium and sodium channels, J. Neurophysiol. 80 (1998) 1558.
[30] A.L. Byzov, A.Yu. Trifonov, L.M. Chailahian, K.W. Golubtzov, Ampli®cation of graded potentials in horizontal cells of the retina, Vis. Res. 17 (1977) 265.
[31] G.C. Taylor, J.A. Coles, J.C. Eilbeck, Conditions under which Nachannels can boost conduction of small graded
potentials, J. Theoret. Biol. 172 (1995) 379.
[32] O. Bernander, C. Koch, R.J. Douglas, Ampli®cation and linearization of distal synaptic input to cortical pyramidal cells, J. Neurophysiol. 72 (1994) 2743.
[33] E. De Schutter, J.M. Bower, Simulated responses of cerebellar Purkinje cells are independent of the dendritic location of granule cell synaptic inputs, Proc. Nat. Acad. Sci. USA 91 (1994) 4736.
[34] J.A. White, N.S. Sekar, A.R. Kay, Errors in persistent inward currents generated by space-clamp errors: a modeling study, J. Neurophysiol. 73 (1995) 2369.
[35] R. Lipowsky, T. Gillessen, C. Alzheimer, Dendritic Nachannels amplify EPSPs in hippocampal CA1 pyramidal
cells, J. Neurophysiol. 76 (1996) 2181.
[36] E.P. Cook, D. Johnston, Active dendrites reduce location-dependent variability of synaptic input trains, J. Neurophysiol. 78 (1997) 2116.
[37] E.P. Cook, D. Johnston, Voltage-dependent properties of dendrites that eliminate location-dependent variability of synaptic input, J. Neurophysiol. 81 (1999) 535.
[38] E. De Schutter, Dendritic voltage and calcium-gated channels amplify the variability of postsynaptic responses in Purkinje cell model, J. Neurophysiol. 80 (1998) 504.
[39] H. Takagi, R. Sato, M. Mori, E. Ito, H. Suzuki, Roles of A- and D-type K channels in EPSP integration at a model dendrite, Neurosci. Lett. 254 (1998) 165.
[40] J.W. Mozrzymas, M. Bartoszkiewicz, The discrete nature of biological membrane conductance, channel interaction through electrolyte layers and the cable equation, J. Theoret. Biol. 162 (1993) 371.
[41] H.P. Larsson, S.J. Kleene, H. Lecar, Noise analysis of ion channels in non-space-clamped cables: estimates of channel parameters in olfactory cilia, Biophys. J. 72 (1997) 1193.
[42] F. Andrietti, G. Bernardini, Segmented andÔequivalentÕrepresentation of the cable equation, Biophys. J. 46 (1984) 615.
[43] S.M. Baer, C. Tier, An analysis of a dendritic neuron model with an active membrane site, J. Math. Biol. 23 (1986) 137.
[44] S.M. Baer, J. Rinzel, Propagation of dendritic spikes mediated by excitable spines: a continuum theory, J. Neurophysiol. 65 (1991) 874.
[45] J. Bell, M. Holmes, Model of the dynamics of receptor potential in a mechano-receptor, Math. Biosci. 110 (1992) 139.
[46] T.I. Toth, V. Crunelli, Eects of tapering geometry and inhomogeneous ion channel distribution in a neuron model, Neurosci. 84 (1998) 1223.
[47] J. Magee, D. Johnston, Characterization of single voltage-gated Naand Ca2channels in apical dendrites of rat
CA1 pyramidal neurons, J. Physiol. (Lond.) 487 (1995) 67.
[48] N.H. Sabah, K.N. Leibovic, Subthreshold oscillatory responses of the Hodgkin±Huxley cable model for the squid gaint axon, Biophys. J. 9 (1969) 1206.
[49] R. Llinas, C. Nicholson, Electrophysiological properties of dendrites and somata in alligator Purkinje cells, J. Neurophysiol. 34 (1971) 532.
[50] R. FitzHugh, Computation of impulse initiation and saltatory conduction in a myelinated nerve ®ber, Biophys. J. 2 (1962) 11.
[51] A.L. Hodgkin, A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (Lond.) 117 (1952) 500.
[52] R.J. MacGregor, A model for responses to activation by axodendritic synapses, Biophys. J. 8 (1968) 305. [53] E.L. Madsen, Theory for a test of the electric cable model of myelinated axon and saltatory conduction, J. Theoret.
Biol. 67 (1977) 203.
[54] P.J. Basser, Cable equation for a myelinated axon derived from its microstructure, Med. Biol. Eng. Comput. 31 (1993) S87.
[55] J. Keizer, G.D. Smith, S. Ponce-Dawson, J.E. Pearson, Saltatory propagation of Ca2 waves by Ca2 sparks,
[56] M.R. Bennett, W.G. Gibson, R.R. Poznanski, Extracellular current ¯ow and potential during quantal transmission from varicosities in a smooth muscle syncytium, Philos. Trans. R. Soc. 342 (1993) 89.
[57] R.R. Poznanski, J. Bell, Math. Biosci., this issue, p. 123.
[58] R. FitzHugh, Dimensional analysis of nerve models, J. Theoret. Biol. 40 (1973) 517.
[59] R.R. Poznanski, Membrane voltage changes in passive dendritic trees: a tapering equivalent cylinder model, IMA J. Math. Appl. Med. Biol. 5 (1988) 113.
[60] J.D. Evans, G.C. Kember, Analytical solutions to a tapering multicylinder somatic shunt cable model for passive neurones, Math. Biosci. 149 (1998) 137.
[61] N.H. Sabah, R.A. Spangler, Repetitive response of the Hodgkin±Huxley model for the squid gaint axon, J. Theoret. Biol. 29 (1970) 155.
[62] A. Mauro, F. Conti, F. Dodge, R. Schor, Subthreshold behavior and phenomenological impedance of the squid giant axon, J. Gen. Physiol. 55 (1970) 497.
[63] J. Schiller, G. Major, H.J. Koester, Y. Schiller, NMDA spikes in basel dendrites of cortical pyramidal neurons, Nature 404 (2000) 285.
[64] C.R. French, P. Sah, K.J. Buckett, P.W. Gage, A voltage-dependent persistent sodium current in mammalian hippocampal neurons, J. Gen. Physiol. 95 (1990) 1139.
[65] A.L. Hodgkin, The optimum density of sodium channels in an unmyelinated nerve, Philos. Trans. R. Soc. Lond. B 270 (1975) 297.
[66] F.J. Sigworth, E. Neher, Single sodium channel currents observed in cultured rat muscle cells, Nature 287 (1980) 497.
[67] W. Stuhmer, B. Methfessel, B. Sakmann, M. Noda, S. Numa, Patch clamp characterization of sodium channels expressed from rat brain cDNA, Eur. Biophys. J. 14 (1987) 131.
[68] N.H. Sabah, K.N. Leibovic, The eect of membrane parameters on the properties of the nerve impulse, Biophys. J. 12 (1972) 1132.
[69] H. Hayashi, S. Ishizuka, Chaotic nature of bursting discharge in theOnchidiumpacemaker neuron, J. Theoret. Biol. 156 (1992) 269.
[70] L.A. Gabel, E.S. Nisenbaum, Biophysical characterization and functional consequences of a slowly inactivating potassium current in neostriatal neurons, J. Neurophysiol. 79 (1998) 1989.
[71] J.C. Magee, Dendritic hyperpolarization-activated currents modify the integrative properties of hippocampal CA1 pyramaidal neurons, J. Neurosci. 18 (1998) 7613.
[72] H.C. Tuckwell, Instruction to Theoretical Neurobiology. Linear Cable Theory and Dendritic Structure, vol. 1, Cambridge University, Cambridge, 1988.
[73] D. Durand, The somatic shunt cable model for neurons, Biophys. J. 46 (1984) 645.
[74] M.B. Jackson, Cable analysis with the whole-cell patch clamp: theory and experiment, Biophys. J. 61 (1992) 756. [75] J.D. Evans, G.C. Kember, G. Major, Techniques for obtaining analytical solutions to the multicylinder somatic
shunt cable model for passive neurones, Biophys. J. 63 (1992) 350.
[76] J.D. Evans, The multiple equivalent cylinder model, in: R.R. Poznanski (Ed.), Modeling in the Neurosciences: From Ionic Channels to Neural Networks, Harwood Academic Publishers, Amsterdam, 1999 (Chapter 5). [77] G. Major, J.D. Evans, J.J.B. Jack, Solutions for transients in arbitrarily branching cables, Biophys. J. 65 (1993)
423.
[78] W. Pogorzelski, Integral Equations and Their Applications, vol. 1, Pergamon, Oxford, 1966.
[79] R.R. Poznanski, Analysis of a postsynaptic scheme based on a tapering equivalent cable model, IMA J. Math. Appl. Med. Biol. 7 (1990) 175.
[80] D.A. Homan, J.C. Magee, C.M. Colbert, D. Johnston, Kchannel regulation of signal propagation in dendrites
of hippocampal pyramidal neurons, Nature 387 (1997) 869.
[81] A. Nicoll, A. Larkman, C. Blakemore, EPSPs in rat neocortical pyramidal neurones in vitro are prolonged by NMDA receptor-mediated currents, Neurosci. Lett. 143 (1992) 5.
[82] C. Alzheimer, P.C. Schwindt, W.E. Crill, Model gating of Nachannels as a mechanism of persistent Nacurrent
