Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol166.Issue2.Aug2000:

Theoretical analysis of the ampli®cation of synaptic potentials

by small clusters of persistent sodium channels in dendrites

R.R. Poznanski

a,*

, J. Bell

b,1 a

Advanced Research Laboratory, Hitachi, Ltd., Hatoyama, Saitama, 350-0395, Japan b

Department of Mathematics, State University of New York, Bu€alo, NY 14214-3093, USA Received 26 October 1999; received in revised form 27 May 2000; accepted 1 June 2000

Abstract

We extend on the work developed by R.R. Poznanski and J. Bell from a linearized somatic persistent sodium current source to a non-linear representation of the dendritic Na‡_{P current source associated with a} small number of persistent sodium channels. The main objective is to investigate the modulation in the ampli®cation of excitatory postsynaptic potentials (EPSPs) in dendrites studded with persistent sodium channels. The relation between membrane potential (V) and persistent sodium current density (INaP) is approximated heuristically with a sigmoidal function and the resultant cable equation is solved analytically using a regular perturbation expansion and GreenÕs function techniques. The transient simulated (non-evoked) response is found as a result of current injection in the form of synaptically induced voltage change located at a distance from the recording site in a cable with a uniform distribution of ion channel densities per unit length of cable (the so-called `hot-spots') and with the conductance of each hot-spot (i.e., number of channels per hot-spot) assumed to be a constant. The results show an ampli®cation in the observed EPSPs to be compatible with the experimentally derived estimates, and in addition a saturation in the ampli®cation is observed indicating an optimum number of ionic channels. Ó _{2000 Elsevier Science Inc.}

All rights reserved.

Keywords: Dendritic Na‡_{P channels; Optimum density; Non-linear ionic cable theory; Analytical solutions;}

Perturbation expansion; Comparison methods; Neuronal modeling

*_{Corresponding author. Tel.: +81-492 96 6111; fax: +81-492 96 6006.} E-mail address:poznan@harl.hitachi.co.jp (R.R. Poznanski). 1

Present address: Department of Mathematics, UMBC, Baltimore, MD 21250, USA.

0025-5564/00/$ - see front matter Ó _{2000 Elsevier Science Inc. All rights reserved.}

1. Introduction

Classical non-linear cable theory (see Refs. [1±11]) assumes voltage-dependent conductances are distributed continuously along the entire length of the cable, which although suitable for studies of axonal membrane may not be applicable for sparse distributions of voltage-dependent ionic channels in the dendrites of neurons (see Refs. [12,13]). To compensate for this caveat in the classical theory, Poznanski and Bell [14] introduced a dendritic cable model based on the idea that discrete loci of voltage-dependent ion channels or hot-spots constitute active point sources of transmembrane current, imposed on a homogeneous (non-segmented) leaky cable structure with each hot-spot assumed to occupy an in®nitesimal region containing a single or a large number of voltage-dependent ionic channels. The approach is new and di€ers from the computational ap-proach of Steinberg [15] in that hot-spots of non-inactivating sodium channels are imposed on a passive cable rather than on an active cable described by the Hodgkin±Huxley equations [1].

Experimental studies have shown that the subthreshold ampli®cation of synaptic potentials is mediated by persistent sodium channels [16±19]. Assuming the persistent sodium current density (INaP) ¯ows through ionic channels that are distributed at a number of discrete locations along a cable of length (L), the aim of this paper is to show how a non-linear I±V relation for the per-sistent sodium current (INaP) e€ects the modulation of EPSPs. The problem is not new as Baginskas and Gutman [20,21] investigated the propagation of synaptic potentials in non-linear cables and dendritic structures. The novelty rests on the utilization of an ionic cable model with a discreteclustering of ionic channels. The major signi®cance of this approach is that it provides an alternative method for obtaining results which can be subsequently tested with the more common approach of approximating the distributed, continuous membrane of the neuron with a discrete set of interconnected compartments (see Refs. [22±28]). The perturbative technique that we shall utilize was used in a neurophysiological context by Tuckwell [29,30].

2. Cable equation for discretely imposed persistent sodium channels

LetVbe the depolarization (i.e., membrane potential less the resting potential assumed to be ®xed and uniform) in mV, and INaP be 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 of diameter d with INaP occurring at discrete points along the cable as depicted in Fig. 1, satis®es the following equation:

CmVt ˆ …d=4Ri†VxxÿV=Rm‡ XN

iˆ1

INaP…x;t;V†d…xÿxi† ‡I…x;t†d…xÿx0†; t>0; …1†

where Ix;t_† is the applied current density per unit membrane surface of cable in A/cm, x the distance in cm, t the time in s, d the diameter of the cable in cm, Cm …ˆcm=pd† the membrane capacitance (F/cm2_{), R}

positioned (and characterizes the in®nitesimal nature of our hot-spots). Subscriptsxandtindicate partial derivatives with respect to these dimensional variables.

Eq. (1) can be cast in terms of non-dimensional space and time variables, X _ˆx=k and

T _ˆt=sm, respectively, where k_ˆRmd=4Ri†

1=2

and sm_ˆRmCm are, respectively, the space and time constants. Thus Eq. (1) becomes

VT ˆVXX ÿV ‡

XN

iˆ1

…Rm=k†INaP…X;T;V†d…X ÿXi†

‡ …Rm=k†I…X;T†d…X ÿX0†; T >0; 0<X <LˆL=k; …2†

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 andN_{denotes the number of persistent sodium channels in each hot-spot}

whereXi_ˆxi=kandX0ˆx0=krepresent loci along the cable of ionic current and synaptic current, respectively, expressed in terms of the dendritic space constant. Eq. (2) will be used in the sub-sequent perturbation analysis (see Section 4).

Composite EPSP input is the number of synapses impinging at equal electrotonic distances on the dendritic tree, modeled as a single synapse at a particular point along the equivalent cable, with a peak current amplitude, multiplied by the maximum peak current amplitude for a single synapse at that electrotonic distance. Exact values governing the mapping between the tree and equivalent cable can be measured (see, [31,32]), but geometrically, a profusely branched neuron with anywhere from 50 to 500 synapses on dendrites impinging distally at equal electrotonic distance from the soma would correspond approximately to a single synapse on an equivalent cable. Hence, a single synaptic input would imply anywhere between 50 and 500 synapses. Therefore the present model implicitly considers the e€ects of a multiple number of synapses impinging on the dendrites via a single location on the equivalent dendritic cable.

3. A heuristic approximation of the dendritic persistent sodium current

The persistent sodium current density at discrete loci along the cable associated with only a few persistent sodium channels can be approximated heuristically from a non-linear (instantaneous) input I±V(iNaP) relationship as obtained by French et al. [33] for an ensemble average or mac-roscopic current measured from the somata of dissociated hippocampal cells as shown in Fig. 2. In practice, however, the value of the membraneI±VrelationINaPalong the cable at loci (xi) at a

speci®c time (ti) will be determined from the value of V at that point. Therefore, an equation

analogous to ColeÕs theorem2 should be used to connect INaP, with the iNaP obtained from in-tracellular recording with a patch-pipette. This is because in most cases the input I±Vrelation is less non-linear than the membrane I±V relation as expressed by the relationship (see Ref. [5])

INaP…xi;ti;V† ˆ …Ri=pd2†‰iNaP=…dV0…xi;ti†=diNaPÿdVL…xi;ti†=diNaP†Š; …3†

whereV0 andVL represent (dimensional)Vatxˆ0 andxˆL, respectively. In practice, Eq. (3) is

limited to the use of a dual intracellular recording method to measure the voltage at two distinct points. Thus a simpler approach is to assume the sodium channel gating occurs suciently quickly for it to be regarded as occurring instantaneously. That is, unlike the macroscopic INaP which persists for a long time, estimated by using whole-cell patch clamp recording at the soma, the dendriticINaPcorresponds to a small number of ionic channels and lasts for only a short period of time denoted by Dt0 from t_ˆ0 to t_ˆt0 depending on the stochastic properties of individual channels leading to an average open time. Also at each unique dendritic location associated with the distribution of persistent sodium channels a non-linearity in the membrane I±V relation

2_Cole

follows a relatively similar shape (i.e., sigmoidal) from the somatically recorded instantaneous inputI±Vrelation of the macroscopic current.

Kay et al. [34] estimated approximately 7375 persistent sodium channels on the somata of Purkinje cells based on a `whole-cell' conductance of 118 ns (assuming a maximum single channel conductance ofg<sub>NaP ˆ</sub>16 pS), which is two orders of magnitude larger than the expected number of channels on the somata of hippocampal cells (i.e., approximately 45) based on a whole-cell conductance of only 0.82 ns [33] (assuming a maximum single channel conductance ofg<sub>NaPˆ</sub>18 ps). Unfortunately the assumption that the whole-cell conductance equals the single channel conductance times the number of channels is limited to the somata of neurons as dendrites are rarely perfectly voltage-clamped, so assuming the 45 channels is limited to the soma of hippo-campal neurons, the next problem is to show the approximate number per unit surface of den-drite. As it is experimentally dicult to obtain a true estimate of the whole-cell conductance, the estimate is also assumed to apply to the dendrites, as well as the soma, except we introduce a spatial `scaling' parameter e1_†, e.g. h_ˆ45 and e_ˆ1 (at soma only) based on a `whole-cell' conductance of 0.82 ns, leaving the conductance ®xed at gNaPˆ0:0183=pd nsˆ0:0145ls=cm. Note that the case ofe_ˆ1 is ®ctitious becausex_ˆ0 represents only a point close to the soma and therefore does not violate the assumption ofe1 along the dendritic cable.

Spatial±temporal `scaling' of the persistent sodium current is necessary because, unlike the whole-cell macroscopic current measured at the soma, the current per hot-spot involves only a small number of ionic channels generating a smaller peak amplitude and requiring a smaller time window for opening of channels. Hence, in the analysis below, e will be considered small. As-suming for the entire duration persistent sodium channels are open, no variation in the activation variableFoccurs, so time derivatives with respect to time ofFcan be ignored as would be the case if INaP remained constant during the period the channels remained open. Recent experimental work supports the notion of current sources of excitation along the entire length of a dendrite being activated simultaneously [35], and therefore the assumption that all the persistent sodium ionic channels remain open during t_{2 ‰}0;t0Š is conceivable.

Thus, an instantaneous voltage dependence of the persistent Na‡ _{transmembrane current}

density per membrane surface of cable (lA/cm) evaluated at the end of the channel opening (t0) can be approximated heuristically as

INaP…x;t;V† iNaP…x;Dt0;V† ˆegNaPF‰V…x;t0†ŠDt0; …4†

where the strength (maximum conductance) of persistent sodium ion channel densities is given by (cf. Ref. [36])

gNaPˆgNaP N: …5†

Here N _ˆh=pd is the number (h) of persistent sodium channels per unit membrane surface of cable in cmÿ1_{, and g}

NaPˆ18 pS is the maximum attainable conductance of a single sodium channel measured by Sigworth and Neher [37] and Stuhmer et al. [38]. The parameterescales the whole-cell conductance at the soma (assumed to be ®xed at gNaPˆ0:0145lS=cm, see below) to re¯ect the conductance of a cluster of small numbers of Na‡_{P channels per hot-spot along the}

surface of cable, and Dt0 ‰H…t† ÿH…tÿt0†Š is a parameter `scaling' the time interval of the somatic whole-cell macroscopic current in terms of dendritic channel openings of a few persistent sodium channels, witht0representing the maximum time the cluster of channels remains open and H() denotes the Heaviside-step function.

The activation variableFV_†is represented by the following sigmoidal function:

FV_{† ˆ}1=_f1_‡exp_‰V0_ÿV_†=k_{Šg ÿ}1=_f1_‡ exp_‰V0=k_Šg†‰V _ÿV_NaP0 _Š; 6_†

whereV0_{, V}0

NaP, andkare experimentally determined constants. In order to ®t data obtained by French et al. [33] from dissociated hippocampal cells, the following values provided the best ®t to the data:V0_{ˆ46,kˆ9, andV}0

NaPˆ195 mV. A non-linear representation of the persistent sodium current as expressed by Eq. (6) is shown in Fig. 2 for the experimentally measured somatic whole-cell current, and for the heuristic representation of a dendritic hot-spot current. Note thatINaPas expressed by Eq. (4) is a sink of current since by convention inward current is negative and outward is positive. Combining expression (4) with Eq. (2), our model takes the form

VT _ˆVXX _ÿV _‡egNaP…Rm=k† XN

iˆ1

F_‰VX;T0†ŠDT0d…X ÿXi† ‡ …Rm=k†I…X;T†d…X ÿX0†; …7†

whereDT0 ‰H…T† ÿH…T ÿT0†ŠandT0ˆt0=smwith the potential initially at rest and sealed-end boundary conditions imposed at each end of the neural segment (X_ˆ0,L_{). Although Eq. (7) is in}

mem-brane potential in Eq. (7) by some characteristic value of the memmem-brane potential Vch. A rea-sonable value forVchwould be the resting potential. Though we keep the same notation below for the analysis, the perturbation analysis should be thought of as being done in non-dimensional variables, and the results expressed in dimensional terms.

4. A perturbative expansion for the ampli®cation of EPSPs

Let the depolarization be represented as a membrane potential perturbation from the passive (RC-cable) voltage responseUX;T_†in the form of a perturbative expansion

VX;T_{† ˆ}UX;T_{† ‡}X

rived in Appendix A. Eq. (8) represents a perturbation series expansion at an equilibrium point

UX0;T0† and the correction terms have been derived in Appendix B. There is no restriction in using Eq. (8) to investigate large EPSPs in the distal dendrites as membrane potentials with peak amplitude approaching 10 mV can be adequately represented through UX;T_† as shown in Section 5.

It was found through a heuristic approach thatU(Xi,T0) is an equilibrium point for only a few values of T0 2 ‰0:06;0:12Š, while outside this range of values, the series solution can generate spurious behavior. Thus, the regular perturbation theory method restricts the dendritic membrane (instantaneous)I±Vcurve to a time-window that re¯ects the average open time for single sodium channels during bursts of sustained activity due to non-inactivating sodium channels (see Refs. [39,40]). In other words, all ionic channels having an average opening time which may only last for a few msecs in duration will remain closed for the remainder of the transient response measured by Eq. (8). In evaluating Eq. (8) parameter (e) was so selected to ensure higher order terms could be neglected, and Eq. (8) is approximated to O(e4_{). In most cases this is satisfactory as the e}5_V5 term is negligible in comparison with eV1. If he is not small, then a higher number of terms is needed.

In Fig. 3 we illustrate the ®rst four perturbative terms for the largestehratio possible in order for the alternating series to converge. As is evident from Fig. 3 even in this situation the fourth term is relatively small in comparison to the ®rst term, but still contributes to the overall boosting e€ect. A justi®cation is that ampli®ed EPSPs have a ‹0.5 mV variation in their membrane po-tential that is usually associated with noise (see Fig. 6) so the present results will not be e€ected by truncating higher order terms in the perturbation expansion. The di€erence becomes even more greater when smaller eh values are considered. The evaluation using Mathematica (version 3) software required under 1 min on a Hitachi Flora 370 PC work station if only the ®rst four terms in Green_Õs function are used forN_ˆ300 or less. A greater amount of computing time is required if more than N_ˆ300 hot-spots are assumed and/or greater number of terms are used in GreenÕs function. The ®rst term dominates with other terms decreasing exponentially with time so no signi®cant di€erence in results occurred by truncating GreenÕs function, but at small times

5. Results

The ®rst question of our investigation was to show how the non-linear persistent sodium current ampli®es the EPSP. In Fig. 4 the ampli®ed EPSPs were generated from expression (8) at x_ˆ0 for a cable of (dimensional) lengthL_ˆ1.6 mm and diameterd_ˆ4lm, and uniform spatial distribution of hot-spots from x_ˆ0 to x_ˆL located at length intervals of jL=N where

j_ˆ1;2;. . .N and with the current input assumed to be located atx0ˆ0:6L (i.e., 960lm from end x_ˆ0). In Fig. 4 identical number of hot-spots, but non-identical number of persistent Na‡

channels per hot-spot is assumed. The results clearly indicate that the strength of the synaptic input is an important indicator of the ampli®cation of the synaptic signal because it allows the voltage-dependent ion channels to exert an e€ect at various levels of membrane potential, in accordance with theINaP±Vdynamics. Hence, for large voltage excursions the solution will also be

Fig. 4. The ampli®cation of synaptic potentials mediated by persistent sodium hot-spots distributed uniformly along the cable of lengthLrecorded at the point (xˆ0). In each case the location of synaptic input is atx0ˆ0.6L, and the total number of hot-spots is assumed to be Nˆ10. The following parameters were also used (cf., Ref. [25]): Riˆ200Xcm,Cmˆ1lF=cm

2

a€ected by the synaptic strength of the input. This is of less importance for distal impinging synapses which rarely generate a response at the soma beyond 10 mV. The stability of the series solution is however given in terms of the number of persistent sodium channels per hot-spot (see Table 1). The number of persistent sodium channels per hot-spot are shown in Fig. 4 to amplify the response if a greater number of these ionic channels is considered per hot-spot.

The next question of concern was to know the degree of ampli®cation of the distal synaptic signal as a result of additional number of hot-spots. We assume two persistent sodium channels occupy each hot-spot (i.e., N_ˆ2=pd), although more realistic values may range anywhere be-tween 2 and 10 based on an approximate density of sodium channels per patch [12] assuming that each spot has similar surface area to a patch-clamp pipette. We selected a wide range of hot-spots and investigated the response by computing expression (8) for a cable of lengthL_ˆ1.6 mm and diameterd_ˆ4lm, assuming a uniform distribution of hot-spots fromx_ˆ0 tox_ˆLlocated at length intervals ofjL=N wherej_ˆ1;2;. . .;N, and with the current input assumed to be located at x0ˆ0.6L (i.e., 960 lm from end xˆ0). The results are presented in Tables 2 and 3 with a greater number of hot-spots showing a greater ampli®cation of the distal synaptic signal, in agreement with experimental results and earlier ®ndings presented in [14]. But there is also sat-uration, where further increases to the number of hot-spots distributed in the dendrites produces no further ampli®cation of the EPSP. It was found in [14] that there was no optimal number of spots for greatest ampli®cation to membrane potential (i.e., the greater the number of hot-spots resulted in more enhanced ampli®cation of the EPSP), due to the assumption of linearity in modeling the persistent sodium INaP±V relation, but the result for N ˆ25…ehˆ0:01† and

N _ˆ45eh_ˆ0:006_† presented in Table 2, together with the result for N _ˆ130eh_ˆ0:002_† and

Table 1

Heuristic stability criteria for convergence of Eq. (8)a t0 (ms) Stability criteria

N>1 and not for a single hot-spot location.

Table 2

N _ˆ150eh_ˆ0:001_†presented in Table 3, clearly indicate that the parameterNcan be associated with an optimum number. These results are dependent on both the location of the synaptic input and the strength of the synaptic current input. The data in Tables 2 and 3 only consider a single location and strength of the synaptic input, but changes to these parameters revealed similar results for relatively small voltage excursions not beyond 20 mV at x_ˆ0 for a distal synaptic input.

It is interesting that spurious behavior occurs always near the succession of the saturation period when the ampli®cation declines from the `plateau'. A further reduction in the peak EPSP after the long period of saturation (i.e., for a wide range ofNvalues) cannot be validated with this approach due to the convergence of the perturbation series being dependent onN. This is not a real limitation since changes in model parameters, especially those related to the dynamics of the INaP±Vcurve are intrinsic to the speci®c ionic current in question, so any changes in the dynamics would require new stability criteria which can be heuristically obtained. By replacing the hot-spot terms in Eq. (7) with appropriate approximation of continuously distributed channels, upper and lower bounds on the voltage response can be obtained using comparison methods in the case of a large number of hot-spots (see, Appendix C).

The maximum ampli®cation of 2.05 mV shown in Tables 2 and 3 is dependent on the duration the instantaneous INaP±V curve, namely 0.1sm corresponding to the estimated opening time of persistent sodium channels during sustained bursts of activity as shown in the inset of Fig. 5. The opening time ranges between 1 and 12 ms for pyramidal neurons compared to an average between 0.2 and 0.6 ms for transient sodium channels [39]. The di€erent amplitudes in the ampli®cation of the maximum peak EPSPs are to be expected for di€erent duration in the instantaneous INaP±V curves assuming identical synaptic input strength. In Fig. 5 we illustrate the maximum possible ampli®cation that can occur for a variety of di€erent instantaneous INaP±V curves measured at x_ˆ0 for a synaptic input atx0_ˆ0.6L. It is interesting to note that for the values of time duration t0 so chosen, the percentage in the maximum ampli®cation of the peak EPSPs at the pointxˆ0 fall in the ballpark between 12% and 28% [25] for persistent sodium channels concentrated in the soma and dendrites of CA1 hippocampal pyramidal neurons, respectively. This is shown in Fig. 6 in the case of TTX applied to a dendritic site.

We have also con®rmed the simulation results of Lipowsky et al. [25] that somaticINaP alone has little e€ect on somatic EPSP ampli®cation. Indeed we have further explored the issue and

Table 3

found that a single hot-spot (N_ˆ1) can only amplify the EPSP at x_ˆ0 between 4% and 6% at all locations along the cable for optimum channel densities, i.e. those densities which yield the greatest ampli®cation in the peak EPSP (see Fig. 7). Hence the results show that spatial lo-cation of the hot-spot for the density of Na‡_{P channels are not the reason for the low}

en-hancement of EPSP, but rather the distribution of the hot-spots (i.e., the number of hot-spots). Similar results have been more thoroughly studied with the linearized current in [14], but quantitatively the results di€er from the linearized current possibly due to the introduction of an inductance component as a result of the linearization procedure which may overestimate the actual ampli®cation of the EPSPs in non-linear cables and produce broadening in the time-course of the ampli®ed EPSP not observed in the present study (cf. Figs. 4, 6 and 7). This further leads to the experimental veri®cation of EPSP ampli®cation without the need to include potassium ion channels as has been done by Lipowsky et al. [25]. This supports the experi-mental ®ndings of Jung et al. [41] that sparse distribution of transient Na‡ _{channels, rather}

than K‡ _{channels are responsible for the reduction in the peak amplitude of back-propagating}

action potential trains.

As a consequence of the above results it is interesting to see if an optimum number of persistent sodium channels can be found for a maximum ampli®cation of the EPSP. To achieve maximum ampli®cation each hot-spot must operate at the optimum conductance as allowed by the INaP±V

dynamics. By increasing the number of persistent sodium channels per hot-spot (N_{) or (h) via the}

conductance gNaP, a relatively small number of hot-spots is required to produce sucient satu-ration in the response in order to yield an optimal number of persistent sodium channels. On the other hand, by decreasing the conductance gNaP, an extremely large number of hot-spots is re-quired to produce saturation in the response yielding a greater number of persistent sodium channels as optimum. The answer can be found by patch clamping the dendrites to determine the `true' value of thegNaP. A summary of the results is presented in Table 4. The saturation period of the response is de®ned for values ofNwhere no signi®cant change occurred in the ampli®cation of the EPSP from about 2 mV. An optimum value is chosen with the greatest ampli®cation peak, although in some cases a non-unique value ofNmay occur and so an average value is estimated. The results show that if the peak INaP at the dendritic hot-spot is 2/5, 1/4, 1/12 or 1/24 to the

somatic peakINaP then the optimum number of persistent sodium channels is 52, 88, 252 or 500, respectively.

Alternatively we can choose the most probable conductance of a hot-spot by using the fol-lowing `rule of thumb' for a cable with sealed-ends [42]:

Fig. 7. Maximum ampli®cation of EPSP for a single hot-spot (Nˆ1) placed at (a)xˆ0 withhˆ28, (b)xˆ0.6L with

hˆ32, (c)xˆ0.8L withhˆ35, and (d)xˆLwithhˆ38. Percentages of enhancement from the peak EPSP atxˆ0 are shown, with other parameter values as in Fig. 6.

Table 4

Summary of results showing the saturation range in the ampli®cation of the peak EPSP and optimum number of Na‡_P

channelsa

PeakINaP(pA/cm) N Optimum (hN)

)16.5 (i.e.heˆ0.01) 22±30 52

)10.0 (i.e.heˆ0.006) 40±49 88

)3.25 (i.e.heˆ0.002) 120±135 252 )1.65 (i.e.heˆ0.001) 230±270 500 a

Total number of channelshN_{† ˆ}q_NaPpd3=2Rm=Ri† 1=2

; 9_†

where qNaP is the pore density estimated in [31] to be 1 channel per 10 lm2_{, d ˆ4 lm,}

Rmˆ50;000 X cm2, and Ri ˆ200 X cm (from Ref. [25]). Substituting these parameters into Eq. (9) and multiplying the result by 0.16/0.316 (since Eq. (9) is valid for number of channels over two length constants which is 2k_ˆ0.316 andL_ˆ0.16 cm), we obtain an estimate of 201 persistent sodium channels. Interestingly, by using the same cable parameters and a pore density ofqNa_ˆ9 per 3 lm2 _{[12] for transient sodium channels we obtain an estimate of approximately 20,120} sodium channels. This value constitutes about 1% of the persistent sodium channels estimated above and supports the results obtained earlier (see Ref. [14]). By choosing the optimum number of channels from this estimate we can then select the conductance of the hot-spot by comparing estimates found using the perturbation series approach. For example, optimum (hN)_ˆ201 sug-gests a peakINaPof)5 pA/cm which is about 1/8 of the whole-cell peakINaPand requires a hot-spot conductance ofgNap_ˆ43.5 pS/cm (i.e., eh_ˆ0.003). This conductance suggests an optimum number of persistent sodium channels for the whole neuron (i.e., soma-dendritic axis) to be ap-proximately 1608 or 1407 for the soma. Such values reinforce the view of persistent sodium channels concentrated in greater numbers near the axosomatic region of the neuron (see Ref. [18]). An important issue with regard to the spatial distribution of hot-spots, is how close the discrete distribution of hot-spots approximates a continuous distribution as for example inherited in the classical non-linear cable theory. The results are presented in Table 5 for two di€erent total number of persistent sodium channels distributed uniformly, but discretely, along the cable of length (L). It is clear from Table 5 that if the distance between hot-spots is under 0.05L then the variation from a `continuous' distribution of hot-spots (assumed to be a 0.01L spatial distribution of hot-spots in the ionic cable model) is negligibly small. However, if the distribution of hot-spots is more sparse (i.e., distance between hot-spots increases to more than 0.1L) then the `continuous' and discrete distributions di€er between 1.5% and 4.2% and more depending on the distribution of hot-spots, a greater reduction in the peak amplitude of the membrane potential is evident if the total number of ionic channels is assumed to be small. It should be noted that a continuous distribution not only requires numerical evaluation but also does not allow for a comparison to be made in terms of the total number of ionic channels, unlike an ionic cable model with discrete channel clusters, hence we have assumed a pseudo-continuous representation in the results of Table 5.

6. Discussion

An a priori assumption of the theory is that persistent sodium ion channels along the soma-dendritic axis occur in far less abundance to those found along the axonal axis as recently shown by Safronov [43] for spinal dorsal horn neurons. Although modeling studies have assumed weakly excitable dendrites (see Ref. [44]), as yet there is no experimental veri®cation of the validity of hot-spots or discrete patches of high density congregation of ion channels because the exact densities and their spatial distribution of persistent sodium channels in dendrites of neurons is a dicult task, often based on crude patch-clamp current estimates (see Refs. [12,45]). However, as den-drites are covered with synaptic receptors (absent along the axon membrane) it would devoid the dendritic membrane of space for the positioning of voltage-dependent ionic channels in a con-tinuous fashion. Therefore, the assumption of sodium channels and other voltage-dependent ionic channels as being distributed in discrete patches or hot-spots is a viable theoretical assumption that needs experimental con®rmation.

Imaging data using sodium binding benzofuran isophthalate suggest that sodium channels in dendrites are in sucient densities to sustain action potentials [46], but the sodium action po-tential failure in the distal dendrites is believed to be caused by an in¯ux of K(Ca‡_{) channels}

[47,48] or as a result of di€erent Na‡_/K‡_{permeability ratios [49], rather than a sparse distribution}

of sodium channels. Evidence for a relatively sparse density distribution of voltage-dependent transient sodium channels comes indirectly from the observed decrement in the amplitude of back-propagating action potentials trains due to prolonged inactivation of the sodium channels or the presence of persistent sodium channels [41]. We believe that further experimental studies are necessary to re-examine the spatial distribution of sodium channels in both proximal and distal dendrites in order to verify or disprove our theoretical assumption of sparse distribution of persistent sodium channels in dendrites of hippocampal neurons.

It is interesting to consider the possibility of an optimal number of persistent sodium channels for maximum ampli®cation of the EPSP under a possible nonuniform distribution of channels. The determination of an optimal number under a non-homogeneously distributed assumption can be determined from Eq. (5) by re-de®ning the number of channels as a function of space, i.e.,

gNaP…x† ˆgNaP N…x† …10†

so that at location x_ˆxi there will be N…xi† ˆhi=pd persistent sodium channels, where hi is

there is no compelling reason for nature to optimize conduction velocity with respect to such a simple criteria.

Finally, there are many interesting directions for which the application of ionic cable model with discrete channel clusters can be utilized to answer important questions at both the single neuron level and at the neural network level. One example is the idea of dendritic bistability (see Refs. [6,7,55]). Here, the non-linearity is not sigmoidal but `N-shaped' with two stable points and one unstable point, and so the FV_† governed by Eq. (6) may need to be replaced with a more appropriate polynomial function (see Ref. [5]). Another is the investigation of solitary ion channel distributions along the cable, and the resultant noise arising from the stochastic nature of voltage-dependent ionic channels (see Ref. [56]).

7. Conclusions

A perturbation method was employed to yield analytical solutions for the voltage response to synaptic input along a cable representation of a single neuron with voltage-dependent ionic channels distributed at discrete locations along the somatodendritic axis. We ®xed the number of persistent sodium channels per hot-spot to a particular value by scaling the whole-cell conduc-tance by a parameter (a similar approach was outlined in Refs. [42,57,58] to investigate supra-threshold responses). Although quantitative results were predicted only up to the start of the cessation in the saturation phase of the response, the perturbative method revealed both new and similar conclusions to those obtained with a linear approximation [14]. The non-linear phe-nomena characterizing signal propagation in dendrites were investigated to predict the following conclusions which have emerged from the analysis:

· _{The inclusion of non-linearINaP current sources ampli®es the EPSP as predicted using the}

lin-earized macroscopicINaPcurrent source, but no broadening in the time-course of the ampli®ed EPSP was observed, possibly because of the relative short duration the dendriticINaP current remained active.

· _{A greater number of hot-spots increases the ampli®cation of the EPSP, but saturation in the}

ampli®cation also occurs after a certain number of hot-spots, which was not predicted using the linearizedINaP current, and hence is a strictly non-linear phenomenon.

· _{Increasing the conductance of the persistent sodium channels results in more enhanced}

ampli-®cation to the EPSP and decreases the optimum number of channels.

· _{A rule of thumb con®rms the optimum number of persistent sodium channels to be 201 (and}

transient sodium channels to be 20,190 conferring experimental studies that Na‡P constitute about 1% of the total transient sodium channels found on typical CA1 hippocampal neurons) which yields at the dendritic hot-spot an estimate of the peakINaP to be approximately 1/8 of the somatic whole-cell peakINaP.

· _{A single hot-spot containing a variety of optimum channel densities placed anywhere along the}

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, respectively. R.R.P. also wishes to thank Armantas Baginskas and Soh Hidaka for very helpful comments on the initial draft of the manuscript. J.B. was partially supported by National Science Foundation grant DMS-9706307.

Appendix A. Time course of the EPSP

We examine the continuous nature of the membrane conductivity, neglecting the more realistic discrete nature of leakage channels in biological membranes, although the error in doing so has been shown to be marginal (see Ref. [59]). By taking advantage of linearity and applying GreenÕs function method of solution (see Ref. [60, p. 191]) to Eq. (7), we obtain the following result: Tat locationXto a unit impulse at locationX0at time T_ˆ0, and is given by the solution of the following I±BVP:

GTX;X0;T† ˆGXX…X;X0;T† ÿG…X;X0;T†;

GX;X0;0† ˆd…X ÿX0†:

Note:bis the charge carried by the synaptic current through a unit membrane surface per unit time (lA/cm).

Eq. (A.3) can be simpli®ed further if we let X _ˆ0 (where we are most concerned with mea-suring the ampli®cation of potential),

d_ˆL=p_†21_ÿa_† and KT_{† ˆ}n_0‰Texp_ÿaT_{† ÿ}n₀exp_ÿaT_{† ‡}n₀exp_ÿT_†Š

and make use of the following identity [61]:

X1

The last series converges so rapidly that other than the ®rst term it can be ignored in computing

U…0;T†, hence we can use the following closed form approximation for the voltage-response:

U0;T_†baexp1_†Rm=…kL†† K…T†

On substituting Eq. (8) into Eq. (7) with F‰VŠ governed by expression (6), and equating co-ecients of powers ofe, a sequence of linear equations governing the non-linear perturbations of the voltage from the passive cable voltage response (U) are found. The ®rst few perturbations can be shown via a Taylor expansion ofFto yield a sequence of linear partial di€erential equations

Oe4_†: V4;T ˆV4;XX ÿV4‡gNaP…Rm=k† XN

iˆ1

‰F0U_†V3‡ …F000…U†=3!†V13

‡F00U_†V1V2ŠDT0d…X ÿXi†; …B:4†

where primes denote di€erentiation w.r.t. U. In the above set of equations, the forcing terms are either given below or are known solutions of preceding equations, and so Green_Õs function method can be applied iteratively to ®nd the voltage correction terms in explicit form

F000…V† ˆ6 exp_‰2V0ÿV†=kŠ=hk2f1_‡ exp_‰V0ÿV†=kŠg3i

The Fs in Eqs. (B.5)±(B.8) have been evaluated at the end of the channel opening time (T0) re-¯ecting an instantaneousI±Vmeasurement at each hot-spot location along the dendritic cable.

Appendix C. A qualitative approach to estimating membrane potential

The objective of this appendix is to introduce another method to analyze our non-linear ionic cable model of membrane potential, particularly at the endpoint X_ˆ0. The calculations are formal and not comprehensive below. A more rigorous treatment would include using Riemann± Stieltjes integration theory and a detailed statement of comparison theorems used, and such details will appear elsewhere.

Our main idea below is to relate the source terms making up the discrete hot-spot contributions to a continuously distributedI±Vrelation, then use comparison equation methods to bound the behavior of the potential at X_ˆ0. To be more de®nite, consider any continuous, bounded functionQ(x) de®ned on LPXP0. Shortly we will identifyQ…X† withF‰V…X†Š. Then

where we consider the hot-spot locations {xj} as a uniform partition of our dendritic segment, and

the numberNis suciently large to consider the Riemann sum as a reasonable approximation to the integral. Therefore, this motivates the relation

…1=Dx0_†F_‰VX_{†Š ˆ}N=L_†F_‰VX_{†Š  ÿ}X N

jˆ1

F_‰V_ŠdX _ÿXj_†: C:1_†

Since the hot-spots act to amplify the response atX_ˆ0, the solution to the cable model without hot-spots which is given byU…X;T† in Eq. (A.1) satis®esU…0;T†6V…0;T†. The purpose of the above calculation is to work with an upper bound forV0;T_†. Since we have a large number of hot-spots, a solution to Eq. (7) with the sum given on the right-hand side of Eq. (C.1) being replaced by the left-hand side of Eq. (C.1) should dominate the solution V of Eq. (7). That is,

Z0;T_†PV0;T_†, where Z is the solution to the comparison equation

ZT _ˆZXX _ÿZ_ÿegNaPRmN=kL†F‰ZŠ ‡ …Rm=k†I…T†d…X ÿX0†: …C:2†

Finally, because of the form ofF_‰V_Šgiven in Eq. (6), we can choose positive constantsqandVa,

with 0<Va <V_NaP0 , such thatqV 6F_‰V_Š for allVin 06V 6Va. For the parameter values chosen in the paper,q_0:013 andVa _172 (mV), which is well above the maximum voltage forV0;T_†

obtained with the current stimulus used in the paper. (It is also true that there is a constantm>0 such thatF_‰V_Š6mV for allV P0 (with present parameters forF,m_0:188), but we do not make use of this observation in the calculations below.)

Now letW be the solution to

WT ˆWXX ÿW ÿlqW ‡ …Rm=k†I…T†d…X ÿX0†; …C:3†

where l_ˆegNaPRmN=kL†, and Eqs. (C.2) and (C.3) possess the same sealed end termination conditions as our original problem. The di€erence function D_ˆW _ÿZ then satis®es:

DT ÿDXX ‡ …1_‡lq†Dˆl_fF‰ZŠ ÿqZgP0. So, by comparison theorem methods (see Refs.

[62,63]), DP0 on its domain. This implies that W PZ. Therefore, W0;T_† and U0;T_† should sandwich the membrane potential V…0;T† for T >0. Using GreenÕs function approach of Appendix A yields

Consider solutions to Eq. (C.3). By another comparison theorem argument, if W…k† _{is the}

solution to Eq. (C.3) corresponding to l_ˆl…k†_{;kˆ1;2, and l}…1†_Pl…2†_{, then} W…1†_X;T†6W…2†_{X;T†. Because we assumedN 1, if we decrease the number of hot-spots (or}

For 0<V <Va, since GreenÕs function GP0, we have V0;T_{† ˆ}vN0;T_†PvNÿ1…0;T† P Pv1…0;T†PU…0;T†. Therefore, starting with a few hot-spots, adding hot-spots increases the response. The saturation behavior and other aspects of this model will be explored analytically in more detail elsewhere, where both the dependence onNof the conductance and the membrane capacity can be accounted for.

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