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Mathematical Biosciences 166 (2000) 123±147
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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

b

Advanced Research Laboratory, Hitachi, Ltd., Hatoyama, Saitama, 350-0395, Japan
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 (nonevoked) 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

*

1

Corresponding author. Tel.: +81-492 96 6111; fax: +81-492 96 6006.
E-mail address: poznan@harl.hitachi.co.jp (R.R. Poznanski).
Present address: Department of Mathematics, UMBC, Baltimore, MD 21250, USA.

0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 3 2 - 8

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R.R. Poznanski, J. Bell / Mathematical Biosciences 166 (2000) 123±147

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 approach 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 persistent 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
discrete clustering 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
Let V be 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:
N
X
INaP …x; t; V †d…x ÿ xi † ‡ I…x; t†d…x ÿ x0 †;
Cm Vt ˆ …d=4Ri †Vxx ÿ V =Rm ‡
iˆ1

t > 0;

…1†

where I…x; 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 …ˆ cm =pd† the membrane
capacitance (F/cm2 ), Rm …ˆ rm pd† the membrane resistivity (X cm2 ), Ri …ˆ ri pd 2 =4† the cytoplasmic resistivity (X cm), N the number of hot-spots (dimensionless), and d is the Dirac-delta
function re¯ecting the position along the circumference of the cable where the ionic current is

R.R. Poznanski, J. Bell / Mathematical Biosciences 166 (2000) 123±147

125

Fig. 1. A schematic illustration showing a cable of diameter d (cm) and length L (cm). (a) The arrow above the hot-spot
re¯ects the notion of INaP representing a point source of current applied at an in®nitesimal area on the cable. The
symbol N denotes the number of hot-spots and N denotes the number of persistent sodium channels in each hot-spot
per unit membrane surface of cable, represented schematically as black dots. (b) An equivalent electrical circuit representation of the same cable. Note the addition of a voltage-dependent conductance gNaP at the hot-spot.

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

Eq. (1) can be cast in terms of non-dimensional space and time variables, X ˆ x=k and
T ˆ t=sm , respectively, where k ˆ …Rm d=4Ri †1=2 and sm ˆ Rm Cm are, respectively, the space and
time constants. Thus Eq. (1) becomes
VT ˆ VXX ÿ V ‡

N
X
…Rm =k†INaP …X ; T ; V †d…X ÿ Xi †
iˆ1

‡ …Rm =k†I…X ; T †d…X ÿ X0 †;

T > 0;

0 < X < L ˆ L=k;

…2†

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R.R. Poznanski, J. Bell / Mathematical Biosciences 166 (2000) 123±147

where Xi ˆ xi =k and X0 ˆ x0 =k represent 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 subsequent 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 macroscopic current measured from the somata of dissociated hippocampal cells as shown in Fig. 2.
In practice, however, the value of the membrane I±V relation INaP along 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 theorem 2 should be used to connect INaP , with the iNaP obtained from intracellular recording with a patch-pipette. This is because in most cases the input I±V relation is
less non-linear than the membrane I±V relation as expressed by the relationship (see Ref. [5])
INaP …xi ; ti ; V † ˆ …Ri =pd 2 †‰iNaP =…dV0 …xi ; ti †=diNaP ÿ dVL …xi ; ti †=diNaP †Š;

…3†

where V0 and VL represent (dimensional) V at x ˆ 0 and x ˆ L, 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
dendritic INaP corresponds to a small number of ionic channels and lasts for only a short period of
time denoted by Dt0 from t ˆ 0 to t ˆ t0 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
ColeÕs theorem states that the total membrane current per unit length of cable Im (V0 ) can be estimated from the
experimental input current±voltage relation (I0 ±V0 ) through the relation: Im (V0 ) ˆ (ri /4) I0 (dI0 /dV0 ), where V0 is the

voltage at the point where the micro-electrode current I0 is applied instantaneously. Note, Eq. (3) is a generalization of
ColeÕs theorem to the persistent sodium ionic channel.

R.R. Poznanski, J. Bell / Mathematical Biosciences 166 (2000) 123±147

127

Fig. 2. The somatically recorded input I±V relationship for the persistent sodium current in cultured hippocampal
neurons taken from experimental data in [33, Fig. 9a]. The continuous curve represents the best ®t to data and was
based on a sigmoidal function expressed by Eq. (4) with the eh ˆ 0.024 and gNaP ˆ 0.0145 ls/cm (at the soma), while the
dots correspond to experimental measurement of the macroscopic current. The dashed curve represents one example of
the membrane INaP ±V dynamics expressed by Eq. (4) with the eh ˆ 0.006 and gNaP ˆ 0.0145 ls/cm (at a dendritic hotspot), re¯ecting a ``macropatch'' current of a small number of persistent sodium channels.

follows a relatively similar shape (i.e., sigmoidal) from the somatically recorded instantaneous
input I±V relation 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

ˆ 16 pS), which is two orders of magnitude larger than the expected number
conductance of gNaP

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 of gNaP
ˆ 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 hippocampal neurons, the next problem is to show the approximate number per unit surface of dendrite. 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 …e  1†, e.g. h ˆ 45 and e ˆ 1 (at soma only) based on a `whole-cell'
conductance of 0.82 ns, leaving the conductance ®xed at gNaP ˆ 0:0183=pd ns ˆ 0:0145 ls=cm.
Note that the case of e ˆ 1 is ®ctitious because x ˆ 0 represents only a point close to the soma and
therefore does not violate the assumption of e  1 along the dendritic cable.

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R.R. Poznanski, J. Bell / Mathematical Biosciences 166 (2000) 123±147

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. Assuming for the entire duration persistent sodium channels are open, no variation in the activation
variable F occurs, so time derivatives with respect to time of F can 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; t0 Š 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 † ˆ egNaP F ‰V …x; t0 †ŠDt0 ;

…4†

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

N :
gNaP ˆ gNaP

…5†

0
Š;
F …V † ˆ …1=f1 ‡ exp‰…V 0 ÿ V †=kŠg ÿ 1=f1 ‡ exp‰V 0 =kŠg†‰V ÿ VNaP

…6†

Here N  ˆ h=pd is the number (h) of persistent sodium channels per unit membrane surface of

cable in cmÿ1 , and gNaP
ˆ 18 pS is the maximum attainable conductance of a single sodium
channel measured by Sigworth and Neher [37] and Stuhmer et al. [38]. The parameter e scales the
whole-cell conductance at the soma (assumed to be ®xed at gNaP ˆ 0:0145 lS=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† ÿ H …t ÿ t0 †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, with t0 representing the maximum time the cluster of channels remains open and
H(
) denotes the Heaviside-step function.
The activation variable F …V † is represented by the following sigmoidal function:
where V0 , V0 NaP , and k are 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
0
ˆ 195 mV. A non-linear representation of the persistent sodium
the data: V0 ˆ 46, k ˆ 9, and VNaP
current as expressed by Eq. (6) is shown in Fig. 2 for the experimentally measured somatic wholecell current, and for the heuristic representation of a dendritic hot-spot current. Note that INaP as
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 ˆ VXX ÿ V ‡ egNaP …Rm =k†

N
X
F ‰V …X ; T0 †ŠDT0 d…X ÿ Xi † ‡ …Rm =k†I…X ; T †d…X ÿ X0 †;
iˆ1

…7†

where DT0  ‰H …T † ÿ H …T ÿ T0 †Š and T0 ˆ t0 =sm with the potential initially at rest and sealed-end
boundary conditions imposed at each end of the neural segment (X ˆ 0, L ). Although Eq. (7) is in
terms of non-dimensional space and time variables, we have not non-dimensionalized the mem-

R.R. Poznanski, J. Bell / Mathematical Biosciences 166 (2000) 123±147

129

brane potential in Eq. (7) by some characteristic value of the membrane potential Vch . A reasonable value for Vch would 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 response U …X ; T † in the form of a perturbative expansion
1
X
V …X ; T † ˆ U …X ; T † ‡
ek Vk …X ; T †;
…8†
kˆ1

where Vk …X ; T † is the perturbed voltage (in mV) from the ®rst-order U …X ; T † approximation derived in Appendix A. Eq. (8) represents a perturbation series expansion at an equilibrium point
U …X0 ; 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 U…X ; T † as shown in
Section 5.
It was found through a heuristic approach that U(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±V curve 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 e5 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 largest eh ratio 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 potential 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 GreenÕs function are used for N ˆ 300 or less. A greater amount of computing time is required if
more than N ˆ 300 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
…T < 0:01† an alternative GreenÕs function should be used which converges more rapidly (see
Ref. [14]).

130

R.R. Poznanski, J. Bell / Mathematical Biosciences 166 (2000) 123±147

Fig. 3. The depolarization measured at x ˆ 0 corresponding to the (a) ®rst (i.e. )eV1 (x,t)) and third (i.e. )e3 V3 (x,t)), (b)
second (i.e. )e2 V2 (x,t)) and fourth-order (i.e. )e4 V4 (x,t)) voltage perturbations from the passive (RC-cable) voltage
response (i.e. U …x; t†). The continuous curve corresponds to the ®rst and second terms and the dashed curve corresponds
to the third and fourth terms, both measured at x ˆ 0 in response to a synaptic input current at x0 ˆ 0.6L of strength
b ˆ 4.38 lA/cm, and the total number of hot-spots N ˆ 100, with h ˆ 2, i.e. each hot-spot contains two persistent sodium
channels, and e ˆ 0.001, t0 ˆ 0.1sm , and gNaP ˆ 0.0145 ls/cm. For other parameter values, see Fig. 4.

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 ˆ 0 for a cable of (dimensional) length L ˆ 1.6 mm and diameter d ˆ 4 lm, and uniform spatial
distribution of hot-spots from x ˆ 0 to x ˆ L located at length intervals of jL=N where

R.R. Poznanski, J. Bell / Mathematical Biosciences 166 (2000) 123±147

131

Fig. 4. The ampli®cation of synaptic potentials mediated by persistent sodium hot-spots distributed uniformly along
the cable of length L recorded at the point (x ˆ 0). In each case the location of synaptic input is at x0 ˆ 0.6L, and the
total number of hot-spots is assumed to be N ˆ 10. The following parameters were also used (cf., Ref. [25]):
Ri ˆ 200 X cm, Cm ˆ 1 l F=cm2 , Rm ˆ 50; 000 X cm2 , d ˆ 4 lm, k ˆ 0:158 cm, L ˆ 0.16 cm, a ˆ 0:25, and (a) and (c)
b ˆ 4:38 lA=cm and (b) and (d) b ˆ 8:76 lA=cm. The values of b and a were selected arbitrarily to yield a response at
x ˆ 0 of (a) and (c) 10 mV, and (b) and (d) 20 mV (in the absence of sodium persistent hot-spots), respectively. The value
of gNaP ˆ 0.0145 ls/cm with the number of persistent sodium channels per hot-spot being: (a) and (b) h ˆ 6, and (c) and
(d) h ˆ 18. The dashed curve corresponds to the ampli®cation in the membrane potential with e ˆ 0.001 and t0 ˆ 0.1sm ,
and the continuous curve represents the EPSP without any persistent sodium ion channels present. Inset shows
schematically the position of hot-spots along the cable of length L ˆ 1.6 mm as well as the position of synaptic current
injection.

j ˆ 1; 2; . . . N and with the current input assumed to be located at x0 ˆ 0:6L (i.e., 960 lm from end
x ˆ 0). 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 the INaP ±V dynamics. Hence, for large voltage excursions the solution will also be

132

R.R. Poznanski, J. Bell / Mathematical Biosciences 166 (2000) 123±147

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  ˆ 2=pd), although more realistic values may range anywhere between 2 and 10 based on an approximate density of sodium channels per patch [12] assuming that
each hot-spot has similar surface area to a patch-clamp pipette. We selected a wide range of hotspots and investigated the response by computing expression (8) for a cable of length L ˆ 1.6 mm
and diameter d ˆ 4 lm, assuming a uniform distribution of hot-spots from x ˆ 0 to x ˆ L located
at length intervals of jL=N where j ˆ 1; 2; . . . ; N , and with the current input assumed to be located
at x0 ˆ 0.6L (i.e., 960 lm from end x ˆ 0). 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 saturation, 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
hot-spots for greatest ampli®cation to membrane potential (i.e., the greater the number of hotspots 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 ˆ 25…eh ˆ 0:01† and
N ˆ 45…eh ˆ 0:006† presented in Table 2, together with the result for N ˆ 130…eh ˆ 0:002† and
Table 1
Heuristic stability criteria for convergence of Eq. (8)a

a

t0 (ms)

Stability criteria

0.06sm
0.08sm
0.1sm
0.12sm

e < 0:6…1=Nh†
e < 0:4…1=Nh†
e < 0:32…1=N h†
e < 0:24…1=N h†

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

Table 2
Ampli®cation of the peak EPSP (mV)a
N
5
10
20
25
30
45
50
a

eh ˆ 0.01

0.7
1.2
1.9
2.05
2.0
±
±

eh ˆ 0.006

0.4
0.75
1.3
1.55
1.75
2.0
1.9

Measured from the peak of the EPSP at x ˆ 0 without hot-spots (i.e., 10 mV) as a result of synaptic input located at
x0 ˆ 0.6L and N* ˆ 2 per unit membrane surface of the cable (i.e., h ˆ 2). Evaluated from Eq. (8) up to O(e4 ) at
t0 ˆ 0.1sm . All results shown above are within an error of ‹0.1 mV.

R.R. Poznanski, J. Bell / Mathematical Biosciences 166 (2000) 123±147

133

Table 3
Ampli®cation of the peak EPSP (mV)a
N
50
100
130
150
200
250
300

eh ˆ 0.002

1.15
1.85
1.99
1.85
±
±
±

eh ˆ 0.001

0.62
1.11
1.4
1.5
1.85
1.95
1.80

a

Measured from the peak of the EPSP at x ˆ 0 without hot-spots (i.e., 10 mV) as a result of synaptic input located at
x0 ˆ 0.6L and N ˆ 2 per unit membrane surface of the cable (i.e., h ˆ 2). Evaluated from Eq. (8) up to O(e4 ) at
t0 ˆ 0.1sm . All results shown above are within an error of ‹0.1 mV.

N ˆ 150…eh ˆ 0:001† presented in Table 3, clearly indicate that the parameter N can 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 ˆ 0 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 of N values) cannot be validated with this
approach due to the convergence of the perturbation series being dependent on N. This is not a
real limitation since changes in model parameters, especially those related to the dynamics of the
INaP ±V curve 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 ˆ 0 for a synaptic input at x0 ˆ 0.6L. 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 point x ˆ 0
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 somatic INaP alone
has little e€ect on somatic EPSP ampli®cation. Indeed we have further explored the issue and

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Fig. 5. The percentage in maximum ampli®cation in the peak of the EPSP at x ˆ 0. The values were selected from N sets
which resulted in maximum synaptic ampli®cation for several distinct instantaneous I±V relationships selected (i.e.
t ˆ t0 ) that were heuristically determined as equilibrium points and which re¯ect the opening times of persistent sodium
channels during sustained bursting activity. Inset: Closed (C1 ) and open (C2 ) time histograms of sodium channel during
bursting activity shown (top) from a layer V pyramidal neuron from rat cortex (from Ref. [39]).

found that a single hot-spot (N ˆ 1) can only amplify the EPSP at x ˆ 0 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 location of the hot-spot for the density of Na‡ P channels are not the reason for the low enhancement 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 timecourse 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 experimental ®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

R.R. Poznanski, J. Bell / Mathematical Biosciences 166 (2000) 123±147

135

Fig. 6. (a) Experimentally measured averages of somatically recorded electrotonic potentials (n ˆ 10) before and during
application of TTX to the dendritic region of CA1 hippocampal pyramidal neurons. The ampli®cation of the somatic
potential is clearly visible when both averaged traces were superimposed (right) (from Ref. [25]). (b) A theoretically
obtained ampli®cation of the somatic potential from a peak of 6.1±7.6 mV (i.e. 26% ampli®cation) generated with
N ˆ 20 and e ˆ 0.002 h ˆ 6 with b ˆ 2.6718 lA/cm and t0 ˆ 0.12sm . Inset shows schematically the position of hot-spots
along the cable of length L ˆ 1.6 mm as well as the position of synaptic current injection.

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 saturation 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 required 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 the gNaP . A summary of the results is presented in Table 4. The saturation period of
the response is de®ned for values of N where 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 of N may 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

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Fig. 7. Maximum ampli®cation of EPSP for a single hot-spot (N ˆ 1) placed at (a) x ˆ 0 with h ˆ 28, (b) x ˆ 0.6L with
h ˆ 32, (c) x ˆ 0.8L with h ˆ 35, and (d) x ˆ L with h ˆ 38. Percentages of enhancement from the peak EPSP at x ˆ 0 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
Peak INaP (pA/cm)
)16.5
)10.0
)3.25
)1.65
a

(i.e.
(i.e.
(i.e.
(i.e.

he ˆ 0.01)
he ˆ 0.006)
he ˆ 0.002)
he ˆ 0.001)

N
22±30
40±49
120±135
230±270

Optimum (hN)
52
88
252
500

Based on the results presented in Tables 2 and 3. The optimum number of channels is an average value.

somatic peak INaP 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 following `rule of thumb' for a cable with sealed-ends [42]:

R.R. Poznanski, J. Bell / Mathematical Biosciences 166 (2000) 123±147

Total number of channels …hN † ˆ qNaP pd 3=2 …Rm =Ri †1=2 ;

137

…9†

where qNaP is the pore density estimated in [31] to be 1 channel per 10 lm2 , d ˆ 4 lm,
Rm ˆ 50; 000 X cm2 , and Ri ˆ 200 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 ˆ 0.316 and L ˆ 0.16 cm), we obtain an estimate of 201 persistent
sodium channels. Interestingly, by using the same cable parameters and a pore density of qNa ˆ 9
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) ˆ 201 suggests a peak INaP of )5 pA/cm which is about 1/8 of the whole-cell peak INaP and requires a hotspot conductance of gNap ˆ 43.5 pS/cm (i.e., eh ˆ 0.003). This conductance suggests an optimum
number of persistent sodium channels for the whole neuron (i.e., soma-dendritic axis) to be approximately 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.

Table 5
Peak reduction in the membrane potentiala
D/L
0.01
0.02
0.04
0.05
0.1
0.2
a

N  N ˆ 100=pd (%)

±
0.7
1.0
1.1
2.1
4.2

N*N ˆ 1000/pd (%)

±
0.1
0.2
0.4
1.5
3.6

Measured from the peak of the membrane potential in a cable with ``continuously'' distributed hot-spots assumed to
be spaced discretely every D ˆ 0.01L, where D is the uniform spacing between successive hot-spots in microns.

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6. Discussion
An a priori assumption of the theory is that persistent sodium ion channels along the somadendritic 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 hotspots 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 dendrites 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 continuous 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 potential 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.,

N  …x†
gNaP …x† ˆ gNaP

…10†

so that at location x ˆ xi there will be N  …xi † ˆ hi =pd persistent sodium channels, where hi is
assumed to be nonuniform function with distance along the cable. The non-homogeneity in gNaP
based on a continuous distribution was previously incorporated into models (see Refs. [50±52]).
For a non-homogeneous distribution with greater occurrence on somata and near the primary
branches of large dendrites an appropriate function could be exponentially decaying in the somatofugal direction hi ˆ exp‰cNaP xi Š where cNaP < 0 is a constant. The present theory can also be
extended to examine conduction velocities of synaptic potentials in cable structures, in order to
make predictions about the underlying mechanism of conduction from optically recorded hippocampal neurons grown in culture [53]. An optimal number of sodium channels can be obtained
for maximum conduction velocities as shown numerically in [36] and analytically in [54], although

R.R. Poznanski, J. Bell / Mathematical Biosciences 166 (2000) 123±147

139

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 F …V † 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 voltagedependent 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 conductance by a parameter (a similar approach was outlined in Refs. [42,57,58] to investigate suprathreshold 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 phenomena characterizing signal propagation in dendrites were investigated to predict the following
conclusions which have emerged from the analysis:
· The inclusion of non-linear INaP current sources ampli®es the EPSP as predicted using the linearized macroscopic INaP current source, but no broadening in the time-course of the ampli®ed
EPSP was observed, possibly because of the relative short duration the dendritic INaP 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 linearized INaP 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 peak INaP to be approximately 1/8 of
the somatic whole-cell peak INaP .
· A single hot-spot containing a variety of optimum channel densities placed anywhere along the
cable enhanced the EPSP at x ˆ 0 by 6% suggesting that spatial distribution (i.e., the number of
hot-spots) and not density of persistent sodium channels appears to be the most important factor governing EPSP enhancement in non-linear ionic cables. In fact, only a small increment in
the peak amplitude of EPSPs will result if persistent sodium channels are placed only near the
soma and not spatially distributed along the dendrites.

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R.R. Poznanski, J. Bell / Mathematical Biosciences 166 (2000) 123±147

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:
Z T
‰I…p†G…X ; X0 ; T ÿ p†Š dp; T > 0;
…A:1†
U …X ; T † ˆ …Rm =k†
0

where I…T † ˆ H…T †baT exp…1 ÿ aT † is the synaptic current per unit surface of cable (lA/cm) (a, b
are taken to be constants), and G…X ; X0 ; T † corresponds to GreenÕs function, i.e. response at time
T at location X to a unit impulse at location X0 at time T ˆ 0, and is given by the solution of the
following I±BVP:
GT …X ; X0 ; T † ˆ GXX …X ; X0 ; T † ÿ G…X ; X0 ; T †;
G…X ; X0 ; 0† ˆ d…X ÿ X0 †:

In the case of a single ®nite cable of electrotonic length (L ) with both ends sealed (i.e.
GX …0; X0 ; T † ˆ GX …L ; X0 ; T † ˆ 0 ) a representation which converges fast for large T values is given
by [60]
1
X
cos …npX =L † cos…npX0 =L † exp‰ÿf1 ‡ …np=L †2 gT Š
G…X ; X0 ; T † ˆ exp …ÿT †=L ‡ …2=L †
nˆ1

…A:2†

for T > 0, 0 < X < L , where L ˆ L=k. The evaluation of Eq. (A.1) gives by the following time
course for the EPSP:
"
U …X ; T † ˆ …ba exp…1†Rm =kL † n0 T exp… ÿ aT † ÿ n20 exp… ÿ aT † ‡ n20 exp… ÿ T †
1
X
cos …npX =L † cos …npX0 =L †fnn T exp… ÿ aT † ÿ n2n exp… ÿ aT †
‡2
nˆ1
#

‡ n2n exp‰ ÿ f1 ‡ …np=L †2 gT Šg ;

where n0 ˆ 1=…1 ÿ a† and nn ˆ 1=‰1 ÿ a ‡ …np=L †2 Š.

…A:3†

R.R. Poznanski, J. Bell / Mathematical Biosciences 166 (2000) 123±147

141

Note: b is 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 ˆ 0 (where we are most concerned with measuring the ampli®cation of potential),
d ˆ …L =p†2 …1 ÿ a†

and K…T † ˆ n0 ‰T exp…ÿaT † ÿ n0 exp…ÿaT † ‡ n0 exp…ÿT †Š

and make use of the following identity [61]:
1
X
p
p
p
p
cos …npX0 =L †=…n2 ‡ d† ˆ …p=2 d†fcosh‰ dp…1 ÿ X0 =L †Š= sinh… dp† ÿ 1=…p d†g:
nˆ1

Hence,

"

p
p
U…0; T † ˆ …ba exp…1†Rm =…kL †† K…T † ‡ …L2 =p d†T exp… ÿ aT †fcosh‰ dp…1
p
p
ÿ X0 =L †Š= sinh…p d† ÿ 1=…p d†g ‡ 2…L =p†4
#
1
X
2
2

cos …npX0 =L †f exp‰ ÿ f1 ‡ …np=L † gT Š ÿ exp… ÿ aT †g=…n2 ‡ d† :


nˆ1

…A:4†

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:
h
p
U…0; T †  …ba exp…1†Rm =…kL †† K…T † ‡ …L2 =p d†T exp… ÿ aT †
p
p
p
 fcosh‰p d…1 ÿ X0 =L †Š= sinh…p d† ÿ 1=…p d†g ‡ 2…L =p†4 cos …pX0 =L †
i
…A:5†
 f exp‰ ÿ f1 ‡ …p=L †2 gT Š ÿ exp… ÿ aT †g=…1 ‡ d†2 :
Appendix B. Membrane potential correction terms
On substituting Eq. (8) into Eq. (7) with F ‰V Š governed by expression (6), and equating coecients of powers of e, 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 of F to yield a sequence of linear partial di€erential equations
N
X
O…e†: V1;T ˆ V1;XX ÿ V1 ‡ gNaP …Rm =k† F …U†DT0 d…X ÿ Xi †;
iˆ1

2

N
X
ÿ V2 ‡ gNaP …Rm =k† F 0 …U †V1 DT0 d…X ÿ Xi †;

O…e †:

V2;T ˆ V2;XX

O…e3 †:

V3;T ˆ V3;XX ÿ V3 ‡ gNaP …Rm =k†

iˆ1

N
X
‰F 0 …U †V2 ‡ …F 00 …U †=2!†V12 ŠDT0 d…X ÿ Xi †;
iˆ1

…B:1†
…B:2†
…B:3†

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R.R. Poznanski, J. Bell / Mathematical Biosciences 166 (2000) 123±147
N
X
O…e4 †: V4;T ˆ V4;XX ÿ V4 ‡ gNaP …Rm =k† ‰F 0 …U †V3 ‡ …F 000 …U †=3!†V13
iˆ1

‡ F 00 …U†V1 V2 ŠDT0 d…X ÿ Xi †;

…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 GreenÕs function
method can be applied iteratively to ®nd the voltage correction terms in explicit form
V1 …X ; T † ˆ gNaP …Rm =k†

N
X
fF ‰U…Xi ; T0 †Šgw…X ; Xi ; T †;

…B:5†

V2 …X ; T † ˆ gNaP …Rm =k†

N
X
fF 0 ‰U …Xi ; T0 †ŠV1 …Xi ; T0 †gw…X ; Xi ; T †;

…B:6†

V3 …X ; T † ˆ gNaP …Rm =k†

N
X