PHYSICAL REVIEW E 79, 041904 共2009兲
Cable theory of protein receptor trafficking in a dendritic tree
Paul C. Bressloff
Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA
and Mathematical Institute, University of Oxford, 24-29 St. Giles’, Oxford OX1 3LB, United Kingdom
共Received 7 November 2008; published 3 April 2009兲
We develop an application of linear cable theory to protein receptor trafficking in the surface membrane of
a neuron’s dendritic tree. We assume that receptors diffuse freely in the dendritic membrane but exhibit periods
of confined motion through interactions with small mushroomlike protrusions known as dendritic spines. We
use cable theory to determine how receptor trafficking depends on the geometry of the dendritic tree and
various important biophysical parameters such as membrane diffusivity, the density of spines, the strength of
diffusive coupling between dendrites and spines, and the rates of constitutive recycling of receptors between
the surface of spines and intracellular pools. We also use homogenization theory to determine corrections to
cable theory arising from the discrete nature of spines.
DOI: 10.1103/PhysRevE.79.041904
PACS number共s兲: 87.16.A⫺, 87.19.L⫺, 87.10.⫺e
I. INTRODUCTION
Recent fluorescent recovery after photobleaching and
single-particle tracking experiments have revealed that neurotransmitter receptors undergo periods of free diffusion
within the dendritic membrane of a neuron interspersed with
periods of restricted motion in confinement domains that coincide with synapses 关1–6兴. Most excitatory synapses in the
brain are located within dendritic spines, which are small,
submicrometer membranous extrusions that protrude from a
dendrite. Typically spines have a bulbous head which is connected to the parent dendrite through a thin spine neck. Confinement of receptors occurs due to the geometry of the spine
and through interactions with scaffolding proteins and cytoskeletal elements within the postsynaptic density 共PSD兲,
which is the protein-rich region at the tip of the spine head.
Moreover, receptors within a spine may be internalized by
endocytosis and then either reinserted into the surface membrane via exocytosis or degraded 关7兴. It follows that under
basal conditions, the steady-state receptor concentration
within a synapse is determined by a dynamical equilibrium
in which the various receptor fluxes into and out of the spine
are balanced. Activity-dependent changes in one or more of
these fluxes can then modify the number of receptors in the
spine and thus alter the strength or weight of a synapse. This
is particularly significant, since there is now a large body of
experimental evidence suggesting that the activity-dependent
regulation of the trafficking of ␣-amino-3-hydroxy5-methyl-4-isoxazole-propionic acid 共AMPA兲 receptors,
which mediate the majority of fast excitatory synaptic transmission in the central nervous system, plays an important
role in synaptic plasticity 关8–10兴.
In a previous paper 关11兴, we analyzed a two-dimensional
diffusion model of protein receptor trafficking on a cylindrical dendritic membrane containing a population of dendritic
spines, which acted as spatially localized traps. We treated
the transverse intersection of a spine and a dendrite as a
small, partially absorbing boundary and used singular perturbation theory to analyze the steady-state distribution of receptors in the dendrite and spines. Using a combination of
analysis and numerical simulations, we showed that under
1539-3755/2009/79共4兲/041904共16兲
normal physiological conditions the variation in receptor
concentration around the circumference of the dendrite is
negligible, thus justifying a reduction to a one-dimensional
model. In the reduced model one can ignore the spatial extent of each spine and represent the density of spines along
the dendrite as a discrete sum of Dirac delta functions 关12兴.
This is motivated by the observation that the spine neck,
which forms the junction between a synapse and its parent
dendrite, varies in radius from ⬃0.02 to 0.2 ␮m, which is
typically an order of magnitude smaller than the spacing between spines 共⬃0.1– 1 ␮m兲 and the circumference of the
dendritic cable 共⬃1 ␮m兲; see Ref. 关13兴. In other words, the
disklike region or hole forming the junction between a spine
and the dendritic cable is relatively small, and can therefore
be neglected in the one-dimensional model. In the full twodimensional model of a dendritic cable, however, one can no
longer treat the spines as pointlike objects due to the fact that
the Green’s function associated with two-dimensional diffusion has a logarithmic singularity 关11兴.
More recently we have carried out a further simplification
of the one-dimensional model by treating the spine density
and biophysical properties of the spines as continuous functions of spatial position along the dendrite 关14兴. One of the
major advantages of this continuum approximation is that we
can adapt many methods and results from linear cable theory.
The latter was originally developed by Rall in order to describe the large-scale electronic structure of dendritic trees
共see the collection of papers in Ref. 关15兴兲, and has subsequently been used to analyze linearized reaction-diffusion
equations describing the diffusion and buffering of calcium
ions and other second messengers in the dendritic cytoplasm
关16,17兴. Cable theory has proven to be a very powerful tool
for quantifying how dendritic structure influences the electrophysiological properties of a neuron, and has led to many
new insights into the possible computational role of dendrites and spines 关18兴.
In this paper, we present a major application of cable
theory, namely, analyzing the role of surface diffusion and
dendritic structure on the trafficking of neurotransmitter receptors across multiple synapses. In particular, we use cable
theory to quantify how the distribution of receptors depends
on the geometry of the tree and various important biophysi-
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©2009 The American Physical Society
PHYSICAL REVIEW E 79, 041904 共2009兲
PAUL C. BRESSLOFF
cal parameters such as membrane diffusivity, the density of
spines, the strength of diffusive coupling between dendrites
and spines, and the rates of constitutive recycling of receptors between the surface of spines and intracellular pools. As
we have illustrated elsewhere 关14兴, the development of such
a theory is important, since it provides a quantitative framework for interpreting biophysical experiments concerned
with global aspects of receptor trafficking, such as inferring
relaxation rates from surface-inactivation experiments. It
also allows the construction of biophysical and computational models of synaptic plasticity that take into account
diffusion-mediated changes in the number of synaptic receptors following chemical or electrical stimulation. Although
our model and choice of parameters are mainly based on
experimental data of AMPA receptor trafficking, the underlying theory is applicable to the more general problem of
membrane diffusion and its role in protein receptor trafficking.
The structure of the paper is as follows. The continuum
model is introduced in Sec. II and steady-state solutions are
analyzed in Sec. III. We show that the steady-state equation
is given by a linear cable equation, which allows us to identify an effective space constant for receptor diffusion and to
derive an analog of the “equivalent-cylinders” representation
of a branched dendritic cable 关19,20兴. That is, the steadystate solution can be obtained by reducing a branched cable
to a single equivalent cable under the constraint that at each
node the diameters d1 , d2 of the two daughter branches are
related to the diameter of the parent branch according to the
rule 冑d = 冑d1 + 冑d2. Such a constraint will not be satisfied by
most dendritic geometries and the reduction cannot be extended to time-dependent solutions, so we consider a more
general approach to handling branching structures in Sec. V.
In Sec. IV we consider a discrete set of spines whose mean
spacing is much smaller than the space constant of the corresponding continuum model so that there is a separation of
spatial scales. We then use a multiscale homogenization procedure to analyze small-scale fluctuations in the steady-state
receptor concentration, following along similar lines to a recent analysis of the cable equation for membrane voltage in
a heterogeneous dendrite 关21兴. In particular, we show that the
continuum approximation is reasonable provided that the diffusive coupling between spines and the parent dendrite is
sufficiently weak. In Sec. V we use Laplace transforms to
calculate the Green’s function for the full time-dependent
model, which takes into account the internal receptor dynamics of spines. The Green’s function determines the timedependent dendritic receptor concentration in response to
both somatic and intracellular sources of receptors. We also
perform an asymptotic analysis of the Green’s function in
order to extract the asymptotic rate of relaxation to the
steady state. Finally, we show how to construct the Green’s
function on an arbitrary dendritic tree using the so-called
sum-over-trips formalism developed originally by Abbott
et al. 关22兴 for electrically passive dendritic trees 共see also
Refs. 关23,24兴兲, and extended more recently to quasiactive
dendrites by Coombes et al. 关25兴.
II. CONTINUUM MODEL OF RECEPTOR TRAFFICKING
Consider a single uniform dendritic cable of length L and
circumference l as shown in Fig. 1共a兲. Let U共x , t兲, with
(a)
J(x,t)
U(x,t)
Isoma
x=0
x=L
(b)
σdeg
k
J(x,t)
C(x,t)
R(x,t)
σ
σrec
spine membrane
intracellular pool
FIG. 1. Continuum model of protein receptor trafficking along a
dendrite. 共a兲 Single dendritic cable. The concentration U共x , t兲 of
surface receptors at time t varies with distance x from the soma. A
surface receptor current Isoma is injected at the somatic end of the
cable. The current J共x , t兲 determines the rate at which receptors flow
between the dendrite and an individual spine at x. 共b兲 Simplified
model of a spine showing constitutive recycling with an intracellular pool. Surface receptors of concentration R共x , t兲 are internalized
at a rate k and are either reinserted into the spine surface from an
intracellular pool of C共x , t兲 receptors at a rate ␴rec or sorted for
degradation at a rate ␴deg. There is also a local production of intracellular receptors at a rate ␴.
0 ⱕ x ⱕ L, denote the concentration of receptors within the
surface of the dendrite at time t and position x, where x
denotes axial distance from the soma. Following Ref. 关14兴,
we model the one-dimensional diffusive transport of surface
receptors along the dendritic cable according to the equation
⳵ 2U
⳵U
= D 2 − ␳共x兲J共x,t兲.
⳵x
⳵t
共2.1兲
The first term on the right-hand side of Eq. 共2.1兲 represents
the Brownian diffusion of receptors along the surface of the
cable, whereas the second term represents the flux of receptors from the dendrite to a local population of dendritic
spines, with ␳共x兲 denoting the spine density and J共x , t兲 representing the number of receptors per unit time flowing into
a single spine at x. The density ␳ satisfies the normalization
condition 兰L0 ␳共x兲dx = N / l, where N is the total number of
spines on the dendrite. Equation 共2.1兲 is supplemented by
boundary conditions at the ends of the cable:
D
冏 冏
⳵U
⳵x
=−
x=0
Isoma
,
l
D
冏 冏
⳵U
⳵x
= 0.
共2.2兲
x=L
Here Isoma represents a current source 共number of surface
receptors per unit time兲 at the boundary x = 0 adjacent to the
soma arising from fast somatic exocytosis 关26兴. The distal
end of the cable at x = L is taken to be closed.
Denoting the concentration of surface receptors in an individual spine at x by R共x , t兲, we take the receptor current
J共x , t兲 to be of the form
J共x,t兲 = ⍀+U共x,t兲 − ⍀−R共x,t兲,
共2.3兲
where ⍀⫾ are effective hopping rates into and out of the
spine. In previous work 关11,12兴, we have taken ⍀⫾ = ⍀, with
⍀ representing the effects of the spine neck on restricting the
flow of receptors 关4兴. However, there are additional factors
that confine receptors within a spine such as interactions with
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PHYSICAL REVIEW E 79, 041904 共2009兲
CABLE THEORY OF PROTEIN RECEPTOR TRAFFICKING…
scaffolding proteins and cytoskeletal elements. Such effects
can be incorporated into our simplified single-spine model
by taking the rate of exit from a spine to be smaller than the
rate of entry; that is, ⍀− ⬍ ⍀+. Substitution of Eq. 共2.3兲 into
Eq. 共2.1兲 then gives
⳵ 2U
⳵U
= D 2 − ␳共x兲关⍀+U共x,t兲 − ⍀−R共x,t兲兴.
⳵x
⳵t
共2.4兲
The dynamics of the receptor concentration R共x , t兲 within
an individual spine is determined by the current across the
junction with the dendritic cable, and the various forms of
local trafficking within a spine. A detailed model of singlespine dynamics has been presented elsewhere 关27,28兴. Here
we follow Refs. 关11,12兴 and consider a simplified model in
which we treat the spine as a single homogeneous compartment; see Fig. 1共b兲. We assume that surface receptors within
a spine are endocytosed at a rate k and stored in a corresponding intracellular pool 关7,29,30兴. Intracellular receptors
are either reinserted into the surface via exocytosis at a rate
␴rec or degraded at a rate ␴deg. Denoting the number of receptors in an intracellular pool at 共x , t兲 by C共x , t兲, we then
have the pair of equations
A
⳵R
= ⍀+U − ⍀−R − kR + ␴recC,
⳵t
⳵C
= − ␴recC − ␴degC + kR + ␴ .
⳵t
共2.5兲
共2.6兲
The first two terms on the right-hand side of Eq. 共2.5兲 constitute the current J共x , t兲 of Eq. 共2.3兲. Since J共x , t兲 is the
number of receptors per unit time flowing across the junction
between the dendritic cable and the spine, it is necessary to
multiply the left-hand side by the surface area A of the spine
in order to properly conserve receptor numbers. We also allow for a local source of intracellular receptors by including
the inhomogeneous term ␴ on the right-hand side of Eq.
共2.6兲. This represents the local accumulation of new 共rather
than recycled兲 receptors within the intracellular pool supplied, for example, by the targeted delivery of intracellular
receptors from the soma 关31,32兴, or possibly by local receptor synthesis 关33–35兴. Note that all parameters in Eqs.
共2.4兲–共2.6兲 could themselves be functions of x and t, although in this paper we will restrict ourselves to the case of
identical spines with time-independent properties.
The model given by Eqs. 共2.4兲–共2.6兲 treats the surface of
the dendritic cable as an effective one-dimensional medium,
in which variations in the dendritic receptor concentration
around the circumference of the cable are neglected. This can
be justified by considering a reduction from a full twodimensional model of surface diffusion in which the junction
between each spine and the dendritic surface is treated as a
small, partially absorbing trap 关11,12兴. Such a reduction
leads to Eq. 共2.4兲 with the spine density given by a sum of
Dirac delta functions:
N
␳共x兲 =
1
兺 ␦共x − x j兲,
l j=1
共2.7兲
where x j is the location of the jth spine. Following Ref. 关14兴,
we will make an additional simplification by taking the spine
density ␳ and the various biophysical parameters characterizing individual spines 共⍀⫾ , k , ␴rec , ␴deg , ␴兲 to be continuous
functions of x. We expect this continuum approximation to
be reasonable in the case of a large number of closely spaced
spines such that neighboring spines share similar properties
共perhaps after spatially averaging over local clusters of
spines兲. In Sec. IV we will show how corrections to this
continuum approximation can be obtained using a multiplescale homogenization procedure, following along similar
lines to the recent analysis of voltage and conductance
changes in a heterogeneous dendrite 关21兴.
Although we focus on the dynamics of receptor trafficking in this paper, it is useful to explain briefly how our biophysical model can be incorporated into more physiological
descriptions of synaptic function. Each excitatory synapse on
the postsynaptic side can be identified with the PSD located
within the spine head. Assuming a uniform concentration R
of receptors in the spine, the total number of synaptic receptors is aR, where a is the area of the PSD. The arrival of an
action potential at the presynaptic terminal leads to the release of chemical neurotransmitters, which bind to receptors
within the PSD. This changes the configurational state of
each receptor, resulting in the transient opening of an ion
channel and the flow of ions through the cell membrane.
Assuming that all synaptic receptors bind to neurotransmitter
and have identical conductance states gS, the total change in
synaptic conductance is gSaR. The maximal conductance
change is one measure of the strength of a synapse. Of
course, this is an oversimplification since there can be different classes of receptors and the conductance of a receptor
may be modified by interactions with scaffolding proteins
and other protein complexes within the PSD. Nevertheless,
even in this simplified model, we see that the strength of a
synapse depends on a number of factors including the concentration of receptors within a spine, the area of the PSD,
and the conductance state of the receptors. All of these are
potential targets for chemical signaling cascades that induce
activity-dependent changes in the strength of a synapse.
Since there is growing experimental evidence that changes in
synaptic strength are associated with changes in the number
of synaptic AMPA receptors 关8–10兴, it follows that the cable
theory of receptor trafficking also has applications to biophysical and computational models of synaptic plasticity; see
Ref. 关14兴.
III. STEADY-STATE CABLE EQUATION
In this section we calculate the steady-state receptor concentration U on a branched dendrite, in which each branch is
treated as a uniform cable with identical uniformly distributed spines. Note that as one proceeds away from the soma,
dendrites tend to become thinner. This is partially taken into
account by taking daughter branches to be thinner than their
parent branch. However, one could also extend the analysis
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PHYSICAL REVIEW E 79, 041904 共2009兲
PAUL C. BRESSLOFF
to the case of tapering dendrites, in which the diameter of a
cable decreases smoothly with distance along the cable
关36,37兴.
A. Single uniform cable
We begin by considering a single branch as shown in Fig.
1. In the case of a constant current Isoma at the end x = 0,
solutions of Eqs. 共2.4兲–共2.6兲 converge to a unique steady
state obtained by setting all time derivatives to zero. Equations 共2.5兲 and 共2.6兲 imply that in the steady state
C共x兲 =
kR共x兲 + ␴
,
␴rec + ␴deg
R共x兲 =
⍀+U共x兲 + ⌳␴
,
⍀− + k共1 − ⌳兲
共3.1兲
where
⌳=
tor diffusion and transport. 共The background term R̄ plays an
analogous role to the membrane reversal potential in a dendrite that is uniformly stimulated by background synaptic
currents with voltage-independent conductances.兲 The general solution of Eq. 共3.4兲 is given by
U共x兲 = A cosh共␥x兲 + B sinh共␥x兲 + R̄,
with the constants A , B determined by boundary conditions
共2.2兲. In anticipation of our analysis of branching structures,
we generalize these boundary conditions by introducing the
notion of a diffusive impedance 共borrowing terminology
from corresponding studies of the linear cable equation for
membrane voltage 关17,38兴兲. First, we define a diffusive receptor current I共x兲 along the cable 关as distinct from the diffusive current J共x兲 between dendrite and spines兴 according to
␴rec
.
␴rec + ␴deg
I = − lD
Substituting Eq. 共3.1兲 into the steady-state version of diffusion equation 共2.4兲 gives
d 2U
D 2 − ␳共x兲⍀̄关U共x兲 − R̄兴 = 0,
dx
共3.2兲
⍀+k共1 − ⌳兲
,
⍀− + k共1 − ⌳兲
R̄ =
⍀− ⌳␴
.
⍀+ k共1 − ⌳兲
d 2U
− ␥2U共x兲 = − ␥2R̄,
dx2
where
共3.3兲
where
␥=
冑 冑
␳0⍀̄
=
D
共3.4兲
⍀̄
.
l⌬D
Isoma = N⍀̄
冋冕
L
0
册
1
=
lD␥
I共0兲 = Isoma,
⌬
lD⍀̄
共3.9兲
U共L兲 − R̄ = ZLI共L兲,
共3.10兲
where ZL is a terminal diffusive impedance due to a possible
diffusive coupling with secondary branches at x = L 共see below兲. We recover the closed or Neumann boundary condition
of Eq. 共2.2兲 by taking the infinite-impedance limit ZL → ⬁.
On the other hand, the zero-impedance limit ZL → 0 corresponds to the Dirichlet boundary condition U共L兲 = R̄.
A straightforward calculation shows that the steady-state
receptor concentration for boundary conditions 共3.10兲 is
given by 关38兴
U共x兲 = ZIsoma
Z sinh共␥关L − x兴兲 + ZL cosh共␥关L − x兴兲
+ R̄.
Z cosh共␥L兲 + ZL sinh共␥L兲
共3.11兲
If the boundary at x = L is also closed as in Eq. 共2.2兲, then
U共x兲dx/L − R̄ .
U共x兲 = ZIsoma
This implies that the total number of receptors entering the
dendrite from the soma is equal to the mean number of receptors hopping from the dendrite into the N spines.
Equation 共3.4兲 is identical in form to the steady-state
cable equation describing membrane voltage and electrical
current flow in passive dendrites 关15,17,19,20兴 with ␥−1 ⬅ ␰
reinterpreted as an effective space constant for surface recep-
冑
共3.8兲
is the characteristic diffusive impedance of the cable. We
then introduce a Robin boundary condition at the end x = L so
that
共3.5兲
Integrating Eq. 共3.4兲 with respect to x and using boundary
conditions 共2.2兲 yields the conservation condition
共3.7兲
I共x兲 = − Z−1关A sinh共␥x兲 + B cosh共␥x兲兴,
Z=
One can view ⍀̄ as an effective spine-neck hopping rate and
R̄ as an effective spine receptor concentration. Equation 共3.2兲
is supplemented by boundary conditions 共2.2兲. Since the
steady-state receptor concentration within spines, R共x兲, is a
simple monotonic function of the dendritic receptor concentration U共x兲, we will focus on solutions of the latter.
In the case of a uniform spine density, ␳共x兲 = ␳0 = 1 / l⌬,
with ⌬ = L / N as the mean spacing between spines, Eq. 共3.2兲
reduces to the simpler form 关14兴
⳵U
.
⳵x
It follows that the steady-state current is given by
where
⍀̄ =
共3.6兲
cosh共␥关x − L兴兲
+ R̄,
sinh共␥L兲
共3.12兲
whereas for an open boundary at x = L we have
U共x兲 = ZIsoma
sinh共␥关L − X兴兲
+ R̄.
cosh共␥L兲
共3.13兲
In the limit L → ⬁, Eq. 共3.11兲 reduces to U共x兲 = ZIsomae−␥x
+ R̄, showing that the concentration approaches the back-
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PHYSICAL REVIEW E 79, 041904 共2009兲
CABLE THEORY OF PROTEIN RECEPTOR TRAFFICKING…
350
200
(a)
300
160
250
140
120
σ = 10-3 s-1
ξ [µm]
concentration [µm-2]
180
100
80
60
σdeg
200
150
100
40
σ=0
20
0
(b)
Ω
D
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
distance from soma [mm]
50
0
1
0.1
1
parameter value [x baseline]
10
FIG. 2. 共a兲 Steady-state concentration of surface receptors in the dendritic membrane as a function of distance x from the soma. Solid
curves show concentration profiles for diffusivity D = 0.1 ␮m2 s−1 and dashed curves are for D = 0.45 ␮m2 s−1. Lower two curves are for a
zero background concentration, whereas upper two curves are for a nonzero background concentration. The length and circumference of the
cable are L = 1 mm and l = 1 ␮m. N = 1000 identical spines are distributed uniformly along the cable with density ␳0 = 1 ␮m−2. The spine
parameters are as follows: surface area A = 1 ␮m2, rate of endocytosis k = 10−3 ␮m2 s−1, rate of recycling ␴rec = 10−3 s−1, rate of degradation
␴deg = 10−5 s−1, and hopping rates ⍀⫾ = ⍀ = 10−3 ␮m2 s−1. The values of most parameters are based on experimental measurements, including the diffusivity 关3,4兴, the rates of constitutive recycling 关7,39–41兴, and the size and density of spines 关13兴, whereas the hopping rate ⍀ is
estimated using a simple model of diffusion within the spine neck 关14兴. The somatic current is taken to be Isoma = 0.1 s−1 so that the
maximum number of synaptic receptors per spine is consistent with experimental observations 关42,43兴. 共b兲 Space constant ␰ as a function of
the diffusivity D, the diffusive coupling ⍀, and the rate of degradation ␴deg.
ground concentration R̄ at a rate ␥. Thus one can view the
space constant ␰ as determining the effective range of the
diffusive transport of surface receptors from the soma to synaptic targets along the dendrite. It follows that there must be
an additional mechanism for supplying distal spines with receptors in the steady state, that is, spines at locations x Ⰷ ␰.
This is taken into account in our model by including the local
source term ␴ in Eq. 共2.6兲, which maintains the steady-state
background concentration R̄; see Eq. 共3.3兲. Examples of
steady-state receptor profiles are shown in Fig. 2共a兲, and the
parameter dependence of the space constant ␰ is shown in
Fig. 2共b兲.
Of course the presence of a source of receptors at the
soma tends to bias proximal spines 共spines adjacent to the
soma兲, particularly when the local source term ␴ is small
关see Fig. 2共a兲兴. As we have highlighted elsewhere 关12,14兴,
this would seem to contradict the notion of synaptic democracy, whereby all synapses of a neuron have a similar capacity for influencing the postsynaptic response regardless of
location along a dendritic tree 关44–46兴. Indeed, it has been
found experimentally that there is actually an increase in
receptor numbers at more distal synapses 关47兴, resulting in a
distance-dependent variation in synaptic conductance consistent with somatic equalization. Such behavior could be incorporated into our model by dropping the assumption of
identical spines distributed uniformly along the cable. This is
also consistent with experimental data indicating that there is
considerable amount of heterogeneity in the properties of
spines within a single neuron. For example, spine morphology ranges from small filopodial protrusions to large mushroomlike bulbs, and properties such as the surface area of a
spine and spine density tend to vary smoothly from proximal
to distal locations along the dendrite 关13,48,49兴. We will return to the issue of heterogeneities in Sec. IV.
B. Branched cable
Let us now consider a branched cable as shown in Fig. 3.
It consists of one primary branch of length L0 and circumference l0 splitting into two secondary branches with lengths
L1 , L2 and circumferences l1 , l2. All branches are assumed to
have the same background concentration R̄. A current Isoma is
injected at the somatic end of the primary branch, whereas
the terminal ends of the two secondary branches are closed.
By continuity of the surface receptor concentration at the
branch point xB, we can take the corresponding surface concentration UB to be single valued. It follows from setting x
= 0 in Eq. 共3.12兲 that
UB = Z jI j coth共␥ jL j兲 + R̄,
j = 1,2,
共3.14兲
where Z j and ␥−1
j are the characteristic impedance and space
constant of the jth secondary branch, and I j is the corresponding current flowing into the branch at point xB. On the
other hand, the membrane potential in the primary branch is
I1
UB
Isoma
L0, l0
I2
L1 l1
L2, l2
FIG. 3. Schematic diagram of a branched cable consisting of a
primary branch with length L0 and circumference l0, and two secondary branches with lengths L1 , L2 and circumferences l1 , l2. The
surface receptor concentration UB at the branch point is continuous,
and the diffusive current I0 flowing through the junction in the
primary branch splits into two currents I1 , I2.
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PHYSICAL REVIEW E 79, 041904 共2009兲
PAUL C. BRESSLOFF
given by Eq. 共3.11兲 with the terminal impedance ZL = 共UB
− R̄兲 / I0, where I0 is the current flowing into xB from the
primary branch. Conservation of receptors implies that I0
= I1 + I2. Dividing the current conservation equation by UB
− R̄ and using Eq. 共3.14兲 then yields
approaches have been developed within the more general
context of time-dependent solutions of the linear cable equation on a tree. In Sec. V we will show how the sum-overtrips method can be extended to our protein trafficking model
by using Laplace transforms, which can then be used to derive the steady-state solution on an arbitrary dendritic tree.
1
1
.
= 兺
ZL j=1,2 Z j coth共␥ jL j兲
IV. DISCRETE SPINES AND HOMOGENIZATION
共3.15兲
Following along similar lines to Rall 关19兴, we can now
derive the analog of the equivalent-cylinders conditions for
receptor trafficking in a branched cable. First, we assume
that ␥1L1 = ␥2L2. 共In the context of standard cable theory this
would mean that both secondary cables have the same electrotonic length.兲 Second, we assume that
1
1
1
=
+ ,
Z0 Z1 Z2
共3.16兲
that is the characteristic impedances of the three branches are
matched. It then follows that ZL = Z0 coth共␥1L1兲. Substituting
this expression for ZL into Eq. 共3.11兲 and performing some
algebra leads to the result
U共x兲 = Z0Isoma
cosh共␥关Leff − x兴兲
+ R̄,
sinh共␥Leff兲
共3.17兲
where Leff = L0 + ␥1L1 / ␥. Equation 共3.17兲 describes the concentration in a single unbranched cable of effective length
Leff and a closed-end boundary condition. In other words, the
branched cable has been reduced to a single equivalent cylinder whose other properties are those of the primary branch.
Finally, substituting Eq. 共3.9兲 into impedance matching condition 共3.16兲 and assuming that D, ⍀̄, and ⌬ are the same in
all branches gives the following relationship between the diameters of the primary and secondary branches:
1/2
1/2
d1/2
0 = d1 + d2 .
共3.18兲
Thus, there is one significant difference between the steadystate trafficking model and the corresponding cable model of
membrane voltage, namely, that the diameters have to satisfy
a d1/2 rule rather than Rall’s d3/2 rule 关19兴.
One could now use the above procedure to collapse a
whole dendritic tree to a single equivalent cylinder provided
that the following conditions are met: 共1兲 All branches have
the same background concentration R̄. 共2兲 All terminal
branches have the same boundary conditions and are at the
same effective distance from the soma 共after rescaling the
length of each branch in the tree according to Li → ␥iLi兲. 共3兲
At every branch point the characteristic impedances must be
matched, which leads to the d1/2 rule when all branches have
the same diffusivity D, hopping rate ⍀̄, and spine spacing ⌬.
In general these three conditions will not hold so that one
needs an alternative procedure for obtaining the steady-state
solution on a tree. Two classes of iterative procedures have
been developed within the context of linear cable theory applied to electrically passive and quasiactive dendritic membranes, one based on the graphical calculus of Butz and
Cowan 关38,50–52兴 and the other based on the so-called sumover-trips formalism of Abbott et al. 关22–25兴. Both of these
As we have already indicated, there are several forms of
heterogeneity that can modify our calculation of the steady
state. Suppose that the various parameters 共k , ␴rec , ␴deg , ␴兲
associated with constitutive recycling are uniform along the
cable. This still allows both the density ␳ and spine-dendrite
coupling ⍀⫾ to be x dependent. Indeed, the experimentally
observed proximal-to-distal variation in the structure and distribution of spines 关13,48,49兴 could be incorporated by taking ␳共x兲 and ⍀⫾共x兲 to be continuous functions of x, and this
is straightforwardly handled using the continuum model.
However, there is also a heterogeneity occurring on a much
smaller spatial scale due to the discrete nature of spines, that
is, when the spine density is given by Eq. 共2.7兲. There is
clearly a separation of length scales between the typical
spine spacing ⌬ of 1 ␮m and the typical space constant ␰ of
100 ␮m 共see Fig. 2兲; that is, L ⱖ ␰ Ⰷ ⌬. This suggests that
taking into account the discrete nature of spines will lead to
small-scale fluctuations in the steady-state dendritic receptor
concentration. We will analyze such fluctuations for a single
cable using a multiple-scale homogenization procedure 关53兴,
following along similar lines to a recent analysis of the cable
equation for voltage and conductance changes along a heterogeneous dendrite 关21兴. For simplicity, we will ignore heterogeneities on large spatial scales so that the homogenized
dendrite reduces to the uniform cable analyzed in Sec. III A.
Introducing the small dimensionless parameter ␧ = ⌬ / ␰,
we rewrite Eq. 共3.2兲 in the form
冉 冊 冉 冊冋
d 2U
x
x
⍀̄
2 =␳
dx
␧
␧
U共x兲 − R̄
冉 冊册
x
␧
= 0,
共4.1兲
where
N
N
j=1
j=1
␳共y兲 = l−1 兺 ␦共␧y − x j兲 = ␳0␰ 兺 ␦共y − y j兲
and y j = x j / ␧. We have made explicit the fact that the coupling ⍀̄ and background concentration R̄ may vary between
neighboring spines. As a further simplification, however, we
will assume for the moment that ⍀̄ and R̄ are uniform and
that the spines are evenly spaced with x j = j⌬ and y j = j␰.
Equation 共4.1兲 then becomes
冋 冉 冊册
d 2U
x
¯
2 = ⌫ + ⌬⌫
dx
␧
where
¯⌫ = ␰−2,
冉
关U共x兲 − R̄兴 = 0,
N
⌬⌫共y兲 = ¯⌫ ␰ 兺 ␦共y − j␰兲 − 1
j=1
such that ⌬⌫共y兲 is a ␰-periodic function of y.
041904-6
共4.2兲
冊
共4.3兲
PHYSICAL REVIEW E 79, 041904 共2009兲
CABLE THEORY OF PROTEIN RECEPTOR TRAFFICKING…
The basic idea of multiscale homogenization is to expand
the solution of Eq. 共4.2兲 as power series in ␧, with each term
in the expansion depending explicitly on the “slow” 共macroscopic兲 variable x and the “fast” 共microscopic兲 variable y
= x / ␧:
U共x,y兲 = U0共x,y兲 + ␧U1共x,y兲 + ␧2U2共x,y兲 + ¯ , 共4.4兲
where U j共x , y兲, with j = 0 , 1 , . . ., are ␰ periodic in y. The perturbation series expansion is then substituted into Eq. 共4.2兲
with x , y treated as independent variables so that derivatives
with respect to x are modified according to ⳵x → ⳵x + ␧−1⳵y.
This generates a hierarchy of equations corresponding to successive powers of ␧,
2
⳵ 2U 0
= 0,
⳵ y2
共4.5兲
⳵ 2U 0 ⳵ 2U 1
+
= 0,
⳵ x ⳵ y ⳵ y2
共4.6兲
d 2U 1 ¯
= ⌫U1 ,
dx2
which has the unique solution U1 = 0 for no-flux boundary
condition 共4.10兲. Thus the leading-order corrections arising
from small-scale fluctuations in the spine density occur at
O共␧2兲. In order to calculate U2, we first subtract the averaged
Eq. 共4.11兲 from Eq. 共4.7兲 to obtain
⳵ 2U 2
= ⌬⌫共y兲关U0共x兲 − R̄兴.
⳵ y2
共4.7兲
冖
共4.8兲
f共z兲dz −
0
冕
d具Un典共L兲
dx
冏
= 0,
n ⱖ 0.
x=L
共4.10兲
Equation 共4.5兲 implies that U0 is independent of y, since
U0 should remain bounded in the limit ␧ → ⬁. Equation 共4.6兲
and boundedness of U1 then imply that U1 is also independent of y. Spatial averaging can now be performed in order
to determine the differential equations satisfied by U0 and
U1. Taking the spatial average of Eq. 共4.7兲 with U0 = 具U0典
gives
d 2U 0 ¯
= ⌫关U0 − R̄兴.
dx2
共4.11兲
We have exploited the fact that U2 is ␰ periodic in y so
具⳵2U2 / ⳵y 2典 = 0. Equation 共4.11兲 together with boundary condition 共4.10兲 is precisely the homogeneous equation analyzed
in Sec. III A with ␳ → 具␳典 = ␳0. Similarly, spatially averaging
Eq. 共4.8兲 for n = 1 shows that U1 satisfies the equation
y
f共z兲dz
0
冔
共4.14兲
冕冖
y⬘
⌬⌫共z兲dzdy ⬘ .
0
Spatially averaging this equation in order to express ␹共0兲 in
terms of 具␹典 and multiplying through by 关U0共x兲 − R̂兴 finally
gives
⌬U2共x,y兲 ⬅ U2共x,y兲 − 具U2典共x兲
= 关U0共x兲 − R̄兴
共4.9兲
F共y兲dy.
0
冏
冓冕
冖冖
y
␰
Since the boundary conditions at the ends x = 0 , L are independent of ␧, we take
冏 冏
y
y
at O共␧n兲, with n ⱖ 1. Define the spatial average of a
␰-periodic function F共y兲, denoted by 具F典, according to
Is
= − ␦n,0,
lD
x=0
冕
0
⳵ 2U n
⳵2Un+1 ⳵2Un+2 ¯
+
2
= 关⌫ + ⌬⌫共y兲兴Un
+
⳵ x2
⳵x ⳵ y
⳵ y2
d具Un典
dx
f共z兲dz ⬅
␹共y兲 = ␹共0兲 +
−1
1
具F典 =
␰
y
for any integrable function f. Another integration with respect to y shows that
at powers ␧ , ␧ , 1 and
−2
共4.13兲
It follows that U2共x , y兲 = 关U0共x兲 − R̄兴␹共y兲 with d2␹共y兲 / dy 2
= ⌬⌫共y兲 and ␹ as a ␰-periodic function of y. Integrating once
with respect to y gives ␹⬘共y兲 = ␹⬘共0兲 + 兰0y ⌬⌫共z兲dz. We can
eliminate the unknown ␹⬘共0兲 by spatially averaging with respect to y and using 具␹⬘典 = 0. This gives ␹⬘共y兲 = 养0y ⌬⌫共z兲dz
with
0
⳵ 2U 0
⳵ 2U 1 ⳵ 2U 2 ¯
+
2
= 关⌫ + ⌬⌫共y兲兴关U0 − R̄兴
+
⳵ x2
⳵ x ⳵ y ⳵ y2
共4.12兲
0
y⬘
⌬⌫共z兲dzdy ⬘ . 共4.15兲
0
It remains to determine the equation satisfied by 具U2典. Spatially averaging Eq. 共4.8兲 for n = 2 gives
d2具U2典 ¯
= ⌫具U2典 + 具⌬⌫共y兲U2共x,y兲典.
dx2
共4.16兲
Substituting Eq. 共4.15兲 into Eq. 共4.16兲 and reordering the
resulting multiple integral yields the result
d2具U2典 ¯
= ⌫具U2典 −
dx2
冓冉冖
冊冔
2
y
⌬⌫共z兲dz
0
关U0共x兲 − R̄兴.
共4.17兲
Finally, writing 具U典 = U0 + ␧具U2典 + ¯ we obtain the homogenized equation 共see also 关21兴兲
d2具U典
= ⌫␧关具U典 − R̄兴,
dx2
where
⌫␧ = ¯⌫ − ␧2
冓冉冖
2
y
0
冊冔
共4.18兲
⌬⌫共z兲dz
+ O共␧3兲.
共4.19兲
The above analysis shows that there are two sources of
O共␧2兲 corrections to solution 共3.13兲 of the homogeneous den-
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PHYSICAL REVIEW E 79, 041904 共2009兲
PAUL C. BRESSLOFF
冉 冓冉冖
冊 冔冊
V. GREEN’S FUNCTIONS AND LAPLACE TRANSFORMS
drite. First, the space constant is increased according to
␰ → ␰␧ = ␰ 1 +
␧2
¯
2⌫
2
y
⌬⌫共z兲dz
0
+ O共␧3兲,
共4.20兲
Second, there are small-scale fluctuations in the dendritic
receptor concentration of the form
⌬U共x,y兲
= ␧2
具U典共x兲
冖冖
y
0
y⬘
⌬⌫共z兲dzdy ⬘ + O共␧3兲.
共4.21兲
0
In this section we use Laplace transforms to construct
Green’s functions for the full time-dependent model given by
Eqs. 共2.4兲–共2.6兲 and its extension to an arbitrary dendritic
tree. It will be convenient to fix the units of length to be
1 ␮m and set A = 1, where A is the surface area of a spine.
共R and C then have the same dimensions.兲 For the sake of
illustration, we will assume that the receptor concentrations
are in steady state with Isoma共t兲 = 0 for t ⬍ 0. We then determine variations about the steady state induced by a timedependent somatic current Isoma共t兲 = I共t兲 for t ⱖ 0.
It is straightforward to calculate the integrals in Eqs. 共4.20兲
and 共4.21兲 for a periodic spine density 关21兴:
冓冉冖
⌬⌫共z兲dz
0
冖冖
y
0
y⬘
冊冔
冋
2
y
⌬⌫共z兲dzdy ⬘ =
0
=
1
,
12␰2
共4.22兲
册
1
y
y2
− 2−
.
12
2␰ 2␰
共4.23兲
Hence, ␰␧ ⬇ ␰共1 + ␧2 / 24兲 and fluctuations in the dendritic receptor concentration vary between −␧2 / 12 at spines and
␧2 / 24 between spines. Clearly, the discrete nature of spines
only has a small effect on the receptor concentration provided that ␧ Ⰶ 1. In terms of physiological parameters 关see
Eq. 共3.5兲兴,
␧=
冑冑
⌬
l
⍀̄
.
D
A. Single uniform cable
Consider a single uniform dendritic cable as shown in Fig.
1. The linear response to the time-dependent current I共t兲 is
given by Eqs. 共2.4兲–共2.6兲 with ␴ = 0 and the initial conditions
U共x , 0兲 = R共x , 0兲 = C共x , 0兲 = 0. Laplace transforming Eqs.
共2.4兲–共2.6兲 with Ũ共x , s兲 = 兰⬁0 e−stU共x , t兲dt, etc., gives
D
1
d2Ũ
2 − sŨ共x,s兲 = ␳0关⍀+Ũ共x,s兲 − ⍀−R̃共x,s兲兴 − Ĩ共s兲␦共x兲,
dx
l
共5.1兲
and
sR̃共x,s兲 = 关⍀+Ũ共x,s兲 − ⍀−R̃共x,s兲兴 − kR̃共x,s兲 + ␴recC̃共x,s兲,
共5.2兲
共4.24兲
Since the mean spine spacing ⌬ and the circumference of the
dendritic cable l are both on the order of 1 ␮m, it follows
that small ␧ corresponds to a weak effective coupling between the spines and parent dendrite; that is, ⍀̄ Ⰶ D. The
latter condition will hold if ⍀⫾ Ⰶ D , k共1 − ⌳兲 Ⰶ D 关see Eq.
共3.3兲兴, which is certainly the case for the basal parameter
values used in Fig. 2.
The above multiscale homogenization method can be extended to the case of randomly rather than periodically distributed spines, as well as nonuniform coupling ⍀̄, provided
that the resulting heterogeneous medium is ergodic 关54兴.
That is, the result of averaging over all realizations of the
ensemble of spine distributions is equivalent to averaging
over the length L of the dendrite in the infinite-L limit. If
such an ergodic hypothesis holds and L is sufficiently large
so that boundary terms can be neglected, then the above
analysis carries over with 具·典 now denoting ensemble averaging. Examples of how to evaluate integrals such as those
appearing in Eqs. 共4.20兲 and 共4.21兲 for randomly distributed
spines are presented in Ref. 关21兴. Finally, note that our analysis is easily extended to the full time-dependent model by
including a term D−1 ⳵ U0 / ⳵t on the right-hand side of Eq.
共4.5兲 and a term D−1 ⳵ Un / ⳵t on the right-hand side of Eq.
共4.8兲, and performing ␧ series expansions of R共x , t兲 and
C共x , t兲. Indeed, the steps of the analysis are almost identical
to those of the steady state if the homogenization procedure
is carried out after Laplace transforming Eqs. 共2.4兲–共2.6兲
along the lines of Sec. V.
sC̃共x,s兲 = − ␴recC̃共x,s兲 − ␴degC̃共x,s兲 + kR̃共x,s兲.
共5.3兲
Equation 共5.1兲 is supplemented by the closed boundary conditions
dŨ共0,s兲
= 0,
dx
dŨ共L,s兲
= 0.
dx
共5.4兲
Note that we have incorporated the source term at the soma
into the diffusion equation itself, rather than in the boundary
condition at x = 0. It is convenient to rewrite Eqs. 共5.2兲 and
共5.3兲 as the matrix equation
共M − sI兲
冉 冊 冉
R̃共x,s兲
=−
⍀+Ũ共x,s兲
C̃共x,s兲
0
冊
,
共5.5兲
with
M=
冉
− k − ⍀−
k
冊
␴rec
.
− ␴rec − ␴deg
共5.6兲
Let ␮⫾ denote the eigenvalues of the matrix −M as
␮⫾ = 21 共k + ⍀− + ␴rec + ␴deg兲
⫾
1
2
冑共k + ⍀− − ␴rec − ␴deg兲2 + 4␴reck.
Solving Eq. 共5.5兲 for R̃ , C̃ in terms of Ũ shows that
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PHYSICAL REVIEW E 79, 041904 共2009兲
CABLE THEORY OF PROTEIN RECEPTOR TRAFFICKING…
冉 冊
R̃共x,s兲
C̃共x,s兲
= − 共M − sI兲−1
冉
⍀+Ũ共x,s兲
␴/s
冉
冊
冊
⌶共s兲 =
H̃共x,y;s兲 =
冋
册
共5.10兲
and Ĩ共x , s兲 is the Laplace transform of an effective current
density at the soma:
1
Ĩ共x,s兲 = Ĩ共s兲␦共x兲.
l
共5.11兲
Let G̃共x , y ; s兲 denote the Laplace-transformed Green’s
function, which satisfies the equation
d2G̃
dx
2
− ⌶共s兲G̃共x,y;s兲 = −
␦共x − y兲
D
共5.12兲
,
together with the same boundary conditions as Ũ共x , s兲 for
fixed y. A standard calculation yields
G̃共x,y;s兲 =
cosh兵冑⌶共s兲关兩x − y兩 − L兴其
2D冑⌶共s兲 sinh关冑⌶共s兲L兴
cosh兵冑⌶共s兲关x + y − L兴其
+
2D冑⌶共s兲 sinh关冑⌶共s兲L兴
共5.13兲
It follows from Eqs. 共5.9兲 and 共5.12兲 that the solution in
Laplace space is given by
Ũ共x,s兲 =
冕
L
G̃共x,y;s兲Ĩ共y,s兲dy.
共5.14兲
0
Inverting Eq. 共5.14兲 using the convolution theorem shows
that
冕冕
t
U共x,t兲 =
0
L
G共x,y;t − t⬘兲I共y,t⬘兲dydt⬘ ,
共5.15兲
0
with G共x , y ; t兲 = L−1G̃共x , y ; s兲 as the time-dependent Green’s
function. Substituting Eq. 共5.14兲 into Eq. 共5.8兲 and applying
the convolution theorem then gives the pair of equations
冕冕
t
R共x,t兲 =
0
L
0
H共x,y;t − t⬘兲I共y,t⬘兲dydt⬘ ,
共5.16兲
共5.18兲
⍀ +k
G̃共x,y;s兲.
共s + ␮+兲共s + ␮−兲
共5.19兲
The Green’s functions G, H, and K determine the linear
response of the receptor concentrations to a time-dependent
change in the somatic current. More generally, they can be
used to calculate the time course of variations in the receptor
concentrations following a rapid change in one or more trafficking parameters during the induction of synaptic plasticity,
for example. They also have an important probabilistic interpretation. For example, G共x , x0 ; t兲 can be interpreted as the
probability density 共per unit length of cable兲 that a single
labeled receptor is at position x in the dendritic membrane at
time t given that it was injected at position x0 at time t = 0.
关Simply set I共y , t兲 = l−1␦共y − x0兲␦共t兲.兴 Similarly, the probability densities for the receptor to be in the surface of a spine or
within an intracellular pool are determined by the Green’s
functions H共x , y ; t兲 and K共x , y ; t兲, respectively.
Having calculated the Laplace-transformed Green’s functions 关see Eqs. 共5.13兲, 共5.18兲, and 共5.19兲兴, one can easily
recover the steady-state results of Sec. III A by taking I共t兲
= Isoma with Isoma as a constant so that Ĩ共s兲 = Isoma / s. The
steady-state solution 共if it exists兲 is given by U共x兲
= lims→0 sŨ共x , s兲. Applying this to Eq. 共5.14兲 and using Eqs.
共5.11兲 and 共5.13兲, we find that
U共x兲 =
.
共5.17兲
⍀+共s + ␴rec + ␴deg兲
G̃共x,y;s兲,
共s + ␮+兲共s + ␮−兲
K̃共x,y;s兲 =
共5.9兲
s + ␴rec + ␴deg
s ␳ 0⍀ +
1 − ⍀−
+
D
D
共s + ␮+兲共s + ␮−兲
K共x,y;t − t⬘兲I共y,t⬘兲dydt⬘ .
The Green’s functions H共x , y ; t兲 and K共x , y ; t兲 are obtained
by inverting the Laplace transforms
Substituting for R̃共x , s兲 into Eq. 共5.1兲 then gives the equation
Ĩ共x,s兲
d2Ũ
− ⌶共s兲Ũ共x,s兲 = −
,
dx2
D
L
0
0
s + ␴rec + ␴deg
⍀+Ũ共x,s兲
=
. 共5.8兲
共s + ␮+兲共s + ␮−兲
k
where
冕冕
t
C共x,t兲 =
Isoma cosh兵冑⌶共0兲关x − L兴其
Isoma
.
G̃共x,0;0兲 =
l
l D冑⌶共0兲 sinh关冑⌶共0兲L兴
共5.20兲
It is easy to show that ⌶共0兲 = ␥2 with ␥ defined by Eq. 共3.5兲
so that we recover steady-state solution 共3.12兲 in the case
␴ = 0 = R̄. The corresponding steady-state solutions R共x兲 and
C共x兲 of Eq. 共3.1兲 are obtained by taking the appropriate s
→ 0 limit in the Laplace-transformed version of Eqs. 共5.16兲
and 共5.17兲.
Although one cannot write down a simple inversion formula for the Laplace-transformed Green’s functions given by
Eqs. 共5.13兲, 共5.18兲, and 共5.19兲, fast Fourier transforms provide an efficient method for performing the inversions numerically. Alternatively, in the case of a simple dendritic geometry such as a single cable, it is straightforward to
generate the time-dependent Green’s functions directly by
solving Eqs. 共2.4兲–共2.6兲 in the time domain. In Fig. 4, we
show example plots of U共x , t兲 = G共x , y ; t兲, R共x , t兲 = H共x , y ; t兲,
and C共x , t兲 = K共x , y ; t兲 as functions of x for y = L / 2 and various times t. These solutions are obtained by numerically
solving Eqs. 共2.4兲–共2.6兲 for ␴ = 0, Isoma = 0 and the initial conditions U共x , 0兲 = ␦共x − y兲 and R共x , 0兲 = 0 = C共x , 0兲. We find that
041904-9
PHYSICAL REVIEW E 79, 041904 共2009兲
PAUL C. BRESSLOFF
0.06
t = 200 s
t = 1000 s
t = 2000 s
t = 6000 s
0.05
0.04
0.03
0.02
0.01
0
-3
14 x 10
(b)
16
14
(c)
12
10
12
10
8
6
4
2
0
20 40 60 80 100 120 140 160 180 200
distance from soma [µm]
intracellular receptors [µm-1]
(a)
0.07
0
x 10-3
18
receptors in spine [µm-1]
receptors in dendrite [µm-1]
0.08
0
8
6
4
2
0
0
20 40 60 80 100 120 140 160 180 200
distance from soma [µm]
20 40 60 80 100 120 140 160 180 200
distance from soma [µm]
FIG. 4. Time-dependent Green’s functions determining response to a surface current injection at the center of a dendritic cable of length
L = 200 ␮m and circumference l = 1 ␮m. All other parameters are as in Fig. 2. 共a兲 Green’s function G共x , L / 2 , t兲 for surface receptors in
dendrite plotted as a function of x and various times t. 共b兲 Green’s function H共x , L / 2 , t兲 for surface receptors in spines. 共c兲 Green’s function
K共x , L / 2 , t兲 for receptors in intracellular pools.
G共x , L / 2 ; t兲 exhibits subdiffusive behavior, in the sense that
its rate of spread across the dendrite is slower than pure
diffusion and this cannot be accounted for by a simple rescaling of the diffusivity; see Fig. 5 and Sec. V B. Such an
effect is a direct consequence of the diffusive coupling between the dendrite and spines, which act as spatially localized traps. These traps become more effective when there is
an asymmetry between the rates of hopping between the dendrite and spines; that is, ⍀− ⬍ ⍀+. As mentioned in Sec. II,
this could be due to receptors interacting with scaffolding
proteins and cytoskeletal elements within the postsynaptic
density located in the spine head. We illustrate the effect of
asymmetric coupling in Fig. 6, where ⍀− is reduced by a
factor of 10 compared to that in Fig. 4. It can be seen that the
Green’s function G is more spatially localized and significantly reduced in amplitude as t increases.
In Fig. 7 we plot time courses for the Green’s functions
after integrating with respect to position x along the cable,
which yields the probabilities of finding a labeled receptor in
different states. It can be seen that the probability PU共t兲
⬅ 兰L0 G共x , L / 2 ; t兲dx that the receptor is in the dendritic surface
at time t, given that it was in the surface at position x = L / 2 at
time t = 0, is a monotonically decreasing function of time. On
the other hand, the corresponding probabilities to be in a
spine or an intracellular pool, PR共t兲 ⬅ 兰L0 H共x , L / 2 ; t兲dx and
PC共t兲 ⬅ 兰L0 K共x , L / 2 ; t兲dx, respectively, are unimodal functions of t. The total probability PU共t兲 + PR共t兲 + PC共t兲 is itself a
0.05
0.04
0.03
Ω = 10-3
t = 200 s
t = 1000 s
t = 2000 s
t = 6000 s
0.02
0.01
0
0
20 40 60 80 100 120 140 160 180 200
distance from soma [µm]
(b)
0.08
0.07
0.06
Ω = 10-4
0.05
0.04
0.03
0.02
0.01
0
0
20 40 60 80 100 120 140 160 180 200
distance from soma [µm]
receptors in dendrite [µm-1]
0.06
0.09
0.09
(a)
0.07
receptors in dendrite [µm-1]
receptors in dendrite [µm-1]
0.08
monotonically decreasing function of t due to receptor degradation. One of the interesting features of the time courses
shown in Fig. 7 is that they exhibit processes occurring on
different temporal scales. In particular, for the basal parameter values used in Fig. 7共a兲, there is a relatively rapid
change over a time scale of around 30 min in which the
probabilities appear to converge to a quasi-steady-state, and
a much slower decay over a time scale of many hours. 关For
our particular choice of parameters, this quasi-steady-state is
PU共t兲 = PR共t兲 = PC共t兲 = 1 / 3, which would be the true steady
state if ␴deg = 0.兴 Increasing the rate of degradation ␴deg
makes the asymptotic decay faster so that the probabilities do
not reach a quasi-steady-state; see Fig. 7共b兲. Interestingly, the
existence of multiple time scales might help resolve conflicting experimental results regarding the effective rate of constitutive recycling. A variety of optical, biochemical, and
electrophysiological studies of AMPA receptors in hippocampal neurons 关7,39,40兴 indicate that the rate of constitutive
recycling is relatively fast 共around 30 min兲, whereas a recent
photoinactivation study of Adesnik et al. 关26兴 suggests that
while recovery of surface receptors at the soma is fast, recovery of AMPA receptors at dendritic synapses is much
slower 共⬃16 h兲. Our mathematical analysis suggests that
there could be both fast and slow components of recovery
following inactivation of receptors, even when the rates of
constitutive recycling are relatively fast. Additional time
scales are introduced if the binding of receptors to scaffolding proteins is taken into account 关14兴.
0.08
(c)
0.07
0.06
Ω=0
0.05
0.04
0.03
0.02
0.01
0
0
20 40 60 80 100 120 140 160 180 200
distance from soma [µm]
FIG. 5. Plots of time-dependent Green’s function G共x , L / 2 , t兲 for different values of the diffusive coupling ⍀. 共a兲 Same as Fig. 4共a兲 with
⍀ = 10−3 ␮m2 s−1. 共b兲 ⍀ = 10−4 ␮m2 s−1. 共c兲 Zero diffusive coupling between dendrite and spines 共pure diffusion兲.
041904-10
PHYSICAL REVIEW E 79, 041904 共2009兲
CABLE THEORY OF PROTEIN RECEPTOR TRAFFICKING…
(a)
0.07
0.06
t = 200 s
t = 1000 s
t = 2000 s
t = 6000 s
0.05
0.04
0.03
0.02
0.01
0
0 20 40 60 80 100 120 140 160 180 200
distance from soma [µm]
x 10-3
12
intracellular receptors [µm-1]
16
receptors in spine [µm-1]
receptors in dendrite [µm-1]
0.08
(b)
14
12
10
8
6
4
2
0
0
x 10-3
(c)
10
8
6
4
2
0
20 40 60 80 100 120 140 160 180 200
distance from soma [µm]
0
20 40 60 80 100 120 140 160 180 200
distance from soma [µm]
FIG. 6. Same as Figs. 4共a兲–4共c兲 except that ⍀− = 0.1⍀+ = 10−4 ␮m2 s−1.
B. Large-t behavior
A useful aspect of Laplace transforms is that the small-s
limit provides information about the asymptotic behavior of
time-dependent solutions in the large-t limit. The large-t behavior of the time-dependent Green’s function G共x , y ; t兲 determines the asymptotic rate at which the dendritic receptor
concentration U共x , t兲 relaxes to the steady state in the case of
a constant somatic current. It also determines the asymptotic
decay of the probabilities shown in Fig. 7. For the sake of
illustration, we set y = 0 and consider a semi-infinite cable
whose Laplace-transformed Green’s function is obtained by
taking the limit L → ⬁ in Eq. 共5.13兲:
G̃共x,0;s兲 =
1
D冑⌶共s兲
e−
冑⌶共s兲x
,
M̃ n共s兲 ⬅
冕
⬁
x ⱖ 0.
共5.22兲
The small-s behavior of M̃ n共s兲 can be used to determine the
large-t behavior of M n共t兲 by invoking the following theorem
关55兴:
Theorem 1. 共Strong Tauberian theorem for the Laplace
transform兲 If f共t兲 ⱖ 0, f共t兲 is ultimately monotonic as t → ⬁,
F is slowly varying at infinity, and 0 ⬍ ␣ ⬍ ⬁, then each of
the relations
f̃共s兲 =
冕
⬁
e−st f共t兲dt ⬃ F共1/s兲s−␣
as s → 0
and
xnG共x,0;t兲dx.
共5.21兲
In particular, M 0共t兲 is the probability PU共t兲 that an individual
receptor is located within the dendritic membrane at time t
given that it was injected at x = 0 at time t = 0; see Sec. V A.
Taking Laplace transforms shows that
f共t兲 ⬃
t␣−1F共t兲
⌫共␣兲
as t → ⬁
implies the other. Here ⌫共␣兲 denotes the gamma function.
First, note that ⌶共s兲 = 0 is a cubic with roots s = −␭ j, where
j = 1 , 2 , 3, so that we can write
⌶共s兲 =
共s + ␭1兲共s + ␭2兲共s + ␭3兲
.
D共s + ␮+兲共s + ␮−兲
1
1
0.8
U
0.7
0.6
0.5
0.8
0.5
0
0.4
0.7
10
20
time [103 s]
30
R
0.3
C
0.2
1
(b)
0.9
probability
probability
(a)
probability
1
0.9
probability
n!
.
D⌶1+n/2共s兲
0
0
U
0.6
0.5
0
0.5
0.4
0
10
20
time [103 s]
30
R
0.3
0.2
0.1
0
xnG̃共x,s兲dx =
0
We will characterize the long-time behavior in terms of moments of the Green’s function defined according to
M n共t兲 =
冕
⬁
C
0.1
0
1000
2000
3000
4000
time [s]
5000
6000
0
0
1000
2000
3000
time [s]
4000
5000
6000
FIG. 7. Time courses of probabilities PU共t兲 , PR共t兲 , PC共t兲 associated with dendritic surface, spines, and intracellular stores, respectively.
共a兲 Same parameters as in Fig. 4 with ␴deg = 10−5 s−1. 共b兲 Same as 共a兲 except for a faster rate of degradation ␴deg = 10−3 s−1. Insets show same
plots over a longer time interval.
041904-11
PHYSICAL REVIEW E 79, 041904 共2009兲
PAUL C. BRESSLOFF
Let us order the roots by taking 0 ⱕ ␭1 ⬍ ␭2 ⬍ ␭3 and define
the function H共s兲 according to H共s兲 = ⌶共s − ␭1兲 / s. Using the
Tauberian theorem together with L−1 f̃共s + a兲 = e−atL−1 f̃共s兲, we
then find that
册
n ! tn/2
,
⬃ e−␭1t
⌫共1 + n/2兲H共0兲1+n/2D
t → ⬁, 共5.23兲
We find that ␭1 ⬍ ␮− ⬍ ␭2 ⬍ ␮+ ⬍ ␭3 so H共0兲 ⬎ D−1.
The zeroth-order moment varies asymptotically as M 0共t兲
⬃ e−␭1t / DH共0兲. If the spines were decoupled from the dendrite 共⍀⫾ = 0兲, then ␭1 = 0 and H共0兲 = 1 / D so that M 0共t兲 = 1
for all t. In other words, the receptor injected into the dendritic surface at t = 0 remains in the surface. On the other
hand, if ⍀⫾ ⬎ 0 and the rate of degradation ␴deg ⬎ 0, then
␭1 ⬎ 0 so that the probability of being in the surface decays
exponentially with time, reflecting the fact that the receptor
is eventually degraded. A third possibility is that ⍀⫾ ⬎ 0 and
␴deg = 0. In this case, ␭1 = 0 but H共0兲 ⬎ 1 / D so that M 0共t兲
→ 关DH共0兲兴−1 ⬍ 1 as t → ⬁. It follows that 1 − 关DH共0兲兴−1 is the
asymptotic probability that the receptor is in a spine or an
intracellular pool rather than in the dendritic membrane.
Given that M 0共t兲 ⬍ 1, it is useful to define the conditional
moments
t → ⬁.
共5.24兲
The first moment M1共t兲 determines the progress of the receptor along the dendritic cable, whereas ⌬M共t兲 ⬅ M2共t兲
− M1共t兲2 determines the variance in its position. Of course,
the net motion along the cable is not due to ballistic transport
but is a consequence of diffusion with a reflecting boundary
at x = 0. In the limit t → ⬁ we have
M1共t兲 ⬃ 2
冑
t
,
␲H共0兲
0.12
2.4
0.08
2.2
0
2
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
σdeg [10-3 s-1]
1
1.8
FIG. 8. 共Color online兲 Plot of asymptotic relaxation rate ␭1
共filled circles兲 and scaling factor H共0兲D 共filled squares兲 as a function of the rate of degradation ␴deg. All other parameter values are
as in Fig. 2.
共␭2 − ␭1兲共␭3 − ␭1兲
.
D共␮+ − ␭1兲共␮− − ␭1兲
n ! tn/2
M n共t兲
⬃
,
Mn共t兲 =
M 0共t兲 ⌫共1 + n/2兲H共0兲n/2
2.8
0.04
with
H共0兲 =
0.16
H(0)D
冋
3
2.6
λ1 [10-3 s-1]
M n共t兲 = e−␭1tL−1
n!
1+n/2
s
H共s兲1+n/2D
0.2
C. Sum-over-trips on dendritic trees
Another useful feature of Laplace transforms is that they
provide an efficient method for evaluating the Green’s function on a more complex dendritic tree structure such as the
one shown schematically in Fig. 9. Let T denote the set of all
branches in the tree, which are labeled by the index i 苸 T. We
assume 共i兲 each branch is a uniform cable, 共ii兲 all terminal
nodes are closed, and 共iii兲 continuity of receptor concentration and conservation of current at all branch nodes. It will
be convenient to choose coordinates such that a branch node
is at the point x = 0 on all branches radiating from that node.
Applying Laplace transforms to the receptor concentrations
in the jth branch as outlined in Sec. V A leads to an equation
of the form
Ĩ j共X,s兲
d2Ũ j
,
2 − Ũ j共X,s兲 = −
dX
D
0 ⱕ X ⱕ L j共s兲, 共5.26兲
where we have performed the rescalings x → X = ␥ j共s兲x and
L j → L j共s兲 = ␥ j共s兲L j, with ␥ j共s兲 = 冑⌶ j共s兲 and ⌶ j共s兲 defined by
Eq. 共5.10兲 for j-dependent spine and cable parameters. The
diffusivity D is taken to be the same in all branches, which is
reasonable given the size of receptors compared to the surface area of even small dendrites. As in the single cable case,
冉 冊
T
B
4
t
⌬M共t兲 ⬃ 2 −
.
␲ H共0兲
共5.25兲
The H共0兲 factors take into account the fact that the dendritic
spines act as traps for the diffusing receptor, thus reducing
the mean and variance of its displacement. In Fig. 8 we illustrate how both the asymptotic relaxation rate ␭1 and the
scale factor H共0兲 vary with the rate of degradation ␴deg. The
relaxation rate ␭1 is a monotonically increasing function of
␴deg, and its range of values is consistent with the asymptotic
time courses shown in the insets of Fig. 7. Moreover,
DH共0兲 → 3 as ␴deg → 0, consistent with the quasi-steady-state
shown in Fig. 7共a兲.
dendrites
S
soma
FIG. 9. Schematic diagram of a branched dendritic tree showing
a branch node B, a terminal node T, and the special terminal node S
adjoining the soma.
041904-12
PHYSICAL REVIEW E 79, 041904 共2009兲
CABLE THEORY OF PROTEIN RECEPTOR TRAFFICKING…
Ĩ共s兲
Ĩ j共X,s兲 =
l
␦„X − L0共s兲…␦ j,0 ,
2pn
m
共5.27兲
2pm−1
with the main branch labeled by i = 0. Continuity of receptor
concentration at a branch node requires
Ũi共0,s兲 = Ũ j共0,s兲
冏 冏
⳵ Ũ j
⳵X
= 0,
z j共s兲 = l jD␥ j共s兲,
共5.29兲
X=0
where the sum is over all branches j connected to the node.
Note that z j共0兲 is the inverse of the characteristic impedance
Z j defined by Eq. 共3.9兲. Finally, we have the boundary condition
冏 冏
⳵ Ũ j
⳵X
= 0,
共5.30兲
X=L j
for all terminal branches j including the main branch.
The general solution of Eq. 共5.26兲 can be written in the
form
Ũi共X,s兲 =
兺
j苸T
冕
L j共s兲
Gij共X,Y ;s兲Ĩ j共Y,s兲dY ,
共5.31兲
0
where the Green’s function Gij共X , Y ; s兲 satisfies the homogeneous equation
1
d2Gij共X,Y ;s兲
− Gij共X,Y ;s兲 = − ␦i,j␦共X − Y兲, 共5.32兲
2
dX
D
with the same boundary conditions as Ũi共X , s兲 for fixed j , Y.
Following the sum-over-trips method of Abbott et al.
关22,23兴, particularly the version applied recently to quasiactive dendrites 关25兴, it can be shown that for a general tree
the corresponding Green’s function has an infinite series expansion in terms of the corresponding Green’s function
G⬁共X兲 for an infinite cable:
Gij共X,Y ;s兲 =
兺 Atrip共s兲G⬁„Ltrip共i, j,X,Y,s兲…,
(b)
±1
共5.28兲
for all pairs 共i , j兲 radiating from the node. Similarly, current
conservation requires that
兺j z j共s兲
n
(a)
Ĩ j共X , s兲 represents the Laplace transform of a current density
which takes into account the current Isoma at the soma and the
local supply of receptors in intracellular pools. If we ignore
the latter, then
共5.33兲
FIG. 10. Schematic diagram showing amplitude factors picked
up by a trip on reaching 共a兲 a branch node and 共b兲 a terminal node.
at a branch or terminal node. A trip is always reflected back
at a terminal node, whereas at a branch node it may be transmitted to another branch or reflected back. A trip may pass
through the points 共X , i兲 and 共Y , j兲 an arbitrary number of
times as long as it starts at 共X , i兲 and ends at 共Y , j兲.
For each trip, the associated amplitude Atrip is calculated
according to the following rules:
共1兲 Initially take Atrip共s兲 = 1.
共2兲 For every branch node at which the trip passes from
an initial segment m to a different segment n, n ⫽ m, multiply
Atrip共s兲 by a factor of 2pn共s兲; see Fig. 10共a兲.
共3兲 For every branch node at which the trip is reflected
back along the same segment m, multiply Atrip共s兲 by a factor
of 2pm共s兲 − 1; see Fig. 10共a兲.
共4兲 For every closed 共open兲 terminal node, multiply
Atrip共s兲 by a factor of +1 共−1兲; see Fig. 10共b兲.
共5兲 Multiply Atrip共s兲 by an additional factor of 2 if a trip
starts or ends on a closed terminal node. 共This factor is usually not mentioned explicitly in the sum-over-trips rules, but
see Ref. 关24兴.兲
Here the factor pm is given by
pm共s兲 =
where
Ũi共x,s兲 =
共5.34兲
and Ltrip共i , j , X , Y , s兲 is the length 共in rescaled coordinates兲 of
a path along the tree starting at point X on branch i and
ending at point Y on branch j. The sum in Eq. 共5.33兲 is
restricted to a set of paths or trips that are constructed using
the following rule 关22兴:
A trip from 共X , i兲 to 共Y , j兲 may start out in either direction
along branch i but it can subsequently change direction only
兺n zn共s兲
,
共5.35兲
where the sum is over all branches n radiating from a given
branch node and zm共s兲 is defined according to Eq. 共5.29兲.
In order to obtain the solution in the time domain, it is
first necessary to express Eq. 共5.31兲 in physical coordinates:
trips
e−兩X兩
,
G⬁共X兲 =
2D
zm共s兲
兺
j苸T
冕
Lj
G̃ij共x,y;s兲Ĩ j共y,s兲dy,
共5.36兲
0
with
G̃ij共x,y;s兲 =
Gij„␥i共s兲x, ␥ j共s兲y;s…
.
␥ j共s兲
共5.37兲
Applying the inverse Laplace transform then gives the general solution for the surface receptor concentration in an arbitrary dendritic tree:
041904-13
PHYSICAL REVIEW E 79, 041904 共2009兲
PAUL C. BRESSLOFF
G̃10共x,y;0兲 =
X
Isoma
共5.42兲
Y
FIG. 11. Simple configuration consisting of one main branch
and two semi-infinite secondary branches. Two shortest trips starting at X and ending at Y are shown.
Ui共x,t兲 =
兺
j苸T
冕
t
Gij共x,y;t − t⬘兲I j共y,t⬘兲dydt⬘ ,
共5.38兲
0
with Gij共x , y ; t兲 = L−1G̃ij共x , y ; s兲. The time-dependent Green’s
function Gij共x , y ; t兲 can be interpreted as the probability that a
receptor is at location x in the surface of the ith branch at
time t, given that it was initially injected into the surface at
location y of the jth branch at time t = 0. Moreover, as in the
case of a single cable 共Sec. V A兲, the steady-state Green’s
function is given by G̃ij共x , y ; 0兲 with ␥ j共0兲 = 冑⌶共0兲 as the
inverse space constant of the jth branch; see Eq. 共3.5兲. Having constructed the Green’s function for dendritic surface
receptors, it is straightforward to write down the corresponding Green’s functions for receptors in spines and intracellular
stores using Eqs. 共5.18兲 and 共5.19兲:
H̃ij共x,y;s兲 =
deg
⍀+,i共s + ␴rec
i + ␴i 兲
G̃ij共x,y;s兲,
共s + ␮+,i兲共s + ␮−,i兲
p0共0兲 e−共␥1x+␥0y兲 + e−关␥1x+␥0共2L0−y兲兴
.
D␥0 1 − 关2p0共0兲 − 1兴e−2␥0L0
共5.39兲
Note that if z0共0兲 = z1共0兲 + z2共0兲, which is equivalent to impedance matching condition 共3.16兲, then p0 = 1 / 2 and we recover the equivalent-cylinders result of Sec. III B.
For more complex tree configurations, it is possible to
carry out partial summations over trips in Eq. 共5.33兲 but the
resulting expression tends to be unwieldy 关24兴. On the other
hand, for sufficiently large s, one can truncate the series to
include a relatively small number of trips and then carry out
a numerical inversion of the Laplace transform 共see 关23,25兴
for further discussions of numerical implementations兲. This
will determine the behavior of the solution up to some finite
time t. The sum-over-trips method is less useful if one is
interested in the long-time asymptotic behavior of solutions,
that is, the small-s behavior of the Laplace-transformed
Green’s function. It is then more useful to use an alternative
approach based on integral equations 关56兴. That is, one can
solve the original diffusion-trapping equations on each
branch separately using Green’s theorem, which yields a set
of coupled integral equations involving the unknown solutions at the branch nodes of the tree. The latter are then
determined self-consistently by imposing current conservation at each branch node and using Laplace transforms. This
generates a closed expression for the Laplace-transformed
Green’s function in the form of a continued fraction, which
can then be used to extract the long-time behavior of solutions. It can also be used to calculate the mean first passage
time for a single receptor to reach a synaptic target from the
soma 关12兴. The continued fraction has a naturally recursive
form, which is particularly useful for analyzing self-similar
structures such as Cayley trees 关56兴.
VI. DISCUSSION
K̃ij共x,y;s兲 =
⍀+,iki
G̃ij共x,y;s兲.
共s + ␮+,i兲共s + ␮−,i兲
共5.40兲
As an illustrative example, consider the configuration
shown in Fig. 11, in which the main branch j = 0 of finite
length L0 bifurcates into two semi-infinite secondary
branches labeled j = 1 , 2. A trip starting at point X in branch 1
and ending at point Y in the main branch has to pass through
the branch point exactly once, picking up a factor of 2p0.
There are two basic trips, one that goes straight to Y with
Ltrip = X + Y, and the other that reflects once off the terminal
node before ending at Y so that Ltrip = X + 2L0 − Y. Each of
these two trips is associated with an infinite set of other trips
involving q reflections at the terminal and branching nodes,
q ⱖ 1, thus resulting in an additional trip length 2qL0 and an
additional amplitude factor 共2p0 − 1兲q. Summing over q gives
G10共X,Y ;s兲 =
p0共s兲 e−共X+Y兲 + e−关X+2L0共s兲−Y兴
.
D 1 − 关2p0共s兲 − 1兴e−2L0共s兲
共5.41兲
It follows that the steady-state Green’s function is given by
In this paper we showed how the trafficking of protein
receptors along the surface of a spiny dendritic tree can be
described in terms of a continuum model that is similar in
structure to the linear cable equation for the electrical potential in quasiactive dendritic membranes. This allowed us to
apply Green’s function and transform methods in order to
calculate the steady-state and time-dependent receptor concentrations along single and branched dendritic cables. In
particular, we derived the Green’s function for the diffusive
transport of receptors within the dendritic membrane and
used this to explore the effects of trapping within spines. We
showed how trapping leads to subdiffusive behavior and that
relaxation to the steady state involves multiple time scales
arising from the receptor dynamics within spines. We also
used homogenization theory to derive corrections to the continuum model due to the discrete nature of spines, and established that such corrections are small provided that the diffusive coupling between dendrites and spines is sufficiently
weak.
In conclusion, we have shown how cable theory provides
a general theoretical framework for exploring the role of lateral membrane diffusion in the transport of protein receptors
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PHYSICAL REVIEW E 79, 041904 共2009兲
in neurons. There are a number of ways in which our work
may be extended. The first concerns a more detailed singlespine model in which interactions between receptors and
scaffolding proteins within the PSD are taken into account
关27,28兴. This introduces nonlinearities into the single-spine
dynamics associated with the kinetics of receptor binding to
scaffolding proteins. In order to apply Laplace-transform
methods it is then necessary to linearize the single-spine dynamics, which is a reasonable approximation if the binding
sites are unsaturated, for example. Another important extension is to consider the effects of noise. In our continuum
model we take the state variable of a single spine to be a
receptor concentration and formulate the single-spine dynamics in terms of a system of kinetic equations. This implicitly assumes that the number of receptors within a spine
is sufficiently large; otherwise random fluctuations about the
mean receptor number may become significant. Typically the
size of fluctuations varies as 1 / 冑N, where N is the number of
receptors. One way to take into account the inherent stochasticity or “intrinsic noise” arising from fluctuations in receptor
number is to replace the kinetic equations by a corresponding
master equation 关57兴, which describes the temporal evolution
of the probability distribution for the receptors within the
spine. Performing some form of small fluctuation expansion
would lead to a stochastic version of our continuum model,
which could then be analyzed using Green’s-function techniques along the lines of the stochastic cable equation 关58兴.
Finally, it would be interesting to develop a higherdimensional version of our diffusion-trapping model in
which molecules are free to diffuse throughout the volume of
a dendrite but can become trapped by spines on the membrane surface. A recent experimental and computational
study of molecular diffusion within the dendrites of Purkinje
cells suggests that the temporary confinement in spines leads
to anomalous diffusion on short time scales and reduced normal diffusion on longer time scales 关59兴.
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ACKNOWLEDGMENT
This work was supported by the National Science Foundation 共Grant No. DMS 0813677兲.
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PAUL C. BRESSLOFF
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