1Department of Physiology, University College London, London WC1E 6BT, United Kingdom; and 2Abteilung Zellphysiologie, Max-Planck-Institut für Medizinische Forschung, D-69120 Heidelberg, Germany
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ABSTRACT |
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Vetter, Philipp, Arnd Roth, and Michael Häusser. Propagation of Action Potentials in Dendrites Depends on Dendritic Morphology. J. Neurophysiol. 85: 926-937, 2001. Action potential propagation links information processing in different regions of the dendritic tree. To examine the contribution of dendritic morphology to the efficacy of propagation, simulations were performed in detailed reconstructions of eight different neuronal types. With identical complements of voltage-gated channels, different dendritic morphologies exhibit distinct patterns of propagation. Remarkably, the range of backpropagation efficacies observed experimentally can be reproduced by the variations in dendritic morphology alone. Dendritic geometry also determines the extent to which modulation of channel densities can affect propagation. Thus in Purkinje cells and dopamine neurons, backpropagation is relatively insensitive to changes in channel densities, whereas in pyramidal cells, backpropagation can be modulated over a wide range. We also demonstrate that forward propagation of dendritically initiated action potentials is influenced by morphology in a similar manner. We show that these functional consequences of the differences in dendritic geometries can be explained quantitatively using simple anatomical measures of dendritic branching patterns, which are captured in a reduced model of dendritic geometry. These findings indicate that differences in dendritic geometry act in concert with differences in voltage-gated channel density and kinetics to generate the diversity in dendritic action potential propagation observed between neurons. They also suggest that changes in dendritic geometry during development and plasticity will critically affect propagation. By determining the spatial pattern of action potential signaling, dendritic morphology thus helps to define the size and interdependence of functional compartments in the neuron.
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INTRODUCTION |
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The propagation of action
potentials in the dendritic tree has been the subject of great interest
for over a century, beginning with the speculations of
Ramón y Cajal (1904) on the directional flow of
signals in neurites. During the past few decades, evidence has
accumulated to suggest that under most conditions, the sodium action
potential (AP), the output signal of the neuron, is initiated in the
axon and retrogradely invades the dendritic tree, a process known as
backpropagation (reviewed by Stuart et al. 1997b
).
Backpropagation has been demonstrated in a variety of different cell
types in vitro, and evidence for backpropagation in vivo has been
provided by recordings in both anesthetized (Helmchen et al.
1999
; Svoboda et al. 1997
, 1999
) and awake
animals (Buzsaki and Kandel 1998
). APs can also be
initiated in the dendrites under conditions of intense synaptic
stimulation (Chen et al. 1997
; Golding and
Spruston 1998
; Kamondi et al. 1998
;
Martina et al. 2000
; Stuart et al. 1997a
;
Turner et al. 1991
), and the spread of these dendritic APs toward the soma is known as forward propagation. Understanding the
factors that determine the efficacy of AP propagation in dendrites is
important since AP propagation has key consequences for the integration
of synaptic input and the induction of synaptic plasticity (Johnston et al. 1996
; Linden 1999
). In
particular, backpropagation of the AP provides a mechanism whereby
information about neuronal output is signaled backward to active
synapses to trigger changes in synaptic strength (Linden
1999
; Stuart et al. 1997b
).
A striking finding that has emerged from recent studies is that under
similar experimental conditions, systematic differences in
backpropagation exist between neuronal types (reviewed by Stuart et al. 1997b). In dopamine neurons (Häusser et al.
1995
) and mitral cells (Bischofberger and Jonas
1997
; Chen et al. 1997
), the somatic AP
propagates nondecrementally to the distal dendrites. On the other hand,
many neuronal types show decremental conduction, with the most extreme
example being cerebellar Purkinje cells, where the AP is reduced to a
few millivolts in amplitude at 100 µm from the soma
(Llinás and Sugimori 1980
; Stuart and
Häusser 1994
). A comparable diversity has also been
observed for the efficacy of forward propagation of dendritically
initiated APs toward the soma (Chen et al. 1997
;
Golding and Spruston 1998
; Kamondi et al.
1998
; Llinás and Sugimori 1980
;
Martina et al. 2000
; Schwindt and Crill
1998
; Stuart et al. 1997a
).
Explanations for these systematic differences in the efficacy of
propagation have concentrated on differences in the densities of
dendritic voltage-gated channels (Magee et al. 1998;
Stuart et al. 1997b
). Theoretical work on AP propagation
in axons has demonstrated that propagation is dependent on axonal
morphology (Goldstein and Rall 1974
; Joyner et
al. 1980
; Lüscher and Shiner 1990
;
Manor et al. 1991
; Parnas and Segev 1979
;
Ramon et al. 1975
). The wide variety of dendritic
arborizations among different neuronal types suggests that differences
in dendritic morphology will contribute to determining the extent of
backpropagation and forward propagation of APs in dendrites. This is a
difficult issue to address quantitatively as cable theory does not
provide an analytical solution describing action potential propagation
in arbitrarily branched cable structures and systematic experimental
manipulations of dendritic morphology are not yet possible. We have
therefore examined this question with simulations of AP propagation in
a wide range of realistic dendritic geometries. By inserting the same
set of passive and active parameters in each cell type, we have
isolated the effect of geometry alone (Mainen and Sejnowski
1996
). We show that dendritic geometry plays a key role in
determining the efficacy of both forward and backward propagation of
APs. These results have important implications for modulation of
backpropagation and for the role of the dendritic AP as an associative
signal in different neurons.
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METHODS |
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Dendritic geometries
Detailed three-dimensional reconstructions of 42 neurons were
used. Two rat Purkinje cells, two rat neocortical layer 5 pyramidal neurons, and four rat substantia nigra dopamine neurons were filled with biocytin and digitally reconstructed using a ×100 oil immersion objective (1.4 NA) on a Zeiss Axioplan (Zeiss, Oberkochen, Germany) in
conjunction with Neurolucida software (MicroBrightField, Colchester, VT). Three rat layer 5 pyramidal neurons were from G. Stuart and N. Spruston, and one from D. Smetters; three guinea pig Purkinje cells
were from M. Rapp; rat CA1/CA3 pyramidal cells, and DG interneurons and
granule cells were obtained from the Duke-Southampton Neuronal Morphology Archive (www.neuro.soton.ac.uk). Reconstructions were inspected carefully, and only those without apparent errors in connectivity or dendritic diameters were used. All dendrites were divided into compartments with a maximum length of 7 µm. Spines were
incorporated where appropriate by scaling membrane capacitance and
conductances (Holmes 1989; Shelton 1985
).
Simulations
Simulations were performed using the NEURON simulation
environment (Hines and Carnevale 1997) on a Pentium II
PC running Red Hat Linux and on a Silicon Graphics Origin 2000. The
time step for the simulations was 25 µs. The structure of the active
model was based on recent modeling studies (Mainen and Sejnowski
1996
; Mainen et al. 1995
; Rapp et al.
1996
). Two Hodgkin-Huxley-type conductances
(gNa and
gK) were inserted into the soma,
dendrites, and spines at uniform densities. For simplicity, we did not
explore the consequences of using spines with different excitability
from dendritic shafts (see Baer and Rinzel 1991
). The
model was tuned by attaching a synthetic axon (Mainen et al.
1995
) to five neocortical pyramidal cell geometries in which
backpropagating APs were initiated by somatic current injection. Active
and passive membrane parameters were then optimized to reproduce
experimental data on AP backpropagation from these neurons in slices
taken from rats aged P26-30 (Stuart et al. 1997a
). The
uniform passive parameters of the model were Ri = 150
cm,
Cm = 1 µF/cm2,
Rm = 12 k
cm2,
which reproduced experimental values for membrane time constant and
input resistance to within 10% (Stuart and Spruston
1998
; Stuart et al. 1997a
). The standard values
for gNa and
gK were 35 and 30 pS/µm2, respectively, and the channel models
were identical to those used by Mainen and Sejnowski
(1996)
(available from www.cnl.salk.edu/simulations/). Channel
kinetics and densities were adjusted for a nominal temperature of
37°C using a Q10 of 2.3. All results shown in
the figures and tables were obtained using this standard model. In some
cases, the standard gNa and
gK were replaced by corresponding
channel models from a different study (Paré et al.
1998
). Simulations using a nonuniform distribution of
the A-type K+ channel were based on previous
studies (Hoffman et al. 1997
; Migliore et al.
1999
). The A-type K+ channel replaced the
K+ channel used in the standard model, and its
density was scaled to increase linearly fivefold from the soma to the
most distal dendrites of each neuron, with the initial density being
480 pS/µm2 (Hoffman et al. 1997
;
Migliore et al. 1999
).
The morphology of the soma and proximal dendrites may itself affect the
invasion of the soma by the axonally initiated AP. Therefore a somatic
AP waveform (amplitude, 96 mV; half-width, 0.6 ms) obtained from one of
the pyramidal neuron morphologies with the standard model was used as a
voltage-clamp command at the soma of all other neurons to simulate
backpropagation. This ensured a fair comparison across all neurons so
that backpropagation begins from the same initial conditions in each
morphology (simulations with a synthetic axon attached to all
morphologies, allowing each to generate its own AP, produced very
similar results). The dendritically initiated AP was generated using a
biexponential synaptic conductance (rise = 0.2 ms,
dec = 1.7 ms,
gmax = 50 nS) near the apical nexus of
a neocortical pyramidal cell (622 µm from the soma; see Fig. 6). This
waveform was used as a voltage-clamp command to examine propagation of
a dendritic AP, again to ensure that propagation begins from the same
initial conditions. In this way, AP propagation was isolated from AP
initiation, conditions for which also depend on morphology
(Segev and Rall 1998
). Similar results were obtained using a somatic AP waveform, or a backpropagating dendritic AP waveform.
Measurements
The rate of increase of membrane area as a function of distance
from the soma (dA/dx) was approximated by
A/
x, where
A is the
spine-adjusted total membrane area at a distance interval (x, x +
x) from the soma. The
discretization
x was 1 µm (
x = 0.25 µm for Purkinje cells). All distances were measured along the
dendrites. The somatodendritic Na+ channel
density threshold for full AP propagation into all dendritic branches,
gNa,thresh, was determined using the
bisection algorithm (Press et al. 1992
). The radius of
equivalent cables was calculated according to the following equation
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(1) |
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RESULTS |
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Backpropagation depends on dendritic geometry
We designed a simple active model to reproduce experimental data
on AP backpropagation in a single cell type to then apply it to a range
of different morphologies. We used a model of neocortical pyramidal
cells since extensive data on backpropagation are available for this
cell type. Furthermore these neurons fall in the middle of the range of
backpropagation efficacies (Stuart et al. 1997b). The
model was based on previous studies (Mainen and Sejnowski 1996
; Mainen et al. 1995
; Rapp et al.
1996
) and incorporated a uniform density of
Na+ and K+ channels in the
soma and dendrites with a high density of Na+
channels in the axon to ensure axonal initiation of the AP. This simple
model is able to reproduce the experimental data in neocortical pyramidal cells remarkably well (cf. Figs. 1B and 2A of Stuart et al. 1997a
).
To isolate the effect of dendritic morphology, the same model was then
inserted into morphologies of different cell types, and backpropagation
of a somatic AP waveform was simulated. With identical channel types
and densities, the different morphologies produced very different
patterns of backpropagation (Fig. 1), which reproduced the experimentally observed results in these neurons
(Häusser et al. 1995; Llinás and
Sugimori 1980
; Stuart and Häusser 1994
;
Stuart and Sakmann 1994
; Stuart et al.
1997a
). Thus the AP spread effectively into all dendritic
branches of substantia nigra dopamine neurons while propagating
decrementally in the apical dendrite of neocortical pyramidal cells and
failing to effectively invade the dendritic tree of Purkinje cells.
Interestingly, even within the same neuron a wide range of AP
amplitudes could be observed at the same physical distances from the
soma (Fig. 1E). As the density of Na+
and K+ channels is uniform in this model, the
diversity in propagation into different dendritic branches must be a
consequence of the morphology. Taken together, these findings indicate
that dendritic geometry determines the interaction between a given
Na+ channel density and backpropagation.
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Sensitivity of backpropagation to channel density in different morphologies
To investigate the relationship between backpropagation and
voltage-gated channel density in different morphologies, we simulated backpropagation over a wide range of Na+ and
K+ channel densities (from 0 to 200 pS/µm2). These simulations were carried out in
a large sample of different neuronal types, including the neurons
studied in the preceding text as well as hippocampal CA1 and CA3
pyramidal neurons, dentate gyrus (DG) granule cells, and hippocampal
interneurons. First, we varied the Na+ channel
density while keeping the K+ channel density
constant. Each neuronal type had a characteristic relation between
dendritic Na+ channel density and backpropagation
(quantified by measuring the AP amplitude 200 µm from the soma; Fig.
2A). With
no dendritic Na+ channels, the attenuation of the
AP waveform was very different in different neuronal types (see Table
1). This indicates that the dendritic
morphologies filter the AP waveform to different extents even when
backpropagation is passive. When Na+ channel
density was increased, an approximately sigmoid relationship was
observed between density and propagation efficacy in most cell types.
The half-maximum of these curves was typically around 20-40
pS/µm2 (see Table 1), within the physiological
range of dendritic Na+ channel densities
(Johnston et al. 1996). In striking contrast, backpropagation in Purkinje cells was essentially insensitive to
increases in Na+ channel density over a wide
range with a sharp threshold leading to full backpropagation being
reached only at very high densities (mean half-maximum, 92.8 ± 23.3 pS/µm2).
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We next varied the density of dendritic K+ channels while maintaining a constant Na+ channel density. As expected, when K+ channel density was increased, backpropagation became less effective. Although backpropagation was less sensitive to changes in dendritic K+ channel density than changes in Na+ channel density, there was a characteristic relationship between K+ channel density and backpropagation in each neuronal type (Fig. 2B). Over the range studied (from 0 to 200 pS/µm2), backpropagation in Purkinje cells and dopamine neurons showed little sensitivity to changes in K+ channel density, while backpropagation in pyramidal neurons and dentate gyrus granule cells was clearly affected (Fig. 2B).
In the same set of neurons, we also examined the sensitivity of backpropagation to changes in the relative densities of dendritic Na+ and K+ channels. This was done by systematically varying the ratio of densities over a wide range (0.1-10) while keeping the total voltage-gated channel density fixed. As shown in Fig. 2C, most neurons showed a sigmoid relationship with backpropagation efficacy as the density of Na+ channels relative to K+ channels was increased. However, the half-maximum of the curve depended strongly on morphology and spanned a wide range (Table 1). Again Purkinje cells were particularly insensitive to changes in dendritic channel densities. These findings demonstrate that, as expected, backpropagation depends on the relative density of dendritic voltage-gated Na+ and K+ channels. However, the same ratio of channel densities produces very different degrees of backpropagation in different morphologies. Furthermore the effect of changing the relative density depends critically on the dendritic morphology.
On the basis of these simulations, we defined an index of backpropagation that reflects the different sensitivities of the morphologies to channel densities. As backpropagation is most sensitive to Na+ channel density, we determined the minimum Na+ channel density required for full backpropagation of the AP (peak voltage >0 mV) into all dendritic branches. The threshold Na+ channel density (gNa,thresh) varied widely across cell types (Fig. 3). Purkinje cells required nearly five times higher Na+ channel density than dopamine neurons to sustain full backpropagation into all dendrites. We therefore used gNa,thresh as a measure that captures the efficacy of propagation in a given neuron.
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Morphological determinants of backpropagation
Which morphological features are most important for determining
the efficacy of backpropagation? Unfortunately no analytical theory
exists to predict whether propagation will be successful given a
dendritic morphology and a set of realistic channel densities and
kinetics (Jack et al. 1983). Attempts to formulate such
a theory have so far succeeded only for simple geometries, using highly
simplified models of excitability (Pastushenko et al.
1969
; Pauwelussen 1982
). We therefore examined
correlations of individual geometric parameters with our functional
index of backpropagation, gNa,thresh,
and with the amplitude of the AP 200 µm from the soma (AP200) using the standard parameters of the
model. The simplest morphological features, such as mean dendritic
diameter, total physical or electrotonic length of the dendrites,
dendritic taper or flare, and total dendritic membrane area only
correlated relatively weakly with
gNa,thresh or
AP200 (|r|
0.6). The number of
dendritic branchpoints, however, showed a strong relationship with the
functional parameters across the population (Fig.
4A; r = 0.81 and
0.73 for gNa,thresh and
AP200, respectively). The range of branching densities was reflected in striking differences in the distribution of
dendritic membrane area with distance x from the soma
(dA/dx, Fig. 4B). Both
gNa,thresh and
AP200 were strongly correlated with the maximum
slope of dA/dx across the different morphologies
(Fig. 4C). Combining features of the
dA/dx distribution with another geometric
parameter (e.g., the maximum slope normalized to the number of
branchpoints or mean dendritic diameter) further strengthened correlations with gNa,thresh and
AP200, such that correlation coefficients of
|r| > 0.9 were obtained.
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We tested the robustness of these correlations in several ways. First,
each of the passive parameters were changed over a wide range
(Ri: 70-200 cm,
Cm: 0.7-1.5
µF/cm2, Rm:
5-50 k
cm2). Second, different models for the
Hodgkin-Huxley gNa and
gK conductances (taken from
Paré et al. 1998
) were incorporated. The
qualitative behavior of the models remained robust under these different conditions, as did the ranking of correlations between morphology and backpropagation; values of individual correlations varied by less than 20%. Finally, as a nonuniform distribution of
A-type K+ channels has recently been demonstrated
to play an important role in regulating backpropagation in hippocampal
pyramidal neurons (Hoffman et al. 1997
) and mitral cells
(Schoppa and Westbrook 1999
), we incorporated a
nonuniform distribution of an A-type K+ channel
model (Migliore et al. 1999
) into the simulations based on experimental and modeling studies (Hoffman et al.
1997
; Migliore et al. 1999
). Even with this
nonuniform channel distribution, strong correlations were observed
between dendritic morphology and backpropagation (e.g.,
r = 0.91 for gNa,thresh and
the maximum slope of dA/dx). Taken together,
these findings suggest that geometric parameters alone can be used to
estimate the relative efficacy of backpropagation in a given dendritic morphology.
Which measure provides the best functional link between the details of
dendritic morphology and backpropagation efficacy? Simulations of AP
propagation in axons (Goldstein and Rall 1974; Lüscher and Shiner 1990
; Manor et al.
1991
) have shown that propagation across a branchpoint depends
on the ratio of the input impedances of the parent and daughter
branches
the impedance mismatch (Goldstein and Rall
1974
). If this ratio equals one, there is no change in propagation of the AP as it approaches the branchpoint. However, if it
is greater than one, i.e., if the combined input impedance of the
daughter branches is lower than the input impedance of the parent
branch, AP propagation may fail. For branches of uniform diameter and
semi-infinite length, the impedance mismatch is given by Rall's
geometric ratio (GR)
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(2) |
However, dendritic branches in realistic dendritic trees are finite in
length, and the branchpoints are not the only sites contributing to the
impedance mismatch along a dendrite. Usually a succession of several
branchpoints, connected by sections with taper or flare, lies within
the spatial extent of a dendritic AP. Thus it is difficult to predict
the relative importance of various local maxima in the impedance
mismatch in determining the fate of the AP except by explicit
simulation of AP propagation (Figs. 1-3) (Lüscher and
Shiner 1990; Manor et al. 1991
). To obtain a
better predictor of AP propagation, we simplified the analysis of
impedance mismatches by constructing reduced models of the dendritic
architecture (Segev 1992
), transforming each dendritic morphology into a single unbranched equivalent cable (Fig.
5A) (Clements and
Redman 1989
; Fleshman et al. 1988
; Ohme
and Schierwagen 1998
). The impedance mismatch calculated in
this cable approximates the mean impedance mismatch seen by an AP
wavefront propagating in the original dendritic morphology. The shape
of the impedance mismatch distribution (Fig. 5B) proved to
be closely related to the efficacy of backpropagation. In particular,
the cumulative impedance mismatch (Fig. 5B,
inset) is a remarkably good predictor of both
gNa,thresh (r = 0.94;
Fig. 5C) and AP200 (r =
0.89).
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Forward propagation of dendritic APs depends on dendritic geometry
Under some circumstances, it is possible to initiate
Na+ APs in dendrites. These dendritic APs
propagate with variable efficacy to the soma in different cell types
(Chen et al. 1997; Golding and Spruston
1998
; Kamondi et al. 1998
; Martina et al.
2000
; Schwindt and Crill 1998
; Stuart et
al. 1997a
). We investigated the influence of morphology on the
extent of this forward propagation by comparing propagation of a
dendritic AP in our set of neurons. The same dendritic AP waveform
propagated to very different extents in different neurons with an
identical distribution of voltage-gated channels (Fig.
6). In dopamine neurons and pyramidal
neurons, propagation of the dendritic AP was very effective, while in
Purkinje cells the dendritic AP was rapidly attenuated.
|
The spread of a dendritic AP is likely to depend on its site of origin
in the dendritic morphology, as previously shown for spread of signals
in passive dendrites (Rall 1964; Zador et al. 1995
). We therefore systematically examined the spread of the dendritic AP from all locations in each dendritic tree. To quantify the
extent of forward propagation, we measured the distance at which the
dendritic AP was reduced to half its original amplitude when traveling
toward the soma. This "half-decay distance" depended strongly on
the site of origin of the dendritic AP and on the cell type (Fig.
7). The dendritic AP could propagate for
hundreds of micrometers toward the soma from many dendritic locations
in both dopamine and neocortical pyramidal neurons. In contrast, for
nearly all locations in the cerebellar Purkinje cell, propagation was
limited to less than 50 µm, being restricted to the branchlets close
to the initiation site. As shown in Table
2, the mean half-decay distance of the
dendritic AP calculated over all sites of origin depended strongly on
cell type. These findings indicate that the local geometry as well as
the overall structure of the dendritic tree are important determinants
of forward propagation of dendritic APs.
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To examine the relationship between dendritic Na+
channel density and forward propagation, we determined
gNa,thresh (this time defined as the
threshold gNa for full propagation to
the soma) for forward propagating APs. As for backpropagation,
gNa,thresh depended on cell type (Fig.
8A), with Purkinje cells again
requiring the highest density of Na+ channels to
ensure full propagation. The sequence of propagation efficacies was
slightly different from that observed for backpropagation, consistent
with the asymmetry of the dendritic architecture. To determine if the
underlying principles established for backpropagation also hold for
propagation in the forward direction, we constructed equivalent cables
from the point of view of the origin of the dendritic AP. Again the
cumulative impedance mismatch in the equivalent cable was an accurate
predictor of propagation efficacy (Fig. 8B). Thus as for
backpropagation, a reduced model of dendritic geometry (Segev
1992) is able to provide a tight functional link between
morphology and the efficacy of forward propagation.
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DISCUSSION |
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We have shown that dendritic morphology plays an important role in
determining both forward propagation and backpropagation of APs in
dendrites. Recent work has highlighted the importance of dendritic
voltage-gated channels in regulating the efficacy of AP propagation.
Our findings complement this work and demonstrate that morphological
features act in concert with dendritic voltage-gated channels to
generate the observed diversity of AP propagation in dendritic trees of
different neuronal types. Although there are clear differences in
channel densities and properties between neurons (Johnston et
al. 1996; Llinás 1988
), we show that
dendritic geometry determines how significant these differences are for neuronal function. These findings have important consequences for our
understanding of how different neurons use dendritic APs as
computational signals.
The link between dendritic morphology and propagation
Our findings demonstrate that some dendritic morphologies are
remarkably resistant to AP propagation. Purkinje cells, for example, do
not show effective propagation even with relatively high densities of
voltage-gated Na+ channels. This indicates that,
as predicted by Rall (1964), the poor AP backpropagation
observed experimentally in Purkinje cells is primarily due to their
distinctive morphology in conjunction with the relatively low channel
densities (Stuart and Häusser 1994
) and narrow AP
widths (Stuart et al. 1997b
) found in these neurons. On
the other hand, we have also identified dendritic geometries
such as
those of dopamine neurons
that are very favorable for propagation,
requiring only very low Na+ channel densities for
effective propagation into the distal dendrites. These findings suggest
that the diversity in backpropagation efficacy observed experimentally
(Stuart et al. 1997b
) may be a consequence of the
diversity in dendritic morphologies in different neurons. As forward
propagation is also influenced by dendritic geometry in a similar
manner, we predict that the experimentally observed differences in
forward propagation may also result in part from differences in
dendritic structure. Although our simulations have focused on APs
mediated by Na+ channels, it is likely that
propagation of dendritic calcium spikes (Amitai et al.
1993
; Helmchen et al. 1999
; Kim and
Connors 1993
; Larkum et al. 1999b
;
Llinás and Sugimori 1980
; Schiller et al.
1997
; Schwindt and Crill 1998
; Seamans et
al. 1997
; Yuste et al. 1994
) is also regulated
by morphology over a wide range.
By correlating morphological features with AP propagation, we have shown that the number of dendritic branchpoints is a critical variable for determining propagation efficacy. An even more accurate predictor of propagation efficacy is the rate of increase in dendritic membrane area, which is determined by the number of branchpoints and the relationship between the diameter of parent and daughter dendrites at branchpoints. Based on this strong correlation, we can make predictions about the relative efficacy of propagation for several cell types in which propagation has not been measured directly (e.g., granule cells and interneurons in the dentate gyrus). This measure should also permit similar predictions to be made for any dendritic morphology.
The link between the distribution of membrane area and propagation can
be understood by noting that the functions describing the rate of
increase in membrane area and the radius of the equivalent cable are
closely related. For real dendrites, this leads to distributions of
similar shape (compare Figs. 4B and 5A). The
profile of the equivalent cable, in turn, directly determines its
impedance mismatch profile (Fig. 5B), which approximates the
mean impedance mismatch at the same electrotonic location in the
original morphology (the impedance mismatch profile would be preserved
exactly for branches of semi-infinite length: compare Eq. 2
and Eq. 1) (Clements and Redman 1989;
Fleshman et al. 1988
). Interestingly, the cumulative impedance mismatch in the equivalent cable provided the best predictor of backpropagation and forward propagation efficacy in 42 dendritic morphologies. This demonstrates that the equivalent cable
transformation is a useful tool for predicting propagation of
suprathreshold signals in active dendritic trees; this is remarkable
given that it involves numerous approximations (Clements and
Redman 1989
; Fleshman et al. 1988
). This result
also suggests that the fate of a dendritic AP is usually not decided at
a single branchpoint but rather depends on the accumulation of
unfavorable impedance mismatches over many branchpoints (Manor
et al. 1991
).
Our finding that dendritic morphology has a significant impact on the
propagation of APs in dendrites is in agreement with recent
experimental evidence. Kim and Connors (1993) have shown that the efficacy of AP backpropagation in layer 5 pyramidal neurons is
correlated with the number of apical oblique branches and the diameter
of the apical trunk. In CA1 pyramidal neurons, a combination of calcium
imaging and electrophysiological recordings has demonstrated that
frequency-dependent attenuation of backpropagating APs is particularly severe in distal dendritic regions with extensive branching (Callaway and Ross 1995
; Spruston et
al. 1995
). Finally, backpropagation in thalamocortical relay
neurons appears to be less effective in dendrites that exhibit
branching (Williams and Stuart 2000
), again consistent
with a key role for dendritic branching in determining propagation.
Modulation of AP propagation in dendrites
The densities of functional voltage-gated channels are subject to
modulation by neurotransmitters, which can in turn affect backpropagation (Johnston et al. 1999). Our findings
demonstrate that dendritic geometry places limits on the ability of
backpropagation to be modulated in different neurons. As shown in Fig.
2, altering the density of dendritic Na+ and
K+ channels over the same range can have
strikingly diverse consequences on backpropagation in different
morphologies. We predict that propagation of APs in different neurons,
and in subregions of individual dendritic trees, will have different
sensitivities to modulation of voltage-gated channels as a consequence
of variations in morphology. In particular, neurons with an
intermediate degree of dendritic branching, and dendritic regions
exhibiting rapid increases in branching such as the apical tuft, should
display the greatest sensitivity to modulation. This is testable
experimentally as the functional density of different channel types can
be varied pharmacologically to determine which morphologies are most
sensitive to modulation. The consequences of use-dependent activation
and inactivation of voltage-gated channels should also depend on
dendritic morphology. Consistent with this idea, frequency-dependent
attenuation of APs in dendrites (Callaway and Ross 1995
;
Spruston et al. 1995
), which depends in part on a
reduction in the effective Na+ channel density
(Colbert et al. 1997
; Jung et al. 1997
),
is more pronounced in CA1 pyramidal neurons than in cortical pyramidal neurons (Stuart et al. 1997a
), which exhibit less branching.
Dendritic geometry is not static but can change dramatically both
during development and in adulthood (Bailey and Kandel
1993; McAllister 2000
; Purves and Hadley
1985
). In particular, a substantial increase in dendritic
branching has been shown to be associated with neuronal maturation
(Altman 1972
; Berry and Bradley 1976
; Kasper et al. 1994
; Wu et al. 1999
) and
with activation of messenger pathways known to be involved in synaptic
plasticity (Nedivi et al. 1998
; Wu and Cline
1998
; Yacoubian and Lo 2000
). Our findings suggest that this increase in dendritic complexity will reduce backpropagation unless compensated by increases in voltage-gated channel densities. Consistent with this idea, there is substantial experimental evidence demonstrating that channel densities increase during development (Huguenard et al. 1988
;
MacDermott and Westbrook 1986
; O'Dowd et al.
1988
) in parallel with the changes in morphological complexity.
Finally our results point to an important role for dendritic
spines in regulating AP propagation in dendrites (Baer and
Rinzel 1991; Jaslove 1992
). As spines can
contribute more than 50% of the dendritic membrane area, the
relationship between membrane area and propagation efficacy (Fig. 4)
suggests that the changes in spine density that can occur during
development (Gould et al. 1990
; Harris et al.
1992
) and synaptic plasticity (Engert and Bonhoeffer
1999
; Maletic-Savatic et al. 1999
) will also
modulate the extent of propagation. This effect should be particularly pronounced in neurons that also exhibit a high degree of dendritic branching. It is therefore interesting to note that neurons with minimal branching, such as dopamine neurons and interneurons, tend to
be aspiny, while Purkinje cells, which exhibit a high degree of
branching, have a very high spine density.
Implications for dendritic computation
Recent work has demonstrated that the backpropagating AP
acts as a retrograde signal to dendritic synapses indicating that the
axon has fired. This provides a coincidence detection mechanism that
links postsynaptic APs and presynaptic activity to trigger synaptic
plasticity (Häusser et al. 2000; Linden
1999
; Stuart et al. 1997b
). Our results show
that since dendritic morphology limits the extent of propagation of
dendritic APs, it defines the spatial range over which associations
between synaptic inputs and APs can take place. In particular, we
demonstrate that highly branching dendritic geometries do not permit
strong coupling between axonal output and distal synapses, and thus in
these neurons, the backpropagating AP cannot act as a global
associative signal (Häusser et al. 2000
;
Larkum et al. 1999b
; Linden 1999
;
Stuart et al. 1997b
). Since such geometries are also
poor substrates for forward propagation of APs (Figs. 6-8), synaptic
integration in these neurons is far more dependent on local
associations between inputs. Indeed such dendritic trees may be adapted
to keep associations between inputs more localized to increase the
number of independent sites of integration (Mel 1993
).
On the other hand, dendritic morphologies that favor propagation and
are sensitive to modulation of propagation allow the associativity
between output and input to be tuned over a wide range.
Our findings suggest that morphology will also influence the
interaction between backpropagating APs and dendritically initiated APs. The initiation of dendritic APs requires strong and temporally synchronous synaptic input (Golding and Spruston 1998;
Schiller et al. 1997
; Stuart et al.
1997a
), properties consistent with their role as coincidence
detectors. However, in cortical pyramidal neurons, pairing
backpropagating APs with distal synaptic input can substantially lower
the threshold for initiation of dendritic APs; this can in turn lead to
burst firing in the axon (Helmchen et al. 1999
;
Larkum et al. 1999a
,b
). By limiting the spatial spread of backpropagating and forward propagating APs, dendritic geometry should therefore play an important role in determining the sensitivity of individual neurons to coincident synaptic input as well as in
defining the relationship between dendritic APs and neuronal output via
the axon. Taken together, these considerations indicate that the
tremendous diversity in dendritic morphology may have direct
consequences for the computational strategies used by different neurons.
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ACKNOWLEDGMENTS |
---|
We thank J.J.B. Jack and I. Segev for helpful discussions, M. Hines for advice and modifications to NEURON, and D. Attwell, G. Borst, B. Clark, M. Farrant, and M. Larkum for comments on the manuscript. M. Häusser and A. Roth thank the Crete Course in Computational Neuroscience for hospitality. A. Roth thanks B. Sakmann for support.
This work was supported by the Wellcome Trust, the European Community, and the Max-Planck-Gesellschaft. P. Vetter was funded by the Wellcome Trust 4-year PhD Programme in Neuroscience.
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FOOTNOTES |
---|
* P. Vetter and A. Roth contributed equally to this work.
Address for reprint requests: M. Häusser, Dept. of Physiology, University College London, Gower Street, London WC1E 6BT, UK (E-mail: m.hausser{at}ucl.ac.uk).
Received 28 July 2000; accepted in final form 23 October 2000.
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REFERENCES |
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