1Section of Neurobiology, School of Medicine and 2Department of Computer Science, Yale University, New Haven, Connecticut 06510; and 3Department of Medical Physiology, Panum Institute, University of Copenhagen, Blegdamsvej, Copenhagen 2200, Denmark
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ABSTRACT |
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Shen, Gongyu Y., Wei R. Chen, Jens Midtgaard, Gordon M. Shepherd, and Michael L. Hines. Computational Analysis of Action Potential Initiation in Mitral Cell Soma and Dendrites Based on Dual Patch Recordings. J. Neurophysiol. 82: 3006-3020, 1999. In olfactory mitral cells, dual patch recordings show that the site of action potential initiation can shift between soma and distal primary dendrite and that the shift is dependent on the location and strength of electrode current injection. We have analyzed the mechanisms underlying this shift, using a model of the mitral cell that takes advantage of the constraints available from the two recording sites. Starting with homogeneous Hodgkin-Huxley-like Na+-K+ channel distribution in the soma-dendritic region and much higher sodium channel density in the axonal region, the model's channel kinetics and density were adjusted by a fitting algorithm so that the model response was virtually identical to the experimental data. The combination of loading effects and much higher sodium channel density in the axon relative to the soma-dendritic region results in significantly lower "voltage threshold" for action potential initiation in the axon; the axon therefore fires first unless the voltage gradient in the primary dendrite is steep enough for it to reach its higher threshold. The results thus provide a quantitative explanation for the stimulus strength and position dependence of the site of action potential initiation in the mitral cell.
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INTRODUCTION |
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A critical factor in the integrative mechanisms of
single neurons is the site of action potential initiation. Classical
studies identified the site to be the axon hillock-initial segment
(Eccles 1957; Edwards and Ottoson 1958
;
Fuortes et al. 1957
). Recent experiments in the tissue
slice using dual recordings from soma and distal dendrite have
confirmed this model for the cortical pyramidal cell and several other
types of neuron (Stuart and Sakmann 1994
).
In the classical model dendrites could be active or passive. Active
properties were suggested to have several possible roles, including
generation of action potentials that could propagate in an orthodromic
direction to the soma and axon hillock (Andersen 1960),
boosting of excitatory synaptic responses by local "hot spots"
(Spencer and Kandel 1961
), or enhancing activation of
presynaptic dendrites (Rall and Shepherd 1968
). A wide
variety of evidence subsequently accumulated for such active properties
(Benardo and Prince 1982
; Benardo et al.
1982
; Herreras et al. 1989
; Turner et al.
1989
, 1991
).
These active dendritic properties pose a problem for the classical
model in that distal dendritic synaptic excitation is able to stimulate
the axon hillock without first giving rise to dendritic action
potentials (see Mainen et al. 1995). Dual patch
recordings have documented the presence of active properties and have
emphasized their role in promoting back-propagation of action
potentials into the dendrites, where they can contribute to synaptic
plasticity and learning (Stuart et al. 1997b
). However,
the role of active properties in orthodromic dendritic responses is
controversial (Chen et al. 1997
; Golding and
Spruston 1998
; Stuart and Sakman 1994
;
Stuart et al. 1997a
). Although it is accepted that they can boost local synaptic or active responses, it has been argued that
they are too slow to participate in full-blown action potential initiation in advance of the axon hillock-initial segment
(Stuart et al. 1997a
).
An opportunity to test this hypothesis was presented by the mitral cell
of the mammalian olfactory bulb. The mitral cell is distinguished by a
long primary dendrite ending in a tuft that receives all of the
excitatory synaptic input to the cell. Dual patch recordings from the
soma and the distal dendrite showed that when the tuft is excited by
weak synaptic input, the action potential arises in the soma region but
with stronger synaptic excitation the site of action potential
initiation shifts to the distal dendrite (Chen et al.
1997). A similar shift can be brought about by synaptic
inhibition in the basal dendrites and soma. These results indicated
that a reassessment is needed of the mechanisms underlying the
different sites of action potential initiation and their functional
significance for the integrative actions of the neuron.
A further advantage of the mitral cell is that computational methods
already have been applied to physiological data to gain insight into
the functional properties of its dendrites (Bhalla and Bower
1993; Rall and Shepherd 1968
; Shepherd
and Brayton 1979
). Here we analyze dual electrode recordings
from the mitral cell soma and distal primary dendrite
(Bischofberger and Jonas 1997
; Chen et al.
1997
), under conditions of strong or weak depolarizing current
injection into the soma or distal dendrite electrode. The dual
recordings have facilitated development of a model for the intervening
dendritic segment that greatly restricts the parameter subspace. We
make precise superpositions of experimental and model activity and show
that the model gives simulations that are nearly identical to the
experimental recordings. We identify the parameters of the model that
are highly constrained by the data. We describe several tools for
carrying out the parameter optimization within the framework of this
highly constrained model. In judging goodness of fit, we emphasize the
importance of sensitivity to critical parameters. The results with
model perturbations give insights into the qualitative features that
are responsible for the shift in the site of action potential
initiation under varying conditions of stimulus location and stimulus
intensity. This has enabled us to explore the factors controlling
action potential initiation sites in this neuron under these
conditions. A brief report has appeared in abstract form (Shen
et al. 1998
).
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METHODS |
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Physiological recordings
The experimental methods followed those in Chen et al.
(1997). Briefly, slices were prepared from rat olfactory bulb.
Mitral cells and their primary dendrites were visualized under infrared differential interference contrast (DIC) microscopy. Dual patch recordings were made with one electrode on or near the soma and the
other at a distal site on the primary dendrite
200 µm away. Simultaneous recordings were made from the two sites combined with
depolarizing current injection into either soma or dendrite. For each
cell, four stimulation protocols were used successively. The four
protocols were: weak current injection into soma, strong current
injection into soma, weak current injection into primary dendrite, and
strong current injection into primary dendrite. With the dual recording
sites at the soma and distal dendrite, this gave experimental data for
a total of eight recordings for a given neuron that provided the basis
for the simulations.
Cell morphology
Mitral cells in the mammalian main olfactory bulb (MOB) have
several salient morphological characteristics (Mori
1987; Shepherd and Greer 1998
). The cell body
emits a single primary dendrite and several secondary dendrites. The
primary dendrite is a long, relatively unbranched, process of uniform
diameter extending several hundred micrometers until it arborizes at
the distal end into a tuft of several thin and varicose processes. The
dendritic tuft branches receive synaptic inputs from olfactory nerve
fibers and also participate in dendrodendritic interactions with the
dendrites of periglomerular cells. The secondary dendrites are
interconnected with granule cell interneurons by numerous reciprocal
dendrodendritic synapses. The cell body tapers into an axon hillock
which gives rise to an initial segment and a myelinated axon.
In this study, we incorporated these basic morphological features into a canonical model. The model (Fig. 1) consists of a cell body (Soma), a primary dendrite (Pri), two dendritic tuft branches (Tuft), two secondary dendrites (Sec), an axon hillock (AH), an axonal initial segment (IS), and a length of axon consisting of five nodes of Ranvier (Node) and five myelin segments (Internode). The size of the soma and length and diameter of the primary dendrite were measured under the microscope. Other morphological parameters were obtained by fitting the passive charging parts of the experimental data. The numbers and sizes of the tuft and secondary dendritic branches covered a wide range. Our canonical model lumped them together in the form of two tuft branches and two secondary dendrites, and their electrotonic effect was parameterized by an effective diameter. One of the aims of this study is to investigate the sensitivity of the simulations to the parameters of these canonical parts outside the inter-electrode segment. The morphological parameter values are summarized in Table 1.
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Passive electrotonic parameters
The passive parameters of our model had three sets of different
values for soma and dendrites, nodes and initial segment, and
myelinated internodes respectively. In a recently published paper
(Stuart and Spruston 1998), a well-fitted passive
parameter model using realistic cell morphology strongly indicated a
much lower value of Ri = 70
·cm and higher value of Cm = 1.4 µf/cm2 for neocortical pyramidal neurons than
those used by other simulation studies (Mainen et al.
1995
; Rapp et al. 1996
). Our fitting results yielded similar results of Ri and
Cm in the case of the primary dendrite
length and recording site constrained by the experimental data. The
Ri and
Cm were from 55 to 70
·cm and 1.0 to 1.5 µf/cm2, respectively, when the diameters
of the primary dendrites were in the range of 2.2-4.0 µm. The
specific membrane resistivity (Rm) was
~25,000-40,000
·cm2. The nodes and initial segment
were assumed to have a large leakage specific membrane conductance and
were accordingly set to 1,000
· cm2
(Black et al. 1990
). The myelinated internodes were
wrapped by 50 turns of oligodendroglia cell membrane so that the
Rm and the Cm were reduced by 100 times compared
with those of the nodes. The fitting result was relatively insensitive
to the change of the number of turns. The values of the best fitted
passive parameters are summarized in Table
2.
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Distributions of ion channels
The soma and primary dendritic membranes have the same density
of sodium channels as shown by experimental analysis of channel densities in patches from the soma and the primary dendrites of rat
mitral cells (Bischofberger and Jonas 1997). The model
therefore assumed a uniform distribution of sodium and potassium
channels in the soma and primary dendrite that define the part of the
mitral cell that was under direct observation by the dual electrode
sites. For initial exploration, we have extended this assumption to
other parts of the neuron, including the axon hillock, primary
dendritic tuft, and secondary dendrites. The simulations showed that
the model depended strongly on the assumptions about the channel
distribution and density of the dendrite between the two recording
locations, and was relatively insensitive to details outside this
region (see RESULTS).
With regard to the axon, experimental studies with dual electrode
recordings in neocortical and hippocampal neurons have shown that
initiation of the action potential occurs at sites 20-30 µm away
from the cell body (Stuart et al. 1997a). Theoretical studies confirmed this result (Mainen et al. 1995
;
Rapp et al. 1996
) if the density of sodium channels in
the axon is much higher than in the soma and dendrites. We therefore
approximated the transition from soma to node channel density as a
discontinuity at some position (denoted x) along the initial segment.
Further parameterization, e.g., the distance over which the change
occurs, was found to be not important over the scale of the 20-µm
length of the initial segment. The densities of sodium and potassium channels of the myelinated internodes in this model were assumed to be zero.
Membrane channel kinetics
This study has focused on the initiation of single action
potentials, for which the fast sodium and delayed potassium channels play the essential roles. We consider the possible contributions of the
other voltage-gated channels in the discussion. The ionic equations
below were a variant of the Hodgkin-Huxley (Hodgkin and Huxley
1952) formalism, taken from Mainen et al.
(1995)
. The sodium and potassium channel densities are shown in
Table 3, and the rate function parameters
are shown in Table 4.
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Numerical methods
Simulations were carried out with the NEURON program
(Hines and Carnevale 1997). The NEURON model code along
with data used to generate the figures in this paper is available from
the NEURONDB model database (Shepherd et al. 1998
) at
http://www.senselab. yale.edu. The fitting algorithm employed by the
program was Brent's (1976)
principal axis method
(PRAXIS). The particular forms of the fitness functions we use with
PRAXIS are discussed in the APPENDIX and have a very large
impact on performance. In most simulations we used the first-order
implicit integration method with dt = 25 µs. The
numerical error because of this time step results in a uniform shift in
peak location of the action potentials under our stimulus conditions of
approximately dt. Error due to the spatial discretization
(compartmentalization) is orders of magnitude less than the error due
to dt. The very low spatial error is partly due to the large
number of compartments we used for the primary dendrite and initial
segment. In the primary dendrite, 35 compartments were used to allow
for accurate electrode placement. In the initial segment, 10 compartments were used to allow high resolution of the location of the
discontinuous increase in channel density (see preceding text). In
retrospect we simultaneously could have had more precise control over
position and greater efficiency (fewer required compartments) if we had
divided these sections into proximal and distal components. Furthermore
our use of three compartments in the soma and hillock and nine
compartments for each internode, though giving a more pleasing
appearance to our space plots (see following text) was also numerically
unnecessary
a single compartment in each of those sections gives
identical results.
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RESULTS |
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Experimental recordings from soma and dendrite
Dual recordings were made from soma (or near soma) and a site on
the distal primary dendrite in a population of 20 cells, with current
injection into either soma or dendrite. In all cells there was an
action potential recorded in the dendrite, in response to current
injection in the soma, which followed the soma action potential. When
current was injected into the dendrite, the interval between soma and
dendritic action potential decreased, the interval decreasing with
increasing dendritic current strength. In one case the dendritic action
potential occurred first. We have focused our simulations on this as
providing the most difficult case to model and as most similar to the
previous demonstration of a shift in action potential site with
synaptic excitation of the dendritic tuft (Chen et al.
1997). Although, in general, current injection induced less
dramatic action potential shifts, the closer quantitative control over
the stimulus makes the current injection results the preferred place to
start in carrying out quantitative simulations. We also have carried
out simulations of the more common current injection result showing
only limited decrease in the action potential interval. Although not
shown here, data and parameters for such a cell are included with the
model code mentioned in Numerical methods.
The experimental results are illustrated in Fig.
2. Simultaneous dual recordings were made
in the whole cell mode from the soma and from a site ~300 µm away
on the primary dendrite. Depolarizing current first was injected into
the distal dendritic site, and the changes in membrane potential were
recorded at both sites. With weak current injection, at just above
threshold intensity for action potential generation, there was an
initial slow depolarization of the cell. Although the amplitude was, as
expected, higher at the site of current injection in the dendrite
(- - -) than at the soma (), the soma eventually gave rise first to
an action potential, followed closely by the dendritic site. This
illustrates the classical model (see INTRODUCTION) in which
the action potential arises first at the soma despite the location of
excitatory input in the distal dendrites.
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With increasing intensity of depolarizing current, the dendritic action potential shifted earlier, until, in this cell, it came to arise first. With the larger depolarization shown in Fig. 2B, the action potential latencies decreased at both sites, but the latencies of action potential onset, peak and repolarization can be seen to be slightly earlier at the dendritic site. The current intensity increase, from 0.5 to 0.8 nA, was presumably well within the physiological range of depolarization of this cell.
Basic steps in simulating the experimental data: the classical model
The measured passive parameters of the model included the interelectrode length and the diameter of the interelectrode segment of the primary dendrite; the fitted parameters included Cm, Rm, and Ri, based on data from other cells and estimates of the capacitance and resistance of the electrodes, as explained in METHODS. Estimates were made of the conductance loads outside the interelectrode segment, which included the primary dendrite and tuft distal to the dendritic recording site, the lumped basal dendrites emanating from the soma, and the axon emerging from the soma and axon hillock. Details of the methods for constructing and testing the passive model are described fully in the APPENDIX.
Multiple tests explored the effects of the passive parameters on fits to the early phases of the experimental responses during the time period when the injected current passively charged the cell (see APPENDIX for full details). A typical fit is illustrated for the prolonged charging transient labeled c in Fig. 3A, for the case of weak current injection into the dendritic recording site. With reasonable estimates of the passive parameters, the passive charging transients could be fitted virtually exactly to the experimental data. The long period of passive charging for the weak currents was a significant constraint governing these multiple fits. The most sensitive parameters were the diameter of the interelectrode dendritic segment and the membrane time constant (Rm × Cm). By contrast, the fit was relatively insensitive to estimates for the conductance loads outside the interelectrode dendritic segment. This supported the premise of using a canonical approach to building the parts of the model lying outside the interelectrode dendritic segment (see METHODS).
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A similar approach was used to estimate the active parameters
responsible for the generation of the action potential. As described in
METHODS, a modification of the Hodgkin-Huxley model was
used. The simulations began with estimates for the critical parameters of the model based on our own (Shepherd and Brayton
1979) and other studies (Cooley and Dodge 1966
;
Mainen et al. 1995
) applying the Hodgkin-Huxley
formalism to action potential generation in the mammal. The fitting
routines (see METHODS) then led rather quickly to a
reasonable approximation of the action potential responses for all four
testing situations.
To refine the model to remove the remaining differences between experimental and simulated responses, we focused on three critical regions of the action potential: the onset, peak, and repolarizing phase (Fig. 3). We found that the most critical region was the onset (Fig. 3, o), where the key factors were the precise latency of the onset and the relatively sharp rise at that point in the membrane potential. Next was the action potential spike peak (sp), where the key factors were the peak amplitude and latency. Least sensitive was the repolarization phase (r). The strategy used to successively refine the fit is discussed in the APPENDIX.
These methods allowed the simulations to remove most of the differences in these three critical regions. As a result, as shown in Fig. 3A, the model was able to reproduce virtually exactly the full sequence of the experimental recording for the case of weak current injection eliciting an action potential first in the soma region followed by the distal dendritic site, as in the classical model. The closeness of the fit extended to the use of the same current intensity (0.5 nA) for generating the nearly identical response, reflecting a close similarity of the input resistance of the model to the cell.
Stronger dendritic current injection initiates dendritic action potentials
The simulation of the classical model with weak current in the dendrites required that the density of sodium channels in the dendrites be low enough to prevent the dendrite firing first. At the same time, to satisfy the four simultaneous test simulations, the density of sodium channels had to be sufficient for the dendrite to fire earlier with stronger dendritic current injection. Figure 3B shows that this was in fact the case. Stronger injected current in the model dendrite resulted in a larger and faster rising depolarization at the dendritic site, leading to the firing of the dendritic before the somatic action potential. The soma action potential simulation was nearly exact, whereas the dendritic simulation showed small differences: the onset was slightly later, the peak was higher (by 1.2 mV) and earlier (by 0.1 ms), and the hyperpolarizing phase was smaller (by 3.6 mV) and less peaked but recovered more slowly than in the experimental recording (see - - -). These small differences were the largest between model and experiment seen in our set of four simulations. One explanation may be that this amount of current is approaching the limits of the ranges for some of the voltage-dependent parameters of the HH model. The increased current of 0.8 nA was the same value as in the experiments, again suggesting a close similarity of the input resistance of the model to the cell. The difference in timing between the action potential peaks in the soma and distal dendrite as a function of stimulus intensity is plotted in Fig. 4 for both the experimental data and the model.
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The fact that with increased dendritic current injection the dendritic
action potential precedes the somatic action potential implies that the
dendritic action potential is forward propagating through the dendrites
under these conditions (Chen et al. 1997). This
interpretation is supported by the results of patch recordings that
indicate that voltage-dependent Na+ channels are
present along the entire extent of the primary dendrite (Bischofberger and Jonas 1997
). On the other hand,
because the initial segment has a lower threshold for action potential
generation, it is also possible to consider that this is a case of
forward triggering of the somatic action potential (see
DISCUSSION).
Soma current injection initiates backpropagating action potentials
The other two simulations in the test set modeled the responses to current injection in the soma. With weak current injection (Fig. 5A), there was a slow passive charging of the membrane, similar to that in response to dendritic current injection but with lower amplitudes. This presumably reflected the lower input resistance at the soma because of the larger conductance load of the basal dendrites plus axon. When the passive depolarization reached threshold the model generated an action potential which, in accord with the classical model of a lower threshold in the initial segment, occurred first at the soma recording site. It can be seen in Fig. 5A that the simulations of the soma and dendritic action potentials were nearly identical to the experimental data. This was especially true of the onset (o) and repolarization (rp) regions; the only slight deviation was a higher peak (by 2.5 mV) of the model dendritic action potential. Comparison with Fig. 3, A and B, shows that when current injection was into the dendrites, it gave rise to a high level of local depolarization from which the dendritic action potentials arose. If this is taken into account, the dendritic action potentials seem to be lower in amplitude in those cases as well as in the case of somatic current injection.
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Given these results, it was expected that stronger current injection into the soma would give a similar result with a shorter latency, and this was the case as shown in Fig. 5B. It can be seen that the simulations were virtually identical during both the passive charging phase and the action potentials, including the three key regions of the action potentials; the only minor variations occurred in the trough of the hyperpolarization and during the slow return to baseline. The current strengths (0.4 and 0.8 nA) for the simulations were identical to those in the experiments.
Training the model on different data subsets
The results described thus far show that a model with a single set of parameters was able to simulate four basic types of activation (action potential responses to weak and strong injected currents through dendritic and somatic recording electrodes) and give results that were in most respects nearly identical to the experimental data. The importance of simultaneous fitting of multiple protocols can be seen in Fig. 6, in which optimizing only one protocol, e.g., weak dendritic current injection protocol (C), allows a wide variation in action potential timing for the other protocols (A, B, and D). Training on the strong dendritic current injection case (not shown) allows even greater variation in the other protocols. Training on pairs of protocols reduced the variation in timing between data and model for the untrained protocols but this variation was still substantial. The results suggest that the delayed response to weak dendritic current injection provides the greatest constraints for the four simultaneous simulations illustrated in Figs. 3 and 5.
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A similar analysis of the responses to current injection in the soma gave similar results overall to those for dendritic current injection. However, the response to weak soma stimulation did not give as large a global constraint on the model as the response to weak primary dendrite stimulation (data not shown). By contrast, the response to strong soma stimulation gave a slightly better constraint on the model than the response to strong dendritic current injection.
These results illustrate the value of developing a well-constrained model through the use of multiple simultaneous simulation sets. Simulations such as those in Fig. 6 with only one set gave results that were qualitatively similar to the experimental results, but it is only with all four members of the set that the model was sufficiently constrained to give a nearly precise simulation of all the data.
Analysis of sodium and potassium channel density
We next examined the sensitivity of the fit to several model
assumptions. We began with the sodium channel density as the critical
parameter in action potential generation. Bischofberger and
Jonas (1997) have measured the sodium peak current density under voltage-clamp conditions from a series of patches taken from
sites along the primary dendrite and found a uniform value of 9.0 ± 1.7 pA/µm2. Patch voltage-clamp simulations
of our model for the family of parameters we found that fit our four
data protocols gave peak current densities ranging from 7.1 to 9.0 pA/µm2. The peak specific conductance
gna (i.e.,
na·m3·h)
during the voltage-clamp pulse was 124 pS/µm2
in our model, whereas it was 90 ± 17 pS/µm2 in the experiments of
Bischofberger and Jonas (1997)
. The difference is partly
caused by the difference of the sodium reversal potential. Our model
used a +60 mV value, while the patch recording experiment used +90 mV.
Simulations showed that changes in reversal potentials for
Na+, over the range of +40 to +110 mV, and
K+, over the range of
110 to
70 mV, could be
compensated for by proportional changes in the channel densities. Other
causes may involve differences of the peak current density and the peak
conductance density under different recording conditions, errors in the
estimate of the area of the recording electrode tips, cell
morphologies, or nonoptimal fitting of our models.
With regard to k, the value in
the model of 42 pS/µm2 differs significantly
from the experimental value of 500 pS/µm2 of
Bischofberger and Jonas (1997)
. However, the latter was
the average of a data set that showed a very large variance, within which some values were similar to or even less than the model value. It
is therefore quite possible that the model value reflects accurately
the value for its particular cell.
The question of the appropriate value of
k had little direct bearing on
our study of the mechanisms of action potential initiation, which are
almost solely dependent on
na.
However, it was of interest to gain some insight into the
k parameter space with higher
values. Excellent dual action potential fits could in fact be obtained
for any single protocol. For simultaneous fits to the four protocols
(cf. Figs 3 and 5), using either a fixed value of
k = 250 pS/µm2 (which required initial manual
adjustment of other parameters to find a suitable starting value for
automatic parameter search with PRAXIS) or incremental increases of
k (the iterations ending at
k = 250 pS/µm2) produced fits with an error value that
was six times larger than our best fit with
k = 42 pS/µm2. Over most of this range these runs fit
well the onset and rise of the action potentials (dependent on
na), but diverged as expected
in the repolarization phase dominated by
k.
The model's single Hodgkin-Huxley-like K+
conductance is, of course, inadequate to capture quantitatively the
complex combination of potassium channels with widely varying time
courses and kinetics that underlie an entire action potential. To gain
some insight into this question, we tested a significantly simpler form
(12 instead of 19 parameters) for the sodium and potassium channel gating states, consisting of the simplest possible two state Boltzman functions for m, h, and n (2 parameters each),
along with two parameter log linear rate functions (defined by the
rates at 70 and +30 mV) for each gating state. The log linear
voltage-sensitive rates, although not particularly representable as a
simple function for an Eyring rate theory energy barrier, have a
phenomenological simplicity on a par with the "two-state" gating
state abstraction. This model was able to fit the eight action
potentials of the four protocols with the same accuracy as the
Hodgkin-Huxley-like model, but with a value for
k of 95 pS/µm2.
Initiation site of the somatically recorded action potential
A study employing locally applied TTX and patch recording on
hippocampal subicular pyramidal neurons has suggested that the action
potential is initiated at an axonal site 30-60 µm beyond the
hillock-initial segment (Colbert and Johnston 1996).
Computer modeling studies with detailed axonal structures have shown
that a much higher density of sodium channels in the initial segment was necessary to reproduce the forward and backward propagation of the
action potential along the dendritic-somatic-axonal axis (Luscher and Larkum 1998
; Mainen et al.
1995
; Rapp et al. 1996
). Although soma-dendritic
loading effects always result in the spike occurring first in the
distal part of the initial segment (see following text), it is of
interest to test in our mitral cell model how nonuniform channel
density in the initial segment affects the model's ability to
reproduce the experimental results.
For this purpose, we tested different distributions of sodium channel density along the initial segment. A site x along the initial segment was designated at which the sodium channel density changed from the low soma/dendritic value to the high axonal node value (Fig. 7A). As the location of the abrupt change was systematically varied from proximal (x = 0) to distal (x = 1) end, the sodium and potassium channel densities on either side were allowed to vary to find the best fit. With increasing x, the axonal sodium channel density needed to maintain good simulations increased slowly up to ~0.6 (Fig. 7D). A good fit could be achieved over the range from x = 0 to x = 0.7 (Fig. 7B) but could not be attained when there was no increased channel density in the initial segment, i.e., when x = 1 (Fig. 7C).
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Spatial distribution of forward and backward propagating action potentials
The foregoing results made it clear that there are complex
spatial relations governing the forward and backward propagation of an
action potential. To analyze these relations, the simulations from Fig.
3 for the responses to a weak and strong depolarizing current step
applied to the primary dendrite were replotted as a function of spatial
distribution along the primary dendrite-soma-axon axis for different
instants of time (cf. Mainen et al. 1995). As can be
seen (Fig. 8A,
left), with weak dendritic current injection, although the
primary dendrite began to be depolarized earlier than the soma and
axon, the initial segment quickly passed both the dendrite and soma and
took the lead. The action potential of the initial segment propagated
faster to the first node than to the dendrite and further depolarized
the more distal nodes. Figure 8A, right (2nd
curve from top) showed that after the axon had begun to
repolarize the dendritic action potential was just reaching its peak.
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With strong depolarizing current to the dendrite (Fig. 8B), the action potential reached its peak in the distal dendrite before the soma and axon (curve 8 from the bottom). When it did start, the action potential in the initial segment rose faster than in the soma, where it was depolarized by axial currents from both the dendrite and the initial segment. Thus the initial segment fired earlier than the soma even though the action potential was propagating in the orthograde direction from the dendrite. The action potential in the dendrite declined before the soma and axon (Fig. 8B, left and right, top curves); the repolarization is obviously slowed by the continuing depolarization spreading from the soma.
Threshold analysis
The analysis illustrated in Fig. 8 suggested that the spatial gradients of initial membrane depolarization along the dendrite are critical in determining the sites of action potential initiation. For more detailed analysis, we focused on the spatial sequence of action potential (AP) initiation, i.e., the sites where the membrane potential first passes AP threshold. For this purpose, we chose several instants of time around AP threshold, as shown in the recording in Fig. 9A (time points 1-4). Figure 9B shows the spatial distribution of membrane potential in response to low current injection, C for high current injection. Two subthreshold curves for high current (- - -) from C also are drawn in B and vice versa for easy comparison of the different gradients. This makes clear the important point that, in B, with low current injection, the potential gradient is small and the lower soma AP threshold is crossed (see curves 3' to 4') before the higher dendritic AP threshold. In C, with high current injection, the steeper gradient along the dendrite allows the higher dendritic AP threshold to be crossed first (curves 3-4).
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Are the thresholds at these sites fixed levels of the membrane
potential, independent of the stimuli? To answer this question, we
focused on the AP threshold of the soma by shifting the action potentials initiated there by the different stimulus conditions so that
they overlapped as precisely as possible in the onset region of the
action potential (Fig. 10). Except for
the high dendritic current case, the trajectories in this region were
virtually identical (Fig. 10A, ). The AP threshold of
the soma is therefore independent of stimulus condition in these cases.
The soma response to high dendritic current seems to deviate slightly
from the others by having a lower AP threshold (Fig.
10A, dashed curve s); however, this is due to the fact
that it is being driven by the earlier rising action potential in the
dendrite (dashed curve d). Analysis of the gating states (data not
shown) showed that Na+ activation and inactivation and
K+ activation were similar for the cases in which threshold
was similar, as expected, whereas in the high current case there was lower Na+ activation, indicating that the
membrane was well below its intrinsic action potential threshold, being
driven by the earlier and larger amplitude dendritic action potential.
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We carried out a similar analysis of the AP threshold at the dendritic
recording site (Fig. 10B). This analysis was limited to
dendritic current injection because, as in the classical model, soma
current injection elicits a somatic action potential that always drives
the dendrite. For the cases of low and high current injection into the
dendrite, the action potential onsets were very similar (Fig.
10B, ). The faster rise of the dendritic action potential (dashed curve d) is due to axial current from the soma action
potential that leads it (dashed curve s). These studies thus indicate
that the action potential threshold is at a fixed membrane potential
level independent of stimulus conditions in the dendrite as well as the soma.
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DISCUSSION |
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The experimental results show that the site of action potential
generation can shift from soma to distal dendrite in response to
differing strengths of distal dendritic excitation of the mitral cell,
extending the findings with synaptic excitation of the distal tuft
(Chen et al. 1997) to the case of depolarizing current
injection into the distal dendrite. The results thus add to the
evidence gathered from a variety of neurons that there can be multiple sites of action potential generation within the axon-soma-dendritic axis of a neuron under varying conditions of excitation (see
INTRODUCTION).
Although the ability of a neuron to support shifts in the site of
action potential initiation is thus in line with growing evidence, a
more difficult problem is to explain two counterintuitive results. One,
as expressed by Mainen et al. (1995), is "the
surprising ability of dendritic Na+ channels to
substantially boost back-propagation of action potentials while failing
to support dendritic initiation." This is the classical model, as
also seen in the mitral cell with weak dendritic stimulation. However,
the mitral cell shows in addition that full-blown action potentials
indeed are elicited in the distal primary dendrite of the mitral cell
with stronger dendritic excitation. This implies a relatively narrow
range of distribution of voltage gated channels in the
axon-soma-dendritic axis. This narrow range would not be possible to
specify without the aid of rigorous modeling. Constructing a model,
however, presents a severe challenge, because of the problem of
adequately constraining the model with regard to the distribution of
active properties.
The aim of this study was to build on previous modeling approaches
(Mainen et al. 1995) to overcome these problems by
taking advantage of a combination of dual recording sites and the
simplified geometry of the mitral cell and exploring how the resulting
model could point the way toward better insights into the mechanisms underlying the shifting sites of action potential initiation. We first
will evaluate the strengths and limitations of the modeling approach
used here and then discuss the insights that the model has provided.
Strengths and limitations of the modeling approach
As already indicated, the strengths of the present approach are several.
First is the simple geometry of the mitral cell. Few neurons present such unusual features as the relatively unbranched nature of the primary dendrite, the uniform diameter of several micrometers through its length, the relatively long length of 300-500 µm, and the restriction of excitatory synaptic input to its distal tuft; combined, these features present unique advantages not only for experimental analysis but also for simplified models.
Second was the use of dual patch recording electrodes, one at or near the soma and the other on the distal dendrite. The availability of data from dual recording sites up to several hundred micrometers apart gives constraints on the parameters of a model that are not available from a single recording site. We found that, given its simple geometry, the intervening length of soma-dendrite could be defined physiologically relatively closely. The cell thus reduced to a well-defined part, the soma-dendrite axis, and the parts lying outside that were undefined: distal tuft, secondary dendrites, and axon initial segment and axon. The passive fit of the simulation was most sensitive to the combination of internal resistivity, membrane capacitance, and primary dendrite diameter within the part of the soma-dendritic axis between the two electrodes. This meant that the other parts of the neuron could be characterized in canonical terms without need for detailed representation of their morphology or physiological properties. This was particularly applicable for the present case in which the sites of stimulation of the cell and of the model were limited to one or other of the two electrode sites.
A third advantage was to use, as a stimulus, DC injection into soma or distal dendrite, which gave a more quantitatively defined input in comparison with synaptic excitation. Extension of the model to the case of synaptic excitation in the distal tuft, where synaptic excitation occurs, will require more detailed modeling of the distal tuft lying beyond the distal electrode. These studies are in progress (unpublished data).
A fourth advantage of the present modeling study was that information
was available on the density of Na+ channels
along the length of the primary dendrite (Bischofberger and
Jonas 1997). This provided a valuable reference against which to calibrate the validity of the estimates arising from the model.
Establishment of well-constrained models
Given these advantages, our first aim was to generate simulations
of the experimental results that were well constrained. One of the most
striking results of this study was the great advantage of the use of
dual recording sites, which quickly led to a model for the intervening
soma-dendritic region that greatly restricted the parameter subspace in
a way that was consistent with the experimental data of this study and
of the studies of Chen et al. (1997) and Bischofberger and Jonas (1997)
. We were able to make
precise superposition of experimental and model activity and show that
the model gave simulations that were nearly identical to the
experimental recordings. The most critical parameter combination is the
product of channel density and Na+ peak channel
open probability (m3·h)
during the rise of the action potential; the variance of this combination was very low for all of our best parameter sets. In assessing the active properties underlying action potential initiation, we have followed longstanding practice in modeling the fast
Na+ and delayed rectifier
K+ conductances using the classical
Hodgkin-Huxley model with parameters adapted from squid axon to mammal
(Cooley and Dodge 1966
; Mainen et al.
1995
; Traub and Llinas 1979
). Although other
ionic conductances could be incorporated easily into NEURON
simulations, the phenomenological action potential that we are fitting
would not allow us to dissect out individual contributions of those
channels, and the simulation would be greatly overparameterized.
Comparison of sodium and potassium densities between data and model
The density distribution of the sodium channel conductance is
critical in the initiation and propagation of action potentials. The
classical initiation site of an action potential is in the axon-hillock
region whether the depolarizing stimulation is applied at the soma or
dendrites (Stuart and Sakmann 1994). This requires a
high ratio of axonal to dendritic sodium density; good fits required an
approximate ratio of 40, which is in the range of many other simulation
studies (Mainen et al. 1995
; Rapp et al. 1996
). The absolute values of the peak current density and the peak Na+ channel conductance in the primary
dendrite and soma of the model were consistent with independent
patch-clamp data (Bischofberger and Jonas 1997
).
The discrepancy between the optimized
k value and the average
steady-state patch values measured by Bischofberger and Jonas (1997)
, although not of direct relevance to our study of action potential initiation, may be discussed from several not mutually exclusive perspectives.
As noted in RESULTS, the data for steady-state potassium
current at different locations and different cells presented by
Bishofberger and Jonas (1997) in their Fig. 3 shows a
large variance, with some of the values at or below the value of
k in our model. Converting from
these measurements of K+ current to estimates of
membrane within the patch to obtain estimates of specific
K+ conductance may have underestimated the true
specific K+ conductance. Alternatively, our
action potential data may have come from a cell that had a particularly
low potassium channel density. Statistical samples of action potential
fits and patch recording measurements would be needed to help resolve
this issue.
The inability to discover parameters in our model that gave a good fit
to the data when k = 500 pS/µm2 does not prove that such a parameter set
does not exist. Any conclusion that the model cannot fit the data with
such a large value of
k is
balanced by the relative ease we found in obtaining excellent dual
action potential fits for any single protocol. Unfortunately we do not
yet have insight into what the four protocol (8 action potential), 22 parameter error function looks like. We might be able to increase our
confidence in the negative result (or reject it) by repeating the
optimization process from a larger statistical sample of starting
points if we knew what the correct statistical analysis would be.
Full action potential initiation can occur in dendrites
The conditions for initiating an action potential at different
sites in different neurons can be quite different (Chen et al.
1997; Golding and Spruston 1998
; Stuart
et al. 1997a
; Turner et al. 1991
). Our study
shows that in mitral cells under moderately strong depolarizing current
injection in a distal dendrite, the distal dendrite is able to initiate
action potentials earlier than the soma and the axon initial segment.
This confirms and extends a previous dual patch recording study
demonstrating the same property in response to moderately strong distal
excitatory synaptic input (Chen et al. 1997
); that study
also found that inhibitory synaptic input near the soma could suppress
the somatic action potential, yielding an isolated full-size dendritic
action potential. Similar results of dendritic action potential
initiation also have been reported in pyramidal neurons (Golding
and Spruston 1998
; Turner et al. 1991
). The
present model also can generate an isolated dendritic action potential
with the soma suppressed by the inhibitory input in the secondary
dendrites (data not shown).
Other experimental and model studies on pyramidal cells have failed to
find that distal dendrites can initiate an action potential earlier
than the soma (Mainen et al. 1995; Rapp et al.
1996
; Stuart and Sakmann 1994
). Factors that may
account for this include the finding that pyramidal neuron apical
dendrites have a much lower Na+ conductance
(approximately one-third) than mitral cell primary dendrites
(Bischofberger and Jonas 1997
; Stuart and Sakmann
1994
). The shift is a sensitive consequence of the combination
of dendritic potential gradient, threshold inhomogeneity, and
relatively robust dendritic action potential.
In addition, in the pyramidal neuron the density of the transient
K+ channel increases progressively from the soma
to the distal dendrites. This IA
current damps the distal action potential initiation effectively (Hoffman et al. 1997). In related model studies, the
peak current densities were set even lower than the experimental ones
(Mainen et al. 1995
; Rapp et al. 1996
).
In our simulation, the action potential of the distal primary dendrite
can take the lead only after the stimulation strength exceeds a certain
value (0.6 nA).
Mechanism of action potential shift
As noted earlier, the problem in many types of neuron has been to explain active back propagation but no forward propagation under all stimulus strengths; the additional problem in the mitral cell is to account for a shift to forward propagation with higher stimulus strengths.
The model embodies a quantitative explanation for the stimulus strength and position dependence of the site of action potential initiation in the mitral cell. In the model, the relative strengths of loading effects and sodium channel density in the axon compared with the soma-dendritic region results in significantly lower "voltage threshold" for spike initiation in the axon. Because of this, the axon generally fires first unless the voltage gradient in the primary dendrite is steep enough for it to reach its higher threshold. As shown by the spatial analysis in Fig. 10, the earliest initiation site of the action potential is determined by two major factors, the difference between the axonal threshold and the distal dendritic threshold, and the voltage gradient along the primary dendrite when the depolarizing current is delivered at the distal dendrite.
For purposes of qualitative interpretation of experiments, the site of action potential initiation has been defined as the location of the earliest peak of the action potential. However, this earliest location of the peak is the causal result of several more fundamental factors, the first of which is arguably more in keeping with the simplest meaning of the word, "initiation." That is, the location of the first peak is determined by the spatial pattern of threshold crossing (the location of the earliest threshold crossing is perhaps a more natural definition of the site of initiation except that the concept of threshold itself is somewhat ambiguous), followed by the spatially dependent rise time of the spike determined by local cable load and local membrane channel density. We are able to account for the shift in first peak location in all our fits by noting that the time between threshold crossing in the primary and soma electrode sites is greater for high current injection (higher voltage gradient) than for low (almost isopotential for low current injection) and that the faster rise time at the soma end of the axon action potential may or may not overcome this difference in threshold crossing time thus ending in an earlier soma peak.
Although the strong depolarizing stimulus can elicit an action
potential in the dendrite earlier than in the soma, it does not mean
that the axonal action potential is caused by the orthograde propagation of a full-size dendritic action potential at the axon. In
fact, the initial segment of the axon generates an action potential earlier than the somatic action potential (see Fig. 8B and
Fig. 9). Whether the dendritic depolarizing stimulus is strong or weak, the initial segment always fires earlier than the soma. That means the
axon does not necessarily need the orthograde propagation of a
full-size dendritic action potential to trigger it but only a moderate
depolarization caused by the dendritic action potential. This
initiation property of the axonal action potential differs from the
dendrite the later action potential of which is elicited by a full-size
retrograde action potential from the axon. This result is in accord
with a triple electrode patch recording study of action potential
initiation in pyramidal neurons (Stuart et al. 1997a).
Threshold analysis
The spatial plots (Fig. 9) showed a distinct difference in the potential gradient along the dendrite associated with action potential initiation at different sites with different current strengths. We hypothesized that it is this gradient of potential along the dendrite that is the primary factor in determining the site of action potential initiation. This interpretation assumes that action potential threshold is a relatively constant intrinsic property, independent of stimulus intensity. A competing hypothesis is that the threshold is significantly dependent on the strength of the injected current.
To discriminate between these hypotheses, we examined the action potential onsets and found that they were very similar under the different stimulation protocols (Fig. 10). In cases where the onset showed a difference, it could be shown that this was due to current into the recording site from more active sites where a full-fledged action potential was already occurring. This was true at both somatic and dendritic recording sites. We therefore concluded that the action potential thresholds were relatively fixed in both axonal region and dendrite and that the gradient was the most critical element in determining the site of action potential initiation.
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APPENDIX |
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Fitting passive parameters
The linear response of the cell around rest, i.e., the first 4 ms after onset of low current injection in which transfer impedance is symmetric, was used to determine the acceptable parameter ranges for seven "passive" parameters: primary electrode resistance and capacitance, specific membrane capacitance and conductance (around rest), axial resistivity, and effective dendritic tuft and secondary dendrite diameters. The latter two served as generalized load parameters for regions outside the interval between the electrodes. Because of transfer impedance symmetry and the greater duration of this symmetry for the low current injection data, we fit only the pair of recordings in response to dendritic low current injection. After fitting the dendritic low current injection data, all cellular parameters were held constant, and the two soma electrode parameters were adjusted to fit the soma membrane potential in response to soma low current injection data. Thus fitting two curves with seven parameters followed by fitting one curve with two parameters, yielded values for those nine parameters which fit the linear response of all eight data curves.
The basic fitting algorithm we employed was the Principal Axis method
(PRAXIS, Brent 1973), a variant of the conjugate
gradient method. Conjugate gradient methods (Press et al.
1993
) have the theoretical property of finding the minimum of a
quadratic form in N dimensions with a sequence of just N + 1, one-dimensional minimizations. PRAXIS improves the practical
performance by never accidentally confining the search space to a
parameter subspace, using information embodied in the previous
conjugate gradient steps, and by more efficiently navigating curved
valleys with very steep sides. The reason for the term "principal
axis" is because the conjugate gradients found in the previous search
sequence are used to determine an orthogonal set of conjugate
directions, i.e., the principal axes of the quadratic form which
locally approximates the search space.
PRAXIS finds a local minimum, and there is no guarantee that this is the global minimum. Our impression of the form of our seven passive parameter error function is that the minimum is more like a connected multidimensional subspace as opposed to an egg-carton-like set of disconnected minima. The reason for this impression is that visualization of the three-dimensional shells corresponding to fixed error values and holding the four other parameters constant always has a form reminiscent of an ellipsoid with very large ratios of principal axis length, analogous to the surface of a flexible ruler which is very long, much less wide and very thin. The long axis often shows a gentle curvature. The visualization of the constant error shells was achieved for three parameter subspaces by recording the path taken by the fitting algorithm from several hundred to 1,000 random (factor of 10 from our standard best fit) start points and plotting the point for each path as it crosses the shell (error) boundary. Several hundred random points on a shell clearly indicate the shape of that shell. Furthermore all (1,000) of our five parameter subspace fits (electrode Re and Ce held constant) returned the same minimum.
Because it was, of course, impossible to explore exhaustively a high dimensional space, we employed the technique of starting the fitting process at random points in parameter space and recording the path taken by the PRAXIS algorithm to the local minimum. This allowed us to compare performance with different forms of the error (fitness) function. It also allowed us to collect a large number of parameter points with their fitness values to determine parameter sensitivity, which will be discussed in a future paper. This latter method was compared with the more direct sensitivity method of holding one parameter constant while allowing the other six to vary to find the best fit and plotting the best fit error versus the value of that parameter.
For the passive fits, we used the square norm between data points and
simulation results (treated as continuous curves) as our error
function. A simulation is a highly complex function of the parameters,
and we investigated two functions for parameter preconditioning in
hopes they would scale the error function in a way that would in some
sense be a better approximation to a quadratic form. Without
preconditioning, the large scale differences of our parameters e.g.,
Rm = 30,000 ·cm2 versus
Cm = 1 µf/cm2,
causes PRAXIS to work very slowly or not at all. The first
preconditioner was merely to scale each fit parameter to a value on the
order of one. The second was to fit the logarithms of the parameters. This latter has the advantage of removing scale from consideration but
is limited to parameter ranges that do not include 0.
For our passive fits, the use of log scaling was generally more
efficient than rough normalization scaling by a factor of two up to an
order of magnitude. However, the performance of perfect normalization
scaling was slightly better than log scaling. Our impression is that
the efficiency of normalization scaling increases with the closeness of
all scale factors to the final resulta not very useful property.
The path to minimum is complex with quick descents followed by occasional wandering with very little improvement. We see no obvious heuristic for stopping. After the initial fast improvement, the velocity along the fit path in parameter space is reasonably steady with short periods of faster movement during the period of abrupt improvement.
The change in the error function as parameters were individually varied was most sensitive to electrode resistance, internal resistivity, membrane capacitance, and primary dendrite diameter. Note that these parameters applied to the part of the soma-dendritic axis between the two electrodes. Passive parameters that could be varied over a wide range without increasing the error function significantly were soma area, membrane resistivity, and electrode capacitance. These were the findings with single parameter variation. Any of the passive parameters could covary with other parameters over an entire order of magnitude while remaining very close to the minimum error value.
Fitting the action potential
It was often the case with action potential fits that qualitative features that we judged to be physiologically important, such as details of the fast onset of the action potential, were not well fitted. Unfortunately, giving these features very high weight caused PRAXIS either to become quickly trapped in a local minimum with a very large error value or move to a region of parameter space with even more serious qualitative discrepancies in other portions of the action potential. We settled for the purposes of this study on the following strategy for managing fits involving 4 density parameters and 16 rate parameters that allowed the discovery of several hundreds of extremely precise fits for single protocol data (soma and dendrite action potential recordings) and several tens of very excellent fits for all four protocols (8 data curves).
The first step was to use several rough action potential features of single protocol data to move to regions of parameter space that gave action potentials with reasonable qualitative shape. These features consisted of peak time and amplitude, action potential width, and a few points in the onset and hyperpolarizing regions.
We then switched to the less biased square norm (useless for fitting when the simulation action potential is not already somewhat similar to the data) to refine the fit. This always returned a reasonably good fit, with the caveat that features we judged important would often have systematic differences with the data. As mentioned, the prime example of this was the very fast rise in soma action potential. Our earlier simulations always had much more rounded onsets that nevertheless contributed only a small amount to the total square norm error value. We have not concerned ourselves with qualitative discrepancies in the repolarization phase because of our focus on action potential initiation and the obvious conceptual inadequacy of our generic sodium and potassium channels.
Our very rough qualitative knowledge of the effect of individual rate parameters would suggest a diagnosis for the cause of the qualitative discrepancy which could be tested by manually forcing a large change in one or two of the rate parameters. This would invariably increase the fit error by several orders of magnitude, but holding those manually adjusted parameters constant and refitting reconverged to a new minimum that was often superior in terms of that qualitative feature.
With a single protocol fit in hand, all four protocols were simultaneously fit with the square norm. After a fit a subset of the parameters was randomly modified by up to a factor of 2, and the four protocols were refit, stopping when two principal axis search sequences did not reduce the error. This step is repeated to obtain a family of good fits and for this model and data yielded a fit similar in quality to our best fit in approximately 1 of every 20 attempts.
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ACKNOWLEDGMENTS |
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This work was supported by grants from the National Institute of Mental Health, the National Aeronautics and Space Administration, the National Institute on Deafness and Other Communication Disorders, the National Institute on Aging, the National Institute on Alcohol Abuse and Alcoholism (Human Brain Project), and the Office of Naval Research (MURI) (G. M. Shepherd); PAO YU-KONG Scholarship for Chinese Students Studying Abroad (G. Y. Shen); NIH Grant NS-11613 (M. L. Hines); the Carlsberg Foundation, the Danish Medical Research Council, the Faculty of Health Sciences (Copenhagen University), and the Danish Medical Association's Research Fund (J. Midtgaard); and NIH Grant DC-03918 (W. R. Chen).
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FOOTNOTES |
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Address for reprint requests: M. L. Hines, Section of Neurobiology, PO Box 208001, Yale University, New Haven, CT 06510.
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 11 May 1999; accepted in final form 7 July 1999.
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REFERENCES |
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