Computational Analysis of Action Potential Initiation in Mitral Cell Soma and Dendrites Based on Dual Patch Recordings

Gongyu Y. Shen,1 Wei R. Chen,1 Jens Midtgaard,3 Gordon M. Shepherd,1 and Michael L. Hines1,2

 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


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

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.


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

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).


    METHODS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

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.



View larger version (12K):
[in this window]
[in a new window]
 
Fig. 1. Canonical model of the mitral cell of the olfactory bulb. Electrode recording and injection sites in soma and dendrite are indicated. Model extends to 5 node-internode pairs. See METHODS for details. Pri, primary dendrite; Sec, secondary dendrite; AH, axon hillock; IS, initial segment.


                              
View this table:
[in this window]
[in a new window]
 
Table 1. Morphological parameter values of the canonical mitral cell

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 Omega ·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 Omega ·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 Omega ·cm2. The nodes and initial segment were assumed to have a large leakage specific membrane conductance and were accordingly set to 1,000 Omega  · 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.


                              
View this table:
[in this window]
[in a new window]
 
Table 2. Values of the passive electrical properties of the best model

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.
<IT>i</IT><SUB><IT>na</IT></SUB><IT>=</IT><IT><A><AC>g</AC><AC>&cjs1171;</AC></A></IT><SUB><IT>na</IT></SUB><IT>·</IT><IT>m</IT><SUP><IT>3</IT></SUP><IT>·</IT><IT>h</IT><IT>·</IT>(<IT>V</IT><IT>−</IT><IT>e</IT><SUB><IT>na</IT></SUB>) <IT>e</IT><SUB><IT>na</IT></SUB><IT>=60 mV</IT>

<IT>i</IT><SUB><IT>k</IT></SUB><IT>=</IT><IT><A><AC>g</AC><AC>&cjs1171;</AC></A></IT><SUB><IT>k</IT></SUB><IT>·</IT><IT>n</IT><SUP><IT>4</IT></SUP><IT>·</IT>(<IT>V</IT><IT>−</IT><IT>e</IT><SUB><IT>k</IT></SUB>) <IT>e</IT><SUB><IT>k</IT></SUB><IT>=</IT>−<IT>90 mV</IT>

<FR><NU>d<IT>m</IT></NU><DE><IT>d</IT><IT>t</IT></DE></FR><IT>=&agr;</IT><SUB><IT>m</IT></SUB><IT>·</IT>(<IT>1−</IT><IT>m</IT>)<IT>−&bgr;</IT><SUB><IT>m</IT></SUB><IT>·</IT><IT>m</IT>

<FR><NU>d<IT>n</IT></NU><DE><IT>d</IT><IT>t</IT></DE></FR><IT>=&agr;</IT><SUB><IT>n</IT></SUB><IT>·</IT>(<IT>1−</IT><IT>n</IT>)<IT>−&bgr;</IT><SUB><IT>n</IT></SUB><IT>·</IT><IT>n</IT>

<FR><NU>d<IT>h</IT></NU><DE><IT>d</IT><IT>t</IT></DE></FR><IT>=</IT><FR><NU><IT>h</IT><SUB><IT>∞</IT></SUB><IT>−</IT><IT>h</IT></NU><DE><IT>&tgr;</IT><SUB><IT>h</IT></SUB></DE></FR>

<IT>h</IT><SUB><IT>∞</IT></SUB><IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>1+</IT><IT>e</IT><SUP>(<IT>V</IT><IT>−&ugr;<SUB>1/2</SUB></IT>)<IT>/k</IT></SUP></DE></FR>

&tgr;<SUB><IT>h</IT></SUB><IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>&agr;</IT><SUB><IT>h</IT></SUB><IT>+&bgr;</IT><SUB><IT>h</IT></SUB></DE></FR>

&agr;=<FR><NU><IT>A</IT><IT>·</IT>(<IT>V</IT><IT>−&ugr;<SUB>1/2</SUB></IT>)</NU><DE><IT>1−</IT><IT>e</IT><SUP><IT>−</IT>(<IT>V</IT><IT>−&ugr;<SUB>1/2</SUB></IT>)<IT>/k</IT></SUP></DE></FR>

&bgr;=<FR><NU>−<IT>A</IT><IT>·</IT>(<IT>V</IT><IT>−&ugr;<SUB>1/2</SUB></IT>)</NU><DE><IT>1−</IT><IT>e</IT><SUP>(<IT>V</IT><IT>−&ugr;<SUB>1/2</SUB></IT>)<IT>/k</IT></SUP></DE></FR>


                              
View this table:
[in this window]
[in a new window]
 
Table 3. Distribution of ion channel densities


                              
View this table:
[in this window]
[in a new window]
 
Table 4. Parameters of kinetic equations of Na+ and K+ currents

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.


    RESULTS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

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.



View larger version (11K):
[in this window]
[in a new window]
 
Fig. 2. Simultaneous dual patch recordings from a mitral cell in response to depolarizing current steps delivered to the primary dendrite (d) 300 µm from the soma recording site. A: action potentials in the soma (s) and primary dendrite evoked with a weak current step (0.5 nA, 10 ms). Soma initiated an action potential first and then the primary dendrite. B: action potentials in the soma and primary dendrite evoked with a strong depolarizing current step (0.8 nA, 10 ms). Primary dendrite initiated an action potential first and then the soma. Electrode resistance compensation was determined visually.

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).



View larger version (12K):
[in this window]
[in a new window]
 
Fig. 3. Simulation of simultaneous dual patch recordings of responses at the soma and primary dendrite, for the case of dendritic current stimulation. Dendritic recording site on the model was 300 µm from the soma, which also was used as the stimulating electrode. A: with weak current injection (0.5 nA, 10 ms), the soma initiated an action potential first. Note the cross-over after which the soma took the lead. Peak time of the somatic action potential was 0.25 ms earlier than that of the dendritic counterpart. Key points for simulating the action potential were the onset (o), spike peak (sp), and repolarization (r). Charging transient (c) was fit separately with the passive model (see APPENDIX). Diagram of the mitral cell shows the stimulation protocols. B: strong current injection (0.8 nA, 10 ms) shifted the site of action potential initiation to the dendritic site in the model. These and the following simulations in Fig. 5 use the best fit parameters detailed in Tables 1-4.

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.



View larger version (13K):
[in this window]
[in a new window]
 
Fig. 4. Plot of the simulated relative shift of the peak time of the somatic and dendritic action potentials with change in injected current. Ordinate shows the subtraction of the somatic peak time from the dendritic peak time. A negative value indicates that the somatic peak time is earlier than that of the dendrite. Abscissa is the depolarizing current step amplitude. ×, experimental values. Experimental resolution was 50 µs. For the simulations underlying this figure, high resolution and accuracy of the peak difference were obtained using a variable time step method with absolute local voltage error tolerance = 10-5 mV, instead of the standard first-order fixed time step.

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.



View larger version (12K):
[in this window]
[in a new window]
 
Fig. 5. Simulation of simultaneous dual patch recordings from the soma and dendrite with the depolarizing current steps delivered to the soma. Soma initiated an action potential in the model with both a weak depolarizing current step (A: 0.4 nA, 10 ms) and a strong depolarizing current step (B: 0.8 nA, 10 ms). Same parameters as in Fig. 3.

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.



View larger version (36K):
[in this window]
[in a new window]
 
Fig. 6. Simulations of models that were constrained only by 1 stimulation protocol. These model parameters were obtained by fitting the models only under the constraints of the experimental responses to a weak depolarizing current step (C: 0.5 nA, 10 ms) injected in the primary dendrite. C: by construction, this simulation is very close to the data. Simulations under the other 3 stimulation protocols (A, B, and D) showed considerable variations.

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., <A><AC>g</AC><AC>&cjs1171;</AC></A>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 <A><AC>g</AC><AC>&cjs1171;</AC></A>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 <A><AC>g</AC><AC>&cjs1171;</AC></A>k had little direct bearing on our study of the mechanisms of action potential initiation, which are almost solely dependent on <A><AC>g</AC><AC>&cjs1171;</AC></A>na. However, it was of interest to gain some insight into the <A><AC>g</AC><AC>&cjs1171;</AC></A>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 <A><AC>g</AC><AC>&cjs1171;</AC></A>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 <A><AC>g</AC><AC>&cjs1171;</AC></A>k (the iterations ending at <A><AC>g</AC><AC>&cjs1171;</AC></A>k = 250 pS/µm2) produced fits with an error value that was six times larger than our best fit with <A><AC>g</AC><AC>&cjs1171;</AC></A>k = 42 pS/µm2. Over most of this range these runs fit well the onset and rise of the action potentials (dependent on <A><AC>g</AC><AC>&cjs1171;</AC></A>na), but diverged as expected in the repolarization phase dominated by <A><AC>g</AC><AC>&cjs1171;</AC></A>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 <A><AC>g</AC><AC>&cjs1171;</AC></A>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).



View larger version (27K):
[in this window]
[in a new window]
 
Fig. 7. Influence of sodium channel density distribution along the initial segment on the properties of the neuronal excitation. Models were constrained by all 4 stimulation protocols. A: diagram of the mitral cell. Normalized length (1) of the initial segment was divided into 2 parts by point x, increasing in value from the axon hillock (x = 0). Sodium channel density from 0 to x was set to the same as the soma and dendrites; the sodium channel density from x to 1 was set to the same as the nodes of Ranvier (i.e., for x = 1, the entire initial segment belonged to the soma). Axon consisted of nodes and myelinated segments, plus the portion (x to 1) of the initial segment with high sodium channel density. B: superposition of the data and simulations when x = 0.7. All parameters were fixed except Na+ and K+ channel density in the axon and soma-dendrite. C: data and simulations when x = 1. Weak depolarizing current step (0.4 nA, 10 ms) failed to evoke a somatic action potential, even though the nodes generated several action potentials, because of the long distance from the 1st node and the heavy conductance load of the soma and dendrites. down-arrow , voltage transients caused by axonal spiking. D: plot of nodal sodium channel density for different values of x that give the best fits to the 4 data curves for low current injection (c.f. B and C).

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.



View larger version (38K):
[in this window]
[in a new window]
 
Fig. 8. Spatial distributions of action potentials for weak and strong depolarization of the distal dendrite (same simulation as in Fig. 3). Inset: diagram of the mitral cell (n, node; m, myelinated internode; i, initial segment; s, soma; p, primary dendrite; T; dendritic tuft). Depolarizing current steps were delivered at the primary dendrite 300 µm from the soma (down-arrow ). A: spatial response at selected times to weak depolarizing current step (0.5 nA, 10 ms). Left: space plot of the upstroke of the action potential; the right one that of the repolarizing phase. B: spatial response to strong depolarizing current step (0.8 nA, 10 ms, see text).

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).



View larger version (29K):
[in this window]
[in a new window]
 
Fig. 9. Membrane potential vs. location for low (0.5 nA) and high (0.8 nA) current injection into the dendrite at 4 instants of time near action potential onset. A: recordings with critical time points for the voltage vs. position plots in B and C, where the node locations are marked by asterisks, and the node-myelin region is reduced in length scale by a factor of 100 to highlight the short 45 µm length of soma, hillock, initial segment. (Soma is isopotential but there can be very large voltage gradients in the initial segment.) B: spatial distribution of membrane potential response to low current injection in the dendrite, C to high current injection. Two subthreshold curves for high current (dashed lines) from C are also drawn in B and vice versa for easy gradient comparison. In B, with low current injection, the gradient is small and the lower soma action potential threshold is crossed before the higher dendritic threshold. In C, with high current injection, the high gradient allows the higher dendritic action potential threshold to be crossed first.

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, left-arrow ). 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.



View larger version (21K):
[in this window]
[in a new window]
 
Fig. 10. Comparison of action potential onsets. A: somatic voltage curves for the 4 cases (soma and dendrite stimulation, high and low current injection protocols) are shifted in time so that the apparent threshold for each occurs at arbitrary time 0. In addition, the dendritic recorded action potential for high dendritic current injection is shown (d, dashed line), and the corresponding soma response (labeled s) also is drawn with a dashed line. Except for high current injection in the dendrite, the soma voltage trajectories are very similar in the action potential onset region prior to the fast rise (arrow). B: soma and dendritic recordings of the responses to dendritic high current injection are shifted with respect to the soma and dendritic recordings of the responses to low current so that the apparent dendritic action potential onset occurs at arbitrary time 0. Dendritic voltage trajectories around threshold (arrow) are similar.

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, down-arrow ). 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.


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

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 <A><AC>g</AC><AC>&cjs1171;</AC></A>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 <A><AC>g</AC><AC>&cjs1171;</AC></A>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 <A><AC>g</AC><AC>&cjs1171;</AC></A>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 <A><AC>g</AC><AC>&cjs1171;</AC></A>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.


    APPENDIX
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

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 Omega ·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 result---a 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.


    ACKNOWLEDGMENTS

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).


    FOOTNOTES

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.


    REFERENCES
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

0022-3077/99 $5.00 Copyright © 1999 The American Physiological Society