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Estimated distribution of specific membrane resistance in hippocampal CA1 pyramidal neuron
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Estimated distribution of specific membrane resistance in hippocampal CA1 pyramidal neuron Toshiaki Omori a,⁎, Toru Aonishi a,b , Hiroyoshi Miyakawa c , Masashi Inoue c , Masato Okada a,d a
RIKEN Brain Science Institute 2-1 Hirosawa, Wako, Saitama 351-0198, Japan Department of Computational Intelligence and Systems Science, Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, 4259 Nagatsudacho, Modori-ku, Yokohama, Kanagawa 226-8502, Japan c Laboratory of Cellular Neurobiology, School of Life Sciences, Tokyo University of Pharmacy and Life Sciences, 1432-1 Horinouchi, Hachioji, Tokyo 192-0392, Japan d Department of Complexity Science and Engineering, Graduate School of Frontier Sciences, The University of Tokyo, Transdisciplinary Sciences Bldg., 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8561, Japan b
A R T I C LE I N FO
AB S T R A C T
Article history:
It has been suggested that dendritic membrane properties play an important role in a
Accepted 28 September 2006
synaptic integration. In particular, the specific membrane resistance, one of membrane
Available online 17 November 2006
properties, has been reported to be non-uniformly distributed in a single neuron, although the spatial distribution of the specific membrane resistance is still unclear. To reveal its non-
Keywords:
uniformity in dendrite, we estimated the spatial distribution of specific membrane
Multi-compartment model
resistance in a single neuron, based on voltage imaging data, observed optically in
Single neuron
hippocampal CA1 slices. As the optically recorded data, we used bi-directional propagations
Dendrite
of subthreshold excitatory postsynaptic potentials in dendrite, which were not be
Passive membrane properties
reproduced numerically with uniform-specific membrane resistance. By numerical
Synaptic integration
simulations for multi-compartment models with non-uniformity of specific membrane resistance, we estimated that the distribution obeys a step function; the optically recorded data were consistently reproduced for the distribution with a steep decrease in the specific membrane resistance at the distal apical dendrite, which occurs 300−500 μm away from the soma. In the estimated distribution, the specific membrane resistance at the distal side is less than about 103 Ωcm2, whereas the resistance at the proximal side is greater than about 104 Ωcm2. This result implies that the specific membrane resistance decreases drastically at the distal apical dendrite in hippocampal CA1 pyramidal neuron. © 2006 Elsevier B.V. All rights reserved.
⁎ Corresponding author. Amari Research Unit, RIKEN Brain Science Institute, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan. Fax: +81 48 467 9693. E-mail address: [email protected] (T. Omori). URL: http://www.brain.riken.jp/labs/mns/omori/ (T. Omori). 0006-8993/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.brainres.2006.09.095
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1.
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Introduction
Dendritic membrane properties and neuronal morphology play important roles in the information integration in a single neuron (Gulledge et al., 2005; Häusser et al., 2000; Kath, 2005; Krichmar et al., 2002; Mainen and Sejnowski, 1996; Vetter et al., 2001). In particular, regarding dendritic membrane properties, it has been suggested that a specific membrane resistance, one of passive membrane properties, is non-uniformly distributed in a single neuron (Inoue et al., 2001; Stuart and Spruston, 1998), and its functionality has been investigated theoretically so far (London et al., 1999). Since this passive membrane property is essential for membrane response to synaptic inputs at each site of the dendrite and also for interactions between postsynaptic potentials, the non-uniformity of specific membrane resistance would be one of major determinants of synaptic integration. In the Rall's linear cable theory (Rall, 1959; Rall and Agmon-Snir, 1998), the effects of morphology on neuronal electrical properties were analytically investigated. For example, it was shown that dendrites with branches could be reduced to an equivalent single cylinder if their compartments fulfill certain conditions, such as a uniform-specific membrane resistance and the 3/2 power law for the diameters of compartments. However, realistically, the specific membrane resistance takes non-uniformly distribution in dendrite, thus it is not enough to dealing with a single neuron model with just a uniform-specific membrane resistance in order to reveal synaptic integration in dendrites. Numerical study using multi-compartment models enables us to investigate a realistic situation in detail, which takes a non-uniformity of the specific membrane resistance into account. It is hard to clarify the spatial properties of the specific membrane resistance in dendrite, since a direct measurement of the distal dendrite via patch electrodes is difficult to be performed. Numerical simulations reproducing observable data are powerful methods for estimating properties which are hard to be measured directly. Stuart and Spruston made a double whole-cell recording of the soma and the proximal dendrite of cortical layer 5 pyramidal neurons (Stuart and Spruston, 1998). They applied a hyperpolarizing current pulse to either the soma or the proximal dendrite and recorded the time course of membrane potential on the counterpart. They showed by numerical simulations that a mathematical model hypothesizing that the specific membrane resistance of the dendrite was lower than that of the soma could consistently reproduce the results obtained from the double whole-cell recording. However, the specific membrane resistance of each dendritic part, including the distal dendrite, has not been clarified in detail, since it is hard to apply the whole-cell recording to the distal dendrite. To go beyond the limitation of recording with electrodes or pipettes, Inoue et al. performed an optical recording (Inoue et al., 2001), which is applicable to measurements for the distal dendrite, and measured not only subthreshold excitatory postsynaptic potentials (EPSPs) propagated from the apical dendrite to the soma in hippocampal CA1 slices (anterograde propagation), but also subthreshold EPSPs propagated from the soma to the apical dendrite (retrograde propagation). Comparing the results of optical
recording with the responses of mathematical models under the assumption that the specific membrane resistance is spatially uniform, they estimated one value of specific membrane resistance that reproduces the anterograde propagation and another one that reproduces the retrograde propagation. As numerical results, the estimated value of specific membrane resistance which reproduces anterograde propagation was much different from that which reproduces retrograde propagation. The discrepancy between those two values may have been due to the assumption that the specific membrane resistance is uniform in the dendrite, since the specific membrane resistance is suggested to be non-uniformly distributed. To reveal non-uniform passive membrane property, we estimated a distribution of the specific membrane resistance in a hippocampal CA1 pyramidal neuron including the distal apical dendrite, by means of numerical simulations using multi-compartment models. We used the optically recorded data of the anterograde propagation and the retrograde propagation of the EPSPs, and chose from candidate functions the one which reproduced the results of both the anterograde and retrograde propagations in the most accurate manner.
2.
Results
2.1.
Numerical simulations using a stylized model
In the first part of this section, we used the stylized model for hippocampal CA1 pyramidal neuron, which was used in numerical simulations by Inoue et al. (Fig. 1(a)). To estimate the distribution of specific membrane resistance in a hippocampal CA1 pyramidal neuron, we performed numerical simulations of EPSP propagation for three kinds of distributions—a constant as uniform case (Fig. 1(c)), and also a linear function (Fig. 1(d)) and a step function (Fig. 1(e)) as nonuniform cases—and compared the results of these simulations with the EPSP propagation observed by the optical method.
2.1.1.
Uniform-specific membrane resistance
We considered a case where a distribution of specific membrane resistance is uniform within a single neuron (Rm(x) = R0) to compare this case with non-uniform cases. This is the same distribution used in the estimation done by Inoue et al. (2001). Results of numerical simulations with optimal fitting parameters in constant cases (R0 = 7 kΩcm2, the specific intracellular resistance, Ri = 252 Ωcm) are shown in Fig. 2(a). The left column shows the results for anterograde propagation of an EPSP, whereas the right column shows the results for retrograde propagation of an EPSP. As shown in the left column of Fig. 2(a), the attenuation of an anterograde-propagating EPSP obtained by the optimal model is steeper than that measured by the optical method. Conversely, as shown in the right column of Fig. 2(a), the attenuation of a retrograde-propagating EPSP obtained by the optimal model is less steep than that measured by optical method. Even in the model with the optimal parameters, there was a discrepancy between the numerical results and optical data for those parameter values.
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that the specific membrane resistance obeys a linear function of the distance from the soma: 8 ! > x−h < R1 1−k ; x>h Rm ðxÞ ¼ ð2Þ Lapical −h > : otherwise R1 ; where x is the distance from the soma to each compartment, and Lapical is the distance along compartments between the soma and the most distal compartment in the apical dendrite, and k (0≤k<1) is a constant proportional to the slope of the distribution. In Eq. (2), the specific membrane resistance of a compartments within θ μm of the soma is indicated by R1. The case where k=0 corresponds to a uniform-specific membrane resistance. When the specific membrane resistance obeys a linear function of the distance from the soma, the model has four parameters: three for specific membrane resistance (R1, θ, k) and one for the specific intracellular resistance (Ri). To look for a set of optimal values of parameters under the assumption that the specific membrane resistance is a linear function of distance from the soma, we obtained the dependence of a summed error for both directional propagations on values of parameters: Elinear ðh;kÞ ¼ min R1 ;Ri
Fig. 1 – Multi-compartment models of pyramidal neurons in the hippocampal CA1 region used in numerical simulations: (a) stylized model (Inoue et al. (2001)); (b) 3D model (Magee and Cook (2000)). Functions used as distribution of specific membrane resistance in numerical simulations (c–e); (c) constant; (d) linear function; (e) step function.
To investigate the parameter dependency of this model, we evaluated the discrepancy between the results obtained from optical recording and responses of the model with corresponding values of parameters R0 and Ri. This evaluation was performed by using the summed error Econst (R0, Ri), consisting of an error for the anterograde propagation and that for the retrograde propagation: Econst ðR0 ; Ri Þ ¼
X
econst ðR0 ; Ri Þ y
ð1Þ
yafa;rg
(R0, Ri) is defined in where the error for each propagation, econst y Eq. (9) given in the Experimental procedure (see Section 4). We found that the region where the summed error was between 25 and 30 was optimal. As shown in Fig. 3(a), the summed error was a smooth function of parameters R0 and Ri. This implies that the parameter sensitivity of the model is small, and thus that the discrepancy shown in Fig. 2(a) could be evident over a wide range of parameters.
Linear function
Next we considered the case where the specific membrane resistance changes gradually in a single neuron. We assumed
elinear ðR1 ;h;k;Ri Þ y
ð3Þ
yafa;rg
In Eq. (3), to visualize the four-dimensional parameter space, the summed error is minimized with respect to R1 and Ri and is expressed as a function of θ and k. The range of values of R1 and Ri used in minimization was 5 kΩcm2 ≤ R1 ≤ 150 kΩcm2 and 5 Ωcm ≤ Ri ≤ 100 Ωcm. The obtained dependence is shown in Fig. 3(b). We found the error to be between 25 and 30 in the optimal region. We also found that the summed error showed small changes for different values of the parameters in Fig. 3(b). This implies that parameter sensitivity of the model is small. To make the dependence clearer, the summed error as a function of k is shown in Fig. 3(c), which represents a onedimensional cross section of Fig. 3(b). From Fig. 3(c) we found that steeper gradients k of the specific membrane resistance gives smaller errors. This dependence shows that nonuniformity of the specific membrane resistance affects the propagation of the EPSPs, although the obtained smallest error for the linear function has a value similar to that obtained for the uniform-specific membrane resistance. To reveal the behavior in the optimal case for the linear function, in Fig. 2(b) we show the result obtained by numerical simulations with optimal parameters. As shown in the left column of Fig. 2(b), the attenuation of anterograde-propagating EPSP obtained in the optimal case is steeper than that shown by the optical data. Conversely, as revealed in the right column of Fig. 2(b), the attenuation of retrograde-propagating EPSP obtained in the optimal case is less steep than that shown by the optical data. We can see in Fig. 2(b) that even with the linear function model with the optimal parameters, there was a discrepancy like that seen in Fig. 2(a) with the uniform model.
2.1.3. 2.1.2.
X
Step function
As a third case, we investigated the case of a step function of distance from the soma, where the specific membrane resistance changes more steeply, compared with the linear
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Fig. 2 – Summary of the results of numerical simulations with the best-fitting parameters in each of three cases; (a) constant-specific membrane resistance, (b) a linear function, (c) a step function. The time courses of membrane potential obtained by optical recording are shown by solid lines, while those obtained by numerical simulations are shown by dashed lines. The left column shows results for anterograde propagation of an EPSP, and the right column shows results for retrograde propagation of an EPSP. The depth of the color of each line shows a position where the membrane potential was observed. A black line corresponds to a time course of the membrane potential in an apical dendrite compartment at 330 μm from the soma. Dark gray, medium gray, and light gray lines respectively correspond to compartments at 220, 110, and 0 μm from the soma. As shown in these figures, the best model which could reproduce both anterograde and retrograde propagations of EPSPs was the step function model. The details of parameter dependency of each model were given in the following figures.
function. We assumed that the distribution of the specific membrane resistance obeys the following step function: Rm ðxÞ ¼ R3 þ ðR2 −R3 ÞΘðx−hÞ
ð4Þ
where R2 and R3 are positive constants and Θ(x) is a unit function which takes one for x > θ, and zero otherwise. To determine the optimal values of the parameters for the step function, we investigated the dependence of the summed error for both the anterograde and retrograde propagations. A set of parameters consists of three parameters concerning the specific membrane resistance (R2, R3, and θ), and the specific intracellular resistance Ri. To visualize the four-dimensional parameter space, in Fig. 4(a) we plot the dependence of the following summed error, Estep (R2, R3), a minimized summed error with respected to Ri and θ: Estep ðR2 ;R3 Þ ¼ min h;Ri
X yafa;rg
estep ðR2 ;R3 ;h;Ri Þ: y
ð5Þ
Minimization was performed for 0 ≤ θ ≤ 600 μm and 10 Ωcm ≤ Ri ≤ 200 Ωcm. In Fig. 4(a), we see that the optimal error between 5 and 10 was obtained for log10 R2 ≲ 3 and log10 R3 ≳ 4. One of the optimal results of numerical simulation for the step function are shown in Fig. 2(c), where we see that the results obtained by optical recording for both anterograde and retrograde propagations were reproduced closely by numerical simulations. We set the values of parameters in Fig. 2(c) as follows: θ = 325 μm, R2 = 100.5 Ωcm2, R3 = 106 Ωcm2, Ri = 80 Ωcm. One of characteristic points for estimated optimal parameters was that the specific membrane resistance at the distal dendrite was about a tenth of that at the proximal dendrite. In order to reveal the spatial profile of the specific membrane resistance, we need to clarify what values of threshold θ reproduce the results measured by the optical method. Then we investigated dependence of the summed error on threshold θ. We show the dependence of the summed
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Although the results obtained by numerical simulations for the step function closely reproduced those obtained by the optical method, it is still unclear what level of a change of the specific membrane resistance is needed to reproduce the results obtained by optical methods. To reveal the appropriate
Fig. 3 – Parameter dependency of two models whose distributions of specific membrane resistances were (a) constant and (b, c) linear function. (a) Dependence of the summed error on specific membrane resistance R0 and specific intracellular resistance Ri, obtained for the constant-specific membrane resistance. (b) Dependence of the summed error on k and θ, obtained for the linear function. The value of the summed error is indicated by a specific color. Lighter colors correspond to smaller summed errors. (c) One-dimensional cross section of panel (b). The errors of these models are larger than that of step function model shown in Fig. 4.
error on threshold and specific intracellular resistance Ri in Fig. 4(b). In Fig. 4(b), we see that an optimal region is obtained for θ ∼ 325 μm.
Fig. 4 – Parameter dependency of a model whose specific membrane resistance obeyed a step function. (a) Dependence of summed error on specific membrane resistances R2 and R3; (b) dependence of summed error on threshold θ and specific intracellular resistance Ri. The error of this model was smaller than ones of constant and linear function models shown in Fig. 3. (c) Dependence of summed error for sigmoid function on slope of change β. The validity of step function as distribution of specific membrane resistance is also confirmed by fitting using sigmoid function.
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level of the slope, we performed numerical simulations for the case in which the specific membrane resistance obeys a sigmoid function of the distance from the soma: Rm ðxÞ ¼ R3 þ
R2 −R3 1 þ exp½−bðx−hÞ
ð6Þ
propagations. These results show that constant and linear functions are inadequate, but that a steeper change of the specific membrane resistance is needed. Thus the step function is appropriate as the distribution of the specific membrane resistance in single hippocampal CA1 pyramidal neurons.
where the slope around the threshold is denoted by β (>0) [μm− 1]. We notice that Rm(x) corresponds to the step function when β → ∞. The values of other parameters were set as follows: θ = 325 μm, R2 = 5 Ωcm2, R3 = 300 kΩcm2 and Ri = 80 Ωcm. The dependence of the summed error on the slope β is shown in Fig. 4(c). We see that the summed error initially decreases and approaches to a constant value as β increases. These results shown in Figs. 4(a), (b) and (c) imply that specific membrane resistance decreases rapidly about 300 μm from the soma, and that the distal dendrite is much leakier than the proximal dendrite. That is, R2 is less than about 103 Ωcm2 and R3 is greater than about 104 Ωcm2.
2.1.4. Comparison of results for different distributions of specific membrane resistance To estimate the distribution of the specific membrane resistance in a hippocampal CA1 pyramidal neuron, we compared the results of the stylized model for three different distributions of the specific membrane resistance. For uniform-specific membrane resistance, a discrepancy existed between the results obtained by the optical methods and those obtained by the numerical simulations, even with the optimal set of parameters, as shown in Fig. 2(a). This shows that uniform-specific membrane resistance is not appropriate, and non-uniformity of the specific membrane resistance should be needed to be considered. However, for the linear function, a case of non-uniform-specific membrane resistance with a relatively moderate change, a discrepancy also exists between the results obtained by numerical simulations and those measured by the optical method, even with optimal parameters, as shown in Fig. 2(b). Assuming that the distribution of the specific membrane resistance obeys a step function, the results obtained by the optical method are consistently reproduced by the numerical simulations, as shown in Fig. 2(c). In fact, numerical simulations with the step function could give smaller summed error than those with the constant and the linear function, as shown in Figs. 3 and 4. The scale relation between true membrane potentials and optically observed data might be non-linearly deformed, so we need to evaluate the validity of our results by using other measures. The peak time is a robust measure against such a scale deformation. Here we also estimated a peak time from onset of the EPSP. The peak time of the EPSP for the anterograde propagation optically observed at 0 μm from the soma may be about 20 ms from its onset, and that for the retrograde propagation observed at 330 μm from the soma may be about 13 ms from its onset, as shown by the solid lines of Fig. 2. Corresponding peak times obtained by numerical simulations in Fig. 2 are about 15.75 ms (constant), 15 ms (linear function), 21.5 ms (step function) for the anterograde propagation, and 17.5 ms (const) 14.75 ms (linear function) 14 ms (step function) for the retrograde propagation. Comparing these results, we find that peak times in the step case are most similar to those of the voltage imaging for both
Fig. 5 – Similar results using the stylized model were also obtained by using the 3D model. Dependence of summed error between results obtained by optical method and those obtained by numerical simulation using 3D model for step function; (a) constant-specific membrane resistance, (b) and (c) step function as distribution of specific membrane resistance.
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The dependence of the summed error for both the anterograde and retrograde propagations on parameters of the distribution shown in Fig. 4(b) implies that the specific membrane resistance changes abruptly around θ ∼ 325 μm, which may correspond to the distal dendrite. These results show that a steep decrease in the specific membrane resistance occurs in the distal dendrite. The slope of the decrease was investigated by assuming that the specific membrane resistance obeys a sigmoid function of distance from the soma. By numerical simulation, we found that as shown in Fig. 4(c), a smaller error is obtained with larger values of β. These results imply that the specific membrane resistance decreases drastically at the distal dendrite.
2.2.
Numerical simulations using a 3D model
To verify the generality of results obtained above, in the second half of this section, we performed numerical simulations using another multi-compartment model: a 3D model reconstructed by Magee and Cook (2000), which is described by the absolute position and direction of each compartment, as
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shown in Fig. 1(b). Numerical simulations using a second multi-compartment model would be necessary to clarify whether the results for the multi-compartment model shown in the previous subsection are general features of single pyramidal neurons in the hippocampal CA1 region, or limited features restricted to a specific hippocampal CA1 pyramidal neuron. As well as numerical simulations for the stylized model in Subsection 2.1, we considered three kinds of distributions of the specific membrane resistance, a constant, a linear function, and a step function, to estimate the distribution of the specific membrane resistance.
2.2.1.
Uniform-specific membrane resistance
We performed numerical simulations for a 3D model of a hippocampal CA1 pyramidal neuron with constant-specific membrane resistance. To find optimal values of the parameters R0 and Ri, the dependence of the summed error on R0 and Ri for both the anterograde and retrograde propagations in case of the constant-specific membrane resistance, is shown in Fig. 5(a). We found from results shown in Fig. 5(a) that there is an optimal area and that the error in the optimal area was
Fig. 6 – Time courses of membrane potential obtained by numerical simulations using other sets of optical data instead of learning data. If the estimated distribution is a general feature of single pyramidal neurons in the hippocampal CA1 region, it could reproduce other sets of optical data. Four data sets indicated by solid lines in these figures were obtained by optical recording for four different slices. The dashed lines superimposed on these experimental results show the results reproduced by an optimal model estimated from the single data set shown in Fig. 2. Upper: results for anterograde propagation of EPSPs in two slices. Lower: results for retrograde propagation of EPSPs in another two slices. The specific membrane resistance was assumed to obey a step function of distance from the soma.
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between 15 and 20. The optimal values of the parameters were Ri = 58 Ωcm and R0 = 8 kΩcm2. Even with the optimal values of the parameters, the results measured by the optical method were not reproduced in numerical simulations with the constant-specific membrane resistance (data not shown), which is a similar tendency to stylized model with constantspecific membrane resistance, shown in Fig. 2(a). The optimal value of the specific membrane resistance R0 was about 10 kΩcm2, which is similar to the value obtained for the stylized model shown in Fig. 3(a). The dependence of the summed error obtained for the 3D model shown in Fig. 5(a) is similar to that obtained for the stylized model, shown in Fig. 3(a).
2.2.2.
Linear function
As a first case of the non-uniform case, we performed numerical simulations using the 3D model with a linear function as the distribution of the specific membrane resistance. We found that the optimal error for the linear function was between 15 and 20. We also found that larger k values had smaller errors (data not shown). The dependence of the summed error on k is similar to that obtained using the stylized model for the linear function.
2.2.3.
Step function
We show the dependence of the error on R2 and R3 for the step function in Fig. 5(b). In Fig. 5(b), we see that an optimal summed error was obtained between 0 and 5 for log10 R2 ≲ 3 and log10 R3 ≳ 4. The results in Fig. 5(b) are similar to the results obtained by numerical simulations for the step function using the stylized model shown in Fig. 4(a). The dependence of the summed error on θ and Ri is shown in Fig. 5(c). We found that the summed error decreased for 400 μm ≤ θ ≤ 550 μm. This was similar to Fig. 4(b) obtained for the stylized model.
2.3. Comparison between results for different multi-compartment models In the estimation of the parameters, we used a set of optical data recorded from only one slice in vitro. According to learning theory and system identification (Hertz et al., 1991), however, over-fitting might occur when parameters are estimated from a small set of learning data. Thus, to avoid over-fitting in our estimation, whether the estimated distribution holds for the other optical data set recorded from several slices must be confirmed. For this purpose, we also showed the results obtained by numerical simulations using other data sets from two different slices for each directional EPSP propagation. If the estimated distribution is a general feature of single pyramidal neurons in the hippocampal CA1 region, our estimated distribution should reproduce other sets of optical data. We used the 3D model and set the same values of parameters for the two slices: θ = 400 μm, R2 = 102.25 Ωcm2, R3 = 105 Ωcm2, and Ri = 40 Ωcm. These values are included in the optical area in Figs. 5(b) and (c). The results for the anterograde and retrograde propagations in other slices are shown in Fig. 6. As shown there, we found that EPSP propagations observed using the optical method in these different slices were reproduced well in numerical simulations. This reveals that the estimated distribu-
tion might not be a limited feature of a specific data set, but a general feature of hippocampal CA1 pyramidal neurons.
3.
Discussion
We estimated that the specific membrane resistance drastically decreases in the apical dendrite in the hippocampal CA1 pyramidal neuron by numerical simulations using multicompartment models. In the estimated distribution, the steep decrease in the specific membrane resistance occurs around 300–500 μm away from the soma, suggesting that the specific membrane resistance in the distal area is less than that in the proximal area. Since optically observed propagation of the EPSP was reproduced with the step function for both the stylized model and the 3D model, the estimated distribution with a steep change could be a general feature of hippocampal CA1 pyramidal neurons. A previous study of the double whole-cell recordings in CA1 slices indicated that the CA1 pyramidal neurons have leaky apical dendrites (Golding et al., 2005). The estimation in the present paper was based on the data of the optical recording, which has spatial resolution potentially higher than that of the double whole-cell recording. Thus the result of our estimation is consistent with the leaky apical dendrites of CA1 pyramidal neurons found in the previous study. Moreover, our results further indicate that the change of the specific membrane resistance in the apical dendrite of the CA1 neurons is steep. Since the hyperpolarization-activated cation channel is active near the resting state of neurons, the low value of the specific membrane resistance at distal dendrite might be due to Ih (Stuart and Spruston, 1998). In the known distribution of Ih, however, the hyperpolarization-activated cation conductance changes gradually rather than steeply, and therefore could not explain the stepwise change found in the present study. Thus another conductance, like leak conductance, may be associated with the estimated stepwise change in the distribution of specific membrane resistance. The spatial profile of specific membrane resistance should have a marked effect on synaptic integration, since the specific membrane resistance at each part of the dendrite has an influence on the generation of a PSP, as well as on the spatio-temporal interaction between PSPs (Gulledge et al., 2005). Backpropagating action potentials are known to play an important role in synaptic integration and synaptic plasticity, and the spatial profile of the specific membrane resistance may affect the behavior of backpropagating action potentials. To reveal computational role of the non-uniform-specific membrane resistance, the effect of the non-uniformity of the specific membrane resistance on backpropagating action potential should be clarified. We left this problem as a future work. Stuart and Spruston (1998) have compared the results of double whole-cell recordings from layer 5 cortical pyramidal neurons with the results of numerical simulations and showed that numerical simulations for a sigmoid function reproduce the results obtained by whole-cell recording, and that the estimated position of change in the specific membrane resistance was around 400 μm from the soma. For the
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layer 5 cortical neuron, this position may correspond to a middle part of the apical trunk. In our estimation based on optical recording, the estimated decrease position was around 300 μm to 500 μm away from the soma, in the apical dendrite of the hippocampal CA1 pyramidal neuron. Our estimation for hippocampal CA1 pyramidal neurons suggests that the specific membrane resistance of the distal dendrite is less than that of the proximal dendrite. It is known that the proximal dendrite of a hippocampal CA1 pyramidal neuron has inputs from CA3 via Schaffer collateral, whereas the distal dendrite has inputs from the entorhinal cortex via the perforant path. This may suggest that there is some functional association of non-uniform membrane properties with two kinds of synaptic inputs to a hippocampal CA1 pyramidal neuron.
4.
Experimental procedure
Numerical simulations were performed for a single-neuron model with morphological data by using the NEURON simulator (Hines and Carnevale, 1997). Two multi-compartment models were used in numerical simulations: that used by Inoue et al. (2001) and that used by Magee and Cook (2000). In the model used by Inoue et al., the neuronal morphology was described by a stylized model (Hines and Carnevale, 1997); morphological information was described only by links between compartments. In the model used by Magee and Cook, the morphology was described using a 3D model: morphological information was described by absolute positions and directions in addition to the links. The stylized model used by Inoue et al. (2001) consists of 130 apical dendrite compartments, 135 basal dendrite compartments, and 14 somatic compartments (Fig. 1(a)). The 3D model used by Magee and Cook (2000) consists of 412 dendritic compartments and one somatic compartment (Fig. 1(b)). These two models have similar dendritic structure; in both models the compartment farthest from the soma on the dendrosomatic axis is about 600 μm away from soma on the corresponding axis. We used the values of diameters and length of each compartment used in Inoue et al. (2001) and in Magee and Cook (2000). The leak current is assumed to be given to each compartment for both models. The parameters in these models were the specific membrane resistance Rm [Ωcm2], the specific intracellular resistance Ri [Ωcm], the specific membrane capacitance C [μF/cm2]. We set the specific membrane capacitance of the compartments of the apical dendrite, CA, and that of the basal dendrite, CB, to CA = CB = 1.6 μF/cm2, and that of the somatic compartments, CS, to CS = 1 μF/cm2 (the large value of specific membrane conductance for the dendrite was assumed in order to take into account the effect of the dendritic spine (Spruston et al., 1999; Inoue et al., 2001)). We performed numerical simulations which reproduced the results of optical recordings of propagations of subthreshold EPSPs observed in hippocampal CA1 slices by Inoue et al. (2001). In numerical simulations, we consider an anterograde and a retrograde propagations of the subthreshold EPSPs observed optically by Inoue et al. (2001). We briefly review the methods of the optical recording performed by Inoue et al.
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(2001). Subthreshold synaptic response was induced by electrically stimulating presynaptic fibers. The stimulus was given to either the stratum lacunosum moleculare or the stratum oriens. The obtained responses were averaged over 16–32 trials. CNQX components were isolated by subtracting responses in a solution with APV, CNQX and bicuculline from those in a solution with only APV and bicuculline. In our study we used time courses of CNQX components as data of propagating EPSPs. To reproduce a propagation of the subthreshold EPSPs from the apical dendrite to the soma, we gave a stimulus to compartments in the apical dendrite, which is 330 μm away from the soma. To reproduce a propagation from the soma to the apical dendrite, we gave a stimulus to the somatic compartments. In our numerical simulations, to mimic an EPSP observed in the optical recordings, we performed voltage clamping in the multi-compartment models by giving the following current as external input to the corresponding compartments, ext ðVi ðtÞ−Vyopt ðtÞÞ I ext y;i ðtÞ ¼ g
ð8Þ
where the command inputs for voltage clamps are the time sequence data of EPSPs optically recorded at their positions in neurons. Vopt is a time course of voltage imaging data. The y direction of an EPSP is indicated by y ∈{a, r} (y = a for the anterograde propagation, y = r for the retrograde propagation), and recorded position was expressed by j ∈ {0, 1, 2, 3} (j = 0, 1, 2, 3 for 0 μm, 110 μm, 220 μm, 330 μm from the soma, respectively). For simulations of the anterograde propagation, we used Vopt a,3 (t) as the command inputs to apical dendritic compartments which is 330 μm away from the soma. For simulations of the retrograde propagation, we used Vopt r,0 (t) as command inputs to somatic compartments. For both propagations, we measured time courses of membrane potentials at (t, x), four positions: the membrane potential at the soma, Vsim 0 and the membrane potentials at three compartments in the apical dendrite: one in a compartment at 110 μm from the (t, x); one in a compartment at 220 μm from the soma, Vsim 1 (t, x); and one in a compartment at 330 μm from soma, Vsim 2 (t, x). x Denotes a vector composed of parathe soma, Vsim 3 meters describing a distribution function of the specific membrane resistance and specific intracellular resistance in the dendrite. We calculated the following squared error between the time course obtained by numerical simulations and that obtained by the optical measurements: ey ðxÞ ¼
N 3 t −1 X X
2 opt sim Vy;i ðnsÞ−Vy;i ðns;xÞ
ð9Þ
n¼0 i¼0
where τ is the sampling period and Nt is the total amount of sampled data. The suffix i denotes an observed position as (t, x), and the suffix opt refers to the optical defined in V sim i recording data. To find the optimal values of parameters consistently satisfying both anterograde and retrograde propagations, we evaluate a discrepancy between results obtained from optical recording and responses of the model with given values of parameters x by using the following summed error: EðxÞ ¼
X yafa;rg
ey ðxÞ:
ð10Þ
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By using the summed error, we clarified the differences in behavior between multi-compartment models with different distributions of the specific membrane resistance. We performed numerical simulations for three kinds of distributions, a constant as uniform case (Fig. 1(c)), and also a linear function (Fig. 1(d)) and a step function (Fig. 1(e)) as nonuniform cases, and determined which distribution of specific membrane resistance best reproduced both the anterograde and retrograde propagations of an EPSP.
Acknowledgments This work was partially supported by Grant-in-Aid for Scientific Research for Young Scientists (B) (Nos. 17700250 (T.O.) and 18700299 (T.A.)), Grant-in-Aid for Scientific Research on Priority Areas (Nos. 18020007 (M.O.) and 18079003 (M.O.)), and Grant-in-Aid for Scientific Research (B) (No. 17300096 (H.M.)), and Grant-in-Aid for Scientific Research (C) (No. 16500093 (M.O.)) from MEXT and JSPS of Japan. T.O. was partially supported by Research Fellowship from Japan Society for the Promotion of Science for Young Scientists.
REFERENCES
Golding, N.L., Mickus, T.J., Katz, Y., Kath, W.L., Spruston, N., 2005. Factors mediating powerful voltage attenuation along CA1 pyramidal neuron dendrites. J. Physiol. (Lond.) 568, 69–82. Gulledge, A.T., Kampa, B.M., Stuart, G.J., 2005. Synaptic integration in dendritic trees. J. Neurobiol. 64, 75–90. Häusser, M., Spruston, N., Stuart, G.J., 2000. Diversity and dynamics of dendritic signaling. Science 290, 739–744.
Hertz, J., Krogh, A., Palmer, R.G., 1991. Introduction to the Theory of Neural Computation. Perseus Books, Massachusetts. Hines, M.L., Carnevale, N.Y., 1997. The NEURON simulation environment. Neural Comput. 9, 1179–1209. Inoue, M., Hashimoto, Y., Kudo, Y., Miyakawa, H., 2001. Dendritic attenuation of synaptic potentials in the CA1 region of rat hippocampal slices detected with an optical method. Eur. J. Neurosci. 13, 1711–1721. Kath, W.L., 2005. Computational modeling of dendrites. J. Neurobiol. 64, 91–99. Krichmar, J.L., Nasuto, S.J., Scorcioni, R., Washington, S.D., Ascoli, G.A., 2002. Effects of dendritic morphology on CA3 pyramidal cell electrophysiology: a simulation study. Brain Res. 941, 11–28. London, M., Meunier, C., Segev, I., 1999. Signal transfer in passive dendrites with nonuniform membrane conductance. J. Neurosci. 19, 8219–8233. Magee, J.C., Cook, E.P., 2000. Somatic EPSP amplitude is independent of synapse location in hippocampal pyramidal neurons. Nat. Neurosci. 13, 895–903. Mainen, Z.F., Sejnowski, T.J., 1996. Influence of dendritic structure on firing pattern in model neocortical neurons. Nature 382, 363–366. Rall, W., 1959. Branching dendritic trees and motoneuron membrane resistivity. Exp. Neurol. 2, 503–532. Rall, W., Agmon-Snir, H., 1998. Cable theory for dendritic neurons, In: Koch, C., Segev, I. (Eds.), Methods in Neuronal Modeling, From Ions to Networks, 2nd ed. MIT Press, pp. 27–92. Spruston, N., Stuart, G., Häusser, M., 1999. Dendritic integration. In: Stuart, G., Spruston, N., Häusser, M. (Eds.), Dendrites. Oxford University Press, pp. 231–270. Stuart, G., Spruston, N., 1998. Determinants of voltage attenuation in neocortical pyramidal neuron dendrites. J. Neurosci. 18, 3501–3510. Vetter, P., Roth, A., Häusser, M., 2001. Propagation of action potentials in dendrites depends on dendritic morphology. J. Neurophysiol. 85, 926–937.
a v a i l a b l e a t w w w. s c i e n c e d i r e c t . c o m
w w w. e l s e v i e r. c o m / l o c a t e / b r a i n r e s
Research Report
Estimated distribution of specific membrane resistance in hippocampal CA1 pyramidal neuron Toshiaki Omori a,⁎, Toru Aonishi a,b , Hiroyoshi Miyakawa c , Masashi Inoue c , Masato Okada a,d a
RIKEN Brain Science Institute 2-1 Hirosawa, Wako, Saitama 351-0198, Japan Department of Computational Intelligence and Systems Science, Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, 4259 Nagatsudacho, Modori-ku, Yokohama, Kanagawa 226-8502, Japan c Laboratory of Cellular Neurobiology, School of Life Sciences, Tokyo University of Pharmacy and Life Sciences, 1432-1 Horinouchi, Hachioji, Tokyo 192-0392, Japan d Department of Complexity Science and Engineering, Graduate School of Frontier Sciences, The University of Tokyo, Transdisciplinary Sciences Bldg., 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8561, Japan b
A R T I C LE I N FO
AB S T R A C T
Article history:
It has been suggested that dendritic membrane properties play an important role in a
Accepted 28 September 2006
synaptic integration. In particular, the specific membrane resistance, one of membrane
Available online 17 November 2006
properties, has been reported to be non-uniformly distributed in a single neuron, although the spatial distribution of the specific membrane resistance is still unclear. To reveal its non-
Keywords:
uniformity in dendrite, we estimated the spatial distribution of specific membrane
Multi-compartment model
resistance in a single neuron, based on voltage imaging data, observed optically in
Single neuron
hippocampal CA1 slices. As the optically recorded data, we used bi-directional propagations
Dendrite
of subthreshold excitatory postsynaptic potentials in dendrite, which were not be
Passive membrane properties
reproduced numerically with uniform-specific membrane resistance. By numerical
Synaptic integration
simulations for multi-compartment models with non-uniformity of specific membrane resistance, we estimated that the distribution obeys a step function; the optically recorded data were consistently reproduced for the distribution with a steep decrease in the specific membrane resistance at the distal apical dendrite, which occurs 300−500 μm away from the soma. In the estimated distribution, the specific membrane resistance at the distal side is less than about 103 Ωcm2, whereas the resistance at the proximal side is greater than about 104 Ωcm2. This result implies that the specific membrane resistance decreases drastically at the distal apical dendrite in hippocampal CA1 pyramidal neuron. © 2006 Elsevier B.V. All rights reserved.
⁎ Corresponding author. Amari Research Unit, RIKEN Brain Science Institute, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan. Fax: +81 48 467 9693. E-mail address: [email protected] (T. Omori). URL: http://www.brain.riken.jp/labs/mns/omori/ (T. Omori). 0006-8993/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.brainres.2006.09.095
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1.
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Introduction
Dendritic membrane properties and neuronal morphology play important roles in the information integration in a single neuron (Gulledge et al., 2005; Häusser et al., 2000; Kath, 2005; Krichmar et al., 2002; Mainen and Sejnowski, 1996; Vetter et al., 2001). In particular, regarding dendritic membrane properties, it has been suggested that a specific membrane resistance, one of passive membrane properties, is non-uniformly distributed in a single neuron (Inoue et al., 2001; Stuart and Spruston, 1998), and its functionality has been investigated theoretically so far (London et al., 1999). Since this passive membrane property is essential for membrane response to synaptic inputs at each site of the dendrite and also for interactions between postsynaptic potentials, the non-uniformity of specific membrane resistance would be one of major determinants of synaptic integration. In the Rall's linear cable theory (Rall, 1959; Rall and Agmon-Snir, 1998), the effects of morphology on neuronal electrical properties were analytically investigated. For example, it was shown that dendrites with branches could be reduced to an equivalent single cylinder if their compartments fulfill certain conditions, such as a uniform-specific membrane resistance and the 3/2 power law for the diameters of compartments. However, realistically, the specific membrane resistance takes non-uniformly distribution in dendrite, thus it is not enough to dealing with a single neuron model with just a uniform-specific membrane resistance in order to reveal synaptic integration in dendrites. Numerical study using multi-compartment models enables us to investigate a realistic situation in detail, which takes a non-uniformity of the specific membrane resistance into account. It is hard to clarify the spatial properties of the specific membrane resistance in dendrite, since a direct measurement of the distal dendrite via patch electrodes is difficult to be performed. Numerical simulations reproducing observable data are powerful methods for estimating properties which are hard to be measured directly. Stuart and Spruston made a double whole-cell recording of the soma and the proximal dendrite of cortical layer 5 pyramidal neurons (Stuart and Spruston, 1998). They applied a hyperpolarizing current pulse to either the soma or the proximal dendrite and recorded the time course of membrane potential on the counterpart. They showed by numerical simulations that a mathematical model hypothesizing that the specific membrane resistance of the dendrite was lower than that of the soma could consistently reproduce the results obtained from the double whole-cell recording. However, the specific membrane resistance of each dendritic part, including the distal dendrite, has not been clarified in detail, since it is hard to apply the whole-cell recording to the distal dendrite. To go beyond the limitation of recording with electrodes or pipettes, Inoue et al. performed an optical recording (Inoue et al., 2001), which is applicable to measurements for the distal dendrite, and measured not only subthreshold excitatory postsynaptic potentials (EPSPs) propagated from the apical dendrite to the soma in hippocampal CA1 slices (anterograde propagation), but also subthreshold EPSPs propagated from the soma to the apical dendrite (retrograde propagation). Comparing the results of optical
recording with the responses of mathematical models under the assumption that the specific membrane resistance is spatially uniform, they estimated one value of specific membrane resistance that reproduces the anterograde propagation and another one that reproduces the retrograde propagation. As numerical results, the estimated value of specific membrane resistance which reproduces anterograde propagation was much different from that which reproduces retrograde propagation. The discrepancy between those two values may have been due to the assumption that the specific membrane resistance is uniform in the dendrite, since the specific membrane resistance is suggested to be non-uniformly distributed. To reveal non-uniform passive membrane property, we estimated a distribution of the specific membrane resistance in a hippocampal CA1 pyramidal neuron including the distal apical dendrite, by means of numerical simulations using multi-compartment models. We used the optically recorded data of the anterograde propagation and the retrograde propagation of the EPSPs, and chose from candidate functions the one which reproduced the results of both the anterograde and retrograde propagations in the most accurate manner.
2.
Results
2.1.
Numerical simulations using a stylized model
In the first part of this section, we used the stylized model for hippocampal CA1 pyramidal neuron, which was used in numerical simulations by Inoue et al. (Fig. 1(a)). To estimate the distribution of specific membrane resistance in a hippocampal CA1 pyramidal neuron, we performed numerical simulations of EPSP propagation for three kinds of distributions—a constant as uniform case (Fig. 1(c)), and also a linear function (Fig. 1(d)) and a step function (Fig. 1(e)) as nonuniform cases—and compared the results of these simulations with the EPSP propagation observed by the optical method.
2.1.1.
Uniform-specific membrane resistance
We considered a case where a distribution of specific membrane resistance is uniform within a single neuron (Rm(x) = R0) to compare this case with non-uniform cases. This is the same distribution used in the estimation done by Inoue et al. (2001). Results of numerical simulations with optimal fitting parameters in constant cases (R0 = 7 kΩcm2, the specific intracellular resistance, Ri = 252 Ωcm) are shown in Fig. 2(a). The left column shows the results for anterograde propagation of an EPSP, whereas the right column shows the results for retrograde propagation of an EPSP. As shown in the left column of Fig. 2(a), the attenuation of an anterograde-propagating EPSP obtained by the optimal model is steeper than that measured by the optical method. Conversely, as shown in the right column of Fig. 2(a), the attenuation of a retrograde-propagating EPSP obtained by the optimal model is less steep than that measured by optical method. Even in the model with the optimal parameters, there was a discrepancy between the numerical results and optical data for those parameter values.
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that the specific membrane resistance obeys a linear function of the distance from the soma: 8 ! > x−h < R1 1−k ; x>h Rm ðxÞ ¼ ð2Þ Lapical −h > : otherwise R1 ; where x is the distance from the soma to each compartment, and Lapical is the distance along compartments between the soma and the most distal compartment in the apical dendrite, and k (0≤k<1) is a constant proportional to the slope of the distribution. In Eq. (2), the specific membrane resistance of a compartments within θ μm of the soma is indicated by R1. The case where k=0 corresponds to a uniform-specific membrane resistance. When the specific membrane resistance obeys a linear function of the distance from the soma, the model has four parameters: three for specific membrane resistance (R1, θ, k) and one for the specific intracellular resistance (Ri). To look for a set of optimal values of parameters under the assumption that the specific membrane resistance is a linear function of distance from the soma, we obtained the dependence of a summed error for both directional propagations on values of parameters: Elinear ðh;kÞ ¼ min R1 ;Ri
Fig. 1 – Multi-compartment models of pyramidal neurons in the hippocampal CA1 region used in numerical simulations: (a) stylized model (Inoue et al. (2001)); (b) 3D model (Magee and Cook (2000)). Functions used as distribution of specific membrane resistance in numerical simulations (c–e); (c) constant; (d) linear function; (e) step function.
To investigate the parameter dependency of this model, we evaluated the discrepancy between the results obtained from optical recording and responses of the model with corresponding values of parameters R0 and Ri. This evaluation was performed by using the summed error Econst (R0, Ri), consisting of an error for the anterograde propagation and that for the retrograde propagation: Econst ðR0 ; Ri Þ ¼
X
econst ðR0 ; Ri Þ y
ð1Þ
yafa;rg
(R0, Ri) is defined in where the error for each propagation, econst y Eq. (9) given in the Experimental procedure (see Section 4). We found that the region where the summed error was between 25 and 30 was optimal. As shown in Fig. 3(a), the summed error was a smooth function of parameters R0 and Ri. This implies that the parameter sensitivity of the model is small, and thus that the discrepancy shown in Fig. 2(a) could be evident over a wide range of parameters.
Linear function
Next we considered the case where the specific membrane resistance changes gradually in a single neuron. We assumed
elinear ðR1 ;h;k;Ri Þ y
ð3Þ
yafa;rg
In Eq. (3), to visualize the four-dimensional parameter space, the summed error is minimized with respect to R1 and Ri and is expressed as a function of θ and k. The range of values of R1 and Ri used in minimization was 5 kΩcm2 ≤ R1 ≤ 150 kΩcm2 and 5 Ωcm ≤ Ri ≤ 100 Ωcm. The obtained dependence is shown in Fig. 3(b). We found the error to be between 25 and 30 in the optimal region. We also found that the summed error showed small changes for different values of the parameters in Fig. 3(b). This implies that parameter sensitivity of the model is small. To make the dependence clearer, the summed error as a function of k is shown in Fig. 3(c), which represents a onedimensional cross section of Fig. 3(b). From Fig. 3(c) we found that steeper gradients k of the specific membrane resistance gives smaller errors. This dependence shows that nonuniformity of the specific membrane resistance affects the propagation of the EPSPs, although the obtained smallest error for the linear function has a value similar to that obtained for the uniform-specific membrane resistance. To reveal the behavior in the optimal case for the linear function, in Fig. 2(b) we show the result obtained by numerical simulations with optimal parameters. As shown in the left column of Fig. 2(b), the attenuation of anterograde-propagating EPSP obtained in the optimal case is steeper than that shown by the optical data. Conversely, as revealed in the right column of Fig. 2(b), the attenuation of retrograde-propagating EPSP obtained in the optimal case is less steep than that shown by the optical data. We can see in Fig. 2(b) that even with the linear function model with the optimal parameters, there was a discrepancy like that seen in Fig. 2(a) with the uniform model.
2.1.3. 2.1.2.
X
Step function
As a third case, we investigated the case of a step function of distance from the soma, where the specific membrane resistance changes more steeply, compared with the linear
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Fig. 2 – Summary of the results of numerical simulations with the best-fitting parameters in each of three cases; (a) constant-specific membrane resistance, (b) a linear function, (c) a step function. The time courses of membrane potential obtained by optical recording are shown by solid lines, while those obtained by numerical simulations are shown by dashed lines. The left column shows results for anterograde propagation of an EPSP, and the right column shows results for retrograde propagation of an EPSP. The depth of the color of each line shows a position where the membrane potential was observed. A black line corresponds to a time course of the membrane potential in an apical dendrite compartment at 330 μm from the soma. Dark gray, medium gray, and light gray lines respectively correspond to compartments at 220, 110, and 0 μm from the soma. As shown in these figures, the best model which could reproduce both anterograde and retrograde propagations of EPSPs was the step function model. The details of parameter dependency of each model were given in the following figures.
function. We assumed that the distribution of the specific membrane resistance obeys the following step function: Rm ðxÞ ¼ R3 þ ðR2 −R3 ÞΘðx−hÞ
ð4Þ
where R2 and R3 are positive constants and Θ(x) is a unit function which takes one for x > θ, and zero otherwise. To determine the optimal values of the parameters for the step function, we investigated the dependence of the summed error for both the anterograde and retrograde propagations. A set of parameters consists of three parameters concerning the specific membrane resistance (R2, R3, and θ), and the specific intracellular resistance Ri. To visualize the four-dimensional parameter space, in Fig. 4(a) we plot the dependence of the following summed error, Estep (R2, R3), a minimized summed error with respected to Ri and θ: Estep ðR2 ;R3 Þ ¼ min h;Ri
X yafa;rg
estep ðR2 ;R3 ;h;Ri Þ: y
ð5Þ
Minimization was performed for 0 ≤ θ ≤ 600 μm and 10 Ωcm ≤ Ri ≤ 200 Ωcm. In Fig. 4(a), we see that the optimal error between 5 and 10 was obtained for log10 R2 ≲ 3 and log10 R3 ≳ 4. One of the optimal results of numerical simulation for the step function are shown in Fig. 2(c), where we see that the results obtained by optical recording for both anterograde and retrograde propagations were reproduced closely by numerical simulations. We set the values of parameters in Fig. 2(c) as follows: θ = 325 μm, R2 = 100.5 Ωcm2, R3 = 106 Ωcm2, Ri = 80 Ωcm. One of characteristic points for estimated optimal parameters was that the specific membrane resistance at the distal dendrite was about a tenth of that at the proximal dendrite. In order to reveal the spatial profile of the specific membrane resistance, we need to clarify what values of threshold θ reproduce the results measured by the optical method. Then we investigated dependence of the summed error on threshold θ. We show the dependence of the summed
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Although the results obtained by numerical simulations for the step function closely reproduced those obtained by the optical method, it is still unclear what level of a change of the specific membrane resistance is needed to reproduce the results obtained by optical methods. To reveal the appropriate
Fig. 3 – Parameter dependency of two models whose distributions of specific membrane resistances were (a) constant and (b, c) linear function. (a) Dependence of the summed error on specific membrane resistance R0 and specific intracellular resistance Ri, obtained for the constant-specific membrane resistance. (b) Dependence of the summed error on k and θ, obtained for the linear function. The value of the summed error is indicated by a specific color. Lighter colors correspond to smaller summed errors. (c) One-dimensional cross section of panel (b). The errors of these models are larger than that of step function model shown in Fig. 4.
error on threshold and specific intracellular resistance Ri in Fig. 4(b). In Fig. 4(b), we see that an optimal region is obtained for θ ∼ 325 μm.
Fig. 4 – Parameter dependency of a model whose specific membrane resistance obeyed a step function. (a) Dependence of summed error on specific membrane resistances R2 and R3; (b) dependence of summed error on threshold θ and specific intracellular resistance Ri. The error of this model was smaller than ones of constant and linear function models shown in Fig. 3. (c) Dependence of summed error for sigmoid function on slope of change β. The validity of step function as distribution of specific membrane resistance is also confirmed by fitting using sigmoid function.
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level of the slope, we performed numerical simulations for the case in which the specific membrane resistance obeys a sigmoid function of the distance from the soma: Rm ðxÞ ¼ R3 þ
R2 −R3 1 þ exp½−bðx−hÞ
ð6Þ
propagations. These results show that constant and linear functions are inadequate, but that a steeper change of the specific membrane resistance is needed. Thus the step function is appropriate as the distribution of the specific membrane resistance in single hippocampal CA1 pyramidal neurons.
where the slope around the threshold is denoted by β (>0) [μm− 1]. We notice that Rm(x) corresponds to the step function when β → ∞. The values of other parameters were set as follows: θ = 325 μm, R2 = 5 Ωcm2, R3 = 300 kΩcm2 and Ri = 80 Ωcm. The dependence of the summed error on the slope β is shown in Fig. 4(c). We see that the summed error initially decreases and approaches to a constant value as β increases. These results shown in Figs. 4(a), (b) and (c) imply that specific membrane resistance decreases rapidly about 300 μm from the soma, and that the distal dendrite is much leakier than the proximal dendrite. That is, R2 is less than about 103 Ωcm2 and R3 is greater than about 104 Ωcm2.
2.1.4. Comparison of results for different distributions of specific membrane resistance To estimate the distribution of the specific membrane resistance in a hippocampal CA1 pyramidal neuron, we compared the results of the stylized model for three different distributions of the specific membrane resistance. For uniform-specific membrane resistance, a discrepancy existed between the results obtained by the optical methods and those obtained by the numerical simulations, even with the optimal set of parameters, as shown in Fig. 2(a). This shows that uniform-specific membrane resistance is not appropriate, and non-uniformity of the specific membrane resistance should be needed to be considered. However, for the linear function, a case of non-uniform-specific membrane resistance with a relatively moderate change, a discrepancy also exists between the results obtained by numerical simulations and those measured by the optical method, even with optimal parameters, as shown in Fig. 2(b). Assuming that the distribution of the specific membrane resistance obeys a step function, the results obtained by the optical method are consistently reproduced by the numerical simulations, as shown in Fig. 2(c). In fact, numerical simulations with the step function could give smaller summed error than those with the constant and the linear function, as shown in Figs. 3 and 4. The scale relation between true membrane potentials and optically observed data might be non-linearly deformed, so we need to evaluate the validity of our results by using other measures. The peak time is a robust measure against such a scale deformation. Here we also estimated a peak time from onset of the EPSP. The peak time of the EPSP for the anterograde propagation optically observed at 0 μm from the soma may be about 20 ms from its onset, and that for the retrograde propagation observed at 330 μm from the soma may be about 13 ms from its onset, as shown by the solid lines of Fig. 2. Corresponding peak times obtained by numerical simulations in Fig. 2 are about 15.75 ms (constant), 15 ms (linear function), 21.5 ms (step function) for the anterograde propagation, and 17.5 ms (const) 14.75 ms (linear function) 14 ms (step function) for the retrograde propagation. Comparing these results, we find that peak times in the step case are most similar to those of the voltage imaging for both
Fig. 5 – Similar results using the stylized model were also obtained by using the 3D model. Dependence of summed error between results obtained by optical method and those obtained by numerical simulation using 3D model for step function; (a) constant-specific membrane resistance, (b) and (c) step function as distribution of specific membrane resistance.
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The dependence of the summed error for both the anterograde and retrograde propagations on parameters of the distribution shown in Fig. 4(b) implies that the specific membrane resistance changes abruptly around θ ∼ 325 μm, which may correspond to the distal dendrite. These results show that a steep decrease in the specific membrane resistance occurs in the distal dendrite. The slope of the decrease was investigated by assuming that the specific membrane resistance obeys a sigmoid function of distance from the soma. By numerical simulation, we found that as shown in Fig. 4(c), a smaller error is obtained with larger values of β. These results imply that the specific membrane resistance decreases drastically at the distal dendrite.
2.2.
Numerical simulations using a 3D model
To verify the generality of results obtained above, in the second half of this section, we performed numerical simulations using another multi-compartment model: a 3D model reconstructed by Magee and Cook (2000), which is described by the absolute position and direction of each compartment, as
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shown in Fig. 1(b). Numerical simulations using a second multi-compartment model would be necessary to clarify whether the results for the multi-compartment model shown in the previous subsection are general features of single pyramidal neurons in the hippocampal CA1 region, or limited features restricted to a specific hippocampal CA1 pyramidal neuron. As well as numerical simulations for the stylized model in Subsection 2.1, we considered three kinds of distributions of the specific membrane resistance, a constant, a linear function, and a step function, to estimate the distribution of the specific membrane resistance.
2.2.1.
Uniform-specific membrane resistance
We performed numerical simulations for a 3D model of a hippocampal CA1 pyramidal neuron with constant-specific membrane resistance. To find optimal values of the parameters R0 and Ri, the dependence of the summed error on R0 and Ri for both the anterograde and retrograde propagations in case of the constant-specific membrane resistance, is shown in Fig. 5(a). We found from results shown in Fig. 5(a) that there is an optimal area and that the error in the optimal area was
Fig. 6 – Time courses of membrane potential obtained by numerical simulations using other sets of optical data instead of learning data. If the estimated distribution is a general feature of single pyramidal neurons in the hippocampal CA1 region, it could reproduce other sets of optical data. Four data sets indicated by solid lines in these figures were obtained by optical recording for four different slices. The dashed lines superimposed on these experimental results show the results reproduced by an optimal model estimated from the single data set shown in Fig. 2. Upper: results for anterograde propagation of EPSPs in two slices. Lower: results for retrograde propagation of EPSPs in another two slices. The specific membrane resistance was assumed to obey a step function of distance from the soma.
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between 15 and 20. The optimal values of the parameters were Ri = 58 Ωcm and R0 = 8 kΩcm2. Even with the optimal values of the parameters, the results measured by the optical method were not reproduced in numerical simulations with the constant-specific membrane resistance (data not shown), which is a similar tendency to stylized model with constantspecific membrane resistance, shown in Fig. 2(a). The optimal value of the specific membrane resistance R0 was about 10 kΩcm2, which is similar to the value obtained for the stylized model shown in Fig. 3(a). The dependence of the summed error obtained for the 3D model shown in Fig. 5(a) is similar to that obtained for the stylized model, shown in Fig. 3(a).
2.2.2.
Linear function
As a first case of the non-uniform case, we performed numerical simulations using the 3D model with a linear function as the distribution of the specific membrane resistance. We found that the optimal error for the linear function was between 15 and 20. We also found that larger k values had smaller errors (data not shown). The dependence of the summed error on k is similar to that obtained using the stylized model for the linear function.
2.2.3.
Step function
We show the dependence of the error on R2 and R3 for the step function in Fig. 5(b). In Fig. 5(b), we see that an optimal summed error was obtained between 0 and 5 for log10 R2 ≲ 3 and log10 R3 ≳ 4. The results in Fig. 5(b) are similar to the results obtained by numerical simulations for the step function using the stylized model shown in Fig. 4(a). The dependence of the summed error on θ and Ri is shown in Fig. 5(c). We found that the summed error decreased for 400 μm ≤ θ ≤ 550 μm. This was similar to Fig. 4(b) obtained for the stylized model.
2.3. Comparison between results for different multi-compartment models In the estimation of the parameters, we used a set of optical data recorded from only one slice in vitro. According to learning theory and system identification (Hertz et al., 1991), however, over-fitting might occur when parameters are estimated from a small set of learning data. Thus, to avoid over-fitting in our estimation, whether the estimated distribution holds for the other optical data set recorded from several slices must be confirmed. For this purpose, we also showed the results obtained by numerical simulations using other data sets from two different slices for each directional EPSP propagation. If the estimated distribution is a general feature of single pyramidal neurons in the hippocampal CA1 region, our estimated distribution should reproduce other sets of optical data. We used the 3D model and set the same values of parameters for the two slices: θ = 400 μm, R2 = 102.25 Ωcm2, R3 = 105 Ωcm2, and Ri = 40 Ωcm. These values are included in the optical area in Figs. 5(b) and (c). The results for the anterograde and retrograde propagations in other slices are shown in Fig. 6. As shown there, we found that EPSP propagations observed using the optical method in these different slices were reproduced well in numerical simulations. This reveals that the estimated distribu-
tion might not be a limited feature of a specific data set, but a general feature of hippocampal CA1 pyramidal neurons.
3.
Discussion
We estimated that the specific membrane resistance drastically decreases in the apical dendrite in the hippocampal CA1 pyramidal neuron by numerical simulations using multicompartment models. In the estimated distribution, the steep decrease in the specific membrane resistance occurs around 300–500 μm away from the soma, suggesting that the specific membrane resistance in the distal area is less than that in the proximal area. Since optically observed propagation of the EPSP was reproduced with the step function for both the stylized model and the 3D model, the estimated distribution with a steep change could be a general feature of hippocampal CA1 pyramidal neurons. A previous study of the double whole-cell recordings in CA1 slices indicated that the CA1 pyramidal neurons have leaky apical dendrites (Golding et al., 2005). The estimation in the present paper was based on the data of the optical recording, which has spatial resolution potentially higher than that of the double whole-cell recording. Thus the result of our estimation is consistent with the leaky apical dendrites of CA1 pyramidal neurons found in the previous study. Moreover, our results further indicate that the change of the specific membrane resistance in the apical dendrite of the CA1 neurons is steep. Since the hyperpolarization-activated cation channel is active near the resting state of neurons, the low value of the specific membrane resistance at distal dendrite might be due to Ih (Stuart and Spruston, 1998). In the known distribution of Ih, however, the hyperpolarization-activated cation conductance changes gradually rather than steeply, and therefore could not explain the stepwise change found in the present study. Thus another conductance, like leak conductance, may be associated with the estimated stepwise change in the distribution of specific membrane resistance. The spatial profile of specific membrane resistance should have a marked effect on synaptic integration, since the specific membrane resistance at each part of the dendrite has an influence on the generation of a PSP, as well as on the spatio-temporal interaction between PSPs (Gulledge et al., 2005). Backpropagating action potentials are known to play an important role in synaptic integration and synaptic plasticity, and the spatial profile of the specific membrane resistance may affect the behavior of backpropagating action potentials. To reveal computational role of the non-uniform-specific membrane resistance, the effect of the non-uniformity of the specific membrane resistance on backpropagating action potential should be clarified. We left this problem as a future work. Stuart and Spruston (1998) have compared the results of double whole-cell recordings from layer 5 cortical pyramidal neurons with the results of numerical simulations and showed that numerical simulations for a sigmoid function reproduce the results obtained by whole-cell recording, and that the estimated position of change in the specific membrane resistance was around 400 μm from the soma. For the
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layer 5 cortical neuron, this position may correspond to a middle part of the apical trunk. In our estimation based on optical recording, the estimated decrease position was around 300 μm to 500 μm away from the soma, in the apical dendrite of the hippocampal CA1 pyramidal neuron. Our estimation for hippocampal CA1 pyramidal neurons suggests that the specific membrane resistance of the distal dendrite is less than that of the proximal dendrite. It is known that the proximal dendrite of a hippocampal CA1 pyramidal neuron has inputs from CA3 via Schaffer collateral, whereas the distal dendrite has inputs from the entorhinal cortex via the perforant path. This may suggest that there is some functional association of non-uniform membrane properties with two kinds of synaptic inputs to a hippocampal CA1 pyramidal neuron.
4.
Experimental procedure
Numerical simulations were performed for a single-neuron model with morphological data by using the NEURON simulator (Hines and Carnevale, 1997). Two multi-compartment models were used in numerical simulations: that used by Inoue et al. (2001) and that used by Magee and Cook (2000). In the model used by Inoue et al., the neuronal morphology was described by a stylized model (Hines and Carnevale, 1997); morphological information was described only by links between compartments. In the model used by Magee and Cook, the morphology was described using a 3D model: morphological information was described by absolute positions and directions in addition to the links. The stylized model used by Inoue et al. (2001) consists of 130 apical dendrite compartments, 135 basal dendrite compartments, and 14 somatic compartments (Fig. 1(a)). The 3D model used by Magee and Cook (2000) consists of 412 dendritic compartments and one somatic compartment (Fig. 1(b)). These two models have similar dendritic structure; in both models the compartment farthest from the soma on the dendrosomatic axis is about 600 μm away from soma on the corresponding axis. We used the values of diameters and length of each compartment used in Inoue et al. (2001) and in Magee and Cook (2000). The leak current is assumed to be given to each compartment for both models. The parameters in these models were the specific membrane resistance Rm [Ωcm2], the specific intracellular resistance Ri [Ωcm], the specific membrane capacitance C [μF/cm2]. We set the specific membrane capacitance of the compartments of the apical dendrite, CA, and that of the basal dendrite, CB, to CA = CB = 1.6 μF/cm2, and that of the somatic compartments, CS, to CS = 1 μF/cm2 (the large value of specific membrane conductance for the dendrite was assumed in order to take into account the effect of the dendritic spine (Spruston et al., 1999; Inoue et al., 2001)). We performed numerical simulations which reproduced the results of optical recordings of propagations of subthreshold EPSPs observed in hippocampal CA1 slices by Inoue et al. (2001). In numerical simulations, we consider an anterograde and a retrograde propagations of the subthreshold EPSPs observed optically by Inoue et al. (2001). We briefly review the methods of the optical recording performed by Inoue et al.
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(2001). Subthreshold synaptic response was induced by electrically stimulating presynaptic fibers. The stimulus was given to either the stratum lacunosum moleculare or the stratum oriens. The obtained responses were averaged over 16–32 trials. CNQX components were isolated by subtracting responses in a solution with APV, CNQX and bicuculline from those in a solution with only APV and bicuculline. In our study we used time courses of CNQX components as data of propagating EPSPs. To reproduce a propagation of the subthreshold EPSPs from the apical dendrite to the soma, we gave a stimulus to compartments in the apical dendrite, which is 330 μm away from the soma. To reproduce a propagation from the soma to the apical dendrite, we gave a stimulus to the somatic compartments. In our numerical simulations, to mimic an EPSP observed in the optical recordings, we performed voltage clamping in the multi-compartment models by giving the following current as external input to the corresponding compartments, ext ðVi ðtÞ−Vyopt ðtÞÞ I ext y;i ðtÞ ¼ g
ð8Þ
where the command inputs for voltage clamps are the time sequence data of EPSPs optically recorded at their positions in neurons. Vopt is a time course of voltage imaging data. The y direction of an EPSP is indicated by y ∈{a, r} (y = a for the anterograde propagation, y = r for the retrograde propagation), and recorded position was expressed by j ∈ {0, 1, 2, 3} (j = 0, 1, 2, 3 for 0 μm, 110 μm, 220 μm, 330 μm from the soma, respectively). For simulations of the anterograde propagation, we used Vopt a,3 (t) as the command inputs to apical dendritic compartments which is 330 μm away from the soma. For simulations of the retrograde propagation, we used Vopt r,0 (t) as command inputs to somatic compartments. For both propagations, we measured time courses of membrane potentials at (t, x), four positions: the membrane potential at the soma, Vsim 0 and the membrane potentials at three compartments in the apical dendrite: one in a compartment at 110 μm from the (t, x); one in a compartment at 220 μm from the soma, Vsim 1 (t, x); and one in a compartment at 330 μm from soma, Vsim 2 (t, x). x Denotes a vector composed of parathe soma, Vsim 3 meters describing a distribution function of the specific membrane resistance and specific intracellular resistance in the dendrite. We calculated the following squared error between the time course obtained by numerical simulations and that obtained by the optical measurements: ey ðxÞ ¼
N 3 t −1 X X
2 opt sim Vy;i ðnsÞ−Vy;i ðns;xÞ
ð9Þ
n¼0 i¼0
where τ is the sampling period and Nt is the total amount of sampled data. The suffix i denotes an observed position as (t, x), and the suffix opt refers to the optical defined in V sim i recording data. To find the optimal values of parameters consistently satisfying both anterograde and retrograde propagations, we evaluate a discrepancy between results obtained from optical recording and responses of the model with given values of parameters x by using the following summed error: EðxÞ ¼
X yafa;rg
ey ðxÞ:
ð10Þ
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By using the summed error, we clarified the differences in behavior between multi-compartment models with different distributions of the specific membrane resistance. We performed numerical simulations for three kinds of distributions, a constant as uniform case (Fig. 1(c)), and also a linear function (Fig. 1(d)) and a step function (Fig. 1(e)) as nonuniform cases, and determined which distribution of specific membrane resistance best reproduced both the anterograde and retrograde propagations of an EPSP.
Acknowledgments This work was partially supported by Grant-in-Aid for Scientific Research for Young Scientists (B) (Nos. 17700250 (T.O.) and 18700299 (T.A.)), Grant-in-Aid for Scientific Research on Priority Areas (Nos. 18020007 (M.O.) and 18079003 (M.O.)), and Grant-in-Aid for Scientific Research (B) (No. 17300096 (H.M.)), and Grant-in-Aid for Scientific Research (C) (No. 16500093 (M.O.)) from MEXT and JSPS of Japan. T.O. was partially supported by Research Fellowship from Japan Society for the Promotion of Science for Young Scientists.
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