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A cell model study of calcium influx mechanism regulated by calcium-dependent potassium channels in Purkinje cell dendrites
Journal of Neuroscience Methods 129 (2003) 115 /127 www.elsevier.com/locate/jneumeth
A cell model study of calcium influx mechanism regulated by calciumdependent potassium channels in Purkinje cell dendrites Koji Chono a,1, Hiroshi Takagi b, Shozo Koyama b, Hideo Suzuki c, Etsuro Ito a,d,* a
Division of Biological Sciences, Graduate School of Science, Hokkaido University, North 10, West 8, Kita-ku, Sapporo 060-0810, Japan b Department of Physiology-2, Shinshu University School of Medicine, Asahi 3-1-1, Matsumoto 390-8621, Japan c Department of Physics, School of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku ku, Tokyo 169-8555, Japan d Division of Innovative Research, Creative Research Initiative ‘Sousei’ (CRIS), Hokkaido University, North 10, West 8, Kita-ku, Sapporo 060-0810, Japan Received 23 April 2003; received in revised form 26 May 2003; accepted 9 June 2003
Abstract The present study was designed to elucidate the roles of dendritic voltage-gated K channels in Ca2 influx mechanism of a rat Purkinje cell using a computer simulation program. First, we improved the channel descriptions and the maximum conductance in the Purkinje cell model to mimic both the kinetics of ion channels and the Ca2 spikes, which had failed in previous studies. Our cell model is, therefore, much more authentic than those in previous studies. Second, synaptic inputs that mimic stimulation of parallel fibers and induce sub-threshold excitability were simultaneously applied to the spiny dendrites. As a result, transient Ca2 responses were observed in the stimulation points and they decreased with the faster decay rate in the cell model including high-threshold Ca2 -dependent K channels than in those excluding these channels. Third, when a single synaptic input was applied into a spiny dendrite, Ca2 -dependent K channels suppressed Ca2 increases at stimulation and recording points. Finally, Ca2-dependent K channels were also found to suppress the time to peak Ca2 values in the recording points. These results suggest that the opening of Ca2 -dependent K channels by Ca2 influx through voltage-gated Ca2 channels hyperpolarizes the membrane potentials and deactivates these Ca2 channels in a negative feedback manner, resulting in local, weak Ca2 responses in spiny dendrites of Purkinje cells. # 2003 Elsevier B.V. All rights reserved. Keywords: Ca2 channel; Dendrite; K channel; Multi-compartment model; Purkinje cell; Simulation
1. Introduction Studies of cerebellar Purkinje cells have shown that Ca2 spikes are produced locally in dendritic regions, differently from Na spikes produced in somatic regions (Llina´s and Sugimori, 1980a,b; Stuart and Hausser, 1994). Two kinds of excitatory synaptic inputs, one from parallel fibers and the other from a climbing fiber, generate the dendritic Ca2 spikes in a Purkinje cell (Lev-Ram et al., 1992; Miyakawa et al., 1992a;
* Corresponding author. Tel.: /81-11-706-2615; fax: /81-11-7064448. E-mail address: [email protected] (E. Ito). 1 Present address: Applied Pharmacology Research, Pharmacology Laboratories, Institute for Drug Discovery Research, Yamanouchi Pharmaceutical Co., Ltd, Tsukuba 305-8585, Japan. 0165-0270/03/$ - see front matter # 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0165-0270(03)00194-8
Takagi et al., 1992). Ca2 spikes induced by stimulation of parallel fibers are confined to small parts of the spiny dendrites (Eilers et al., 1995), whereas those occur in the proximal dendrite by stimulation of a climbing fiber (Miyakawa et al., 1992a). Associative stimulations of parallel fibers and a climbing fiber induce long-term depression (LTD) in Purkinje cells (Ito et al., 1982), which is built by interaction of signal transduction systems possibly including some immediate early genes between the spiny dendrites connected by parallel fibers and the proximal dendrite activated by inputs of a climbing fiber (Ito, 2001). Such compartmentalization of Ca2 spikes and local changes in intracellular Ca2 concentration ([Ca2]i) in Purkinje cell dendrites are dynamically modulated by voltage-gated K channels (Midtgaard et al., 1993; Midtgaard, 1994; Takagi, 2000). Even though the
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physiological and pharmacological profiles of dendritic K channels are well characterized in Purkinje cells (Housegard and Midtgaard, 1988; Rudy, 1988), the limitations of voltage- and space-clamp techniques employing a somatic electrode to measure ionic currents in dendrites have so far prevented us from elucidating the detailed mechanisms of dendritic K channels in the regulation of [Ca2]i at spiny dendrites (Spruston et al., 1993). In dendrites of hippocampal CA1 pyramidal cells, however, combined experimental and computational approaches revealed many patterns of biophysical behaviors of dendritic K channels (e.g. Ca2-dependent K channels, A-type K (KA) channels, and Dtype K (KD) channels) (Sah and Bekkers, 1996; Hoffman et al., 1997; Takagi et al., 1998, 2000, 2003). Recently, a similar success was acquired in Purkinje cells by these approaches, showing that dendritic Ca2-spike generation is regulated by low voltage-gated (T-like) Ca2 channels and KD channels (Watanabe et al., 1998; Miyasho et al., 2001). Despite such cumulative evidence, the physiological roles of K channels in the regulation of [Ca2]i at spiny dendrites of Purkinje cells have not yet been elucidated. In the present study, to elucidate the physiological roles of voltage-gated K channels in the regulation of Ca2 influx at spiny dendrites of rat Purkinje cells, we applied a computer program to a typical multi-compartment model for the Purkinje cells. In particular, we investigated whether or not Ca2-dependent K channels play the major role in regulating the decay time course of transient Ca2 inward current that is induced by sub-threshold synaptic inputs to these dendrites. The following five features are emphasized. (1) Dendritic function can be analyzed, which is limited when conventional electrophysiological and Ca2 imaging techniques are used because of technical difficulties. (2) We improve the channel descriptions and the maximum conductance in the Purkinje cell model to mimic both the kinetics of ion channels and the Ca2 spikes, which failed in previous studies. (3) The most important channel in the dendritic Ca2 responses can be determined from various K channels. (4) The dynamics of [Ca2]i can be examined in addition to that of Ca2 currents. (5) Effects of weak synaptic inputs to induce sub-threshold excitability on [Ca2]i are particularly noted, because they participate in the initiation of LTD (Eilers and Konnerth, 1997).
2. Materials and methods 2.1. Morphology and passive membrane parameters Computer simulation was made with a simulation program, NEURON (ver. 4.3.1) (Hines and Carnevale, 1997), on a personal computer. The simulations were
Fig. 1. Multi-compartment model for Purkinje cell and action potentials by current injection. (A) Multi-compartment model. (B) Repetitive Na -spike firings recorded in the soma by a current injection of /2.3 nA for 1000 ms into the soma. (C) Expanded presentation of repetitive Na spikes in the soma by injection of current of /1.7, /2.0, or /2.3 nA.
run with a time step of 25 ms. The morphology of the cell model was based on a detailed reconstruction of a Golgi-stained Purkinje cell from an adult rat (Fig. 1A) (Shelton, 1985). The model contained 1089 compartments (soma: 1 compartment, smooth dendrites: 85 compartments, and spiny dendrites: 1003 compartments). A specific membrane capacitance of 0.9 mF/ cm2 was used for the soma and smooth dendrites (Hines and Carnevale, 1997), whereas that of 1.5 mF/cm2 was used for the spiny dendrites (De Schutter and Bower,
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g(V ; [Ca2 ]i ; t) gm(V ; t)p h(V ; t)q z([Ca2 ]i ; t)r ; @m=@t am (1 m) bm m; @h=@t ah (1 h) bh h; @z=@t (z z)=tz ; am (V ) A=[Bexpf(V C)=Dg]; bm (V ) E=[F expf(V C)=Dg]; ah (V ) A=[Bexpf(V G)=Hg]; bh (V ) E=[F expf(V G)=Hg]; z 1=(1A=[Ca2 ]i ); tz B:
1994a,b; Miyasho et al., 2001). Membrane resistivity was set to 10000 V cm2 and specific intracellular resistivity set to 250 V cm (Miyasho et al., 2001). Alpha synaptic inputs were applied to spiny dendrites in our cell model. Here, a synapses refer to the shape of the function (an a function with an exponential form) used to model the membrane conductance change in the postsynaptic site of neuron resulting from the binding of neurotransmitter released from the pre-synaptic neuron. The time course and amplitude of this synaptic input were expressed as a conductance with a single-exponential time course of the form gmax(t /t )exp{ /(t /t )/t }, where t /0 at the onset of the synaptic conductance; gmax is a maximum value of the synaptic conductance; t is time to reach gmax. These values were set as t /0.1 ms, gmax /1.68 mS/cm2 to induce sub-threshold excitability. When simultaneous synaptic inputs were applied into some spiny dendrites, nine points were selected in the spiny dendrites to mimic responsive points recorded by stimulation of parallel fibers (Eilers et al., 1995).
The descriptions of three channels, delayed rectifier K (Kdr) channel, anomalous rectifier K (Kar) channel, and persistent K (KM) channel, were identical to those used in Miyasho’s model (Miyasho et al., 2001). The descriptions of fast Na channel, persistent Na channel, KA channel, KD channel, high-threshold Ca2-dependent K (KC) channel, low-threshold Ca2-dependent K (K2) channel, T-type Ca2 channel, P-type Ca2 channel, and class-E Ca2 channel were slightly modified from those in Miyasho’s model. The rationale for these modifications is given in Section 3, and their details are discussed in Section 4. To describe the Ca2-dependent gating of Ca2-dependent K channels and to update correctly the equilibrium potential of Ca2 in physiological conditions, it was necessary to calculate [Ca2]i. Because any crucial factors required to describe intracellular Ca2 diffusion, buffering release, and extrusion are not yet known, we did not attempt to realistically simulate intracellular Ca2 dynamics. Instead, we assumed that Ca2, which
2.2. Ion channel kinetics We incorporated 12 types of voltage-gated ion channels into our cell model. To describe the kinetics of these channels, we slightly modified the channel descriptions from those in a previous model (Miyasho et al., 2001), which had been determined on the basis of the parameters offered by De Schutter and Bower (1994a,b). The equations and parameters used in our cell model are listed below and in Table 1. Table 1 Parameters for ion channels in the model Channel
Er (mV)
NaF
45
NaP CaT
57 140
CaP CaE
135 135
Kdr Kar KM KA
/85 /30 /85 /85
KD
/85
KC
/85
K2
/97.5
Factor
p, q, r
A
m h m m h m m h * * * m h m h m z m z
3 1 3 2 1 1 1 1
20.88 /9 24.48 3.75 0.0714 8.5 2.6 0.0025
4 1 1 1 1 2 1 2
1.4 0.0175 8.5 0.0015 am /7.5 400 a /25.0 20
B
C
D
E
F
G
H
1 1 1 1 1 1 1 1
58 90 68 50 79 /8 7 32
/3 6 /3 /7.5 4.45 /12.5 /8 8
/26.8 4.5 /30.8 0.5151 0.07905 35 0.18 0.19
1 1 1 1 1 1 1 1
67 35 77 51 85 74 26 42
20 /10 20 4.4 /4 14.5 4 /10
1 1 1 1 / 10 / 10
27 50 17 89 / / / /
/12 8 /12.5 8 / / / /
0.49 1.3 35 0.0055 0.11 / 0.075 /
1 1 1 1 0 / 0 /
30 13 99 83 /35 / 5 /
4 /10 14.5 /8 14.9 / 10 /
Er, reversal potentials; A /H , parameters used in methods; *, the descriptions of these three channels, Kdr, Kar, and KM, were identical to those used in Miyasho’s model (see Miyasho et al., 2001 for details); NaF, fast Na ; NaP, persistent Na ; CaT, T-type Ca2 ; CaP, P-type Ca2 ; CaE, class-E Ca2 ; Kh, anomalous rectifier; Kdr, delayed rectifier; KM, persistent K ; KA, A-type K ; KD, D-type K ; KC, high-threshold K ; K2, low-threshold K .
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enters through Ca2 channels instantaneously, diffused within a thin (0.1 mm thick) sub-membrane shell and that there were two Ca2 extrusion mechanisms: the Ca2 pump and single-exponential decay mechanism. The change in [Ca2]i was expressed as the ratio of Ca2 concentrations between non-stimulated model and stimulated model under the same parameter conditions. 2.3. Statistical analysis
injection of /2.3 nA (Fig. 1B, C). The input impedance of our cell model with a realistic morphology was calculated to be 16.7 MV under normal conditions and to be 22.7 MV when Cs-sensitive K (Kar) channels were removed. Note here that only Kar channels are open around the resting membrane potential. These values are in good agreement with those measured electrophysiologically (Llina´s and Sugimori, 1980a; Rapp et al., 1994). 3.2. Ca2 and Na spikes
Data were expressed as mean9/S.E.M. Differences between groups were examined for statistical significance by the Student t-test. A P-value less than 0.05 denoted the presence of a statistically significant difference.
3. Results 3.1. Reconstruction of experimental results To find the optimum set of parameters to reconstruct the electric properties of real Purkinje cells without loss of calculation time, we started with Miyasho’s model and explored the parameter space in the vicinity of his values. Although Miyasho’s model was suitable for reproducing the depolarization-induced firing pattern of Purkinje cells, there was a need to improve the activation and inactivation profiles of some channels, for example Kdr and KM channels. We therefore modified the channel descriptions, as mentioned in Section 2 (Table 1), and further changed the maximum conductance (gmax) of T-type Ca2 channel, P-type Ca2 channel, class-E Ca2 channel, KA channel, and KD channel (Table 2), until we obtained qualitatively realistic responses. For example, repetitive Na -spike firings could be obtained in the soma by a current Table 2 Maximal channel conductance (mS/cm2) Channel
Soma
Dendrite
NaF NaP CaT CaP CaE Kdr Kar KM KA KD KC K2 Leak
10 1 0 0.5 0 36 0.3 0.04 15 0 0 0 0.21
0 0 0.15 0.4 0.4 0.6 0 0.01 8 9 60 0.39 0.21
Abbreviations for channels are the same as in Table 1.
Ca2 and Na spikes were repetitively produced locally in the dendrites and soma in our cell model, respectively, by a current injection of /2.3 nA for 1000 ms into the soma (Fig. 2). Although P-type Ca2 channel is responsible for Ca2-spike firings in dendrites (Miyakawa et al., 1992a; Watanabe et al., 1998), no information from previous experiments is available to allow determination of the conductance distribution (i.e. gmax) of the P-type Ca2 channels. We therefore assumed that in our cell model the P-type Ca2 channel was distributed uniformly over the dendrites, as in Miyasho’s model. The density distributions of other ion channels were also uniform over the dendrites in our cell model. 3.3. Stability of intracellular Ca2 concentration Because a numerical integration method is used as an algorithm in the present simulation program, NEURON, we must consider that the time for membrane potential and [Ca2]i can reach stable values (see Hines and Carnevale, 1997 for details). For example, the membrane potentials of the soma, smooth dendrites, and spiny dendrites became /77.08, /77.06 mV (n /7; the error value is /0.01), and /77.05 mV (n /7; the error value is /0.01), respectively, 2.5 s after onset of simulation (Fig. 3A). We therefore started to simulate the changes in [Ca2]i in spiny dendrites 2.5 s after onset of simulation (Fig. 3B). Even in this condition, the Ptype Ca2 channel was found to function well in the spiny dendrites (Fig. 3C), in comparison with the experimental data (Regan et al., 1991; Mintz et al., 1992a,b). 3.4. Changes in intracellular Ca2 concentration in spiny dendrites by a synaptic inputs We examined the changes in [Ca2]i by a synaptic inputs, which were simultaneously given to nine points in spiny dendrites extended from one smooth dendrite (Fig. 4A). As mentioned in Sections 1 and 2, the synaptic inputs used here were set to be weak to mimic their Ca2 responses recorded by stimulation of parallel fibers (Eilers et al., 1995), so that no Na2 spikes but
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Fig. 2. Locally produced Ca2 and Na spikes in dendrites and soma. (A, B) Ca2 and Na currents recorded in the soma by a current injection of /2.3 nA for 1000 ms into the soma. (C, D) Ca2 and Na currents recorded in the dendrite by a current injection of / 2.3 nA for 1000 ms into the soma.
excitatory postsynaptic potentials were observed (data not shown). Transient increases in [Ca2]i were observed in these nine points and these responses decreased in exponential fashion, whereas no responses were observed in any other spiny dendrites (Fig. 1B). The decay time of [Ca2]i from the peak to 1/e was 180.79/ 17.7 ms in these nine points (Fig. 4B). These Ca2
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Fig. 3. Stability of membrane potentials and intracellular Ca2 concentration in stimulation process. (A) Changes in membrane potentials of the soma, smooth dendrite, and spiny dendrite after onset of simulation. (B) Change in [Ca2 ]i in the spiny dendrite after onset of simulation. The unit of [Ca2 ]i is arbitrary. (C) Ca2 currents in the spiny dendrite. Following the resting potential ( /77 mV), the membrane potential of soma was stepped to a test potential ranging from /60 to 0 mV at 10 mV increments by a 100 ms pulse. These data were recorded 2.5 s after onset of simulation, showing that P-type Ca2 channels function without any inactivation.
responses remained mainly around the synaptic input points, even if they interfered with each other (Fig. 5). These results completely mimicked the previous data reported by a combination of electrophysiological and Ca2 imaging methods (Eilers et al., 1995), and confirmed the validity of our cell model. Here we could reproduce the transient and local increases in [Ca2]i in Purkinje cell dendrites elicited by synaptic inputs that are equivalent to stimulation of parallel fibers.
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model, KC channels lead the fast decay rate for Ca2 in spiny dendrites (time from peak to 1/e /187.09/18.6 ms (n /9), P B/0.005 vs. the model including none of K channels, Fig. 6B). In contrast, K2 channels and a combination of KC and K2 channels did not evoke such a fast decay rate (K2: time from peak to 1/e /252.99/3.7 ms (n /9); KC/K2: 211.49/26.0 ms (n /9), P /not significant vs. the model including none of K channels, Fig. 6C, D). The differences in the peak values of Ca2 responses between the models including or excluding KC channels will be compared later (see Fig. 7). On the other hand, K channels independent of Ca2 currents (e.g. Kdr, KA, KD, and KM channels) were not involved in the fast decay rate for Ca2 (Kdr: time from peak to 1/e /256.49/4.7 ms; KA: 252.39/3.5 ms; KD: 251.59/3.4 ms; KM: 253.69/3.6 ms, n /9 each, P /not significant vs. the model including none of K channels, Fig. 6E /H). 3.6. Effects of Ca2-dependent K channels on Ca2 propagation in dendrites by single a synaptic input
Fig. 4. Changes in intracellular Ca2 concentration in spiny dendrites by a synaptic inputs. (A) Nine synaptic input points (S1 /S9) in the spiny dendrites extended from one smooth dendrite and another spiny dendrite (O) without a synaptic input. (B) Changes in [Ca2 ]i in the spiny dendrites. The synaptic inputs (t /0.1 ms, gmax /1.68 mS/cm2) were applied 2.5 s after onset of simulation to S1 /S9 simultaneously, but not to O, to evoke sub-threshold excitability. The Ca2 change is presented as a ratio of Ca2 concentrations between non-stimulated model and stimulated model under the same parameters.
3.5. Effects of K channels on regulation of intracellular Ca2 concentration in spiny dendrites When all voltage-gated K channels were excluded from the spiny dendrites in our cell model, the decay rate was slower than otherwise (time from peak to 1/e / 252.29/3.8 ms (n /9), P B/0.005, Fig. 6A). We thus examined which voltage-gated K channel acts in such a fast decay-rate mechanism for [Ca2]i. In our cell
To examine the effects of Ca2-dependent K channels on Ca2 propagation in dendrites, we applied a single a synaptic input to a single spiny dendrite in some models including all K channels, KC and K2 channels, KC channels, or none of K channels (Fig. 7). Here, we should note the two Ca2-dependent K channels, i.e., not only KC channels but also K2 channels, because of the following two reasons, even though KC channels act more effectively than K2 channels in the decay of Ca2 responses (Fig. 6B, C). (1) Both these two channels are expressed by the same channel kinetics description. (2) The absolute values for the Ca2 sensitivities for these channels are unknown, even though their relative difference was taken account upon simulation. We selected five types of dendrites from our cell model and applied a synaptic input into a spiny dendrite in each model (Fig. 7A1, B1, C1, D1, and E1). The peak values of Ca2 observed in the stimulation point and some recording points, whose adjacent distances were 2 /6 mm, showed that KC channels suppressed the increases in [Ca2]i at all the points of stimulation and recording in all dendrites (14.5% decrease in the KC channels-including models in comparison with none of K channels at the stimulation point) (Fig. 7A2, B2, C2, D2, E2). Interestingly, Ca2 level was maintained at the first recording point and decreased gradually at the sequent recording points in a fashion of reciprocal or exponential to the distance (Fig. 7A2, B2, C2, D2, and E2). The time to reach peak [Ca2]i was compared in the five dendrite models (Fig. 7A3, B3, C3, D3, and E3). In some models, KC channels delayed the time for [Ca2]i increase, in other words, KC channels suppressed the propagation of Ca2 in dendrites. At the synaptic input
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Fig. 5. Pseudo-color presentation of Ca2 concentration changes in spiny dendrites by a synaptic inputs. The protocol is the same as in Fig. 4. Numerals represent time (ms); 0 is the timing of the synaptic inputs.
points in all the dendrite models, [Ca2]i reached to peak values in about 10 ms. 3.7. Effects of Ca2-dependent K channels on activation and inactivation gating variables of P-type and T-type Ca2 channels in dendrites by single a synaptic input To elucidate the mechanisms determining the Ca2 dynamics in dendrites, we calculated the factors m (voltage-dependent activation gating variable) and h
(voltage-dependent inactivation gating variable) of Ptype and T-type Ca2 channels. Alpha synaptic inputs were simultaneously given to nine points in spiny dendrites extended from one smooth dendrite (the same positions as shown in Fig. 4A). The changes in the m value of P-type Ca2 channel were compared between the presence and the absence of Ca2-dependent K channels, such as KC and K2 channels, in dendrites. In the absence of these Ca2dependent K channels, the peak value and the integrated value from the onset of stimulation to the
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Fig. 6. Effects of K channels on regulation of intracellular Ca2 concentration in spiny dendrites by a synaptic inputs. (A) Changes in [Ca2 ]i in the cell model without any voltage-gated K channels. The protocol is the same as in Fig. 4. (B) The model including high-threshold Ca2 dependent K channels. (C) The model including low-threshold Ca2 -dependent K channels. (D) The model including both high-threshold and low-threshold Ca2 -dependent K channels. (E) The model including delayed rectifier K channels. (F) The model including persistent K channels. (G) The model including A-type K channels. (H) The model including D-type K channels.
time point returning to the resting state for the m value were 16 and 41% larger than the cases when the Ca2dependent K channels existed in dendrites, respectively (Fig. 8A and B). When the same comparisons were performed for T-type Ca2 channel, the peak value and the integrated value for the m value were 4 and 8% larger in the absence of Ca2-dependent K
channels than those in their presence in dendrites, respectively (Fig. 8C and D). Furthermore, we compared the changes in the h value of T-type Ca2 channel in the presence of Ca2dependent K channels in dendrites with those in the absence of them. In the absence of Ca2-dependent K channels, its value in the resting state was significantly
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Fig. 7. Effects of Ca2 -dependent K channels on peak values of Ca2 concentration changes and time to reach these peak values in spiny dendrites by single a synaptic input. (A1, B1, C1, D1, E1) Five dendrite models selected from Fig. 1A. ‘S’ represents the synaptic input point; ‘R1’ to ‘R6’ show the recording points. (A2, B2, C2, D2, E2) Peak values of changes in [Ca2 ]i recorded at the synaptic input and recording points. All, KC/K2, KC and none represent the dendrite models including all K channels, both high-threshold (KC) and low-threshold (K2) Ca2 -dependent K channels, high-threshold (KC) Ca2 -dependent K channels, and none of K channels, respectively. (A3, B3, C3, D3, E3) Time to peak Ca2 values recorded at the synaptic input and recording points.
larger than that in the presence of these channels (Fig. 8E, F). This fact possibly depended on the hyperpolarization of membrane potential because of the absence of Ca2-dependent K channels. Shortly after the onset of stimulation, the h value of T-type Ca2 channel was extremely increased in the absence of Ca2-dependent K channels in comparison with the case of their presence. Note here that inactivation of P-type Ca2 channels must not be taken into account (Table 1).
3.8. Effects of high-threshold Ca2-dependent K channels on activation and inactivation gating variables of Ca2 channels in dendrites shortly after application of single a synaptic input We calculated the changes in the membrane potential of spiny dendrites of the cell model including only KC channels and in that excluding voltage-gated K channels, in the case of simultaneous a synaptic inputs
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Fig. 8. Effects of Ca2 -dependent K channels (KC and K2) on activation and inactivation gating variables of P-type and T-type Ca2 channels in spiny dendrites by single a synaptic inputs. The protocol is the same as in Fig. 4. (A) Change in the activation gating variable (m ) of T-type Ca2 channel in the presence of Ca2 -dependent K channels. (B) Change in the m value of T-type Ca2 channel in the absence of Ca2 -dependent K channels. (C) Change in the m value of P-type Ca2 channel in the presence of Ca2 -dependent K channels. (D) Change in the m value of P-type Ca2 channel in the absence of Ca2 -dependent K channels. (E) Change in the inactivation gating variable (h ) of T-type Ca2 channel in the presence of Ca2 -dependent K channels. (F) Change in the h value of T-type Ca2 channel in the absence of Ca2 -dependent K channels.
to nine points (Fig. 4A). The average membrane potentials 1 ms after synaptic inputs recorded at these nine points were /63.3 and /29.6 mV in the cell model with only KC channels and in that without any K channels, respectively. Then, using these values, we calculated the factors m and h in the above two cell
models. The m values of P-type Ca2 channel were 2.5 /10 3 and 2.0 /10 1 in the cell model with only KC channels and in that without any K channels, respectively. Note here that inactivation of P-type Ca2 channels must not be taken into account (Table 1). The m values of T-type Ca2 channel were 0.61 and 1.00 in
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the cell model with only KC channels and in that without any K channels, respectively. The h values of T-type Ca2 channel were 2.3 /10 2 and 1.5 /10 6 in the cell model with only KC channels and in that without any K channels, respectively. In addition, the m values of class-E Ca2 channel were 1.3 /10 2 and 5.3 /10 1 in the cell model with only KC channels and in that without any K channels, respectively. The h values of class-E Ca2 channel were 1.1 /10 1 and 7.2 /10 3 in the cell model with only KC channels and in that without any K channels, respectively.
4. Discussion We studied the physiological roles of K channels in the regulation of Ca2 influx at spine dendrites of rat Purkinje cells, when a synaptic inputs were applied to evoke sub-threshold excitability, using the computer simulation program, NEURON. These synaptic inputs were equivalent to stimulation of parallel fibers and known to take part in the induction of LTD. First, we improved the channel descriptions and the maximum conductance in the cell model to mimic both the kinetics of ion channels and the Ca2 spikes, which had failed in the previous simulation studies (Figs. 1 /3). Then, our simulation results showed that the local Ca2 influx in the spiny dendrites of Purkinje cells is regulated by action of Ca2-dependent K channels, particularly high-threshold Ca2-dependent K (KC) channels (Figs. 4 /8). 4.1. Deactivation of voltage-gated Ca2 channels by opening of Ca2-dependent K channels Because the change in the peak value of [Ca2]i at the synaptic input point corresponds to that in the peak value of Ca2 current, suppression of the peak values of [Ca2]i in the dendrite models including Ca2-dependent K channels (Figs. 6 and 7) suggests that these Ca2-dependent K channels suppress Ca2 current. Furthermore, the decay rate of [Ca2]i was fast in the cell model including Ca2-dependent K channels (Fig. 6). Taken together with the data that the time to reach the peak values of [Ca2]i is about 10 ms (Fig. 7), these results suggest that voltage-gated Ca2 channels are deactivated by the opening of Ca2-dependent K channels. In other words, the following scheme of events can be proposed (Fig. 8). (1) Synaptic inputs open voltage-gated Ca2 channels in spiny dendrites of Purkinje cell. (2) Ca2 enters into the dendrites through Ca2 channels. (3) Before the [Ca2]i reaches its peak value (B/10 ms), Ca2-dependent K channels are activated. (4) Ca2-dependent K channels hyperpolarize the dendrite membrane. (5) Finally, voltage-gated Ca2 channels are deactivated.
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Event (3) above is possible, because previous experiments showed activation of Ca2-dependent K current within 10 ms in depolarized Purkinje cells (Shimoda et al., 1994). We proved events (4) and (5) in Sections 3.7 and 3.8. The values of activation and inactivation gating variables of Ca2 channels observed shortly after stimulation clearly showed that voltage-gated Ca2 channels in the cell model including KC channels are strongly suppressed even 1 ms after synaptic inputs, confirming that Ca2-dependent K channels hyperpolarize the dendrite membrane, which is followed by rapid deactivation of voltage-gated Ca2 channels. Further, the deactivation, but not the inactivation, of P-type and T-type Ca2 channels plays the major role in the Ca2 dynamics in dendrites. An influence by the inactivation of T-type Ca2 channels is small. Such importance of the deactivation mechanism in P-type and T-type Ca2 channels can be also recognized by the result that the activation gating variables of these Ca2 channels changed shortly after the onset of stimulation, that is the onset of membrane potential changes (Fig. 8). 4.2. Synaptic plasticity regulated by Ca2 influx due to opening of Ca2-dependent K channels Such regulation of Ca2 influx by opening of Ca2dependent K channels can be applied to explain the initiation mechanism of long-term potentiation (LTP). For example, when an inhibitor of afterhyperpolarization channels, which are considered to be equivalent to Ca2-dependent K channels, is applied to CA1 pyramidal neurons of the hippocampus, weak tetanus stimulation, which usually failed to evoke LTP, could evoke LTP (Bekkers, 2000; Sah and Bekkers, 1996; note here that recent work also revealed that small-conductance Ca2-dependent K channels modulate membrane excitability in rat Purkinje cells due to afterhyperpolarization (Cingolani et al., 2002).). The initiation of LTP in dendrites of pyramidal neurons requires the generation of back-propagation of action potentials in dendritic regions (Magee and Johnston, 1997). The amount of Ca2 influx through voltagegated Ca2 channels activated by action potentials was much larger than that through NMDA receptors (Miyakawa et al., 1992b). Taken together, the above initiation of LTP is considered to be due to inhibition of Ca2-dependent K channels, suggesting that weak tetanus stimulation can introduce enough amount of Ca2 into dendrites if Ca2-dependent K channels do not work. 4.3. Improvement of cell model The parameters used for ion channels in the present study were modified from those of Miyasho’s model (Miyasho et al., 2001), which had been founded on De
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Schutter and Bower’s model (De Schutter and Bower, 1994a,b). The reasons of these modifications are as follows. (1) Although the kinetics of activation and inactivation of ion channels included in De Schutter and Bower’s model mimicked well the experimental results, the above model failed to reproduce Ca2 spikes in dendrites (see Miyasho et al., 2001 for details), which can be recorded even in the presence of tetrodotoxin (Llina´s and Sugimori, 1980b; Housegard and Midtgaard, 1988; Lev-Ram et al., 1992; Watanabe et al., 1998). (2) Miyasho’s model improves this issue, but the reconstruction of the above kinetics of activation and inactivation of ion channels is not always adequate (Miyasho et al., 2001). We therefore improved the channel descriptions of fast Na channel, persistent Na channel, KA channel, KD channel, KM channel, KC channel, K2 channel, Ttype Ca2 channel, P-type Ca2 channel, and class-E Ca2 channel (Table 1) and the maximum conductance (gmax) of T-type Ca2 channel, P-type Ca2 channel, class-E Ca2 channel, KA channel, and KD channel (Table 2) to satisfy the above two issues: the reproduction of Ca2 spikes and the reconstruction of kinetics of activation and inactivation of ion channels. Here the effects of tetrodotoxin was taken into account as g NaP /0 mS/cm2 and g NaF /0 mS/cm2 in our cell model. Thus, our cell model is much more authentic than those of De Schutter and Bower and Miyasho. 4.4. Roles of A-type and D-type K channels We have previously shown that cooperative work by two types of K channels, KA and KD channels, is necessary for the propagation of excitatory postsynaptic potential (EPSP) in dendrites (Takagi et al., 1998, 2000). For example, the co-existence of KA and KD channels is necessary to produce stable EPSPs during the highfrequency synaptic simulation necessary for induction of LTP. However, it should be noted that information integration in dendrites is controlled by not only EPSPs but also changes in [Ca2]i (Takagi, 2000). Therefore, our present results, considered together with the previous ones, showed for the first time that Ca2dependent K channels, KA channels, and KD channels work cooperatively and integrate information in dendrites. In conclusion, we analyzed the effects of K channels on Ca2 influx in spiny dendrites of rat Purkinje cells by weak synaptic inputs. The results showed that the opening of Ca2-dependent K channels by Ca2 influx through voltage-gated Ca2 channels hyperpolarizes the membrane potentials and deactivates these Ca2 channels in a negative feedback manner, resulting in local, weak Ca2 responses in spiny dendrites of Purkinje cells.
Acknowledgements This work was partly supported by Grants-in-Aid (Nos. 13210006 and 15014201) for Scientific Research on Priority Areas from the Ministry of Education, Culture, Sports, Science and Technology of Japan and grants from the Inamori Foundation and the Brain Science Foundation to E.I.
References Bekkers JM. Distribution of slow AHP channels on hippocampal CA1 pyramidal neurons. J Neurophysiol 2000;83:1756 /9. Cingolani LA, Gymnopoulos M, Boccaccio A, Stocker M, Pedarzani P. Developmental regulation of small-conductance Ca2 -activated K channel expression and function in rat Purkinje neurons. J Neurosci 2002;22:4456 /67. De Schutter E, Bower JM. An active membrane model of the cerebellar Purkinje cell. I. Simulation of current clamps in slice. J Neurophysiol 1994a;71:375 /400. De Schutter E, Bower JM. An active membrane model of the cerebellar Purkinje cell. II. Simulation of synaptic responses. J Neurophysiol 1994b;71:401 /19. Eilers J, Konnerth A. Dendritic signal integration. Curr Opin Neurobiol 1997;7:385 /90. Eilers J, Augustine GJ, Konnerth A. Subthreshold synaptic Ca2 signaling in fine dendrites and spines of cerebellar Purkinje neurons. Nature 1995;373:155 /8. Hines ML, Carnevale NT. The NEURON simulation environment. Neural Comput 1997;9:1179 /209. Hoffman DA, Magee JC, Colbert CM, Johnston D. K channel regulation of signal propagation in dendrites of hippocampal pyramidal neurons. Nature 1997;387:869 /75. Housegard J, Midtgaard J. Intrinsic determinants of firing pattern in Purkinje cells of turtle cerebellum in vitro. J Physiol 1988;402:731 / 49. Ito M. Cerebellar long-term depression: characterization, signal transduction, and functional roles. Physiol Rev 2001;81:1143 /95. Ito M, Sakurai M, Tongroach P. Climbing fiber induced depression of both mossy fiber responsiveness and glutamate sensitivity of cerebellar Purkinje cells. J Physiol 1982;342:113 /34. Lev-Ram V, Miyakawa H, Lasser-Ross N, Ross WN. Calcium transients in cerebellar Purkinje neurons evoked by intracellular stimulation. J Neurophysiol 1992;68:1167 /77. Llina´s R, Sugimori M. Electrophysiological properties of in vitro Purkinje cell somata in mammalian cerebellar slices. J Physiol 1980a;305:171 /95. Llina´s R, Sugimori M. Electrophysiological properties of in vitro Purkinje cell dendrites in mammalian cerebellar slices. J Physiol 1980b;305:197 /213. Magee JC, Johnston D. A synaptically controlled, associative signal for Hebbian plasticity in hippocampal neurons. Science 1997;10:209 /13. Midtgaard J. Processing of information from different source: spatial synaptic integration in the dendrites of vertebrate CNS neurons. Trend Neurosci 1994;17:166 /73. Midtgaard J, Lasser-Ross N, Ross WN. Spatial distribution of Ca2 influx in turtle Purkinje cell dendrites in vitro: role of a transient outward current. J Neurophysiol 1993;70:2455 /69. Mintz IM, Adams ME, Bean BP. P-type calcium channels in rat central and peripheral neurons. Neuron 1992a;9:85 /95.
K. Chono et al. / Journal of Neuroscience Methods 129 (2003) 115 /127 Mintz IM, Venema VJ, Swiderek KM, Lee TD, Bean BP, Adams ME. P-type calcium channels blocked by the spider toxin omega /AgaIVA. Nature 1992b;27:827 /9. Miyakawa H, Lev-Ram V, Lasser-Ross N, Ross WN. Calcium transients evoked by climbing fiber and parallel fiber synaptic inputs in guinea pig cerebellar Purkinje neurons. J Neurophysiol 1992a;68:1178 /89. Miyakawa H, Ross WN, Jaffe D, Callaway JC, Lasser-Ross N, Lisman JE, Johnston D. Synaptically activated increases in Ca2 concentration in hippocampal CA1 pyramidal cells are primarily due to voltage-gated Ca2 channels. Neuron 1992b;9:1163 /73. Miyasho T, Takagi H, Suzuki H, Watanabe S, Inoue M, Kudo Y, Miyakawa H. Low-threshold potassium channels and a lowthreshold calcium channel regulate Ca2 spike firing in the dendrites of cerebellar Purkinje neurons: a modeling study. Brain Res 2001;891:106 /15. Rapp M, Segev I, Yarom Y. Physiology, morphology and detailed passive models of guinea-pig cerebellar Purkinje cells. J Physiol 1994;474:101 /18. Regan LJ, Sah DW, Bean BP. Ca2 channels in rat central and peripheral neurons: high-threshold current resistant to dihydropyridine blockers and omega-conotoxin. Neuron 1991;6:269 /80. Rudy B. Diversity and ubiquity of K channels. Neuroscience 1988;25:729 /49. Sah P, Bekkers JM. Apical dendritic location of slow afterhyperpolarization current in hippocampal pyramidal neurons: implications for the integration of long-term potentiation. J Neurosci 1996;16:4537 /42. Shelton DP. Membrane resistivity estimated for the Purkinje neuron by means of a passive computer model. Neuroscience 1985;14:111 / 31.
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Shimoda M, Takagi H, Kurata S, Yoshioka T, Natori S. Inhibition of the Ca2 -activated K -channel by sapecin B, an insect antibacterial protein. FEBS Lett 1994;339:59 /62. Spruston N, Jaffe DB, Williams SH, Johnston D. Voltage- and space clamp errors associated with the measurement of electronically remote synaptic events. J Neurophysiol 1993;70:781 /802. Stuart G, Hausser M. Initiation and spread of sodium action potentials in cerebellar Purkinje cells. Neuron 1994;13:703 /12. Takagi H. Roles of ion channels in EPSP integration at neuronal dendrite. Neurosci Res 2000;37:167 /71. Takagi H, Takimizu H, Yoshioka T, Suzuki N, Ito E, Kudo Y. Delayed appearance of G-protein coupled signal transduction system in developing cerebellar Purkinje cell dendrites. Neurosci Res 1992;15:206 /12. Takagi H, Sato R, Mori M, Ito E, Suzuki H. Roles of A- and D-type K channels in EPSP integration at a model dendrite. Neurosci Lett 1998;254:165 /8. Takagi H, Sato R, Mori M, Matsumoto T, Saito M, Ito E, Suzuki H. Time-sharing contributions of A- and D-type K channels to the integration of high-frequency sequential excitatory postsynaptic potentials at a model dendrite in rats. Neurosci Lett 2000;289:169 / 72. Takagi H, Kodama K, Saito M, Suzuki H. Presynaptic K channel modulation is a crucial ionic basis of neuronal damage induced by ischemia in rat hippocampal CA1 pyramidal neurons. Zool Sci 2003;20:7 /11. Watanabe S, Takagi H, Miyasho T, Inoue M, Kirino Y, Kudo Y, Miyakawa H. Differential roles of two types of voltage-gated Ca2 channels in the dendrites of rat cerebellar Purkinje neurons. Brain Res 1998;791:43 /55.
A cell model study of calcium influx mechanism regulated by calciumdependent potassium channels in Purkinje cell dendrites Koji Chono a,1, Hiroshi Takagi b, Shozo Koyama b, Hideo Suzuki c, Etsuro Ito a,d,* a
Division of Biological Sciences, Graduate School of Science, Hokkaido University, North 10, West 8, Kita-ku, Sapporo 060-0810, Japan b Department of Physiology-2, Shinshu University School of Medicine, Asahi 3-1-1, Matsumoto 390-8621, Japan c Department of Physics, School of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku ku, Tokyo 169-8555, Japan d Division of Innovative Research, Creative Research Initiative ‘Sousei’ (CRIS), Hokkaido University, North 10, West 8, Kita-ku, Sapporo 060-0810, Japan Received 23 April 2003; received in revised form 26 May 2003; accepted 9 June 2003
Abstract The present study was designed to elucidate the roles of dendritic voltage-gated K channels in Ca2 influx mechanism of a rat Purkinje cell using a computer simulation program. First, we improved the channel descriptions and the maximum conductance in the Purkinje cell model to mimic both the kinetics of ion channels and the Ca2 spikes, which had failed in previous studies. Our cell model is, therefore, much more authentic than those in previous studies. Second, synaptic inputs that mimic stimulation of parallel fibers and induce sub-threshold excitability were simultaneously applied to the spiny dendrites. As a result, transient Ca2 responses were observed in the stimulation points and they decreased with the faster decay rate in the cell model including high-threshold Ca2 -dependent K channels than in those excluding these channels. Third, when a single synaptic input was applied into a spiny dendrite, Ca2 -dependent K channels suppressed Ca2 increases at stimulation and recording points. Finally, Ca2-dependent K channels were also found to suppress the time to peak Ca2 values in the recording points. These results suggest that the opening of Ca2 -dependent K channels by Ca2 influx through voltage-gated Ca2 channels hyperpolarizes the membrane potentials and deactivates these Ca2 channels in a negative feedback manner, resulting in local, weak Ca2 responses in spiny dendrites of Purkinje cells. # 2003 Elsevier B.V. All rights reserved. Keywords: Ca2 channel; Dendrite; K channel; Multi-compartment model; Purkinje cell; Simulation
1. Introduction Studies of cerebellar Purkinje cells have shown that Ca2 spikes are produced locally in dendritic regions, differently from Na spikes produced in somatic regions (Llina´s and Sugimori, 1980a,b; Stuart and Hausser, 1994). Two kinds of excitatory synaptic inputs, one from parallel fibers and the other from a climbing fiber, generate the dendritic Ca2 spikes in a Purkinje cell (Lev-Ram et al., 1992; Miyakawa et al., 1992a;
* Corresponding author. Tel.: /81-11-706-2615; fax: /81-11-7064448. E-mail address: [email protected] (E. Ito). 1 Present address: Applied Pharmacology Research, Pharmacology Laboratories, Institute for Drug Discovery Research, Yamanouchi Pharmaceutical Co., Ltd, Tsukuba 305-8585, Japan. 0165-0270/03/$ - see front matter # 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0165-0270(03)00194-8
Takagi et al., 1992). Ca2 spikes induced by stimulation of parallel fibers are confined to small parts of the spiny dendrites (Eilers et al., 1995), whereas those occur in the proximal dendrite by stimulation of a climbing fiber (Miyakawa et al., 1992a). Associative stimulations of parallel fibers and a climbing fiber induce long-term depression (LTD) in Purkinje cells (Ito et al., 1982), which is built by interaction of signal transduction systems possibly including some immediate early genes between the spiny dendrites connected by parallel fibers and the proximal dendrite activated by inputs of a climbing fiber (Ito, 2001). Such compartmentalization of Ca2 spikes and local changes in intracellular Ca2 concentration ([Ca2]i) in Purkinje cell dendrites are dynamically modulated by voltage-gated K channels (Midtgaard et al., 1993; Midtgaard, 1994; Takagi, 2000). Even though the
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physiological and pharmacological profiles of dendritic K channels are well characterized in Purkinje cells (Housegard and Midtgaard, 1988; Rudy, 1988), the limitations of voltage- and space-clamp techniques employing a somatic electrode to measure ionic currents in dendrites have so far prevented us from elucidating the detailed mechanisms of dendritic K channels in the regulation of [Ca2]i at spiny dendrites (Spruston et al., 1993). In dendrites of hippocampal CA1 pyramidal cells, however, combined experimental and computational approaches revealed many patterns of biophysical behaviors of dendritic K channels (e.g. Ca2-dependent K channels, A-type K (KA) channels, and Dtype K (KD) channels) (Sah and Bekkers, 1996; Hoffman et al., 1997; Takagi et al., 1998, 2000, 2003). Recently, a similar success was acquired in Purkinje cells by these approaches, showing that dendritic Ca2-spike generation is regulated by low voltage-gated (T-like) Ca2 channels and KD channels (Watanabe et al., 1998; Miyasho et al., 2001). Despite such cumulative evidence, the physiological roles of K channels in the regulation of [Ca2]i at spiny dendrites of Purkinje cells have not yet been elucidated. In the present study, to elucidate the physiological roles of voltage-gated K channels in the regulation of Ca2 influx at spiny dendrites of rat Purkinje cells, we applied a computer program to a typical multi-compartment model for the Purkinje cells. In particular, we investigated whether or not Ca2-dependent K channels play the major role in regulating the decay time course of transient Ca2 inward current that is induced by sub-threshold synaptic inputs to these dendrites. The following five features are emphasized. (1) Dendritic function can be analyzed, which is limited when conventional electrophysiological and Ca2 imaging techniques are used because of technical difficulties. (2) We improve the channel descriptions and the maximum conductance in the Purkinje cell model to mimic both the kinetics of ion channels and the Ca2 spikes, which failed in previous studies. (3) The most important channel in the dendritic Ca2 responses can be determined from various K channels. (4) The dynamics of [Ca2]i can be examined in addition to that of Ca2 currents. (5) Effects of weak synaptic inputs to induce sub-threshold excitability on [Ca2]i are particularly noted, because they participate in the initiation of LTD (Eilers and Konnerth, 1997).
2. Materials and methods 2.1. Morphology and passive membrane parameters Computer simulation was made with a simulation program, NEURON (ver. 4.3.1) (Hines and Carnevale, 1997), on a personal computer. The simulations were
Fig. 1. Multi-compartment model for Purkinje cell and action potentials by current injection. (A) Multi-compartment model. (B) Repetitive Na -spike firings recorded in the soma by a current injection of /2.3 nA for 1000 ms into the soma. (C) Expanded presentation of repetitive Na spikes in the soma by injection of current of /1.7, /2.0, or /2.3 nA.
run with a time step of 25 ms. The morphology of the cell model was based on a detailed reconstruction of a Golgi-stained Purkinje cell from an adult rat (Fig. 1A) (Shelton, 1985). The model contained 1089 compartments (soma: 1 compartment, smooth dendrites: 85 compartments, and spiny dendrites: 1003 compartments). A specific membrane capacitance of 0.9 mF/ cm2 was used for the soma and smooth dendrites (Hines and Carnevale, 1997), whereas that of 1.5 mF/cm2 was used for the spiny dendrites (De Schutter and Bower,
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g(V ; [Ca2 ]i ; t) gm(V ; t)p h(V ; t)q z([Ca2 ]i ; t)r ; @m=@t am (1 m) bm m; @h=@t ah (1 h) bh h; @z=@t (z z)=tz ; am (V ) A=[Bexpf(V C)=Dg]; bm (V ) E=[F expf(V C)=Dg]; ah (V ) A=[Bexpf(V G)=Hg]; bh (V ) E=[F expf(V G)=Hg]; z 1=(1A=[Ca2 ]i ); tz B:
1994a,b; Miyasho et al., 2001). Membrane resistivity was set to 10000 V cm2 and specific intracellular resistivity set to 250 V cm (Miyasho et al., 2001). Alpha synaptic inputs were applied to spiny dendrites in our cell model. Here, a synapses refer to the shape of the function (an a function with an exponential form) used to model the membrane conductance change in the postsynaptic site of neuron resulting from the binding of neurotransmitter released from the pre-synaptic neuron. The time course and amplitude of this synaptic input were expressed as a conductance with a single-exponential time course of the form gmax(t /t )exp{ /(t /t )/t }, where t /0 at the onset of the synaptic conductance; gmax is a maximum value of the synaptic conductance; t is time to reach gmax. These values were set as t /0.1 ms, gmax /1.68 mS/cm2 to induce sub-threshold excitability. When simultaneous synaptic inputs were applied into some spiny dendrites, nine points were selected in the spiny dendrites to mimic responsive points recorded by stimulation of parallel fibers (Eilers et al., 1995).
The descriptions of three channels, delayed rectifier K (Kdr) channel, anomalous rectifier K (Kar) channel, and persistent K (KM) channel, were identical to those used in Miyasho’s model (Miyasho et al., 2001). The descriptions of fast Na channel, persistent Na channel, KA channel, KD channel, high-threshold Ca2-dependent K (KC) channel, low-threshold Ca2-dependent K (K2) channel, T-type Ca2 channel, P-type Ca2 channel, and class-E Ca2 channel were slightly modified from those in Miyasho’s model. The rationale for these modifications is given in Section 3, and their details are discussed in Section 4. To describe the Ca2-dependent gating of Ca2-dependent K channels and to update correctly the equilibrium potential of Ca2 in physiological conditions, it was necessary to calculate [Ca2]i. Because any crucial factors required to describe intracellular Ca2 diffusion, buffering release, and extrusion are not yet known, we did not attempt to realistically simulate intracellular Ca2 dynamics. Instead, we assumed that Ca2, which
2.2. Ion channel kinetics We incorporated 12 types of voltage-gated ion channels into our cell model. To describe the kinetics of these channels, we slightly modified the channel descriptions from those in a previous model (Miyasho et al., 2001), which had been determined on the basis of the parameters offered by De Schutter and Bower (1994a,b). The equations and parameters used in our cell model are listed below and in Table 1. Table 1 Parameters for ion channels in the model Channel
Er (mV)
NaF
45
NaP CaT
57 140
CaP CaE
135 135
Kdr Kar KM KA
/85 /30 /85 /85
KD
/85
KC
/85
K2
/97.5
Factor
p, q, r
A
m h m m h m m h * * * m h m h m z m z
3 1 3 2 1 1 1 1
20.88 /9 24.48 3.75 0.0714 8.5 2.6 0.0025
4 1 1 1 1 2 1 2
1.4 0.0175 8.5 0.0015 am /7.5 400 a /25.0 20
B
C
D
E
F
G
H
1 1 1 1 1 1 1 1
58 90 68 50 79 /8 7 32
/3 6 /3 /7.5 4.45 /12.5 /8 8
/26.8 4.5 /30.8 0.5151 0.07905 35 0.18 0.19
1 1 1 1 1 1 1 1
67 35 77 51 85 74 26 42
20 /10 20 4.4 /4 14.5 4 /10
1 1 1 1 / 10 / 10
27 50 17 89 / / / /
/12 8 /12.5 8 / / / /
0.49 1.3 35 0.0055 0.11 / 0.075 /
1 1 1 1 0 / 0 /
30 13 99 83 /35 / 5 /
4 /10 14.5 /8 14.9 / 10 /
Er, reversal potentials; A /H , parameters used in methods; *, the descriptions of these three channels, Kdr, Kar, and KM, were identical to those used in Miyasho’s model (see Miyasho et al., 2001 for details); NaF, fast Na ; NaP, persistent Na ; CaT, T-type Ca2 ; CaP, P-type Ca2 ; CaE, class-E Ca2 ; Kh, anomalous rectifier; Kdr, delayed rectifier; KM, persistent K ; KA, A-type K ; KD, D-type K ; KC, high-threshold K ; K2, low-threshold K .
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enters through Ca2 channels instantaneously, diffused within a thin (0.1 mm thick) sub-membrane shell and that there were two Ca2 extrusion mechanisms: the Ca2 pump and single-exponential decay mechanism. The change in [Ca2]i was expressed as the ratio of Ca2 concentrations between non-stimulated model and stimulated model under the same parameter conditions. 2.3. Statistical analysis
injection of /2.3 nA (Fig. 1B, C). The input impedance of our cell model with a realistic morphology was calculated to be 16.7 MV under normal conditions and to be 22.7 MV when Cs-sensitive K (Kar) channels were removed. Note here that only Kar channels are open around the resting membrane potential. These values are in good agreement with those measured electrophysiologically (Llina´s and Sugimori, 1980a; Rapp et al., 1994). 3.2. Ca2 and Na spikes
Data were expressed as mean9/S.E.M. Differences between groups were examined for statistical significance by the Student t-test. A P-value less than 0.05 denoted the presence of a statistically significant difference.
3. Results 3.1. Reconstruction of experimental results To find the optimum set of parameters to reconstruct the electric properties of real Purkinje cells without loss of calculation time, we started with Miyasho’s model and explored the parameter space in the vicinity of his values. Although Miyasho’s model was suitable for reproducing the depolarization-induced firing pattern of Purkinje cells, there was a need to improve the activation and inactivation profiles of some channels, for example Kdr and KM channels. We therefore modified the channel descriptions, as mentioned in Section 2 (Table 1), and further changed the maximum conductance (gmax) of T-type Ca2 channel, P-type Ca2 channel, class-E Ca2 channel, KA channel, and KD channel (Table 2), until we obtained qualitatively realistic responses. For example, repetitive Na -spike firings could be obtained in the soma by a current Table 2 Maximal channel conductance (mS/cm2) Channel
Soma
Dendrite
NaF NaP CaT CaP CaE Kdr Kar KM KA KD KC K2 Leak
10 1 0 0.5 0 36 0.3 0.04 15 0 0 0 0.21
0 0 0.15 0.4 0.4 0.6 0 0.01 8 9 60 0.39 0.21
Abbreviations for channels are the same as in Table 1.
Ca2 and Na spikes were repetitively produced locally in the dendrites and soma in our cell model, respectively, by a current injection of /2.3 nA for 1000 ms into the soma (Fig. 2). Although P-type Ca2 channel is responsible for Ca2-spike firings in dendrites (Miyakawa et al., 1992a; Watanabe et al., 1998), no information from previous experiments is available to allow determination of the conductance distribution (i.e. gmax) of the P-type Ca2 channels. We therefore assumed that in our cell model the P-type Ca2 channel was distributed uniformly over the dendrites, as in Miyasho’s model. The density distributions of other ion channels were also uniform over the dendrites in our cell model. 3.3. Stability of intracellular Ca2 concentration Because a numerical integration method is used as an algorithm in the present simulation program, NEURON, we must consider that the time for membrane potential and [Ca2]i can reach stable values (see Hines and Carnevale, 1997 for details). For example, the membrane potentials of the soma, smooth dendrites, and spiny dendrites became /77.08, /77.06 mV (n /7; the error value is /0.01), and /77.05 mV (n /7; the error value is /0.01), respectively, 2.5 s after onset of simulation (Fig. 3A). We therefore started to simulate the changes in [Ca2]i in spiny dendrites 2.5 s after onset of simulation (Fig. 3B). Even in this condition, the Ptype Ca2 channel was found to function well in the spiny dendrites (Fig. 3C), in comparison with the experimental data (Regan et al., 1991; Mintz et al., 1992a,b). 3.4. Changes in intracellular Ca2 concentration in spiny dendrites by a synaptic inputs We examined the changes in [Ca2]i by a synaptic inputs, which were simultaneously given to nine points in spiny dendrites extended from one smooth dendrite (Fig. 4A). As mentioned in Sections 1 and 2, the synaptic inputs used here were set to be weak to mimic their Ca2 responses recorded by stimulation of parallel fibers (Eilers et al., 1995), so that no Na2 spikes but
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Fig. 2. Locally produced Ca2 and Na spikes in dendrites and soma. (A, B) Ca2 and Na currents recorded in the soma by a current injection of /2.3 nA for 1000 ms into the soma. (C, D) Ca2 and Na currents recorded in the dendrite by a current injection of / 2.3 nA for 1000 ms into the soma.
excitatory postsynaptic potentials were observed (data not shown). Transient increases in [Ca2]i were observed in these nine points and these responses decreased in exponential fashion, whereas no responses were observed in any other spiny dendrites (Fig. 1B). The decay time of [Ca2]i from the peak to 1/e was 180.79/ 17.7 ms in these nine points (Fig. 4B). These Ca2
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Fig. 3. Stability of membrane potentials and intracellular Ca2 concentration in stimulation process. (A) Changes in membrane potentials of the soma, smooth dendrite, and spiny dendrite after onset of simulation. (B) Change in [Ca2 ]i in the spiny dendrite after onset of simulation. The unit of [Ca2 ]i is arbitrary. (C) Ca2 currents in the spiny dendrite. Following the resting potential ( /77 mV), the membrane potential of soma was stepped to a test potential ranging from /60 to 0 mV at 10 mV increments by a 100 ms pulse. These data were recorded 2.5 s after onset of simulation, showing that P-type Ca2 channels function without any inactivation.
responses remained mainly around the synaptic input points, even if they interfered with each other (Fig. 5). These results completely mimicked the previous data reported by a combination of electrophysiological and Ca2 imaging methods (Eilers et al., 1995), and confirmed the validity of our cell model. Here we could reproduce the transient and local increases in [Ca2]i in Purkinje cell dendrites elicited by synaptic inputs that are equivalent to stimulation of parallel fibers.
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model, KC channels lead the fast decay rate for Ca2 in spiny dendrites (time from peak to 1/e /187.09/18.6 ms (n /9), P B/0.005 vs. the model including none of K channels, Fig. 6B). In contrast, K2 channels and a combination of KC and K2 channels did not evoke such a fast decay rate (K2: time from peak to 1/e /252.99/3.7 ms (n /9); KC/K2: 211.49/26.0 ms (n /9), P /not significant vs. the model including none of K channels, Fig. 6C, D). The differences in the peak values of Ca2 responses between the models including or excluding KC channels will be compared later (see Fig. 7). On the other hand, K channels independent of Ca2 currents (e.g. Kdr, KA, KD, and KM channels) were not involved in the fast decay rate for Ca2 (Kdr: time from peak to 1/e /256.49/4.7 ms; KA: 252.39/3.5 ms; KD: 251.59/3.4 ms; KM: 253.69/3.6 ms, n /9 each, P /not significant vs. the model including none of K channels, Fig. 6E /H). 3.6. Effects of Ca2-dependent K channels on Ca2 propagation in dendrites by single a synaptic input
Fig. 4. Changes in intracellular Ca2 concentration in spiny dendrites by a synaptic inputs. (A) Nine synaptic input points (S1 /S9) in the spiny dendrites extended from one smooth dendrite and another spiny dendrite (O) without a synaptic input. (B) Changes in [Ca2 ]i in the spiny dendrites. The synaptic inputs (t /0.1 ms, gmax /1.68 mS/cm2) were applied 2.5 s after onset of simulation to S1 /S9 simultaneously, but not to O, to evoke sub-threshold excitability. The Ca2 change is presented as a ratio of Ca2 concentrations between non-stimulated model and stimulated model under the same parameters.
3.5. Effects of K channels on regulation of intracellular Ca2 concentration in spiny dendrites When all voltage-gated K channels were excluded from the spiny dendrites in our cell model, the decay rate was slower than otherwise (time from peak to 1/e / 252.29/3.8 ms (n /9), P B/0.005, Fig. 6A). We thus examined which voltage-gated K channel acts in such a fast decay-rate mechanism for [Ca2]i. In our cell
To examine the effects of Ca2-dependent K channels on Ca2 propagation in dendrites, we applied a single a synaptic input to a single spiny dendrite in some models including all K channels, KC and K2 channels, KC channels, or none of K channels (Fig. 7). Here, we should note the two Ca2-dependent K channels, i.e., not only KC channels but also K2 channels, because of the following two reasons, even though KC channels act more effectively than K2 channels in the decay of Ca2 responses (Fig. 6B, C). (1) Both these two channels are expressed by the same channel kinetics description. (2) The absolute values for the Ca2 sensitivities for these channels are unknown, even though their relative difference was taken account upon simulation. We selected five types of dendrites from our cell model and applied a synaptic input into a spiny dendrite in each model (Fig. 7A1, B1, C1, D1, and E1). The peak values of Ca2 observed in the stimulation point and some recording points, whose adjacent distances were 2 /6 mm, showed that KC channels suppressed the increases in [Ca2]i at all the points of stimulation and recording in all dendrites (14.5% decrease in the KC channels-including models in comparison with none of K channels at the stimulation point) (Fig. 7A2, B2, C2, D2, E2). Interestingly, Ca2 level was maintained at the first recording point and decreased gradually at the sequent recording points in a fashion of reciprocal or exponential to the distance (Fig. 7A2, B2, C2, D2, and E2). The time to reach peak [Ca2]i was compared in the five dendrite models (Fig. 7A3, B3, C3, D3, and E3). In some models, KC channels delayed the time for [Ca2]i increase, in other words, KC channels suppressed the propagation of Ca2 in dendrites. At the synaptic input
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Fig. 5. Pseudo-color presentation of Ca2 concentration changes in spiny dendrites by a synaptic inputs. The protocol is the same as in Fig. 4. Numerals represent time (ms); 0 is the timing of the synaptic inputs.
points in all the dendrite models, [Ca2]i reached to peak values in about 10 ms. 3.7. Effects of Ca2-dependent K channels on activation and inactivation gating variables of P-type and T-type Ca2 channels in dendrites by single a synaptic input To elucidate the mechanisms determining the Ca2 dynamics in dendrites, we calculated the factors m (voltage-dependent activation gating variable) and h
(voltage-dependent inactivation gating variable) of Ptype and T-type Ca2 channels. Alpha synaptic inputs were simultaneously given to nine points in spiny dendrites extended from one smooth dendrite (the same positions as shown in Fig. 4A). The changes in the m value of P-type Ca2 channel were compared between the presence and the absence of Ca2-dependent K channels, such as KC and K2 channels, in dendrites. In the absence of these Ca2dependent K channels, the peak value and the integrated value from the onset of stimulation to the
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Fig. 6. Effects of K channels on regulation of intracellular Ca2 concentration in spiny dendrites by a synaptic inputs. (A) Changes in [Ca2 ]i in the cell model without any voltage-gated K channels. The protocol is the same as in Fig. 4. (B) The model including high-threshold Ca2 dependent K channels. (C) The model including low-threshold Ca2 -dependent K channels. (D) The model including both high-threshold and low-threshold Ca2 -dependent K channels. (E) The model including delayed rectifier K channels. (F) The model including persistent K channels. (G) The model including A-type K channels. (H) The model including D-type K channels.
time point returning to the resting state for the m value were 16 and 41% larger than the cases when the Ca2dependent K channels existed in dendrites, respectively (Fig. 8A and B). When the same comparisons were performed for T-type Ca2 channel, the peak value and the integrated value for the m value were 4 and 8% larger in the absence of Ca2-dependent K
channels than those in their presence in dendrites, respectively (Fig. 8C and D). Furthermore, we compared the changes in the h value of T-type Ca2 channel in the presence of Ca2dependent K channels in dendrites with those in the absence of them. In the absence of Ca2-dependent K channels, its value in the resting state was significantly
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Fig. 7. Effects of Ca2 -dependent K channels on peak values of Ca2 concentration changes and time to reach these peak values in spiny dendrites by single a synaptic input. (A1, B1, C1, D1, E1) Five dendrite models selected from Fig. 1A. ‘S’ represents the synaptic input point; ‘R1’ to ‘R6’ show the recording points. (A2, B2, C2, D2, E2) Peak values of changes in [Ca2 ]i recorded at the synaptic input and recording points. All, KC/K2, KC and none represent the dendrite models including all K channels, both high-threshold (KC) and low-threshold (K2) Ca2 -dependent K channels, high-threshold (KC) Ca2 -dependent K channels, and none of K channels, respectively. (A3, B3, C3, D3, E3) Time to peak Ca2 values recorded at the synaptic input and recording points.
larger than that in the presence of these channels (Fig. 8E, F). This fact possibly depended on the hyperpolarization of membrane potential because of the absence of Ca2-dependent K channels. Shortly after the onset of stimulation, the h value of T-type Ca2 channel was extremely increased in the absence of Ca2-dependent K channels in comparison with the case of their presence. Note here that inactivation of P-type Ca2 channels must not be taken into account (Table 1).
3.8. Effects of high-threshold Ca2-dependent K channels on activation and inactivation gating variables of Ca2 channels in dendrites shortly after application of single a synaptic input We calculated the changes in the membrane potential of spiny dendrites of the cell model including only KC channels and in that excluding voltage-gated K channels, in the case of simultaneous a synaptic inputs
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Fig. 8. Effects of Ca2 -dependent K channels (KC and K2) on activation and inactivation gating variables of P-type and T-type Ca2 channels in spiny dendrites by single a synaptic inputs. The protocol is the same as in Fig. 4. (A) Change in the activation gating variable (m ) of T-type Ca2 channel in the presence of Ca2 -dependent K channels. (B) Change in the m value of T-type Ca2 channel in the absence of Ca2 -dependent K channels. (C) Change in the m value of P-type Ca2 channel in the presence of Ca2 -dependent K channels. (D) Change in the m value of P-type Ca2 channel in the absence of Ca2 -dependent K channels. (E) Change in the inactivation gating variable (h ) of T-type Ca2 channel in the presence of Ca2 -dependent K channels. (F) Change in the h value of T-type Ca2 channel in the absence of Ca2 -dependent K channels.
to nine points (Fig. 4A). The average membrane potentials 1 ms after synaptic inputs recorded at these nine points were /63.3 and /29.6 mV in the cell model with only KC channels and in that without any K channels, respectively. Then, using these values, we calculated the factors m and h in the above two cell
models. The m values of P-type Ca2 channel were 2.5 /10 3 and 2.0 /10 1 in the cell model with only KC channels and in that without any K channels, respectively. Note here that inactivation of P-type Ca2 channels must not be taken into account (Table 1). The m values of T-type Ca2 channel were 0.61 and 1.00 in
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the cell model with only KC channels and in that without any K channels, respectively. The h values of T-type Ca2 channel were 2.3 /10 2 and 1.5 /10 6 in the cell model with only KC channels and in that without any K channels, respectively. In addition, the m values of class-E Ca2 channel were 1.3 /10 2 and 5.3 /10 1 in the cell model with only KC channels and in that without any K channels, respectively. The h values of class-E Ca2 channel were 1.1 /10 1 and 7.2 /10 3 in the cell model with only KC channels and in that without any K channels, respectively.
4. Discussion We studied the physiological roles of K channels in the regulation of Ca2 influx at spine dendrites of rat Purkinje cells, when a synaptic inputs were applied to evoke sub-threshold excitability, using the computer simulation program, NEURON. These synaptic inputs were equivalent to stimulation of parallel fibers and known to take part in the induction of LTD. First, we improved the channel descriptions and the maximum conductance in the cell model to mimic both the kinetics of ion channels and the Ca2 spikes, which had failed in the previous simulation studies (Figs. 1 /3). Then, our simulation results showed that the local Ca2 influx in the spiny dendrites of Purkinje cells is regulated by action of Ca2-dependent K channels, particularly high-threshold Ca2-dependent K (KC) channels (Figs. 4 /8). 4.1. Deactivation of voltage-gated Ca2 channels by opening of Ca2-dependent K channels Because the change in the peak value of [Ca2]i at the synaptic input point corresponds to that in the peak value of Ca2 current, suppression of the peak values of [Ca2]i in the dendrite models including Ca2-dependent K channels (Figs. 6 and 7) suggests that these Ca2-dependent K channels suppress Ca2 current. Furthermore, the decay rate of [Ca2]i was fast in the cell model including Ca2-dependent K channels (Fig. 6). Taken together with the data that the time to reach the peak values of [Ca2]i is about 10 ms (Fig. 7), these results suggest that voltage-gated Ca2 channels are deactivated by the opening of Ca2-dependent K channels. In other words, the following scheme of events can be proposed (Fig. 8). (1) Synaptic inputs open voltage-gated Ca2 channels in spiny dendrites of Purkinje cell. (2) Ca2 enters into the dendrites through Ca2 channels. (3) Before the [Ca2]i reaches its peak value (B/10 ms), Ca2-dependent K channels are activated. (4) Ca2-dependent K channels hyperpolarize the dendrite membrane. (5) Finally, voltage-gated Ca2 channels are deactivated.
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Event (3) above is possible, because previous experiments showed activation of Ca2-dependent K current within 10 ms in depolarized Purkinje cells (Shimoda et al., 1994). We proved events (4) and (5) in Sections 3.7 and 3.8. The values of activation and inactivation gating variables of Ca2 channels observed shortly after stimulation clearly showed that voltage-gated Ca2 channels in the cell model including KC channels are strongly suppressed even 1 ms after synaptic inputs, confirming that Ca2-dependent K channels hyperpolarize the dendrite membrane, which is followed by rapid deactivation of voltage-gated Ca2 channels. Further, the deactivation, but not the inactivation, of P-type and T-type Ca2 channels plays the major role in the Ca2 dynamics in dendrites. An influence by the inactivation of T-type Ca2 channels is small. Such importance of the deactivation mechanism in P-type and T-type Ca2 channels can be also recognized by the result that the activation gating variables of these Ca2 channels changed shortly after the onset of stimulation, that is the onset of membrane potential changes (Fig. 8). 4.2. Synaptic plasticity regulated by Ca2 influx due to opening of Ca2-dependent K channels Such regulation of Ca2 influx by opening of Ca2dependent K channels can be applied to explain the initiation mechanism of long-term potentiation (LTP). For example, when an inhibitor of afterhyperpolarization channels, which are considered to be equivalent to Ca2-dependent K channels, is applied to CA1 pyramidal neurons of the hippocampus, weak tetanus stimulation, which usually failed to evoke LTP, could evoke LTP (Bekkers, 2000; Sah and Bekkers, 1996; note here that recent work also revealed that small-conductance Ca2-dependent K channels modulate membrane excitability in rat Purkinje cells due to afterhyperpolarization (Cingolani et al., 2002).). The initiation of LTP in dendrites of pyramidal neurons requires the generation of back-propagation of action potentials in dendritic regions (Magee and Johnston, 1997). The amount of Ca2 influx through voltagegated Ca2 channels activated by action potentials was much larger than that through NMDA receptors (Miyakawa et al., 1992b). Taken together, the above initiation of LTP is considered to be due to inhibition of Ca2-dependent K channels, suggesting that weak tetanus stimulation can introduce enough amount of Ca2 into dendrites if Ca2-dependent K channels do not work. 4.3. Improvement of cell model The parameters used for ion channels in the present study were modified from those of Miyasho’s model (Miyasho et al., 2001), which had been founded on De
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Schutter and Bower’s model (De Schutter and Bower, 1994a,b). The reasons of these modifications are as follows. (1) Although the kinetics of activation and inactivation of ion channels included in De Schutter and Bower’s model mimicked well the experimental results, the above model failed to reproduce Ca2 spikes in dendrites (see Miyasho et al., 2001 for details), which can be recorded even in the presence of tetrodotoxin (Llina´s and Sugimori, 1980b; Housegard and Midtgaard, 1988; Lev-Ram et al., 1992; Watanabe et al., 1998). (2) Miyasho’s model improves this issue, but the reconstruction of the above kinetics of activation and inactivation of ion channels is not always adequate (Miyasho et al., 2001). We therefore improved the channel descriptions of fast Na channel, persistent Na channel, KA channel, KD channel, KM channel, KC channel, K2 channel, Ttype Ca2 channel, P-type Ca2 channel, and class-E Ca2 channel (Table 1) and the maximum conductance (gmax) of T-type Ca2 channel, P-type Ca2 channel, class-E Ca2 channel, KA channel, and KD channel (Table 2) to satisfy the above two issues: the reproduction of Ca2 spikes and the reconstruction of kinetics of activation and inactivation of ion channels. Here the effects of tetrodotoxin was taken into account as g NaP /0 mS/cm2 and g NaF /0 mS/cm2 in our cell model. Thus, our cell model is much more authentic than those of De Schutter and Bower and Miyasho. 4.4. Roles of A-type and D-type K channels We have previously shown that cooperative work by two types of K channels, KA and KD channels, is necessary for the propagation of excitatory postsynaptic potential (EPSP) in dendrites (Takagi et al., 1998, 2000). For example, the co-existence of KA and KD channels is necessary to produce stable EPSPs during the highfrequency synaptic simulation necessary for induction of LTP. However, it should be noted that information integration in dendrites is controlled by not only EPSPs but also changes in [Ca2]i (Takagi, 2000). Therefore, our present results, considered together with the previous ones, showed for the first time that Ca2dependent K channels, KA channels, and KD channels work cooperatively and integrate information in dendrites. In conclusion, we analyzed the effects of K channels on Ca2 influx in spiny dendrites of rat Purkinje cells by weak synaptic inputs. The results showed that the opening of Ca2-dependent K channels by Ca2 influx through voltage-gated Ca2 channels hyperpolarizes the membrane potentials and deactivates these Ca2 channels in a negative feedback manner, resulting in local, weak Ca2 responses in spiny dendrites of Purkinje cells.
Acknowledgements This work was partly supported by Grants-in-Aid (Nos. 13210006 and 15014201) for Scientific Research on Priority Areas from the Ministry of Education, Culture, Sports, Science and Technology of Japan and grants from the Inamori Foundation and the Brain Science Foundation to E.I.
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