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The role of Ca2+-dependent cationic current in generating gamma frequency rhythmic bursts: modeling study
PII: S 0 3 0 6 - 4 5 2 2 ( 0 2 ) 0 0 5 3 7 - 7
Neuroscience Vol. 115, No. 4, pp. 1127^1138, 2002 H 2002 IBRO. Published by Elsevier Science Ltd All rights reserved. Printed in Great Britain 0306-4522 / 02 $22.00+0.00
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THE ROLE OF Ca2þ -DEPENDENT CATIONIC CURRENT IN GENERATING GAMMA FREQUENCY RHYTHMIC BURSTS: MODELING STUDY T. AOYAGI,a;b Y. KANG,b;c N. TERADA,d T. KANEKOb;e and T. FUKAIb;f a
Department of Applied Analysis and Complex Dynamical Systems, Kyoto University, Sakyoku, Kyoto, 606-8501, Japan b c d
Department of Applied Mathematics and Physics, Kyoto University, Sakyoku, Kyoto 606-8501, Japan e
f
CREST, Japan Science and Technology (JST), Saitama, Japan
Department of Physiology, Kyoto University, Sakyoku, Kyoto 606-8501, Japan
Department of Morphological Brain Science, Kyoto University, Sakyoku, Kyoto 606-8501, Japan
Department of Information-Communication Engineering, Tamagawa University, Machida, Tokyo 194-8610, Japan
Abstract
Baudry et al., 1997) and short-term memory (TallonBaudry et al., 1998). Doublet/triplet spikes seen during FRBs appeared to be generated through an enhancement of a hump-like depolarizing afterpotential (DAP) (Kang and Kayano, 1994; Gray and McCormick, 1996; Steriade et al., 1998; Brumberg et al., 2000). Ca2þ -dependent cationic current (Icat ) has been found to underlie such a DAP as well as slow DAPs in cortical pyramidal cells (Caeser et al., 1993; Haj-Dahmane and Andrade, 1997; Kang et al., 1998). A recent study argued against the involvement of Ca2þ -dependent conductances in generating FRBs, based on the observation that bursts of spikes persisted after extracellular Ca2þ was substituted with Mn2þ (Brumberg et al., 2000). FRBs, however, appeared to be replaced by slow rhythmic bursts ( 6 20 Hz, see ¢gs. 6B and 9B of Brumberg et al., 2000) in such a Ca2þ -free condition. The slow rhythmic bursts were quite similar to those seen in intrinsically bursting (IB) neurons (Connors et al., 1982; McCormick et al., 1985; Chagnac-Amitai et al., 1990), in which a persistent Naþ current (INaP ) gen-
Fast rhythmic bursts (FRBs) of doublet/triplet spikes repeating at 25^70 Hz (Kang and Kayano, 1994; Gray and McCormick, 1996; Steriade et al., 1998; Brumberg et al., 2000) may play a crucial role in increasing the coherence among di¡erent subsets of neurons (Gray et al., 1989; Gray et al., 1997), as re£ected in a gammaband EEG rhythm (Tallon-Baudry et al., 1997; TallonBaudry et al., 1998). It is believed that the gamma-band EEG is engaged in cognitive process (Gray and McCormick, 1996; Gray et al., 1989, 1997; Tallon-
*Correspondence to: Y. Kang, Division of Brain Research, Research Institute of Health Sciences, Health Sciences University of Hokkaido, 1757 Kanazawa, Ishikari-Tobetsu, Hokkaido 0610293, Japan. Tel.: +81-13323-1402; fax: +81-13323-1402. E-mail address: [email protected] (Y. Kang). Abbreviations : AHP, afterhyperpolarization ; DAP, depolarizing afterpotential ; FRB, fast rhythmic burst; Icat , cationic current ; INaP , persistent Naþ current ; ISK , small-conductance Kþ current ; IB, intrinsically bursting; ISI, interspike interval; Kd , dissociation constant; SK, small-conductance Kþ . 1127
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erates bursts only at frequencies lower than 20 Hz (Azouz et al., 1996; Jensen et al., 1996). Furthermore, during such a slow burst seen in a Ca2þ -free condition, the FRB neurons displayed even more robust bursts quite similar to those observed in IB neurons in a similar Ca2þ -free condition (Azouz et al., 1996; Jensen et al., 1996). Also in a recent simulation study with INaP (Wang, 1999), the frequencies of FRBs remained lower than 40 Hz. These results seem to indicate that the ionic mechanism enabling FRBs is no longer active in the Ca2þ -free condition, but a Ca2þ -independent mechanism, presumably INaP , remains active to underlie such slow bursts. Therefore, the persistence of slow bursts in a Ca2þ -free condition does not necessarily indicate the non-involvement of Ca2þ -dependent conductances in the generation of FRBs, especially at frequencies higher than 40 Hz. On the other hand, since the reversal potential of Ca2þ -dependent Icat has been shown to be approximately 340 mV, the cationic channel seems suitable for generating an inward current following spike repolarization below 340 mV (Kang et al., 1998) to produce not only a long-lasting DAP but also a hump-like DAP. In the present study, we have mathematically examined the possible involvement of Ca2þ -dependent Icat in generating the hump-like DAP and FRBs. We have investigated whether FRBs ranging from 20 to 70 Hz can be achieved in a single-compartment model through successive generation of DAP and afterhyperpolarization (AHP) by di¡erential activation of Ca2þ -dependent Icat and smallconductance potassium current (ISK ), respectively.
EXPERIMENTAL PROCEDURES
Our model of the FRB neuron has a single compartment, in which only ionic currents essential to generate FRBs are included. The membrane potential V obeys the current^balance equation dV ¼ 3I Na 3I K 3I Ca 3I cat 3I SK 3I leak þ I app dt where the membrane capacitance Cm = 1 WF/cm2 and Iapp is an applied current. The leak current is given by Ileak = gleak (V3Eleak ) with gleak = 0.13 mS/cm2 and Eleak = 368.8 mV. The permeability ratio for the leak current is PK :PNa :PCl = 1:0.04:0.45. The voltage-dependent currents INa , IK and ICa (Hodgkin and Huxley, 1952; Wang, 1988) are described by the standard Hodgkin^Huxley formalism. Thus, a gating variable x satis¢es a ¢rst-order kinetics, dx/dt = xx [K(V)(13x)3 L(V)x] = xx [xr (V)3x]/dx (V), where xx is the temperature factor. The fast spike-generating sodium current is given as INa = gNa m3Na hNa (V3ENa ) with Km = 30.1(V+25)/{exp(30.1(V+25)) 31}, Lm = 4exp(3(V+50)/12), Kh = 0.07exp(30.1(V+42)) and Lh = 1/{exp(30.1(V+12))+1}. The delayed recti¢er potassium current is given as IK = gK m4K (V3EK ) with Km = 30.01(V+26)/ {exp(30.1(V+26))31}, Lm = 0.125exp(3(V+36)/25). The highvoltage-activated calcium current is described as ICa = m2Ca IGHK with Km = 1.6/{1+exp(30.072(V35))}, Lm = 0.02(V+8.69)/ {exp((V+8.69)/5.36)31} (Kay and Wong, 1987). Here, the Goldman^Hodgkin^Katz current IGHK is given as IGHK = Pmax V([Ca2þ ]i 3[Ca2þ ]o h)/(13h) with h = exp(32FV/RT). The parameter values are gNa = 130, gK = 35 (mS/cm2 ), ENa = 60, EK = 397 (mV), F = 9.6485U104 C/mol, T = 293.1 K, R = 8.3145 J/(KWmol) and Pmax = 0.01 WA/(WMWmVWcm2 ). For all temperature factors, a single value of xx = 10 is assumed. In addi-
tion to these voltage-dependent currents, two calcium-dependent currents are included in the model; calcium-dependent Icat (Kang et al., 1998) and ISK (Kohler et al., 1996). These currents are described as Iy = gy my (V3Ey ), where y = SK or cat. The kinetics for the gating variable my is described as dmy /dt = with mr ([Ca2þ ]i ) = [Ca2þ ]i / {mr ([Ca2þ ]i )3my }/dm ([Ca2þ ]i ) ([Ca2þ ]i +Kd;y ) and dm ([Ca2þ ]i ) = 8y /([Ca2þ ]i +Kd;y ). There are two functionally important di¡erences between the two currents. One is the reversal potential: Ecat = 342 mV and ESK = EK = 397 mV, where the reversal potential of the cationic current Ecat is determined by the permeability ratio PNa /PK = 0.16 (Kang et al., 1998). The other is the dissociation constant, which indicates a half-activation calcium concentration: Kd;cat = 15 and Kd;SK = 0.4 (WM). Other parameter values are given as gSK = 0.85 and gcat = 108 (mS/cm2 ) unless otherwise stated, and 8cat = 8SK = 2.8 WMWms, where 8 is an inverse of the association rate constant of the Ca2þ -binding channels. The gating characteristics of the ionic currents, such as the voltagedependent steady states and the activation/inactivation time constants of gating variables, and the calcium-dependent nature of Icat and ISK , are shown in Fig. 1. For the calcium dynamics, we take three processes into account: the entry via ICa , a fast bu¡ering and a slow pump extrusion. The resultant equations are as follows: d½Ca2þ i ¼ dt
R I Ca þ k3 ½BOc 3kþ ½Ca2þ i ½Bð13Oc Þ3gpump
½Ca2þ i ½Ca2þ i þ K m;pump
and dOc ¼ 3k3 Oc þ kþ ½Ca2þ i ð13Oc Þ dt where Oc is the ratio of the bu¡er molecules occupied by Ca2þ (Robertson et al., 1981). The total concentration of bu¡er molecules [B] is assumed to be constant (([B] = 30 WM). Other parameter values are given as k3 = 0.3 ms31 , kþ = 0.1 ms31 WM31 , Km;pump = 0.75 WM, gpump = 3.6 WM ms31 and R = 0.027. The reversal potentials used in the model are determined from the distribution of ions across the membrane: [Naþ ]o = 150, [Naþ ]i = 14, [Kþ ]o = 3, [Kþ ]i = 140, [Cl3 ]o = 100, [Cl3 ]i = 7 and [Ca2þ ]o = 2.1 (in mM). The model was simulated on a UNIX Workstation, using the Runge-Kutta Fehlberg method with adaptive time steps. In addition, as a tool for calculation of bifurcation diagrams, AUTO (Doedel, 1981) was used for identifying the type of bifurcations and attempting the fast^slow analysis.
Cm
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RESULTS
Generation of hump-like DAPs and epileptiform bursts In response to a short current pulse, the model neuron displays an action potential followed by a hump-like DAP (Fig. 2A). This DAP is similar to those seen in the previous electrophysiological studies (Kang and Kayano, 1994; Gray and McCormick, 1996; Brumberg et al., 2000; Kang, 1997; Haj-Dahmane and Andrade, 1997). The amplitude or the peak level of the DAP is determined independently by the following three parameters: the reversal potential for Icat (Ecat ), the maximum conductance (gcat ) and the dissociation constant of Ca2þ dependent cationic channels (Kd;cat ) for Ca2þ . Throughout the whole simulation except in the case of Fig. 2B, Ecat in the standard extracellular and intracellular ionic environments was set to 342 mV based on the previous study (Kang et al., 1998).
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B
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Fig. 1. Characteristics of gating variables. (A) Steady-state activation and inactivation are displayed as functions of the membrane potential for three voltage-dependent currents, INa , IK and ICa . (B) The voltage dependences of the time constants are shown for INa , IK and ICa . (C and D) Steady-state activation and time constants are shown for ISK and Icat as functions of the intracellular Ca2þ concentration.
With decreasing the gcat , the DAP decreases in amplitude (green trace in Fig. 2A) and disappears leaving the AHP alone (red trace in Fig. 2A). When the Ecat is shifted in the depolarizing direction (e.g. 320 mV) by increasing the permeability ratio (PNa /PK ) from 0.16 to a larger value ( 9 1), the same short current pulse triggers a tonic burst (Fig. 2B). Similar tonic bursts were reported in bag-cell neurons that express the Ca2þ -dependent non-selective cationic channels (PNa /PK = 1) (Wilson et al., 1996). If the extracellular Kþ concentration ([Kþ ]o ) is increased from 3 to 5.5 mM, the Ecat determined by the Goldman^Hodgkin^Katz equation is shifted from 342 to 340 mV, and the apparent peak level of the DAP is positively shifted. This results in a burst of spikes in the model neuron (Fig. 2C). The burst is very similar to the epileptiform activity induced in cortical pyramidal cells by a similar increase in [Kþ ]o (Connors et al., 1982; Kang et al., 1998), suggesting an involvement of this cationic current in generating the
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epileptiform activity as well. A reduction of the maximum conductance of ISK (gSK ) also leads to a generation of the epileptiform burst (Fig. 2D) in the model. Thus, incorporating Ca2þ -dependent small-conductance Kþ (SK) and cationic channels with di¡erent permeability ratio reproduces a variety of known ¢ring patterns. Induction of FRBs An in vitro study (Kang and Kayano, 1994) previously demonstrated that FRBs could be slowly induced in regular spiking pyramidal cells through a repetitive application of strong depolarizing current pulses, as reproduced in Fig. 3A. This observation has recently been con¢rmed both in in vivo (Steriade et al., 1998) and in in vitro (Brumberg et al., 2000) studies. Strong membrane depolarization would increase intracellular concentration of Ca2þ which may in turn induce cascades of signal transduction in pyramidal cells, resulting in a modulation of
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Fig. 2. The burst ¢ring generated by varying the parameters of the cationic channel and SK channel. (A) Following an action potential triggered by a short current pulse, hump-like DAPs with di¡erent amplitudes are produced by increasing gcat . (B) Changing Ecat from 342 mV to 320 mV triggers a tonic burst. (C) Raising [Kþ ]o also triggers a transient burst, which is quite similar to the epileptiform burst after discharge induced in cortical pyramidal cells by a similar increase in [Kþ ]o . (D) Decreasing gSK triggers a transient burst.
Kd;cat or gcat , presumably expressed as di¡erent molecular conformations of cationic channels. In the present simulation study, therefore, FRBs were induced in the regular spiking model neuron either by an increase in gcat or by a decrease in Kd;cat . As illustrated in Fig. 3B and C (top traces), in response to a long depolarizing current pulse, the model neuron displays a train of single spikes, each of which is followed by a DAP, similar to that seen in cortical pyramidal cells (Fig. 3A1). When gcat is increased leaving Kd;cat unchanged, the model neuron displays trains of doublet/triplet/quintet spikes in response to the same current pulse (Fig. 3B), quite similar to the chattering or the FRB pattern in real neurons (Fig. 3A2) (Kang and Kayano, 1994; Gray and McCormick, 1996; Steriade et al., 1998; Brumberg et al., 2000). When Kd;cat is decreased leaving gcat unchanged, the model neuron also displays a variety of FRBs (Fig. 3C). Thus, changes in gcat and in Kd;cat play crucial roles in inducing various patterns of FRBs in the regular spiking model neurons responding to a current pulse with the same intensity. It is of essential importance to clarify what determines the FRB pattern to be doublet, triplet or quartet spikes, etc. To this end, we have addressed how each burst is
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terminated during FRBs. Figure 4 displays the transients of free and bu¡ered Ca2þ and the activation variables of the Ca2þ -dependent currents seen during the triplet spikes. As seen in Fig. 4B, the Ca2þ that enters the cell body during each action potential is rapidly bu¡ered and a large and sharp decrease in the free Ca2þ transient is followed by a small and slow decay. Therefore, due to the large Kd;cat (14^16 WM), Icat is only transiently activated when free [Ca2þ ]i is as high as the peak level, as revealed by the behavior of mcat (Fig. 4C). The activation of Icat triggers the spikes following the ¢rst one in each burst. On the other hand, ISK shows a prolonged activation due to the small Kd;SK (0.4 WM), and a temporal summation of mSK occurs during the triplet spikes. Thus, the prolonged and increasing activation of ISK may overwhelm the transient and non-increasing activation of Icat to terminate the burst. When the voltage-dependent Ca2þ current is blocked (Fig. 4D), the model neuron exhibits only periodic single spikes rather than bursts in response to a depolarizing step current. Only a fast AHP generated by a delayed Kþ current is seen because ISK is no longer activated in this condition. On the basis of some standard mathematical techniques (Guckenheimer and Holmes, 1983; Rinzel and
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Fig. 3. Induction of FRBs in regular spiking neurons. (A1) A depolarizing pulse evoked a train of spikes with a pattern of phasic-tonic ¢ring in a layer VI pyramidal cell of the cat motor cortex in an in vitro slice experiment (Kang and Kayano, 1994). (A2) 10^20 repetitive injections of strong depolarizing current pulses slowly induced FRBs in the same neuron as in (A1). (B) and (C) In response to a 250 ms current pulse with 2.0 WA/cm2 , FRBs with two to ¢ve intraburst spikes are evoked in the two regular spiking model neurons with di¡erent values of gcat and Kd;cat (B, 94 mS/cm2 and 15 WM; C, 108 mS/cm2 and 16.4 WM), either by increasing gcat (B) or by decreasing Kd;cat (C). The second blue traces in (B) and (C) are identical (gcat = 108 mS/cm2 and Kd;cat = 15 WM). Note the di¡erences in the voltage trajectory underlying the burst of four to ¢ve spikes: The bottom trace in (B) decays more slowly than in (C). Accordingly, the ISI between the fourth and ¢fth spikes in each burst is more prolonged in (B) than in (C), as exempli¢ed in Fig. 6.
Ermentrout, 1998), we conducted a bifurcation analysis of the bursting mechanism in the present model. The analysis revealed that the transition from a resting state to the rhythmic ¢ring with singlet spikes is a saddle-node type. We also attempted the fast^slow analysis to identify the bursting mechanism, in which the activation of SK, mSK , was chosen as a representative variable in slow subsystem. Figure 4E shows that the stable periodic solution describing the repetitive spiking state in each cycle of bursting disappears via the saddle node of periodics bifurcation, while the resting state of bursting terminates via the saddle-node bifurcation. The coexistence between a depolarized periodic solution and a lower resting state in fast subsystem is essential for the bursting mechanism. At the beginning of each repetitive ¢ring state, mSK is su⁄ciently small and the neuron (activated by the applied current with 3 WA/cm2 ) is in a periodic solution which is a unique stable state for mSK W0. The accumulation of the intracellular calcium gradually activates the SK current (mSK C1.0) during the repetitive ¢ring until it is terminated via the annihilation of the stable periodic solution with an unstable one. At this point, the membrane potential shows a transition to the stable resting
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state. During the resting state, the intracellular calcium concentration decreases gradually due to the extrusion by the Ca2þ pump and so does mSK . The gradual decrease of mSK then terminates the quiescent period via the disappearance of the resting state. The above transitions between the repetitive ¢ring and resting states are repeated. The bursting mechanism such as presented here is referred to as type IV bursting (Bertram et al., 1995) or fold/fold cycle bursting (Izhikevich, 2000) in the literature. Thus the fast^slow analysis provides a simple mathematical explanation for the bursting mechanism of the present neuron model, although slight discrepancy is seen in Fig. 4E between the trajectory of the fast^slow analysis and that of the full dynamics in the higher dimensional state space. The e¡ects of a somatic persistent sodium current on the burst generation It is intriguing to see whether the slow bursting observed in a Ca2þ -free condition (Brumberg et al., 2000) can be achieved by our neuron model when it
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Fig. 4. Typical behavior of the model neuron during triplet ¢ring (Iapp = 3 WA/cm2 , Kd;cat = 12.3 WM). (A) Time evolution of the membrane potential is shown. (B) Time evolution is shown for calcium concentration ([Ca2þ ]i ) and Ca2þ -bound bu¡er concentration ([Ca-B]). (C) The activation of SK current and Ca2þ -dependent Icat is shown in terms of mSK and mcat . (D) If the high-voltage-activated calcium current ICa is blocked (Pmax = 0), the model neuron ceases to ¢re triplet spikes and exhibits only periodic single spikes. (E) Fast^slow analysis of the model neuron, in which the activation of mSK is chosen as a representative variable in slow subsystem. The voltage of the stable (unstable) rest state is indicated by the solid (dotted) line. The stable (solid circles) and unstable (open circles) periodic states are represented by the minimum and maximum voltage of these solutions. The projection of the triplet bursting orbit in (A) is superimposed.
includes INaP . To this end, we added INaP to our singlecompartment model in which slow bursts of doublet spikes were not produced without INaP . Nevertheless, as shown in Fig. 5A, a reduction of the voltage-dependent Ca2þ current lowered the bursting frequency partly due to a less activation of Icat , similar to the slow burst observed in the Ca2þ -free condition. Thus, Icat is likely involved in modifying such slow bursts that are caused by INaP . Due to a di¡erence in the activation mechanism of INaP and Icat , i.e. the former is voltage-activated while the latter is Ca2þ -activated, these currents may show qualitatively di¡erent behaviors during bursting. To examine the di¡erences, we have compared the behavior of the two currents during triplet/quartet ¢ring. Since INaP is kept active as long as the membrane potential is 10^15 mV positive to the resting membrane potential and 10 mV negative to the spike threshold (Crill, 1996), the current remains activated not only during each burst, but also in the intervals between the bursts (Fig. 5B and C, upper traces). Then, an increase of the applied current intensity necessarily increases the intensity of INaP in the interburst intervals as much as during bursts, and may consequently have induced more bursts. On the other hand, [Ca2þ ]i can be kept low in the interburst intervals, irrelevant to the applied current intensity (see Fig. 4B).
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Therefore, Icat is easily deactivated in those intervals (Fig. 5B and C, lower traces) so that it may have little e¡ect on the interburst interval. These di¡erences in nature between INaP and Icat underlie the qualitative differences between the INaP -based and Icat -based models in the bursting behavior. In the former, the number of spikes per burst is necessarily increased with an increase in the applied current intensity (Fig. 5D), which is also consistent with the behavior of the INaP -based two-compartment model (Wang, 1999). As will be shown later, in the latter the number of spikes per burst can remain unchanged in a wide range of the bursting frequency if the values of parameters are adjusted appropriately. Modulation of FRBs by gcat and Kd;cat Hereafter we again concentrate on the model neuron without INaP . It is important to examine whether the present model can by any mean account for the changes in the in vitro ¢ring pattern from regular spiking to doublet bursting (Kang and Kayano, 1994). As previously shown, gcat and Kd;cat play crucial roles in inducing FRBs. As illustrated in Fig. 6A and B, in response to a current pulse with an intensity of 3 WA/cm2 , increasing gcat or decreasing Kd;cat dramatically and systematically changes the FRB pattern, i.e. the number of intraburst
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Fig. 5. E¡ect of a persistent sodium current on FRBs. The persistent sodium current modeled as INaP = gNaP mr (V)(V3ENa ) with mr (V) = 1/1+exp[3(V+45)/5]) (French et al., 1990; Wang, 1999) is included in the present single-compartment model. The maximum conductance is given as gNaP = 0.1 mS/cm2 and that of a delayed Kþ current was modi¢ed as gK = 85 mS/cm2 . (A) In the presence of INaP , a reduction of the voltage-dependent Ca2þ current can result in a lower bursting frequency. Here, Iapp = 0.34 WA/cm2 . (B^D) Typical behaviors of INaP and Icat which underlie DAPs responsible for generating FRBs. Iapp = 0.5 WA/cm2 (B), 2 WA/cm2 (C) and 3 WA/cm2 (D). The model neuron displays triplet ¢ring in (B) and (C), whereas it displays quartet ¢ring in (D).
spikes, the interburst frequency and respective interspike intervals (ISIs). The two parameters, gcat and Kd;cat , play similar roles in modulating the FRB pattern of the present model. Thus, the changes of ¢ring pattern similar to those observed in vitro can be induced in the present model by moderate changes of gcat or Kd;cat . In reality, either or both of the parameters may be modulated through synaptic inputs. Relationship between gcat and Kd;cat for achieving FRBs in the entire gamma frequency range Both in vivo and in vitro studies demonstrated that the interburst frequency of FRBs increased almost linearly from 20 to 70 Hz with an increase in the current intensity of intracellularly injected depolarizing current pulses, leaving the intraburst frequency unchanged (Gray and McCormick, 1996; Brumberg et al., 2000). In the following simulation, we have examined if the frequency of FRBs in the model neuron may range over the entire gamma frequency band with varying the intensity of de-
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polarizing current pulses. As shown in Fig. 7A, with increasing the intensity of current pulses, FRBs of stable doublet ¢ring are induced. The interburst frequency is almost linearly increased with an increase in the intensity of depolarizing current pulses, and consequently ranges from less than 20 to more than 70 Hz (Fig. 7B). The ISI between the ¢rst and second spikes of the ¢rst doublet initially decreases with an increase in the intensity of current pulses, but remains almost constant with further increases (Fig. 7B). When the ISI of the doublet is decreased to be less than 3 ms, the bursting frequency exceeds 25 Hz. De¢ning the gamma-band FRBs as having interburst frequencies of 25^70 Hz (Gray and McCormick, 1996; Brumberg et al., 2000; Gray et al., 1989) and intraburst ISIs shorter than 3 ms (Gray and McCormick, 1996), we have searched combinations of the gcat and Kd;cat values with which the interburst frequency of FRBs ranges over the entire gamma-band. When gcat is plotted against Kd;cat , there exists only a narrow area where the gamma-band FRBs can be induced (Fig. 7C). The optimal Kd;cat value ranges
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Fig. 6. Di¡erential e¡ects of Kd;cat and gcat on the FRB pattern. The number of intraburst spikes, burst frequency and intraburst ISIs are plotted against gcat (A) and Kd;cat (B). The ISIs between the ¢rst and second spikes, between the second and third spikes, between the third and fourth spikes, and between the fourth and ¢fth spikes are shown with blue diamonds, green diamonds, orange diamonds, and sky blue diamonds, respectively. The current intensity was ¢xed at 3 WA/cm2 . With an increase in gcat or with a decrease in Kd;cat , the number of intraburst spikes and the bursting frequency are increased. The burst frequency displays discontinuous jumps simultaneously with increases in the number of intraburst spikes, while the frequency is almost linearly increased as long as this number remains constant. Note longer ISIs between the third and fourth spikes in (A) than in (B), and also note the absence of FRBs of quintet spikes in (A).
approximately between 14 and 18 WM. For a Kd;cat of 15 WM, an optimal gcat for inducing gamma-band FRBs is found to be 108 mS/cm2 . When gcat is changed leaving the Kd;cat (15 WM) unchanged, neither a larger nor a smaller gcat outside the narrow area achieves such FRBs. With a larger gcat (a blue cross labeled as d in Fig. 7C), bursts of abortive triplet spikes are induced when the current intensity is increased, and consequently higher frequency FRBs (V70 Hz) are not achieved (Fig. 7D). On the other hand, with a smaller gcat (a red cross labeled as e in Fig. 7C), bursts of doublet spikes with intraburst ISIs longer than 3 ms are induced by a decrease of the current intensity, and consequently lower frequency FRBs (V25 Hz) are not achieved (Fig. 7E). E¡ects of a raised [K þ ]o on gamma-band bursts In a recent in vitro experiment (Gray and McCormick, 1996), FRBs were induced in a regular spiking neuron by increasing [Kþ ]o from 3 to 5 mM, and this e¡ect was attributed entirely to the attenuation of AHP due to the positive shift of the reversal potential of Kþ currents (EK ). We have examined the e¡ects of increasing [Kþ ]o on the ¢ring pattern in our model neuron. When Ecat
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together with EK and Eleak are allowed to shift with a raised [Kþ ]o according to the Goldman equation, FRBs are e⁄ciently induced in the regular spiking model neuron (Fig. 8A and B). However, when only EK and Eleak are allowed to shift leaving Ecat unchanged, FRB cannot be induced although the ¢ring frequency consequently becomes higher than the control (Fig. 8A and C). Although the results shown above do not necessarily imply that the attenuation of AHP alone cannot induce such FRBs as observed in the in vitro study (Brumberg et al., 2000), they demonstrate that the experimental result can be easily accounted for by the present model. Such a reduction of AHP by increasing [Kþ ]o is di¡erent from the muscarine-induced suppression of AHP discussed in the previous study (Wang, 1999). The present model suggests that Kþ is involved in the generation of FRBs as a charge carrier, presumably through the cationic channel.
DISCUSSION
INaP -based model vs. Icat -based model A previous simulation study proposed that the gamma-band FRB was primarily generated, without
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Fig. 7. A tight reciprocal relationship between gcat and Kd;cat achieving FRBs in the entire gamma frequency range. (A) In a model neuron with a combination of gcat (108 mS/cm2 ) and Kd;cat (15 WM) (a black cross labeled with a in (C)), doublet spikes during FRBs remain stable regardless of the decrease or increase in the interburst interval (IBI). Thus, FRBs throughout the entire gamma frequency band can be achieved as the intensity of depolarizing current pulses is changed. (B) Bursting frequency (red circles) and intraburst ISI (blue triangles) are plotted against the intensity of depolarizing current pulses. With increasing the intensity of current pulses, the bursting frequency is linearly increased. When the bursting frequency exceeds 25 Hz, ISIs are decreased to be less than 3 ms and remain almost constant. (C) Only combinations of Kd;cat and gcat within a narrow area (a green narrow band) in the Kd ^g plane ful¢ll the criteria of FRBs with stable doublet spikes throughout the entire gamma frequency band. (D) and (E) The two combinations of Kd;cat and gcat indicated with a blue cross (Kd;cat = 15 WM, gcat = 125 mS/cm2 ; labeled as d) and a red one (Kd;cat = 15 WM, gcat = 100 mS/cm2 ; labeled as e) lie outside the optimal narrow area and can not achieve high- and low-frequency FRBs, respectively. In fact, the triplet and doublet spikes (lower traces in (D) and (E), respectively) induced at d and e in the Kd ^g plane for an increased and a decreased current intensity, respectively, have intraburst ISIs longer than 3 ms.
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Fig. 8. E¡ects of raised [Kþ ]o on FRBs. (A) Control responses with regular spiking pattern to two di¡erent current pulses are shown for [Kþ ]o = 3 mM, gcat = 58 mS/cm2 . (B) Following an increase in [Kþ ]o from 3 to 5 mM, Ecat together with EK and Eleak were positively shifted. This resulted in the generation of FRBs with doublet spikes. (C) By contrast, FRBs were not generated when only EK and Eleak were positively shifted, but Ecat was left unchanged to exclude the e¡ect of raised [Kþ ]o on cation channels. Note that a positive shift of EK by 13 mV was insu⁄cient to induce FRBs, while a positive shift of Ecat only by 2 mV could generate FRBs.
Ca2þ -activated channels, by a ‘ping-pong’ interplay between action potentials in the somatic compartment and INaP in the dendritic compartment (Wang, 1999). In such a two-compartment model neuron, the somatic action potential propagated backward into the dendrite to activate INaP , which in turn generated a DAP and led to the generation of bursts. The bursting pattern of the model neuron was determined by the degree of inhibition of a muscarine-sensitive Kþ current, while the oscillation period was primarily determined by a slowly inactivating Kþ current. In terms of generating FRBs, our model has several advantages over the INaP -based FRB model. First, multiple compartments are not necessary in our model. In our model, DAPs, which are essential for the generation of bursts, are produced by a Ca2þ -activated Icat in the soma. On the contrary, in the INaP -based model the ping-pong mechanism between the somatic and dendritic compartments is essential for generating bursts of multiple spikes. Wherever, in the extensive apical dendrites of pyramidal neurons, a synaptic input caused membrane depolarization, spikes were ¢rst generated at axon hillock or initial segment and thereafter they propagated backward into the original synaptic input site (Stuart and Sakmann, 1994). Then, the electrotonic separation or distance between the two sites generating spikes and the subthreshold depolarization underlying FRBs would largely a¡ect the behavior of spiking. Since a large electrotonic distance may serve as a low-pass ¢lter, the electrotonic interactions between the soma and dendrites would be signi¢cantly suppressed at the interburst frequencies higher than the cut-o¡ frequency. Therefore, it is di⁄cult for the ping-pong mechanism to achieve FRBs beyond the cut-o¡ frequency
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(V10^30 Hz: see the simulations for neocortical neurons by Douglas and Martin, 1990), which was in fact the case with a two-compartment model ( 6 40 Hz, Wang, 1999). By contrast, the frequency of FRB in our singlecompartment model can range up to 70 Hz, as has been observed in in vivo (Gray et al., 1989, 1997; Gray and McCormick, 1996; Friedman-Hill et al., 2000; Maldonado et al., 2000) and in vitro (Brumberg et al., 2000) experiments. The crucial role of 40^60 Hz FRBs has been found in recordings from the visual cortex of the cat and monkey (Gray et al., 1989; Gray and Singer, 1989; Friedman-Hill et al., 2000; Maldonado et al., 2000) and from the prefrontal cortex of the monkey (Mikami et al., 2001). In contrast, however, to the previous two-compartment model (Wang, 1999), it is di⁄cult for the present model to achieve FRBs at frequencies lower than 25 Hz without INaP . Second, in the INaP -based two-compartment model, the number of intraburst spikes almost linearly increased with an increase in the intensity of current pulses (Wang, 1999), easily leading to a tonic ¢ring without achieving interburst frequencies higher than 40 Hz. This is because the activation of INaP increases with membrane depolarization produced by increasing the intensity of depolarizing current pulses (Fig. 5B, C and D). This could be another reason why it is di⁄cult for the INaP -based model to have the maximum frequency of FRBs higher than 40 Hz. By contrast, in the present single-compartment model, the number of intraburst spikes can remain constant irrespective of an increase in the intensity of current pulses, and consequently stable FRBs are achieved throughout the entire gamma frequency band (Fig. 7B). This situation is realized when Icat is deacti-
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The role of Ca2+-dependent cationic current
vated in the interburst intervals (Fig. 5B^D). In previous in vitro and in vivo studies (Brumberg et al., 2000; Gray and McCormick, 1996; Mikami et al., 2001) except one in vivo study (Steriade et al., 1998), the number of intraburst spikes remained almost constant in spite of changes in the interburst frequencies. These experimental results can be easily accounted for by the present model. Third, cholinergic induction of FRB (Gray and McCormick, 1996; Brumberg et al., 2000) or gamma oscillation (Buhl et al., 1998) is in favor of our model, because it has been shown that INaP is depressed by activation of muscarinic receptor (Mittmann and Alzheimer, 1998) while there are many studies demonstrating the enhancement of Ca2þ -dependent Icat by activation of muscarinic receptors (Caeser et al., 1993; Constanti and Libri, 1992; Fraser and MacVicar, 1996; Guerineau et al., 1995). Cation channels with a low Ca2þ a⁄nity Recently, the dissociation constant of Ca2þ -dependent small-conductance potassium channels (Kd;SK ) for Ca2þ has been reported to be 0.3^0.4 WM (Xia et al., 1998). By contrast, Kd;cat (14^18 WM; Fig. 7C) has been predicted to be approximately 30-fold larger than the Kd;SK in the present simulation study. We have found no condition to achieve FRBs in the entire gamma frequency range with any Kd;cat smaller than 14 WM. Such a large di¡erence in the Ca2þ a⁄nity is suitable for activating Icat and ISK successively. The fast bu¡ering and the slow pump extrusion of intracellular free Ca2þ , which is rapidly increased by a Ca2þ in£ux during action potentials through voltage-gated Ca2þ channels, create a biphasic Ca2þ transient following each action potential. Since the activation time constant of a Ca2þ -dependent channel is inversely proportional to the sum of Kd and [Ca2þ ]i , the transient activation of Icat with a large Kd precedes that of ISK with a small Kd following each action potential (Fig. 4C). Hence a hump-like DAP followed by an AHP is generated. An enhancement of this DAP by increasing gcat or decreasing Kd;cat triggers doublet/triplet spikes. Such a burst of spikes is terminated by the growth of
1137
the hyperpolarizing e¡ect of ISK during the burst (Fig. 4C) and, thus, FRBs can be induced in our model under the suitable combinations of gcat and Kd;cat (Fig. 7). The biphasic Ca2þ transient was normally seen underneath the plasma membrane of neurons (HernandezCruz et al., 1990). Although channels with such a low Ca2þ a⁄nity as Kd;cat (see Fig. 7C) are not known yet, it is known that large-conductance Kþ channels change the Ca2þ a⁄nity from the low micromolar range to the submicromolar range depending on the membrane depolarization (Sun et al., 1999). Similarly, changes in Kd;cat , but with yet unknown mechanisms, might switch the ¢ring pattern from regular spiking to FRBs, as indicated in Fig. 3C. Such a switching of the ¢ring pattern has been demonstrated to occur when strong depolarization was repetitively applied to cortical pyramidal neurons (Kang and Kayano, 1994; Kang, 1997; Brumberg et al., 2000). For understanding the role of FRB in generating the gamma-band EEG rhythm, it is very important to address how the FRB emerges and how the pacemaker FRB neuron can bring the coherence onto di¡erent subsets of neurons. The present study has revealed the key role of Ca2þ -dependent Icat in generating and controlling FRBs. A preliminary simulation of a small network among this type of model neurons suggests that the degree of synchrony in neuronal ¢ring instantaneously and dramatically changes at the critical points where the FRB pattern changes (Fig. 6A and B). Extending the present model towards a large-scale network model, we may be able to see how the spatio-temporal pattern of coherence among cortical pyramidal cells is governed by FRB pattern or, for instance, by the Ca2þ sensitivity of the Icat . Such spatio-temporal patterns of the coherence among neurons may provide additional information to those conveyed by rate coding.
Acknowledgements1This work was supported by a Grant-inAid for Scienti¢c Research on Priority Areas (A) and by CREST (Core Research for Evolutional Science and Technology) of Japan Science and Technology (JST).
REFERENCES
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Dynamic properties of corticothalamic neurons and lacal cortical interneurons generating fast rhythmic (30^40 Hz) spike bursts. J. Neurophysiol. 79, 483^490. Stuart, G.L., Sakmann, B., 1994. Active propagation of somatic action potentials into neocortical pyramidal cell dendrites. Nature 367, 69^72. Sun, X-P., Schlichter, L.C., Stanley, E.F., 1999. Single-channel properties of BK-type calcium-activated potassium channels at a cholinergic presynaptic nerve terminal. J. Physiol. 518, 639^651. Tallon-Baudry, C., Bertrand, O., Delpuech, C., Pernier, J., 1997. Oscillatory gamma-band (30^70 Hz) activity induced by a visual search task in humans. J. Neurosci. 17, 722^734. Tallon-Baudry, C., Bertrand, O., Peronnet, F., Pernier, J., 1998. Induced gamma-band activity during the delay of a visual short-term memory task in humans. J. Neurosci. 18, 4244^4254. Wang, X-J., 1988. Calcium coding and adaptive temporal computation in cortical pyramidal neurons. J. Neurophysiol. 79, 1549^1566. Wang, X-J., 1999. Fast burst ¢ring and short-term synaptic plasticity: a model of neocortical chattering neurons. Neuroscience 89, 347^362. Wilson, G.F., Richardson, F.C., Fisher, T.E., Olivera, B.M., Kaczmarek, L.K., 1996. Identi¢cation and characterization of a Ca(2+)-sensitive nonspeci¢c cation channel underlying prolonged repetitive ¢ring in aplysia neurons. J. Neurosci. 16, 3661^3671. Xia, X.M., Fakler, B., Rivard, A., Wayman, G., Johnson-Pais, T., Keen, J.E., Ishii, T., Hirschberg, B., Bond, C.T., Lutsenko, S., Maylie, J., Adelman, J.P., 1998. Mechanism of calcium gating in small-conductance calcium-activated potassium channels. Nature 395, 503^507. (Accepted 31 July 2002)
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Neuroscience Vol. 115, No. 4, pp. 1127^1138, 2002 H 2002 IBRO. Published by Elsevier Science Ltd All rights reserved. Printed in Great Britain 0306-4522 / 02 $22.00+0.00
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THE ROLE OF Ca2þ -DEPENDENT CATIONIC CURRENT IN GENERATING GAMMA FREQUENCY RHYTHMIC BURSTS: MODELING STUDY T. AOYAGI,a;b Y. KANG,b;c N. TERADA,d T. KANEKOb;e and T. FUKAIb;f a
Department of Applied Analysis and Complex Dynamical Systems, Kyoto University, Sakyoku, Kyoto, 606-8501, Japan b c d
Department of Applied Mathematics and Physics, Kyoto University, Sakyoku, Kyoto 606-8501, Japan e
f
CREST, Japan Science and Technology (JST), Saitama, Japan
Department of Physiology, Kyoto University, Sakyoku, Kyoto 606-8501, Japan
Department of Morphological Brain Science, Kyoto University, Sakyoku, Kyoto 606-8501, Japan
Department of Information-Communication Engineering, Tamagawa University, Machida, Tokyo 194-8610, Japan
Abstract
Baudry et al., 1997) and short-term memory (TallonBaudry et al., 1998). Doublet/triplet spikes seen during FRBs appeared to be generated through an enhancement of a hump-like depolarizing afterpotential (DAP) (Kang and Kayano, 1994; Gray and McCormick, 1996; Steriade et al., 1998; Brumberg et al., 2000). Ca2þ -dependent cationic current (Icat ) has been found to underlie such a DAP as well as slow DAPs in cortical pyramidal cells (Caeser et al., 1993; Haj-Dahmane and Andrade, 1997; Kang et al., 1998). A recent study argued against the involvement of Ca2þ -dependent conductances in generating FRBs, based on the observation that bursts of spikes persisted after extracellular Ca2þ was substituted with Mn2þ (Brumberg et al., 2000). FRBs, however, appeared to be replaced by slow rhythmic bursts ( 6 20 Hz, see ¢gs. 6B and 9B of Brumberg et al., 2000) in such a Ca2þ -free condition. The slow rhythmic bursts were quite similar to those seen in intrinsically bursting (IB) neurons (Connors et al., 1982; McCormick et al., 1985; Chagnac-Amitai et al., 1990), in which a persistent Naþ current (INaP ) gen-
Fast rhythmic bursts (FRBs) of doublet/triplet spikes repeating at 25^70 Hz (Kang and Kayano, 1994; Gray and McCormick, 1996; Steriade et al., 1998; Brumberg et al., 2000) may play a crucial role in increasing the coherence among di¡erent subsets of neurons (Gray et al., 1989; Gray et al., 1997), as re£ected in a gammaband EEG rhythm (Tallon-Baudry et al., 1997; TallonBaudry et al., 1998). It is believed that the gamma-band EEG is engaged in cognitive process (Gray and McCormick, 1996; Gray et al., 1989, 1997; Tallon-
*Correspondence to: Y. Kang, Division of Brain Research, Research Institute of Health Sciences, Health Sciences University of Hokkaido, 1757 Kanazawa, Ishikari-Tobetsu, Hokkaido 0610293, Japan. Tel.: +81-13323-1402; fax: +81-13323-1402. E-mail address: [email protected] (Y. Kang). Abbreviations : AHP, afterhyperpolarization ; DAP, depolarizing afterpotential ; FRB, fast rhythmic burst; Icat , cationic current ; INaP , persistent Naþ current ; ISK , small-conductance Kþ current ; IB, intrinsically bursting; ISI, interspike interval; Kd , dissociation constant; SK, small-conductance Kþ . 1127
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erates bursts only at frequencies lower than 20 Hz (Azouz et al., 1996; Jensen et al., 1996). Furthermore, during such a slow burst seen in a Ca2þ -free condition, the FRB neurons displayed even more robust bursts quite similar to those observed in IB neurons in a similar Ca2þ -free condition (Azouz et al., 1996; Jensen et al., 1996). Also in a recent simulation study with INaP (Wang, 1999), the frequencies of FRBs remained lower than 40 Hz. These results seem to indicate that the ionic mechanism enabling FRBs is no longer active in the Ca2þ -free condition, but a Ca2þ -independent mechanism, presumably INaP , remains active to underlie such slow bursts. Therefore, the persistence of slow bursts in a Ca2þ -free condition does not necessarily indicate the non-involvement of Ca2þ -dependent conductances in the generation of FRBs, especially at frequencies higher than 40 Hz. On the other hand, since the reversal potential of Ca2þ -dependent Icat has been shown to be approximately 340 mV, the cationic channel seems suitable for generating an inward current following spike repolarization below 340 mV (Kang et al., 1998) to produce not only a long-lasting DAP but also a hump-like DAP. In the present study, we have mathematically examined the possible involvement of Ca2þ -dependent Icat in generating the hump-like DAP and FRBs. We have investigated whether FRBs ranging from 20 to 70 Hz can be achieved in a single-compartment model through successive generation of DAP and afterhyperpolarization (AHP) by di¡erential activation of Ca2þ -dependent Icat and smallconductance potassium current (ISK ), respectively.
EXPERIMENTAL PROCEDURES
Our model of the FRB neuron has a single compartment, in which only ionic currents essential to generate FRBs are included. The membrane potential V obeys the current^balance equation dV ¼ 3I Na 3I K 3I Ca 3I cat 3I SK 3I leak þ I app dt where the membrane capacitance Cm = 1 WF/cm2 and Iapp is an applied current. The leak current is given by Ileak = gleak (V3Eleak ) with gleak = 0.13 mS/cm2 and Eleak = 368.8 mV. The permeability ratio for the leak current is PK :PNa :PCl = 1:0.04:0.45. The voltage-dependent currents INa , IK and ICa (Hodgkin and Huxley, 1952; Wang, 1988) are described by the standard Hodgkin^Huxley formalism. Thus, a gating variable x satis¢es a ¢rst-order kinetics, dx/dt = xx [K(V)(13x)3 L(V)x] = xx [xr (V)3x]/dx (V), where xx is the temperature factor. The fast spike-generating sodium current is given as INa = gNa m3Na hNa (V3ENa ) with Km = 30.1(V+25)/{exp(30.1(V+25)) 31}, Lm = 4exp(3(V+50)/12), Kh = 0.07exp(30.1(V+42)) and Lh = 1/{exp(30.1(V+12))+1}. The delayed recti¢er potassium current is given as IK = gK m4K (V3EK ) with Km = 30.01(V+26)/ {exp(30.1(V+26))31}, Lm = 0.125exp(3(V+36)/25). The highvoltage-activated calcium current is described as ICa = m2Ca IGHK with Km = 1.6/{1+exp(30.072(V35))}, Lm = 0.02(V+8.69)/ {exp((V+8.69)/5.36)31} (Kay and Wong, 1987). Here, the Goldman^Hodgkin^Katz current IGHK is given as IGHK = Pmax V([Ca2þ ]i 3[Ca2þ ]o h)/(13h) with h = exp(32FV/RT). The parameter values are gNa = 130, gK = 35 (mS/cm2 ), ENa = 60, EK = 397 (mV), F = 9.6485U104 C/mol, T = 293.1 K, R = 8.3145 J/(KWmol) and Pmax = 0.01 WA/(WMWmVWcm2 ). For all temperature factors, a single value of xx = 10 is assumed. In addi-
tion to these voltage-dependent currents, two calcium-dependent currents are included in the model; calcium-dependent Icat (Kang et al., 1998) and ISK (Kohler et al., 1996). These currents are described as Iy = gy my (V3Ey ), where y = SK or cat. The kinetics for the gating variable my is described as dmy /dt = with mr ([Ca2þ ]i ) = [Ca2þ ]i / {mr ([Ca2þ ]i )3my }/dm ([Ca2þ ]i ) ([Ca2þ ]i +Kd;y ) and dm ([Ca2þ ]i ) = 8y /([Ca2þ ]i +Kd;y ). There are two functionally important di¡erences between the two currents. One is the reversal potential: Ecat = 342 mV and ESK = EK = 397 mV, where the reversal potential of the cationic current Ecat is determined by the permeability ratio PNa /PK = 0.16 (Kang et al., 1998). The other is the dissociation constant, which indicates a half-activation calcium concentration: Kd;cat = 15 and Kd;SK = 0.4 (WM). Other parameter values are given as gSK = 0.85 and gcat = 108 (mS/cm2 ) unless otherwise stated, and 8cat = 8SK = 2.8 WMWms, where 8 is an inverse of the association rate constant of the Ca2þ -binding channels. The gating characteristics of the ionic currents, such as the voltagedependent steady states and the activation/inactivation time constants of gating variables, and the calcium-dependent nature of Icat and ISK , are shown in Fig. 1. For the calcium dynamics, we take three processes into account: the entry via ICa , a fast bu¡ering and a slow pump extrusion. The resultant equations are as follows: d½Ca2þ i ¼ dt
R I Ca þ k3 ½BOc 3kþ ½Ca2þ i ½Bð13Oc Þ3gpump
½Ca2þ i ½Ca2þ i þ K m;pump
and dOc ¼ 3k3 Oc þ kþ ½Ca2þ i ð13Oc Þ dt where Oc is the ratio of the bu¡er molecules occupied by Ca2þ (Robertson et al., 1981). The total concentration of bu¡er molecules [B] is assumed to be constant (([B] = 30 WM). Other parameter values are given as k3 = 0.3 ms31 , kþ = 0.1 ms31 WM31 , Km;pump = 0.75 WM, gpump = 3.6 WM ms31 and R = 0.027. The reversal potentials used in the model are determined from the distribution of ions across the membrane: [Naþ ]o = 150, [Naþ ]i = 14, [Kþ ]o = 3, [Kþ ]i = 140, [Cl3 ]o = 100, [Cl3 ]i = 7 and [Ca2þ ]o = 2.1 (in mM). The model was simulated on a UNIX Workstation, using the Runge-Kutta Fehlberg method with adaptive time steps. In addition, as a tool for calculation of bifurcation diagrams, AUTO (Doedel, 1981) was used for identifying the type of bifurcations and attempting the fast^slow analysis.
Cm
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RESULTS
Generation of hump-like DAPs and epileptiform bursts In response to a short current pulse, the model neuron displays an action potential followed by a hump-like DAP (Fig. 2A). This DAP is similar to those seen in the previous electrophysiological studies (Kang and Kayano, 1994; Gray and McCormick, 1996; Brumberg et al., 2000; Kang, 1997; Haj-Dahmane and Andrade, 1997). The amplitude or the peak level of the DAP is determined independently by the following three parameters: the reversal potential for Icat (Ecat ), the maximum conductance (gcat ) and the dissociation constant of Ca2þ dependent cationic channels (Kd;cat ) for Ca2þ . Throughout the whole simulation except in the case of Fig. 2B, Ecat in the standard extracellular and intracellular ionic environments was set to 342 mV based on the previous study (Kang et al., 1998).
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A
B
1.0
Time constant (ms)
m∞ or h∞
0.8 0.6 m Na hNa mK m Ca
0.4 0.2 0.0 -100
-50
C
V(mV)
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Fig. 1. Characteristics of gating variables. (A) Steady-state activation and inactivation are displayed as functions of the membrane potential for three voltage-dependent currents, INa , IK and ICa . (B) The voltage dependences of the time constants are shown for INa , IK and ICa . (C and D) Steady-state activation and time constants are shown for ISK and Icat as functions of the intracellular Ca2þ concentration.
With decreasing the gcat , the DAP decreases in amplitude (green trace in Fig. 2A) and disappears leaving the AHP alone (red trace in Fig. 2A). When the Ecat is shifted in the depolarizing direction (e.g. 320 mV) by increasing the permeability ratio (PNa /PK ) from 0.16 to a larger value ( 9 1), the same short current pulse triggers a tonic burst (Fig. 2B). Similar tonic bursts were reported in bag-cell neurons that express the Ca2þ -dependent non-selective cationic channels (PNa /PK = 1) (Wilson et al., 1996). If the extracellular Kþ concentration ([Kþ ]o ) is increased from 3 to 5.5 mM, the Ecat determined by the Goldman^Hodgkin^Katz equation is shifted from 342 to 340 mV, and the apparent peak level of the DAP is positively shifted. This results in a burst of spikes in the model neuron (Fig. 2C). The burst is very similar to the epileptiform activity induced in cortical pyramidal cells by a similar increase in [Kþ ]o (Connors et al., 1982; Kang et al., 1998), suggesting an involvement of this cationic current in generating the
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epileptiform activity as well. A reduction of the maximum conductance of ISK (gSK ) also leads to a generation of the epileptiform burst (Fig. 2D) in the model. Thus, incorporating Ca2þ -dependent small-conductance Kþ (SK) and cationic channels with di¡erent permeability ratio reproduces a variety of known ¢ring patterns. Induction of FRBs An in vitro study (Kang and Kayano, 1994) previously demonstrated that FRBs could be slowly induced in regular spiking pyramidal cells through a repetitive application of strong depolarizing current pulses, as reproduced in Fig. 3A. This observation has recently been con¢rmed both in in vivo (Steriade et al., 1998) and in in vitro (Brumberg et al., 2000) studies. Strong membrane depolarization would increase intracellular concentration of Ca2þ which may in turn induce cascades of signal transduction in pyramidal cells, resulting in a modulation of
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Fig. 2. The burst ¢ring generated by varying the parameters of the cationic channel and SK channel. (A) Following an action potential triggered by a short current pulse, hump-like DAPs with di¡erent amplitudes are produced by increasing gcat . (B) Changing Ecat from 342 mV to 320 mV triggers a tonic burst. (C) Raising [Kþ ]o also triggers a transient burst, which is quite similar to the epileptiform burst after discharge induced in cortical pyramidal cells by a similar increase in [Kþ ]o . (D) Decreasing gSK triggers a transient burst.
Kd;cat or gcat , presumably expressed as di¡erent molecular conformations of cationic channels. In the present simulation study, therefore, FRBs were induced in the regular spiking model neuron either by an increase in gcat or by a decrease in Kd;cat . As illustrated in Fig. 3B and C (top traces), in response to a long depolarizing current pulse, the model neuron displays a train of single spikes, each of which is followed by a DAP, similar to that seen in cortical pyramidal cells (Fig. 3A1). When gcat is increased leaving Kd;cat unchanged, the model neuron displays trains of doublet/triplet/quintet spikes in response to the same current pulse (Fig. 3B), quite similar to the chattering or the FRB pattern in real neurons (Fig. 3A2) (Kang and Kayano, 1994; Gray and McCormick, 1996; Steriade et al., 1998; Brumberg et al., 2000). When Kd;cat is decreased leaving gcat unchanged, the model neuron also displays a variety of FRBs (Fig. 3C). Thus, changes in gcat and in Kd;cat play crucial roles in inducing various patterns of FRBs in the regular spiking model neurons responding to a current pulse with the same intensity. It is of essential importance to clarify what determines the FRB pattern to be doublet, triplet or quartet spikes, etc. To this end, we have addressed how each burst is
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terminated during FRBs. Figure 4 displays the transients of free and bu¡ered Ca2þ and the activation variables of the Ca2þ -dependent currents seen during the triplet spikes. As seen in Fig. 4B, the Ca2þ that enters the cell body during each action potential is rapidly bu¡ered and a large and sharp decrease in the free Ca2þ transient is followed by a small and slow decay. Therefore, due to the large Kd;cat (14^16 WM), Icat is only transiently activated when free [Ca2þ ]i is as high as the peak level, as revealed by the behavior of mcat (Fig. 4C). The activation of Icat triggers the spikes following the ¢rst one in each burst. On the other hand, ISK shows a prolonged activation due to the small Kd;SK (0.4 WM), and a temporal summation of mSK occurs during the triplet spikes. Thus, the prolonged and increasing activation of ISK may overwhelm the transient and non-increasing activation of Icat to terminate the burst. When the voltage-dependent Ca2þ current is blocked (Fig. 4D), the model neuron exhibits only periodic single spikes rather than bursts in response to a depolarizing step current. Only a fast AHP generated by a delayed Kþ current is seen because ISK is no longer activated in this condition. On the basis of some standard mathematical techniques (Guckenheimer and Holmes, 1983; Rinzel and
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Fig. 3. Induction of FRBs in regular spiking neurons. (A1) A depolarizing pulse evoked a train of spikes with a pattern of phasic-tonic ¢ring in a layer VI pyramidal cell of the cat motor cortex in an in vitro slice experiment (Kang and Kayano, 1994). (A2) 10^20 repetitive injections of strong depolarizing current pulses slowly induced FRBs in the same neuron as in (A1). (B) and (C) In response to a 250 ms current pulse with 2.0 WA/cm2 , FRBs with two to ¢ve intraburst spikes are evoked in the two regular spiking model neurons with di¡erent values of gcat and Kd;cat (B, 94 mS/cm2 and 15 WM; C, 108 mS/cm2 and 16.4 WM), either by increasing gcat (B) or by decreasing Kd;cat (C). The second blue traces in (B) and (C) are identical (gcat = 108 mS/cm2 and Kd;cat = 15 WM). Note the di¡erences in the voltage trajectory underlying the burst of four to ¢ve spikes: The bottom trace in (B) decays more slowly than in (C). Accordingly, the ISI between the fourth and ¢fth spikes in each burst is more prolonged in (B) than in (C), as exempli¢ed in Fig. 6.
Ermentrout, 1998), we conducted a bifurcation analysis of the bursting mechanism in the present model. The analysis revealed that the transition from a resting state to the rhythmic ¢ring with singlet spikes is a saddle-node type. We also attempted the fast^slow analysis to identify the bursting mechanism, in which the activation of SK, mSK , was chosen as a representative variable in slow subsystem. Figure 4E shows that the stable periodic solution describing the repetitive spiking state in each cycle of bursting disappears via the saddle node of periodics bifurcation, while the resting state of bursting terminates via the saddle-node bifurcation. The coexistence between a depolarized periodic solution and a lower resting state in fast subsystem is essential for the bursting mechanism. At the beginning of each repetitive ¢ring state, mSK is su⁄ciently small and the neuron (activated by the applied current with 3 WA/cm2 ) is in a periodic solution which is a unique stable state for mSK W0. The accumulation of the intracellular calcium gradually activates the SK current (mSK C1.0) during the repetitive ¢ring until it is terminated via the annihilation of the stable periodic solution with an unstable one. At this point, the membrane potential shows a transition to the stable resting
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state. During the resting state, the intracellular calcium concentration decreases gradually due to the extrusion by the Ca2þ pump and so does mSK . The gradual decrease of mSK then terminates the quiescent period via the disappearance of the resting state. The above transitions between the repetitive ¢ring and resting states are repeated. The bursting mechanism such as presented here is referred to as type IV bursting (Bertram et al., 1995) or fold/fold cycle bursting (Izhikevich, 2000) in the literature. Thus the fast^slow analysis provides a simple mathematical explanation for the bursting mechanism of the present neuron model, although slight discrepancy is seen in Fig. 4E between the trajectory of the fast^slow analysis and that of the full dynamics in the higher dimensional state space. The e¡ects of a somatic persistent sodium current on the burst generation It is intriguing to see whether the slow bursting observed in a Ca2þ -free condition (Brumberg et al., 2000) can be achieved by our neuron model when it
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Fig. 4. Typical behavior of the model neuron during triplet ¢ring (Iapp = 3 WA/cm2 , Kd;cat = 12.3 WM). (A) Time evolution of the membrane potential is shown. (B) Time evolution is shown for calcium concentration ([Ca2þ ]i ) and Ca2þ -bound bu¡er concentration ([Ca-B]). (C) The activation of SK current and Ca2þ -dependent Icat is shown in terms of mSK and mcat . (D) If the high-voltage-activated calcium current ICa is blocked (Pmax = 0), the model neuron ceases to ¢re triplet spikes and exhibits only periodic single spikes. (E) Fast^slow analysis of the model neuron, in which the activation of mSK is chosen as a representative variable in slow subsystem. The voltage of the stable (unstable) rest state is indicated by the solid (dotted) line. The stable (solid circles) and unstable (open circles) periodic states are represented by the minimum and maximum voltage of these solutions. The projection of the triplet bursting orbit in (A) is superimposed.
includes INaP . To this end, we added INaP to our singlecompartment model in which slow bursts of doublet spikes were not produced without INaP . Nevertheless, as shown in Fig. 5A, a reduction of the voltage-dependent Ca2þ current lowered the bursting frequency partly due to a less activation of Icat , similar to the slow burst observed in the Ca2þ -free condition. Thus, Icat is likely involved in modifying such slow bursts that are caused by INaP . Due to a di¡erence in the activation mechanism of INaP and Icat , i.e. the former is voltage-activated while the latter is Ca2þ -activated, these currents may show qualitatively di¡erent behaviors during bursting. To examine the di¡erences, we have compared the behavior of the two currents during triplet/quartet ¢ring. Since INaP is kept active as long as the membrane potential is 10^15 mV positive to the resting membrane potential and 10 mV negative to the spike threshold (Crill, 1996), the current remains activated not only during each burst, but also in the intervals between the bursts (Fig. 5B and C, upper traces). Then, an increase of the applied current intensity necessarily increases the intensity of INaP in the interburst intervals as much as during bursts, and may consequently have induced more bursts. On the other hand, [Ca2þ ]i can be kept low in the interburst intervals, irrelevant to the applied current intensity (see Fig. 4B).
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Therefore, Icat is easily deactivated in those intervals (Fig. 5B and C, lower traces) so that it may have little e¡ect on the interburst interval. These di¡erences in nature between INaP and Icat underlie the qualitative differences between the INaP -based and Icat -based models in the bursting behavior. In the former, the number of spikes per burst is necessarily increased with an increase in the applied current intensity (Fig. 5D), which is also consistent with the behavior of the INaP -based two-compartment model (Wang, 1999). As will be shown later, in the latter the number of spikes per burst can remain unchanged in a wide range of the bursting frequency if the values of parameters are adjusted appropriately. Modulation of FRBs by gcat and Kd;cat Hereafter we again concentrate on the model neuron without INaP . It is important to examine whether the present model can by any mean account for the changes in the in vitro ¢ring pattern from regular spiking to doublet bursting (Kang and Kayano, 1994). As previously shown, gcat and Kd;cat play crucial roles in inducing FRBs. As illustrated in Fig. 6A and B, in response to a current pulse with an intensity of 3 WA/cm2 , increasing gcat or decreasing Kd;cat dramatically and systematically changes the FRB pattern, i.e. the number of intraburst
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Fig. 5. E¡ect of a persistent sodium current on FRBs. The persistent sodium current modeled as INaP = gNaP mr (V)(V3ENa ) with mr (V) = 1/1+exp[3(V+45)/5]) (French et al., 1990; Wang, 1999) is included in the present single-compartment model. The maximum conductance is given as gNaP = 0.1 mS/cm2 and that of a delayed Kþ current was modi¢ed as gK = 85 mS/cm2 . (A) In the presence of INaP , a reduction of the voltage-dependent Ca2þ current can result in a lower bursting frequency. Here, Iapp = 0.34 WA/cm2 . (B^D) Typical behaviors of INaP and Icat which underlie DAPs responsible for generating FRBs. Iapp = 0.5 WA/cm2 (B), 2 WA/cm2 (C) and 3 WA/cm2 (D). The model neuron displays triplet ¢ring in (B) and (C), whereas it displays quartet ¢ring in (D).
spikes, the interburst frequency and respective interspike intervals (ISIs). The two parameters, gcat and Kd;cat , play similar roles in modulating the FRB pattern of the present model. Thus, the changes of ¢ring pattern similar to those observed in vitro can be induced in the present model by moderate changes of gcat or Kd;cat . In reality, either or both of the parameters may be modulated through synaptic inputs. Relationship between gcat and Kd;cat for achieving FRBs in the entire gamma frequency range Both in vivo and in vitro studies demonstrated that the interburst frequency of FRBs increased almost linearly from 20 to 70 Hz with an increase in the current intensity of intracellularly injected depolarizing current pulses, leaving the intraburst frequency unchanged (Gray and McCormick, 1996; Brumberg et al., 2000). In the following simulation, we have examined if the frequency of FRBs in the model neuron may range over the entire gamma frequency band with varying the intensity of de-
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polarizing current pulses. As shown in Fig. 7A, with increasing the intensity of current pulses, FRBs of stable doublet ¢ring are induced. The interburst frequency is almost linearly increased with an increase in the intensity of depolarizing current pulses, and consequently ranges from less than 20 to more than 70 Hz (Fig. 7B). The ISI between the ¢rst and second spikes of the ¢rst doublet initially decreases with an increase in the intensity of current pulses, but remains almost constant with further increases (Fig. 7B). When the ISI of the doublet is decreased to be less than 3 ms, the bursting frequency exceeds 25 Hz. De¢ning the gamma-band FRBs as having interburst frequencies of 25^70 Hz (Gray and McCormick, 1996; Brumberg et al., 2000; Gray et al., 1989) and intraburst ISIs shorter than 3 ms (Gray and McCormick, 1996), we have searched combinations of the gcat and Kd;cat values with which the interburst frequency of FRBs ranges over the entire gamma-band. When gcat is plotted against Kd;cat , there exists only a narrow area where the gamma-band FRBs can be induced (Fig. 7C). The optimal Kd;cat value ranges
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Fig. 6. Di¡erential e¡ects of Kd;cat and gcat on the FRB pattern. The number of intraburst spikes, burst frequency and intraburst ISIs are plotted against gcat (A) and Kd;cat (B). The ISIs between the ¢rst and second spikes, between the second and third spikes, between the third and fourth spikes, and between the fourth and ¢fth spikes are shown with blue diamonds, green diamonds, orange diamonds, and sky blue diamonds, respectively. The current intensity was ¢xed at 3 WA/cm2 . With an increase in gcat or with a decrease in Kd;cat , the number of intraburst spikes and the bursting frequency are increased. The burst frequency displays discontinuous jumps simultaneously with increases in the number of intraburst spikes, while the frequency is almost linearly increased as long as this number remains constant. Note longer ISIs between the third and fourth spikes in (A) than in (B), and also note the absence of FRBs of quintet spikes in (A).
approximately between 14 and 18 WM. For a Kd;cat of 15 WM, an optimal gcat for inducing gamma-band FRBs is found to be 108 mS/cm2 . When gcat is changed leaving the Kd;cat (15 WM) unchanged, neither a larger nor a smaller gcat outside the narrow area achieves such FRBs. With a larger gcat (a blue cross labeled as d in Fig. 7C), bursts of abortive triplet spikes are induced when the current intensity is increased, and consequently higher frequency FRBs (V70 Hz) are not achieved (Fig. 7D). On the other hand, with a smaller gcat (a red cross labeled as e in Fig. 7C), bursts of doublet spikes with intraburst ISIs longer than 3 ms are induced by a decrease of the current intensity, and consequently lower frequency FRBs (V25 Hz) are not achieved (Fig. 7E). E¡ects of a raised [K þ ]o on gamma-band bursts In a recent in vitro experiment (Gray and McCormick, 1996), FRBs were induced in a regular spiking neuron by increasing [Kþ ]o from 3 to 5 mM, and this e¡ect was attributed entirely to the attenuation of AHP due to the positive shift of the reversal potential of Kþ currents (EK ). We have examined the e¡ects of increasing [Kþ ]o on the ¢ring pattern in our model neuron. When Ecat
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together with EK and Eleak are allowed to shift with a raised [Kþ ]o according to the Goldman equation, FRBs are e⁄ciently induced in the regular spiking model neuron (Fig. 8A and B). However, when only EK and Eleak are allowed to shift leaving Ecat unchanged, FRB cannot be induced although the ¢ring frequency consequently becomes higher than the control (Fig. 8A and C). Although the results shown above do not necessarily imply that the attenuation of AHP alone cannot induce such FRBs as observed in the in vitro study (Brumberg et al., 2000), they demonstrate that the experimental result can be easily accounted for by the present model. Such a reduction of AHP by increasing [Kþ ]o is di¡erent from the muscarine-induced suppression of AHP discussed in the previous study (Wang, 1999). The present model suggests that Kþ is involved in the generation of FRBs as a charge carrier, presumably through the cationic channel.
DISCUSSION
INaP -based model vs. Icat -based model A previous simulation study proposed that the gamma-band FRB was primarily generated, without
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Fig. 7. A tight reciprocal relationship between gcat and Kd;cat achieving FRBs in the entire gamma frequency range. (A) In a model neuron with a combination of gcat (108 mS/cm2 ) and Kd;cat (15 WM) (a black cross labeled with a in (C)), doublet spikes during FRBs remain stable regardless of the decrease or increase in the interburst interval (IBI). Thus, FRBs throughout the entire gamma frequency band can be achieved as the intensity of depolarizing current pulses is changed. (B) Bursting frequency (red circles) and intraburst ISI (blue triangles) are plotted against the intensity of depolarizing current pulses. With increasing the intensity of current pulses, the bursting frequency is linearly increased. When the bursting frequency exceeds 25 Hz, ISIs are decreased to be less than 3 ms and remain almost constant. (C) Only combinations of Kd;cat and gcat within a narrow area (a green narrow band) in the Kd ^g plane ful¢ll the criteria of FRBs with stable doublet spikes throughout the entire gamma frequency band. (D) and (E) The two combinations of Kd;cat and gcat indicated with a blue cross (Kd;cat = 15 WM, gcat = 125 mS/cm2 ; labeled as d) and a red one (Kd;cat = 15 WM, gcat = 100 mS/cm2 ; labeled as e) lie outside the optimal narrow area and can not achieve high- and low-frequency FRBs, respectively. In fact, the triplet and doublet spikes (lower traces in (D) and (E), respectively) induced at d and e in the Kd ^g plane for an increased and a decreased current intensity, respectively, have intraburst ISIs longer than 3 ms.
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Fig. 8. E¡ects of raised [Kþ ]o on FRBs. (A) Control responses with regular spiking pattern to two di¡erent current pulses are shown for [Kþ ]o = 3 mM, gcat = 58 mS/cm2 . (B) Following an increase in [Kþ ]o from 3 to 5 mM, Ecat together with EK and Eleak were positively shifted. This resulted in the generation of FRBs with doublet spikes. (C) By contrast, FRBs were not generated when only EK and Eleak were positively shifted, but Ecat was left unchanged to exclude the e¡ect of raised [Kþ ]o on cation channels. Note that a positive shift of EK by 13 mV was insu⁄cient to induce FRBs, while a positive shift of Ecat only by 2 mV could generate FRBs.
Ca2þ -activated channels, by a ‘ping-pong’ interplay between action potentials in the somatic compartment and INaP in the dendritic compartment (Wang, 1999). In such a two-compartment model neuron, the somatic action potential propagated backward into the dendrite to activate INaP , which in turn generated a DAP and led to the generation of bursts. The bursting pattern of the model neuron was determined by the degree of inhibition of a muscarine-sensitive Kþ current, while the oscillation period was primarily determined by a slowly inactivating Kþ current. In terms of generating FRBs, our model has several advantages over the INaP -based FRB model. First, multiple compartments are not necessary in our model. In our model, DAPs, which are essential for the generation of bursts, are produced by a Ca2þ -activated Icat in the soma. On the contrary, in the INaP -based model the ping-pong mechanism between the somatic and dendritic compartments is essential for generating bursts of multiple spikes. Wherever, in the extensive apical dendrites of pyramidal neurons, a synaptic input caused membrane depolarization, spikes were ¢rst generated at axon hillock or initial segment and thereafter they propagated backward into the original synaptic input site (Stuart and Sakmann, 1994). Then, the electrotonic separation or distance between the two sites generating spikes and the subthreshold depolarization underlying FRBs would largely a¡ect the behavior of spiking. Since a large electrotonic distance may serve as a low-pass ¢lter, the electrotonic interactions between the soma and dendrites would be signi¢cantly suppressed at the interburst frequencies higher than the cut-o¡ frequency. Therefore, it is di⁄cult for the ping-pong mechanism to achieve FRBs beyond the cut-o¡ frequency
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(V10^30 Hz: see the simulations for neocortical neurons by Douglas and Martin, 1990), which was in fact the case with a two-compartment model ( 6 40 Hz, Wang, 1999). By contrast, the frequency of FRB in our singlecompartment model can range up to 70 Hz, as has been observed in in vivo (Gray et al., 1989, 1997; Gray and McCormick, 1996; Friedman-Hill et al., 2000; Maldonado et al., 2000) and in vitro (Brumberg et al., 2000) experiments. The crucial role of 40^60 Hz FRBs has been found in recordings from the visual cortex of the cat and monkey (Gray et al., 1989; Gray and Singer, 1989; Friedman-Hill et al., 2000; Maldonado et al., 2000) and from the prefrontal cortex of the monkey (Mikami et al., 2001). In contrast, however, to the previous two-compartment model (Wang, 1999), it is di⁄cult for the present model to achieve FRBs at frequencies lower than 25 Hz without INaP . Second, in the INaP -based two-compartment model, the number of intraburst spikes almost linearly increased with an increase in the intensity of current pulses (Wang, 1999), easily leading to a tonic ¢ring without achieving interburst frequencies higher than 40 Hz. This is because the activation of INaP increases with membrane depolarization produced by increasing the intensity of depolarizing current pulses (Fig. 5B, C and D). This could be another reason why it is di⁄cult for the INaP -based model to have the maximum frequency of FRBs higher than 40 Hz. By contrast, in the present single-compartment model, the number of intraburst spikes can remain constant irrespective of an increase in the intensity of current pulses, and consequently stable FRBs are achieved throughout the entire gamma frequency band (Fig. 7B). This situation is realized when Icat is deacti-
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vated in the interburst intervals (Fig. 5B^D). In previous in vitro and in vivo studies (Brumberg et al., 2000; Gray and McCormick, 1996; Mikami et al., 2001) except one in vivo study (Steriade et al., 1998), the number of intraburst spikes remained almost constant in spite of changes in the interburst frequencies. These experimental results can be easily accounted for by the present model. Third, cholinergic induction of FRB (Gray and McCormick, 1996; Brumberg et al., 2000) or gamma oscillation (Buhl et al., 1998) is in favor of our model, because it has been shown that INaP is depressed by activation of muscarinic receptor (Mittmann and Alzheimer, 1998) while there are many studies demonstrating the enhancement of Ca2þ -dependent Icat by activation of muscarinic receptors (Caeser et al., 1993; Constanti and Libri, 1992; Fraser and MacVicar, 1996; Guerineau et al., 1995). Cation channels with a low Ca2þ a⁄nity Recently, the dissociation constant of Ca2þ -dependent small-conductance potassium channels (Kd;SK ) for Ca2þ has been reported to be 0.3^0.4 WM (Xia et al., 1998). By contrast, Kd;cat (14^18 WM; Fig. 7C) has been predicted to be approximately 30-fold larger than the Kd;SK in the present simulation study. We have found no condition to achieve FRBs in the entire gamma frequency range with any Kd;cat smaller than 14 WM. Such a large di¡erence in the Ca2þ a⁄nity is suitable for activating Icat and ISK successively. The fast bu¡ering and the slow pump extrusion of intracellular free Ca2þ , which is rapidly increased by a Ca2þ in£ux during action potentials through voltage-gated Ca2þ channels, create a biphasic Ca2þ transient following each action potential. Since the activation time constant of a Ca2þ -dependent channel is inversely proportional to the sum of Kd and [Ca2þ ]i , the transient activation of Icat with a large Kd precedes that of ISK with a small Kd following each action potential (Fig. 4C). Hence a hump-like DAP followed by an AHP is generated. An enhancement of this DAP by increasing gcat or decreasing Kd;cat triggers doublet/triplet spikes. Such a burst of spikes is terminated by the growth of
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the hyperpolarizing e¡ect of ISK during the burst (Fig. 4C) and, thus, FRBs can be induced in our model under the suitable combinations of gcat and Kd;cat (Fig. 7). The biphasic Ca2þ transient was normally seen underneath the plasma membrane of neurons (HernandezCruz et al., 1990). Although channels with such a low Ca2þ a⁄nity as Kd;cat (see Fig. 7C) are not known yet, it is known that large-conductance Kþ channels change the Ca2þ a⁄nity from the low micromolar range to the submicromolar range depending on the membrane depolarization (Sun et al., 1999). Similarly, changes in Kd;cat , but with yet unknown mechanisms, might switch the ¢ring pattern from regular spiking to FRBs, as indicated in Fig. 3C. Such a switching of the ¢ring pattern has been demonstrated to occur when strong depolarization was repetitively applied to cortical pyramidal neurons (Kang and Kayano, 1994; Kang, 1997; Brumberg et al., 2000). For understanding the role of FRB in generating the gamma-band EEG rhythm, it is very important to address how the FRB emerges and how the pacemaker FRB neuron can bring the coherence onto di¡erent subsets of neurons. The present study has revealed the key role of Ca2þ -dependent Icat in generating and controlling FRBs. A preliminary simulation of a small network among this type of model neurons suggests that the degree of synchrony in neuronal ¢ring instantaneously and dramatically changes at the critical points where the FRB pattern changes (Fig. 6A and B). Extending the present model towards a large-scale network model, we may be able to see how the spatio-temporal pattern of coherence among cortical pyramidal cells is governed by FRB pattern or, for instance, by the Ca2þ sensitivity of the Icat . Such spatio-temporal patterns of the coherence among neurons may provide additional information to those conveyed by rate coding.
Acknowledgements1This work was supported by a Grant-inAid for Scienti¢c Research on Priority Areas (A) and by CREST (Core Research for Evolutional Science and Technology) of Japan Science and Technology (JST).
REFERENCES
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NSC 5874 12-11-02
Cyaan Magenta Geel Zwart