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1Department of Physiology and Zlotowski Center for Neuroscience, Faculty of Health Sciences, Ben-Gurion University, Be’er-Sheva, Israel; and 2Department of Physiology, Institute of Medical Sciences, Hebrew University-Hadassah Faculty of Medicine, Jerusalem, Israel
Submitted 26 February 2006; accepted in final form 21 June 2006
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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; Schwartzkroin 1975; for a detailed description of firing and bursting patterns, see Fig. 1 in Su et al. 2001). A large body of evidence now indicates that the propensity of a neuron to burst depends not only on its constitution, i.e., the nature and properties of ionic conductances expressed in its plasma membrane, but also on its environment, i.e., the ionic composition of the milieu in which it is embedded. Thus regular firing pyramidal cells readily convert to a bursting mode when the extracellular concentrations of Ca2+ ([Ca2+]o) or K+ ([K+]o) decrease or increase, respectively (Jensen et al. 1994; Su et al. 2001). Such changes in extracellular ion composition accompany neuronal activity (Heinemann et al. 1977), which may explain why the propensity for bursting of pyramidal cells increases with the level of activity in their surrounds (Harris et al. 2001). Interestingly, blocking Ca2+ currents or Ca2+-activated K+ currents (Azouz et al. 1996; Jensen et al. 1994; Su et al. 2001) does not increase the propensity for bursting. Given that the bursting mode plays important roles in electrical signaling, normal and abnormal neuronal synchronization, and induction of long-term synaptic plasticity (Izhikevich et al. 2003; Lisman 1997; Yaari and Beck 2002), it is important to understand how constitution and environment interact in regulating this discharge mode. To date, most theoretical studies of intrinsic bursting have focused on the former factor.
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; Jung et al. 2001; Larkum et al. 1999; Metz et al. 2005; Wong and Prince 1981). In line with these experimental observations, many models of Ca2+ current-dependent bursting suggest a "ping-pong" interplay between fast Na+ and K+ currents in the soma and slow Ca2+ and K+ (mostly Ca2+-dependent) currents in the apical dendrites (Mainen and Sejnowski 1996; Pinsky and Rinzel 1994; Traub and Miles 1991; Traub et al. 1991, 1994; Warman et al. 1994). In adult CA1 pyramidal cells, however, bursting behavior persists after almost complete truncation of the apical dendrites (Yue et al. 2005). Therefore the mechanism of bursting in the latter neurons may be different from the ping-pong mechanism, which depends on the integrity of apical dendrites. We study the mechanism by which changes in [Ca2+]o modulate transitions between regular firing and bursting in adult CA1 pyramidal cell by combining electrophysiological, computational, and analysis techniques.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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All animal experiments were conducted in accordance with the guidelines of the Animal Care Committees of the Hebrew University. Adult male Sabra rats were decapitated under deep isoflurane anesthesia, and transverse hippocampal slices (400 µm) were prepared with a vibrating microslicer (Leica) and transferred to a storage chamber perfused with oxygenated (95% O2-5% CO2) artificial cerebrospinal fluid (ACSF) containing (in mM) 124 NaCl, 3.5 KCl, 2 MgCl2, 1.6 CaCl2, 26 NaHCO3, and 10 D-glucose, pH 7.4, osmolarity 305 mosM, where they were maintained at room temperature. Slices were placed one at a time in an interface chamber (33.5°C) and perfused with oxygenated ACSF. To truncate the apical dendrites of CA1 pyramidal cells, a deep cut was made in stratum radiatum close to and parallel to stratum pyramidale using a broken pipette or a razor blade chip propelled by a micromanipulator (Yue et al. 2005). Truncation was confirmed by recording field potentials in stratum pyramidale (see following text) before and after cutting. Orthodromic field potentials evoked by stimulating in stratum radiatum disappeared after cutting, while antidromic field potentials evoked by stimulating in alveus remained large (Yue et al. 2005). The slices were allowed to recover in the chamber for 
1 h before initiating a recording session.
Electrophysiological recordings
Intracellular recordings were obtained using sharp glass microelectrodes containing 4 M K+-acetate (90–110 M
). An active bridge circuit in the amplifier (Axoclamp 2B, Axon Instruments, Foster City, CA) allowed simultaneous injection of current and measurement of membrane potential. The bridge balance was carefully monitored and adjusted before each measurement. The intracellular signals were filtered on-line at 10 kHz, digitized at a sampling rate of 10 kHz, and stored by a personal computer using a data-acquisition system (Digidata 1322A) and pCLAMP software (Axon Instruments).
Drugs
Stock solutions of 4
-phorbol 12,13-dibutyrate (PDB; 10 mM), riluzole (10 mM), and phenytoin (100 mM) were prepared in dimethyl sulfoxide (DMSO) and stored at –20°C. They were usually diluted at 1:1,000 when added to the ACSFs. Control ACSFs contained equal amounts of DMSO (0.001%), which by itself had no effects on the measured parameters. All other drugs were added to the ACSFs from aqueous stock solutions. Chemicals and drugs were obtained from Sigma (Petach-Tikva, Israel).
Cell model
The somatic, single-compartment model was represented by coupled differential equations according to the Hodgkin-Huxley-type scheme. We constructed the model in two stages. In the first stage, we introduced only the ionic currents that are involved in firing dynamics in 0 [Ca2+]o, at which it is simpler to analyze. In the second state, we added voltage-gated Ca2+ and Ca2+-activated K+ currents, to explore their influence on bursting behavior.
MODEL FOR 0 [CA2+]o.
The model includes the currents that are known to exist in the soma and proximal dendrites: the transient Na+ current (INa) and the delayed rectifier K+ current (IKdr) that generate spikes, and the muscarinic-sensitive K+ current (IM) that contributes the slow variable necessary for bursting (Bertram et al. 1995; Yue and Yaari 2004, 2006). A model with these three currents only is the minimal model that allows bursting. We added the persistent sodium current (INaP) because we wanted to focus on its contribution to bursting (Su et al. 2001; Yue et al. 2005). The A-type K+ current (IA) is included as well even though its density is much higher in the apical dendrites than in the soma (Hoffman et al. 1997). The current balance equation is (Borg-Graham 1999)
 | (1) |
3(V)h(V – VNa), INaP(V) = gNaPp(V)(V – VNa), IKdr(V, n) = gKdrn4(V – VK), IA(V, b) = gAa3(V)b(V – VK), IM(V, z) = gMz(V – VK). The conductances and reversal potentials are: gNa = 35 mS/cm2, gNaP varies between 0 and 0.41 mS/cm2, gKdr = 6 mS/cm2, gA = 1.4 mS/cm2, gM = 1 mS/cm2, VNa = 55 mV, VK = –90 mV. The kinetics equations and parameters are listed in Table 1. The cell model for zero [Ca2+]o has five dynamical variables: V, h, n, b, and z.
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p, the half-maximum potential of INaP; see Table 1). In addition, we added three more ionic currents: the high-threshold Ca2+ current (ICa), and two Ca2+-activated K+ currents, namely, the fast Ca2+-activated K+ current (IC), which contributes to rapid spike repolarization (Storm 1987), and the slow Ca2+-activated K+ current (IsAHP), which mediates a slow afterhyperpolarization (AHP) and spike frequency adaptation (Madison and Nicoll 1984). This model includes, therefore the most importantcurrents known in somata and proximal axons of CA1 cells. The current balance equation in this variation of the model is
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([Ca2+]i)c(V – VK), IsAHP(V, q) = gsAHPq(V – VK). The conductances and reversal potentials are: gCa is typically between 0 and 0.2 mS/cm2, gC = 10 mS/cm2, gsAHP = 5 mS/cm2, VCa = 120 mV. The kinetics equations and parameters are listed in Table 2. The dynamics of the calcium concentration inside the cell, [Ca2+]i are
 | (3) |
= 0.13 cm2/(msxµA), 
Ca = 13 ms. The variable [Ca2+]i in our model is dimensionless (Traub et al. 1994). It is proportional to the Ca2+ concentration in a thin internal cylindrical shell adjacent to the membrane.
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; Thompson and Wong 1991). The hyperpolarization-activated cationic current (Ih) (Magee 1998; Vasilyev and Barish 2002) was not included in the model because its density in the soma is much lower than in the apical dendrites (Magee 1998) and because the activation kinetics of the largest component of Ih are faster than that of IM (Maccaferri and McBain 1996; Magee 1998; Spain et al. 1987) and cannot support bursting via a mechanism based on two slow variables (Bertram et al. 1995). The apamin-sensitive, small-conductance (SK) Ca2+-activated K+ current (IAHP), which can be evoked under voltage-clamp conditions in CA1 pyramidal cells (Gu et al. 2005; Stocker et al. 1999), was not included in the model. The reason for this exclusion is that in current-clamp recordings this current does not contribute appreciably to spike frequency adaptation and the medium AHP (Gu et al. 2005).
NUMERICAL METHODS.
Simulations were performed using the fourth-order Runge-Kutta method with a time step of 0.05 ms implemented as a C program or within the software package XPPAUT (Ermentrout 2002), that was used also for computing bifurcation diagrams.
STIMULATION.
We analyzed the firing patterns of the neuron model in response to two types of stimuli, namely, brief and prolonged square positive current pulses. In the former case, pulse duration was 3 ms, so it could evoke a spike response without interfering with spike afterpotentials (Su et al. 2001; Yue and Yaari 2004). In the latter case, pulse duration was very long, allowing neuronal dynamics to reach steady-state behavior (mathematically, converging to an attractor). There were also conditions in which the neuron model fired spontaneously without application of positive currents.
FIRING PATTERNS.
Depending on the model parameters, the neuron model might fire in one of two patterns when stimulated with prolonged positive current pulses. 1) Regular, tonic firing, in which neurons fired solitary spikes in a periodic manner. Often, two distinct types of tonic firing were observed: low-frequency tonic firing at frequencies of up to 
5 Hz; and high-frequency tonic firing, in which neurons fired continuously at 100 Hz. 2) Rhythmic bursting, in which sequential bursts were separated by periods of neuronal quiescence, typically lasting several hundreds of milliseconds. Rhythmic bursting was either periodic or aperiodic (chaotic) (Terman 1992).
BURSTING.
In this study, we defined a burst as a tight cluster (inter-spike interval in the order of 10 ms) of two or more spikes generated alone (as in the case of bursts evoked by brief stimuli) or distinctly separated from the following spikes (as in the cases of spontaneous bursting or bursting evoked by prolonged stimuli). In the cases of prolonged or spontaneous activity, we quantified bursting behavior at long times (after the dynamics has converges to an attractor), namely after transient effects of the initial conditions have decayed. The average number of intraburst spikes (NS) was computed by averaging the number of spikes per burst during a 1.5-s period 1 s after stimulus onset (i.e., after the dynamics has converged to an attractor) and rounding up this average to the closest integer. Doublet states were particular cases of periodic bursting states in which the neuron model fired exactly two spikes in a burst, namely, NS = 2 (Mandelblat et al. 2001). Clearly, in regular spiking cells NS = 1.
Fast-slow analysis
We used the fast-slow method (Bertram et al. 1995; Hoppensteadt and Izhikevich 1997; Izhikevich 2000, 2006; Rinzel and Ermentrout 1998) to study the bursting mechanism, to determine the role of the currents INaP and IM in the dynamics, and to determine the necessary conditions for bursting. This method has been applied successfully to analyze periodic bursting of various biophysical models of neurons (e.g., Mandelblat et al. 2001). We applied the method for the case [Ca2+]o = 0 (and thus p = –47 mV and gCa = 0) because it has simpler dynamics. The analysis is based on separating the dynamical variables of the system into two subsystems, "fast" and "slow." The model has five dynamic variables: V, h, n, b, and z. The first four variables are considered to be "fast" and belong to the fast subsystem. The variable z is considered to be "slow." The analysis is exact in the limit z 
, where z is the time constant of z. It describes an approximation of the dynamics for large but finite z. In the first stage of the analysis, the bifurcation diagram of the fast subsystem was computed with the slow variable z considered as a parameter. In the second stage, the dynamics of z itself were computed using the time-averaged values of the fast subsystem.
If the dynamics of the fast subsystem converges to a stable rest state (fixed point), V is determined by its value for the fixed point of the fast subsystem for the instantaneous value of z, denoted by Vrest,fast(z), whereas z evolves slowly (Table 1). If z[Vrest,fast(z)] > z, z increases slowly, and if z[Vrest,fast(z)] < z, z decreases slowly. If, for a certain value of z denoted by zFP, z[Vrest,fast(zFP)] = zFP, namely, the line representing the stable rest state, Vrest,fast(z), intersects with z(V), the activation carve of z (Table 1), the point [Vrest,fast(zFP), zFP] represents a fixed point of the full system. It is stable if dVrest,fast(z)/dz < [dz(V)/dV]–1. The dynamics of the fast subsystem may converge also to a stable tonic firing state (limit cycle), where Tz is the time period of the cycle that depends on z. The approximation that z is much larger than the other time scales in the system, and, specifically, z > Tz, means that z does not vary significantly during one cycle of the fast subsystem. Therefore we can average the dynamical equation for dz/dt (Table 1) over one cycle of the fast, tonic firing to obtain (Bertram et al. 1995; Izhikevich 2006; Mandelblat et al. 2001)
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...
denotes time average over a period Tz and Vequiv is defined implicitly by the equation
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dz/dt by dz/dt. Therefore the dynamics of the slow variable z is determined, on average, by the right hand side of Eq. 5. If z[Vequiv(z)] > z, z increases slowly, and if z[Vequiv(z)] < z, z decreases slowly. We consider the case where, for a certain value of z denoted by zLC, the line representing Vequiv(z) intersects with the activation carve of z, z(V), namely z[Vequiv(zLC)] = zLC. In this case, the point (Vequiv(zLC), zLC) represent a limit cycle of the full system. It is stable if dVequiv(z)/dz < [dz(V)/dV]–1. In particular, the limit cycle is stable if the slope of Vequiv(z) at z = zLC is negative. |
RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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We have recorded from 24 CA1 pyramidal cells after truncation of their apical dendrites. When activated with brief (3–4 ms) depolarizing current pulses, truncated neurons displayed normal spikes followed by a distinct, albeit variable in size, afterdepolarization (ADP; Fig. 1A, left). The mean (±SE) resting potential, apparent input resistance, and spike amplitude of these neurons were –67.8 ± 0.5 mV (range, –75 to –61 mV), 40.1 ± 2.0 M (range, 20.5 to 48.7 M), and 92.1 ± 5.7 mV (range, 71.7 to 103.2 mV), respectively. These values are well within the values obtained for intact neurons (Yue and Yaari 2004; Yue et al. 2005) and indicate that despite the damage inflicted by the cut and loss of most apical dendrites, the electrophysiological properties of the axo-soma are well preserved. When activated with prolonged (200 ms) depolarizing current pulses of increasing intensities, the neurons discharged repetitively and displayed spike frequency adaptation (Fig. 1B, left). Like in intact neurons (Su et al. 2001; Yue et al. 2005), approximately half of the neurons were nonbursters and the other half were mostly high-threshold bursters (bursting only in response to long depolarizing current pulses of 2–3 times threshold intensity) with a small subset of low-threshold bursters (bursting in response to threshold-straddling stimuli).
In all 24 truncated neurons examined, lowering [Ca2+]o from 2 to nominally 0 mM caused a progressive increase in the spike ADP. In 19 of these neurons (79.2%), the increase in spike ADP culminated in a high-frequency burst of three to six spikes as their minimal response to threshold depolarization (Fig. 1A). Lowering [Ca2+]o also reduced spike frequency adaptation (Fig. 1B). These effects are similar to those described previously in intact neurons (Azouz et al. 1996; Su et al. 2001). Thus functional apical dendrites are not required for the expression of bursting behavior in low [Ca2+]o.
In nominally Ca2+-free ACSF, many of the neurons converted to bursting mode manifested rhythmic bursting at their native resting potential (see following text, Fig. 4A) or at more depolarized potentials. Increasing the depolarizing current intensity enhanced the frequency of bursting without strongly affecting the shape of individual bursts. However, at the higher frequencies of bursting, the number of intraburst spikes progressively decreased with time (Fig. 1B, right).
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A previous study in intact CA1 pyramidal cells concluded that the augmented spike ADPs and bursting behavior in low [Ca2+]o are driven predominantly by persistent Na+ current (INaP) (Su et al. 2001). To extrapolate this finding to truncated neurons, we tested the effects of the INaP blocker riluzole (Yue et al. 2005) on bursting induced by low [Ca2+]o in five such neurons. Riluzole (10 µM) blocks INaP in CA1 pyramidal cells almost completely while exerting a lesser effect on the transient Na+ current (Spadoni et al. 2002; Urbani et al. 2000; Yue et al. 2005). Representative results are shown in Fig. 2. Adding 10 µM riluzole to the ACSF abolished low Ca2+-induced bursting (Fig. 2A) and converted rhythmic bursting during prolonged depolarizations to regular firing (Fig. 2B). Similar results were obtained (data not shown) with 100 µM phenytoin (n = 3) and 5 µM PDB (n = 2), which also block INaP (Cantrell et al. 1996; Chao and Alzheimer 1995; Yue et al. 2005). Together, these data strongly suggest that bursting in low [Ca2+]o is driven predominantly by INaP.
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Because inactivation of INaP in CA1 pyramidal cells is very slow (in the order of seconds) (French et al. 1990), burst termination likely involves activation of an outward K+ current. In nominally Ca2+-free ACSF, Ca2+-activated K+ currents are inoperative. Hence a likely candidate for burst termination in this condition is the M-type K+ current (IM), previously shown to activate during the spike ADP and repolarize the neuron back to its resting potential (Yue and Yaari 2004). We tested this notion using linopirdine and XE991, selective blockers of IM (Schnee and Brown 1998; Wang et al. 1998). Representative results from one truncated neuron bathed in Ca2+-free ACSF are shown in Fig. 3. When activated by brief depolarizing pulses, the neuron generated a burst of three spikes (Fig. 3A, left). Adding 10 µM linopirdine to the Ca2+-free ACSF markedly prolonged the burst and delayed repolarization of the neuron for several hundreds of milliseconds (Fig. 3A, middle). Both the primary burst and the underlying plateau potentials were readily suppressed by subsequent addition of 10 µM riluzole to the ACSF (Fig. 3A, right), suggesting that they are driven by INaP. Similar results were obtained when the neuron was stimulated with prolonged depolarizing current pulses that induced rhythmic bursting (Fig. 3B, left). Under the influence of linopirdine, rhythmic bursting converted to a plateau depolarization that lasted throughout the period of stimulation (Fig. 3B, middle). Again, adding 10 µM riluzole to the ACSF suppressed the initial burst responses and the plateau depolarizations and imposed a regular firing mode (Fig. 3B, right). Similar results were obtained in five truncated neurons treated with 10 µM linopirdine and two truncated neurons treated with 3 µM XE991. Together these data suggest that bursting in truncated neurons bathed in Ca2+-free ACSF is due to interplay between INaP and IM.
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We have constructed a model of CA1 pyramidal cells to further understand the mechanisms underlying their somatic bursting. The results of the model are compared with the experimental results from the following aspects. First, can the model account for the variant firing patterns observed experimentally as its parameters are varied? Second, do the propensity for bursting and NS vary in a similar manner in the model and in the experiments as Iapp, gNaP, or gM vary? Finally, are the influences of gCa, gC, and [Ca2+]o on the propensity for bursting and NS similar in the model and in the experiments? We have included in the model neuronal dynamics with only two time scales, namely that of spikes (a few milliseconds) and that of bursting (100 ms). Experimentally, we found that many CA1 pyramidal cells also exhibited very slow dynamics (>1 s) that may cause rhythmic bursting to change eventually to repetitive spikes (Figs. 1B, middle, and 3B, left, 2 top traces). The very slow dynamics, however, often reached a steady state. In those cases, the neurons displayed rhythmic bursting continuously, as shown in Fig. 4A. Therefore we omitted the very slow dynamics from the model, allowing it also to manifest rhythmic bursting (Fig. 4B).
Fast-slow analysis of the necessary conditions for bursting and the roles of INaP and IM
We have shown in the preceding text that two currents, namely INaP and IM, play essential roles in bursting in truncated CA1 pyramidal cells bathed in 0 [Ca2+]o. The INaP is considered here to be nonactivating (at the relevant time scales), and its activation variable, p, is considered to be instantaneous and equal to p(V). The activation variable of IM, z, is relatively slow, with time constant z = 75 ms. Therefore we used the fast-slow method in the condition that z is the only slow variable.
To illuminate the contribution of INaP to bursting, we carried out the analysis for four values of gNaP: 0 (Fig. 5A), 0.2 mS/cm2 (Fig. 5B), 0.3 mS/cm2 (Fig. 5C), and 0.41 mS/cm2 (Fig. 5D). In all panels, the bifurcation diagrams of the fast subsystem are computed with z considered as a parameter. The steady state (fixed point; thin black line) is stable for large z. This stable rest state coalesces with an unstable state and ceases to exist in a saddle-node bifurcation. The rest state is stable again for negative values of z (negative z values do not have physiological meaning, but they are important for a complete mathematical analysis). At the z value where the high rest state gains its stability (a Hopf bifurcation that is out of the scale in Fig. 5, A and B), an oscillatory state (limit cycle) emerges, corresponding to tonic, periodic firing. This oscillatory state extends toward the right. For gNaP = 0 (A), the oscillatory state is the only stable state for small (positive) z values, and the rest state is the only stable state for larger z values. There is a tiny regime of bistability where both states are stable, but for our description, it can be ignored. For gNaP = 0.2 mS/cm2 (B), the bistable regime, in which both the rest state and the oscillatory state exist and are stable, is more extended. For gNaP = 0.3 mS/cm2 (C), the oscillatory solution extends toward the right, coalesces with an unstable periodic state, and this unstable state coalesces with another stable oscillatory state with a larger amplitude. For gNaP = 0.41 mS/cm2 (D), the Hopf bifurcation point is shifted to the right, and there is bistability between a rest state and a depolarized plateau. The stable oscillatory state with large amplitude almost disappears, and the amplitude of the remaining oscillatory state is small.
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1. BISTABILITY OF THE FAST SUBSYSTEM.
The bursting mechanism in our model depends on the bistability of the fast subsystem. To obtain bistability, there should be strong-enough inward current that is active in steady state at membrane potentials around spike threshold, namely [gNam3(V)h(V) + gNaPp(V)]x(V – VNa) should be large enough for V around spike threshold. In our model, this inward current can be generated by INaP or the window INa. In addition, the minimal voltage of the fast subsystem during the tonic state should be depolarized enough (above the thin dotted black line representing the unstable rest state). Therefore the kinetics of the currents underlying spike depolarization and repolarization (INa and IKdr in our model) should be fast enough (Bertram et al. 1995). Indeed, a previous model in which h and n were 3.7 times slower than in the present model that did not display bursting behavior (Golomb and Amitai 1997). If the kinetics of these variables are too fast, however, the fast subsystem will exhibit a high plateau depolarization As a result, the neuronal voltage will oscillate between a rest state and a high plateau depolarization instead of exhibiting bursts of spikes.
2. STRENGTH OF gNaP. The effect of INaP on the tonic state is larger than its effect on the rest state because this current is activated at depolarized membrane potentials (Table 1). As gNaP increases, the tonic state is shifted more to the right(toward larger values of z) than the rest state. Hence the bistable regime expands as gNaP increases. This regime may even be produced by increasing gNaP if it had not existed for gNaP = 0. This means that elevating gNaP may induce bursting behavior and increase NS. Increasing gNaP also shifts the Hopf bifurcation of the depolarized plateau potential to the right (compare Fig. 5, A–D). Therefore when gNaP is very large, the active phase of the burst becomes a high plateau depolarization or a sequence of low-amplitude fast spikes converging to a high plateau depolarization rather than full-blown spikes (Fig. 5D).
3. ACTIVATION CURVE OF IM.
To allow bursting behavior the activation curve of IM, z(V), should be located between the rest state curve and the curve of Vequiv (Bertram et al. 1995). If z(V) is shifted toward more depolarized levels, (z, the half-maximum potential of IM, increases), the neuron will fire tonically. If z(V) is shifted toward more hyperpolarized levels (z decreases), the neuron will be quiescent at rest.
4. STRENGTH OF gM. IM is proportional to gMxz. For gM = 0 (equivalent to z = 0 in Fig. 5, A–D), the neuron fires tonically or reaches a high plateau depolarization when gNaP is very large. Increasing gM is equivalent to compressing the bifurcation diagram of the fast subsystem along the z axis in Fig. 5, without modifying the kinetics of the slow variable z. Therefore as gM increases, the slow variable z spends less time in the oscillatory state and NS decreases. When NS decreases to 1, the neuron fires in a tonic pattern. For very large gM, the rest state of the fast subsystem intersects with the activation curve of z, and the neuron becomes quiescent at rest.
SUMMARY OF NECESSARY CONDITIONS FOR BURSTING.
Our analysis reveals four necessary conditions for bursting. All these conditions demand that certain parameters will be in particular ranges. 1) The kinetics of the variables underlying spike generation (inactivation of INa and activation of IKdr) should be fast enough (otherwise, there will be no bistability and the neuron will fire continuously) but not be too fast (to prevent a plateau depolarization). 2) The fast subsystem should show bistability, and gNaP supports bistability. Therefore gNaP should be strong enough to generate bistability unless this bistability has already been generated by the window INa. If gNaP is too strong, however, the neuron model will manifest very fast spiking with small amplitude or a plateau depolarization. 3)The half-maximum potential of IM, z, should not be too depolarized (at which condition the neuron will fire tonically) or too hyperpolarized (at which condition the neuron will be at resting potential). 4) gM should be strong enough (otherwise, the neuron will fire in a regular mode or will display a high plateau depolarization) but not too strong (otherwise, the neuron will fire in a regular mode or will not fire at all).
The analysis is carried out in the limit that z is much larger than all the other time constants, even though in reality it is not extremely large (z = 75 ms). Therefore it is necessary to examine whether the conditions for bursting obtained using the analysis are still valid for this value of z. In the following subsections, we turn back to the full system and study it using numerical simulations. We also examine whether the model can account for the variant firing patterns of CA1 pyramidal cells.
Effects of varying gNaP
CA1 pyramidal cells perfused with Ca2+-free ACSF display a diversity of firing patterns, ranging from regular firing to spontaneous rhythmic bursting (Figs. 1–3 and 4A) (Azouz et al. 1996; Su et al. 2001). We first tested the hypothesis that a variation in the density of persistent Na+ channels may generate such diversity. We assumed that in this condition p = -47 mV and examined the consequences of increasing gNaP. When gNaP was set to zero, the neuron fired in a regular mode during sustained depolarization (Fig. 6A). As gNaP was raised, the propensity for bursting increased, so that at gNaP = 0.08 mS/cm2, the neuron became a periodic burster when strongly depolarized (Fig. 6B) and at gNaP = 0.18 mS/cm2, it burst-fired periodically in response to all suprathreshold stimuli (Fig. 6C). Further increasing gNaP to 0.3 mS/cm2 increased the burstiness of the neuron, so it now fired a burst also in response to brief stimuli (Fig. 6D). Changing VL from –70 to –62 mV destabilized the rest state and caused spontaneous rhythmic bursting when gNaP was high (Fig. 4B). For gNaP = 0, the neuron was quiescent even for VL = –62 mV.
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To further characterize the neuron model behavior, we have drawn a map of NS in bursts evoked by brief (3 ms; Fig. 7A) and prolonged (Fig. 7B) stimuli in the parameter plane of gNaP and Iapp. All other parameters were fixed as in the reference parameter set. Brief pulses evoked only a single spike if gNaP was zero or small. As gNaP increased, NS increased as predicted by the fast-slow analysis. For very large gNaP, the neuron fired many fast and short-amplitude spikes, almost converging to a high plateau, before going back to rest (Fig. 7A, top right). The borders of the regimes of 4, 5, 6, etc. spikes were almost independent of Iapp when it was not too large. This is consistent with the common experimental observation that when Iapp is brief, NS is independent of Iapp. The map for bursts evoked by prolonged current pulses also showed that NS increases with gNaP (Fig. 7B). The parameter Iapp had a larger impact on NS in comparison with the case of brief pulses, so that increasing it could raise NS by one or two spikes.
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COMPARING EXPERIMENTAL RESULTS, MODELING, AND THEORY. Increasing gNaP increases NS and eventually generates bursts with fast, low-amplitude spiking both in the analysis (Fig. 5) and the simulations (Fig. 7). Similarly, blocking gNaP decreases NS and eventually eliminates bursting in response to prolonged current pulses in the experiments (Fig. 2), the simulations (Figs. 6 and 7) and the mathematical analysis (Fig. 5). The same dependency NS on gNaP is found experimentally and computationally in response to brief pulses. Furthermore, in response to prolonged pulses, the number of spikes in the first burst is larger than the number of spikes in subsequent bursts in both experiments and simulations.
Role of gM
Our data (Fig. 3), together with previous data about intact neurons (Yue and Yaari 2004, 2006), show that selective block of IM markedly enhances the burstiness of CA1 pyramidal cells. These data suggested that IM normally counteracts the depolarizing drive furnished by INaP. Hence, it is expected that NS will increase with INaP and decrease with IM. The situation, however, is more complicated because other states appear in the model, as we describe in the following text.
We explored the effects of varying gNaP and gM at a given Iapp. In Fig. 8, we present maps of the various states as a function of gM and gNaP. For brief pulses (Fig. 8A), bursting states appeared at intermediate gNaP and gM values that were not too small. In this bursting regime, NS increased with gNaP and decreased with gM. Three other types of patterns appeared for gNaP values above the regimes of single bursts or spikes, denoted by I–III in Fig. 8A: I, a fast burst of spikes with decaying amplitudes followed by a high plateau; II, sustained, high-frequency firing; III, an irregular burst with many fast, low-amplitude spikes. The patterns in regimes I and II coexisted with the rest states, namely, the system was bistable in these parameter regimes.
The map obtained for neuronal responses to prolonged stimuli (Fig. 8B) was qualitatively similar to the previous map (Fig. 8A), although there were some notable differences. First, there were parameter regimes having NS = 1 (regular spiking behavior) that markedly differed in their firing rates. The firing rate in the regime on the right side of the map, denoted by "1slow," was 5–10 Hz, whereas the firing rate in the regime on the left side of the map, denoted by "1fast," was in the order 100 Hz. Second, for gNaP = 0, it was possible to obtain a burst of two spikes in a restricted gM domain. For larger gNaP values, there were two "fingers" in the map where NS was 2. The finger that extended to the right showed bursting with frequency in the order of 5 Hz. A second finger extending from gNaP = 0 to higher gNaP values (adjacent to the "1fast" regime) displayed spiking at a high rate (in the order of 100 Hz) in a doublet manner: the interspike interval alternated between larger and smaller values. Between the two fingers (for gM values larger 0.6 mS/cm2), the neuron exhibited normal bursting for intermediate gNaP values. Three types of patterns, analogous to the patterns obtained for brief pulses, are denoted by I–III in Fig. 8B.
To conclude, for both types of stimuli, regular bursts with large NS appear in a restricted parameter regime with intermediate values of gNaP and gM, composed of diagonal bands of constant NS. In this regime, NS indeed increases with gNaP and decreases with gM. Several irregular patterns appear outside of this regime.
For a given NS, f generally decreased as gM increased; At gM values at which NS decreased by 1, f increased abruptly (Fig. 8C). At and above a critical gM value (3.4 mS/cm2), the neuron became quiescent (f = 0). As expected from the fast-slow analysis (Table 1, Eqs. 4 and 5, Fig. 5), f depends linearly on 1/z for large z (Fig. 8D). Jumps in f occur at moderate values of z at every z value for which NS increases by 1.
COMPARING EXPERIMENTAL RESULTS, MODELING, AND THEORY.
We find that both in the experiment and in the simulations: 1) blocking gM increases the NS evoked by brief stimuli (Figs. 3A, middle and left, and 8A) and eventually transfers the neuron to a bistable tonic firing mode (Figs. 3A, third panel from left, and 8A). 2) Blocking gNaP after the gM blockage causes the neuron to fire only one spike in response to a brief pulse (Figs. 3A, right, and 8A). 3) In response to a prolonged pulse, blocking gM can transfer the neuron to a high plateau depolarization state (Figs. 3B, middle and left; and 8B). 4) If gNaP is also blocked, the neuron fires in a regular mode or becomes quiescent (Figs. 3B, right; and 8B). The conclusions of the analysis that are demonstrated in the simulations (Fig. 8) are 1) gM should be strong enough, but not too strong, to obtain bursting, 2) NS decreases with gM, and 3) f is proportional to 1/z.
Note that the firing frequency of the real neuron under the effects of both linopirdine and riluzole is of order 10 Hz (Fig. 3B), which corresponds to the firing pattern denoted by 1slow in Fig. 8B and not to 1fast. In contrast, in the cases in which the neuron bursts in "intact" cells and blocking gM leads to high plateau, additional blockade of gNaP in the model leads to fast tonic firing. This means that in this respect, the specific example in Fig. 3B is different from what is shown in Fig. 8B. In reality, but not in the model, the high plateau depolarization eventually decreases and the neuron repolarizes (compare Fig. 3B, middle, with Fig. 8B, I). This is because the model does not include dynamical processes with very slow time scale, such as slow inactivation of INaP or of INa (e.g., Fleidervish et al. 1996; French et al. 1990; Mickus et al. 1999).
Effects of varying [Ca2+]o
The analyses described in the preceding text were done in conditions of 0 [Ca2+]o. Here we use the model to interpret intriguing experimental evidence (Su et al. 2001) that reducing [Ca2+]o, but not blocking Ca2+ currents and Ca2+-activated K+ currents, may transfer a nonbursting cell into a burster. Clearly raising [Ca2+]o will modify the neuronal dynamics by introducing voltage-gated Ca2+ currents and Ca2+-activated K+ currents. In addition, raising [Ca2+]o will shift the voltage dependence of INaP activation back toward more positive potentials (Li and Hatton 1996; Yue et al. 2005). Figure 9, A—C, illustrates the firing patterns of the neuron model obtained for three sets of parameters (p is the half-maximum potential of the persistent sodium current): gCa = 0.08 mS/cm2, p = –41 mV (Fig. 9A), gCa = 0.05 mS/cm2, p = –44 mV (Fig. 9B), and gCa = 0.02 mS/cm2, p = –46 mV (Fig. 9C). The parameter set in Fig. 9A represents parameter values of physiological [Ca2+]o, whereas those in Fig. 9BC represent parameters values of reduced [Ca2+]o. The firing pattern of the neuron model obtained for the set gCa = 0, p = –47 mV, corresponding to [Ca2+]o = 0, has already been presented in Fig. 6D. It can be seen that reducing [Ca2+]o augments burstiness in this model.
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p and a second parameter related to the Ca2+-gated currents or Ca2+-activated K+ currents, are shown in Fig. 10. Increasing gCa with all Ca2+-activated K+ currents blocked (gC = gsAHP = 0) augmented NS for both brief and prolonged stimuli (Fig. 10, A and B) and weakly reduced f (Fig. 10C). To assess the effects of Ca2+-activated K+ conductances, we set gCa = 0.08 mS/cm2 and computed NS as a function of p and either gC or gsAHP. As expected, increasing gC suppressed burstiness and decreased NS for a specific value of p (Fig. 10, D and E); f increased with gC (Fig. 10F). Increasing gsAHP within the range of the parameter we used (from 0 to 20 mS/cm2) did not affect NS substantially (data not shown). This finding is expected because the IsAHP is slow, and the burst is terminated by IM before IsAHP becomes strong enough to have a considerable effect.
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The increase in burstiness on reducing [Ca2+]o resulted mainly from the decrease in p. However, the decrease of gCa per se also could affect NS. Although in most cases blocking gCa did not change NS (e.g., Fig. 9D and points 9A and 9D in Fig. 10H), there were also conditions in which blocking gCa transferred the system to a higher level of burstiness (for example, when gCa = 0.2 mS/cm2 and p = –45 mV, Fig. 10H). Similarly, although blocking gC did not lead to bursting for our reference parameter set (Fig. 9E), it could increase NS for lower values of
0.6 mS/cm2) to the regime of normal bursting was very complex and involved chaotic dynamics. The dotted line denotes the gNaP value used for C. The solid circles denoted by I–III in each map represent the parameter sets of the firing patterns shown right to the maps. The time course of the applied current is plotted below the voltage time courses. C: dependence of the bursting frequency f on gM. Parameters: Iapp = 1 µA/cm2, gNaP = 0.25 mS/cm2, 
