INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

One of the most important cellular properties exhibited by thalamic relay neurons is a voltage-dependent, transient, low-threshold Ca²⁺ conductance, leading to depolarization via Ca²⁺ entry through T-type Ca²⁺ channels (Crunelli et al. 1987; Jahnsen and Llinás 1984a,b). This Ca²⁺ current thus is known as I_T, and it produces a low-threshold spike referred to here as the "Ca²⁺ spike." I_T can be activated by a depolarizing input, such as an excitatory postsynaptic potential (EPSP), but only if the membrane already has been hyperpolarized sufficiently for 50-100 ms (Jahnsen and Llinás 1984a). This is because at more depolarized potentials, I_T is inactivated and a period of hyperpolarization is required to remove the inactivation or deinactivate I_T. In regard to its transient nature, the low-threshold Ca²⁺ conductance underlying I_T behaves much like the Na⁺ conductance underlying the conventional action potential except the voltage sensitivity of I_T operates in a more hyperpolarized range and its kinetics are slower.

The importance of I_T derives from the fact that the large, depolarizing Ca²⁺ spike that it produces usually reaches threshold for activating conventional Na⁺/K⁺ action potentials, producing a brief burst of firing (Jahnsen and Llinás 1984a,b). This is known as the burst mode of firing and reflects the response of a relay cell to depolarizing inputs when the cell is initially hyperpolarized. When the cell is depolarized initially so that I_T is inactive, the cell responds to the same depolarizing inputs with a steady stream of unitary action potentials, and this is known as the tonic mode of firing (Huguenard and McCormick 1992; Jahnsen and Llinás 1984a,b; McCormick and Huguenard 1992b; Steriade and Llinás 1988). Both firing modes are a ubiquitous property of relay cells throughout the mammalian thalamus, and both firing modes have been seen during both in vitro and in vivo recording, the latter including recording from awake, behaving animals (Ghazanfar and Nicolelis 1997; Guido and Weyand 1995; Guido et al. 1992, 1995; Sherman and Guillery 1996). It is now clear that the different patterns of firing represented by burst and tonic firing provide different types of relays of information to cortex. Because the firing mode is determined by the inactivation state of I_T at the time an activating, depolarizing input, such as an EPSP, arrives at a relay cell, it is of great interest to understand how I_T behaves.

Since the first descriptions of I_T and burst firing in thalamic neurons, which involved intracellular recording in current-clamp mode (Jahnsen and Llinás 1984a,b), much of the quantitative and systematic study of I_T has involved voltage-clamp recording, often in acutely dissociated cells and usually in rodents (Coulter et al. 1989; Hernández-Cruz and Pape 1989). The main advantage of this approach is that it permits a more complete description of the voltage dependency and kinetics of a voltage-dependent conductance, such as that associated with I_T. Such data have enabled the development of Hodgkin-Huxley-like models that reproduce many properties of Ca²⁺ spikes (Crunelli et al. 1989; Destexhe et al. 1998; Huguenard and McCormick 1992, 1994; McCormick and Huguenard 1992; Wang et al. 1991; Williams et al. 1997). However, among those aspects that have not been studied systematically in current-clamp mode, experimentally and including comparison with theory, is the nature of threshold for Ca²⁺-spike generation.

The main purpose of the present study is to do this for relay cells of the lateral geniculate nucleus, the thalamic relay of retinal input to cortex, using current-clamp recording from in vitro slice preparations. We have found that the sensitivity for generation of Ca²⁺ spikes is surprisingly high, with nearly all-or-none threshold behavior, and that these spikes and their associated bursts of action potentials occur with quite long latency near threshold. We compare our experimental data with simulations from a cellular model based on published voltage-clamp data. Also, our recordings are from the cat thalamus and the model is derived from voltage-clamp studies of rodent thalamus, so our comparisons also serve to test the generality of the behavior of I_T across species.

	METHODS

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Slice preparation

All intracellular recordings were made from relay neurons of the cat's lateral geniculate nucleus in an in vitro thalamic slice preparation. Young animals (4-8 wk old) of either sex were handled in compliance with approved animal protocols. Briefly, animals were anesthetized deeply with a mixture of ketamine (25 mg/kg) and xylazine (2 mg/kg) and mounted in a stereotaxic device. We then opened a rectangular area of skull overlying the lateral geniculate nucleus and removed a block of tissue containing the lateral geniculate nucleus. After placing this block in oxygenated cold slicing solution (see next section), we killed the animal with an overdose of pentobarbital sodium. Thalamic slices (400-500 µm thick) were cut in a coronal or sagittal plane with a vibrating tissue slicer and placed in a holding chamber for 2 h before recording. Individual slices were transferred to an interface type recording chamber and continuously superfused with warm oxygenated physiological solution (see next section). The tissue was maintained at 33°C for all recordings.

Solutions

The slicing solution was used throughout the tissue preparation until the slice was transferred to the holding chamber. It contained (in mM) 2.5 KCl, 1.25 NaH₂PO₄, 10.0 MgCl₂, 0.5 CaCl₂, 26.0 NaHCO₃, 11.0 glucose, and 234.0 sucrose. The physiological solution used in the holding and recording chamber for intracellular recording contained (in mM) 126.0 NaCl, 2.5 KCl, 1.25 NaH₂PO₄, 2.0 MgCl₂, 2.0 CaCl₂, 26.0 NaHCO₃, and 10.0 glucose; and it was gassed with a mixture of 95% O₂-5% CO₂ to a final pH of 7.4. In some experiments, the Na⁺ channel blocker, tetrodotoxin (TTX; 0.5-1 µM), was added to the bath to block conventional Na⁺/K⁺ action potentials. The intracellular recording electrodes were filled with a solution that was either 3 M KAc or 1 M KAc with 2-5% neurobiotin.

Electrophysiological recordings and data analysis

We obtained intracellular recordings in current-clamp mode from the geniculate relay cells using sharp electrodes. Recording electrodes were pulled to an impedance of 40-80 M at 100 Hz when filled with the aforementioned solution. An Axoclamp 2A amplifier was used in bridge mode to enable current-clamp recordings. During the recordings, we adjusted an active bridge circuit to balance the drop in potential produced by passing current through the recording electrode. Current protocols were generated using either AxoData or pClamp on a laboratory computer, and the data were stored digitally with a temporal resolution of 0.2 ms. Cells were held at different initial holding potentials by injecting current into the cell (i.e., the holding current). The duration of holding current before evoking responses assured that any depolarizing sag due to I_h would reach an equilibrium, leading to a stable membrane voltage for a sufficient time to create a stable level of I_T inactivation. Responses, including Ca²⁺ spikes, then were evoked by depolarizing current steps on top of the initial holding current, the steps ranging from 10 to 3,000 pA and having a duration of 200-1,000 ms. After >10 s of holding current, additional current steps were given on top of the holding current at a rate ranging from 0.1 to 0.3 Hz to ensure a stable initial holding potential before the depolarizing steps.

We adopted a set of minimum requirements to judge an intracellular recording as acceptable. These included having a resting membrane potential more negative than 50 mV and action potentials that reached at least 10 mV; most overshot 0 mV. We routinely monitored and balanced the bridge during the recordings, and we also measured the input resistance by determining the slope of the linear portion of the I-V relationship.

In some experiments for which we had neurobiotin in the recording electrode, we iontophoresed this dye into the cell with depolarizing current steps (200-500 pA in steps of 200-400 ms at 0.5-2 Hz for 1-3 min). This was done at the end of electrophysiological recording. After each such iontophoretic injection, we processed the slice with a standard protocol to reveal the neurobiotin (Zhan and Troy 1997) and assess the morphology of the labeled cell with the light microscope.

Model

Our experimental observations (see RESULTS) demonstrated that threshold behavior and excitability of Ca²⁺ spikes were similar in the presence or absence of TTX (Hernández-Cruz and Pape 1989; Jahnsen and Llinás 1984a,b). Thus we chose for our computations a minimal Hodgkin-Huxley type of model that neglects the primary currents involved in generating and shaping action potentials. Our model includes those currents that we believe capture the essence of our observed activation of Ca²⁺ spikes. Because our main goal for modeling here is qualitative understanding of threshold and latency phenomena under current clamp mode, we sought to obtain semiquantitative agreement, rather than detailed quantitative fits, of simulated results with experimental data.

Exploratory simulations were performed with the computer program Cclamp, developed by Huguenard and McCormick (1994). The basic behaviors of low-threshold excitability were obtained using the "I_T" case and "blocking" several voltage-dependent currents until we identified our candidate model. Thus we developed a minimal model for generation of Ca²⁺ spikes that mimics our experimental results.

The current balance equation is

(1)

where I_T is the "T type" low-threshold Ca²⁺ current, I_A is a transient K⁺ current, the leakage components (I_K-leak and I_Na-leak) are ohmic, and I_app represents any current injected into the cell; V is membrane potential (in millivolts); t is time (in milliseconds); and C is total capacitance, equal to 290 pF, corresponding to a cell model with a surface membrane area of 29,000 µm². We used the formulations for I_T and I_A found in the computer program Cclamp of Huguenard and McCormick (1994) and based on their earlier voltage-clamp data (summarized in McCormick and Huguenard 1992). The model uses the Goldman-Hodgkin-Katz formulation for I_T as

(2)

where P_T is the maximum permeability of an open channel (30 cm³/s), z = 2, Ca_int and Ca_ext are the concentrations of Ca²⁺ inside and outside the cell, respectively (assumed fixed in our model at 50 nM and 2 mM, respectively); F, R, and T are Faraday's constant, the gas constant, and absolute temperature, respectively (Hille 1992). The transient K⁺ current is given by

(3)

with the reversal potential V_K = 105 mV and g_A = 2 µS unless stated otherwise. The leakage currents are given by

(4)

and

(5)

where g_Na-leak = 2.65 nS, g_K-leak = 7 nS, and V_Na = 45 mV. The general form for the gating dynamics of the voltage-gated channels is

(6)

where x = m_T, h_T, m_A, or h_A with

(7)

The specific parameter values (in millivolts) are _mT = 60.5, k_mT = 6.2, _hT = 84, k_hT = 4.03, _mA = 60, k_mA = 8.5, _hA = 78, k_hA = 6, and the "time-constant" functions are

(8)

(9)

(10)

(11)

(12)

We adjusted the gating rates to 33.5°C from Cclamp's set conditions of 23.5°C by using a temperature correction factor, , of 3.

All computed results shown here were obtained with the above minimal model using the software XPPAUT (found at http://www. pitt.edu/~phase/). For numerical integration we used the fourth-order, adaptive-step Runge-Kutta method in XPPAUT (with error tolerance, 10⁵). Computations were performed on a Linux/Unix Pentium II workstation.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Experimental observations

We obtained intracellular recordings in the current clamp mode from a total of 42 neurons from the cat's lateral geniculate nucleus. All appeared to be relay cells on the basis of readily evoked Ca²⁺ spikes. Also we injected with tracer and recovered a subset of five of these cells after the recording session, and their anatomic properties clearly distinguished them as relay cells and not interneurons (Friedlander et al. 1981; Guillery 1966; Sherman and Friedlander 1988). Detailed measurements of resting potential and input resistance were made for a subset of 35 neurons. For these, resting potential was 60.1 ± 4.6 (SD) mV (with a range of 50 to 69 mV). The input resistances tended to be slightly higher as the cells were hyperpolarized. At rest, the these values were 31.3 ± 14.7 M, whereas at 20 mV hyperpolarized to rest, they were 41.5 ± 16.0 M; this difference was statistically significant (P < 0.001 on a paired t-test). Of the 42 cells, 20 were studied extensively before TTX (0.5-1 µM) was added to the bath to block sodium channels and thus action potential generation; 15 were studied briefly before TTX application and then extensively after; and the remaining 7 cells were studied extensively both before and after TTX application.

Figure 1 illustrates two of the main phenomena we observed for a typical cell from our sample. The cell in Fig. 1A was held at the indicated initial membrane potential at which the voltage-dependent Ca²⁺ conductance underlying the low-threshold spike is deinactivated substantially. Figure 1A, top traces, shows the voltage responses to application of rectangular current injection (indicated below the traces) in the presence of TTX, which was applied to show evoked Ca²⁺ spikes without action potentials. With incremental 10-pA steps of current injection, we saw ohmic responses for smaller injections until a voltage threshold was reached, at which point that current injection and all larger ones evoked a Ca²⁺ spike. Interestingly, the amplitude of the Ca²⁺ spike evoked by the smallest suprathreshold current injection was effectively as large as all Ca²⁺ spikes evoked from larger current injections. The input resistance of this cell was ~60 M, and thus each incremental 10-pA current would produce 0.6-mV depolarization. This means that the full-blown Ca²⁺ spike was evoked as a nearly all-or-none event over a range of 1 mV in membrane potential (see DISCUSSION for further consideration of this nearly all-or-none behavior). This nearly all-or-none behavior of the Ca²⁺ spike is shown more dramatically in the Fig. 1A, bottom, in which the evoked Ca²⁺ spikes are overlapped. These overlapped traces also show that the evoked suprathreshold responses exhibit three components: an initial ohmic response (arrow 1), a slow depolarization (arrow 2), and the Ca²⁺ spike itself (arrow 3), although the first two phases are best seen with smaller depolarizing inputs and may be difficult to see with larger ones (see traces 5 and 6 in Fig. 2B). Figure 1, C and D, shows this same nearly all-or-none behavior for two other geniculate relay cells with TTX application. Figure 1B shows that a similar effect was seen in the same cell as shown in Fig. 1A before TTX was applied. Fewer traces are illustrated in Fig. 1B than in Fig. 1A for clarity because the evoked action potentials tend to obscure the key features, but when all incremental 10-pA current injection steps are analyzed, the result is exactly as predicted from Fig. 1A with action potentials present.

View larger version (25K): [in this window] [in a new window]   Fig. 1. Activation of low-threshold Ca2+ spikes from relay cells in an in vitro slice preparation through the cat's lateral geniculate nucleus. Cells were recorded intracellularly in current-clamp mode. Membrane potentials shown indicate the initial holding potentials. A: responses of 1 cell in the presence of 1 µM TTX. Top: original responses. Activation was achieved by injecting depolarizing rectangular current pulses (indicated below the traces) starting at 200 pA and incremented in 10-pA steps. Ohmic responses are evoked from smaller currents; the longest latency Ca2+ spike evoked is from the smallest suprathreshold current injection, and this latency decreased monotonically as the amplitude of the current injection was increased. Bottom: same evoked Ca2+ spikes as shown top but shifted in time so that they are superimposed. Numbered arrows indicate 3 different phases of the response to the current injections: 1 points to the initial, ohmic response; 2 points to a slower voltage-dependent depolarization; and 3 points to the Ca2+ spike. Note that the Ca2+ spikes overlap almost perfectly, indicating their nearly all-or-none nature. B: same cell as in A before application of TTX. Top: original recordings; bottom: traces adjusted in time to overlap the evoked Ca2+ spikes. Latencies of evoked bursts of action potentials decreased with increasing amplitudes of current injection. To avoid extensive overlap in the bottom traces, only 3 burst responses are shown. C and D: 2 further examples with 1 µM TTX application as in A. Although numbered arrows are not shown for these examples, the 3 phases of the evoked response are clearly visible as in A. Scale marks represent 100 ms and 10 mV and those in A also apply to B.

The second main phenomenon we observed is the dramatic decrease in the latency of evoked Ca²⁺ spikes as current injection steps are increased from the first suprathreshold injection. This is seen both in the Ca²⁺ spike latencies (Fig. 1, A, C, and D) and in the latency of action potentials riding the crests of the Ca²⁺ spikes (Fig. 1B). In both cases, the latency shift exceeds 150 ms. Interestingly, as can be seen in the overlapped traces of Fig. 1, all Ca²⁺ spikes seem to activate from near the same voltage, as if a simple threshold phenomenon was involved similar to that for an action potential. The slow depolarization leading to the Ca²⁺ spike (indicated by arrow 2 in Fig. 1A) has a lower slope for smaller current injections, and this is why it takes longer to reach the threshold to activate the Ca²⁺ spike with smaller current injections. Modeling results described in the following text suggest that this slower depolarization indicated by arrow 2 in Fig. 1A is caused by activation of I_T that is too small to be regenerative. The outward ohmic current nearly balances the slowly growing inward I_T during this phase of extended latency. The response becomes regenerative when the threshold is reached to activate the nearly all-or-none Ca²⁺ spike.

NEARLY ALL-OR-NONE NATURE OF CA²⁺ SPIKES. Figure 2 further illustrates for the same cell as shown in Fig. 1, A and B, the nearly all-or-none nature of the Ca²⁺ spike during TTX application. Figure 2A shows the peak evoked voltage as a function of injected current when the cell initially is held at one of three different initial holding membrane potentials. At a holding potential of 59 mV (the resting potential for this cell), there is insufficient deinactivation of the low-threshold Ca²⁺ conductance for activation of Ca²⁺ spikes. The result is that current injection from these holding potentials evokes only an ohmic response, thereby producing a gradual, smooth increase in evoked potential with injected current (Fig. 2B, traces 1-3). However, at initial holding potentials of 77 and 87 mV, at which the Ca²⁺ conductance is significantly deinactivated, Ca²⁺ spikes are evoked by suprathreshold depolarizing current injections. Thus we see initial ohmic responses for small current injections until a threshold is reached, at which point a sudden, dramatic increase in evoked membrane voltage is seen (as in Fig. 1). This is because Ca²⁺ spikes are now evoked. After this threshold is reached, larger current injections have little effect on the amplitude of the evoked Ca²⁺ spike (Fig. 2B, traces 4-6). Note also that the Ca²⁺ spikes evoked from the 87-mV holding potential are slightly larger. This is because I_T is more deinactivated when the initial holding potential is 87 mV. Also, a larger current step is required from 87 mV to get to the activation threshold for I_T so that this response curve is shifted to the right compared with the curve starting at 77 mV.

View larger version (25K): [in this window] [in a new window]   Fig. 2. Relationship between current injection (400-ms pulses) and the evoked responses as in Fig. 1A for the same cell in the presence of 1 µM TTX. A: current injection amplitude vs. peak evoked voltage defined as the voltage difference between the peak voltage response and the membrane potential before the current pulse. Shown are 3 curves representing 3 different initial holding potentials as shown. Two of the curves ( and ) involved holding potentials (77 and 87 mV) that were sufficiently hyperpolarized that Ca2+ spikes were evoked once their activation threshold was crossed. Other curve ( and ) involved a holding potential at rest (59 mV) that was too depolarized for generation of Ca2+ spikes, and thus mainly ohmic responses were seen. Numbered arrows reflect data points from representative traces in B. B: representative traces as indicated for generation of the plots in A. C: current injection vs. amplitude of evoked response for 11 other cells. Initial holding potentials for these cells were 80 to 90 mV, ensuring fairly complete deinactivation of the underlying Ca2+ conductance, and injected current was in the form of square wave pulses of 400- to 1,000-ms duration.

View larger version (35K): [in this window] [in a new window]   Fig. 3. Relationship between current injection (400-ms pulses) and the evoked action potential responses for same cell as in Fig. 1B. A: current injection amplitude vs. total number of action potentials evoked. Curves with  and  reflect burst firing, at least initially, whereas the curves with  and  reflect tonic firing. B: representative traces for data points indicated in A. Note that the small current injection of trace 4 evokes only a single burst, that the larger injection of trace 5 evokes an initial burst followed after ~70 ms by a single tonic action potential, and that the even larger injection of trace 6 evokes an initial burst followed immediately by sustained tonic firing. C: current injection amplitude vs. initial firing frequency, which was calculated from the 1st 6 action potentials evoked by the current injection to provide a clearer comparison of burst vs. tonic firing.

View larger version (25K): [in this window] [in a new window]   Fig. 4. Comparison between burst and tonic firing evoked by current injections without TTX. Curves with  represent burst firing, and those with  represent tonic firing. Unlike the plots in Fig. 3 in which the abscissas are the amplitude of the depolarizing pulse from the holding potential, the abscissas in these plots are the total current injected (i.e., the sum of the current used to move the cell from rest to the initial holding potential and the depolarizing current step). A: same data as plotted in Fig. 3A, except that the abscissa is recalculated to reflect the total current injection. Of particular interest is the range of near-threshold current injections, which show that burst firing always can be evoked with less current injection than is the case for tonic firing. For much larger depolarizing currents, which evoked strong tonic firing even after an initial burst, the curves largely overlap. Note that the activating thresholds for burst firing are hyperpolarized with respect to those for tonic firing. B and C: data plotted as in A for 2 additional cells, showing only the range of near-threshold current injections. Again, bursts are evoked with less current injection than is the case for tonic firing.

LATENCY OF LOW-THRESHOLD CA²⁺ ACTIVATION. The latency of the evoked Ca²⁺ spike becomes considerably shorter with greater current injection (see Fig. 1). This appears to be related to the rate of rise of the initial depolarizing response before a near-threshold voltage is reached to initiate the Ca²⁺ spikes. Figure 5, A and B, plots the relationship between current injection and these latencies for the same cell as shown in Fig. 1, A and B. From both initial holding potentials (77 and 87 mV) during TTX application, the latency of Ca²⁺ spike was reduced gradually as the current injections increased, from 186 to 25 ms at a holding potential of 77 mV and from 139 to 24 ms at a holding potential of 87 mV (Fig. 5A). A similar effect is seen before TTX application when the latency to the first action potential peak riding the crest of a Ca²⁺ spike is plotted against injected current (Fig. 5B). These latencies were reduced from 143 to 4 ms at a holding potential of 77 mV and from 164 to 7 ms at a holding potential of 83 mV. The latency reduction was sharp for initial suprathreshold current injections and became more gradual with larger current injections.

View larger version (21K): [in this window] [in a new window]   Fig. 5. Relationship between current injection and response latency. A: effect of current injection on response latency during 1 µM TTX application for same cell as shown in Fig. 1A. Latency is measured from the initiation of the depolarizing pulse to the peak of the evoked low-threshold Ca2+ spike (LTS). Relationships are shown for the 2 initial holding potentials shown. B: relationships for the same cell as in A before TTX application for the 2 initial holding potentials shown before TTX application. Latency was measured from the start of the current injection to the peak of 1st action potential of the evoked burst. C: effect of current injection on response latency of Ca2+ spike for population of additional 11 cells in the presence of 1 µM TTX.

View larger version (12K): [in this window] [in a new window]   Fig. 6. Activation of Ca2+ spikes with brief depolarizing pulses (5-ms duration). Shown are 4 superimposed traces, including responses to the largest 2 subthreshold injections and to the smallest 2 suprathreshold injections. No TTX was present for these responses. Ca2+ spikes were evoked well after the cessation of the current injection, and the latency of the evoked Ca2+ spikes dropped dramatically as the suprathreshold injection was increased in amplitude.

VARIATION OF CA²⁺ SPIKE AMPLITUDE WITH DEINACTIVATION LEVEL. The preceding results show that the Ca²⁺ spike is activated effectively in a nearly all-or-none manner. Thus with 10-pA current injection steps, a depolarizing input activates only an ohmic response if subthreshold and a full-blown Ca²⁺ if suprathreshold. We cannot rule out the possibility that partial Ca²⁺ spikes might have been seen if we used smaller injection increments, but the fact that we never saw this in any of our cells suggests that the incremental range of current injections over which partial Ca²⁺ spikes might be activated is much less than 10 pA and thus 1 mV of depolarization for a cell with an input resistance of 50 M. In theory (see DISCUSSION), graded responses may be expected, although probably with quite a steep dependence on stimulus amplitude, which is qualitatively what we have seen experimentally.

View larger version (27K): [in this window] [in a new window]   Fig. 7. Effect of deinactivation on Ca2+ spike amplitude. A: relationship between current injection and peak evoked response in presence of 1 µM TTX as in Fig. 2. Different initial holding membrane potentials are shown and were achieved with constant hyperpolarizing currents. Series of depolarizing current pulses incremented by 10 pA and with a duration of 300 ms was injected to evoke ohmic responses (subthreshold currents) or Ca2+ spikes (suprathreshold currents). Curves with  represent activation of Ca2+ spikes, and those with  indicate only ohmic responses. B: representative series of traces from the data plotted in A for activation of Ca2+ spikes from different initial holding membrane potentials. Note that for the relatively depolarized initial holding potential (72 mV), graded amplitudes of response are seen that may or may not involve Ca2+ spikes (see text for details). C: relationship between the initial holding membrane potential and the maximum Ca2+ spike amplitude for same cell as is illustrated in Fig. 1A. Ordinate represents the amplitude of the Ca2+ spike evoked from just suprathreshold current injections. LTS amplitude is defined here as the voltage difference between the peak of the Ca2+ spike and the level of the ohmic response remaining 100 ms after the Ca2+ spike ended. , 6 initial membrane potentials from which the plots in A are taken. D: relationship between initial holding membrane potential and normalized amplitude of evoked Ca2+ spike for 11 other relay cells.

Theoretical observations

Our minimal model with only two voltage-dependent currents, I_T and I_A, shows the basic features of generation of Ca²⁺ spikes as seen experimentally (Fig. 8; compare with Figs. 1 and 7). These features include a nearly constant amplitude of the evoked Ca²⁺ spike for different current steps from a given holding level, a dramatic effect on latency of the evoked Ca²⁺ spikes over a narrow range of suprathreshold current injection amplitudes, increasing gradation and decreasing amplitude of the Ca²⁺ spike with less hyperpolarized holding levels. A response for a suprathreshold stimulus shows an early small depolarization (the initial ohmic response, see arrow 1 in Fig. 1A), followed by a phase of slow depolarization (see arrow 2 in Fig. 1A), and finally the Ca²⁺ spike (see arrow 3 in Fig. 1A). To evoke a Ca²⁺ spike from the model, as in the experimental data, the initial holding membrane potential must be hyperpolarized adequately so that I_T has been deinactivated sufficiently. Under these conditions, the peak amplitude of the Ca²⁺ spike is nearly independent of the amplitude of the suprathreshold current step. When the simulation starts with more I_T deinactivation (Fig. 8, A and B), the Ca²⁺ spike appears in a nearly all-or-none fashion. If the holding potential is less hyperpolarized, meaning more of the I_T is inactivated, responsiveness may become more graded (Fig. 8C). The behavior is thus not strictly all or none, as becomes apparent when the level of I_T inactivation is relatively high, and this correlates with the experimental observation that graded responses can be obtained from relatively depolarized holding potentials (see trace 3 of Fig. 10; see also DISCUSSION).

View larger version (22K): [in this window] [in a new window]   Fig. 8. Simulated membrane potential time courses showing responses to depolarizing current steps for minimal model of relay cell without action potential currents but with the K+ "leak" conductance and IA present. Three different initial holding membrane potentials are shown. Below the voltage traces is shown the time course of the 400-ms current injection. In all cases, as in the experimental data, the minimum suprathreshold current injection evoked the Ca2+ spike with the longest latency, and this latency reduced to asymptotic levels as the amplitude of current injections increased (see Figs. 1, 2, 8, and 9). Current injections are given as Ip = Ip, o + j*Ip;j = 1,2. A: initial holding potential of 90 mV, corresponding to holding current of 258 pA with Ip, o = 48 pA and Ip = 10 pA. , relative peak values of IT (using the holding membrane potential as a reference line) and their times of occurrence. B: initial holding potential of 85 mV, corresponding to holding current of 220 pA with Ip, o = 20 pA and Ip = 16.7 pA. C: initial holding potential of 80 mV, corresponding to holding current of 188 pA with Ip, o = 10 pA and Ip = 12 pA. Response amplitude diminishes as holding potential increases to less negative levels (A-C), as in experimental data from Figs. 11 and 12. Range of latencies is largest and gradation of response amplitude is smallest for the most negative holding potential (A).

DISSECTING THE MODEL'S CA²⁺ SPIKE. To understand the effect of just suprathreshold activating currents on the latency of Ca²⁺ spikes, we have dissected in Fig. 9 the time course of the response in Fig. 8A to the minimum suprathreshold current injection, which evoked the longest-latency Ca²⁺ spike. Also shown is the time course of h_T (- - -), which is the fraction of channels not inactivated: the larger is h_T, the more I_T is deinactivated, and as expected depolarization during the Ca²⁺ spike causes h_T to drop with a time scale of tens of milliseconds. Figure 9B shows the interplay between the different currents during this response. The early depolarization originates from the combination of injected current and leakage current (I_leak  I_app;  - ), which is initially inward. This current approaches zero at around 83 mV, and the membrane would come to rest there if no other currents were involved. However, I_T (  ), beginning from near zero at the (deactivated) holding state, continues to grow slowly. Although the leakage and injected current combination becomes outward, this is more than offset by the slowly growing I_T, which is inward. Until relatively late in the response, these components remain small and nearly cancel each other. Thus while the net current () is depolarizing during period of slow depolarization before initiation of the Ca²⁺ spike, its time course is not monotonic. It jumps up, then decreases to a minimum, becoming quite small and thereby stalling the depolarization. I_T and membrane voltage continue to grow slowly together until the membrane voltage enters the regime for regenerative activation of I_T. The nonlinear voltage dependent gating of I_T determines the voltage range in which the sharply rising upstroke of the Ca²⁺ spike occurs.

View larger version (15K): [in this window] [in a new window]   Fig. 9. Time courses of variables and individual ionic currents during a simulated response of a Ca2+ spike as in Fig. 8. Response shown is to a just-suprathreshold current injection (97 pA), so the Ca2+ spike has a long latency. Initial holding membrane potential is 95 mV achieved with a holding current of 300 pA. A: time courses of membrane voltage (V, ) and inactivation variable (hT, - - -) to a depolarizing current pulse starting at 100 ms. Ordinate reflects relative values for V and hT. Membrane potential initially relaxes with a membrane time constant (of ~30 ms) to a more slowly rising phase before the Ca2+ spike begins abruptly. Some inactivation of IT occurs (hT decreases) during the initial depolarization before the initiation of the Ca2+ spike, and the inactivation then proceeds rapidly during the upstroke of the Ca2+ spike. B: membrane currents during response shown in A; inward currents are plotted down. IT (  ) grows slowly and then very fast to initiate the Ca2+ spike. IA (- - -) makes little contribution until the Ca2+ spike begins. Combined stimulating and leakage currents (Iapp and Ileak, respectively;  - ) is initially inward then becomes outward, opposing IT. Sum of currents with negative sign () is proportional to dV/dt, and it thus has a local flat minimum during period before the Ca2+ spike is initiated. Horizontal  corresponds to 0 current.

View larger version (31K): [in this window] [in a new window]   Fig. 10. Simulated membrane potential time courses showing responses to depolarizing current steps as in Fig. 8 showing effects of IA. Initial membrane potential is 91.7 mV, which is achieved with a holding current of 272 pA. A: simulations with all voltage-dependent K+ currents suppressed, including IA. As in the experimental data shown in Fig. 1, the responses to subthreshold current steps are ohmic and to suprathreshold steps are Ca2+ spikes. Increasing the suprathreshold current injections significantly reduces the latency of the evoked Ca2+ spike with little effect on its amplitude. B: simulations similar to A, except that IA is present with a varying inactivation time constant (see Eq. 11). Numbered arrows indicate simulations with the normal time constant (1), twice the time constant (2), and 10 times the time constant (10). These changes in IA affect the amplitude but not the latency of the evoked Ca2+ spikes. Thus the latency effect in this model does not depend on IA.

View larger version (16K): [in this window] [in a new window]   Fig. 11. Trajectories of Ca2+ spikes in the voltage-current phase plane. Shown are responses of a reduced model having IT and Ileak but without IA to various current injection steps from an initial holding membrane potential of 91.5 mV. A: series of curves of membrane voltage vs. membrane current, the latter calculated from C*dV/dt, where C is capacitance, each curve has a clockwise motion (arrows) with time as a parameter. Initial holding state is at the far left. B: single curve () redrawn from A. Shown for comparison are the current-voltage trajectories of the individual components of the total current, which are IT (- - ) and Ileak  Iapp ( - ); these currents are plotted with reversed sign. At each voltage, the sum of components equals C*dV/dt. Note that the rapid phase of the upstroke is nearly equal to the regenerative IT. , equal time increments (5 ms) along the trajectory of the Ca2+ spike; their wide spacing during the upstroke indicates that it is very rapid.