INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

The temporal discharge pattern of central neurons is an important element of signal processing and information transfer. Cortical neurons have traditionally been grouped into three broad classes according to their discharge patterns in response to depolarizing current injection: regular spiking, fast spiking, and intrinsic bursting (Connors and Gutnick 1990; Connors et al. 1982; McCormick et al. 1985). Both regular and fast spiking cells respond to depolarizing current with a repetitive discharge of action potentials but differ in that regular spiking neurons show significant frequency adaptation in their firing pattern compared with the consistent discharge frequency of fast spiking cells. However, the discharge pattern of intrinsic bursting neurons is quite distinct in generating a phasic burst followed by a tonic discharge of action potentials. Several studies have focused on distinguishing morphological and electrophysiological characteristics of neurons exhibiting these three patterns of spike output (Franceschetti et al. 1995; Jensen et al. 1994; Mason and Larkman 1990; Nuñez et al. 1993; Schwindt et al. 1997; Williams and Stuart 1999). A fourth discharge pattern consisting of rhythmic spike bursts in the -frequency range (20-80 Hz) has now been identified in cortical as well as sub-cortical and medullary neurons (Brumburg et al. 2000; Gray and McCormick 1996; Lemon and Turner 2000; Lo et al. 1998; Paré et al. 1995; Steriade et al. 1998; Turner et al. 1994). This pattern differs from intrinsic bursting cells by exhibiting a continuous and nonadapting series of spike bursts during current injection (Gray and McCormick 1996; Steriade et al. 1998; Turner et al. 1994).

Gamma frequency discharge is thought to be important to several aspects of signal processing and neuronal synchronization (Gray and McCormick 1996; Gray and Singer 1989; Lisman 1997; Ribary et al. 1991), yet comparatively few studies have examined the mechanisms underlying burst output at such a high-frequency. Gamma frequency bursting in hippocampus is known to involve extensive interneuronal synaptic circuitry (Buzsáki and Chrobak 1995; Stanford et al. 1998; Traub et al. 1998). Other studies have revealed that backpropagating dendritic spikes contribute to burst discharge by generating a depolarizing afterpotential (DAP) at the soma (Mainen and Sejnowski 1996; Turner et al. 1994; Wang 1999; Williams and Stuart 1999). The amplitude of the DAP can be augmented by a persistent sodium current (I_NaP) (Brumburg et al. 2000; Franceschetti et al. 1995; Wang 1999) or dendritic Ca²⁺ current (Magee and Carruth 1999; Williams and Stuart 1999). Alternatively, the amplitude of the DAP can be influenced by dendritic morphology because the dendrite-to-soma current flow increases with the relative dendritic to somatic surface area and decreases with axial resistance (Mainen and Sejnowski 1996; Quadroni and Knofnel 1994). Lemon and Turner (2000) recently described a novel mechanism of "conditional spike backpropagation" that modulates DAP amplitude and produces a -frequency oscillatory burst discharge in pyramidal neurons of the electrosensory system.

Electrosensory lateral line lobe (ELL) pyramidal cells are principal output cells in the medulla that respond to AM of electric fields detected by peripheral electroreceptors (Bastian 1981; Shumway 1989). Several studies have described the properties of burst discharge in ELL pyramidal cells (Bastian and Nguyenkim 2001; Gabbiani and Metzner 1999; Gabbiani et al. 1996; Lemon and Turner 2000; Metzner et al. 1998; Rashid et al. 2001; Turner and Maler 1999; Turner et al. 1994, 1996). Signal detection analysis has shown that ELL pyramidal cells generate burst discharge in relation to specific signal features, such as up or down strokes in the external electric field (Gabbiani and Metzner 1999; Gabbiani et al. 1996; Metzner et al. 1998). Further, significant progress has been made in identifying how conditional backpropagation generates an oscillatory pattern of spike bursts in ELL pyramidal cells in vitro. Pyramidal cell spike bursts are initiated when a Na⁺ spike backpropagating over the initial 200 µm of apical dendrites generates a somatic DAP (Turner et al. 1994). A frequency-dependent broadening of dendritic spikes potentiates the DAP until a high-frequency spike doublet is triggered at the soma (Lemon and Turner 2000). The short inter-spike interval (ISI) of the doublet falls within the dendritic refractory period and blocks spike backpropagation, removing the dendritic depolarization that drives the burst. Repetition of this conditional process of backpropagation groups repetitive spike discharges into bursts in the -frequency range. A key issue that remains in understanding the mechanism of ELL burst discharge is the identity of factor(s) underlying the frequency-dependent broadening of dendritic spikes that drives burst discharge.

Our present knowledge of spike discharge in ELL pyramidal cells and the simple mechanism underlying conditional backpropagation provides an excellent opportunity to model a form of -frequency burst discharge and test hypotheses about burst generation. This study presents a detailed compartmental model of an ELL pyramidal cell that is based on extensive electrophysiological and morphological data. We establish the distribution and complement of ion channels that are necessary to fit various aspects of Na⁺ spike discharge and spike backpropagation to physiological data. However, the resulting model neuron fails to reproduce the change in dendritic spike repolarization and somatic afterpotentials required to induce -frequency bursting. Hence we test potential ionic mechanisms that could underlie burst discharge. Our results show that cumulative inactivation of a dendritic K⁺ current is necessary and sufficient to generate a burst discharge that is remarkably similar to that found in ELL pyramidal cells in vitro. Some of this work has been previously reported in abstract form (Doiron et al. 2000).

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

The compartmental model we use in this investigation builds on the earlier one introduced in Doiron et al. (2001). Simulations are performed with the software package NEURON (Hines and Carnevale 1997), which uses a central difference algorithm (Crank-Nicholson) to integrate forward in time. The time step for all simulations is 0.025 ms, well below the time scale of the synaptic and ionic responses present in the ELL (Berman and Maler 1999; Berman et al. 1997).

Cell morphology and discretization

Pyramidal cell somata are located within a pyramidal cell body layer, a distinct lamina in the ELL. Basal and apical dendrites emanate from the ventral and dorsal aspects of the cell soma, respectively; the basal dendrites receive electrosensory afferent input while the apical dendrites receive feedback input (Berman and Maler 1999). There are two classes of pyramidal cells, basilar and nonbasilar (lacking a basal dendrite), both of which generate -frequency oscillatory spike bursts that depend on conditional backpropagation into the apical dendrites (Lemon and Turner 2000; Turner et al. 1994). The present model is focused on the activity associated with basilar pyramidal cells to allow future analysis of the effect of electrosensory afferent input on burst discharge.

Figure 1 shows our two-dimensional spatial compartmentalization of a basilar pyramidal neuron based on detailed spatial measurements of confocal images of a Lucifer yellow-filled neuron (Berman et al. 1997). The model contains 153 compartments with lengths ranging from 0.8 to 669.2 µm and diameters spanning from 0.5 to 11.6 µm. The total model cell surface area is 65,706 µm², comparable to other pyramidal cell models (Koch et al. 1995). Longer compartments are further subdivided to guarantee that no isopotential segment is of length >25 µm, ensuring sufficient computational precision. This results in a total spatial discretization of the full model cell into 312 isopotential regions. For simplicity, the initial segment and soma are represented as a single compartment.

View larger version (29K): [in this window] [in a new window]   Fig. 1. Basilar pyramidal cell morphology and ionic channel distribution incorporated into the compartmental model. Typical basilar pyramidal cells have somata of 10-25 µm in diameter; a basilar dendritic trunk of 5-12 µm in diameter extending 200-400 µm distance before branching to form a distal basilar bush (Maler 1979). A thick proximal apical dendritic shaft of ~10 µm in diameter extends dorsally ~200 µm prior to branching in a molecular layer; the apical dendrites continue to branch over a distance of 800 µm. The particular pyramidal cell used for the model was reconstructed from a Lucifer-yellow filled cell (Berman et al. 1997). The distribution of ionic channels in the core model is indicated in 3 boxes according to their location on the soma, proximal apical dendrite, or all dendrites, respectively. An enlargement of the proximal apical dendrite is shown on the left to detail the placement of Na+ and K+ channels in separate proximal dendritic compartments over the region for backpropagation of dendritic Na+ spikes.

The proximal apical dendrite is modeled by ten connected compartments (total length, 200 µm) whose diameter decreases with distance from the somatic compartment (initial diameter of 11.6 mm, final diameter of 6 µm; see Fig. 1, inset). The dendrite then bifurcates to give rise to daughter branches of 6 µm diameter, which further bifurcate with an associated step decrease in diameter, extending a total distance of ~800 µm in an overlying molecular layer. The proximal basilar dendrite is modeled as a single compartment of 7.4 µm diameter and 194 µm in length. The distal extent of the dendrite separates into many compartments that form a bush-like pattern (see Fig. 1). The lengths of dendrites within the basilar bush range from 3 to 56 µm with diameters as thick as 4 µm to as thin as 0.5 µm. As the axial resistivity (R_a) and membrane capacitance per unit area (C_m) are not precisely known for ELL pyramidal cells, R_a is set to 250 cm and C_m to 0.75 µF/cm², both realistic values for vertebrate neurons (Mainen and Sejnowski 1998). The model cell temperature is set to 28°C, similar to the natural habitat of the fish.

Model equations

Each ionic current, I_x, is modeled as a modified Hodgkin/Huxley channel (Hodgkin and Huxley 1952) governed by

(1.1)

(1.2)

(1.3)

h_x is given by a similar description (equations omitted). Here g_max,x is the maximal channel conductance of a generic channel x, while m_x and h_x are the activation and inactivation state variables, raised to powers i and j, respectively. m_,x is the steady state activation curve, and _m,x the activation time constant, both governing the evolution of the variable m_x. The voltage dependence of m_,x is given by a sigmoid relationship (Eq. 1.3) determined by the half voltage V_1/2,m,x and slope parameter k_m,x. The channel reversal potential is E_rev,x. To facilitate parameter fitting, our model follows Koch et al. (1995) in assuming _m,x is voltage independent.

Figure 1 introduces the specific ionic currents and indicates their distribution over the cell. Table 1 gives the parameters stated in Eq. 1.1-1.3 for each current. In this study, "proximal dendrite" refers to the initial 200 µm of the apical dendrite, which is in most pyramidal cells a single nonbranching shaft over this distance. A detailed analysis of basilar dendritic electrophysiology is not available, hence few ionic channels are localized to the basilar dendritic region. The ionic channels (I_x) influence the potential of each compartment according to

(2)

where the sum is over all the ionic currents I_x present in a given compartment as indicated in the distribution of Fig. 1. I_app represents an externally applied current that is always constant in time and injected into the somatic compartment for this study.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

For each ionic channel incorporated into the model, we first justify the chosen distribution over the cell (Fig. 1) and identify the influence of each conductance on cell membrane potential. Finally, the parameters are fit so that model performance matches known experimental recordings. In the case of insufficient experimental evidence from pyramidal cells, values were taken from experimental or computational results reported for other cells.

Channel distribution and kinetics

PASSIVE CURRENT I_LEAK. I_leak represents the classic leak channel with voltage-independent conductance and is present in all compartments. The channel density, g_max,leak is chosen to establish the model input resistance as 76.1 M and the passive membrane time constant as 26.8 ms, similar to values measured in intracellular recordings of pyramidal cells in vitro (Lemon and Turner 2000). The leak reversal potential, E_rev,leak, is set to 70 mV (Koch et al. 1995; Mainen et al. 1995; Rapp et al. 1996). Due to the high-density of K⁺ currents (see following text) the final resting membrane potential (RMP) of the core model is 73.3 mV, a value within the range recorded in pyramidal cells in vitro (Berman and Maler 1998a; Turner et al. 1994).

NA⁺ AND K⁺ CURRENTS. Previous work has established the distribution pattern of immunolabel for both Na⁺ channels and an apteronotid homologue of the Kv3.3 K⁺ channel, AptKv3.3, over the dendritic-somatic axis of pyramidal cells (Rashid et al. 2001; Turner et al. 1994). Electrophysiological studies have further mapped the site for Na⁺ spike initiation and conduction over the soma-dendritic axis (Lemon and Turner 2000; Turner et al. 1994) and identified the kinetic properties of AptKv3.3 K⁺ channels in both native pyramidal cells and when expressed in a heterologous expression system (Rashid et al. 2001). Although other K⁺ channels are incorporated into the model, our existing knowledge of the distribution and properties of Na⁺ and AptKv3.3 K⁺ channels are used when possible to constrain our parameters and improve the representation of action potential waveforms. We begin by matching the distribution and kinetic properties of AptKv3.3 channels to experimental data.

I_AptKv3.3: high-voltage-activated K⁺ channel

AptKv3.3 K⁺ channels are distributed over ELL pyramidal cell somata as well as apical and basal dendrites (Rashid et al. 2001). A dendritic distribution of AptKv3.3 channels is unique in that all previous studies on Kv3 channels have shown a distribution that is restricted to the soma, axon, and presynaptic terminals (Perney and Kaczmarek 1997; Sekirnjak et al. 1997; Weiser et al. 1995). Rashid et al. (2001) also showed that AptKv3.3 K⁺ channels contribute to spike repolarization in both somatic and apical dendritic membranes. This role is particularly relevant in dendritic regions where a reduction in AptKv3.3 current enhances the somatic DAP and lowers burst threshold.

Figure 2 shows the fit of AptKv3.3 current in the model to whole cell recordings of AptKv3.3 K⁺ current when expressed in human embryonic kidney (HEK) cells. AptKv3.3 channels produce an outwardly rectifying current that exhibits a high-threshold voltage for initial activation (more than 20 mV; Fig. 2, A and B). Over a 100-ms time frame, step commands produce little inactivation of AptKv3.3 current in the whole cell recording configuration (Fig. 2A), although inactivation can be recorded over longer time frames (Rashid et al. 2001). As indicated in Table 1, both activation (m_AptKv3.3) and inactivation (h_AptKv3.3) curves are used to describe I_AptKv3.3. The V_1/2 for m_AptKv3.3 is set to 0 mV so as to produce the high-threshold voltage necessary for activation as shown in Fig. 2B. A shallow slope of activation, k_m,AptKv3.3, is required to produce a gradual rise of I_AptKv3.3 with voltage.

View larger version (23K): [in this window] [in a new window]   Fig. 2. AptKv3.3 K+ channel kinetics. A: whole cell K+ currents recorded in tsA 201 HEK cells transiently transfected with AptKv3.3 K+ channels (left). Shown are currents activated by step commands in 10-mV increments from an initial prepotential of 90 mV. Model AptKv3.3 currents (right) in response to an equivalent voltage-clamp protocol as the experimental recordings in A, restricted to the somatic compartment for space-clamp considerations. Simulated AptKv3.3 whole cell currents shown here and in C have calibration bars normalized to the current evoked at 30 mV. We normalize the simulation results because we do not wish to quantitatively compare the simulation to the HEK cell experiments. B: normalized current-voltage plots for both experimental and model AptKv3.3 currents in response to the voltage-clamp protocol shown in A. A high initial voltage for activation of AptKv3.3 current is evident in that whole cell current becomes significant only for voltage steps above 20 mV. C: time expanded plots of the beginning and end of AptKv3.3 whole cell currents in response to the voltage steps shown in A. This shows both the fast activation and deactivation of AptKv3.3 current. See Rashid et al. (2001) for a description of experimental methods.

As reported for Kv3 channels in some expression systems, AptKv3.3 current exhibits an early peak on initial activation followed by a relaxation to a lower amplitude current for the duration of a 100-ms step command (Fig. 2A). The origin of this early peak and relaxation is currently unknown, but it has been proposed to reflect a transient accumulation of extracellular K⁺ when Kv3 channels are expressed at high-density in expression systems (Rudy et al. 1999). Since this has not been firmly established, we incorporate inactivation kinetics (V_{1/2,h,AptKv3.3}, k_h,AptKv3.3, _h,AptKv3.3) that allow the model to reproduce the early transient peak of current (Fig. 2A). An additional characteristic of Kv3 channels is a fast rate of both activation and deactivation (Fig. 2C). The activation time constant, _m,Kv3.3, is chosen to produce rates of activation and deactivation that most closely match the experimental data (Fig. 2C). Because AptKv3.3 channels are found with high prevalence in the soma and dendrites, they are incorporated with the above kinetics over the entire axis of the model cell (Fig. 1). The conductance level is adjusted accordingly to the fit to experimental data (see following text).

Action potential discharge in ELL pyramidal cells

We first treat somatic and dendritic spike discharge separately to determine the necessary descriptions of the model Na⁺ and K⁺ channels required to produce action potential waveforms that match in vitro recordings. Next we focus on the soma-dendritic interaction that gives rise to the DAP that drives burst discharge.

I_Na,s, I_Dr,s, and I_AptKv3.3: somatic spike

Figure 3A illustrates an action potential recorded in the soma of pyramidal cells in an in vitro slice preparation (see also Table 2 for quantitative comparison of model and experimental somatic action potential features). As we are interested in first modeling the somatic spike in the absence of backpropagating spikes, the recording in Fig. 3A was obtained after TTX had been locally applied to the entire dendritic tree. This effectively blocks all spike backpropagation and selectively removes the DAP at the soma (Lemon and Turner 2000). The somatic spike in pyramidal cells is of large amplitude (typically reaching 20-mV peak voltage) and very narrow half-width (width at half-maximal amplitude) of ~0.45 ms (Table 2) (Berman and Maler 1998a; Turner et al. 1994). In the absence of a DAP, a fast afterhyperpolarization (fAHP) that follows the somatic spike is readily apparent (Fig. 3A). A slow Ca²⁺-sensitive AHP (sAHP) follows the fAHP but does not contribute directly to the soma-dendritic interaction that underlies burst discharge (Lemon and Turner 2000).

View larger version (18K): [in this window] [in a new window]   Fig. 3. Fit of somatic and dendritic spikes to experimental data. A: in vitro recording of an electrosensory lateral line lobe (ELL) pyramidal cell somatic action potential under conditions of complete block of all spike backpropagation by dendritic TTX application (Lemon and Turner 2000). This serves to selectively remove the depolarizing afterpotential (DAP) from the somatic recording, unmasking a large fast afterhyperpolarization (fAHP). B and C: superimposed simulations comparing the effects of various combinations of K+ currents on the rate of somatic spike repolarization. All simulations have KA and KB channels present at the soma and Na,d channels have been removed to block backpropagation. All action potentials are produced via a square wave step depolarization for Iapp at the soma (amplitude of 0.3 nA, 0.5 ms in duration). B: insertion of IDr,s at the soma results in a substantial increase in the rate of spike repolarization with an associated decrease in spike height (trace 1 vs. trace 2). C: a series of superimposed simulations to illustrate the effects of combining IDr,s and IAptKv3.3 on spike repolarization. The results show that IAptKv3.3 alone affects primarily the initial falling phase of the spike but only a fraction of the total repolarization (trace 1 vs. 2). The best fit is obtained by combining IDr,s and IAptKv3.3 (trace 4). Trace 3 indicates that subsequent removal of IAptKv3.3 at the soma has a moderate effect on spike repolarization but little effect on spike amplitude. D: in vitro recording of an ELL pyramidal cell dendritic spike (~150 µm). Note the slower rate of rise, lower amplitude, and slower rate of repolarization as compared with the somatic spike in A. E and F: superimposed simulations comparing the effects of various combinations of K+ currents on the rate of dendritic spike repolarization. E: with no dendritic K+ channels (trace 1), the dendritic spike has a long duration with the repolarizing phase interrupted by 2 small positive-going potentials reflecting the passive conduction of spikes generated at the soma (*). Trace 2 shows that IDr,d can account for a large extent of the repolarizing phase of dendritic spikes. F: a comparison of the role for IAptKv3.3 and IDr,d in combination on dendritic spike repolarization. IAptKv3.3 can account for a proportion of spike repolarization (trace 1 vs. 2), but the best fit is obtained when both IDr,d and IAptKv3.3 are present (trace 4). Removal of IAptKv3.3 reveals a final role in adjusting both spike amplitude and repolarization (trace 3). Experimental methods for the recordings in A and D are as in Lemon and Turner (2000). Calibration bars in B and E also apply to C and F, respectively.

ELL pyramidal cell somatic spikes are generated by TTX-sensitive Na⁺ channels (Mathieson and Maler 1988; Turner et al. 1994), which are modeled here as a fast activating and inactivating current, I_Na,s. Rashid et al. (2001) established that Kv3 channels contribute to repolarizing the somatic spike in pyramidal cells. However, local blockade of somatic Kv3 channels using focal ejections of TEA in vitro only blocks a fraction of the total repolarization, indicating the additional involvement of other voltage-dependent K⁺ channels. Pyramidal cells are known to express large conductance (BK) Ca²⁺-activated K⁺ channels in both somatic and dendritic membranes (E. Morales and R. W. Turner, unpublished observations), but they do not appear to contribute to spike repolarization (Noonan et al. 2000; Rashid et al. 2001). A large fraction of the somatic spike repolarization is likely mediated by a small conductance K⁺ channel (<10 pS) found with high prevalence in patch recordings (Morales and Turner, unpublished observations). Although the kinetic properties of this channel have not been established, whole cell currents in pyramidal cell somata indicate that it will have properties consistent with a fast activating and deactivating delayed rectifier-like (Dr) current, referred to here as I_Dr,s. We therefore model somatic spike depolarization and repolarization using a combination of I_Na,s, I_Dr,s, and I_AptKv3.3.

I_Na,s and I_Dr,s are confined to the somatic compartment, which includes both the initial segment and somatic membrane, and represents the site for Na⁺ spike initiation in the cell (Turner et al. 1994). The activation (both I_Na,s and I_Dr,s) and inactivation (only I_Na,s) steady-state conductance curve half voltages, V_1/2, and slopes, k, are similar to values used in previous compartmental models (Koch et al. 1995; Mainen et al. 1995; Rapp et al. 1996). Spike threshold occurs at V_thres > 60 mV, a value close to that reported for pyramidal cell spike threshold in vitro (Berman and Maler 1998a).

Previous compartmental models of mammalian neurons typically incorporate action potential half-widths of 1-2 ms (de Schutter and Bower 1994; Koch et al. 1995; Mainen et al. 1995; Rapp et al. 1996). Given the comparatively narrow half-width of ELL pyramidal cell spikes (Table 2), we choose relatively short I_Na,s and I_Dr,s time constants of activation and inactivation, _m and _h (Table 1), and g_max,Na,s is set to 1.8 S/cm². To compensate the Na⁺ conductance and achieve proper repolarization, g_max,Dr,s is set to a value of 0.7 S/cm². We note that the model density of Na, s and Dr, s channels are comparable to that used in the spike initiation zone of other compartmental models (de Schutter and Bower 1994; Koch et al. 1995; Mainen et al. 1995; Rapp et al. 1996). Figure 3B shows somatic action potentials with I_Na,s alone (trace 1) and both I_Na,s and I_Dr,s (trace 2) inserted in the model. The reduction in spike half-width and amplitude when I_Dr,s is present allows the model somatic spike to better approximate experimental recordings (compare Fig. 3, A and B).

In considering the role of AptKv3.3 K⁺ current, we find that AptKv3.3 channels alone could account for only a fraction of the total somatic spike repolarization. Figure 3C compares the somatic action potential with no I_Dr,s or I_AptKv3.3-mediated repolarization (trace 1) to one with only I_AptKv3.3 current (trace 2). A reduction of spike width by I_AptKv3.3 during the initial falling phase of the action potential with a minimal reduction in peak voltage matches the results of earlier modeling studies that considered the role of Kv3.1 K⁺ channels in spike repolarization (Perney and Kaczmarek 1997; Wang et al. 1998). As shown in Fig. 3C (trace 4), spike repolarization is most closely modeled when I_Dr,s and I_AptKv3.3 are both present at the soma, with AptKv3.3 conductance set to 6.5 S/cm² (Rashid et al. 2001). Removing AptKv3.3 conductance slightly reduces the rate of spike repolarization without significantly affecting action potential amplitude. Such an effect on spike repolarization by AptKv3.3 is also consistent with pharmacological tests in vitro (Rashid et al. 2001).

I_Na,d, I_Dr,d, and I_AptKv3.3: dendritic spike

Many central neurons are known to be capable of actively backpropagating Na⁺ spikes from the soma and over the majority of the dendritic tree (Stuart et al. 1997b; Turner et al. 1991, 1994; Williams and Stuart 1999; see Häusser et al. 2000 for review). Patch-clamp recordings in hippocampal pyramidal cells further indicate that Na⁺ channels are distributed with a relatively uniform density over the entire dendritic tree (Magee and Johnston 1995); a distribution that has been used in other modeling studies (Mainen et al. 1995; Rapp et al. 1996). We have previously determined that Na⁺ channel immunolabel in ELL pyramidal cells is uniformly distributed in the cell body region but exhibits a distinct punctate distribution over the proximal 200 µm of apical dendrites (Turner et al. 1994). This distribution correlated with the distance over which TTX-sensitive spike discharge was recorded, suggesting that the immunolabel correspond to Na⁺ channels involved in spike generation. Electrophysiological recordings further established that Na⁺ spikes are initiated in the somatic region but backpropagate over only ~200 µm of the dendritic tree that can extend as far as 800 µm (Maler 1979; Turner et al. 1994). In this respect, spike backpropagation in ELL pyramidal cell dendrites falls between the active conduction of Na⁺ spikes seen over the entire axis of cortical neuron dendrites and the passive decay of Na⁺ spikes seen over the proximal dendrites of cerebellar Purkinje cells (Häusser et al. 2000; Hoffman et al. 1997; Stuart and Häusser 1994; Stuart et al. 1997a).

Figure 3D shows an ELL pyramidal cell dendritic spike recorded ~150 µm from the soma. The dendritic spike half-width is substantially larger than that of the somatic spike, with the total duration approaching as much as 12 ms in recordings ~200 µm from the soma. Recordings in the slice preparation further indicate that even proximal dendritic spikes (50 µm) exhibit a sharp decrease in the rate of rise and rate of repolarization with respect to somatic spikes. This difference in rate of repolarization leads to a substantial delay in the peak latency of dendritic versus somatic spikes with a rapid increase in this difference beyond ~100 µm (Turner et al. 1994). Modeling this rapid change in spike characteristics in proximal dendrites requires a specific set of parameters for both dendritic Na⁺ and K⁺ conductances.

Given the immunocytochemical data (Turner et al. 1994), the distribution of dendritic Na⁺ channels is confined to five punctate zones of 5 µm in length (200 µm² area) along the proximal apical dendrite that are assigned a high Na⁺ channel density (Fig. 1). The distance between active dendritic zones also increases with distance from the soma. Specifically, active I_Na,d zones 1-3 are separated by 15 µm of passive dendrite, while zones 3-5 are separated by 60 µm of passive dendrite. Lacking a cytochemical localization of I_Dr,d K⁺ channels in dendrites, we co-localized these channels to the five active dendritic zones. As stated in the preceding text, AptKv3.3 channels are incorporated over the entire soma-dendritic axis, although with varying levels of conductance for somatic, proximal apical dendritic, and distal apical and basal dendritic compartments.

We first attempted to equate the kinetic parameters of Na, d and Dr, d channels to those of somatic Na, s and Dr, s, respectively, but with reduced channel densities to produce a lower amplitude dendritic spike, as done in previous studies (Mainen et al. 1995; Traub et al. 1994). However, this led to dendritic spikes with too short of a half-width and delay time to peak incompatible with intracellular dendritic recordings. A first attempt at correcting this discrepancy was raising the model axial resistance, thereby reducing the passive cable propagation of the spike along the dendrite. However, to obtain model agreement with experimental data, R_a had to be set outside acceptable norms by a factor of 10 (Mainen and Sejnowski 1998). This approach was problematic and hence abandoned.

Recently differences in ion channel kinetics have been observed for dendritic Na⁺ and K⁺ channels as compared with those at the soma, indicating precedence for applying differential kinetic properties to somatic versus dendritic ion channels (Colbert et al. 1997; Hoffman et al. 1997; Jung et al. 1997). Since a similar level of analysis is not yet available for ELL pyramidal cells, we explored a broad range of kinetic parameters describing both I_Na,d and I_Dr,d. More successful modeling of experimental results is achieved by increasing the steady-state conductance curve slope, k, and time constant, , for both the activation and inactivation of I_Na,d and I_Dr,d as compared with their somatic counterparts. Simulations using only I_Na,d and I_Dr,d then begin to reproduce the delay in dendritic spike peak as well as the increase in spike half-width, the longer rate of rise, and the slower rate of repolarization when compared with the somatic spike (Fig. 3B). By comparison, modeling the dendritic spike with only I_Na,d and I_AptKv3.3, does not achieve a sufficient rate of spike repolarization (Fig. 3F, trace 1 vs. 2). By combining I_Dr,d with I_AptKv3.3 set to a density of 1 S/cm² in the proximal apical dendrite and a lower density of 0.5 S/cm² in other dendritic compartments a good fit to experimental data is obtained (cf. Fig. 3, D and F, trace 4). Note that incorporation of AptKv3.3 current also reduces dendritic spike amplitude (Fig. 3F; trace 4). This result is consistent with experimental data indicating a slight increase in dendritic spike amplitude following local ejections of 1 mM TEA to block dendritic AptKv3 channels (Rashid et al. 2001).

Soma-dendritic interactions underlying the DAP

Figure 4 illustrates the effects of combining active somatic and dendritic compartments on spike waveforms. Previous intracellular recordings have indicated substantial differences in the duration and peak response of somatic versus dendritic spike waveforms (Fig. 4A). Both experimental recordings in Fig. 4A were obtained in intact slices with full backpropagation of spikes into dendrites. Under these conditions, the somatic spike is followed by a clear DAP that superimposes on the somatic fAHP (Fig. 4A). The DAP is due to the dendritic fast sodium currents, I_Na,d, which boost the backpropagating action potential to elevated voltages in the dendrite, and promotes return electrotonic current flow during the longer duration dendritic spike (Turner et al. 1994). When the model includes active spike discharge in both somatic and dendritic compartments, it successfully reproduces a DAP at the soma that is offset by the fAHP (Fig. 4B). If the g_max of all I_Na,d currents is set to zero, in effect removing the active zones from the apical dendrite, the influence of the DAP at the soma is lost, uncovering a clear fAHP at the soma. Only a low-amplitude passive reflection of the somatic spike occurs in the apical dendrites (Fig. 4C), as recorded in vitro when TTX is focally applied to the dendrites to block active spike backpropagation (Turner et al. 1994).

View larger version (26K): [in this window] [in a new window]   Fig. 4. Generation of a DAP through spike backpropagation. A: superimposed single spike discharge recorded in separate intracellular somatic and dendritic impalements (~200 µm from the soma), aligned temporally at the onset of spike discharge (Lemon and Turner 2000). The somatic DAP generated by the backpropagating spike is enlarged within the inset. The scale bars shown in A also apply to B and C. B: superimposed somatic and dendritic spike simulations (measured 200 µm from the soma) evoked by depolarizing somatic current injection. The somatic DAP is enlarged within the insert for comparison with A. Simulations in B and C also include IKA and IKB at the soma that underlies a late slow AHP (sAHP; not shown). C: superimposed model somatic and dendritic spike responses to somatic current injection with all INa,d removed from the dendritic compartments. The somatic DAP is removed, uncovering the prominent fAHP. Note that only a small passive response is reflected into apical dendrites on the discharge of somatic spikes in the absence of INa,d. D: Plot of the peak voltage as a function of distance from the soma (0 µm) after discharging a somatic spike, with and without active dendritic INa,d zones. The location of active sites in the proximal dendritic region are indicated above the abscissa. The slight deviation from expected passive decay in the proximal dendritic region is due to a reduction in the diameter of the model proximal dendritic shaft.

Spike backpropagation and refractory period

Figure 4D plots how the fitted kinetics of I_Na,d and I_Dr,d affect the backpropagation of action potentials initiated at the soma. Shown is the peak voltage of the response measured at a select number of compartments for both active propagation (g_max,Na,d = 0.6 S/cm²) and passive electronic conduction (g_max,Na,d = 0 S/cm²) in the dendritic compartments. In the proximal apical dendrite (<200 µm), the five active I_Na,d sites incorporated into the model boost the peak of the backpropagating action potential over that measured during passive conduction. In the mid-distal dendrite (>200 µm), the peak voltage decays exponentially in both cases because no active inward currents are believed to boost the voltage beyond this distance (Turner et al. 1994). This decrease in spike amplitude near 200 µm also qualitatively mimics the properties of backpropagating spikes as measured through laminar profile field potential analysis (Turner et al. 1994).

To test the accuracy of fit of channel parameters for I_Na,s, I_Na,d, I_Dr,s, and I_Dr,d, we measure the refractory period of both somatic and dendritic spikes (Fig. 5, A and B). The simulation protocol is equivalent to the condition-test (C-T) interval experiment used by Lemon and Turner (2000). In the model, this consists of first injecting a brief somatic current pulse sufficient to induce a single somatic spike that backpropagates into the dendrites. A second identical current pulse is then applied at a variable time interval following the first pulse to identify relative and absolute refractory periods. At sufficiently long test intervals (Fig. 5, A and B, 8 ms) the second pulse evokes full somatic and dendritic spikes, both identical to the condition responses. At the soma, a reduction in the C-T interval to ~4 ms results in a select blockade of the DAP without significant effect on the somatic spike (Fig. 5A). Somatic spike amplitude remains essentially stable for C-T intervals above 2.5 ms, although C-T intervals below this slightly reduce spike amplitude, given that I_Na,s has not completely recovered from inactivation. An absolute somatic refractory period is evident at a C-T interval of 1.5 ms (Fig. 5A). The effects of a similar series of C-T intervals monitored 200 µm from the soma reveals a relative refractory period for dendritic spikes between 4 and 6 ms, as reflected by a gradual decline in spike amplitude. This reduction in amplitude ends at C-T intervals between 2.0 and 3.0 ms (Fig. 5B), with subsequent failure of the small potential evoked at C-T intervals of ~1.5 ms.

View larger version (15K): [in this window] [in a new window]   Fig. 5. Refractory period of model somatic and dendritic spikes. Spike discharge was evoked with a short-duration depolarization at the soma and a condition-test (C-T) pulse pair presented at varying intervals to determine if the model reproduced the observed differential refractory period between somatic and dendritic regions of pyramidal cells. A: several superimposed C-T stimulus pairs at the soma. The C-T interval is indicated below the traces (arrows). The absolute refractory period of the model somatic spike occurs at 1.5 ms. Note that the somatic DAP is also selectively blocked at a C-T interval of 4 ms (slanted arrows) with no significant change in the somatic spike. B: superimposed C-T stimuli monitored at 200 µm in the dendrite. The dendritic spike exhibits a relative refractory period beginning at ~6 ms as shown by a decline in spike amplitude at progressively shorter C-T intervals. An absolute refractory period at 3 ms is indicated by the final presence of only a passively reflected response associated with the somatic spike. Each of these properties accurately reflect the conditions found in ELL pyramidal cells in vitro.

Each of these properties qualitatively match experimental results in pyramidal cells of an absolute somatic refractory period at ~2 ms, a relative dendritic refractory period between 5 and 7 ms and an absolute dendritic refractory period between 3 and 4 ms (Lemon and Turner 2000). Furthermore a selective block of the somatic DAP over the same range of C-T intervals that reduce dendritic spike amplitude is also a characteristic observed in intact pyramidal cells (Turner et al. 1994).

I_NaPdetermining RMP, nonlinear EPSP boosting, and latency to first spike shifts

A TTX-sensitive and persistent Na⁺ current (I_NaP) (French et al. 1990) has been recorded at both the somatic and dendritic level of ELL pyramidal cells (Berman et al. 2001; Turner et al. 1994). Blocking this current with focal ejections of TTX in vitro results in a clear hyperpolarizing shift in cell resting membrane potential (RMP) and an increase in the latency to discharge a spike at the soma (Turner et al. 1994). There is also a prominent voltage-dependent late component to the dendritic and somatic excitatory postsynaptic potential (EPSP) evoked by stimulation of descending tractus stratum fibrosum (tSF) inputs that terminate in the proximal dendritic region (Berman and Maler 1998c; Berman et al. 1997). Recent work indicates that focal ejection of TTX at the soma selectively blocks this late component of the tSF-evoked EPSP (Berman et al. 2001). Each of these results identifies an important contribution by I_NaP in determining the resting membrane potential (RMP) of the cell, in boosting the tSF synaptic depolarization, and in setting the latency to spike discharge. We determine the model NaP channel parameters through a detailed match to these constraints.

NaP channels are modeled in both the soma and proximal apical dendrite as suggested by experimental and modeling studies of ELL pyramidal cells (Fig. 1) (Berman and Maler 1998c; Turner et al. 1994). In the absence of definitive knowledge of the distribution of dendritic NaP channels, a uniform distribution of NaP channels over the entire proximal apical dendrite is chosen. Local TTX applications in vitro produce quantitatively similar shifts in RMP at both the somatic and dendritic level of pyramidal cells (Turner et al. 1994). We hence choose a relative factor of 3.5 for the ratio of total somatic/dendritic NaP channel density to account for the greater proximal apical dendritic surface area. This produces approximately equal net I_NaP current from both soma and dendrite in response to similar depolarizations.

To set the NaP channel activation parameters, V_1/2 and k, we consider its effects on the model cell RMP. To do so, we apply a small constant depolarizing somatic current injection to the model (I_app < 100 pA), which is insufficient to cause spiking. After a transient period of depolarization, the model membrane voltage settles to a steady-state value, which we label as the "effective" RMP. The shape of the increase in RMP as I_app increases will be determined by the specific NaP distribution. Figure 6A plots this final equilibrium somatic voltage (t >200 ms) as a function of I_app with NaP present in various compartments (see figure legend for description). For small currents (I_app < 50 pA), the rise in RMP is linear (Ohm's law) because at these potentials (V_m < 70 mV) NaP is not significantly activated. However, for larger applied currents (I_app > 60 pA) yielding higher RMP values (V_m > 70 mV), a significant nonlinear RMP increase occurs when NaP is present (trace 1, Fig. 6A). This effect requires a steep subthreshold voltage dependency of activation that forces the half activation of I_NaP, V_1/2,m,NaP, to be set to 58.5 mV and the activation slope parameter, k_m,NaP, to be low, 6 mV. This agrees with the experimental effects of TTX application on pyramidal cell RMP in vitro (Berman and Maler 1998c; Turner et al. 1994) and with experimentally determined V_1/2 (56.92 mV) and k (9.09) voltages for NaP in rat and cat thalamocortical neurons (Parri and Crunelli 1998). The quantitative agreement with experiment is achieved by adjusting g_max,NaP. However, since f-I characteristics (see Fig. 8) are compromised with a significant modification of g_max,NaP, the fit is balanced to produce a realistic RMP, and proper f-I relationship for the model cell.

View larger version (14K): [in this window] [in a new window]   Fig. 6. The effects of somatic or dendritic INaP on resting membrane potential (RMP), excitatory postsynaptic potential (EPSP) amplitude, and spike latency. A: RMP in the model cell plotted as a function of a series of subthreshold depolarizing current injections (Iapp) 200 ms after the onset of the depolarization, at which point dVsoma/dt  0. Several superimposed plots are shown labeled according to the model conditions: control (all parameters are as given in Table 1), somatic gNaP = 0, dendritic gNaP = 0, and both somatic and dendritic gNaP = 0 (see figure legend). Note the linear displacement of potential with no INaP present (trace 4), but a nonlinear increase in membrane voltage with current injection when INaP is present in both somatic and proximal dendritic compartments (trace 1). B: model somatic responses to an evoked EPSP in the proximal dendritic region (200 µm from soma; alpha function,  = 1.5 ms, gmax = 16 nA) for the NaP conditions as labeled in A. Inset: pyramidal cell somatic EPSPs evoked by tractus stratum fibrosum (tSF) inputs to the proximal dendrites before and after focal somatic TTX ejection to block INaP (Berman et al. 2001). C: latency to 1st spike is plotted as a function of suprathreshold depolarizations for the NaP conditions shown in A. Trace 1 best approximates previous experimental latency measurements from ELL pyramidal cells (Berman and Maler 1998a).

As illustrated in Fig. 6A, restricting the NaP distribution to either somatic or dendritic membranes (traces 2 and 3) reveals the nonlinear effect of NaP density in determining the RMP (for I_app > 80 pA the difference between traces 1 and 4 is larger than the sum of the differences between traces 2 and 3 with 4). The nonlinear effect will be treated when we analyze the role of NaP in EPSP boosting (see following text). The near exact quantitative agreement of traces 2 and 3 in Fig. 6A indicate that the effect of NaP on the RMP is not site-specific between soma and proximal dendrite. This is expected since the net NaP currents of both somatic and dendritic membranes are set to be equal (see preceding text). The parameters that model the steady-state characterization of the NaP, V_1/2,m,NaP, and k_m,NaP, are now set; however, the dynamics of the I_NaP current, determined by _m,NaP, remains to be addressed.

I_NaP currents have been shown to boost the amplitude of subthreshold EPSPs in cortical pyramidal cells (Andreasen and Lambert 1999; Lipowsky et al. 1996; Schwindt and Crill 1995; Stafstrom et al. 1985; Stuart and Sakmann 1995); there is recent evidence in ELL pyramidal cells for a similar effect (Berman and Maler 1998c; Berman et al. 2001). To set the time constant of NaP activation, _m,NaP, we consider the EPSP boosting properties of the model I_NaP. The inset in Fig. 6B shows somatic recordings of a distally evoked EPSP under control and somatic TTX conditions. The boost provided by TTX-sensitive currents begins with the fast rising phase of the EPSP, suggesting that NaP channels activate quickly (Fig. 6B; compare the inset control trace to the TTX trace). To match the data, the model _m,NaP is set to a small value ( = 0.3 ms). This fast activation coincides with the treatment of I_NaP in other ionic models (Lipowsky et al. 1996; Wang 1999). Figure 6B shows the model somatic voltage response due to dendritic EPSP activation under the same various NaP distributions considered in Fig. 6A. The EPSP boost observed with all distributions shows that the subthreshold boost of the EPSP by I_NaP is substantial and that indeed the boost begins with the rising edge of the potential, thereby matching experimental data.

The nonlinear nature of I_NaP boosting of EPSPs is apparent in Fig. 6B when we compare EPSP amplitudes when NaP is present in both the somatic and proximal apical dendritic compartments (trace 1) to NaP distributions in only in the somatic (trace 2) or dendritic (trace 3) compartments. Figure 6B shows approximately a 3-mV boost of the model EPSP amplitude when comparing the control case to complete NaP removal (trace 1 as compared with 4). However, with only somatic or dendritic NaP distribution (trace 2 or 3), a boost of <1 mV at the peak of the EPSP is observed. A linear increase in the degree of EPSP enhancement would require that the effects of traces 2 and 3 summate algebraically to produce trace 1. The nonlinear boosting is a result of the steep sigmoidal voltage activation of NaP (plot not shown).

It has been experimentally shown in thalamocortical neurons that I_NaP activation reduces the latency to spike in response to depolarizing current (Parri and Crunelli 1998). We hence further test the fit of NaP parameters by considering the model cell's latency to discharge a spike on membrane depolarization. Figure 6C plots the model latency to first spike as a function of applied somatic suprathreshold depolarizing current under all four NaP conditions described in Fig. 6 (A and B). All traces show that just above the rheobase current (minimum current required to induce spiking) the latency to first spike is long, yet as the input current is increased, the latency decays to shorter values. Comparing the various NaP conditions reveals that increasing I_NaP reduces the rheobase current of the cell, with a full 0.71-nA shift occurring between the condition in which all somatic and dendritic NaP removed (trace 4) as compared with the condition in which I_NaP is distributed over the both somatic and dendritic membranes (trace 1). As a result of the rheobase shift, Fig. 6C shows the latency to first spike at I_app = 0.81 nA with all NaP removed to be 62.4 ms as compared with the control latency of 4.3 ms at the same current.

Figure 6C also indicates that the nonlinear effects of I_NaP already seen on the RMP and EPSP amplitude are even more dramatic on both the rheobase current and latency shift. Partial removal of NaP (traces 2 and 3) only shifts the rheobase and the latency to first spike by slight amounts from the control case (trace 1) when compared with the dramatic shifts observed with a block of both somatic and dendritic I_NaP (trace 4). The quantitative agreement between latency shifts observed when either somatic or dendritic NaP are removed shows that there is no spatial differences of NaP effects on rheobase current or latency as expected by the spatially balanced net ionic current.

Finally, we compare the model spike latency characteristics with experimental measurements from ELL pyramidal cell latency to spike (Berman and Maler 1998b). In ELL pyramidal cells, the transition from long to short latencies to spike does indeed occur over a small depolarizing current interval (~0.1 nA), as in the model control case (trace 1, Fig. 6C). The model results show a power law decay (spike latency ~ 1/). However, experimental work indicates a step decline from long latencies (>20 ms) to a constant latency (~4 ms) for large depolarization (Berman and Maler 1998b). This could not be fit with a power law as in the model results. The model latency decay occurred with all I_NaP parameter sets explored; however, the given fitted parameters (Table 1) produced the best approximation to experimental latency shifts. The discrepancy between the experimental and model latency decay is presently unexplained.

I_Alatency to first spike from hyperpolarized potentials

Previous studies of ELL pyramidal cell electrophysiology (Berman and Maler 1998b; Mathieson and Maler 1988) have suggested the existence of a transient outward current similar to I_A (Connor and Stevens 1971). Both studies observed an increase in spike latency if a depolarization was preceded by membrane hyperpolarization, an effect previously attributed to the activity of an I_A current (Connor and Stevens 1971; McCormick 1991). At this time, patch-clamp recordings have not conclusively shown the possible contribution of an I_A current at the somatic or dendritic level of pyramidal cells (Turner, unpublished observations). Our attempts to simulate the experiments performed in Mathieson and Maler (1988) now show that to reproduce the observed effects of membrane potential on the latency to spike discharge, a low density of I_A-like K⁺ current must be introduced into the model cell.

Because patch-clamp recordings in pyramidal cells have not yet isolated transiently activating and inactivating channels consistent with a traditional A-type K⁺ current, the choice of channel distribution over the model cell must be postulated. A previous compartmental model has shown that altering the distribution of I_A channels over a cell gives only small deviations in observed I_A character, with the exception of cells with large axial resistance (~2,000 cm) (Sanchez et al. 1998). Our model neuron axial resistance (250 cm) is much lower, and hence any chosen channel distribution should not significantly affect cell output. Considering this result, we distribute I_A uniformly over the whole cell (Fig. 1).

The half voltages for the steady-state conductance curves for I_A currents are set to V_1/2,m,A = 75 mV and V_1/2,h,A = 85 mV and the curve slope factors to k_m,A = 4 mV and k_h,A = 2 mV. Other models of I_A (Huguenard and McCormick 1992; Johnston and Miao-Sin Wu 1997; Sanchez et al. 1998) show I_A window currents at more depolarized levels ranging over a larger voltage interval. However, in the absence of definitive I_A characterization in ELL pyramidal cells, we choose to fit the activation/inactivation parameters to ensure that I_A does not affect the model RMP yet that inactivation could be removed with moderate hyperpolarization (in correspondence with the results of Mathieson and Maler 1988). _m,A is set to 10 ms to have sufficiently rapid A-type K⁺ channel activation (Johnston and Miao-Sin Wu 1997). However, the time constant of I_A inactivation, _h,A, is chosen to be 100 ms to produce an appropriate latency to first spike and a subsequent transient increase in spike frequency over the first 200 ms (Doiron et al. 2001).

Figure 7A shows the model cell response without I_A present (g_max,IA = 0) to a depolarizing somatic current injection of 0.6 nA. The model cell is at a resting potential of 73 mV prior to the stimulus. A repetitive firing pattern results with a latency to first spike of 6.05 ms (stimulus onset is at t = 0) and ISI of 9.5 ms. Figure 7B repeats this simulation but with a 50-ms prestimulus hyperpolarization of 11 mV induced via 0.2-nA current injection. Again, repetitive firing occurs, yet with a slightly longer latency to first spike of 12.1 ms followed by a repeating ISI of ~9.5 ms. These results qualitatively match those obtained by Mathieson and Maler (1988) when 1 mM 4-aminopyridine (4-AP, a known I_A channel blocker) was bath applied to ELL pyramidal cells in vitro. However, under control conditions, a substantially longer latency to first spike was observed in ELL recordings when a prestimulus hyperpolarization preceded depolarization (Mathieson and Maler 1988). Furthermore when depolarizing current is applied from a hyperpolarized level, the first ISI is long and then subsequent ISIs slowly shorten (an increase in cell firing rate) in the first 200 ms of the stimulus, presumably due to I_A inactivation.

View larger version (45K): [in this window] [in a new window]   Fig. 7. The effects of IA on latency to spike discharge. All panels show the model response to a particular applied current protocol as indicated below each simulation. Each panel marks the IA channel conductance used in the simulation and the value of the membrane potential at the time of depolarization. A and B: core model response in the absence of IA to equivalent depolarizations from a resting potential of 73 mV (A) or following a prestimulus hyperpolarization to 84 mV (B). C and D: the model response to a similar current injection protocol when IA is inserted into the model. At 73 mV resting potential (C) IA is inactivated, leading to an equivalent response to that shown in A. Note that depolarizing the cell following a prestimulus hyperpolarization (D) results in a substantially longer latency to spike when IA is present (cf. B and D). The set of calibration bars shown in C applies to all panels.

Figure 7, C and D, illustrates simulation protocols identical to those shown in A and B, respectively, yet with I_A channels distributed across the model cell. g_max,IA is fit to match the experimental findings of Matheison and Maler (1988) with a final assigned value of g_max,IA = 1.2 mS/cm². At a resting potential of 73 mV, I_A is substantially inactivated so that depolarizing the somatic compartment via a 0.6-nA current injection results in a latency to first spike and subsequent ISI shift that is nearly identical to the results obtained when I_A is not incorporated into the model (cf. Fi