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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptNIH Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
J Comput Neurosci. Author manuscript; available in PMC Mar 14, 2010.
Published in final edited form as:
PMCID: PMC2837924

Somato-dendritic mechanisms underlying the electrophysiological properties of hypothalamic magnocellular neuroendocrine cells: A multicompartmental model study


Magnocellular neuroendocrine cells (MNCs) of the hypothalamus synthesize the neurohormones vasopressin and oxytocin, which are released into the blood and exert a wide spectrum of actions, including the regulation of cardiovascular and reproductive functions. Vasopressin-and oxytocin-secreting neurons have similar morphological structure and electrophysiological characteristics. A realistic multicompartmental model of a MNC with a bipolar branching structure was developed and calibrated based on morphological and in vitro electrophysiological data in order to explore the roles of ion currents and intracellular calcium dynamics in the intrinsic electrical MNC properties. The model was used to determine the likely distributions of ion conductances in morphologically distinct parts of the MNCs: soma, primary dendrites and secondary dendrites. While reproducing the general electrophysiological features of MNCs, the model demonstrates that the differential spatial distributions of ion channels influence the functional expression of MNC properties, and reveals the potential importance of dendritic conductances in these properties.

Keywords: Hypothalamic neuron, Oxytocin, Vasopressin, Dendritic conductance, Multicompartmental model

1 Introduction

Magnocellular neuroendocrine cells (MNCs) of the hypothalamic paraventricular nucleus (PVN) and supraoptic nucleus (SON) synthesize the neurohormones vasopressin (VP) or oxytocin (OT), which are released from the neurohypophysis into the blood and exert a wide spectrum of actions, including regulation of cardiovascular, reproductive and homeostatic functions. Magnocellular neurons show specific electrophysiological properties and bursting patterns that distinguish them from other neurons of the central nervous system. Among the most distinctive properties of MNCs is their capacity to generate repetitive bursts of action potentials (Poulain and Wakerley 1982). OT-secreting cells demonstrate brief (2–4 s), synchronized high-frequency bursts (40–80 Hz) during reflex milk ejections (Wakerley and Lincoln 1973). VP-secreting cells respond to changes in blood pressure and blood osmolality with a phasic firing pattern, or asynchronous, lower-frequency bursts (8–15 Hz) lasting tens of seconds and separated by silent intervals of similar durations (Brimble and Dyball 1977; Poulain et al. 1977; Poulain and Wakerley 1982).

Oxytocin- and VP-secreting MNCs are quite similar to one another with respect to their ontogenic, morphological and membrane electrical properties. Several voltage- and Ca2+-activated currents are known to contribute to the electrogenic properties of both OT and VP MNCs, including K+, Na+, Ca2+ and non-selective cationic currents (Hatton and Li 1998). Differences in electrophysiological properties between OT-secreting and VP-secreting cells are relatively few and subtle, and include differences in depolarizing after potentials (DAPs), which are more prevalent in, but not exclusive to, VP-secreting cells (Stern and Armstrong 1996); in a sustained outward rectification, found only in OT-secreting cells (Stern and Armstrong 1995; Armstrong and Stern 1997); in an A current-mediated transient outward rectification, which is more pronounced in VP-secreting cells (Stern and Armstrong 1996; Fisher et al. 1998); and in a time-dependent inward rectification, more prominent in OT-secreting cells than in VP-secreting cells (Hirasawa et al. 2003). Nevertheless, the electrophysiological properties of OT- and VP-secreting MNCs are, for the most part, remarkably similar, and they have been used routinely as a fingerprint to distinguish MNCs as a group from other hypothalamic neurons (Hoffman et al. 1991; Tasker and Dudek 1991; Stern and Armstrong 1997).

Experimental data on the differential distribution of ion channels between soma and dendrites have been obtained in other types of neurons (Golding and Spruston 1998; Larkum et al. 1996; Magee and Johnston 1995; Stuart and Sakmann 1994; for review see Migliore and Shepherd 2002), but direct recordings from the dendrites of magnocellular neurons have not been performed to date. It is not known, therefore, what impact the distribution of ionic conductances between the soma and dendrites has on the electrophysiological properties of MNCs. The development of a multicompartmental model of MNCs will allow predictions to be made concerning the role of spatially distributed ion channels in the distinctive electrophysiological properties of these cells under different physiological conditions. Therefore, we have constructed a multicompartmental model based on the Hodgkin-Huxley formalism and Ca2+ dynamics that reproduces the basic electrophysiological properties of MNCs. The model has a simplified but morphologically realistic structure and biophysical parameters. Using this model, we demonstrate how the compartmental distribution of conductances underlying voltage- and Ca2+-dependent currents and intracellular Ca2+ dynamics may contribute to the electrophysiological properties of MNCs. Some of these findings have appeared in abstract form (Komendantov et al. 2002, 2003).

2 Materials and methods

2.1 Model development

2.1.1 Morphological structure and electrotonic properties of the model

Under unstimulated conditions, the MNCs have a relatively simple morphology that is similar for both OT and VP neurons (Stern and Armstrong 1998). According to morphometric analyses of identified OT and VP neurons in the SON (Stern and Armstrong 1998), the soma of OT neurons of diestrous female rats has long axis of 25.7±1.3 μm and a short axis of 15.9±1.1 μm, and the soma of VP neurons has a long axis of 24.6±1.8 μm and a short axis of 15.9± 1.2 μm. Both cell types show a relatively symmetric branching pattern. The dendritic diameter decreases with increasing branch order from 2 μm in the first order branches to 1.5–1 μm in the second order and higher branches. Total length of terminal branches of MNCs is about 800–1,000 μm. Maximal extension of the dendritic trees for VP neurons of diestrous rats is 268±28.3 μm and for OT neurons is 316±32.9 μm rats (Stern and Armstrong 1998). Our multicompartmental model of the hypothalamic MNC includes a soma and two double branching dendrites subdivided into primary and secondary compartments (Fig. 1). The soma was modeled as a cylinder, 15 μm in diameter and 25 μm in length. The primary and secondary dendrites were also modeled as cylinders, 2.0and 1.0 μm in diameter and 50 and 200 μm in length, respectively (see Appendix). This symmetrical dendritic structure of the model was selected for its computational simplicity while retaining a similarity with the morphometric structure of hypothalamic MNCs (Armstrong 1995; Millhouse 1979; Stern and Armstrong 1998). The model equations and parameters are presented in the Appendix and in the Table 1; these were maintained constant between simulated protocols unless otherwise specified.

Fig. 1
Compartmental structure of the magnocellular neuron model. (a) The seven-compartment model of magnocellular neurons. The model features a cylindrical somatic compartment (S) and two symmetrical dendrites, each subdivided into a single cylindrical primary ...
Table 1
Standard model parameters (unless otherwise specified)

Each compartment of the model includes a leak current (IL) as a sum of leaks for Na+ and K+ (IL,Na and IL,K). The description of intercompartmental currents is similar to that used in previous models of midbrain dopaminergic neurons (Canavier 1999; Komendantov and Canavier 2002; Komendantov et al. 2004). The model, with its secondary branching structure and realistic sizes of compartments, has a surface area that approximates the surface area of real MNCs (Stern and Armstrong 1998), which allowed the use of a realistic value of the specific capacitance (Cm =1 μF/cm2). Using leak currents, the input resistance (Ri =976 MΩ), the time constant (54 ms) and the resting membrane potential (−58 mV) were adjusted to match data obtained in in vitro brain slice experiments on magnocellular neurons using patch electrodes (Luther and Tasker 2000). Thus this simplified model with cylindrical compartments maintains a general correspondence between its morphological and electrophysiological properties and those of real hypothalamic neurons (Armstrong 1995; Chevaleyre et al. 2001; Luther and Tasker 2000; Randle et al. 1986; Stern and Armstrong 1998).

2.1.2 Voltage-gated and Ca2+-activated membrane currents

As illustrated in Fig. 1, the model contains several voltage-gated currents: the transient voltage-gated sodium current (INa) (Tanaka et al. 1999), the delayed rectifier potassium current (IK(DR)) (Luther and Tasker 2000), the A-type transient potassium current (IA) (Luther and Tasker 2000), and L- and N-type Ca2+ currents (ICa,L, ICa,N) (Foehring and Armstrong 1996; Joux et al. 2001). It also contains the following Ca2+-dependent currents: a K+ current mediated by large conductance K+ (BK) channels (Dopico et al. 1999), a K+ current mediated by small conductance K+ (SK) channels (Kirkpatrick and Bourque 1996), a non-selective cation (CAN) current (ICAN) (Ghamari-Langroudi and Bourque 2002), and a slowly activated K+ current in OT neurons (Stern and Armstrong 1995, 1997).

The voltage-gated and Ca2+-dependent currents have the following standard generic forms for inactivating currents and non-inactivating currents, respectively:


where Iκ is a current, gκ is a conductance, Eκ is a reversal potential, α is an activation, and γ is an inactivation. For all voltage-gated currents (INa, IK(DR), IA, ICa,L, ICa,N, ISOR), dynamics of activation and inactivation X was described by the equation:


where X(V) is the steady-state activation/inactivation and τx is the time constant. The steady-state activation and inactivation of the voltage-gated currents were described by Boltzmann functions:


where Vhalf,x is the half activation voltage for the gating variable X and θx is the slope factor for that variable.

Distributions of active conductances between compartments were based on direct experimental data and indirect evidence from different types of neurons reported in the literature (Bischofberger and Jonas 1997; Hoffman et al. 1997; Joux et al. 2001; Mainen and Sejnowski 1999; Migliore and Shepherd 2002). Parameters for the time constants and conductances of INa, IA, and IBK were adjusted in order to fit the shape, amplitude, and duration of the action potentials in the model to experimental data from rat MNCs (Greffrath et al. 2004; Luther and Tasker 2000; Stern and Armstrong 1996; Tasker and Dudek 1991; Teruyama and Armstrong 2002).

Na+ and K+ voltage-gated currents

Activation parameters in the Hodgkin-Huxley type equations for INa and IK(DR) were raised to the power of three (m3 and n3, respectively) and the IA activation parameter was raised to the power of four (p4) (see Appendix). For model simplicity, we used the steady state value of activation for the sodium current (m), because we focused on the electrophysiological processes, which are much longer than single spike rising phase. The function for the time constant of inactivation (τh) of this current was selected as the sum of two Boltzmann functions in order to satisfy the dynamics of the sodium current in voltage clamp experiments in freshly isolated SON neurons (Tanaka et al. 1999). The function for the time constant of inactivation of the potassium current (τn) was selected as the sum of three Boltzmann functions in order to provide correspondence of spike shape and duration. The structure of the model provides a good fit to published results from voltage-clamp experiments (Luther and Tasker 2000; Tanaka et al. 1999) and produces spikes with amplitude and duration similar to those observed in MNCs. Steady-state activation and inactivation characteristics of INa, IK(DR) and IA are presented in Fig. 2(a).

Fig. 2
Steady-state characteristics of activation and inactivation of ionic currents in the model. (a) Steady-state characteristics of voltage-gated Na+ and K+ currents. The variables m3 (red) and h (green) are associated with INa, n ...

Ca2+-activated K+ currents

The model includes two Ca2+-activated K+ currents mediated by SK and BK channels (Marty 1981; Marty and Neher 1985; Vergara et al. 1998). Activation of the SK channels is voltage independent and highly sensitive to intracellular Ca2+ (Hille 2001; Sah and Davies 2000); BK channel activation is both voltage-and Ca2+-dependent (Vergara et al. 1998; Dopico et al. 1999). While activation of BK channels contributes to the falling phase of individual action potentials and the generation of the fast after hyperpolarizing potential (AHP) or hyperpolarizing after potential in many types of neurons (Lancaster and Adams 1986; Lancaster and Nicoll 1987; MacDermott and Weight 1982; Womack and Khodakhah 2002), including MNCs (Dopico et al. 1999), SK channel activation is responsible for the generation of the AHP following a train of action potentials (Armstrong et al. 1994; Bourque and Brown 1987; Greffrath et al. 1998; Kirkpatrick and Bourque 1996; Lancaster and Adams 1986; Lancaster and Nicoll 1987; Sah and Bekkers 1996). This apamin-sensitive AHP is intermediate between the fast AHP (~50 ms) and “slow” AHP (lasting >5 s) (Ghamari-Langroudi and Bourque 2004).

Activation of SK current was described by the Hill function:


where [Ca2+]in is the intracellular Ca2+ concentration, Km,SK is the dissociation constant and ϕ is the Hill coefficient. The activation parameters of SK channels (Fig. 2(b)) were taken from data obtained in giant inside-out patches of Xenopus oocytes (Xia et al. 1998) showing that all SK-channel subtypes exhibit similar Ca2+ dose-response relationships, with a half-maximal activation of about 0.3 μM Ca2+ and a Hill coefficient of about 4 (0.32±0.03 μM and 5.0±0.6 for SK3 channels) (see Appendix).

BK channels are activated by an increase in intracellular [Ca2+] and by depolarization (Vergara et al. 1998), and the activation was described by a modified Boltzmann function:


where the half-activation voltage for the gating variable Vhalf,BK([Ca2+]BK) is a logarithmic function of intracellular Ca2+ concentration in the vicinity of the channel (see Appendix). Parameters of these channels in the model were based on data obtained in rat supraoptic MNCs (Dopico et al. 1999) (Fig. 2(c)).

Ca2+-activated non-selective cation current

The model includes a Ca2+-activated nonselective cation (CAN) current, which is present in neuronal cells (Partridge et al. 1994), including MNCs (Ghamari-Langroudi and Bourque 2002). Because CAN channels are usually nearly equally permeable to Na+ and K+ (see Teulon 2000), the CAN current was described in the model as the sum of Na+ current and K+ current with unitary conductance by the following equations:


The unitary conductance


was used in the description of the CAN current:


where gCAN is the maximal conductance for ICAN and mCAN is the activation function for ICAN The Ca2+ and voltage-dependence of mCAN in the model is described as the product of two functions:


One of them is based on the Hill equation:


the other is based on the Boltzmann equation:


The function S([Ca2+]in) reflects the Ca2+-sensitive shift of half-activation:


where a1 =0.138·10−3 mM; b1 =0.5·10−5 mM; a2 = 0.135·10−3 mM; b2=0.7·10−5 mM. This function provides a shift of half-activation of approximately −10 mV in the physiological range of intracellular [Ca2+]. The Ca2+ dependence of half-activation is in correspondence with the findings of Bourque (1986), which showed that application of the Ca2+ channel antagonist Cd2+ strongly reduced the amplitude of the IDAP and shifted the threshold of activation of this current toward a more positive membrane potential by approximately 10–15 mV. A Hill coefficient of 1 did not match the reported difference in the amplitudes of IDAP(ICAN) evoked by 5–8 spikes in control and with application of Cd2+ (see Bourque 1986). Therefore, a Hill coefficient of 2 and a dissociation constant (Kd,CAN) of 300 nM were used in our model CAN current. The activation characteristics of the CAN current are described in Fig. 2(d,e).

SOR current

A slowly activated K+ current underlying the sustained outward rectification (SOR) and rebound depolarization in OT neurons was included in each compartment of the model in order to simulate electrical properties of OT cells (Stern and Armstrong 1995, 1997; Teruyama and Armstrong 2002). Steady-state characteristics of activation of this current are presented in Fig. 2(a) (see also Appendix 1.3.3).

Ca2+ currents

In acutely dissociated magnocellular neurons from the rat SON, high-voltage activated L-, N-, P/Q- and R-type Ca2+ currents, but not the low-voltage activated T-type current, were observed (Joux et al. 2001). The T-type current has been observed in other studies of MNCs (Erickson et al. 1993; Fisher and Bourque 1995), but it appears not to play a prominent role in MNC electrophysiological properties (Luther and Tasker 2000) and, therefore, it was not included in the model. The L- and R-type currents are fast activating and slowly inactivating, whereas N- and P/Q-type currents are activated more slowly and are non-inactivating. For simplicity, we included only two voltage-gated Ca2+ currents, one of each type: the lower threshold and rapidly activating L-type current and the higher threshold and slower N-type current. Following the current/voltage relationships obtained by Joux et al. (2001), activation of the N-type current was described by a single Boltzmann function, whereas activation of the L-type current was described by two Boltzmann functions (see Appendix). According to Joux et al. (2001), L-type calcium channels in MNCs can be formed by either Cav1.2 or Cav1.3. Each channel has its own voltage dependence. Immunocytochemical studies (Joux et al. 2001) showed that both of these subunits are present and have differential space distributions in MNCs. Experimental current-voltage relationships in MNCs were composed of at least two components, being best described by double Boltzmann functions with half-activation potentials around −30 and −10 mV (Joux et al. 2001). In our model we used double Boltzmann functions with half-activation potentials of −27 and −11.4 mV and slope factors of 4.5 and 2.0. These functions provided a good fit to the L-current experimental current/voltage relationship (Fig. 3(a2)). Each compartment has its own activation curve for the L-type current, and features a shoulder caused by summation of Boltzmann functions with different half activation potentials and slope factors (Fig. 2(f)). Parameters for inactivation of the L-type current and the distributions of L- and N-type conductances between compartments were selected to provide a fit to voltage clamp data obtained for the different Ca2+ current types and to reproduce the total Ca2+ current in the cell (Fig. 3). Compartmental specificity of the Ca2+ currents in the model is based on immunohistochemical evidence of the differential somato-dendritic distribution of Cav1.2 and Cav1.3 subunits (L-type channels) and Cav2.2 subunits (N-type channels) in magnocellular neurons described by Joux et al. (2001). In order to take into account the loss of distal dendrites in the dissociated cell preparation (Joux et al. 2001), the current/voltage relationships of the Ca2+ currents in the model were fitted to the experimental data obtained in dissociated cells assuming a loss of 80% of secondary dendrite length. In the model, the weights of the Boltzman functions for the L-type current activation equations in each compartment as well as parameters for inactivation equations were chosen such as to achieve the best agreement with voltage clamp experiments (Joux et al. 2001) (see Appendix).

Fig. 3
Ca2+ currents in the model and in rat MNCs. (a) L-type current. (b) N-type current. (c) Total Ca2+ current. a1, b1 and c1 show L-type, N-type and total Ca2+ current responses, respectively, to a depolarizing step from −80 to 0 mV (a1, b1) or to ...

2.1.3 Intracellular Ca2+ dynamics

The dynamics of intracellular [Ca2+] in each compartment (“i”) is described by the first order equation:


where fCa,i is the endogenous Ca2+ buffer capacity, a measure of how fast [Ca2+] changes in the compartment; ΣICa,j is the total Ca2+ current (subscript j represents the L- or N-type Ca2+ current in the compartment); Ui is the rate constant of Ca2+ uptake by intracellular stores; [Ca2+]r is the intracellular [Ca2+] at rest; di is the diameter of the compartment; and F is the Faraday constant.

The specificity of coupling between Ca2+ channels and Ca2+-activated K+ channels has been demonstrated in hippocampal neurons (Marion and Tavalin 1998), where N-type Ca2+ channels preferentially activate BK channels, while L-type channels activate SK channels. The temporal association of the opening times of N-type channels and BK channels indicates that these channels are in close proximity to one another. Taking into account the results of studies showing that, in hippocampal pyramidal neurons, BK channel distribution decreases with the distance from the soma (Johnston et al. 2000), we did not include BK channels in the secondary dendrites of our model. We assumed that the [Ca2+] in the vicinity of BK channels changed much faster than the bulk [Ca2+] in the compartment. Therefore, additional Ca2+ subcompartments (subdomains) for BK channels were added to the soma and primary dendrites (Fig. 1(b)) according to the following formalism:


where [Ca2+]BK,i is the [Ca2+] that activates BK channels, fCa,BK is the fast endogenous Ca2+ buffering capacity in the vicinity of BK channels, KBK,i is the rate constant of [Ca2+] decay in the subcompartment, and r is the ratio of BK subcompartment volume to the total compartment volume. Here, “i” can be either “s” (soma) or “pd” (primary dendrite). The simulated dynamics of [Ca2+] in the BK subcompartment triggers the activation of the BK current during spike development and plays a role in spike AHP.

2.2 Model implementation

Numerical integrations of simulations were performed using an implicit Runge-Kutta method of order five with a variable time step (Hairer and Wanner 1996). The symmetrical structure of the model allows one to perform numerical computations for three compartments only, the soma, the primary dendrite, and the secondary dendrite. The dynamics of membrane potential and [Ca2+] in the three compartments were described by a total of 39 first-order differential equations. The simulations were run on a 700 MHz R16000 SGI Origin 350 and a 360 MHz R12000 SGI Octane2 workstation.

3 Results

3.1 Action potential generation and calcium dynamics

To date there are no direct recordings from the dendrites of MNCs, so we based the selection of the distributions of conductances on experimental data obtained from other nerve cells. Somatic and dendritic recordings in hippocampal CA1 pyramidal neurons have shown that somatic spikes in these cells are larger than dendritic spikes (Callaway and Ross 1995). Also, a prominent gradient in the density of the A-type current exists in CA1 pyramidal neurons (Hoffman et al. 1997; Golding and Spruston 1998). While in our model the conductance for IK(DR) is distributed uniformly, the conductance for the sodium current is larger in the soma and the proximal dendrites than in the distal dendrites (gNa,s: gNa,pd:gNa,sd =7:5:1); the conductance for the transient potassium current has the opposite distribution (gA,s:gA,pd: gA,sd≈1:5:7). The selected set of conductances provided a spike amplitude and duration similar to those recorded with somatic recordings from MNCs. Magnocellular neurons have action potentials with a fast ascending phase that provides a peak amplitude of 77.8±2.0 mV in VP neurons and 73.2±1.8 mV in OT neurons and a duration at the half-maximum spike amplitude of 1.5±0.1 ms (Armstrong et al. 1994). Figure 4 shows the voltage-gated currents and action potentials generated in the different compartments of the model MNC. In the model, spikes have a peak amplitude of 75 mV and a duration at the half-maximum spike amplitude of 1.2 ms, which are values close to the average values measured experimentally (Armstrong et al. 1994; Bourque and Renaud 1985; Stern and Armstrong 1996; Tasker and Dudek 1991; Teruyama and Armstrong 2002). In the model, action potentials and ionic currents have different characteristics in the soma, primary dendrite and secondary dendrite. Under simulated somatic stimulation with a depolarizing current of +25 pA, the initiation of the somatic spike and the spike in the primary dendrite precedes significantly (by about 0.4 ms) spike initiation in the secondary dendritic compartment (Fig. 4(a)). Also, the amplitude of the spike in the distal dendrites is decreased to 49 mV. The model neuron demonstrates notable differences in the compartmental dynamics of voltage-gated ionic currents with action potential generation in the soma and in the primary and secondary dendrites (Fig. 4(b)) due to the differences in the parameters of each of these compartments (see Section 2.1 and Appendix). Thus, the A-current makes a bigger contribution to membrane repolarization during the descending phase of the action potential in the dendrites than in the soma.

Fig. 4
Simulation of voltage-gated currents and action potentials in different compartments of the MNC model. (a) A single action potential evoked by somatic injection of a suprathreshold depolarizing current and recorded in the soma of a rat SON MNC in a hypothalamic ...

It has been established in calcium imaging studies that the dendrites of different types of nerve cells exhibit larger-amplitude Ca2+ spikes and more rapid Ca2+ accumulation and decay than the soma (Wilson and Callaway 2000; Abel et al. 2004). Some differences in Ca2+ dynamics have also been revealed in different compartments of MNCs during and after evoked bursts with spike frequencies of 5–10 Hz (Roper et al. 2004). Thus, dendritic Ca2+ concentration rises faster and stays higher in the dendrites compared to the soma of MNCs during 2–3 s of the spike discharge, and the dendritic Ca2+ concentration decays more rapidly than the somatic Ca2+ concentration following the spike discharge. In the model, the different volume-to-surface ratios of the soma, primary dendrites and secondary dendrites, as well as the differential spatial distributions of voltage-gated Ca2+ channels and Ca2+ uptake by stores (Table 1) provide specificity of Ca2+ dynamics in the different compartments that are consistent with experimental data. During spike trains, each action potential makes a contribution to the rising Ca2+ concentration, as shown in Fig. 5(a,b).

Fig. 5
Time course of changes in calcium concentration and membrane potential in the model MNC. Somatic potential (Vs) was −59 mV and a depolarizing current injection of 10 pA was applied as shown in the top trace. (a) Time course of the change in membrane ...

3.2 Simulation of the basic electrophysiological properties of MNCs

3.2.1 Transient outward rectification

When hyperpolarized, MNCs respond to depolarizing current injections with an IA-mediated transient outward rectification (TOR) (Bourque 1988; Fisher et al. 1998; Luther and Tasker 2000), which is characterized by a slowing of membrane charging and a delay to the onset of the first action potential (Fig. 6(a)). Under the same simulated protocol, the model MNC shows a similar dampening of the depolarization and delayed onset of spike firing (Fig. 6(b)). The model shows a “notch” in the rising phase of the voltage transient characteristic of the strong TOR generated in MNCs (Tasker and Dudek 1991; Luther and Tasker 2000; Luther et al. 2000). Figure 6(c) depicts the dynamics of the model variables underlying the behavior of INa(m3 and h) and IA(p4 and q) during the time course of Vs from the onset of the depolarizing current to the end of the first spike, as shown in Fig. 6(b) (red trace). The corresponding time span of 150 ms is marked by the blue bar in Fig. 6(b). The relatively fast activation (p variable) of IA and its slow inactivation (q variable) prevent development of the first spike.

Fig. 6
Simulation of transient outward rectification (TOR) in the model MNC. (a) Experimental record of the membrane responses of an MNC in the PVN to a series of incrementally larger depolarizing current pulses (top traces) following steady-state hyperpolarization ...

3.2.2 Depolarizing after potential

Magnocellular neurons exhibit depolarizing after potentials (DAPs) after single spikes or brief trains of spikes (Andrew and Dudek 1984a; Bourque and Renaud 1985). The summation of DAPs causes a small plateau potential and an after-discharge of spikes, which are dependent on the stimulation protocol, the physiological state of the neuron (i.e., from virgin, pregnant, or lactating rats), and the type of magnocellular neuron (i.e., OT or VP neuron) (Stern and Armstrong 1996; Teruyama and Armstrong 2002). The calcium and voltage dependence of the IDAP in MNCs has been established (Bourque 1986), however its ionic mechanisms remain uncertain. There are two possible ionic mechanisms for the DAP in MNCs proposed on the basis of experimental findings. One involves the Ca2+-dependent block of a resting K+ conductance, presumably TASK-1, producing a brief depolarizing shift in the cells’ resting potential (Li and Hatton 1997a,b; Roper et al. 2003). Our model is based on a second possible mechanism, activation of a calcium-dependent, non-specific cation (CAN) current, based on the sensitivity of DAPs and phasic firing in SON neurons to flufenamic acid, a blocker of CAN channels (Ghamari-Langroudi and Bourque 2002) (see Section 4). The main characteristics of CAN channels are their conductance between 18 and 34 pS, their lack of discrimination between monovalent cations, their low permeability to Ca2+, their sensitivity to intracellular [Ca2+], their inhibition by cytosolic ATP, and their blockade by flufenamic acid (for review see Teulon 2000).

The DAP is expressed in the majority (75%) of VP neurons, whereas it is present in only a minority (32%) of OT neurons recorded in the brain slice preparation (Armstrong et al. 1994). The findings of Greffrath et al. (1998) suggest that the induction, maintenance and termination of the plateau potential underlying bursting in MNCs are regulated by the balance between the DAP and the slow AHP. Experimental evidence of a contribution of the slow AHP to burst termination was presented by Ghamari-Langroudi and Bourque (2004). However, the best characterized mechanism for burst termination is an activity-dependent inhibition of the DAP (Brown and Bourque 2004; Brown et al. 2006), which was incorporated in the model of Roper et al. (2004). In slice experiments, about half of the MNCs encountered in the SON displayed a DAP (Armstrong et al. 1994; Li et al. 1995). Only a small proportion (16%) of dissociated MNCs isolated from the rat SON displayed DAPs or spontaneous bursting activity following brief current-evoked trains (Oliet and Bourque 1992). These cells usually retained one to three short (<30 μm) processes. These findings suggest that the dendrites play an important role in DAP generation. Since CAN channel activation depends on intracellular [Ca2+], the distribution of gCAN between model compartments may have a significant influence on the DAP. The values of gCAN,s=45 μS/cm2, gCAN,pd=90 μS/cm2 and gCAN,sd=0 provided the best correspondence between simulated and experimental data (Bourque 1986; Li and Hatton 1997b) (Fig. 7(a–d)). Simulated blockade of this current by setting gCAN,i to zero (Fig. 7(e)) reproduced the results of experiments with application of flufenamic acid to MNCs (Ghamari-Langroudi and Bourque 2002).

Fig. 7
Voltage dependence of the DAP in MNCs. (a) In vitro brain slice recording of a SON MNC (from Li and Hatton 1997a). (a1) Immediately following a train of three spikes elicited by successive current injections the membrane potential was brought to various ...

3.2.3 After hyperpolarizing potential and spike frequency adaptation

OT and VP neurons exhibit a prominent AHP following trains of action potentials evoked with long-lasting stimulation (Andrew and Dudek 1984b; Armstrong et al. 1994; Bourque et al. 1985). The size of the AHP is dependent upon the number of spikes in the train. The application of the SK channel blocker, apamin, strongly inhibits the AHP (Armstrong et al. 1994). Figure 8 shows the AHP and Ca2+ dynamics in the model that match data obtained in experiments with prolonged stimulation of MNCs in slice preparations and simultaneous recordings of membrane potential and [Ca2+] in the soma (Roper et al. 2003). The model neuron is able to simulate AHPs evoked by long-lasting stimulation (Armstrong et al. 1994; Stern and Armstrong 1996; Teruyama and Armstrong 2002) as well as spike frequency adaptation similar to that observed experimentally in MNCs (Stern and Armstrong 1996; Tasker and Dudek 1991; Teruyama and Armstrong 2002). This phenomenon is caused by the SK current underlying the AHP (Fig. 8(c)); the simulated blockade of SK channels eliminated spike frequency adaptation.

Fig. 8
AHP and spike frequency adaptation in the model MNC. (a) A spike train and subsequent AHP elicited by repetitive stimulation (5 ms pulses at 40 Hz for 1 s) in a somatic recording of a MNC (a1) (modified from Roper et al. 2003) and in the soma of the model ...

3.2.4 Sustained outward rectification and rebound depolarization

Magnocellular OT-secreting neurosecretory cells, but not VP-secreting cells, demonstrate a depolarization-activated sustained outward rectification (SOR) (Stern and Armstrong 1995, 1997; Teruyama and Armstrong 2002). The SOR is observed as a sag in the voltage trajectory during hyperpolarizing pulses from a depolarized holding potential, and as a rebound depolarization that supports action potentials following the hyperpolarizing pulses. The SOR is present positive to −75 mVand decreases in amplitude as the hyperpolarizing pulses approach the potassium equilibrium potential (EK). The SOR has been proposed to be the result of the deactivation of a voltage-dependent, non-inactivating K+ current (Stern and Armstrong 1995).

In our model, ISOR is described as a slowly activating K+ current using Hodgkin-Huxley formalism (see Appendix). The steady-state activation of ISOR is described as a Boltzmann function, with the time constant of activation (τm,SOR) equal to 350 ms. Specific conductances for ISOR in all compartments were equal (gSOR,s = gSOR,pd = gSOR,sd = 20 μS/cm2), simulating a uniform distribution of channels. This representation of the SOR current is similar to the one in the single compartment model of Roper et al. (2003), but our set of parameters was selected to provide the best fit to experimental I–V curves (Stern and Armstrong 1995) in our multicompartmental model (Fig. 9). The redistribution of channels between compartments had no significant influence on the SOR characteristics (data not shown).

Fig. 9
SOR generation in the model MNC. (a) Voltage responses to hyperpolarizing current pulses from a subthreshold depolarized membrane potential (−50 mV) in a recorded OT neuron show a time- and voltage-dependent inward rectification and a rebound ...

3.3 Role of the SOR Current in the AHP

Although it was not the purpose of this study to systematically simulate all the differences in the electrophysiological properties of VP and OT neurons, our model suggests that the expression of SOR in MNCs may be a critical factor underlying the differences in the dynamics of AHPs between OT and VP neurons. Figure 10(a) shows experimental records of AHPs in a VP neuron (A1) and an OT neuron (A2) evoked by short depolarizing current pulses (Armstrong et al. 1994). Membrane potential changes in response to the same stimulation protocol in two models of MNCs are presented in Fig. 10(b). All the parameters for the two models were the same, except for the DAP and SOR parameters. The model VP neuron included the IDAP (generated by a CAN current), but no ISOR. The model OT neuron included the ISOR, but no IDAP. The respective OT and VP neuron models simulated distinct AHPs, with differences in time constants of decay and dynamics similar to those observed experimentally (Armstrong et al. 1994; Teruyama and Armstrong 2002). Thus, in the model OT neuron, the τAHP is 188 ms, while in experiments on OT neurons in hypothalamic explants, the τAHP was found to be 257±73 ms; in the model VP neuron the τAHP is 424 ms, while in experiments on VP neurons, the τAHP value was 431±93 ms (Armstrong et al. 1994) (Fig. 10(b)).

Fig. 10
Effect of SOR on the AHP. AHPs were evoked in a real MNC (Modified from Armstrong et al. 1994) and in the model MNC by nine current pulses at 50 Hz. (a) AHPs (arrows) in a VP neuron (a1) and in an OT neuron (a2) recorded in a slice and identified by immunohistochemistry ...

3.4 Effects of compartmental distribution of ion conductances on the electrophysiological properties of the model neuron

Because of the lack of experimental data on the somato-dendritic distribution of ion channels in MNCs and its impact on MNC electrical activity, we studied the effects of the redistribution of some intrinsic ion conductances among the three compartments on the model’s response to typical experimental stimulation protocols. Other than the conductances that were redistributed in the specific simulations (gA, gCAN and gSK), all other parameters of the model were held constant at the values indicated in Table 1.

3.4.1 Distribution of gA

In the model, the simulated A-type channel density increases with the distance from the soma (gA,s:gA,pd:gA,sd≈1:5:7), similar to what has been observed in hippocampal CA1 pyramidal neurons (Hoffman et al. 1997) and in mitral/tufted cells (Bischofberger and Jonas 1997). We found that the compartmental distribution of A-current channels has a marked influence on the frequency of evoked spiking in the model neuron. We tested the effect on spiking of changing the compartmental distribution of the A-current conductance while keeping the total membrane conductance, AsgA,s+2 ApdgA,pd+4AsdgA,sd, constant (where As, Apd, and Asd are the surface areas of the soma, primary dendrites, and secondary dendrites, respectively) (Fig. 11(b)). A similar approach was used for the redistribution of other conductances (gCAN and gSK). Figure 11(a) shows typical high frequency spiking in a VP neuron recorded experimentally in vitro in response to depolarization. Distributions of A-current conductances that were predominantly dendritic (Fig. 11(b1,b6,b7)) produced repetitive spiking activity that is similar to that observed experimentally (Fig. 11(c1, c6,c7)). Somatic, primary dendritic or uniform distributions of gA (Fig. 11(b2–c2,b3–c3,b4–c4,b5–c5)) caused a delay in the onset of each successive action potential and a decrease in the frequency of action potentials, resulting in an evoked spiking activity that differed from that of MNCs recorded experimentally in vitro under the same stimulation protocol (Fig. 11(a)).

Fig. 11
Influence of spatial distribution of A-type channels on spiking activity in the model MNC. (a) Spike discharge evoked by a 30 pA, 500 ms current pulse in a hypothalamic slice recording of a rat SON MNC (see Boudaba et al. 2003). (b) Diagrams showing the ...

3.4.2 Distribution of the DAP conductance (gCAN)

We simulated the expression of CAN channels in the model MNC, which allowed us to reproduce DAPs and spike after-discharges following short depolarizing pulses similar to those observed experimentally (Fig. 12) (Li and Hatton 1997a). With all other conductances and the total CAN conductance maintained constant, the simulated compartmental distribution of CAN channels in the soma and primary dendrites gave the best correspondence with the DAP amplitude and dynamics observed in MNCs recorded in slices (gCAN,s=45 μS/cm2, gCAN,pd=90 μS/cm2 and gCAN,sd=0 (Fig. 12(b1,c1))).

Fig. 12
Influence of spatial distribution of CAN channels on the DAP. (a) Voltage record (top trace) from a real MNC demonstrating progressively larger, longer-lasting DAPs and a spike after discharge following an increasing number of spikes (1, 2 and 4) evoked ...

3.4.3 Distribution of the AHP conductance (gSK)

The possible role of SK channel distribution in AHP generation was also studied in the model MNC. A good correspondence with the AHP recorded in VP neurons in vitro (Armstrong et al. 1994; Teruyama and Armstrong 2002) was obtained in the model (without ISOR) with a compartmental distribution of the SK conductance of gSK,s= 10 μS/cm2, gSK,pd=40 μS/cm2 and gSK,sd=25 μS/cm2, (Fig. 13(c1,d1)) as well as with uniform distribution of the gSK throughout all three compartments (Fig. 13(c2,d2)), with SK channels distributed only in primary and secondary dendrites (Fig. 13(c6,d6)), and with SK channels expressed only in secondary dendrites (Fig. 13(c7,d7)). On the other hand, SK channels located in the soma only (Fig. 13(c3, d3)), in the soma and primary dendrites (Fig. 13(c4,d4)) and in the primary dendrites only (Fig. 13(c5,d5)) produced a much slower decay of the AHP than has been observed in MNCs experimentally (Fig. 13(a,b)). The model’s SK channel distribution (Fig. 13(c1)) was selected based on these data and data generated with other stimulation protocols (Figs. 8(a) and 10(b)) that provided the best match to experimental data.

Fig. 13
Influence of spatial distribution of SK channels on AHPs. (a) The time constant of decay of the AHP in VP neurons recorded in a slice (modified from Teruyama and Armstrong 2002). (b) Voltage record from a recorded VP neuron showing an AHP following a ...

4 Discussion

4.1 Compartmentalization of properties in the model

During the last decade it has become evident that dendrites of nerve cells are dynamic structures that express voltage-gated ion channels and actively participate in the generation of neuronal electrical activity (for review see Häusser et al. 2000). Electrophysiological evidence of active dendritic properties has been obtained in brain structures like the hippocampus and neocortex, where neuronal architecture makes dendrites of nerve cells accessible experimentally. Such direct experimental measurements in the hypothalamus are more difficult because the dendrites of MNCs are interwoven with somata and axons. However, indirect studies have suggested a differential expression of ion currents in the soma and dendrites of MNCs (Bains 2002; Mason and Leng 1984). Therefore, a multicompartmental model of MNCs would be useful for studying the electrophysiological mechanisms that rely on differential somato-dendritic properties in these cells.

We designed a seven-compartment model of the hypothalamic MNC using basic morphological parameters of these neurons, experimentally measured passive electrical properties, and a compartmentalized distribution of ionic currents and Ca2+ dynamics based on experimental observations. The activation and inactivation properties of voltage-gated currents were extracted from available voltage-clamp data from MNCs (Joux et al. 2001; Luther and Tasker 2000; Tanaka et al. 1999), and from other types of mammalian CNS neurons (Hoffman et al. 1997; Migliore and Shepherd 2002). The main criteria for determining the distribution of channels were simulations of appropriate physiological responses for MNCs for different experimental protocols. The model closely reproduces the basic electrophysiological properties of MNCs in vitro. This multi-compartmental model expands on a recently proposed single-compartment model (Roper et al. 2003) to allow us to investigate the putative roles of compartmentally distributed ion currents and intracellular Ca2+ dynamics in shaping the electrophysiological properties of MNCs.

The simulated Ca2+ dynamics in the different parts of our model neuron are close to those observed experimentally in recorded MNCs in amplitude and time constant of decay (Roper et al. 2003, 2004) and are similar to analogous results of Ca2+ imaging studies done in other types of neurons (Abel et al. 2004; Wilson and Callaway 2000). In addition to L- and N-type Ca2+ currents, our model includes two Ca2+ subcompartments for activation of BK channels in the soma and primary dendrites that are separated by barriers from the cytoplasm. The physiological basis of these calcium microdomains is the experimentally observed colocalization of Ca2+ channels and BK channels, which suggests that BK channels are activated by a fast increase in [Ca2+] delivered in close proximity to the channels (Marion and Tavalin 1998; Sun et al. 2003). Thus, changes in [Ca2+] in these structurally and functionally small subcompartments are larger than in the bulk cytoplasm.

4.2 Compartmentalization of the A-current

In the model, IA is most prominent in the secondary dendrites due to the increase in the maximum conductance of this current with distance from the soma (gA,s<gA,pd<gA, sd), similar to that observed in hippocampal pyramidal neurons (Hoffman et al. 1997; Johnston et al. 2000). The Atype current may increase in the dendrites due not only to a corresponding increase in the density of Kv4-family α-subunits that form A-type channels, but also to changes in the distribution of Kv-channel interacting proteins (KChIPs), which may augment IA by a variety of mechanisms (An et al. 2000). For simplicity we did not account for changes in channel properties in different model compartments and showed only how the subcellular distribution of uniform A-type channels influences the spiking characteristics of MNCs (Fig. 11).

The presence of A-type channels in the dendrites of MNCs in PVN and SON has been suggested by the finding that these neurons appear to lose their IA after dissociation and the resulting loss of their dendritic trees (Widmer et al. 1997). Also, experiments with immunocytochemical identification of dissociated MNCs revealed that the A-type current was selectively expressed in OT-neurophysin-positive cells (Widmer et al. 1997). Taking into account that experiments in hypothalamic slices show that A-type channels are expressed in both VP and OT neurons (see Armstrong and Stern 1998), these results, taken together, suggested that A-type channels may be expressed only in the dendritic tree of VP neurons. However, in a series of experiments by Fisher et al. (1998), an A-type current was recorded in both VP and OT neurons following acute isolation and loss of their dendritic trees, implying a soma/primary dendrite localization of the A-type channels in both types of MNCs. In our modeling study, available experimental data on A-current characteristics were taken into account (Luther and Tasker 2000). We found that the best correspondence between model and spiking activity recorded in MNCs was achieved with a primarily dendritic distribution of A-type channels (Fig. 11). While somatic, somato-primary dendritic, primary dendritic only and uniform distributions of A-type channels did not produce spiking patterns similar to that seen in MNCs, both dendritic-only and increasing somatic-to-dendritic distributions produced spike patterns that resembled MNC spiking activity.

4.3 Ionic and compartmental mechanisms of the DAP

There are considerable discrepancies in the experimental data concerning the ionic mechanisms responsible for the DAP in MNCs. Two mechanisms have been proposed and backed up with experimental findings: the blockade of a resting K+ current (Li and Hatton 1997a) and the generation of a Ca2+-dependent, non-selective cation, or CAN, current (Ghamari-Langroudi and Bourque 2002). We selected the CAN current mechanism in our model for the following reasons. First, there is no experimental evidence for the inhibition of resting K+ channels, presumably TASK-1 channels, by intracellular Ca2+, and experimental manipulation of the intracellular [Ca2+] in both directions fails to influence the activity of TASK channels (Duprat et al. 1997; Lenoudakis et al. 1998). These channels have relative voltage independence and rapid activation with little or no inactivation following depolarization, as judged by their opening across the physiological range of membrane potentials (Lesage and Lazdunski 2000; Goldstein et al. 2001). Therefore, these properties of TASK-1 channels make it unlikely that resting K+ channels are responsible for the DAP. On the other hand, there are several lines of evidence that support the role of a CAN current in mediating the DAP. CAN channels exist in the membranes of a variety of mammalian cells, both excitable and non-excitable, and support important cellular responses such as neuronal bursting, secretion, and cardiac rhythmicity (Siemen 1993; Thorn and Petersen 1993; Partridge et al. 1994; Teulon 2000). For example, CAN channels take part in plateau potential generation and persistent spiking in entorhinal cortex neurons (Egorov et al. 2002), in DAPs in myenteric neurons (Vogalis et al. 2002), and in after-depolarization and burst after discharge in rat neocortical neurons (Kang et al. 1998; Aoyagi et al. 2002). ICAN appears to have higher calcium sensitivity than other Ca2+-activated currents, such as IBK and ISK (Zhu et al. 2004). The voltage-dependence of opening of CAN channels recently was reported in several excitable cells, including in rat and human cardiomyocytes (Guinamard et al. 2002, 2004) and in hamster vomeronasal sensory neurons (Liman 2003). We included these voltage-dependent properties in our mathematical description of ICAN (see Appendix). Thus, based on the combined Ca2+ dependence and voltage sensitivity of these channels, we regarded the CAN channel hypothesis (Ghamari-Langroudi and Bourque 2002) to be the most plausible one for the DAP mechanism. Further support of this hypothesis came from the recent finding that ~30% of cell-attached patches on isolated SON MNCs contained nonselective cation channels that were active at rest (Han et al. 2003). Also, recent experimental studies with whole cell recordings in both current and voltage clamp demonstrated that VP cells in the SON express a CAN current, whose activation contributes to the generation of the DAP (Teruyama and Armstrong 2006).

No data are available concerning the distribution of CAN channels in different parts of neurons, however our modeling study allows one to draw the conclusion that this distribution may play an important role in the generation of DAPs in MNCs. In our model, the activity-dependent rise and decay in intracellular [Ca2+] are fastest in the secondary dendrites and slowest in the soma, which is in agreement with spatial [Ca2+] measurements and recordings from MNCs (Roper et al. 2004). The best correspondence to recorded DAPs (Fig. 12(a)) was observed when CAN channels were located in the primary dendrites and soma, with more prominent expression in the primary dendrites (Fig. 12(b1,c1)). Simulated redistribution of CAN channels from the soma to the dendrites leads to an increase in the amplitude of the DAP, but also to an increase in its decay rate (Fig. 12(b4–b6,c4–c6)), which is not consistent with experimental data. A similar result was obtained with a uniform distribution of CAN channels in all three compartments of the cell (Fig. 12(b7,c7)). Redistribution of CAN channels to the soma slightly decreases the amplitude of the DAP and significantly increases its decay time (Fig. 12(b3, c3)), which also is not consistent with the experimental data. These results suggest that the expression of CAN channels in different parts of the cell may be important for burst initiation and persistent spiking activity in MNCs. Since the best match to experimental data was achieved when the CAN channels underlying the DAP were restricted to the primary dendrites and soma, our findings suggest that the majority of acutely dissociated MNCs did not display DAPs and DAP-mediated bursting (Oliet and Bourque 1992) because of the loss of dendrites caused by cell dissociation.

4.4 Ionic and compartmental mechanisms of the AHP

In our model, the AHP is mediated by a SK current, which is activated by elevation of intracellular [Ca2+]. Differences in the compartmental [Ca2+] dynamics provide the potential for differences in AHP generation and decay by changing the ISK conductance among the compartments. The model allows one to fit the AHP to experimental data from magnocellular neurons (Armstrong et al. 1994; Stern and Armstrong 1996; Teruyama and Armstrong 2002) and suggests mechanisms of regulation of the AHP not only by varying channel properties, but also by changing compartmental [Ca2+] dynamics and SK channel distribution.

The finding that dissociated MNCs (i.e., without dendrites or with short primary dendrites) express a Ca2+-dependent K+ conductance (Oliet and Bourque 1992) similar to that found in neurons recorded in slices (Armstrong 1995; Andrew and Dudek 1984b; Bourque and Renaud 1985) suggests that the channels responsible for the AHP are located primarily in the soma and primary dendrites. Our results of changing the compartmental distribution of the SK channels in the model suggest that these channels are probably not located exclusively in the soma or in the soma and primary dendrites, but rather are distributed among the somatic, primary dendritic and secondary dendritic compartments (Fig. 13(a)), since these distributions provide waveforms and decay time constants of the AHP that most closely approximate those of recorded MNCs under different stimulation protocols. Also, our findings show that the different AHP dynamics reported for OT and VP neurons (Armstrong et al. 1994; Teruyama and Armstrong 2002) may be due not only to the differential expression and distribution of SK channels, but also to the selective expression of the SOR current in OT neurons (see Fig. 10).

4.5 Model limitations

Construction of a mathematical model is always limited by available experimental data and adequate complexity. The characteristics of some currents were not available from MNC experimental data. This gap was filled by the adoption of data obtained from other mammalian CNS neurons, as well as by fitting characteristics of currents to obtain realistic model responses. In our model, simple first order differential equations were used for the description of Ca2+ dynamics in each compartment and sub-compartment. Intra- and intercompartmental diffusion was not taken into account. We did not completely explore the parameter space of the model; the analyses were done based on changes of one parameter and, therefore, other possible physiologically plausible solutions may exist. Possible ways to tune parameters of the model include the systematic sampling of the parameter space (Prinz et al. 2003) or evolution strategy with a fitness function based on phase-plane analysis (Achard and De Schutter 2006).

4.6 Conclusions

Our model provides a multicompartmental computational tool that takes into account MNC morphology and electrotonic properties, and includes a set of realistic voltage-gated and Ca2+-activated ionic currents with authentic activation and inactivation characteristics. The model reproduces several of the hallmarks of MNC electrophysiological properties, including TOR, spike-frequency adaptation, AHPs, SOR, DAPs and DAP-mediated after discharges. The multiple compartments and different compartmental distributions of ionic conductances in our model provide a more realistic computational environment than single compartment models of MNCs. In particular, it is possible in our model to take into account differences in electrical properties between the soma and dendrites. Thus, the model allowed us to probe how the expression and spatial distribution of different ion channels influences the spiking properties of these neurons. The model, therefore, provides a framework for making realistic predictions concerning specific electrophysiological properties of MNCs.


We thank Dr. Cherif Boudaba for providing recordings of MNC spiking activity. This work was supported by U.S. Department of Energy Grant DE-FG02-01ER63119 (to the Tulane University Center for Computational Science), and National Institutes of Health grants NS039099 (to J. G. Tasker) and HL063195 (to N.A. Trayanova).


The membrane potential of each compartment of the magnocellular neuron is described by the current balance equation:


where Cm is the specific neuronal membrane capacitance; Isd-pd, Ipd-s, Ipd-sd, and Is-pd are the intercompartmental coupling currents; and Istim is the applied microelectrode current. The subscripts “sd,” “pd,” and “s” indicate secondary (distal) dendritic compartment, primary (proximal) dendritic compartment, and somatic compartment, respectively. Each compartment has its own set of conductances for the ion currents.

A.1 Membrane ion currents

The membrane ion currents in the model were described using the Hodgkin-Huxley formalism. The subscript “i” indicates a compartment (s, pd or sd).

A.1.1 Transient sodium current (based on Tanaka et al. 1999)

A.1.2 Calcium currents (based on Joux et al. 2001)

A.1.2.1 High threshold current with no or very slow inactivation (N-type)

A.1.2.2 Lower threshold (L-type) current

A.1.3 Potassium currents

A.1.3.1 Delayed rectifier current (based on Luther and Tasker 2000)

A.1.3.2 A-current (based on Luther and Tasker 2000)

A.1.3.3 SOR current (for the model OT neuron only; based on Stern and Armstrong 1995)

A.1.3.4 Ca2+-sensitive K+ currents (In the equations, [Ca2+] is in mM)

A. SK current (based on Vergara et al. 1998; Xia et al. 1998)

A. BK current (based on Dopico et al. 1999)

A.1.4 CAN current

A.1.5 Linear leakage current

A.2 Calcium dynamics

A.2.1 Calcium dynamics in the cytoplasm

Subscript “j” represents the L- or N-type Ca2+ current in the compartment.

A.2.2 Calcium dynamics in the BK channel subdomains

Here, “i” can be either “s” (soma) or “pd” (primary dendrite).

A.3 Intercompartmental coupling currents


A.4 Applied current

Istim =IST/(pdsLs).

Contributor Information

Alexander O. Komendantov, Center for Computational Science, Tulane University, New Orleans, LA 70118, USA. Department of Neurobiology and Anatomy, Drexel University College of Medicine, 2900 Queen Lane, Philadelphia, PA 19129, USA.

Natalia A. Trayanova, Center for Computational Science, Tulane University, New Orleans, LA 70118, USA. Department of Biomedical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA.

Jeffrey G. Tasker, Center for Computational Science, Tulane University, New Orleans, LA 70118, USA. Division of Neurobiology, Department of Cell and Molecular Biology, Tulane University, New Orleans, LA 70118, USA. Neuroscience Program, Tulane University, New Orleans, LA 70118, USA.


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