A Computational Model of the Dendron of the GnRH Neuron

Abstract

Gonadotropin-releasing hormone (GnRH) neurons have two major processes that have properties of both dendrites (they receive synaptic input from other neurons) and axons (they actively propagate action potentials to the synaptic terminal). These processes have thus been termed dendrons. We construct a stochastic spatiotemporal model of the dendron of the GnRH neuron, with the goal of studying how stochastic synaptic input along the length of the dendron affects the initiation and propagation of action potentials. We show (1) that synaptic inputs closer to the soma are effective controllers of action potential initiation and electrical bursting and (2) that although the effects on the amplitude and width of propagating action potentials are critically dependent on the timing and location of synaptic input addition, the effects remain small. We conclude that although stochastic synaptic input along the length of the dendron is likely to be a major determinant of action potential initiation, it is an unlikely mechanism for controlling whether or not action potentials reach the synaptic terminal. Thus, the role of synaptic inputs situated along the dendron a long way from the site of action potential initiation remains unclear. We also show that the actions of kisspeptin can result in significant modulation of the amount of calcium released by an action potential at the synaptic terminal. Furthermore, we show that the actions of kisspeptin are greatest when multiple effects operate together and that a kisspeptin-induced increase in firing rate is, by itself, less effective at increasing $\mathrm{Ca}^{2+}$ release than is a combination of an increased firing rate, an increase in $\mathrm{Ca}^{2+}$ influx, and an increase in inositol trisphosphate ($\mathrm{IP}_3$) production. We conclude that the inherent synergies in the various actions of kisspeptin make it a likely candidate for the precise control of $\mathrm{Ca}^{2+}$ transients at the synaptic terminal.

Appendix: The deterministic model of a GnRH neuron introduced in Chen et al. 2013

1.1 Voltage Submodel

The equation for membrane potential ($V$) in the voltage subsystem is

$$\begin{aligned} \frac{\partial V(x,t)}{\partial t}&= -\frac{1}{C_\mathrm{m}}I_\text {ionic} (V,x,t)+ D_v\frac{\partial ^2 V(x,t)}{\partial x^2}, \end{aligned}$$

where $C_\mathrm{m}$ is the membrane capacitance and $I_\text {ionic}$ is the sum of the ionic currents.

The soma is the region $[0, x_1]$, the dendrite between the soma and the iSite is $[x_1, x_2]$, the iSite is $[x_2, x_3]$, and the dendrite elsewhere is $[x_3, x_4]$.

For $x \in [0, x_1]$, the currents in the soma are modeled as

$$\begin{aligned} I_\text {ionic} (V,x,t)&= I_\text {naf}+I_\text {nap}+I_\text {kdr}+I_\text {kir}+I_\text {km}+I_\text {cal} +I_\text {cat}\\&\quad +sI_\text {AHP-SK}+sI_\text {AHP-UCL}+I_\text {App}. \end{aligned}$$

For $x \in [x_2, x_3]$, the currents in the iSite are the same as in the soma, except that we use a higher conductance for $I_\text {naf}$, representing a higher density of $\mathrm{Na}^{+}$ channels in the iSite. We use a $\mathrm{Na}^{+}$ conductance ($g_\text {naf}$) of 410 nS in the iSite, and 150 nS elsewhere (Table 2).

Table 2 Parameter values of the model

For $x \in [x_1, x_2]$ and $x \in [x_3, x_4]$, the currents in the dendrite are modeled as

$$\begin{aligned} I_\text {ionic} (V,x,t) = I_\text {naf}+I_\text {nap}+I_\text {kdr}+I_\text {kir}+I_\text {km}+I_\text {cal}+I_\text {cat}+I_\text {App}. \end{aligned}$$

$I_\text {naf}$ and $I_\text {nap}$ denote the fast, persistent $\mathrm{Na}^{+}$ currents, $I_\text {kdr}$, $I_\text {kir}$, and $I_\text {km}$ denote the delayed rectifier, inward rectifier, and m-type $\mathrm{K}^{+}$ currents, respectively, $I_\text {cal}$ and $I_\text {cat}$ are L-type and T-type $\mathrm{Ca}^{2+}$ currents, $sI_\text {AHP-SK}$ is an SK-type $\mathrm{Ca}^{2+}$-activated $\mathrm{K}^{+}$ current, and $sI_\text {AHP-UCL}$ is a slow $\mathrm{Ca}^{2+}$-activated afterhyperpolarization current. $I_\text {App}$ is a passive membrane leakage current. It may incorporate current from synaptic inputs, although there are no explicit synaptic inputs in our model. All the ion channels and fluxes are modeled as in Lee et al. (2010) and references therein.

We used a Hodgkin–Huxley formalism to model the currents. For example, $I_\mathrm{naf}$ is described as

$$\begin{aligned} I_\text {naf}&= g_\text {naf}M^3_{\text {naf}_\infty }H_\text {naf}\left( V - V_\text {na}\right) , \end{aligned}$$

where $g_\text {naf}$ is the maximum conductance, $M_\text {naf}$ is the activation gating variable, $H_\text {naf}$ is the inactivation gating variable, and $V_\text {na}$ is the reversal potential for $\mathrm{Na}^{+}$. Similarly, equations governing the other voltage-dependent currents are described by

$$\begin{aligned} I_\text {nap}&= g_\text {nap}M_{\text {nap}_\infty } H_{\text {nap}_\infty }\left( V - V_\text {na}\right) , \\ I_\text {kdr}&= g_\text {kdr}N^4_\text {kdr}\left( V - V_k\right) ,\\ I_\text {kir}&= g_\text {kir}N_{\text {kir}_\infty }\left( V - V_k\right) ,\\ I_\text {km}&= g_\text {km}N_\text {km}\left( V - V_k\right) , \\ I_\text {cal}&= g_\text {cal}M^2_{\text {cal}_\infty }\left( V - V_\text {ca}\right) , \\ I_\text {cat}&= g_\text {cat}M^2_{\text {cat}_\infty }H_{\text {cat}_\infty }\left( V - V_\text {ca}\right) . \end{aligned}$$

The gating variables $M_\text {naf}, M_\text {nap}, N_\text {kir}, M_\text {cal}, M_\text {cat}$, and $H_\text {cat}$ are set to their steady-state values, while the gating variables $H_\text {naf}, N_\text {kdr}$, and $N_\text {km}$ are modeled by

$$\begin{aligned} \frac{\mathrm{d}G}{\mathrm{d}t}&= \frac{G_\infty - G}{\tau _G}. \end{aligned}$$

The steady-state functions $H_\text {naf}, N_\text {kdr}$, and $N_\text {km}$ can be found in Lee et al. (2010).

The equation for $sI_\text {AHP-SK}$ is

$$\begin{aligned} sI_\text {AHP-SK}&= g_\text {sk}\frac{c^{n_\text {sk}}}{c^{n_\text {sk}}+K^{n_\text {sk}}_\text {sk}}\left( V-V_k\right) . \end{aligned}$$

The equation for $sI_\text {AHP-UCL}$ is

$$\begin{aligned} sI_\text {AHP-UCL}&= g_\text {ucl}\left( O_\text {ucl}+O^*_\text {ucl}\right) \left( V-V_k\right) , \end{aligned}$$

where $O_\text {ucl}$ and $O^*_\text {ucl}$ are two open states of the channel governed by the kinetic equations of the system introduced in Lee et al. (2010).

1.2 Calcium Submodel

The equations describing the calcium concentration in the cytosol $(c)$ and in the endoplasmic reticulum (ER)$(c_\mathrm{e})$ are as follows:

$$\begin{aligned} \frac{\partial c(x,t)}{\partial t}&= \rho \left( J_\text {in}-J_\text {pm}\right) +J_\text {release}-J_\text {serca}+D_c\frac{\partial ^2 c(x,t)}{\partial x^2}, \\ \frac{\mathrm{d}c_\mathrm{e}(x,t)}{\mathrm{d}t}&= \gamma \left( J_\text {serca}-J_\text {release}\right) , \end{aligned}$$

where $\rho$ is used to scale plasma membrane and ER fluxes, and $\gamma$ is the volume ratio between the ER and the cytosol. $J_\text {in}$, $J_\text {pm}$, $J_\text {release}$, and $J_\text {serca}$ denote the influx via plasma membrane channels, efflux via PMCA and NCX plasma membrane pumps, release of $\mathrm{Ca}^{2+}$ from the ER to cytosol, and $\mathrm{Ca}^{2+}$ pumping from the cytosol to the ER, respectively. We have

$$\begin{aligned} J_\text {in}&= -\alpha \left( I_\text {cal}+I_\text {cat}\right) +\beta IP_3, \\ J_\text {pm}&= V_p\frac{c^2}{c^2+K^2_p}+V_\text {NaCa}\frac{c^4}{c^4+K^4_\text {NaCa}}, \\ J_\text {release}&= \left( K_fP_o+J_\text {er}\right) \left( c_\mathrm{e}-c\right) , \\ J_\text {serca}&= P_\text {rate}\frac{c-a_1c_\mathrm{e}}{a_2+a_3c+a_4c_\mathrm{e}+a_5cc_\mathrm{e}}. \end{aligned}$$

The IPR open probability $(P_\mathrm{o})$ is from Gin et al. (2009):

$$\begin{aligned} P_\mathrm{o}&= \frac{q_{12}q_{32}q_{24}}{q_{12}q_{32}q_{24}+q_{42}q_{23}q_{12}+q_{42}q_{32}q_{12}+q_{42}q_{32}q_{21}}, \end{aligned}$$

where $q_{12}, q_{21}, q_{24}$, and $q_{42}$ are set to their steady-state values, and where $q_{23}$ and $q_{32}$ are given by

$$\begin{aligned} q_{23}(c)&= a_{23}-\left( \frac{V_{23}}{k^2_{23}+c^2}+b_{23}\right) \left( \frac{V_{-23}c^5}{k^5_{-23}+c^5}+b_{-23}\right) , \\ q_{32}(c)&= \left( \frac{V_{32}}{k^3_{32}+c^3}+b_{32}\right) \left( \frac{V_{-32}c^7}{k^7_{-32}+c^7}+b_{-32}\right) . \end{aligned}$$

Since $\mathrm{Ca}^{2+}$ diffusion is orders of magnitude slower than the diffusion of $V$, $\mathrm{Ca}^{2+}$ diffusion was omitted from all our model simulations.