Regulation of Electrical Bursting in a Spatiotemporal Model of a GnRH Neuron

Abstract

Gonadotropin-releasing hormone (GnRH) neurons are hypothalamic neurons that control the pulsatile release of GnRH that governs fertility and reproduction in mammals. The mechanisms underlying the pulsatile release of GnRH are not well understood. Some mathematical models have been developed previously to explain different aspects of these activities, such as the properties of burst action potential firing and their associated Ca²⁺ transients. These previous studies were based on experimental recordings taken from the soma of GnRH neurons. However, some research groups have shown that the dendrites of GnRH neurons play very important roles. In particular, it is now known that the site of action potential initiation in these neurons is often in the dendrite, over 100 μm from the soma. This raises an important question. Since some of the mechanisms for controlling the burst length and interburst interval are located in the soma, how can electrical bursting be controlled when initiated at a site located some distance from these controlling mechanisms? In order to answer this question, we construct a spatio-temporal mathematical model that includes both the soma and the dendrite. Our model shows that the diffusion coefficient for the spread of electrical potentials in the dendrite is large enough to coordinate burst firing of action potentials when the initiation site is located at some distance from the soma.

Appendices

Appendix A: Voltage Submodel

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

$$\frac{\partial V(x,t)}{\partial t} = -\frac{1}{C_m}I_{\mathrm {ionic}} (V,x,t)+ D_v \frac{\partial^2 V(x,t)}{\partial x^2}, $$

where C _m is the membrane capacitance and I _ionic is the sum of the ionic currents.

For x∈[0,x ₁], the currents in the soma are modeled as

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

For x∈[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 _naf , representing a higher density of Na⁺ channels in the iSite. We use a Na⁺ conductance (g _naf ) of 410 nS in the iSite, and 150 nS elsewhere (Table 1).

Table 1 Parameter values of the model. These parameters are either new to this model, or have values taken from Lee et al. (2010), and Duan et al. (2011) (with very minor changes)

For x∈[x ₁,x ₂] and x∈[x ₃,x ₄], the currents in the dendrite are modeled as

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

I _naf and I _nap denote the fast, persistent Na⁺ currents, I _kdr , I _kir , and I _km denote the delayed rectifier, inward rectifier, and m-type K⁺ currents respectively, I _cal and I _cat are L-type and T-type Ca²⁺ currents, $sI_{\mathit{AHP}_{\mathit{SK}}}$ is an SK-type Ca²⁺-activated K⁺ current, and $sI_{\mathit{AHP}_{\mathit{UCL}}}$ is a slow Ca²⁺-activated after hyperpolarization current. I _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), Duan et al. (2011) and references therein.

We used a Hodgkin–Huxley formalism to model the currents. For example, I _naf is described as

$$I_{naf} = g_{naf}M^3_{naf_\infty}H_{naf} (V - V_{na} ), $$

where g _naf is the maximum conductance, M _naf is the activation gating variable, H _naf is the inactivation gating variable, and V _na is the reversal potential for Na⁺. Similarly, equations governing the other voltage-dependent currents are described by

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

The gating variables M _naf ,M _nap ,N _kir ,M _cal ,M _cat , and H _cat are set to their steady-state values, while the gating variables H _naf ,N _kdr , and N _km are modeled by

$$\frac{dG}{dt} = \frac{G_\infty- G}{\tau_G}. $$

The steady-state functions H _naf ,N _kdr , and N _km can be found in Lee et al. (2010) and Duan et al. (2011).

The equation for $sI_{\mathit{AHP}_{\mathit{SK}}}$ is

$$sI_{\mathit{AHP}_{\mathit{SK}}}= g_{sk}\frac{c^{n_{sk}}}{c^{n_{sk}}+K^{n_{sk}}_{sk}} (V-V_k ). $$

The equation for $sI_{\mathit{AHP}_{\mathit{UCL}}}$ is

$$sI_{\mathit{AHP}_{\mathit{UCL}}} = g_{ucl} \bigl(O_{ucl}+O^*_{ucl} \bigr) (V-V_k ), $$

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

Appendix B: Calcium Submodel

The equations describing the calcium concentration in the cytosol (c) and in the endoplasmic reticulum (ER)(c _e ) are as follows:

$$\begin{aligned} \frac{\partial c(x,t)}{\partial t} =& \rho(J_{in}-J_{pm} )+ J_{\mathrm{release}}-J_{\mathrm{serca}}+D_c\frac{\partial^2 c(x,t)}{\partial x^2}, \\ \frac{dc_e(x,t)}{dt} =&\gamma(J_{\mathrm{serca}}-J_{\mathrm {release}} ), \end{aligned}$$

where ρ is used to scale plasma membrane and ER fluxes, and γ is the volume ratio between the ER and the cytosol. J _in , J _pm , J _release, and J _serca denote the influx via plasma membrane channels, efflux via the Ca-ATPase and Na-Ca exchanger (NCX) plasma membrane pumps, release of Ca²⁺ from the ER to cytosol, and Ca²⁺ pumping from the cytosol to the ER, respectively. We have

$$\begin{aligned} J_{in} =&-\alpha(I_{cal}+I_{cat} )+\beta IP_3, \\ J_{pm} =&V_p\frac{c^2}{c^2+K^2_p}+V_{\mathrm{NaCa}} \frac{c^4}{c^4+K^4_{\mathrm{NaCa}}}, \\ J_{\mathrm{release}} =& (K_fP_o+J_{er} ) (c_e-c ), \\ J_{\mathrm{serca}} =&P_{\mathrm{rate}}\frac {c-a_1c_e}{a_2+a_3c+a_4c_e+a_5cc_e}. \end{aligned}$$

The IPR open probability (P _o ) is from Gin et al. (2009):

$$P_o = \frac {q_{12}q_{32}q_{24}}{q_{12}q_{32}q_{24}+q_{42}q_{23}q_{12}+q_{42}q_{32}q_{12}+q_{42}q_{32}q_{21}}, $$

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

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

Since Ca²⁺ diffusion is orders of magnitude slower than the diffusion of V (Keener and Sneyd 2008), Ca²⁺ diffusion was omitted from all our model simulations.

Appendix C: Numerical Method

We used a finite difference method to solve the model equations in MATLAB (MathWorks). We discretized the spatial derivative using the second-order implicit central difference method, and the time derivative using the first order explicit Euler method. For some long time simulations, we also used the method of lines, using the routine ode15s.