Electrical Bursting in a Spatiotemporal Model of a GnRH Neuron 

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Mathematical Models

In the last 15 years, a number of mathematical models of GnRH neurons have been devel-oped. These models have been used to explain different aspects of GnRH neurons, such as the properties of burst electrical firing and their associated Ca2+ transients. Some models were constructed to explain the experimental data obtained from GT1 cells (van Goor et al., 2000; LeBeau et al., 2000; Fletcher and Li, 2009), while others were based on hypothalamic cells (Lee et al., 2010; Duan et al., 2011; Roberts et al., 2006, 2008, 2009; Cserscik et al., 2012). Some are spatially homogeneous (van Goor et al., 2000; LeBeau et al., 2000; Fletcher and Li, 2009; Lee et al., 2010; Duan et al., 2011), whereas others include a model of the dendrite, reflecting the spatiotemporal properties of GnRH neurons (Roberts et al., 2006, 2008, 2009; Cserscik et al., 2012).
This chapter gives a brief review of these mathematical models and shows the importance of mathematical models in studying GnRH neurons. However, none of these models was sufficient for our purposes, as they did not take into account the more recent data of Iremonger and Herbison (2012). As a result, we constructed a new model based partially on the older models, but also incorporating some new elements.

The van Goor model

The van Goor model is a conductance-based model of the electrical activity of a GT1 neuron, based on both experimental and modeling results, by van Goor et al. (2000). The authors ex-plained how the GnRH neuron spiking pattern shifted from a sharp, high amplitude to a broad, low amplitude under sustained membrane depolarization.
This model included fast, voltage-dependent currents: a TTX-sensitive Na+ current (INa), L- and T-type Ca2+ currents (ICaL and ICaT, respectively), a delayed-rectifier-type K+ current (IKDR), an M-like K+ current (IM), an inward rectifier K+ current (Iir), and a Ca2+-carrying, voltage-insensitive inward leak current Id . All of these currents (except INa) were modeled by typical Hodgkin-Huxley equations (Hodgkin and Huxley, 1952). The authors suggested that the Hudgkin-Huxley-like Na+ channel description was unable to accurately explain GT1 cell behavior. The model of INa was hence adapted from Kuo and Bean (1994) with a four-state Markov model. Their modeling study also indicated that the inactivation of the Na+ channel was responsible for the spike amplitude reduction, while the decrease in K+ current activation affected the spike broadening.

The LeBeau model

LeBeau et al. (2000) constructed a quantitative description of the regulation of action potential pacemaking and the associated Ca2+ signaling in GT1 cells. They extended the van Goor model by adding Ca2+ dynamics and Ca2+-sensitive currents. With their theoretical study, LeBeau et al. (2000) reported that interplay between three pacemaker currents, ISOC, ISK, and Id , could explain responses to the various stimuli in experimental tests.
A schematic diagram of the model is given in Figure 3.1. LeBeau et al. (2000) kept all the fast currents exactly the same as used in van Goor et al. (2000), except for the leak current. Based on the experimental evidence from van Goor et al. (1999a), three pacemaker currents, a SK-type Ca2+-activated K+ current (ISK), a store-operated Ca2+ current (ISOC), and an inward leak current (Id ) which is a Ca2+-inactivated non-specific cation current, were added. The SK channel was modeled in the usual way, with a linear dependence on fractional activation by Ca2+ and voltage driving force. The description of ISOC was defined in a similar way, except that it had the inverse fractional activation by shell ER Ca2+ concentration. Id was assumed to be a cyclic adenosine monophosphate (cAMP)-regulated pacemaker current, controlling spon-taneous electrical firing and associated Ca2+ signaling.

The Roberts models

Figure 3.1 Schematic diagram of the LeBeau model. The cell was divided into two compartments, ER and cytosol. Both compartments were further separated into shell and bulk sub-compartments. The model kept all the currents used in van Goor et al. (2000), except for the leak current. The Ca2+ dynamics and three Ca2+-sensitive pacemaker currents ISOC, ISK, and Id were also added. Figure adapted from LeBeau et al. (2000).
To model Ca2+ dynamics, the cell was separated into ER and cytosol. These two compart-ments were further divided into two sub-compartments, shell and bulk; see Figure 3.1. The Ca2+-sensitive currents mentioned above were coupled to the shell compartment’s Ca2+ con-centration. The Ca2+ dynamics included influx via plasma membrane channels ( jin), efflux via PMCA and NCX plasma membrane pumps ( jeff), release and uptake of Ca2+ from the shell and bulk ER ( jrels, jups, jrelb, and jupb, respectively), diffusional exchange in the cytosol and ER (cytex, ERex), and efflux and uptake from the shell and bulk mitochondria ( jmeffs, jmups, jmeffb, and jmupb, respectively).


The Roberts models

Roberts et al. (2006) was the first study to construct multi-compartmental models of hypotha-lamic GnRH neurons. Electrophysiological recordings and neuronal morphology were used to generate computer models, which studied how synaptic input to the dendrite of GnRH neurons would control firing. Figure 3.2 shows the computer renderings of the bipolar and branching GnRH neuron, where each gray shaded cylinder represents individual compartments for the soma, dendrite and axon in the model. Roberts et al. (2006) also reduced the model to a single soma compartment to understand the fundamental behavior of GnRH neurons. The kinetics of the channels were adapted from published models of GT1 cells (van Goor et al., 2000; LeBeau et al., 2000). The model of Roberts et al. (2006) only included the three strongest currents as shown in Table 3.1, the fast Na+ current (INa), delayed rectifier K+ current (IKdr), and L-type Ca2+ current (ICaL), all described by typical Hodgkin-Huxley equations and applied to both soma and axon compartments. Their findings indicated that the cellular mechanisms generat-ing spikes from GT1 cells and cultured neurons may not be the same as those in hypothalamic GnRH neurons. The authors assumed that the dendrite of the GnRH neuron is passive, which suggested that only synapses located on the soma and very proximal dendrites were capable of controlling somatic spiking. Moreover, it suggested that the dendrite may receive extensive synaptic input, but has limited effect on controlling spike firing.
The experimental work of Roberts et al. (2008) revealed voltage-gated Na+ channels in the dendrites of GnRH neurons, and was the first study to suggest the dendrite as the site of action potential initiation. This study also constructed a multi-compartmental, morphologi-cal, conductance-based computer model of action potential generation in GnRH neurons; see (a) (b) Figure 3.2 Computer renderings of the compartmental models of (a) bipolar and (b) branching GnRH neurons used in Roberts et al. (2006, 2008, 2009). The gray shaded cylinders illustrate different com-partments in the model. Figures adapted from Roberts et al. (2006).
Table 3.1 Conductance types used in the three studies: I (Roberts et al., 2006), II (Roberts et al., 2008), and III (Roberts et al., 2009). The dendrite model was described as a passive model in the first study, an active model in the second study and both a passive and an active model in the third model.

1 Introduction
2 Biological Background 
2.1 GnRH in the Reproductive System
2.2 Electrophysiological Properties and Ca2+ Dynamics in GnRH Neurons
2.3 Experimental Data
3 Mathematical Models 
3.1 The van Goor model
3.2 The LeBeau model
3.3 The Roberts models
3.4 The Fletcher and Li model
3.5 The Lee model and the Duan model
3.6 The Cserscik model
3.7 Summary
4 Regulation of Electrical Bursting in a Spatiotemporal Model of a GnRH Neuron 
4.1 Abstract
4.2 Introduction
4.3 Model Description
4.4 Results
4.5 Discussion
5 A Computational Model of the Dendron of the GnRH Neuron 
5.1 Abstract
5.2 Introduction
5.3 Model Description
5.4 Results
5.5 Discussion
6 Conclusions and Future Work 

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