Gonadotropin-releasing hormone (GnRH) neurons exhibit complex patterns of electrical spiking. Action potentials occur in groups, often called bursts, and each burst is closely correlated with an increase in the concentration of intracellular . Each burst is separated by a quiet interval with no action potentials, of length ranging from a few seconds to a few tens of seconds. Although it is unclear exactly what is the physiological function of these patterns of electrical bursting, it is highly likely that they play a role in the regulation of the release of GnRH from the nerve terminal in the median eminence of the hypothalamus. GnRH in turn controls the release from the anterior pituitary of luteinizing hormone and follicle-stimulating hormone, which are crucial for sexual development and fertility. Thus, there is ample motivation for the study of the mechanisms underlying the patterns of electrical spiking seen in GnRH neurons.

Because of the highly complex dynamical phenomena observed in GnRH neurons (and, indeed, many other types of neurons), mathematical models have an important role to play. It is difficult, if not impossible, to attain a detailed understanding of the mechanisms underlying electrical bursting, and other kinds of complex oscillations, using qualitative models alone. Qualitative models (i.e., nonmathematical explanations of the behavior) are important for suggesting underlying structures and possible mechanisms, but, to test whether or not such mechanisms actually work, and to make specific predictions, a mathematical model is necessary.

Neuroendocrine neurons have, in general, some features of particular interest to modelers. The complex electrical bursting behavior is necessarily a combination of at least two different oscillatory processes –one oscillatory process giving the fast electrical spiking, and the other, slower, process, causing the transitions from bursting to nonbursting regions. Although this must be true in general, it has not proven to be an easy task to determine exactly what each of these mechanisms is.

The fast spiking is reasonably well understood in many cell types, being the result of the modulation by the voltage of ionic conductances (usually, and conductances) in the plasma membrane. Models of such fast spiking are usually based, more or less explicitly, on the model of Hodgkin and Huxley (1952). The precise nature of the conductances change, as do the parameter values, but the underlying model structure remains essentially the same.

However, the nature of the slow process (or processes) that causes the transitions between silent and active phases remains elusive, and is almost certainly different in each cell type. Some models use the intrinsic oscillatory dynamics of intracellular as the additional slow process (Chay and Keizer, 1983). Others use mechanisms based on metabolism or mitochondrial transport (Bertram et al., 2004, 2007a,b).

Ultimately, it is not an easy task to determine the exact combination of oscillatory processes that causes bursting oscillations in any particular cell type. Mostly, models strive to present possible hypotheses which can then be tested, leading to the refinement of the models and additional tests. As this cycle of experiment and theory progresses through multiple models, we hope to get closer to an understanding of what is actually happening in the cell.

Here, we present a model of electrical bursting in GnRH neurons, a model that has already assisted us in understanding some of the basic mechanisms that control the length of the burst, and the length of the interburst interval.

Much of the previous modeling work on GnRH neurons has been based not on data from GnRH neurons themselves, but on data from analogous cell lines, such as GT1 cells. Although these early models, thus, have only limited direct applicability to GnRH neurons in slice preparations, they still remain the basis for all subsequent modeling work.

The earliest models of electrical spiking in GT1 cells were those of Van Goor et al. (2000) and LeBeau et al. (2000). These models have the same basic structure as the Hodgkin–Huxley equations. Thus,

where is the membrane capacitance and is the potential difference across the cell membrane. (For an introductory discussion of the Hodgkin–Huxley model and a brief look at the mathematical theory of that model, see Keener and Sneyd (2008).)

In the LeBeau model, there were nine different ionic conductances:

where is a TTX-sensitive current, and are L-type and T-type currents, and , , and are delayed-rectifier, M-type, and inward-rectifier currents. is a -activated current, while is a store-operated channel, that is, a channel that opens when the endoplasmic reticulum is depleted of . Finally, is a nonselective inward cation current.

Each of these currents is modeled as the product of a conductance, some gating variables, and a linear current–voltage curve. Thus, for example,

where is a time-dependent gating variable satisfying the equation

is the channel conductance, and is the Nernst potential of . Many of the gating variables were set at their steady states, for simplicity.

In the model of Van Goor et al. (2000), there was no detailed model of the intracellular dynamics, but this was substantially improved in the subsequent version of LeBeau et al. (2000). In the LeBeau model, spiking was essentially caused by the and currents, with the frequency modulated by the three pacemaker currents, , , and , all three currents being coupled to the intracellular concentration, and thus to the influx through and .

One of the most interesting results of the LeBeau model was their prediction of the existence of a nonselective cation current, , activated by cAMP and inhibited by . Although they were unable to confirm the existence of this current experimentally, the model still demonstrated the components needed to explain the experimental results.

The handling in this model was further improved by Fletcher and Li (2009), who constructed a spatial model of the intracellular , as well as simplifying the currents in the model.

Given the plethora of existing models of GnRH neurons, it is natural to ask why another model is needed. To answer this question, it is important first to make clear exactly which data the model is designed to explain, and what is the purpose of the model. Essentially, we are less concerned with the detailed structure of each individual action potential, but are more interested in the mechanisms that determine the bursting structures such as the interburst interval, or the number of action potentials in each burst.

Our original model (Lee et al., 2010) was also designed to answer these same questions, and was successful (at least partially) in doing so. However, the complexity of our original model is such as to make it difficult to extract and understand the important mechanisms that control the bursting. This motivates the construction of a simpler model which nevertheless incorporates the important control mechanisms. Here, we discuss this simpler version of the model.

Because of this particular focus, our simplified model is not a good quantitative description of a single action potential. In this respect, it can compete neither with the work of Roberts and Suter, nor with the integrated model of Csercsik et al. However, the full model has already led to an increased understanding of behaviors on the longer time scale of the bursts, and has led to the discovery of long-lasting -sensitive hyperpolarising currents which appear to set the interburst interval. Because of this success, it is a worthwhile exercise to simplify the model, so that its underlying structure and behavior can be better understood.