Mathematical model

The suggested model framework of single cell models

The model framework is proposed for a single compartment. The developed single compartment model can serve as a good basis for possible later research focusing on multicompartmental modelling.

Although the modelling of intracellular Ca²⁺ levels can unravel interesting inter-actions [64], at this first stage of model development we do not include the changes of intracellular Ca²⁺ concentration and calcium dependent currents in the model and, as a consequence, we assume a constant reversal potential of Ca²⁺. This sim-plification can be accepted as long as we do not wish to take into account Ca²⁺

dependent currents and exocytosis, and the model provides reasonable results. In addition, possible model simulation results corresponding to intracellular Ca²⁺ lev-els could only be validated with Ca²⁺ imaging, which was not available during the measurements.

Elements of the model

The elements of the model are presented in terms of ionic channels that are taken into account.

1. The presence of tetrodotoxin-sensitive Na⁺ currents has been experimentally confirmed in the case of GT1 cells [19] and embryonic GnRH neurons [100].

Adult GnRH neurons were found to fire Na⁺dependent action potentials [138].

The sodium current in the model will be denoted by IN a. We suppose third order activation and second order inactivation dynamics.

2. The presence of A-type K⁺ transient or rapidly activating/ inactivating con-ductance has been described in the case of GT1 cells [19, 30], in embryonic cultures [100], and in GnRH neurons originating from mice [138, 67]. This current will be denoted by IA in the model. This type of potassium current is quite widely studied in the literature even in the case of GnRH neurons [35], and on hypothalamic neurons in general [159, 112]. These results provide useful initial values for the parameters of this current. Furthermore, literature data indicated that the ovarian hormone estradiol modulates this current in mice GnRH neurons [35] and also in GT1 cells [43].

3. A voltage gated delayed outward rectifier K⁺ channel can be assumed, which contributes to the more slowly activating, sustained component of the outward K⁺ current (IK) - see [100, 30, 138, 19, 67].

4. A non-inactivating M-type K⁺ current (IM) is also taken into account, which is considered a key modulator of neuronal activity in GnRH cells [172].

As stated before, the main perspective of this modeling procedure is the de-scription of GnRH release. Based on the results that underline the importance of calcium oscillations corresponding to hormone release [141, 97], we take into account 3 types of Ca²⁺ conductance to be able to describe the qualitatively different components of the calcium current.

Furthermore, according to the results of Beurrier et al. [13], the interplay of different calcium currents can contribute to periodic bursting behavior which can be of high importance regarding neuroendocrine functions.

5. Low voltage activated (LVA) T-type Ca²⁺ conductance, which is activated in earlier phases of depolarization (IT), has been described in the case of rat [80] and mouse GnRH neurons [67], as well as in GT1 cells [153]. The paper [175] proves that the expression of the T type calcium channels is estradiol dependent in hypothalamic GnRH neurons. As a result, during the preovula-tory LH surge a much altered calcium conductance of the GnRH neurons will contribute to their action potential burst formation.

6. Furthermore, based on the results of Watanabe et al. [160] related to GT1-7 cells, and in vitro experiments [80, 125], we assume ahigh voltage gated (HVA) Ca²⁺ channel representing R and N type conductances (IR)

7. In addition, a HVA long-lasting current (L-type) Ca²⁺ channel is modelled (IL) - see the articles [97, 125] for in vitro results and [153] for GT1 measure-ments.

8. Lastly, two leakage currents corresponding to sodium (IleakN a) and potassium (IleakK) with constant conductance are taken into account.

Several other ionic currents have been shown to appear in GnRH neurons, for example the IQ/H current [138], Ca²⁺ activated potassium currents [27, 46], which are not considered in the model. The reason for this is that the further (especially Ca²⁺ dependent) currents would significantly increase the model complexity, which would lead to a significantly harder solvability of the parameter estimation problem.

Furthermore these currents turned out to be nonessential for the reproduction of the features determined in section 4.2.1. After a simple model has been identified it can easily be extended with the currents omitted in the first step of the modelling process.

Model Equations

The equivalent electric circuit of a one-compartment GnRH neuron model with all the above conductances is shown in Fig, 4.1.

V

Figure 4.1: Parallel conductance model, with conductances representing different ion channels in voltage dependent and independent manner. gN a denotes the sodium conductance, gA, gK andgM denote the A-type, delayed rectifier and M-type potas-sium conductances,gT,gRand gLstand for the conductances related to T-type LVA and the R and L-type HVA calcium currents, gleakN a and gleakK correspond to the voltage independent leakage currents.

The HH type model depicted in Fig, 4.1 can be described by the following

equa-tions:

whereV is the the membrane voltage,C is the membrane capacitance,IN a denotes the sodium current, IA, IK and IM denote the A-type, delayed rectifier and M-type potassium currents, IT, IR and IL stand for the T-type LVA and the R and L-type HVA calcium currents, IleakN a and IleakK for the leakage currents. The mi

and hi variables are the activation and inactivation variables of the corresponding currents. mi∞, hi∞ and τmi/hi denote the steady-state activation and inactivation functions, and the voltage dependent time constants of activation and inactivation variables, which are nonlinear Boltzmann and Gauss -like functions of the membrane potential:

The M-type current has only activation dynamics.

Finally, Iex refers to the external injected current. The currents of ionic channels are given by where the EN a, EK, ECa denote the reversal potentials of the corresponding ions.