Bulletin of Mathematical Biology (2008) 70: 2103–2125DOI 10.1007/s11538-008-9328-z

O R I G I NA L A RT I C L E

Robust Synchrony and Rhythmogenesis in EndocrineNeurons via Autocrine Regulations In Vitro and In Vivo

Yue-Xian Lia,b,∗, Anmar Khadrab

aDepartments of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2,Canada

bDepartments of Zoology, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

Received: 28 September 2007 / Accepted: 29 April 2008 / Published online: 17 September 2008© Society for Mathematical Biology 2008

Abstract Episodic pulses of gonadotropin-releasing hormone (GnRH) are essential formaintaining reproductive functions in mammals. An explanation for the origin of thisrhythm remains an ultimate goal for researchers in this field. Some plausible mechanismshave been proposed among which the autocrine-regulation mechanism has been impli-cated by numerous experiments. GnRH binding to its receptors in cultured GnRH neu-rons activates three types of G-proteins that selectively promote or inhibit GnRH secre-tion (Krsmanovic et al. in Proc. Natl. Acad. Sci. 100:2969–2974, 2003). This mechanismappears to be consistent with most data collected so far from both in vitro and in vivo ex-periments. Based on this mechanism, a mathematical model has been developed (Khadraand Li in Biophys. J. 91:74–83, 2006) in which GnRH in the extracellular space plays theroles of a feedback regulator and a synchronizing agent. In the present study, we showthat synchrony between different neurons through sharing a common pool of GnRH is ex-tremely robust. In a diversely heterogeneous population of neurons, the pulsatile rhythmis often maintained when only a small fraction of the neurons are active oscillators (AOs).These AOs are capable of recruiting nonoscillatory neurons into a group of recruited os-cillators while forcing the nonrecruitable neurons to oscillate along. By pointing out theexistence of the key elements of this model in vivo, we predict that the same mechanismrevealed by experiments in vitro may also operate in vivo. This model provides one plau-sible explanation for the apparently controversial conclusions based on experiments onthe effects of the ultra-short feedback loop of GnRH on its own release in vivo.

Keywords Mathematical modeling · GnRH pulse generator · Synchronization ·Heterogeneous neuronal populations · Coupling via a shared signal · Parametricaveraging

∗Corresponding author.E-mail address: yxli@math.ubc.ca (Yue-Xian Li).

2104 Li and Khadra

1. Introduction

Reproductive function in mammals is controlled by episodic pulses of gonadotropin-releasing hormone (GnRH) that are produced by GnRH neurons in the hypothalamus.These neurons will be referred to as the “GnRH pulse generator” in this paper. The pi-tuitary gonadotrophs that are targets of GnRH action respond only to a GnRH signalwith highly specific period and temporal pattern (Knobil, 1980; Li and Goldbeter, 1989).The absence or malfunction of this pulse generator is associated with several reproduc-tive and developmental diseases (Knobil, 1980). The search for mechanisms underlyingthe GnRH pulse generator has been conducted by researchers with different backgroundsand view points. Potential explanations range from autocrine regulations by GnRH onits own release (Krsmanovic et al., 1991, 1993, 1999, 2003; Woller et al., 1998, 2003)to interactions of a network of GnRH neurons synaptically and/or chemically coupledthrough one or more neurotransmitters (Witkin and Silverman, 1985; Terasawa, 2001;Moenter et al., 2003). Despite the apparent diversity in opinions, it is now generallyrecognized that an ultimate resolution of this puzzle depends on an ultimate answer tothree key questions: (1) What are the cells that directly participate in the pulse generation?(2) Through what mechanism(s) do these cells coordinate or synchronize their secretoryactivities? (3) What is the cellular mechanism that makes the secretion pulsatile?

Consensus seems to have been reached on the answer to the first question. It isnow widely believed that the pulses are produced by GnRH neurons. This was sup-ported by in vivo studies showing that lesion but not deafferentation of the medial basalhypothalamus abolished the pulsatility (Blake and Sawyer, 1974; Krey et al., 1975;Plant et al., 1978). The participation of other types of cells in GnRH pulse generationwas shown unnecessary in experiments using immortalized GnRH cell lines (GT1 cells)(de la Escalera et al., 1992), since these cultures contained only GT1 cells. Similar conclu-sions have been reached in the study of cultured GnRH neurons (Krsmanovic et al., 1991,1999; Terasawa et al., 1999) and from enzymatically dispersed rat hypothalamic explants(Woller et al., 1998). In the hypothalamus, these neurons are scattered throughout sev-eral nuclei within the rostral hypothalamus and preoptic area (Silverman et al., 1994;Rubin and King, 1995).

Important progress has been made in answering the second and the third questionsbased on experiments in cultured GT1 cells and GnRH neurons. Synchronization througha diffusible mediator was first proposed by de la Escalera et al. (1992) based on the obser-vation that GT1 cells on two cell-coated coverslips without direct contact were synchro-nized. The fact that synchronized release of GnRH pulses was observed in enzymaticallydispersed rat hypothalamic tissue (Woller et al., 1998, 2003) supported the roles of adiffusible mediator in coordinating and synchronizing the acutely dissociated cells. Thefinding of GnRH receptors in GT1 cells (Krsmanovic et al., 1991) and of the autocrineinfluences of GnRH on its own release (Krsmanovic et al., 1991, 1993, 2003) furthersuggested that GnRH is itself the diffusible mediator responsible for synchronizing thescattered GnRH neurons. These experiments led to the illustration of the plausible mole-cular mechanism that underlies the GnRH pulsatility by Krsmanovic et al. (2003). Theyfound that the autocrine regulation of GnRH on its own release is mediated by three G-proteins. These G-proteins selectively activate or inhibit GnRH secretion by regulating theintracellular levels of Ca2+ and cAMP. Based on this mechanism, a mathematical modelof the GnRH pulse generator was developed (Khadra and Li, 2006). It demonstrated that

Robust Synchrony and Rhythmogenesis in GnRH Neurons 2105

the autocrine regulation mechanism is viable in generating GnRH pulses and gave supportfor this molecular mechanism in culture conditions.

Experimental data that led to the autocrine regulation mechanism were mostly col-lected in vitro. Whether this mechanism is actually responsible for the GnRH pulse gen-eration in vivo remains undetermined. Although a final resolution of this problem dependscritically on collecting more data, mathematical modeling can provide useful informationon the viability of such a model under in vivo conditions. The present study is an at-tempt at answering this question using a mathematical model. We aim at achieving twogoals. First, we show that autocrine regulation via a hormonal signal in a common poolis very robust in synchronizing diversely heterogeneous populations of GnRH neurons.Here “diversely heterogeneous” means that the key parameters of each model neuron arerandomly chosen from a wide range of values that stretches far outside the oscillatorydomain for a single isolated neuron. We show that a small fraction of actively oscillatingGnRH neurons is often sufficient to recruit many nonoscillatory neurons so as to maintainthe pulse-generating function of the whole population. The robustness of such a mech-anism makes it hard to argue against the possibility that it could also operate under invivo conditions, especially when existing data point to the fact that the key elements thatensure the operation of such a mechanism in vitro are also present in vivo. Second, weapply the model to experimental conditions in vivo and provide sensible explanations forsome in vivo experiments.

The paper is organized as follows. In Section 2, we introduce the model of GnRHpulse generator and a reduced, dimensionless version of it. In Section 3, we introduce thedefinitions of some important concepts that will be used in our study. These include thesynchrony measure, the diversely heterogeneous population, the deviation measure, andthe classification of neurons in a heterogeneous population. The phenomenon of paramet-ric averaging that occurs in this model is explained. The results supporting the extraordi-nary robustness of the pulse generating mechanism and the applicability of this model toGnRH pulse generation in vivo are also discussed. A summary is provided in Section 4together with a discussion of the potential physiological significance of the results.

2. The model

2.1. The single-cell model

The model developed by Khadra and Li (2006) was based on the autocrine regulationmechanism derived from experimental data obtained by Krsmanovic et al. (2003). Thismechanism is schematically summarized in Fig. 1(a). The binding of GnRH in the ex-tracellular medium (abbreviated by G hereafter) to its receptors on the GnRH neuronsactivates three types of G-proteins Gs, Gq, and Gi. This triggers the dissociation of the α

subunits αs, αq, and αi from their respective βγ subunits. αs and αq promote the secretionof GnRH by increasing the two intracellular messengers cAMP and Ca2+, respectively,while αi inhibits GnRH secretion by inhibiting cAMP production. In this model, G playsthe roles of a diffusible mediator as well as a synchronizing agent. The autocrine effectsof G on its own secretion through αs and αq provide two distinct positive feedback loopsthat are essential for triggering the explosive increasing phase of the spike in G, while the

2106 Li and Khadra

Fig. 1 Schematic diagram of the model of GnRH pulse generator (a) by Khadra and Li (2006) based onthe autocrine mechanism proposed by Krsmanovic et al. (2003). Comparison between the full six-variablemodel (b) and the reduced four-variable, dimensionless model (c) reveals no visible difference in theirpulse generating properties. In (b), variations of C and A are not shown. In the lower panels of (b) and (c),S or s are dashed, Q or q are dotted, I or i are solid.

inhibition through αi is crucial for terminating the spike and for holding G at the basallevel for an extended interspike interval (see Fig. 1(b)).

In the full model of GnRH pulse generation, there are six variables: G, C, A, S, Q,

and I (see Eqs. (A.1)–(A.6) in Appendix A), where C and A denote, respectively, thecytosolic Ca2+ and cAMP concentrations while S, Q, and I represent the concentrationsof αs, αq, and αi, respectively. Because the variations of C and A are much faster than theother variables, we simplified the system into a four variable system using quasi-steadystate approximations (Eqs. (B.1)–(B.4) in Appendix B). The dimensionless form of thissimplified system is given below.

dg

dτ= λ

[ν + ηF(s, q, i) − g

], (1)

Robust Synchrony and Rhythmogenesis in GnRH Neurons 2107

ds

dτ= φ

[g4

σ 4 + g4− s

], (2)

dq

dτ= ψ

[g2

ρ2 + g2− q

], (3)

di

dτ= ε

[g2

κ2 + g2− i

], (4)

where g, s, q and i denote the scaled (dimensionless) variables G, S, Q, and I , respec-tively. The function F(s, q, i) and all the scaled parameters in system (1)–(4) are givenin Appendices A and B. Specific choices of the parameter values and the nonlinear func-tions in the model were determined based on fitting the steady state activation/inactivationcurves to available experimental data (Krsmanovic et al., 2003). Simulations of the sim-plified model and the full model demonstrate that the former is a very good approximationof the latter (see Figs. 1(b), (c)).

2.2. The population model

Available data seem to suggest that a single GnRH neuron possesses all the necessaryparts for generating GnRH pulses under ideal conditions (see references cited in Khadraand Li, 2006). Mathematically, a single-cell model is equivalent to a population model ofidentical neurons (Khadra and Li, 2006). However, homogeneity is unrealistic since neu-rons are not identical in their biological environment. Therefore, we focus in this presentstudy on heterogeneous neuronal populations. A dimensionless form of the model of N

heterogeneous GnRH neurons coupled through a common pool of extracellular GnRH isgiven by the following equations (see Appendix B for a detailed dimensioned form).

dg

dτ= 1

N

N∑

n=1

λn

[νn + ηnFn(sn, qn, in) − g

], (5)

dsn

dτ= φn

[g4

σ 4n + g4

− sn

], (6)

dqn

dτ= ψn

[g2

ρ2n + g2

− qn

], (7)

din

dτ= εn

[g2

κ2n + g2

− in

], (n = 1,2, . . . ,N), (8)

where we use the subscript n in the function Fn and the parameters to indicate that theyall have different values. The fact that all neurons share a common extracellular pool of G

implies that the extracellular medium is continuously stirred so that GnRH secretion byeach neuron is diluted and averaged immediately. This can be regarded as an approxi-mation of the perifusion experiments in which the continuous flow through the chambercan cause a stirring effect. A more realistic model of a heterogeneous population in theabsence of stirring would have to consider the geometric distribution of the neurons, thediffusion of g in the medium and the “shell-effect” in the vicinity of the cell surface where

2108 Li and Khadra

g is released. Such a model is beyond the scope of this study and will likely yield quali-tatively similar conclusions, although some details may differ. In this paper, we shall usesystem (5)–(8) to analyze synchrony in heterogeneous populations of GnRH neurons.

3. Results

3.1. The synchrony measure

The model for a heterogeneous population given by Eqs. (5)–(8) represents a system ofN neurons coupled through a shared variable g. Neurons interact with each other throughsecreting g into the pool and responding to changes in g in the pool. Oscillators coupledthrough a common variable have been studied in some other systems (see, e.g., Li etal., 1992). Each neuron is considered an oscillator only when this common variable istaken into account. A general theory of synchrony between heterogeneous populationscoupled in such a way is not available. Studies of this kind are mostly based on numericalsimulations.

Unlike synaptically coupled neuronal networks, even the meaning of the word “syn-chrony” is not clearly defined in this case. Two neurons are considered “synchronized” iftheir membrane potentials vary in synchrony and peak at the same time. Here, the extra-cellular signal g is shared by all neurons. To determine the degree of synchrony, we needto identify one important variable that is internal to each neuron and not shared by others.In this study, we choose the variable i which plays a key role in determining the oscil-lation frequency in a single-cell model. For a heterogeneous population, in differs for allneurons and peaks at different times (see Fig. 2(a)). However, a pulsatile g is generated de-spite of the fact that the variables that are internal to each neuron are not “synchronized.”Therefore, we introduce the following synchrony measure: A heterogeneous populationis considered synchronized if a pulsatile signal g is generated and if the variables in of allthe pulsing neurons in the population peak within the width of the g pulse in each period.Here, the width of a g pulse is defined as the time duration in which the value of g is atleast three times higher than its baseline level. The term actively pulsatile neuron refersgenerally to those neurons that actively participate in the pulse-generation.

3.2. Diversely heterogeneous populations of GnRH neurons

To test the robustness of the autocrine mechanism, heterogeneity in the values of manyselected parameters within their respective oscillatory ranges has been tested in Khadraand Li (2006). The oscillatory range of a parameter is determined by the single-cell model(which is identical to a population model consisting of identical neurons) as follows. Foreach parameter, the range is determined while all the other parameters are kept fixed at astandard set of values (see Table A.2, Appendix B). The oscillatory ranges of three keyparameters are listed in Table 1. We have tested both the uniform and truncated Gaussiandistributions of all parameters within their respective oscillatory ranges, and observed thatsynchronized pulses occurred in all cases (results not shown). This demonstrates that themechanism is robust and insensitive to parameter variations, provided that all parametersare chosen from within their respective oscillatory ranges. In reality, there is no guaranteethat all neurons are active oscillators. Electrophysiological evidence seems to suggest that

Robust Synchrony and Rhythmogenesis in GnRH Neurons 2109

Fig. 2 Pulse generation by a diversely heterogeneous neuron population. (a) 50 model neurons with theparameter κ evenly distributed within its oscillatory range (between the two vertical bars) and to the leftside of the range. The apparent nonuniformity in the distribution is caused by the logarithmic scale. Activeoscillators (AOs), recruited oscillators (ROs), and slaved oscillators (SOs) are denoted, respectively, bygreen diamonds, blue circles, and red triangles in the lower panel. In the middle panel, AOs, ROs, and SOsare plotted in green solid, blue dashed, and red dotted curves respectively. The thick black curve is theaverage of all the AOs. (b) A case of parametric averaging including 50 model neurons whose κ valuesare all distributed outside the oscillatory range, of which 25 are distributed to the left (red triangles anddotted curves) and 25 to the right (blue diamonds and dashed curves).

only a fraction of GnRH neurons participate in the pulse generation (Nunemaker et al.,2001; Moenter et al., 2003). To determine whether the robustness is retained under a sim-ilar condition in the model, we focus on populations that contain neurons with parametervalues distributed far beyond the oscillatory ranges. We call such a population a diverselyheterogeneous population hereafter. The heterogeneity of realistic GnRH neurons is un-likely worse than the cases that are studied in this paper.

3.3. Deviation measure and classification of neurons in a heterogeneous population

Figure 2(a) shows the pulse generation of one diversely heterogeneous population ofmodel GnRH neurons. Of the 50 neurons in the system, 25 are randomly chosen fromthe oscillatory range between the two vertical bars in the lower panel. We call these neu-rons active oscillators (AOs) and plotted them in thin solid curves of green color. The

2110 Li and Khadra

others have κ randomly chosen between 0 and 32 which represent all possible values thatare smaller than the oscillatory range. These neurons are not oscillators themselves whendecoupled from the others and are thus called non-AO (NO) neurons. Simulations of manyheterogeneous populations of the model neurons revealed that different NOs can behavevery differently in the coupled system (see the dashed curves in blue and dotted curvesin red in Fig. 2(a)). Notice that the blue-colored NOs behave very much like an AO whilethe red-colored NOs remain elevated cycle after cycle.

To divide the NOs into different subgroups, we introduce a quantitative criterion namedthe deviation measure. It is designed to differentiate the behavior of an NO from that ofan average AO. For this purpose, we first calculate the average of the i variables of all theAOs (i.e., all the thin solid curves in green in Fig. 2(a)) to yield i(t) which is representedby the thick black curve. The deviation of each neuron in the population from i(t) iscalculated as follows.

dn = 1

T

[∫ T

0

(in(t) − i(t)

)2dt

]1/2

, (9)

where T is the period of the g pulses. Then we calculate the deviation of all the AOsand define dmax = max{dn, for all AOs} as the maximum possible deviation of the AOs.Notice that dn ≤ dmax for each AO and that dn > dmax for each NO.

Using this measure, we can divide NOs into two subtypes. Those that behave similarlyto AOs, and thus considered active participants, are referred to as recruited oscillators(ROs), and those that are passively dragged by the others to oscillate along are calledslaved oscillators (SOs). The ROs are plotted in dashed blue curves and the SOs in dottedred curves in Fig. 2(a). Very often, the boundary between the ROs and SOs is not at allclear. It depends on the parameter that is chosen in the study as well as the extent andthe location of the distribution of NOs (see the lower panel). To separate the two, weintroduce a coefficient r such that those NOs that satisfy dmax < dn ≤ rdmax are definedas ROs, while those that yield dn > rdmax are defined as SOs. In Fig. 2(a), r = 3, whichmeans that every neuron whose deviation is larger than three times dmax is considered anSO. A slightly different choice of the value of r will not change the qualitative nature ofthe results.

Note that the values of the variable in of all the NOs are higher than those of theAOs in Fig. 2(a). This is because the values of the parameter κn for NOs are all chosenfrom a range that is smaller than the oscillatory domain. A lower value of κn means thatthe threshold value of g for activating the production of in is lower. This leads to highervalues of in given that the value of g in the common pool is shared. This specific biastypically leads to smaller amplitude in the pulses of g. In the case when all NOs arechosen from a range of κn values that are larger than the oscillatory range, the in of eachNO will oscillate within a range that is below the green curves thus leading to an increasedamplitude in g. The reason for choosing all the NOs from a range located to one side ofthe oscillatory domain will be explained below. The NOs located closer to the oscillatorydomain are more easily recruited than those that are farther away.

3.4. Parametric averaging

“Parametric averaging” refers to the phenomenon in which coupling between two (condi-tional) oscillators that differ in the value of one parameter results in a coupled system that

Robust Synchrony and Rhythmogenesis in GnRH Neurons 2111

is equivalent to a single oscillator with an averaged value of the parameter. A conditionaloscillator refers to a dynamical system that is nonoscillatory, but is capable of generatingstable limit-cycle type of oscillations when the parameter values are appropriately cho-sen. For simplicity, we use the term “oscillator” to refer indiscriminately to both types ofoscillators. Parametric averaging often occurs when coupling two oscillators that differ inthe values of more than one parameter. In the study of a mixed suspension of two sub-populations of Dictyostelium discoideum amoeba (Li et al., 1992), it was shown that insome cases the mixed population behaves like a pure population with a parameter that isequal to the weighted average of the corresponding parameter of the two subpopulations.In this study, the coupled system is an exact parametric average of the two coupled oscil-lators. Similar kind of parametric averaging has been found in a number of other systemswith different types of coupling including those described by Manor et al. (1997) andCartwright (2000). In Manor et al., for example, the coupled system was shown to rapidlyrelax to a parametrically averaged system.

Coupling through a common chemical pool is a shared feature of the GnRH pulse-generator model and the model of coupled amoeba populations (Li et al., 1992). Paramet-ric averaging is expected to also occur in the GnRH pulse generator. A typical example isshown in Fig. 2(b) where all neurons in the population are NOs. Half of them lies abovethe oscillatory range (the blue diamonds or dashed curves), while the other half lies be-low (the red triangles or dotted curves). Figure 2(b) shows that pulsatile oscillations ing are generated in this population composed of only NOs. An arithmetic average of the50 different values of κ results in a value of κ ≈ 316 which lies within the oscillatoryrange of κ . Figure 2(b) reveals that the interactions between the two types of neuronsswitched the latter group into a pulse generating mode. Coupling through a shared pool offeedback signal is often a strong coupling. Notice that although the red neurons are SOsbased on the deviation measure, their existence is essential for switching the blue neuronsinto ROs. The blue neurons would remain nonoscillatory if the red neurons were to beremoved from the population. This suggests that the pulsatile signal can be generated ina heterogeneous population whose individual members are all NOs when decoupled fromothers. This result is consistent with the important roles of heterogeneity in biologicalsystems as emphasized by Cartwright (2000).

Parametric averaging involving oscillators with parameter values distributed above andbelow the oscillatory range with equal probability supports the robustness of the pulsegenerating mechanism. This result has been established in a number of previous studiesthat are cited above. Therefore, this type of parametric averaging is not the main focusof this study. Instead, we test the robustness of this mechanism further by focusing onheterogeneous populations in which the NOs are picked from one side of the oscillatoryrange. The question we try to answer here is what is the minimal fraction of AOs forgenerating pulsatility in such an extreme case. Figure 2(a) shows a typical example inwhich AOs are picked randomly from the oscillatory range (green diamonds) while NOsare picked from an interval below this range (blue circles and red triangles). The resultindicates that NOs distributed closer to the oscillatory range are more readily recruitedto become an RO. Since the oscillatory range of κ contains values above 1,000 and theNOs are chosen from a domain of smaller values, κ ≈ 301 is located within the oscillatoryrange. Although the parameters κ and ω appear nonlinearly in the equations, the reasonfor why simple arithmetic averaging remains operative is not clearly understood.

2112 Li and Khadra

3.5. A small fraction of AOs is often sufficient for pulse generation

We now focus on the cases in which the NOs follow a one-sided distribution. They arechosen from either above or below the oscillatory domain. The goal is to test how tolerantthe pulse generating mechanism is to a unidirectional deviation of a subpopulation of theGnRH neurons from the oscillatory domain. Such deviations may occur when a disease orsome environmental stress causes the cells to change in a specific way. We selected threeparameters that play crucial roles in the model: κ , ω, and ε. Their definitions are givenin Appendix A. ε characterizes the time scale of the i variable, which must be smallerthan λ, φ, and ψ for oscillations to occur (see Table A.2 in Appendix B). It is, therefore,a very sensitive parameter of the oscillator. κ is the threshold value of g for activating i.Thus, for small κ values, the i variable typically stays at elevated levels (see the red curvesin Fig. 2). The role of ω is defined in Eq. (B.5). It specifies a threshold for the i variablebeyond which the production of cAMP is inhibited. In other words, the sensitivity of theadenylyl cyclase to the inhibition of αi is determined by ω.

The oscillatory ranges for κ , ω, and ε are listed in Table 1 together with the rangesof NO distributions. Three different combinations were tested for each parameter. Theresults are shown in Fig. 3, where the fraction of the total number of actively puls-ing oscillators is plotted as a function of the number of AOs that are in the pop-ulation. Here, the total number of actively pulsing oscillators is defined as the totalnumber of AOs plus ROs. In a nonoscillatory population, even the AOs are quiescent(i.e., all neurons are at steady state), so the number of actively pulsing oscillators iszero.

We calculated the minimum number of AOs required to generate pulsatile g signal.Such a number is called a critical number. When the number of AOs is small, the wholepopulation is inactive and nonoscillatory (see Fig. 3). As we bring one NO at a time ran-domly into the oscillatory range, a discontinuous transition from a nonoscillatory stateto an oscillatory state occurs at a critical number of AOs in each case studied. Such asharp transition from quiescence to pulsatility is reminiscent to the phase transition-likeevent that was discovered by Kuramoto in a heterogeneous population of phase oscilla-tors (Kuramoto, 1984). In Fig. 3, the transition from nonoscillatory to oscillatory behavioris discontinuous since the transition marks the boundary between quiescence and oscil-lation. In all simulations of the present study, pulsatility in g appears to be all-or-none.Small amplitude oscillations were never observed. In Kuramoto’s system, however, thetransition marks the boundary between asynchronous and synchronous oscillations. Thetransition is smooth and the increase in the amplitude of the order parameter is grad-ual.

Table 1 Nine different combinations of parameter ranges for κ , ω and ε tested in the search for thesmallest possible fraction of AOs that produces a pulsatile g signal

κ ω ε

Case AO Range NO Range Case AO Range NO Range Case AO Range NO Range

(a1) [32,1071] [0,32] (b1) [0.00005,0.0209] [0,0.00005] (c1) [0.0022,0.032] [0,0.0022](a2) [32,1071] [0,1] (b2) [0.00005,0.0209] [0,0.00001] (c2) [0.0022,0.032] [0,0.0001](a3) 464.7059 [0,1] (b3) 0.01125 [0,0.00001] (c3) 0.0125 [0,0.0022]

Robust Synchrony and Rhythmogenesis in GnRH Neurons 2113

Fig. 3 Minimum number of AOs in a network of 50 model neurons required for generating pulsatile g.Three parameter ranges are studied for three important parameters: κ , ω, and ε. Detailed choices of theparameter ranges are given in Table 1. The abscissa represents the number of AOs in the population, thatwill be engaged in a network of 50 coupled model neurons, whereas the ordinate denotes the fractionof actively pulsatile neurons (defined as the fraction of AOs plus that of ROs) in a pulsatile population.A quiescent population is represented by a zero value in the ordinate. The number of AOs was increasedfrom 1 to 50.

Simulations of the model reveal that the critical number of AOs for pulse generationis often small. For heterogeneity in the parameters κ and ω, less than 50% of AOs areneeded for the phase transition to take place. In Figs. 3(a1) and (b1), the neurons areuniformly distributed from zero up to the upper limit of the oscillatory domain (see alsoTable 1). About 10% or less of AOs are sufficient to generate the pulsatile signal. In otherwords, broadening the range of distribution of NOs seems to cause a decrease in thiscritical number, thus making it easier to generate oscillations. It is different, however, forthe cases shown in Figs. 3(c1)–(c3), where about 70–90% of neurons must be AOs toproduce oscillaions. In this case, broadening the distribution of NOs results in no changein the critical number (see (c1) and (c2)). This suggests that severely slowing down theactivation of αi subunit may cause oscillation death. This can be regarded as a modelprediction that we hope one can test experimentally in the future. Results in Figs. 3(a2)–(a3) reveal that the recruitment of NOs can occur partially. This means that a significantfraction of NOs are not recruited as ROs when the pulsatile signals are generated. In (a3),complete recruitment occurs only when all neurons are AOs. When distributed values ofω and ε are introduced, the recruitment is complete almost right at the critical number (see(b1)–(b3), (c1)–(c3)). In these studies, we have used the deviation criterion, d ≤ 3dmax, todetermine if an NO neuron is recruited. Furthermore, for each case shown in Fig. 3, thearithmetic average is located within the oscillatory range.

2114 Li and Khadra

Fig. 4 Minimum fraction of AOs required for pulse generation as a function of the population size N .The parameter ranges are taken from Table 1, of which (a2) corresponds to the dotted curve, (b1) relatesto the dashed curve, and (c3) was used for the solid curve.

3.6. Effect of the population size on synchronization

The number of GnRH neurons in the hypothalamus is estimated to be around 2,000 (Gold-smith et al., 1983). They are widely distributed in the hypothalamus and the preoptic re-gions. In the present study, we examined populations consisting of 50 and 100 neurons.To determine whether the total number of neurons in a population, N , causes any changein the results, we investigated the effects of changing N on the critical fraction of AOs re-quired for pulse generation. We found that varying N has little or no effect on the criticalfraction. We anticipate that this trend will persist for even higher values of N . Figure 4demonstrates that the critical fractions fluctuate around a flat level at increasing valuesof N in all three cases studied.

3.7. Application of the model to in vivo experiments

Evidence for the autocrine regulation of GnRH on its own secretion in vivo appearedlong before GT1 cell lines and GnRH neuronal cultures were developed. Hyyppa etal. (1971) found that subcutaneous administration of crude rat hypothalamic extractscontaining GnRH and devoid of any pituitary hormone and sex steroid considerablyreduced GnRH in the hypothalamus of castrated-hypophysectomized rats. They la-beled this action as an “ultrashort feedback loop (UFL).” More in vivo evidence forUFL was obtained in other studies (Corbin and Beattie, 1976; DePaolo et al., 1987;Valenca et al., 1987). Conflicting results in these studies make it difficult to deter-mine whether the feedback was of a positive or negative nature (DePaolo et al., 1987;Todman et al., 2005) and whether the actions are direct or indirect. Recent experiments

Robust Synchrony and Rhythmogenesis in GnRH Neurons 2115

(Todman et al., 2005) provided evidence that this action is direct and electrically excita-tory. Several studies of UFL were based on indirect measurements of pituitary hormones(LH and FSH). Caraty et al. (1990), however, suggested that measuring the variations inthe levels of LH and FSH is not appropriate for determining changes in the GnRH. Thiscontroversy was compounded by the fact that the regimes of GnRH injections in thesestudies appeared arbitrary and inconsistent (DePaolo et al., 1987; Valenca et al., 1987;Caraty et al., 1990). GnRH was administered periodically at different intervals rangingfrom a few hours to days. The resulting changes in the LH and FSH levels were assessedduring the injection period in some studies and immediately after in others. No explana-tion was provided for why these periods were chosen and why injections of other frequen-cies were not tested. We aim here at resolving these problems using computer simulationsof the model. The advantage of using a model is that the effects of GnRH injections onthe GnRH pulses can be accessed directly.

The model for the GnRH pulse generator (Khadra and Li, 2006) involving the au-tocrine mechanism is based on in vitro experiments showing that GnRH activates its ownsecretion through Gs and Gq and inhibits it through Gi (Krsmanovic et al., 2003). In thismodel, GnRH activates its own secretion at low doses and inhibits it at higher doses whenthe steady state response is considered. The fact that some studies show a positive whileothers show a negative effect is not at all contradictory based on this mechanism. Theseemingly controversial studies cited above could be considered as an indirect supportfor a model of the GnRH pulse generator based on autocrine regulations. However, whenthe injection of GnRH is periodic, the result could appear counterintuitive as we shalldemonstrate below. We show that the observations demonstrating that GnRH injectionsdo not alter endogenous GnRH secretion (Caraty et al., 1990) can also be explained bythe model.

To study the effects of GnRH injections, we introduce periodic injections of g usingperiods that closely mimic those adopted in the in vivo experiments cited above. Themodel pulse generator was forced by brief square-wave pulses of GnRH injection at reg-ular intervals. This is realized by adding a forcing term to the right-hand side of Eq. (5).The dose of each injection is represented by the area of the square wave which will be de-noted by R hereafter. Effects of such injections were determined during both the transientperiod immediately following the onset of the injections (called the transient effect), andlong after the transient period (called the steady-state effect). We performed two differentnumerical experiments: (1) varying the doses with fixed periods; and (2) varying periodswith fixed doses. Simulations using other regimes of GnRH administration which yieldedno qualitatively distinguishable results were also done (not shown).

Simulations of the dose response with a fixed period are shown in Fig. 5. The in-jection period is 6T , where T (≈487) is the intrinsic period of the pulse generator inthe absence of forcing. To make the simulations closer to experimental conditions, smallfluctuations are introduced to the onset of each square-wave injection. The left column ofFig. 5 displays the changes in the amplitude (a), period (b), average (c), and nadir (d) ofthe transient (dotted) and steady state (solid) responses of g as a function of the dose R.At doses beyond 100 units, the response amplitude and average increase as R increases(see (a)–(d)). The time course of the steady state responses to four different doses isshown in panels (e) to (h). It is obvious that there exists a one-to-six phase locking in allcases. For doses below 100 units ((e) and (f)), there is no detectable change in the am-plitude of the pulses that are locked with the injection pulses. For higher doses however

2116 Li and Khadra

Fig. 5 Dose-dependence of the pulse generator on exogenous injections of g. A population of 50 hetero-geneous model neurons with three parameters κ , ω and ε distributed within their oscillatory ranges givenin Table 1. The injection period is fixed at six times the natural period of the pulse generator (T ≈ 487).The left column shows (a) amplitude, (b) period, (c) average, and (d) nadir of the transient (dotted curvesfollowing the right vertical axes) and the steady state (solid curves following the left vertical axes) of thepulse generator, g, as a function of the injection dose. The transient response was taken between τ = 0and τ = 2,000, and the steady-state response was taken between τ = 20,000 and τ = 30,000. The rightcolumn shows the typical responses of the pulse generator (lower panels in (e), (f), (g), and (h)) to fourdifferent injection doses (upper panels in (e), (f), (g), and (h)) specified by the arrows in (d).

((g)–(h)), the amplitude of the locked pulses are boosted in a dose-dependent manner.This explains why the average GnRH level is increased. A similar result was obtained forthe transient responses (see the dotted curves in (c)). This result explains why in somein vivo experiments a positive effect of UFL was reported (Corbin and Beattie, 1976;DePaolo et al., 1987; Todman et al., 2005). It appears to contradict the expectation thathigher levels of g should inhibit its secretion through the action of i. This is because theinjection is pulsatile and applied at a much lower frequency than the intrinsic one.

Similar results are obtained for pulsatile forcing with a slightly shorter period (∼5T )when the dose is either small or big (see Figs. 6(a),(c)). At an intermediate dose, however,oscillation death was obtained (Fig. 6(b)). For some combinations of dose and period, al-ternative transitions between oscillations and quiescence could occur (results not shown).This might be related to the fact that the whole population behaves like an averaged sin-gle cell in which bistability between oscillation and steady state occurs as demonstratedin Fig. 4 of Khadra and Li (2006). Bistability provides one plausible explanation for theoccurrence of oscillation death and intermittency at intermediate doses of GnRH injec-tions. Notice that when oscillation death or intermittency occurs, the long term averageof g is greatly reduced. This may explain the observations made in some in vivo experi-ments when periodic GnRH injections were shown to inhibit GnRH secretion (Hyyppa etal., 1971; DePaolo et al., 1987; Valenca et al., 1987).

Responses of the pulse generator model to periodic injections of varying periods atfixed doses (R = 9 and 900) have also been studied. The results are shown in Fig. 7.

Robust Synchrony and Rhythmogenesis in GnRH Neurons 2117

Fig. 6 Three different samples of the time series of g exposed to increasing doses of exogenous GnRH at(a) R = 9, (b) R = 90, and (c) R = 900. The period of injection in all of these simulations is 5 × T andthe arrows in (b) indicate two consecutive moments of injections.

Fig. 7 Period-dependence of the pulse generator on exogenous injections of g. A network of 50 modelneurons with the three parameters ε, κ and ω distributed within their oscillatory ranges given in Table 1,is subjected to periodic injections in g. The period of the transient (dotted curve) and the steady state(solid curve) of the pulse generator are plotted as a function of the period of injections applied at (a) highdose (R = 900) and (b) low dose (R = 9). The transient is calculated as in Fig. 5. The horizontal axes arecalibrated using multiples of the natural period of the pulse generator (≈487). The arrows in (a) and (b)show the value of the natural period.

When the dose is high (Fig. 7(a)), a one-to-one phase-locking occurs for low period (highfrequency) forcing. As the forcing period increases to values significantly larger thanthe intrinsic period T , the pulse generator more or less maintains its intrinsic period asindicated by the horizontal arrow. However, a one-to-m phase locking is possible whenthe forcing period is an integer multiple of the intrinsic period. For example, a one-to-six locking was observed in Figs. 5(e)–(h) while a one-to-five locking was obtained inFigs. 6(a) and (c). When the forcing dose is low, the one-to-one phase locking is notmaintained for some high frequency (low period) injections (see Fig. 7(b)).

2118 Li and Khadra

The results presented here suggest that the GnRH pulse generator shows different re-sponses to periodic GnRH injections with different doses and periods. When high-dose,long-period injections are applied (as has been done in most experimental studies), theaverage level of g increases reflecting a positive feedback action by g on its own secre-tion. At intermediate doses, oscillation death sometimes occurs when the period is a fewtimes longer than the intrinsic one. This may explain why in some experiments, negativefeedback was observed. When the injection dose is small, no obvious changes can be de-tected. This may explain why some researchers claim that GnRH injections do not alterthe pulse generator (Caraty et al., 1990).

4. Discussion

Using a model of the GnRH pulse generator, we studied the robustness of the autocrineregulation mechanism and investigated the applicability of this mechanism to in vivo con-ditions. GnRH neurons in this model interact through secreting and sensing GnRH in acommon pool. As a result, each individual neuron can not be regarded as a distinguish-able oscillator since it does not possess a g variable that belongs only to itself. Under thiscondition, the coupling is usually considered strong as in the model of Li et al. (1992).The system is either pulsatile or quiescent. The shared variable makes it impossible forasynchrony to occur. In other words, asynchrony implies oscillation death. In order tostudy the effects of one-sided heterogeneity in coupled GnRH neurons, we introducedsome concepts concerning the synchronization of neurons coupled in this way. These in-clude a new synchrony measure and a deviation measure. The latter helps in classifyingthe neurons in a heterogeneous population into three subtypes: AO, RO, and SO.

The occurrence of the parametric averaging phenomenon revealed that pulse genera-tion can occur in a heterogeneous population that contains nonoscillatory neurons only. Inthe presence of one-sided heterogeneity, the pulse generating mechanism remains robust.It was shown that only a small fraction of AOs is often sufficient for recruiting enoughNOs to produce pulsatile GnRH signals. This result remains valid for a wide range ofpopulation sizes. Heterogeneity in GnRH neurons has been demonstrated in numerousexperiments (see, e.g., Todman et al., 2005 and the references cited there). Our study re-vealed that this mechanism remains robust and operational even when the NOs follow aone-sided distribution beyond the oscillatory range.

The autocrine mechanism for GnRH pulse generation was established mainly in cul-tured GnRH neurons and cell lines. Whether this mechanism operates in vivo remainsunknown. Although a mathematical model is incapable of providing a proof of a fact thatrequires experimental evidence, it does provide insight for the plausible occurrence ofa similar mechanism in vivo. The three key elements that ensure the occurrence of theGnRH pulses are: (1) the expression of GnRH receptors on the surface of GnRH neu-rons; (2) the existence of a common extracellular pool of GnRH where newly secretedGnRH is rapidly mixed; (3) the activation and/or inhibition of the two second messengersthrough three G-proteins Gs, Gq, and Gi, following the binding of GnRH to its receptors.Since the pulse-generating mechanism involving these three elements is shown to be suf-ficient for generating GnRH pulses in an extraordinarily robust way, it strongly supportsthe prediction that such a mechanism is likely operating under in vivo conditions.

Robust Synchrony and Rhythmogenesis in GnRH Neurons 2119

GnRH neurons in the gonadectomized male mouse (Xu et al., 2004) and cultured em-bryonic hypothalamus (Krsmanovic et al., 1999) have been found to express GnRH-R1receptors. Direct evidence came from Todman et al. (2005) who revealed the expressionof GnRH-R1 mRNA in GnRH neurons of adult female mouse. They demonstrated for thefirst time that GnRH does exert direct influence on GnRH neurons. These data provide di-rect support for the existence of element (1) in vivo. Previous experiments on the existenceof UFL influences of GnRH on its own release also provide support to this conclusion.As to the common extracellular pool of GnRH, the primary plexus of the hypophysealportal system is one potential candidate (Grzegorzewski et al., 1997). This is because itconstitutes the capillaries that supply nutrients to the hypothalamic cells. The heart beat,which is about once per second, is fast enough to provide an efficient mixing of newlyreleased GnRH in this pool. The common pool could simply be the interstitial fluid inthe hypothalamus where newly secreted GnRH is constantly transported in and out of thecapillaries through microcirculation. Regarding the third key element, it is not known ifthe binding of GnRH to its receptors on a mature GnRH neuron in vivo would triggerthe activation of the same three proteins as demonstrated in GnRH cell lines and culturedGnRH neurons in vitro. Any experimental evidence for the existence of the same cascad-ing effects of GnRH binding on the second messengers as demonstrated in Krsmanovic etal. (2003) would bridge the gap between GnRH pulse generating mechanisms in vivo andin vitro.

Motivated by the in vivo experiments that demonstrated UFL influences of GnRH onits own secretion, we carried out studies of the transient and steady-state responses ofthe pulse generator to periodic, square-wave injections of GnRH into the common pool.The results provide one plausible explanation for the seemingly controversial conclusionsreached in experiments by different groups. The model is capable of producing resultsthat are consistent to all known experimental observations. High-dose, long-period injec-tions usually increase the average level of g, thus providing a positive UFL influence. Atintermediate doses, oscillation death sometimes occurs, causing a negative UFL effect.When the dose is small, no obvious changes can be detected. Among the different effectsof periodic GnRH injections, the oscillation death caused by injections with intermedi-ate doses is of special interest. This prediction of the model can be tested by carefullydesigned experimental studies. It could be the reason behind the conclusions reached insome in vivo studies suggesting that the UFL effect is negative (Hyyppa et al., 1971;DePaolo et al., 1987; Valenca et al., 1987). Experimental confirmation of the occurrenceof such a phenomenon would provide further support of the autocrine mechanism forGnRH pulse generation.

The exact role of the electrical activities of the GnRH neurons in pulse generation can-not be answered by the present study. This is because the model completely ignored thefact that different firing patterns can occur in these neurons (Moenter et al., 2003; Nune-maker et al., 2001, 2003; Suter et al., 2000a, 2000b). It was assumed in one of the basichypotheses that electrical fluctuations in the membrane potential at time scales of mil-liseconds are not essential for generating GnRH pulses with a period close to 1 hour. Thishypothesis is supported by substantial direct evidence, including experiments in whichpulsatile GnRH signals were detected in enzymatically dispersed and not electrically cou-pled GnRH neurons (Woller et al., 1998, 2003). The vast differences in the intrinsic timescales of the electrical activities and the GnRH pulses also suggest that the former is un-likely responsible for the latter. The model does take into account the calcium entry from

2120 Li and Khadra

extracellular medium since the intracellular stores would otherwise be depleted eventu-ally in the absence of calcium influx. Only a constant basal influx of calcium entry fromthe extracellular medium is considered in the model. This would roughly correspond tothe case when all neurons are voltage clamped at a constant potential. However, it wouldbe interesting to determine what influences can the membrane electrical activities have onthe pulse generator. A current based model of the membrane electrical activities has beendeveloped previously (LeBeau et al., 2000). A more complete model taking into accountthe interactions between electrical activities and the second messenger pathways activatedby the autocrine regulation of GnRH is being developed.

Acknowledgements

This work was financially supported by NSERC (Natural Sciences and Engineering Re-search Council of Canada) grants to Yue-Xian Li, and partially by MITACS (The Mathe-matics of Information Technology and Complex systems) grant to Leah Keshet.

Appendix A: Nondimensionalization of the original model

We briefly demonstrate here how to derive the two dimensionless models presented inSection 2; namely, systems (1)–(4) and (5)–(8). To achieve this goal, we use the originalmodel presented in Khadra and Li (2006) describing the dynamics of one GnRH neuron.The time evolution of this six-variable model is given by

G = bG + vG(AC)3 − kGG, (A.1)

C = JIN + [� + vCQ] (CER − C) − kCC, (A.2)

A = bA + vAShI

hI + I− kAA, (A.3)

S = vS

G4

K4S + G4

− kSS, (A.4)

Q = vQ

G2

K2Q + G2

− kQQ, (A.5)

I = vI

G2

K2I + G2

− kI I, (A.6)

where KS < KQ < KI represent the threshold levels of G for activating αs, αq and αi,respectively. A detailed table containing all the parameters used in Eqs. (A.1)–(A.6) to-gether with their values and units has been previously published in Khadra and Li (2006)and presented here in Table A.1.

In order to reduce the number of parameters used in the model and for simplificationpurposes, we prefer to use the nondimensionalized form of system (A.1)–(A.6). We shall

Robust Synchrony and Rhythmogenesis in GnRH Neurons 2121

Table A.1 Values of the standard parameters of the model given by system (A.1)–(A.6). Parameters inroman style symbols are obtained by curve fitting to experimental data in Krsmanovic et al. (2003)

Standard Parameter ValuesSymbol Value Symbol Value Symbol Value Symbol Value

JIN 0.2 μM/min bG 0.144 nM/min � 60 min−1 kG 0.6 min−1

kA 60 min−1 kC 5100 min−1 vG 324 (nM)−4 min−1 vC 1200 (nM)−1min−1

CER 2.5 μM bA 1.8 nM/min hI 0.036 nM KS 0.34 nMKQ 21 nM KI 158 nM kS 9 min−1 kQ 9 min−1

kI 0.1125 min−1 vA 150 min−1 vS 23.4 nM/min vQ 22.5 nM/minvI 0.36 nM/min

demonstrate this by applying the following set of substitutions:

g = 1

KS

G, c = vC

kC

C, a = kA

bA

A, s = kS

vS

S,

q = kQ

vQ

Q, i = kI

vI

I, τ = kSt.

Notice that the new set of scaled variables are dimensionless. In this case, Eqs. (A.1)–(A.6) become

dg

dτ= λ

[ν + η(ca)3 − g

], (A.7)

dc

dτ= ζ

[jin + [μ + δt](c0 − c) − c

], (A.8)

da

dτ= ξ

[ι + θs

ω

ω + i− a

], (A.9)

ds

dτ= φ

[g4

σ 4 + g4− s

], (A.10)

dq

dτ= ψ

[g2

ρ2 + g2− q

], (A.11)

di

dτ= ε

[g2

κ2 + g2− i

], (A.12)

where

λ = kG

kS

, ν = bG

kGKS

, η = vG

kGKS

{bAkC

kAvC

}3

, ζ = kC

kS

,

jin = vCJIN

k2C

, μ = �

kC

, δ = vCvQ

kCkQ

, c0 = vCCER

kC

,

ξ = kA

kS

, θ = vAvS

bAkS

, ω = kIhI

vI

, ψ = kQ

kS

,

2122 Li and Khadra

Table A.2 Values of the dimensionless parameters of the model given by system (A.7)–(A.12)

Dimensionless Parameter ValuesSymbol Value Symbol Value Symbol Value Symbol Value Symbol Value

λ 0.067 ν 0.706 η 3.292 ζ 566.67 jin 9.227 × 10−6

μ 0.012 δ 0.588 c0 0.588 ξ 6.67 ι 1θ 216.67 ω 0.01125 φ 1 σ 1 ψ 1ρ 61.765 ε 0.0125 κ 464.706

ρ = KQ

KS

, ε = kI

kS

, κ = KI

KS

.

For a single cell model described by (A.7)–(A.12), the three scaled parameters ι, φ andσ satisfy ι = φ = σ = 1. However, this will no longer be the case in a heterogeneouspopulation of coupled model neurons. Table A.2 shows the values of all the dimensionlessparameters listed above.

Appendix B: Simplification of the model to four variables

It was pointed out in Khadra and Li (2006) that the variations in the two variables C

and A are much faster than the changes in the remaining variables, a fact consistent withexperimental observations. This claim could be verified by using system (A.7)–(A.12) andcomparing the values of the two parameters ξ and ζ with those of the parameters λ, φ, ψ

and ε (the former set of parameters are 2–4 orders of magnitude larger than the latter set).Therefore, one could reduce the original six-variable model to four variables by using thequasi-steady state approximation for the fast variables C and A, as follows:

G = bG + vGF(S,Q, I) − kGG, (B.1)

S = vS

G4

K4S + G4

− kSS, (B.2)

Q = vQ

G2

K2Q + G2

− kQQ, (B.3)

I = vI

G2

K2I + G2

− kI I, (B.4)

where F(S,Q, I) = [C∞(Q)A∞(S, I )]3 and

C∞(Q) = JIN + (� + vCQ)CER

� + kC + vCQ, A∞(S, I ) = bA

kA

+ vA

kA

ShI

hI + I

are the steady state values of C and A obtained from Eqs. (A.2) and (A.3), respectively.Similarly, we could reapply the quasi-steady state approximation once again on the vari-able c and a and obtain system (1)–(4), where F(s, q, i) = [c∞(q)a∞(s, i)]3 and

c∞(q) = Jin + (μ + δq)c0

μ + 1 + δq, a∞(s, i) = ι + θs

ω

ω + i. (B.5)

Robust Synchrony and Rhythmogenesis in GnRH Neurons 2123

In our previous analysis, we have limited ourselves to models depicting the dynamicsof only one GnRH neuron. For the case of N -coupled GnRH neurons sharing a commonpool of GnRH hormone, a new (3N + 1)-variable model based on system (A.1)–(A.6)was presented in Khadra and Li (2006). The time evolution of this model was given by

G = 1

N

N∑

n=1

[bGn + vGnFn(Sn,Qn, In)

] −[

1

N

N∑

n=1

kGn

]

G, (B.6)

Sn = vSn

G4

K4Sn

+ G4− kSnSn, (B.7)

Qn = vQn

G2

K2Qn

+ G2− kQnQn, (B.8)

In = vIn

G2

K2In

+ G2− kInIn, n = 1,2, . . . ,N, (B.9)

where the summation sign in Eq. (B.6) represents the total contribution of all GnRHneurons to the common pool of GnRH hormone. It should be mentioned here that thecommon pool idea was based on the experimental observation that the GnRH hormoneacts as a “diffusible mediator” or as a synchronizing agent. This latter formulation hasbeen extended in a similar fashion to the dimensionless model given by (1)–(4) to obtainEqs. (5)–(8) by using the variable g as the common pool.

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