Robust Synchrony and Rhythmogenesis in Endocrine Neurons via Autocrine Regulations In Vitro and In Vivo

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1 Bulletin of Mathematical Biology (2008) 70: DOI /s z ORIGINAL ARTICLE Robust Synchrony and Rhythmogenesis in Endocrine Neurons via Autocrine Regulations In Vitro and In Vivo Yue-Xian Li a,b,, Anmar Khadra b a Departments of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada b Departments 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 for maintaining reproductive functions in mammals. An explanation for the origin of this rhythm remains an ultimate goal for researchers in this field. Some plausible mechanisms have been proposed among which the autocrine-regulation mechanism has been implicated by numerous experiments. GnRH binding to its receptors in cultured GnRH neurons activates three types of G-proteins that selectively promote or inhibit GnRH secretion (Krsmanovic et al. in Proc. Natl. Acad. Sci. 100: , 2003). This mechanism appears to be consistent with most data collected so far from both in vitro and in vivo experiments. Based on this mechanism, a mathematical model has been developed (Khadra and Li in Biophys.J.91:74 83,2006) in which GnRH in the extracellular space plays the roles of a feedback regulator and a synchronizing agent. In the present study, we show that synchrony between different neurons through sharing a common pool of GnRH is extremely robust. In a diversely heterogeneous population of neurons, the pulsatile rhythm is 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 oscillators while forcing the nonrecruitable neurons to oscillate along. By pointing out the existence of the key elements of this model in vivo, we predict that the same mechanism revealed by experiments in vitro may also operate in vivo. This model provides one plausible explanation for the apparently controversial conclusions based on experiments on the 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 Parametric averaging Corresponding author. address: yxli@math.ubc.ca (Yue-Xian Li).

2 2104 Li and Khadra 1. Introduction Reproductive function in mammals is controlled by episodic pulses of gonadotropinreleasing 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 pituitary gonadotrophs that are targets of GnRH action respond only to a GnRH signal with highly specific period and temporal pattern (Knobil, 1980; Li and Goldbeter, 1989). The absence or malfunction of this pulse generator is associated with several reproductive and developmental diseases (Knobil, 1980). The search for mechanisms underlying the GnRH pulse generator has been conducted by researchers with different backgrounds and view points. Potential explanations range from autocrine regulations by GnRH on its 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 coupled through one or more neurotransmitters (Witkin and Silverman, 1985; Terasawa, 2001; Moenter et al., 2003). Despite the apparent diversity in opinions, it is now generally recognized that an ultimate resolution of this puzzle depends on an ultimate answer to three 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 secretory activities? (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 is now widely believed that the pulses are produced by GnRH neurons. This was supported by in vivo studies showing that lesion but not deafferentation of the medial basal hypothalamus 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 generation was 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 conclusions 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 several 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 questions based on experiments in cultured GT1 cells and GnRH neurons. Synchronization through a diffusible mediator was first proposed by de la Escalera et al. (1992) based on the observation that GT1 cells on two cell-coated coverslips without direct contact were synchronized. The fact that synchronized release of GnRH pulses was observed in enzymatically dispersed rat hypothalamic tissue (Woller et al., 1998, 2003) supported the roles of a diffusible mediator in coordinating and synchronizing the acutely dissociated cells. The finding of GnRH receptors in GT1 cells (Krsmanovic et al., 1991) and of the autocrine influences of GnRH on its own release (Krsmanovic et al., 1991, 1993, 2003) further suggested that GnRH is itself the diffusible mediator responsible for synchronizing the scattered GnRH neurons. These experiments led to the illustration of the plausible molecular mechanism that underlies the GnRH pulsatility by Krsmanovic et al. (2003). They found 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 the intracellular levels of Ca 2+ and camp. Based on this mechanism, a mathematical model of the GnRH pulse generator was developed (Khadra and Li, 2006). It demonstrated that

3 Robust Synchrony and Rhythmogenesis in GnRH Neurons 2105 the autocrine regulation mechanism is viable in generating GnRH pulses and gave support for this molecular mechanism in culture conditions. Experimental data that led to the autocrine regulation mechanism were mostly collected in vitro. Whether this mechanism is actually responsible for the GnRH pulse generation in vivo remains undetermined. Although a final resolution of this problem depends critically on collecting more data, mathematical modeling can provide useful information on the viability of such a model under in vivo conditions. The present study is an attempt at answering this question using a mathematical model. We aim at achieving two goals. First, we show that autocrine regulation via a hormonal signal in a common pool is very robust in synchronizing diversely heterogeneous populations of GnRH neurons. Here diversely heterogeneous means that the key parameters of each model neuron are randomly chosen from a wide range of values that stretches far outside the oscillatory domain for a single isolated neuron. We show that a small fraction of actively oscillating GnRH neurons is often sufficient to recruit many nonoscillatory neurons so as to maintain the pulse-generating function of the whole population. The robustness of such a mechanism makes it hard to argue against the possibility that it could also operate under in vivo conditions, especially when existing data point to the fact that the key elements that ensure the operation of such a mechanism in vitro are also present in vivo. Second, we apply the model to experimental conditions in vivo and provide sensible explanations for some in vivo experiments. The paper is organized as follows. In Section 2, we introduce the model of GnRH pulse generator and a reduced, dimensionless version of it. In Section 3, we introduce the definitions of some important concepts that will be used in our study. These include the synchrony measure, the diversely heterogeneous population, the deviation measure, and the classification of neurons in a heterogeneous population. The phenomenon of parametric averaging that occurs in this model is explained. The results supporting the extraordinary robustness of the pulse generating mechanism and the applicability of this model to GnRH pulse generation in vivo are also discussed. A summary is provided in Section 4 together 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 regulation mechanism derived from experimental data obtained by Krsmanovic et al. (2003). This mechanism is schematically summarized in Fig. 1(a). The binding of GnRH in the extracellular medium (abbreviated by G hereafter) to its receptors on the GnRH neurons activates three types of G-proteins G s,g q, and G i. This triggers the dissociation of the α subunits α s, α q, and α i from their respective βγ subunits. α s and α q promote the secretion of GnRH by increasing the two intracellular messengers camp and Ca 2+, respectively, while α i inhibits GnRH secretion by inhibiting camp production. In this model, G plays the roles of a diffusible mediator as well as a synchronizing agent. The autocrine effects of G on its own secretion through α s and α q provide two distinct positive feedback loops that are essential for triggering the explosive increasing phase of the spike in G, while the

4 2106 Li and Khadra Fig. 1 Schematic diagram of the model of GnRH pulse generator (a) by Khadra and Li (2006) based on the autocrine mechanism proposed by Krsmanovic et al. (2003). Comparison between the full six-variable model (b) and the reduced four-variable, dimensionless model (c) reveals no visible difference in their pulse 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 α i is crucial for terminating the spike and for holding G at the basal level 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, the cytosolic Ca 2+ and camp concentrations while S, Q, and I represent the concentrations of α s, α q, and α i, respectively. Because the variations of C and A are much faster than the other variables, we simplified the system into a four variable system using quasi-steady state approximations (Eqs. (B.1) (B.4) in Appendix B). The dimensionless form of this simplified system is given below. dg dτ = λ[ ν + ηf(s,q,i) g ], (1)

5 Robust Synchrony and Rhythmogenesis in GnRH Neurons 2107 [ ds dτ = φ dq dτ = ψ di dτ = ɛ g 4 σ 4 + g s 4 [ g 2 ρ 2 + g q 2 [ g 2 κ 2 + g 2 i ], (2) ], (3) ], (4) where g, s, q and i denote the scaled (dimensionless) variables G, S, Q, and I, respectively. The function F(s,q,i) and all the scaled parameters in system (1) (4) are given in Appendices A and B. Specific choices of the parameter values and the nonlinear functions in the model were determined based on fitting the steady state activation/inactivation curves to available experimental data (Krsmanovic et al., 2003). Simulations of the simplified model and the full model demonstrate that the former is a very good approximation of the latter (see Figs. 1(b), (c)) The population model Available data seem to suggest that a single GnRH neuron possesses all the necessary parts for generating GnRH pulses under ideal conditions (see references cited in Khadra and Li, 2006). Mathematically, a single-cell model is equivalent to a population model of identical neurons (Khadra and Li, 2006). However, homogeneity is unrealistic since neurons are not identical in their biological environment. Therefore, we focus in this present study on heterogeneous neuronal populations. A dimensionless form of the model of N heterogeneous GnRH neurons coupled through a common pool of extracellular GnRH is given by the following equations (see Appendix B for a detailed dimensioned form). dg dτ = 1 N N [ λ n νn + η n F n (s n,q n,i n ) g ], (5) n=1 [ ds n dτ = φ g 4 ] n σn 4 + s g4 n, (6) [ dq n dτ = ψ g 2 ] n ρn 2 + q g2 n, (7) [ di n dτ = ɛ n g 2 κ 2 n + g2 i n ], (n= 1, 2,...,N), (8) whereweusethesubscriptn in the function F n and the parameters to indicate that they all 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 by each neuron is diluted and averaged immediately. This can be regarded as an approximation of the perifusion experiments in which the continuous flow through the chamber can cause a stirring effect. A more realistic model of a heterogeneous population in the absence of stirring would have to consider the geometric distribution of the neurons, the diffusion of g in the medium and the shell-effect in the vicinity of the cell surface where

6 2108 Li and Khadra g is released. Such a model is beyond the scope of this study and will likely yield qualitatively similar conclusions, although some details may differ. In this paper, we shall use system (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 of N neurons coupled through a shared variable g. Neurons interact with each other through secreting g into the pool and responding to changes in g in the pool. Oscillators coupled through a common variable have been studied in some other systems (see, e.g., Li et al., 1992). Each neuron is considered an oscillator only when this common variable is taken into account. A general theory of synchrony between heterogeneous populations coupled in such a way is not available. Studies of this kind are mostly based on numerical simulations. Unlike synaptically coupled neuronal networks, even the meaning of the word synchrony is not clearly defined in this case. Two neurons are considered synchronized if their membrane potentials vary in synchrony and peak at the same time. Here, the extracellular signal g is shared by all neurons. To determine the degree of synchrony, we need to 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 oscillation frequency in a single-cell model. For a heterogeneous population, i n differs for all neurons and peaks at different times (see Fig. 2(a)). However, a pulsatile g is generated despite of the fact that the variables that are internal to each neuron are not synchronized. Therefore, we introduce the following synchrony measure: A heterogeneous population is considered synchronized if a pulsatile signal g is generated and if the variables i n of all the 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 at least three times higher than its baseline level. The term actively pulsatile neuron refers generally to those neurons that actively participate in the pulse-generation Diversely heterogeneous populations of GnRH neurons To test the robustness of the autocrine mechanism, heterogeneity in the values of many selected parameters within their respective oscillatory ranges has been tested in Khadra and 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. For each parameter, the range is determined while all the other parameters are kept fixed at a standard set of values (see Table A.2, Appendix B). The oscillatory ranges of three key parameters are listed in Table 1. We have tested both the uniform and truncated Gaussian distributions of all parameters within their respective oscillatory ranges, and observed that synchronized pulses occurred in all cases (results not shown). This demonstrates that the mechanism is robust and insensitive to parameter variations, provided that all parameters are chosen from within their respective oscillatory ranges. In reality, there is no guarantee that all neurons are active oscillators. Electrophysiological evidence seems to suggest that

7 Robust Synchrony and Rhythmogenesis in GnRH Neurons 2109 Fig. 2 Pulse generation by a diversely heterogeneous neuron population. (a) 50 model neurons with the parameter κ evenly distributed within its oscillatory range (between the two vertical bars) and to the left side of the range. The apparent nonuniformity in the distribution is caused by the logarithmic scale. Active oscillators (AOs), recruited oscillators (ROs), and slaved oscillators (SOs) are denoted, respectively, by green diamonds, blue circles, and red triangles in the lower panel. In the middle panel, AOs, ROs, and SOs are plotted in green solid, blue dashed, and red dotted curves respectively. The thick black curve is the average of all the AOs. (b) A case of parametric averaging including 50 model neurons whose κ values are all distributed outside the oscillatory range, of which 25 are distributed to the left (red triangles and dotted 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 similar condition in the model, we focus on populations that contain neurons with parameter values distributed far beyond the oscillatory ranges. We call such a population a diversely heterogeneous population hereafter. The heterogeneity of realistic GnRH neurons is unlikely worse than the cases that are studied in this paper Deviation measure and classification of neurons in a heterogeneous population Figure 2(a) shows the pulse generation of one diversely heterogeneous population of model GnRH neurons. Of the 50 neurons in the system, 25 are randomly chosen from the oscillatory range between the two vertical bars in the lower panel. We call these neurons active oscillators (AOs) and plotted them in thin solid curves of green color. The

8 2110 Li and Khadra others have κ randomly chosen between 0 and 32 which represent all possible values that are smaller than the oscillatory range. These neurons are not oscillators themselves when decoupled from the others and are thus called non-ao (NO) neurons. Simulations of many heterogeneous populations of the model neurons revealed that different NOs can behave very differently in the coupled system (see the dashed curves in blue and dotted curves in red in Fig. 2(a)). Notice that the blue-colored NOs behave very much like an AO while the red-colored NOs remain elevated cycle after cycle. To divide the NOs into different subgroups, we introduce a quantitative criterion named the deviation measure. It is designed to differentiate the behavior of an NO from that of an average AO. For this purpose, we first calculate the average of the i variables of all the AOs (i.e., all the thin solid curves in green in Fig. 2(a)) to yield ī(t) which is represented by the thick black curve. The deviation of each neuron in the population from ī(t) is calculated as follows. d n = 1 [ T ( in (t) ī(t) ) ] 1/2 2 dt, (9) T 0 where T is the period of the g pulses. Then we calculate the deviation of all the AOs and define d max = max{d n, for all AOs} as the maximum possible deviation of the AOs. Notice that d n d max for each AO and that d n >d max for each NO. Using this measure, we can divide NOs into two subtypes. Those that behave similarly to 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 called slaved oscillators (SOs). The ROs are plotted in dashed blue curves and the SOs in dotted red curves in Fig. 2(a). Very often, the boundary between the ROs and SOs is not at all clear. It depends on the parameter that is chosen in the study as well as the extent and the location of the distribution of NOs (see the lower panel). To separate the two, we introduce a coefficient r such that those NOs that satisfy d max <d n rd max are defined as ROs, while those that yield d n >rd max are defined as SOs. In Fig. 2(a), r = 3, which means that every neuron whose deviation is larger than three times d max is considered an SO. A slightly different choice of the value of r will not change the qualitative nature of the results. Note that the values of the variable i n of all the NOs are higher than those of the AOs in Fig. 2(a). This is because the values of the parameter κ n for NOs are all chosen from a range that is smaller than the oscillatory domain. A lower value of κ n means that the threshold value of g for activating the production of i n is lower. This leads to higher values of i n given that the value of g in the common pool is shared. This specific bias typically leads to smaller amplitude in the pulses of g. In the case when all NOs are chosen from a range of κ n values that are larger than the oscillatory range, the i n of each NO will oscillate within a range that is below the green curves thus leading to an increased amplitude in g. The reason for choosing all the NOs from a range located to one side of the oscillatory domain will be explained below. The NOs located closer to the oscillatory domain are more easily recruited than those that are farther away Parametric averaging Parametric averaging refers to the phenomenon in which coupling between two (conditional) oscillators that differ in the value of one parameter results in a coupled system that

9 Robust Synchrony and Rhythmogenesis in GnRH Neurons 2111 is equivalent to a single oscillator with an averaged value of the parameter. A conditional oscillator refers to a dynamical system that is nonoscillatory, but is capable of generating stable limit-cycle type of oscillations when the parameter values are appropriately chosen. For simplicity, we use the term oscillator to refer indiscriminately to both types of oscillators. Parametric averaging often occurs when coupling two oscillators that differ in the values of more than one parameter. In the study of a mixed suspension of two subpopulations of Dictyostelium discoideum amoeba (Li et al., 1992), it was shown that in some cases the mixed population behaves like a pure population with a parameter that is equal 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 oscillators. Similar kind of parametric averaging has been found in a number of other systems with different types of coupling including those described by Manor et al. (1997) and Cartwright (2000). In Manor et al., for example, the coupled system was shown to rapidly relax to a parametrically averaged system. Coupling through a common chemical pool is a shared feature of the GnRH pulsegenerator model and the model of coupled amoeba populations (Li et al., 1992). Parametric averaging is expected to also occur in the GnRH pulse generator. A typical example is shown in Fig. 2(b) where all neurons in the population are NOs. Half of them lies above the oscillatory range (the blue diamonds or dashed curves), while the other half lies below (the red triangles or dotted curves). Figure 2(b) shows that pulsatile oscillations in g are generated in this population composed of only NOs. An arithmetic average of the 50 different values of κ results in a value of κ 316 which lies within the oscillatory range of κ. Figure 2(b) reveals that the interactions between the two types of neurons switched the latter group into a pulse generating mode. Coupling through a shared pool of feedback signal is often a strong coupling. Notice that although the red neurons are SOs based on the deviation measure, their existence is essential for switching the blue neurons into ROs. The blue neurons would remain nonoscillatory if the red neurons were to be removed from the population. This suggests that the pulsatile signal can be generated in a heterogeneous population whose individual members are all NOs when decoupled from others. This result is consistent with the important roles of heterogeneity in biological systems as emphasized by Cartwright (2000). Parametric averaging involving oscillators with parameter values distributed above and below the oscillatory range with equal probability supports the robustness of the pulse generating mechanism. This result has been established in a number of previous studies that are cited above. Therefore, this type of parametric averaging is not the main focus of this study. Instead, we test the robustness of this mechanism further by focusing on heterogeneous populations in which the NOs are picked from one side of the oscillatory range. The question we try to answer here is what is the minimal fraction of AOs for generating pulsatility in such an extreme case. Figure 2(a) shows a typical example in which AOs are picked randomly from the oscillatory range (green diamonds) while NOs are picked from an interval below this range (blue circles and red triangles). The result indicates that NOs distributed closer to the oscillatory range are more readily recruited to become an RO. Since the oscillatory range of κ contains values above 1,000 and the NOs are chosen from a domain of smaller values, κ 301 is located within the oscillatory range. Although the parameters κ and ω appear nonlinearly in the equations, the reason for why simple arithmetic averaging remains operative is not clearly understood.

10 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 are chosen from either above or below the oscillatory domain. The goal is to test how tolerant the pulse generating mechanism is to a unidirectional deviation of a subpopulation of the GnRH neurons from the oscillatory domain. Such deviations may occur when a disease or some environmental stress causes the cells to change in a specific way. We selected three parameters that play crucial roles in the model: κ, ω, and ɛ. Their definitions are given in Appendix A. ɛ characterizes the time scale of the i variable, which must be smaller than 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 curves in Fig. 2).Theroleofω is defined in Eq. (B.5). It specifies a threshold for the i variable beyond which the production of camp is inhibited. In other words, the sensitivity of the adenylyl cyclase to the inhibition of α i is determined by ω. The oscillatory ranges for κ, ω, and ɛ are listed in Table 1 together with the ranges of NO distributions. Three different combinations were tested for each parameter. The results are shown in Fig. 3, where the fraction of the total number of actively pulsing oscillators is plotted as a function of the number of AOs that are in the population. Here, the total number of actively pulsing oscillators is defined as the total number 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 is zero. 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 whole population is inactive and nonoscillatory (see Fig. 3). As we bring one NO at a time randomly into the oscillatory range, a discontinuous transition from a nonoscillatory state to an oscillatory state occurs at a critical number of AOs in each case studied. Such a sharp transition from quiescence to pulsatility is reminiscent to the phase transition-like event that was discovered by Kuramoto in a heterogeneous population of phase oscillators (Kuramoto, 1984). In Fig. 3, the transition from nonoscillatory to oscillatory behavior is discontinuous since the transition marks the boundary between quiescence and oscillation. 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, the transition marks the boundary between asynchronous and synchronous oscillations. The transition is smooth and the increase in the amplitude of the order parameter is gradual. Table 1 Nine different combinations of parameter ranges for κ, ω and ɛ tested in the search for the smallest possible fraction of AOs that produces a pulsatile g signal κ ω ɛ Case AO Range NO Range Case AO Range NO Range Case AO Range NO Range (a1) [32, 1071] [0, 32] (b1) [ , ] [0, ] (c1) [0.0022, 0.032] [0, ] (a2) [32, 1071] [0, 1] (b2) [ , ] [0, ] (c2) [0.0022, 0.032] [0, ] (a3) [0, 1] (b3) [0, ] (c3) [0, ]

11 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 ɛ. Detailed choices of the parameter ranges are given in Table 1. The abscissa represents the number of AOs in the population, that will be engaged in a network of 50 coupled model neurons, whereas the ordinate denotes the fraction of 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 increased from 1 to 50. Simulations of the model reveal that the critical number of AOs for pulse generation is often small. For heterogeneity in the parameters κ and ω, less than 50% of AOs are needed for the phase transition to take place. In Figs. 3(a1) and (b1), the neurons are uniformly distributed from zero up to the upper limit of the oscillatory domain (see also Table 1). About 10% or less of AOs are sufficient to generate the pulsatile signal. In other words, broadening the range of distribution of NOs seems to cause a decrease in this critical number, thus making it easier to generate oscillations. It is different, however, for the cases shown in Figs. 3(c1) (c3), where about 70 90% of neurons must be AOs to produce oscillaions. In this case, broadening the distribution of NOs results in no change in the critical number (see (c1) and (c2)). This suggests that severely slowing down the activation of α i subunit may cause oscillation death. This can be regarded as a model prediction 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 significant fraction 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 of ω and ɛ 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 3d max,to determine if an NO neuron is recruited. Furthermore, for each case shown in Fig. 3, the arithmetic average is located within the oscillatory range.

12 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) relates to the dashed curve, and (c3) was used for the solid curve Effect of the population size on synchronization The number of GnRH neurons in the hypothalamus is estimated to be around 2,000 (Goldsmith et al., 1983). They are widely distributed in the hypothalamus and the preoptic regions. 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 change in the results, we investigated the effects of changing N on the critical fraction of AOs required for pulse generation. We found that varying N has little or no effect on the critical fraction. We anticipate that this trend will persist for even higher values of N. Figure 4 demonstrates that the critical fractions fluctuate around a flat level at increasing values of N in all three cases studied Application of the model to in vivo experiments Evidence for the autocrine regulation of GnRH on its own secretion in vivo appeared long before GT1 cell lines and GnRH neuronal cultures were developed. Hyyppa et al. (1971) found that subcutaneous administration of crude rat hypothalamic extracts containing GnRH and devoid of any pituitary hormone and sex steroid considerably reduced GnRH in the hypothalamus of castrated-hypophysectomized rats. They labeled this action as an ultrashort feedback loop (UFL). More in vivo evidence for UFL 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 determine 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

13 Robust Synchrony and Rhythmogenesis in GnRH Neurons 2115 (Todman et al., 2005) provided evidence that this action is direct and electrically excitatory. 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 in the levels of LH and FSH is not appropriate for determining changes in the GnRH. This controversy was compounded by the fact that the regimes of GnRH injections in these studies appeared arbitrary and inconsistent (DePaolo et al., 1987; Valenca et al., 1987; Caraty et al., 1990). GnRH was administered periodically at different intervals ranging from a few hours to days. The resulting changes in the LH and FSH levels were assessed during the injection period in some studies and immediately after in others. No explanation was provided for why these periods were chosen and why injections of other frequencies were not tested. We aim here at resolving these problems using computer simulations of the model. The advantage of using a model is that the effects of GnRH injections on the GnRH pulses can be accessed directly. The model for the GnRH pulse generator (Khadra and Li, 2006) involving the autocrine mechanism is based on in vitro experiments showing that GnRH activates its own secretion through G s and G q and inhibits it through G i (Krsmanovic et al., 2003). In this model, GnRH activates its own secretion at low doses and inhibits it at higher doses when the steady state response is considered. The fact that some studies show a positive while others show a negative effect is not at all contradictory based on this mechanism. The seemingly controversial studies cited above could be considered as an indirect support for a model of the GnRH pulse generator based on autocrine regulations. However, when the injection of GnRH is periodic, the result could appear counterintuitive as we shall demonstrate below. We show that the observations demonstrating that GnRH injections do not alter endogenous GnRH secretion (Caraty et al., 1990) can also be explained by the model. To study the effects of GnRH injections, we introduce periodic injections of g using periods that closely mimic those adopted in the in vivo experiments cited above. The model pulse generator was forced by brief square-wave pulses of GnRH injection at regular 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 denoted by R hereafter. Effects of such injections were determined during both the transient period immediately following the onset of the injections (called the transient effect), and long after the transient period (called the steady-state effect). We performed two different numerical experiments: (1) varying the doses with fixed periods; and (2) varying periods with fixed doses. Simulations using other regimes of GnRH administration which yielded no qualitatively distinguishable results were also done (not shown). Simulations of the dose response with a fixed period are shown in Fig. 5. Theinjection period is 6T,whereT ( 487) is the intrinsic period of the pulse generator in the absence of forcing. To make the simulations closer to experimental conditions, small fluctuations are introduced to the onset of each square-wave injection. The left column of Fig. 5 displays the changes in the amplitude (a), period (b), average (c), and nadir (d) of the 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 is shown in panels (e) to (h). It is obvious that there exists a one-to-six phase locking in all cases. For doses below 100 units ((e) and (f)), there is no detectable change in the amplitude of the pulses that are locked with the injection pulses. For higher doses however

14 2116 Li and Khadra Fig. 5 Dose-dependence of the pulse generator on exogenous injections of g. A population of 50 heterogeneous model neurons with three parameters κ, ω and ɛ distributed within their oscillatory ranges given in 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 curves following the right vertical axes) and the steady state (solid curves following the left vertical axes) of the pulse generator, g, as a function of the injection dose. The transient response was taken between τ = 0 and τ = 2,000, and the steady-state response was taken between τ = 20,000 and τ = 30,000. The right column shows the typical responses of the pulse generator (lower panels in (e), (f), (g), and (h)) to four different 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 for the transient responses (see the dotted curves in (c)). This result explains why in some in 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 that higher levels of g should inhibit its secretion through the action of i. This is because the injection 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, alternative 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 single cell in which bistability between oscillation and steady state occurs as demonstrated in Fig. 4 of Khadra and Li (2006). Bistability provides one plausible explanation for the occurrence of oscillation death and intermittency at intermediate doses of GnRH injections. Notice that when oscillation death or intermittency occurs, the long term average of g is greatly reduced. This may explain the observations made in some in vivo experiments when periodic GnRH injections were shown to inhibit GnRH secretion (Hyyppa et al., 1971; DePaolo et al., 1987; Valenca et al., 1987). Responses of the pulse generator model to periodic injections of varying periods at fixed doses (R = 9 and 900) have also been studied. The results are shown in Fig. 7.

15 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 and the 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 model neurons with the three parameters ɛ, 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) high dose (R = 900) and (b) low dose (R = 9). The transient is calculated as in Fig. 5. The horizontal axes are calibrated 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 (high frequency) forcing. As the forcing period increases to values significantly larger than the intrinsic period T, the pulse generator more or less maintains its intrinsic period as indicated by the horizontal arrow. However, a one-to-m phase locking is possible when the forcing period is an integer multiple of the intrinsic period. For example, a one-tosix locking was observed in Figs. 5(e) (h) while a one-to-five locking was obtained in Figs. 6(a) and (c). When the forcing dose is low, the one-to-one phase locking is not maintained for some high frequency (low period) injections (see Fig. 7(b)).

16 2118 Li and Khadra The results presented here suggest that the GnRH pulse generator shows different responses 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), the average level of g increases reflecting a positive feedback action by g on its own secretion. At intermediate doses, oscillation death sometimes occurs when the period is a few times longer than the intrinsic one. This may explain why in some experiments, negative feedback was observed. When the injection dose is small, no obvious changes can be detected. This may explain why some researchers claim that GnRH injections do not alter the pulse generator (Caraty et al., 1990). 4. Discussion Using a model of the GnRH pulse generator, we studied the robustness of the autocrine regulation mechanism and investigated the applicability of this mechanism to in vivo conditions. GnRH neurons in this model interact through secreting and sensing GnRH in a common pool. As a result, each individual neuron can not be regarded as a distinguishable oscillator since it does not possess a g variable that belongs only to itself. Under this condition, 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 for asynchrony to occur. In other words, asynchrony implies oscillation death. In order to study the effects of one-sided heterogeneity in coupled GnRH neurons, we introduced some concepts concerning the synchronization of neurons coupled in this way. These include a new synchrony measure and a deviation measure. The latter helps in classifying the neurons in a heterogeneous population into three subtypes: AO, RO, and SO. The occurrence of the parametric averaging phenomenon revealed that pulse generation can occur in a heterogeneous population that contains nonoscillatory neurons only. In the 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 enough NOs to produce pulsatile GnRH signals. This result remains valid for a wide range of population sizes. Heterogeneity in GnRH neurons has been demonstrated in numerous experiments (see, e.g., Todman et al., 2005 and the references cited there). Our study revealed that this mechanism remains robust and operational even when the NOs follow a one-sided distribution beyond the oscillatory range. The autocrine mechanism for GnRH pulse generation was established mainly in cultured GnRH neurons and cell lines. Whether this mechanism operates in vivo remains unknown. Although a mathematical model is incapable of providing a proof of a fact that requires experimental evidence, it does provide insight for the plausible occurrence of a similar mechanism in vivo. The three key elements that ensure the occurrence of the GnRH pulses are: (1) the expression of GnRH receptors on the surface of GnRH neurons; (2) the existence of a common extracellular pool of GnRH where newly secreted GnRH is rapidly mixed; (3) the activation and/or inhibition of the two second messengers through three G-proteins G s,g q, and G i, following the binding of GnRH to its receptors. Since the pulse-generating mechanism involving these three elements is shown to be sufficient for generating GnRH pulses in an extraordinarily robust way, it strongly supports the prediction that such a mechanism is likely operating under in vivo conditions.

17 Robust Synchrony and Rhythmogenesis in GnRH Neurons 2119 GnRH neurons in the gonadectomized male mouse (Xu et al., 2004) and cultured embryonic hypothalamus (Krsmanovic et al., 1999) have been found to express GnRH-R1 receptors. Direct evidence came from Todman et al. (2005) who revealed the expression of GnRH-R1 mrna in GnRH neurons of adult female mouse. They demonstrated for the first time that GnRH does exert direct influence on GnRH neurons. These data provide direct support for the existence of element (1) in vivo. Previous experiments on the existence of 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 hypophyseal portal system is one potential candidate (Grzegorzewski et al., 1997). This is because it constitutes 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 newly released GnRH in this pool. The common pool could simply be the interstitial fluid in the hypothalamus where newly secreted GnRH is constantly transported in and out of the capillaries through microcirculation. Regarding the third key element, it is not known if the binding of GnRH to its receptors on a mature GnRH neuron in vivo would trigger the activation of the same three proteins as demonstrated in GnRH cell lines and cultured GnRH neurons in vitro. Any experimental evidence for the existence of the same cascading effects of GnRH binding on the second messengers as demonstrated in Krsmanovic et al. (2003) would bridge the gap between GnRH pulse generating mechanisms in vivo and in vitro. Motivated by the in vivo experiments that demonstrated UFL influences of GnRH on its own secretion, we carried out studies of the transient and steady-state responses of the pulse generator to periodic, square-wave injections of GnRH into the common pool. The results provide one plausible explanation for the seemingly controversial conclusions reached in experiments by different groups. The model is capable of producing results that are consistent to all known experimental observations. High-dose, long-period injections usually increase the average level of g, thus providing a positive UFL influence. At intermediate doses, oscillation death sometimes occurs, causing a negative UFL effect. When the dose is small, no obvious changes can be detected. Among the different effects of periodic GnRH injections, the oscillation death caused by injections with intermediate doses is of special interest. This prediction of the model can be tested by carefully designed experimental studies. It could be the reason behind the conclusions reached in some 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 occurrence of such a phenomenon would provide further support of the autocrine mechanism for GnRH pulse generation. The exact role of the electrical activities of the GnRH neurons in pulse generation cannot be answered by the present study. This is because the model completely ignored the fact that different firing patterns can occur in these neurons (Moenter et al., 2003; Nunemaker et al., 2001, 2003; Suter et al., 2000a, 2000b). It was assumed in one of the basic hypotheses that electrical fluctuations in the membrane potential at time scales of milliseconds are not essential for generating GnRH pulses with a period close to 1 hour. This hypothesis is supported by substantial direct evidence, including experiments in which pulsatile GnRH signals were detected in enzymatically dispersed and not electrically coupled GnRH neurons (Woller et al., 1998, 2003). The vast differences in the intrinsic time scales of the electrical activities and the GnRH pulses also suggest that the former is unlikely responsible for the latter. The model does take into account the calcium entry from