In this section, we consider the following network model of a population of GnRH neurons:

{x}_{j}^{\mathrm{\prime}}=\tau (-{y}_{j}+4{x}_{j}-{x}_{j}^{3}-{\varphi}_{\mathrm{fall}}({\mathrm{Ca}}_{j})),

(6a)

{y}_{j}^{\mathrm{\prime}}=\tau \epsilon {k}_{j}({x}_{j}+{a}_{1}{y}_{j}+{a}_{2}-{\eta}_{j}{\varphi}_{\mathrm{syn}}(\sigma )),

(6b)

{\mathrm{Ca}}_{j}^{\mathrm{\prime}}=\tau \epsilon ({\varphi}_{\mathrm{rise}}({x}_{j})-\frac{{\mathrm{Ca}}_{j}-{\mathrm{Ca}}_{\mathrm{bas}}}{{\tau}_{\mathrm{Ca}}}),

(6c)

{\sigma}^{\mathrm{\prime}}=\tau (\delta \epsilon \sigma -\gamma (\sigma -{\sigma}_{0}){\varphi}_{\sigma}(\frac{1}{N}\sum _{i=1}^{N}{\mathrm{Ca}}_{i}-{\mathrm{Ca}}_{\mathrm{desyn}})),

(6d)

for j=1,\dots ,N, with *N* the number of neurons and

\begin{array}{rl}{\varphi}_{\mathrm{syn}}(\sigma )& =\frac{1}{1+exp(-{\rho}_{\mathrm{syn}}(\sigma -{\sigma}_{\mathrm{on}}))},\\ {\varphi}_{\sigma}(u)& =\frac{1}{1+exp(-{\rho}_{\sigma}u)}.\end{array}

(7)

In system (6a)–(6d), function {\varphi}_{\sigma} is applied to

u=\frac{1}{N}\sum _{i=1}^{N}{\mathrm{Ca}}_{i}-{\mathrm{Ca}}_{\mathrm{desyn}},

which is the difference between the mean calcium level and the desynchronization threshold {\mathrm{Ca}}_{\mathrm{desyn}}. For each j=1,\dots ,N, subsystem (6a)–(6c) (of the same type as system (1a)–(1c)) represents the activity of the *j* th cell. The values of parameters {k}_{j} are chosen randomly using a uniform distribution in the interval [0.8,1.2] to reproduce, as explained in Sect. 3, the variability in the IPI and height of the peaks from one cell to another. The values of the parameters that have been already introduced in Sect. 3 are given in Table 1. Variable *σ* represents a global state of the network and acts on each cell through the term {\eta}_{j}{\varphi}_{\mathrm{syn}}(\sigma ). Its dynamics consists of a very slow linear part (*ε* and *δ* are assumed to be small) and a term that depends on the level of synchronization of the network and acts as a reset mechanism when the network is sufficiently synchronized.

Note that the individual cells ({x}_{j},{y}_{j},{\mathrm{Ca}}_{j}) are coupled only through variable *σ* which depends on the mean calcium concentration. This coupling is different from the one used in most synchronization studies and creates a link between calcium synchronization and higher calcium peaks. Similar global coupling arises in coupled arrays of Josephson junctions [14] as well as in a model of the Belusov–Zhabotinsky reaction with global feedback [15]. However, the specific feature of our coupling is that it is active only during very short periods when the mean calcium level is high.

Parameter {\sigma}_{0} plays the role of a reset value and is chosen smaller than {\sigma}_{\mathrm{on}}. Functions {\varphi}_{\mathrm{syn}} and {\varphi}_{\sigma} are increasing sigmoidal functions with inflection points at {\sigma}_{\mathrm{on}} and 0, respectively, and are both bounded above by 1. Since they play the role of activation functions, parameters {\rho}_{\mathrm{syn}} and {\varphi}_{\sigma} are assumed to be sufficiently large. In the limit {\rho}_{\mathrm{syn}}\to +\mathrm{\infty} (resp. {\rho}_{\sigma}\to +\mathrm{\infty}), {\varphi}_{\mathrm{syn}} (resp. {\varphi}_{\sigma}) converges pointwise to the following Heaviside function with activation point {\sigma}_{\mathrm{on}} (resp. 0):

{\varphi}_{\mathrm{syn}}^{\mathrm{\infty}}(\sigma )=H(\sigma -{\sigma}_{\mathrm{on}})=\{\begin{array}{ll}0& \text{if}\sigma {\sigma}_{\mathrm{on}},\\ 1& \text{if}\sigma \ge {\sigma}_{\mathrm{on}}\end{array}

(8)

(\text{resp.}{\varphi}_{\sigma}^{\mathrm{\infty}}(u)=H(u)=\{\begin{array}{ll}0& \text{if}u0,\\ 1& \text{if}u\ge 0\end{array}).

(9)

### 4.1 Qualitative Study of the Network Model

We now explain how the model can reproduce the alternation of asynchronous phases and episodes of synchronization in the case when {\varphi}_{\mathrm{syn}} and {\varphi}_{\sigma} behave as the Heaviside functions (8) and (9), respectively. We refer to Fig. 4 for a visual help on the *σ* driven transition of a particular cell of the network from the independent regime to the synchronized regime. Let us consider an initial value of *σ* just above {\sigma}_{0}. While \sigma <{\sigma}_{\mathrm{on}}, {\varphi}_{\mathrm{syn}}(\sigma ) is almost zero and each cell (6a)–(6c) (for j=1,\dots ,N) acts as described in Sect. 3. Since the values of parameters {k}_{j} are different, each cell generates a {\mathrm{Ca}}_{j} pattern with its own IPI. As a consequence, the calcium peaks are asynchronous and, as time evolves, the mean calcium level among cells, given by \frac{1}{N}{\sum}_{i=1}^{N}{\mathrm{Ca}}_{i}, remains low. As long as the mean calcium level is smaller than {\mathrm{Ca}}_{\mathrm{desyn}}, the second term of the *σ* dynamics is negligible. Then, since *δ* is assumed to be small, *σ* increases very slowly. This regime corresponds to the orbit in blue shown in panel (a) of Fig. 4 and the blue parts of the time series in panels (c) to (f).

Once the mean calcium level exceeds the threshold value {\sigma}_{\mathrm{on}}, {\varphi}_{\mathrm{syn}}(\sigma ) activates. Let us consider a particular cell, i.e., system (6a)–(6c) for a particular *j*. When {\varphi}_{\mathrm{syn}}(\sigma ) is activated, the {y}_{j} nullcline quickly moves to the right and, provided that {\eta}_{j} is large enough, ends up intersecting the {x}_{j} nullcline on its right branch as shown on panel (b) of Fig. 4. Hence, as long as {\varphi}_{\mathrm{syn}}(\sigma ) is activated, the cell remains in a steady regime. The current point ({x}_{j},{y}_{j}) reaches the vicinity of a singular point on the right branch and remains stationary. Therefore, the corresponding calcium level is higher than usual. Provided that sufficiently many cells are recruited in this process, the mean level quickly becomes higher than {\mathrm{Ca}}_{\mathrm{desyn}}. This corresponds to the red parts of the curves in Fig. 4. Then the reset term of the *σ* dynamics activates, *σ* quickly decreases, crossing back the threshold value {\sigma}_{\mathrm{on}}, to a value near {\sigma}_{0}. Consequently, {\varphi}_{\mathrm{syn}}(\sigma ) is deactivated, and the whole process starts again.

It is worth noticing that all cells recruited in the event (i.e., those corresponding to a large enough value of {\eta}_{j}) were synchronized by the global variable to produce a higher calcium peak than usual. Moreover, they come back to their own pulsatile regime approximately at the same time, starting by a quiescence phase. Hence, all individual calcium levels are at the baseline for a while, before individual peaks rise again unsynchronized, which corresponds to a postexcitatory suppression.

### 4.2 Frequency of Synchronization Episodes

In Sect. 3, we have shown how to specify the parameters of individual cells to obtain the required time traces. In this section, we show how to control the network level parameters {\sigma}_{0}, {\sigma}_{\mathrm{on}} and *δ* to obtain global synchronization with a specified frequency. In Proposition 1, we prove that the evolution of *σ* depends on the ratio {\sigma}_{\mathrm{on}}/{\sigma}_{0} rather than on each of these parameters independently. Proposition 2 gives a formula for the dependence of the frequency of the synchronized peaks on *δ* and {\sigma}_{\mathrm{on}}/{\sigma}_{0}.

**Proposition 1** *For any given* \alpha >0, *the outputs* {\mathrm{Ca}}_{j} *of system* (6a)–(6d) *are invariant under the change of parameter values from* ({\sigma}_{0},{\sigma}_{\mathrm{on}},{\rho}_{\mathrm{syn}}) *to* (\alpha {\sigma}_{0},\alpha {\sigma}_{\mathrm{on}},\frac{{\rho}_{\mathrm{syn}}}{\alpha}).

*Proof* Changing the parameters from ({\sigma}_{0},{\sigma}_{\mathrm{on}},{\rho}_{\mathrm{syn}}) to (\alpha {\sigma}_{0},\alpha {\sigma}_{\mathrm{on}},\frac{{\rho}_{\mathrm{syn}}}{\alpha}) in system (6a)–(6d) yields

{x}_{j}^{\mathrm{\prime}}=\tau (-{y}_{j}+4{x}_{j}-{x}_{j}^{3}-{\varphi}_{\mathrm{fall}}({\mathrm{Ca}}_{j})),

(10a)

{y}_{j}^{\mathrm{\prime}}=\tau \epsilon {k}_{j}({x}_{j}+{a}_{1}{y}_{j}+{a}_{2}-{\eta}_{j}\overline{{\varphi}_{\mathrm{syn}}}(\sigma )),

(10b)

{\mathrm{Ca}}_{j}^{\mathrm{\prime}}=\tau \epsilon ({\varphi}_{\mathrm{rise}}({x}_{j})-\frac{{\mathrm{Ca}}_{j}-{\mathrm{Ca}}_{\mathrm{bas}}}{{\tau}_{\mathrm{Ca}}}),

(10c)

{\sigma}^{\mathrm{\prime}}=\tau (\delta \epsilon \sigma -\gamma (\sigma -\alpha {\sigma}_{0}){\varphi}_{\sigma}(\frac{1}{N}\sum _{i=1}^{N}{\mathrm{Ca}}_{i}-{\mathrm{Ca}}_{\mathrm{desyn}})),

(10d)

where the new function \overline{{\varphi}_{\mathrm{syn}}} is given by

\begin{array}{rl}\overline{{\varphi}_{\mathrm{syn}}}(\sigma )& =\frac{1}{1+exp(-({\rho}_{\mathrm{syn}}/\alpha )(\sigma -\alpha {\sigma}_{\mathrm{on}}))}.\end{array}

(11)

Changing *σ* to *ασ*, and using the relation

\begin{array}{rl}\overline{{\varphi}_{\mathrm{syn}}}(\alpha \sigma )& ={\varphi}_{\mathrm{syn}}(\sigma )=\frac{1}{1+exp(-{\rho}_{\mathrm{syn}}(\sigma -{\sigma}_{\mathrm{on}}))},\end{array}

(12)

one obtains precisely system (6a)–(6d) with former parameters ({\sigma}_{0},{\sigma}_{\mathrm{on}},{\rho}_{\mathrm{syn}}). □

*Remark 1* As explained at the beginning of the section, the value of {\rho}_{\mathrm{syn}} is chosen large enough so that the activation of {\varphi}_{\mathrm{syn}} is almost immediate. It is worth noticing that a change in this value, provided that it remains large (typically greater than 10), does not affect the qualitative features of the model outputs (periodic episodes of synchronization) and has a limited impact on the period between two successive episodes of synchronization. Proposition 1 shows that, for {\rho}_{\mathrm{syn}} large enough and a fixed value of *δ*, the period of the synchronization episodes depends mainly on the ratio {\sigma}_{\mathrm{on}}/{\sigma}_{0}. On the other hand, the value of these parameters can be chosen arbitrarily (provided that {\sigma}_{0}<{\sigma}_{\mathrm{on}}) and the synchronization period can be adjusted by choosing the value of *δ* as proved in Proposition 2.

**Proposition 2** *In the case* {\rho}_{\sigma}=\mathrm{\infty}, *for* *γ* *large enough relative to* *ε* *and* *δ*, *the period between two successive episodes of synchronization in system* (6a)–(6d) *is approximated by*

{T}_{\mathrm{syn}}=\frac{1}{\tau \epsilon \delta}ln\frac{{\sigma}_{\mathrm{on}}}{{\sigma}_{0}}.

(13)

*Remark 2* If the values of all the parameters of system (6a)–(6d), except *δ*, are fixed, we can adjust the synchronization period in the {\mathrm{Ca}}_{j} pattern to any value {T}_{\mathrm{syn}}>0 by choosing

\delta =\frac{1}{\tau \epsilon {T}_{\mathrm{syn}}}ln\frac{{\sigma}_{\mathrm{on}}}{{\sigma}_{0}}

*Proof of Proposition 2* As explained above, each episode of synchronization results in a decrease of *σ*, under the influence of the calcium dependent part of its dynamics. For a value of *γ* large compared to *ε* and *δ*, the *σ* dynamics, in the period when {\varphi}_{\sigma} is active, is much faster than the {\mathrm{Ca}}_{j} dynamics. Since {\rho}_{\sigma}=\mathrm{\infty}, *σ* decreases quickly down to a value very close to the singular point \overline{\sigma} of its dynamics defined by

\delta \epsilon \overline{\sigma}-\gamma (\overline{\sigma}-{\sigma}_{0})=0

i.e.,

\overline{\sigma}=\frac{\gamma {\sigma}_{0}}{\gamma -\delta \epsilon}={\sigma}_{0}+O\left(\frac{\epsilon \delta}{\gamma}\right).

Hence, the time {T}_{\mathrm{syn}} between two successive synchronization episodes is approximately given by the time needed for *σ* to increase from {\sigma}_{0} up to {\sigma}_{\mathrm{on}}. Let us recall that the cells are asynchronous during this phase and

{\varphi}_{\sigma}^{\mathrm{\infty}}(\frac{1}{N}\sum _{i=1}^{N}{\mathrm{Ca}}_{i}-{\mathrm{Ca}}_{\mathrm{desyn}})=0.

It follows that, for \sigma <{\sigma}_{\mathrm{on}}, the *σ* dynamics is given by its linear part: {\sigma}^{\mathrm{\prime}}=\delta \epsilon \sigma. By direct integration, one obtains

{\sigma}_{0}exp(\tau \epsilon \delta {T}_{\mathrm{syn}})={\sigma}_{\mathrm{on}}\phantom{\rule{1em}{0ex}}\iff \phantom{\rule{1em}{0ex}}{T}_{\mathrm{syn}}=\frac{1}{\tau \epsilon \delta}ln\frac{{\sigma}_{\mathrm{on}}}{{\sigma}_{0}}.

(14)

□