The system would in principle be 3-dimensional (cells in a living body), but for simplicity we will consider a population of oscillators placed on a one dimensional space. These oscillators are Hopf oscillators, which exhibit a stable limit cycle.\\

\tab - The dynamic of the protein field produced by these oscillators is controlled by :

•  the presence of sources
•  a degradation process
•  a diffusion process\newline

The evolution of its concentration is described by the following equation :

$latex \displaystyle{\Large \frac{\partial A}{\partial t}(x,t)=\alpha Z(x,t)-\gamma A(x,t)+D\frac{\partial^2 A(x,t)}{\partial x^2} }$

where $latex \displaystyle{\Large Z(x,t)=\sum_{i} Z_{i}(t)\delta(x-x_i)}$

In turn, the oscillators evolve according to the Hopf equation, plus a term that accounts for the coupling with the protein concentration field with a possible delay :

\newline

$latex \displaystyle{\Large \cases {
     \dot Z_i(t) = (\mu_i+i\omega_i) Z_{i}(t) - Z_i(t)|Z_i(t)^2|+e^{i\theta}A(x_{i},t)\cr
      Z(x,t)=\sum\limits_i Z_i(t) \delta (x-x_i)}}$

This accounts for the fact that the oscillators are sensitive to the phase of their neighbours. If the neighbours of a given oscillator are at their maximal amplitude, the concentration around this oscillator will be high and will tend to accelerate, and vice-versa.\\

Equations \ref{3} and \ref{4} form a coupled system describing the population of the cells. The physical problem is to find the steady states of this system, to understand if the cells can synchronize and how they behave as a group.