We consider the case where both ϵ 1 and ϵ 2 are small, that is, a strong time-scale separation between the fast variable v ϵ ∈ R p and the slow one w ϵ ∈ R q, and a fast periodic modulation of the fast drift F ( v, w, ⋅ ).
A Simulated timeseries for the fast variable x and the slow one z.
For system (1), each v plays the role of the fast variable f from (A1 while the other variable linked to v is the slow variable s.
Then a simplified equation, which is independent of the fast variable and possesses the essential features of the system, is highly desirable.
The second variable receives feedback from the fast one and exploits the hysteresis present in the fast variable to create a limit cycle in the plane (( x,y )) (relaxation oscillations).
With the separation of time scale, we can view the state variable of the system as being divided into two parts, the "slow" variable (X^{epsilon }_{t}) and the "fast" variable (Y^{epsilon }_{t}).
For each bursting class found in the unfolding we produced the bifurcation diagram of the fast subsystem using z as bifurcation parameter and plotting the fast variable x as ordinate.
In other words, we are interested in the evolution of the connectivity variable, which evolves on a slow time-scale, under the influence of the external input and some noise added on the fast variable.
Hence, it is advantageous to express the slow flow in terms of only one of the slow variables, m SO or h SI, together with the fast variable V. We choose to work with V and m SO.
The curve L+ is projected vertically (along the fast variable V) onto the lower sheet to obtain the projection curve P(L+), and similarly for the (L-) projection onto the upper sheet.
