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These two parameters β and σ directly control the amount of non-verbal influence be-
tween the agents: they must be high enough to enable coupling, for instance reducing
initial phase-shift between oscillators or compensating phase deviation when u 1
= u 2 .
- The initial phase shift Δφ ini , which makes agents start with a phase shift between
S 1 ( t ini ) and S 2 ( t ini ) at the beginning of each test of the architecture.
Finally, the variables recorded during these tests are the internal states of both agents,
S 1 ( t ) and S 2 ( t ) (see Fig.6 for an example).
Fig. 6. Activations recorded for u 1 =0 . 01, u 2 =0 . 011, β =0 . 85, σ =0 . 05 and Δφ ini =0 . 4.
Despite the initial phase shift and the phase deviation, the two agents synchronise. This is a stable
state of the dyad, it remains until the end of the experiment (5000 time steps).
3.2
Test of Synchrony Emergence
For a given set of parameters, to determine if in-phase synchronisation occurred be-
tween agents, we used a procedure described by Pikovsky, Rosenblum and Kurths in
their reference topic “Synchronisation” [21]. This procedure consists in comparing the
phases of two signals to determine if they are synchronous or not.
First we used the fact that relaxation oscillators can be characterised by their peaks.
There is a peak at time t k when S i ( t k )
0 . 9 β and S i ( t k +1)=0 . Then, we used the
fact that phase can be rebuilt from these peaks [21]. We assign to the time t k the values
of the phase φ ( t k )=2 πk , and for every instants of time t k <t<t k +1 determine the
phase as a linear interpolation between these values (see fig.7):
t − t k
t k +1
φ ( t )=2 πk +2 π
(10)
t k
After that, when the phases of signals are obtained, we consider their difference modulo
2 π (see fig.8). Horizontal plateaus in this graph reflect periods of constant phase-shift
Fig. 7. Signal, Peaks and Phase. In the upper part of the graph, there is the original signal S 1
(shown in fig.6) and the associated re-built phase (we can notice the change of phase slope when
synchronisation occurs). In the lower part of the graph, there are the peaks extracted from S 1 in
order to re-build the phase.
 
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