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3.3.3. Response to external noise: quantized cycling time
The presence of multistability inuences the response of the system to external
stimuli, in particular noise. This response can be modeled by substituting Eq. (3.12)
above by:
d!
i
dt
= "(
4
g(u
i
)!
i
) + 2d(!
e
!
i
) +
i
(t) :
(3.14)
Let us consider the case when all oscillators are conned to the oscillatory region.
In order to establish the eect of noise in a population of such genetic units, we
quantify the histogram of cycling times, analogous to the inter-spike interval (ISI)
histograms used in studies of neural dynamics. We nd that noise contributes to
the establishment of variability and leads to multiple frequencies [Fig. 3.12(a,b)],
even when the oscillators are initially synchronized. The cycling is now quantized,
having either a bimodal [Fig. 3.12(a)] or a polymodal [Fig. 3.12(b)] distribution
of periods. Thus, choosing slightly dierent
1
values, one can eectively switch
between dierent multipeak distributions. The ISI peaks observed are determined
by the probability density to nd phase points near the jumping threshold between
the stochastic version of the attractors revealed by the bifurcation analysis above
[Koseska et al. (2007a)]. The modes in the polymodal histogram might be separated
by almost equal intervals if one of the stochastic attractors dominates over the
others, or by dierent intervals in the opposite case. The same interplay between
attractors disrupts the exponential decay of the peak amplitudes that is typical for
a noisy attractor under the inuence of a periodic signal [Longtin (1995)].
These results indicate that the interplay between intercell signaling and stochas-
ticity might explain the emergence of quantized cycles, a concept that is central in
the research of time-dependent biological processes, such as the cell cycle [Lloyd and
Volkov (1990)]. Clear experimental evidence for quantized cycles has been obtained
0.015
0.015
(a)
(b)
0.01
0.01
0.005
0.005
0
0
0
500
1000
1500
2000
0
500
1000
1500
2000
T
Fig. 3.12. (a) Bimodal ISI distribution for 8 identical oscillators (
1
= 3:3), and (b) polymodal
ISI distribution (
1
= 3:328). The noise intensity is 5
10
7
.
