1.5. STOCHASTIC RESONANCE IN MODELS FROM ELECTRONICS TO BIOLOGY 21
the resonance point of the diffusion with time periodic landscape U. Let us note
that for v−(t) =
V +v
4
+
V −v
4
cos(2πt) and v+(t) = v−(t + π), which corresponds
to the case of periodically switching wells’ depths between
v
2
to
V
2
as in the frozen
landscape case described above. Then the optimal tuning is T (ε) = exp(
μR
ε
) with
μR =
v+V
2
. This optimal rate is equivalent to the optimal rate given by the SPA
coefficient.
The big advantage of the quality measure based on the transition times is its
robustness. Let us therefore consider the reduced model consisting in a two-state
Markov chain with the infinitesimal generator
Q(t) =
−ϕ(t) ϕ(t)
ψ(t) −ψ(t)
,
where ϕ(t) = exp(−
v−(t/T )
ε
) and ψ(t) = exp(−
v+(t/T )
ε
). The distribution of transi-
tion times of this Markov chain is well known (see Chapter 4) and, divided by the
period length, converges to aμ. i The reduced dynamics of the diffusion is captured
by the Markov chain, and the optimization of the quality measure Mh(ε, T ) for the
Markov chain and the diffusion leads to the same resonance points.
Our investigation focuses essentially on two criteria: one concerning the family
of spectral measures, especially the spectral power amplification coefficient, and the
other one dealing with transitions between the local minima of the potential. Many
other criteria for optimal tuning between weak periodic signals in dynamical systems
and stochastic response can be employed (see Chapter 3). The relation between long
deterministic periods and noise intensity usually is expressed in exponential form
T (ε) = exp(
μ
ε
), since the particle needs exponentially large times to cross the barrier
separating the wells. This approach relies on the basic assumption that the barrier
height is bounded below uniformly in time. This assumption which seems natural in
the simple energy balance model of climate dynamics may be questionable in other
situations. If the barrier height becomes small periodically on a scale related to the
noise intensity, the Brownian particle does not need to wait an exponentially long
time to climb it. In this scaling trajectories may appear periodic in the small noise
limit. The modulation is assumed to be slow, but the time dependence does not
have to be assumed exponentially slow in the noise intensity. In a series of papers
[8, 9, 10, 11, 12] and in a monograph [13], Berglund and Gentz study the case in
which the barrier between the wells becomes low twice per period: at time zero the
right-hand well becomes almost flat and at the same time the bottom of the well
and the saddle approach each other; half a period later, the scenario with the roles
of the wells switched occurs. Even in this situation, there is a threshold value for
the noise intensity under which transitions are unlikely and, above this threshold,
trajectories typically exhibit two transitions per period. In this particular situation,
optimal tuning can be described in terms of the concentration of sample paths in
small space-time sets.
1.5. Stochastic resonance in models from electronics to biology
As described in the preceding sections, the paradigm of stochastic resonance
can quite generally and roughly be seen as the optimal amplification of a weak pe-
riodic signal in a dynamical system triggered by random forcing. In this section, we
shall briefly deviate from the presentation of our mathematical approach of optimal
tuning by large deviations methods, illustrate the ubiquity of the phenomenon of
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