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