Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 237

where 1isthe

ratio between the typical time scales of the two variables, a are random point-specific
numbers greater than one, D is the diffusion coefficient of the activator (i.e., of

φ 1 and φ 2 are the fast and slow variables, respectively, = τ φ 1 /τ φ 2

φ 1 ),

ξ gn ( t ) is a Gaussian white (in space and time) noise with unit intensity, and I is a

parameter modulating the noise intensity. When a 1, the deterministic uncoupled

version of the FitzHugh-Nagumo model describes an excitable dynamical system,
characterized by the fact that, when the steady state is weakly perturbed, the system
does not exhibit a monotonic return to the steady state, but a strong transient growth
occurs before relaxation to the rest state (see Chapter 3). Thus, when a

>

> 1, all points

in the domain behave as independent, uncoupled excitable oscillators.
Figure 5.49 shows some examples of numerical solutions of ( 5.103 )and( 5.104 ).
The first column shows the case with weak noise: Excursions are seldom activated
at random points of the field and, because of the spatial coupling, they propagate as
waves across the domain. In this case the φ 1 variable in different cells is correlated

on only a short time scale dictated by the wave celerity, and no correlation occurs
between distant cells. If the noise strength is increased, a pattern instead becomes
evident: Firing (i.e., excursions) of distant cells occurs almost in phase, and a spatial
coherence emerges. Notice that the pattern is not steady but periodical: The field
alternates homogeneous states, when all oscillators are at the peak of the excursion,
and patterned states, when islands of firing elements emerge from a basic rest state.
Finally, the third column shows what happens when the noise is even stronger. In
this case, excursions are continuously activated in the field, no particular patterns
emerge, and the field appears to be random. Thus Fig. 5.49 shows that there is an
optimal noise level able to give rise to stochastic long-range synchronization in the
space-time dynamics of individual cells.
In the third chapter, temporal stochastic coherence was shown to also emerge when
a dynamical system close to a Hopf's bifurcation is randomly perturbed. In this
case the deterministic system has a stable steady state, but it is not necessarily an
excitable system. The role of noise is to induce the system to temporarily cross the
bifurcation point, thereby allowing the dynamics to stochastically explore the limit
cycle. In this way, the noise acts on the system as a precursor of the limit cycle and
unveils the existence of a periodic behavior already in subthreshold conditions before
the bifurcation point is reached. For suitable (intermediate) noise intensities, random
perturbations and the deterministic periodicity associated with the limit cycle can then
cooperate to generate a regular temporal pattern.
A similar behavior can occur also in the spatiotemporal version of the system. Even
in this case the deterministic system is in a (now spatially) homogeneous stable state
close to a bifurcation point where the homogeneous state becomes unstable. When
the control parameter crosses a given threshold, the system evolves deterministically
toward a patterned state. Similar to the temporal case, the effect of noise is to dis-
turb the stable state and to temporarily drive the system across a bifurcation point,