24 1. HEURISTICS

is a function of the external noise intensity, with a maximum of captures of more

distant plankton at some optimal external noise intensity. If experimentally noisy

electric signals improve the sensitivity of the electroreceptors, nature itself should

also provide sources of noise. In [95] it was conjectured that, besides the signal,

such a noise might be produced by the populations of prey animals themselves.

In [42] this conjecture was confirmed by measurements of the noise strength pro-

duced by single Daphnia in the vicinity of a swarm. In the simple quantitative

approaches, quality of tuning is measured by Fisher information, a concept that

may be comparable to the entropy notions in Chapter 3, Section 3.2.

1.5.4. The FitzHugh–Nagumo system. A more detailed modeling of neu-

ral activities of living systems underlies this well known and studied example. It

deals with action potentials and electric currents transmitted through systems of

ion channels provided by the axons in neural networks, triggered by their mutual

interaction and the interaction of the system with the biological environment. Neu-

rons communicate with each other or with muscle cells by means of electric signals.

Each single neuron can be modeled as an excitable dynamical system: in the rest

state characterized by a negative potential gap with respect to the extracellular

environment, no current flows through the membrane of the neuronal cell. If this

threshold potential barrier disappears due to noisy perturbations created by the

environment (neighboring cells, external field), ion channels through the membrane

are opened and currents appear in form of a spike or firing, followed by a deter-

ministic recovery to the rest state. During a finite (refractory) time interval, the

membrane potential is hyperpolarized by the current flow, and any firing impossi-

ble. The theory that captures the above-mentioned features of neuronal dynamics,

including the finite refractory time, is described by the FitzHugh–Nagumo (FHN)

equations (see Kanamaru et al. [63]). In the diffusively coupled form, a system of

N coupled neurons is described by the system of equations (see [63])

τ ˙ u

i

(t) = − vi + ui −

ui

3

3

+ S(t) +

√

ε

˙

W

i

(t) +

1

N

N

j=1

(ui − uj),

˙i(t) v = ui − βvi + γ.

Here ui describes the membrane potential of neuron i, vi a variable describing

whether and to which degree neuron i is in the refractory interval of time after

firing, S describes an external periodic pulse acting on the potential levels, while

W1,...,Wn is a vector of independent Brownian motions. Finally, τ, β, ε and γ

are system parameters. In the infinite particle limit, the system becomes a sto-

chastic partial differential equation. Roughly, total throughput current will be a

function of the model parameters, and stochastic resonance appears as its optimal

value for a suitable parameter choice (synchronization). The paper by Wiesenfeld

et al. [109] reports about a much simplified form of this system, in which action

potentials of single mechanoreceptor cells of the crayfish Procambarus clarkii are

concerned. The mechanosensory system of the crayfish consists of hairs located

on its tailfan, connected to mechanoreceptor cells. Streaming water moves the hair

and so provides the external excitation that causes the mechanoreceptor cell to fire.

Experimentally (see [109]), a piece of tailfan containing the hair and sensory neuron

was extracted and put into a saline solution environment. Then, periodic pressure