# 3. Selforganization in Mathematics

As early as 1952, the English mathematician Alan Turing published a paper entitled “ The chemical basis of morphogenesis” in which he suggests that:

*
"a
system of chemical substances, called morphogens, reacting together and
diffusing through a tissue is adequate to account for the main phenomena of
morphogenesis. Such a system, although it may originally be quite
homogeneous, may later develop a pattern or structure due to an instability
of the homogeneous equilibrium, which is triggered off by random
disturbances."
*

Here Turing already stated the main principles of selforganization. Furthermore,
he suggests that reaction-diffusion processes play an important role in morphogenesis.
He continues to illustrate his point with the aid of a mathematical model, a
set of differential equations the solution of which gives the development of the
concentration pattern of morphogens over time. This mathematical approach has been proven to be very valuable, because it provides a level of
abstraction on
which different dynamic systems can be compared with each other. For instance,
Troy (1978) showed that the dynamics of the Belousov-Zhabotinski
reaction and
the dynamics of an action potential according to the model of Hodgkin and
Huxley are mathematically closely related. More important
is however, that
where thermodynamics cannot tell us what happens after a dynamic
system has
become unstable (**Part
2**), mathematics can!
Differential equations describe
the behaviour of a dynamic system
in a deterministic way, without
referring
to
the microscopic fluctuations, which were shown to be so important
for the
onset of selforganization. Then
how can these deterministic
equations provide
a
meaningful model for selforganization? The answer is
simple: analytical
methods to solve the equations can no longer be applied and
numerical methods
are used. The computational inaccuracies implicit in these methods play
the role
of
fluctuations, perturbing the system from an unstable state and leading it away.
Since Turing, a lot of theoretical models have been
developed explaining
pattern formation during development by processes of
selforganization (Reviews:
Brenner *et al., *1981*; *Meinhardt, 1978). Experimental evidence for
the involvement
of
selforganization in development is still confined to the
**aggregation of****
**
**slime molds** (Robertson and Cohen, 1972;
Tomchik and Devreotes, 1981).

Up till now we have mostly dealt with structures in 'real space'. It
is important to realize that selforganizing processes can also
explain the formation of 'time structures', of which a periodic oscillation
is a simple example. An extensive review of "the geometry of biological time" is
given by Winfree (1980). A dynamic system, governed by simple (nonlinear)
kinetics, is able to display very complicated behaviour: jump transitions
between several steady states, simple and compound oscillations,
non-periodic oscillations *('chaos'),
*bursts (Rossler and Wegman.
1977).

An
important point is, that this complex behaviour is not a property of a small
class of exotic dynamic systems: nearly all nonlinear dynamic systems will
have interesting (chaotic, periodic) regions in their 'parameter space'
(Helleman,1980). There are two important classes of dynamic systems:
*Hamiltonian *systems in which there
are no terms equivalent to 'friction' and motion is pertained indefinitely
without external drive (the harmonic oscillator is an example) and
*dissipative *systems which must be
continuously driven to prevent motion to die out. Only dissipative systems
are relevant for our discussion. Dissipative systems differ from Hamiltonian
ones, in that they possess *
'attractors': *modes of behaviour to which neighbouring solutions are
attracted. This property is a direct consequence of dissipation and it
ensures stability of a certain type of behaviour: any perturbation away from
it, will eventually die out. Behaviour on an attractor can be simple
(equilibrium or steady state; point attractor), periodic (limit cycle or
toroidal attractor) or *chaotic *
('strange attractor'). Although the behaviour of a system on such a strange
attractor may look quite erratic, it represents a highly ordered structure:
the dynamic variables which describe the behaviour of the system are not
allowed to take on any possible value, but instead the behaviour of the
system is limited to a specific volume in
*phase space *(the space spanned by
all the dynamic variables). Inside this volume resides the strange
attractor, which is a *fractal *(cf.
Mandelbrot,1983 for an extensive discussion of fractals): a scale-invariant
structure, consisting of a single trajectory (the
path a system follows in
phase space) which never closes on itself, filling
the restricted phase
volume inhomogeneously (cf. Abraham and Shaw, 1983 for a 'visual'
introduction to *chaos *and strange
attractors).

The behaviour of dissipative dynamic systems can be switched between the
different (stationary, periodic, chaotic) modes by changing the value of a
nonlinearity parameter (usually a 'friction' parameter regulating the level
of dissipation). When the value of such a parameter is gradually increased,
a series of 'bifurcations' is observed. In a bifurcation one at tractor
becomes unstable, while at the same time another attractor (possibly of a
completely different type) appears. As a result, the system changes its
behaviour abruptly. The following scenario is often found. First the system
is stable: any perturbation from this steady state will die away. Increasing
the parameter gradually changes the state, until at a critical value of the
parameter the steady state becomes unstable. At this point the system
'bifurcates': it changes its behaviour abruptly to an oscillatory regime,
which is again stable against perturbations. Further increase of the
parameter leads to a succession of bifurcations in which the complexity of
the oscillations increases. These '**period doublings**' follow each other ever
faster, until at another critical value of the parameter, a
*
chaotic *regime is entered. Here
the behaviour of the system can hardly be distinguished from a stochastic
one.

Depending
on the type of system this chaotic behaviour can be more or less oscillatory
in nature (reviews: Schuster, 1984; Holden, 1986; Olsen and Degn, 1986).
This route to chaos via a converging sequence of period doublings has some
very universal characteristics (Cvitanovic, 1983). There are at least two
alternative routes (Eckman, 1981): a short one with chaos after only three
bifurcations (Ruelle and Takens, 1971) and a route via 'intermittent'
behaviour (Pommeau and Manneville, 1980; Manneville and Pommeau, 1980). When
the parameter is further increased in the chaotic region, a number of
'windows' may appear where the behaviour is again periodic. Clearly, the
system can be made to behave (pseudo-)stochastically or deterministically,
simply by changing an external parameter
value.

In
summary, most nonlinear dynamic systems are capable of producing a wealth of
'time-structures' including chaotic ones.
Transitions among different types of behaviour appear very suddenly
and are governed by the value of one or more parameters relating to the
level of dissipation. When spatial dependence and diffusion are taken into account,
the spontaneous formation of ordered spatial structures out of an initially
homogeneous state is possible. Both temporal and spatial aspects of
selforganization seem to be relevant to brain function and development.