How fast do structures emerge in hypercycle-systems?

Abstract
A general framework for the simulation of reaction-diffusion systems with probabilistic cellular automata is presented. The basic reaction probabilities of the chemical model translate directly into the transition rules of the automaton, thus allowing a clear comparison between simulation results and analytic calculations. This framework is then applied to simulations of hypercycle-systems in up to three dimensions. Furthermore, a new measurement quantity is introduced and applied to the hypercycle-systems in two and three dimensions. It can be shown that this quantity can be interpreted as a measure for the macroscopic order of the hypercycle systems.

arXiv:adap-org/9806003v1 5 Jun 1998
How fast do structures emerge in
hypercycle-systems?
Stephan Altmeyer, Claus Wilke, and Thomas Martinetz
Institut f¨ ur Neuroinformatik
Ruhr-Universit¨ at Bochum
D-44780 Bochum, Germany
Stephan.Altmeyer@neuroinformatik.ruhr-uni-bochum.de
November 19, 2018
Abstract
A general framework for the simulation of reaction-diﬀusion sys-
tems with probabilistic cellular automata is presented. The basic re-
action probabilities of the chemical model translate directly into the
transition rules of the automaton, thus allowing a clear comparison
between simulation results and analytic calculations. This framework
is then applied to simulations of hypercycle-systems in up to three di-
mensions.
Furthermore, a new measurement quantity is introduced and applie d to
the hypercycle-systems in two and three dimensions. It can be sho wn
that this quantity can be interpreted as a measure for the macros copic
order of the hypercycle systems.
1 The automaton
In the past a lot of work was done in the ﬁeld of algorithmic che mistry [1, 2]
and chemistry on cellular automata (CA) [3, 4]. Our goal is to give a com-
plete framework which begins with the determination of the t ransition rules
from basic reaction probabilities and ends with the compari son between the
emergence of spatially ordered structures and a macroscopi c measurement
quantity in hypercycle-systems.
In this paper we can only give a brief introduction of our prob abilistic CA
with asynchronous updating. Our aim was to establish a one-t o-one corre-
spondence between real chemical reactions (deﬁned by their reaction prob-
abilities) and the simulations on a CA (deﬁned by the transit ion rules).
Therefore, we explain the calculation of these transition r ules for a simple
particle production process in detail and then present the t ransition rules
for more complex reaction types without derivation.

1.1 Particle production processes
In our system, particles can be produced through spontaneou s or induced
reactions. The result of both reactions is the change of an em pty cell to an
occupied one. Since we evaluate production processes for em pty cells, we
will usually have the situation that several diﬀerent reacti ons are possible,
resulting in a diﬀerent ﬁnal state for the previously empty ce ll. If we think,
e.g. of spontaneous self-replication, all the nearest neig hbors of the empty
cell are in principle equally likely to reproduce into this c ell. In chemistry
such symmetry breaking phenomena are induced by statistica l ﬂuctuations,
e.g. temperature gradients.
Let pi, i ∈ A , be the probability for a single species i to replicate into the
empty cell by a special reaction, e.g. self-replication. He re A denotes an
index set containing all indices which are necessary to enum erate all species
in the given context and Ni will denote the number of molecules of species
i.
In the case there are several nearest neighbor cells occupie d, we ﬁrst have
to calculate the total probability ptot that at least one arbitrary nearest-
neighbor-molecule will occupy the empty cell. We get
ptot = 1 −
∏
i∈A
(1 − pi)Ni. (1)
From ptot we can calculate the eﬀective probabilities for the nearest n eighbor
molecules to replicate in the given arrangement of species. We will denote
these probabilities with peﬀ
i . In this calculation we have to make sure that
the ratio pi/p j is equal to peﬀ
i /p eﬀ
j , ∀i, j ∈ A . Since the sum over all pi is
not necessarily 1, we introduce a normalization factor ∑
i∈A Nipi.1 We get
peﬀ
i = pi
ptot
∑
i∈A Nipi
. (2)
Note that ptot is the sum of Nipeﬀ
i over all i.
1.1.1 Type I reactions
Reactions of type I,
Mi
pi,m,n
− − − →Mm + Mn,
represent the group of spontaneous reactions with two react ion products.
After some work and with the calculations done above we get:
peﬀ
i,m,n= pi,m,n
1 − ∏
i∈A(1 − pi,m,n)Ni
∑
i∈A Nipi,m,n
. (3)
1The normalization factor is introduced to ensure that the su m over all peﬀ
i and 1 − ptot
is 1.

The corresponding transition rule then reads:
A cell with state zero will change to state n and a nearest-neighbor cell with
state i will change to m with the probability peﬀ
i,m,n.
1.1.2 Type II reactions
Reactions of type II,
Mi + Mj
pi,j,m,n,o
− − − − − →Mm + Mn + Mo,
describe a group of two-body reactions with three reaction p roducts. Obvi-
ously, such reaction processes are more diﬃcult to handle si nce we have to
implement the determination of the correct reaction partne r. The related
quantities are primed.
After some calculations we get:
peﬀ
i,j,m,n,o= p′
i,j,m,n,o
1 − ∏
i∈A(1 − p′
i,j,m,n,o)N ′
i
∑
i∈A Nip′
i,j,m,n,o
. (4)
with
p′
i,j,m,n,o= p(N ′ ̸= 0) N ′
j
N ′ pi,j,m,n,o. (5)
p(N ′ ̸= 0) denotes the probability that there is at least one molecu le to
react with. For the derivation of p′
i,j,m,n,owe have implicitly used the idea
of an attractive force between the molecules which means tha t the reactive
molecule therefore ”jumps” to a reaction partner. On the oth er hand, if we
do not assume such an attractive force, which means a molecul e can also
”jump” into a hole, we have to substitute p′
i,j,m,n,owith N ′
j/N ′
max.
Then we can write down the transition rule:
The probability that a cell in state zero, with a nearest-neig hbor cell in state
i and with a ”reaction” cell in state j will change to o, while the nearest-
neighbor cell changes to m and the ”reaction” cell changes to n, is peﬀ
i,j,m,n,o.
1.1.3 Particle decay
The decay process,
Mi
pdec
i
− − →0,
can change an occupied cell into an empty one. The transition rule descrip-
tion is rather easy:
The probability that a cell in state i will change to a cell in st ate zero is pdec
i .

1.2 Diﬀusion
The diﬀusion is an additional process. Since we use an asynchr onous au-
tomaton, an often used diﬀusion algorithm for CA [5] is not nec essary.
Instead, we use a more natural algorithm that can be divided i nto three
subprocesses:
1. An arbitrary cell of the CA is chosen.
2. One of it’s nearest neighbor cells is determined at random .
3. The states of the two cells are interchanged.
1.3 Automaton update
We use asynchronous updating, since this seems to be the more natural way
[6]. The basic processes are then executed in the following w ay:
Choose an arbitrary cell at random. If the cell is occupied we use the decay-
transition rule. Otherwise the transition rule for reactio ns of type I or of
type II or a combination of both are used.
The reaction processes and the diﬀusion process are then comb ined to the
full simulation in the following way. Let k be the total number of cells in
a CA and d the ratio of the number of basic processes and the number of
diﬀusion steps. Then a CA-time-step t consists of k basic processes and dk
diﬀusion steps. It is important that the basic processes and t he diﬀusion
steps are mixed up very well.
Note that t is independent of the system’s size, since we scale the numbe r
internal processes with it.
2 Simulation
We have simulated systems with (symmetrical) hypercycles [ 7, 8], which
means we set
pi,j,k=
{
pself if i = j = k,
0 else (6)
and
pi,j,m,n,o=
{
pcat if i = m = o, j = n and i = j + 1 (cyclic) ,
0 else (7)

2.1 Stable patterns
In this part of the paper we will show the well-known spiral wa ves [4] in
two dimensions and the emergence of a stable scroll wave stru ctures [9] in
three dimensions. In the latter case we have used a two-dimen sional spiral
as initial condition. We will see the system going through di ﬀerent unstable
patterns. For reasons of clear visualization, in the pictur es we show only
one of the six species present in the simulation.
The three-dimensional analog of a spiral wave is a scroll wav e [9]. As we
can see in Fig. 3, even in the two dimensional case, structure s can be very
complex. Since there are more degrees of freedom in three spa tial dimen-
sions, the structures arising from non-ordered initial con ditions are much
more complex. Therefore, we use a 2d spiral as a symmetry-bre aking initial
condition for our simulations to simplify the emergent stru ctures.
We can see the spiral’s growth in the z-direction (see Fig. 1, 2). As a matter
of fact, the reproduction activity of a species becomes high if it is adjacent to
it’s ”supporter”. Therefore, in a layer above, molecules ar e ﬁrst created over
the small region where their own and their ”supporter” are pr esent. Since
the spirals rotate in the ”supporter” direction, we can see t he phenomenon
that the new spirals, growing on top of the old ones, are phase shifted in
the rotation direction. This continues until the space in z- direction is oc-
cupied. The resulting structure looks like a rotating screw . When all the
available space is occupied, the molecules cannot replicat e freely any more.
Since space is rare than, the molecules have to compete for em pty space to
replicate. For this reason, the molecules begin to arrange t hemselves in a
diﬀerent way so that the ”free” screw begins to change to a cone structure.
This cone structure is stable so that a real scroll wave canno t be seen.
The whole re-arrangement process takes much more time than t he occupa-
tion at the beginning. The pictures show the emergence of the cone struc-
ture. At the top of the cone the emergence of other small cones can be seen
so that the whole structure becomes more complex.
2.2 A microscopic measurement quantity
Now we will introduce a measurement quantity with which we ca n measure
the arrangement of the molecules in our automaton. The deﬁni tion of this
order-quantity, called α k, is rather easy:
For every molecule, we take the nearest neighbor molecules. If an adjacent
molecule is a ”supporter”, the variable S is incremented by one. Otherwise
the variable N is incremented by one. If a cell is not occupied, nothing is
done. With these microscopic values we can deﬁne the macrosc opic quantity

Figure 1: The ﬁrst two pictures show the system’s behavior sh ortly after the
”injection” of the 2d spiral. We can see the emergence of a rot ating screw.
The left picture was taken at t = 150 and the right one at t = 250.
Figure 2: If the space is nearly completely occupied the mole cules cannot
replicate freely and have to compete for empty space. Theref ore the struc-
ture changes from a screw to a cone-like pattern.

Figure 3: The three pictures show the time dependent behavio r of the sys-
tem. The times when the pictures have been taken are (from lef t to right):
t = 0, t = 120 and t = 9000. See also Fig. 4 and text for further information.
α k:
α k = kS
N + kS , (8)
where k ≥ 0 is a tuning constant. The larger k, the more sensitive is α k on
S.
Since both N, S ≥ 0, the quantity α k ranges between 0 and 1. We will later
see that α k is even constant for large t. (Obviously, if α k is constant, α k′,
with another k′ ̸= k, is also constant.)
Moreover, since we do not take into account empty cells, α k is independent
of the system’s concentration and therefore independent of pdec.
For our simulations we have used k = 10 and applied α 10 to both the two
and three dimensional case.
Note that there are a lot of possible microscopic/macroscop ic quantities
which all can describe the system from diﬀerent points of view .
2.3 Two dimensions
For the two dimensional case we have used two diﬀerent initial conditions.
In both cases the parameter set is pdec = 0 . 1, pself = 0 . 05, pcat = 0 . 9 and
d = 1.
First, the hypercycle-system was initialized at random, se e Fig. 3. We
have seen α 10 converging rapidly to a constant value at the very beginning .
After t = 120, however, α 10 changes only in very small amounts. In compar-
ison, the global structure of the system also changes very sl owly, see Fig. 4.
It seems that the system’s rough structure is determined at t he very begin-
ning and will then only be reﬁned, later on. This is very inter esting, since
there are a lot of possible stable patterns. However, the dec ision is made
very early.
For the asymptotic value of α 10 we have found α 10 ≈ 0. 72, independent

Figure 4: The picture shows the concentrations of the six hyp ercycle species
(solid lines) and the value of α 10 versus time for a system with random initial
condition.
Figure 5: The three pictures show the time dependent behavio r of the sys-
tem. The times when the pictures have been taken are (from lef t to right):
t = 0, t = 140 and t = 550. See also Fig. 6 and text for further information.
of the number of species in the hypercycle and of the system’s size. Fur-
thermore, α k depends only weakly on d for d < 1. The next step was to
examine a system where only one stable ﬁnal pattern exits. Th erefore, we
have used a special initial condition which is displayed in F ig. 5. The ﬁnal
structure will be of course a single rotating spiral.
What we have seen was α 10 changing comparably to the transformation of
the triangular structure to the spiral structure. Since the major part of this
transformation is done at the beginning, α also increases fast during the ﬁrst
simulation steps and then increases slower and slower. The s ystem behaves
similar to the case with random initialization, although th e convergence of
α k to it’s asymptotic value is signiﬁcantly slower. But what is more surpris-
ing, the asymptotic value of α 10 is again the same, α 10 ≈ 0. 72.

Figure 6: The picture shows the concentrations of the six hyp ercycle species
(solid lines) and the value of α 10 versus time for a system with ordered
triangles as initial condition.
We will now go a step beyond and apply α 10 on our three dimensional sim-
ulator.
2.4 Three dimensions
As we have seen above (Fig. 1, 2) in three dimensions there are several
diﬀerent unstable structures, beginning with a single two di mensional spiral
as initial condition and ending with a stable cone-structur e.
In Fig. 7 we can see α 10 starting with α 10 ≈ 0. 72. But then we see α k
increasing in the time interval when the spiral occupies the empty space by
growing in the z-direction. After that, when the available s pace is occupied,
new boundaries are present and α 10 decreases to the former value.
This is a bit surprising since the cone structures are quite d iﬀerent and at the
ﬁrst glance there seems to be no reason why the macroscopic or der should
be similar.
3 Conclusions
In this paper we have shown pictures of the transformation fr om two di-
mensional patterns to three dimensional ones and have there fore seen the
system walking through diﬀerent states of arrangement. But w hat is more
interesting is the fact that our macroscopic quantity α k (here: α 10) is quite
constant after a short time and has even the same asymptotic v alue in dif-
ferent settings.
In all our simulations, especially in two dimensions, we hav e seen α 10 con-

Figure 7: The picture shows the concentrations of the six hyp ercycle species
(solid lines) and the value of α 10 versus time for a system in three spatial
dimensions. Note that a single two dimensional spiral was us ed as initial
condition.
verging rapidly to the ﬁxed value, α 10 ≈ 0. 72. Thereof we can deduce that
such catalytic systems have the urge to determine their ﬁnal structures at
the very beginning. Later on structures are only reﬁned.
Although the hypercycles have been studied very well in the p ast years some
new features can still be found. We think that the deﬁnition o f microscopic
quantities and their interpretation in a macroscopic sense is one of the right
ways to learn more about catalytic systems which can be even m ore complex
than such ”simple” symmetric hypercycles.
4 Outlook
It seems to be interesting for future work to study whether st ability against
parasites is possible in three dimensions, similar to the tw o-dimensional case
[4, 10]. Furthermore, the behavior of α k in such parasite-systems might be
interesting.
References
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ﬁcial Life II . Santa Fe Institut, Addison-Wesley, 1992.
[3] M. Gerhardt, H. Schuster, and J. J. Tyson. A cellular auto maton de-
scribing the formation of spatially ordered structures in c hemical sys-
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[4] M. Boerlijst and P. Hogeweg. Self-structuring and selec tion: Spiral
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[6] I. Harvey and T. Bossomaier. Time out of joint: Attractor s in ran-
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dashed curve denoted as MFPT (mean First Passage Time) repre sents exponential approximation with MFPT substituted int o

Nondecay probability of a metastable state: almost exact analytical description in wide range of noise intensity

Abstract
This paper presents a complete description of noise-induced decay of a metastable state in a wide range of noise intensity. Recurrent formulas of exact moments of decay time valid for arbitrary noise intensity have been obtained. The nondecay probability of a metastable state was found to be really close to the exponent even for the case when the potential barrier height is comparable or smaller than the noise intensity.

arXiv:adap-org/9807001v2 21 Jul 1999
Nondecay probability of a metastable state: almost exact an alytical description
in wide range of noise intensity
Svetlana P. Nikitenkova
Nizhny Novgorod State Technical University, Applied Mathe matics Dpt.,
24 Minin str., Nizhny Novgorod, 603155, Russia. E-mail: spn @waise.nntu.sci-nnov.ru
Andrey L. Pankratov
Institute for Physics of Microstructures of RAS, GSP 105,
Nizhny Novgorod, 603600, RUSSIA. E-mail: alp@ipm.sci-nno v.ru
This paper presents a complete description of noise-induce d decay of a metastable state in a
wide range of noise intensity. Recurrent formulas of exact m oments of decay time valid for arbitrary
noise intensity have been obtained. The nondecay probabili ty of a metastable state was found to
be really close to the exponent even for the case when the pote ntial barrier height is comparable or
smaller than the noise intensity.
PACS numbers: 05.40.+j
I. INTRODUCTION
Overdamped Brownian motion in a ﬁeld of force (Markov process) is a model widely used for description of noise-
induced transitions in diﬀerent polystable systems. In many practic al tasks (e.g. tasks of Josephson electronics,
kinetics of chemical reactions and so on) it is enough to know the pro bability of transition or even only a time scale
of transition. When the transition occurs over a potential barrier high enough in comparison with noise intensity,
the probability of transition is a simple exponent ∼ exp(−t/τ ) [1], where τ is the mean transition time. In this case
the mean transition time gives complete information about the proba bility evolution. The boundaries of validity of
exponential approximation of the probability were previously studie d in [2], [3]. In [2] authors extended the Mean
First Passage Time to the case of ”radiation” boundary condition an d for two barrierless examples demonstrated good
coincidence between exponential approximation and numerically obt ained probability. In a more general case the
exponential behavior of observables was demonstrated in [3] for r elaxation processes in systems having steady states.
Using the approach of ”generalized moment approximation” the aut hors of [3] obtained the exact mean relaxation time
to steady state and for particular example of a rectangular poten tial well demonstrated good coincidence of exponential
approximation with numerically obtained observables. The considere d in [3] example of the rectangular well does not
have a potential barrier, and the authors of that paper suppose d that their approach (and the corresponding formulas)
should also give good approximation in tasks with diﬀusive barrier cros sing for a wide range of noise intensity.
In the frame of this paper we consider a diﬀerent case than in [2], [3]: w e present investigation of nondecay
probability of a metastable state. We treat the decay as a transitio n of Markov process trajectory outside the region
of a metastable state. We consider namely the probability to ﬁnd a re lization of Markov process in a given interval,
but not the probability to pass the boundary for the ﬁrst time (Firs t Passage Time formalizm), since usually in
experiment we can measure only the ﬁrst one. In most of applied tas ks no absorbing boundaries can be introduced
and the precision of the used equipment allows to register the only fa ct of leaving the considered domain, but the
boundary may be crossed many times during the transition (inﬁnite n umber of times for Markov process, since this
process is not diﬀerentiable).
Using the approach proposed by Malakhov [4], [5], that requires only k nowledge of the behavior of a potential at
±∞, we have decomposed the nondecay probability into a set of moment s (cumulants), obtained recurrent formulas for
these moments and approximately summarized them into the require d probability. The obtained nondecay probability
demonstrates exponential behavior with a good precision even in th e case of a small potential barrier in comparison
with noise intensity.
II. MAIN EQUATIONS AND SET UP OF THE PROBLEM
Consider a process of Brownian diﬀusion in a potential proﬁle Φ (x). Let a coordinate x(t) of the Brownian particle
described by the probability density W (x, t ) at initial instant of time has a ﬁxed value x(0) = x0 within the interval
(c, d ), i.e. the initial probability density is the delta function: W (x, 0) = δ(x − x0), x0 ∈ (c, d ).

In this case the one-dimensional probability density W (x, t ) is the transition probability density from the point x0
to the point x: W (x, t ) = W (x, t ; x0, 0). It is known that the probability density W (x, t ) of the Brownian particle in
the overdamped limit satisﬁes to the Fokker–Planck equation (FPE) :
∂W (x, t )
∂t = − ∂G(x, t )
∂x = 1
B
{ ∂
∂x
[ dϕ (x)
dx W (x, t )
]
+ ∂ 2W (x, t )
∂x 2
}
. (1)
with the delta-shaped initial distribution. Here B = h/kT , G(x, t ) is the probability current, h is the viscosity (in
computer simulations we put h = 1), T is the temperature, k is the Boltzmann constant and ϕ (x) = Φ (x)/kT is
the dimensionless potential proﬁle. In this paper we restrict ourse lves by the case of metastable potentials, i.e. we
consider an overdamped Brownian motion in a potential ﬁeld ϕ (x) in systems, having metastable states, such that
ϕ (−∞) = + ∞ and ϕ (+∞) = −∞. This leads to the following boundary conditions: G(−∞, t ) = W (+∞, t ) = 0.
Note, that the results obtained may be generalized for potentials o f arbitrary types, e.g. for such that ϕ (±∞) = ∞.
It is necessary to ﬁnd the probability P (x0, t ) of a Brownian particle, located at the point x0 (t = 0) within the
interval (c, d ) to be at the time t > 0 inside the considered interval: P (x0, t ) =
d∫
c
W (x, t )dx. Further we for simplicity
will call the probability P (x0, t ) as nondecay probability. We suppose, that c and d are arbitrary chosen points of an
arbitrary potential proﬁle ϕ (x) and boundary conditions at these points may be arbitrary: W (c, t ) ≥ 0, W (d, t ) ≥ 0.
In this case there is the possibility for a Brownian particle to come bac k in the interval ( c, d ) after crossing boundary
points.
III. MOMENTS OF DECAY TIME
Consider the nondecay probability P (x0, t ). We can decompose this probability to the set of moments. On the
other hand, if we know all moments, we can in some cases construct a probability as the set of moments. Thus,
analogically to moments of the First Passage Time [6]- [8] we can introdu ce moments of decay time τn(c, x 0, d ) (or,
generally, moments of transition time, see [9], where it was performe d for the probability Q(x0, t ) = 1 − P (x0, t )):
τn(c, x 0, d ) =< t n >=
∞∫
tn ∂P (x0,t )
∂t dt
P (x0, ∞) − P (x0, 0) . (2)
Here we can formally denote the derivative of the probability divided b y the normalization factor as w(x0, t ) and
thus introduce the probability density of decay time w(x0, t ) in the following way [9]:
w(x0, t ) = ∂P (x0, t )
∂t
[P (x0, ∞) − P (x0, 0)] . (3)
It is important to mention that the moments of decay (transition) t ime (2) is a generalization of the well-known
moments of the First Passage Time for the case of arbitrary bound ary conditions (see discussion in [9]). For example,
in the considered case of the potential ϕ (x) (such that ϕ (−∞) = + ∞ and ϕ (+∞) = −∞) the moments of decay time
coincide with the corresponding moments of the First Passage Time, if a reﬂecting boundary at the point c and an
absorbing boundary at the point d are introduced. On the other hand, if we consider the decay of met astable state
as transition over a barrier top, and compare mean decay time obta ined via approach discussed in the present paper
(case of a smooth potential without absorbing boundary) and the mean First Passage Time (MFPT) of the absorbing
boundary located at the barrier top, we get two times diﬀerence be tween these time characteristics even in the case of
a high potential barrier in comparison with the noise intensity. This is d ue to the fact, that the MFPT does not take
into account the backward probability current and therefore is se nsitive to the location of an absorbing boundary. For
the considered situation, if we will move the boundary point down fro m the barrier top, the MFPT will increase up to
two times and tend to reach value of the corresponding mean decay time, which is less sensitive to the location of the
boundary point over a barrier top. Such weak dependence of the m ean decay time from the location of the boundary
point at the barrier top or further is intuitively obvious: much more t ime should be spent to reach the barrier top
(activated escape) than to move down from the barrier top (dyna mic motion).
The required moments of decay time may be obtained via the approac h proposed by Malakhov [4], [5]. This
approach is based on the Laplace transformation method of the FP E (1). Following this approach, one can introduce
the function H(x, s ) ≡ s ˆG(x, s ), where ˆG(x, s ) =
∞∫
G(x, t )e− stdt is the Laplace transformation of the probability
current, and expand it in the power series in s:

H(x, s ) ≡ s ˆG(x, s ) = H0(x) + sH1(x) + s2H2(x) + . . . (4)
It is possible to ﬁnd the diﬀerential equations for Hn(x) (see [4], [5]; dH0(x)/dx = 0):
dH1(x)
dx = δ(x − x0),
d2Hn(x)
dx2 + dϕ (x)
dx
dHn(x)
dx = BHn− 1(x), n = 2, 3, 4, . . .
(5)
Using the boundary conditions W (+∞, t ) = 0 and G(−∞, t ) = 0, one can obtain from (5) H1(x) = 1( x − x0) and
H2(x) = −B
x∫
−∞
e− ϕ (v)
∞∫
v
eϕ (y)1(y − x0)dydv,
Hn(x) = −B
x∫
−∞
e− ϕ (v)
∞∫
v
eϕ (y)Hn− 1(y)dydv, n = 3, 4, 5, . . .
(6)
Why did we calculate this recurrent formula for the functions Hn(x)? The matter is, that from formula (2) (taking
the integral by parts and Laplace transforming it using the proper ty P (x0, 0) − s ˆP (x0, s ) = ˆG(d, s ) − ˆG(c, s ) together
with the expansion (4)) one can get the following expressions for mo ments of decay time:
τ1(c, x 0, d ) = −(H2(d) − H2(c)),
τ2(c, x 0, d ) = 2( H3(d) − H3(c)),
τ3(c, x 0, d ) = −2 · 3(H4(d) − H4(c)), . . .
τn(c, x 0, d ) = ( −1)nn!(Hn+1(d) − Hn+1(c)).
(7)
One can represent the n-th moment in the following form:
τn(c, x 0, d ) = n!τn
1 (c, x 0, d ) + rn(c, x 0, d ). (8)
This is a natural representation of τn(c, x 0, d ) due to the structure of recurrent formulas (6), which is seen fr om the
particular form of the ﬁrst and the second moments for the case c = −∞ (c < x 0 < d). From the recurrent formulas
(6), (7) one can obtain:
τ1(−∞, x 0, d ) = B



d∫
−∞
e− ϕ (x)dx ·
∞∫
x0
eϕ (v)dv −
d∫
x0
e− ϕ (x)
x∫
x0
eϕ (v)dvdx


 . (9)
τ2(−∞, x 0, d ) = 2 B2
{
[τ1(−∞, x 0, d )]2 +
+
d∫
−∞
e− ϕ (x)dx ·
∞∫
x0
eϕ (v)
v∫
d
e− ϕ (u)
∞∫
u
eϕ (z)dzdudv − (10)
−
d∫
x0
e− ϕ (x)
x∫
x0
eϕ (v)
v∫
d
e− ϕ (u)
∞∫
u
eϕ (z)dzdudvdx


 .
Using the approach, applied in the paper by Shenoy and Agarwal [10] for analysis of moments of the First Passage
Time, it can be demonstrated, that in the limit of a high barrier ∆ ϕ ≫ 1 (∆ ϕ = ∆ Φ/kT is the dimensionless barrier
height) the remainders rn(c, x 0, d ) in formula (8) may be neglected. For ∆ ϕ ≈ 1, however, a rigorous analysis should
be performed for estimation of rn(c, x 0, d ). Let us suppose, that the remainders rn(c, x 0, d ) may be neglected in wide
range of parameters and further we will check numerically when our assumption is valid.
The cumulants of decay time æ n [11], [8] are much more useful for our purpose to construct the pr obability
P (x0, t ), that is the integral transformation of the introduced probabilit y density of decay time w(x0, t ) (3). Unlike
the representation via moments, the Fourier transformation of t he probability density (3) - the characteristic function
- decomposed into the set of cumulants may be inversely transform ed into the probability density.
Analogically to representation for moments (8), similar representa tion can be obtained for cumulants æ n:
æ n(c, x 0, d ) = ( n − 1)!æ n
1 (c, x 0, d ) + Rn(c, x 0, d ). (11)

It is known that the characteristic function Θ( x0, ω ) =
∞∫
w(x0, t )ejωt dt (j = √−1) can be represented as the set of
cumulants (w(x0, t ) = 0 for t < 0):
Θ( x0, ω ) = exp
[ ∞∑
n=1
æ n(c, x 0, d )
n! (jω )n
]
. (12)
In the case, when the remainders Rn(c, x 0, d ) in (11) (or rn(c, x 0, d ) in (8)) may be neglected, the set (12) may be
summarized and inverse Fourier transformed:
w(x0, t ) = e− t/τ
τ , (13)
where τ is the mean decay time [4], [5] ( τ(c, x 0, d ) ≡ τ1 ≡ æ 1):
τ(c, x 0, d ) = B



d∫
x0
eϕ (x)
x∫
c
e− ϕ (v)dvdx +
∞∫
d
eϕ (x)dx
d∫
c
e− ϕ (v)dv


 . (14)
This expression is a direct transformation of formula (9), where c is arbitrary, such that c < x 0 < d.
Probably, similar procedure was previously used (see [7], [10], [12], [13]) for summation of the set of moments of the
First Passage Time, when exponential distribution of the First Pass age Time probability density was demonstrated
for the case of a high potential barrier in comparison with noise inten sity.
IV. NONDECAY PROBABILITY EVOLUTION
Integrating probability density (13), taking into account deﬁnition (3), we get the following expression for the
nondecay probability P (x0, t ) ( P (x0, 0) = 1, P (x0, ∞) = 0):
P (x0, t ) = exp( −t/τ ), (15)
where mean decay time τ is expressed by (14). Probability (15) represents a well-known exp onential decay of a
metastable state with a high potential barrier [1]. Where is the bound ary of validity of formula (15) and when can we
neglect by reminders rn and Rn in formulas (8),(11)? To answer this question we have considered th ree examples of
potentials having metastable states and compared numerically obta ined nondecay probability P (x0, t ) =
d∫
c
W (x, t )dx
with its exponential approximation (15). We used the usual explicit d iﬀerence scheme to solve the FPE (2), supposing
the reﬂecting boundary condition G(cb, t ) = 0 ( cb < c ) far above the potential minimum and the absorbing one
W (db, t ) = 0 ( db > d ) far below the potential maximum, instead of boundary conditions a t ±∞, such that the
inﬂuence of phantom boundaries at cb and db on the process of diﬀusion was negligible.
The ﬁrst considered system is described by the potential Φ (x) = ax2 − bx3. We have taken the following particular
parameters: a = 2, b = 1 that leads to the barrier height ∆ Φ ≈ 1. 2, c = −2, d = 2 a/ 3b, and kT = 0 . 5; 1; 3.
The corresponding curves of the numerically simulated probability an d its exponential approximation are presented
in Fig.1. In the worse case when kT = 1 the maximal diﬀerence between the corresponding curves is 3 . 2%. For
comparison, there is also presented a curve of exponential appro ximation with the mean First Passage Time (MFPT)
of the point d for kT = 1 (dashed line). One can see, that in the latter case the error is sig niﬁcantly larger.
The second considered system is described by the potential Φ (x) = ax4 −bx5. We have taken the following particular
parameters: a = 1, b = 0 . 5 that leads to the barrier height ∆ Φ ≈ 1. 3, c = −1. 5, d = 4 a/ 5b, and kT = 0 . 5; 1; 3. The
corresponding curves of the numerically simulated probability and its exponential approximation are presented in
Fig.2. In the worse case ( kT = 1) the maximal diﬀerence between the corresponding curves is 3 . 4%.
The third considered system is described by the potential Φ (x) = 1 − cos(x) − ax. This potential is multistable. We
have considered it in the interval [ −10, 10], taking into account three neighboring minima. We have taken a = 0 . 85
that leads to the barrier height ∆ Φ ≈ 0. 1, c = −π − arcsin(a), d = π − arcsin(a), x0 = arcsin(a), and kT = 0. 1; 0. 3; 1.
The corresponding curves of the numerically simulated probability an d its exponential approximation are presented
in Fig.3. In diﬀerence with two previous examples, this potential was c onsidered in essentially longer interval and
with smaller barrier. The diﬀerence between curves of the numerica lly simulated probability and its exponential
approximation is larger. Nevertheless, the qualitative coincidence is good enough.

Finally, we have considered an example of metastable state without p otential barrier: Φ (x) = −bx3, where b = 1,
x0 = −1, d = 0, c = −3 and kT = 0 . 1; 1; 5. By dashed curve an exponential approximation with the MFP T of the
point d for kT = 1 is presented. It is seen, that even for such example the expone ntial approximation (with the mean
decay time (14)) gives an adequate description of the probability ev olution and that this approximation works better
for larger noise intensity.
V. CONCLUSION
In the present paper the decay of metastable states, described by the model of Markov process, has been considered.
Recurrent formulas of exact moments of decay time, valid for arbit rary noise intensity, have been obtained. Some
concrete examples of metastable states have been analysed nume rically, and the time evolution of the nondecay
probability of a metastable state is found to be really close to the exp onent even for the case when the potential
barrier height is comparable or smaller than the noise intensity if the e xact mean decay time (14) is substituted into
the factor of exponent.
For all investigated examples, the exponential approximation gives an adequate behavior of the probability. This
approximation may be used in a wide range of parameters, enough fo r solution of many practical tasks, but it is
necessary to remark, that the exponential approximation may lea d to a signiﬁcant error in the case of extremely large
noise intensity, and in the case when the noise intensity is small, the po tential is tilted, and the barrier is absent
(purely dynamical motion slightly modulated by noise perturbations) .
VI. ACKNOWLEDGMENTS
Authors wish to thank Prof. V.Belykh, Prof. P.Talkner, Prof. P.Jun g and Dr. W.Nadler for discussions and
constructive comments. This work has been supported by the Rus sian Foundation for Basic Research (Project N 96-
02-16772-a, Project N 96-15-96718 and Project N 97-02-1692 8), by Ministry of High Education of Russian Federation
(Project N 3877) and in part by Grant N 98-2-13 from the Interna tional Center for Advanced Studies in Nizhny
Novgorod.
[1] C.W.Gardiner, Handbook of Stochastic methods (Springe r-Verlag, 1985).
[2] A.Szabo, K.Schulten and Z.Schulten, Journ. Chem. Phys. , 72, 4350 (1980).
[3] W.Nadler and K.Schulten, Journ. Chem. Phys., 82, 151 (1985).
[4] A.N.Malakhov and A.L.Pankratov, Physica C 269, 46 (1996).
[5] A.N.Malakhov, CHAOS 7, 488 (1997).
[6] L.A.Pontryagin, A.A.Andronov and A.A.Vitt, Zh. Eksp. T eor. Fiz. 3, 165 (1933) [translated by J.B.Barbour and repro-
duced in ”Noise in Nonlinear Dynamics”, 1989, edited by F.Mo ss and P.V.E. McClintock (Cambridge University Press,
Cambridge) Vol. 1, p.329].
[7] P.Hanggi, P.Talkner and M.Borkovec, Rev.Mod.Phys., 62, 251 (1990).
[8] H.Risken, The Fokker-Planck equation (Springer Verlag , Berlin, 1985).
[9] A.L.Pankratov, Physics Letters A 234, 329 (1997).
[10] S.R.Shenoy and G.S.Agarwal, Phys. Rev. A, 29, 1315 (1984).
[11] A.N.Malakhov, Cumulant analysis of random Non-Gaussi an processes and its transformations (Sovetskoe Radio, Mos cow,
1978, in Russian).
[12] R.Roy, R.Short, J.Durnin and L.Mandel, Phys. Rev. Lett ., 45, 1486 (1980).
[13] K.Lindenberg and B.West, J. Stat. Phys., 42, 201 (1986).

0.60
0.80
1.00
/G33/G0B/G57/G0C
kT = 3
kT = 1
kT = 0.5
FIG. 1. Evolution of the nondecay probability for the potent ial Φ (x) = ax2 − bx3 for diﬀerent values of noise intensity; the

0.60
0.80
1.00
/G33/G0B/G57/G0C
FIG. 2. Evolution of the nondecay probability for the potent ial Φ (x) = ax4 − bx5 for diﬀerent values of noise intensity.

0.0 20 .0 40 .0 60 .0 80 .0 100 .0
0.60
0.80
1.00
/G33/G0B/G57/G0C
FIG. 3. Evolution of the nondecay probability for the potent ial Φ (x) = 1 − cos(x) − ax for diﬀerent values of noise intensity.

0.0 3.0 6.0 9.0 12 .0 15 .0
0.60
0.80
1.00
/G33/G0B/G57/G0C
kT = 0 .1
kT = 1
kT = 5
FIG. 4. Evolution of the nondecay probability for the potent ial Φ (x) = − bx3 for diﬀerent values of noise intensity; the


Development and Evolution of Neural Networks in an Artificial Chemistry

Abstract
We present a model of decentralized growth for Artificial Neural Networks (ANNs) inspired by the development and the physiology of real nervous systems. In this model, each individual artificial neuron is an autonomous unit whose behavior is determined only by the genetic information it harbors and local concentrations of substrates modeled by a simple artificial chemistry. Gene expression is manifested as axon and dendrite growth, cell division and differentiation, substrate production and cell stimulation. We demonstrate the model's power with a hand-written genome that leads to the growth of a simple network which performs classical conditioning. To evolve more complex structures, we implemented a platform-independent, asynchronous, distributed Genetic Algorithm (GA) that allows users to participate in evolutionary experiments via the World Wide Web.

arXiv:adap-org/9807003v1 17 Jul 1998
Development and Evolution of Neural Networks in an Artiﬁcia l
Chemistry
Jens C. Astor ∗ and Christoph Adami
Computation and Neural Systems and Kellogg Radiation Laboratory
California Institute of Technology
Pasadena, CA, 91125 (USA)
Abstract
We present a model of decentralized growth for Artiﬁ-
cial Neural Networks (ANNs) inspired by the develop-
ment and the physiology of real nervous systems. In
this model, each individual artiﬁcial neuron is an au-
tonomous unit whose behavior is determined only by
the genetic information it harbors and local concentra-
tions of substrates modeled by a simple artiﬁcial chem-
istry. Gene expression is manifested as axon and den-
drite growth, cell division and diﬀerentiation, substrate
production and cell stimulation. We demonstrate the
model’s power with a hand-written genome that leads
to the growth of a simple network which performs clas-
sical conditioning. To evolve more complex structures,
we implemented a platform-independent, asynchronous,
distributed Genetic Algorithm (GA) that allows users to
participate in evolutionary experiments via the World
Wide Web .
1 Introduction
Ever since computational neuroscience was born with the
introduction of the abstract neuron by McCulloch and
Pitts in 1943 [1], we have witnessed a gap between the
mathematical modeling of neurons—inspired by Turing’s
notions of universal computation—and the physiology of
biological neurons and the networks they form. The cur-
rent state of aﬀairs reﬂects this dichotomy: neurophysio-
logical simulation test beds [2] cannot solve engineering
problems, while sophisticated ANN models [3] do not
explain the miracle of biological information processing.
Compared to real nervous systems, classical ANN
models have a serious shortcoming owing to the fact
that they are engineered to solve particular classiﬁca-
tion problems, and analyzed according to standard the-
ory based mainly on statistics and global error reduc-
tion. As such, they can hardly be considered universal.
Hence, such models deﬁne the network architecture a
priori which is in most cases a ﬁxed structure of homo-
geneous computation units.
Some models support problem-dependent network
∗⋆On leave from Dept. of Medical Informatics, University
of Heidelberg/School of Technology, Heilbronn, Germany,
jastor@stud.fh-heilbronn.de
changes during simulation [4, 3]. In these models, global
decisions lead to network structures adapted to the prob-
lem at hand. Other approaches try to shape networks
for a particular problem by evolving ANNs either di-
rectly [5], or indirectly via a growth process [6]. More
recently, approaches like Ref. [7] include a kind of ar-
tiﬁcial chemistry which allows a more natural develop-
ment. Still, in these models neurons are unevolvable
homogeneous structures in a more or less ﬁxed architec-
ture which, we believe, limits their relevance to natural
nervous systems.
In this paper we investigate the idea that interesting
information-processing structures can be grown from a
model which follows four basic principles of molecular
and evolutionary biology, listed below. While models for
ANNs currently exist that implement a selection of them,
the inclusion of all four opens the possibility that, given
enough evolutionary time, novel structures can emerge
that are comparable to natural nervous systems.
• Coding. The model should encode ANNs in such way
that evolutionary principles can be applied.
• Development. The model should be capable of
growing an ANN by a completely decentralized de-
velopmental process, based exclusively on the cell and
its interactions.
• Locality. Each neuron must act autonomously and
be determined only by its genetic code and the state
of its local environment.
• Heterogeneity. The model must have the capabil-
ity to describe diﬀerent, heterogeneous neurons in the
same ANN.
One of the key features of a model implementing those
principles will be the absence of explicit activity func-
tions, learning rules, or connection structures. Rather,
such characteristics should emerge in the adaptive pro-
cess and lead to ANNs with open architectures and more
universal artiﬁcial neurogenesis.
While keeping in mind that the model is not designed
to reproduce real neural systems, we posit that an ad-1

herence to the fundamental tenets of molecular and evo-
lutionary principles—albeit in an artiﬁcial medium—
represents the most promising unexplored avenue in the
search for intelligent information-processing structures.
2 Model
In this section we introduce our model of neurogenesis
starting with the artiﬁcial physics and biochemistry, and
go on to explain how local gene expression ultimately
results in information-processing structures. This gene
expression takes place exclusively in artiﬁcial neurons
which are embedded in a tissue-like structure. As this
model is inspired by the concepts of molecular cell biol-
ogy, we use the nomenclature of this science unabashedly
while issuing the caveat that they are analogical in na-
ture only.
2.1 Artiﬁcial Physics
The physical world is a two-dimensional grid of
hexagons. Each such site harbors certain concentrations
of substrates, measured as percentage values of satura-
tion between 0 and 1. As cells are equidistant in a hexag-
onal lattice, the diﬀusion of substrate k in cell i can be
modeled discretely as
Cik(t + 1) = D
6∑
j=1
(
Cik(t) − CNi,jk(t)
)
(1)
where Cik(t) is the concentration of substrate k in site
i, D is a diﬀusion coeﬃcient ( D < 0. 5 to avoid substrate
oscillation), and Ni,j represents the jth neighbor of grid
element i.
Figure 1: Hexagonal grid with boundary elements. Diﬀusion
occurs from a local concentration peak at grid element N .
Accordingly, a local concentration of substrate will dif-
fuse through the tissue under conservation of mass. The
tissue itself is surrounded by special boundary elements
which absorb substrates (Figure 1), thus modeling diﬀu-
sion in inﬁnite space. We caution at this point that the
hexagons are sites that may harbor cells, but otherwise
only represent a convenient equidistant discretization of
space to facilitate the distribution of chemicals via dif-
fusion.
2.2 Artiﬁcial Biochemistry
We distinguish four diﬀerent classes of substrates:
• External proteins : Diﬀusive substrates which can
be produced by neurons if expressed.
• Internal proteins : Produced by neurons, but non-
diﬀusive as they cannot cross cell membranes.
• Cell-type proteins : Each neural cell harbors an
external protein that deﬁnes its type. Like any exter-
nal protein it is diﬀusive.
• Neurotransmitter: Special type of internal pro-
tein used for directed information exchange between
neurons.
2.3 Artiﬁcial Cell
Cell types We distinguish three kinds of neurons on
the cellular level: actuator cells, sensor cells, and com-
mon neurons. The ﬁrst two types are special versions of
the third and are used as interface to a (simulated) en-
vironment to which the network adapts and on which it
computes. These neurons can be excited to a real-valued
level between 0 and 1, and take part in the information
transfer via dendritic or axonal connections, respectively
(see below). Each type of cell is also characterized by its
own cell-type protein, which it produces continuously at
a certain rate. These cell-type proteins diﬀuse over the
tissue (Figure 1) and therefore signal cell existence to
other cells. They can be compared to growth factors
known from the development of real nervous systems.
Actuator and sensor cells do not carry genetic infor-
mation; they are used solely as interfaces to the environ-
ment (input-output units, see Section 2.4.) They rep-
resent sources and sinks of signal. Consequently, their
behavior is hardwired and does not depend on transcrip-
tion as for common neurons. The latter can receive a
ﬂux of neurotransmitter from dendrites with a particu-
lar weight. However, this does not imply an automatic
stimulation of activity unless such behavior is explicitly
encoded in the neuron’s genome. Table 1 summarizes
the cell types and how they interact with other compu-
tational elements used in our model.
Genetic code and gene expression in artiﬁcial
neurons Each neural cell carries a genome which com-
pletely encodes its behavior. Genomes consist of genes
which can be viewed as a genetic program that can ei-
ther be executed ( expressed) or not, depending on a gene
condition (akin to the regulator/operator genes in the2

Type [1] [2] [3] [4] [5] [6]
Neuron x x x x x x
Sensor x x x x
Actuator x x x x
Grid element x
Boundary element
Table 1: Features of diﬀerent cell types building the ar-
tiﬁcial organic tissue: [1] participates in diﬀusion, [2]
can be stimulated, [3] depends on gene expression, [4]
can have axons, [5] can have dendrites, [6] produces a
diﬀusible cell-type protein.
Condition Description
ADD [EP] Φ new=Φ before + [EP]
MUL [eNT] Φ new=Φ before × [eNT]
SUP [CTP0] suppresses gene if cell not of type CTP0
AND [IP] fuzzy AND: Φ new = max([IP], Φ before)
Table 2: Condition atoms and their interpretation. Con-
dition atoms build a gene condition, which is obtained by
evaluating its condition atoms in the given order. Here,
[XY] means ‘the current local concentration of substrate
XY inside of the cell to which the gene condition belongs’.
Φ is the evaluation result of this gene condition.
Jacob-Monod-model). A gene condition is a combina-
tion of several condition atoms, usually related to local
concentrations of substrates. The expression of a gene
can result in diﬀerent behaviors such as the production of
a protein, cell division, axon/dendrite growth, cell stim-
ulation, etc. Figure 2 clariﬁes the structure of the ge-
netic code. Thus, gene conditions model the inﬂuence
of external concentrations on the expression level of the
gene, i.e., they model activation and suppression sites.
To evaluate a gene condition, each element of its chain of
Condition 3
Expression 1 Expression 2
Condition 1 Condition 2
Gene 2
Gene 1
Gene 4
Gene 3
Gene 2
N
Genome
Figure 2: Genetic structure of neural cells. Local con-
centrations of substrates (condition atoms) trigger gene
expression.
condition atoms contributes to obtain a real-valued ex-
pression level between 0 and 1, describing the strength
with which the respective gene expression will take place.
t=0 cycles:
t=10 cycles:
t=6 cycles:
t=8 cycles:
Figure 3: Development of the network for classical condi-
tioning.

Expression Command Description Inﬂuence of
Condition Value
PRD[XY] produce substrate XY production quantity
GDR[XY] grow dendrite following gradient of XY growing probability
GRA[XY] grow axon following gradient of XY growing probability
SPL[CTPx] divide to CTPx-type cell probability
EXT excitatory stimulus increase rate
INH inhibitory stimulus decrease rate
MOD+ increase connection weights strengthening factor
MOD- decrease connection weights weakening factor
Table 3: Overview of expression commands. Growing axons/dendrit es follow the substrate gradient until the local
maximum is reached, then connect to the cell (if it exists). Strengt hening/weakening is a percentage increase/decrease
of connection weights, determined by the product of the last neur otransmitter inﬂux at each connection and the value
of the gene condition. The cell-type protein assigned to a cell division command determines the future type of the
oﬀspring cell. In this example, the daughter will be of type CTPx and therefore produce cell-type protein CTPx
continuously.
Consider for example substrates with local concen-
trations [ep0, ip, ep1]=(0.3, 0.5, 0.5 ). Then, the
evaluation of gene condition ADD[ep0] ADD[ip] MUL[ep1]
would lead to an overall expression level 0.4 (this value is
modded back into range between 0 and 1 if it falls outside
it). Table 2 illustrates a few examples of such conditions,
while Table 3 gives an overview about the diﬀerent gene
expression commands. The evaluation result of the gene
condition has diﬀerent meanings depending on the gene
expression command.
2.4 Simulation of the Artiﬁcial Organism
The tissue of cells produced by gene expression and cell
growth is termed the artiﬁcial organism . It receives in-
put from the environment (the outside world) and can
act on it by signaling to the environment via its actu-
ators. In the simplest case, thus, the organism receives
and generates patterns.
A simulation always starts by creating sensor and ac-
tuator cells. Their number is determined only by the
complexity of the outside world and is not coded for in
the genome. In other words, these cells really represent
possible signals and actuations in the world, not actual
signals and actuations performed by the organism. An
organism chooses to receive input or perform an actua-
tion by connecting to these cells. If needed, an additional
reinforcement cell can be created. This is a special sen-
sor cell (with its own cell-type protein) used to provide a
reinforcement signal from the world about the behavior
of the organism. Whether or not this signal is used is
determined by the organism’s genome. Furthermore, at
the start of each simulation, one initial neuron is placed
in the center of the grid. After initialization, the simula-
tion can begin. Input from the world is provided to the
sensor cells, diﬀusion of produced cell-type proteins and
external proteins takes place, and neurons execute their
genetic code synchronously.
Depending on its gene expression, a neuron starts
growing axons and dendrites, produces oﬀspring cells
and might initiate cell diﬀerentiation. Gene expressions
may lead to protein production cascades, stimulation,
and ultimately information exchange between neurons.
After every simulation cycle the network’s ‘ﬁtness’ is de-
termined by comparing any inputs and outputs to what
is expected in this particular world, producing a real-
valued reinforcement signal between 0 (punishment) and
1 (reward). This signal can be used by the organism if
a reinforcement sensor is present and if the organism
chooses to connect to it.
3 An Example Genome
Figure 3 documents the development of a simple ANN
from a hand-written genome. Starting from a single ini-
tial neuron, cell division takes place and connections (ax-
ons and dendrites) start to grow along the gradient of dif-
fusing cell-type proteins. After a while, sensors, neurons
and actuator cells are connected in a particular manner.
In fact, this network displays conditioned reﬂex behav-
ior as in Pavlov’s classical experiment [10]. Suppose the
sensor on the lower left side in Figure 3 is stimulated
(active) at the sound of a bell. Further, suppose the
upper left sensor is an optical stimulus representing the
presence (or absence) of food. Finally, let us imagine
that the actuator on the right side triggers a salivary
gland if food is present. This behavior is the hardwired
unconditioned reﬂex. The above network can learn to
associate the reﬂex with a condition: the sound of the
bell. If presence of food and the ringing of the bell are
associated repeatedly, the network will learn to trigger
the gland even if only the bell rings. If the bell rings
after the conditioning without the presence of food, the
association will gradually, but steadily, weaken. Such a4

1. NNY(ip) SUP(cpt) ANY(spt0) -> SPL(acpt0) PRD(ip) SPL(ac pt2) GDR(spt0) DFN(NT1)
2. NNY(ip) SUP(acpt0) ANY(spt1) ANY(cpt) -> PRD(ip) GDR(sp t1) GRA(cpt) DFN(NT1)
3. ANY(spt1) SUP(acpt2) NNY(ip) ANY(apt0) -> SPL(acpt1) GD R(spt1) PRD(ip) GRA(apt0)
4. ANY(acpt2) SUP(acpt1) ANY(spt0) NNY(ip) -> GRA(acpt2) G DR(spt0) GDR(cpt) PRD(ip)
5. ANY(ip) -> PRD(ip)
6. NSUP(cpt) NSUP(acpt1) ADD(eNT) -> EXT
7. SUP(acpt1) ADD(NT1) MUL(eNT) -> EXT
8. ADD(eNT) -> PRD(ip1)
9. ADD(ip1) -> PRD(ip2)
10. SUP(cpt) ADD(NT1) MUL(ip2) -> PRD(ep)
11. SUP(cpt) ADD(ep) -> EXT
Figure 4: Genome for development and behavior of network exhibitin g classical conditioning
behavior can be modelled using diﬀerent kinds of cell-
types. The “C” cell is activated if the network is in
the conditioned state, which means that acoustical and
optical stimuli have been present together before. Cell
“E” is activated if the acoustical stimulus is currently
present and the network is in the conditioned state at
the same time. If so, cell “G” representing the trigger of
the salivary gland is activated. Of course, cell “G” is also
activated if only food is present. This is the “hardwired”
reﬂex. A schematical drawing of the network is shown in
Figure 5. The genome which encodes the development
and behavior of this network is shown in Fig. 4.
Bell
Food
E
G
C eNT
Sensor cells Actuator
eNT
eNT
spt0
acpt1
acpt2 apt0
eNT
acpt0eNT
spt1
NT-switch
NT1
eNT
cpt
NT1
Figure 5: A schematical representation of the network for
classical conditioning. The types of neurotransmitter use d
are shown next to the axons. The cell-type protein used by
each cell is indicated near the cell body.
It is beyond the scope of this paper to go into the
details of this genome and its function (see [8, 9] for a
more thorough description). However, the explanation
that follows still gives an idea of the type of information
necessary to grow networks with particular characteris-
tics.
The genome consists of 11 genes, each of which has
its condition (left-hand side) and its expression (right-
hand side). Genes 1 to 4 control cell division into the
diﬀerent types that are needed, as well as the growth of
axons and dendrites. The ﬁrst gene is only expressed by
the initial cell, and only if no internal protein ip is present
(the sequence NNY(ip) SUP(cpt) ). In addition, gene 1 is
only expressed if it senses nonzero concentrations of cell-
type protein spt0 (ANY(spt1) ANY(spt0) ) which is emitted
by one of the sensor cells and has diﬀused. Under these
circumstances the initial cell will divide and produce oﬀ-
spring of type acpt0 and acpt2 (SPL(acpt2) SPL(acpt0) ),
grow a dendrite that follows the gradient of the sensor
protein spt0 (GDR(spt0)), and produce the internal pro-
tein ip. Once ip is produced, this will continue to happen
(gene 5) which prevents that gene 1 can ever be turned on
again. Genes 2 to 4 work just as gene 1, but for other cell
types. While gene 5 takes care of the hardwired-reﬂex
stimulation, genes 10 and 11 control the conditioning
(expressed only by cell-type cpt). If food is present and
the bell rings, gene 10 is expressed. It produces certain
amounts of external protein ep. The concentration of ep
inﬂuences cell stimulation (gene 11 exhibits stimulation
via EXT). Due to diﬀusion, ep diminishes over time, so
the conditioning decreases accordingly. As pathways to
the “C” cell are not equally long, a production cascade of
internal proteins in genes 8, 9 and 10 is necessary that
delays the input of the acoustical sensor. The behav-
ior of the resulting phenotype network is documented in
Figures 6, 7 and 8.
4 WWW-based Genetic Algorithm:
Community of Artiﬁcial Organisms
While it is not diﬃcult to write genomes which lead
to simple networks with desired characteristics, one of
the main features of the system is its evolvability. Cer-
tainly, the search space for such genomes is immense,
and it is unreasonable to hope that interesting genomes
can be found without massive parallelism. Rather than
choosing to implement this system on supercomputers,
we opted to allow users on the Internet to donate their
CPU time by participating in a global evolutionary ex-
periment.
Using Sun’s JavaTM technology, we developed an asyn-
chronous, distributed GA system which allows a massive
parallel search for new genomes based on evolutionary
principles [8, 9]. It consists mainly of a central server5

t=0..19
t
1.0
0.0
t
1.0
0.0
t
1.0
0.0
t
1.0
0.0
t
0.0
1.0
Food
Bell
E-Cell
Gland
C-CellFood
Bell
Sensor cells
Actuator
C
E
G
Figure 6: First, only food is present. This triggers the gland because of the hardwired reﬂex, while the “C”-cell and “E”-cell
remain inactivated. Later, only the bell signal is present. Due to the fact that the network is not yet conditioned, none o f the
cells becomes active as a result.
t=16..38
t
1.0
0.0
t
0.0
1.0
t
1.0
0.0
t
1.0
0.0
t
1.0
0.0
Bell
Food
C-Cell
E-Cell
Gland
Food
Bell
C
E
G
Sensor cells
Actuator
Figure 7: Both sensors, food and sound, are stimulated. The ANN become s conditioned (“C”-cell) and the gland is triggered
due to the presence of food.
t=50..64
t
1.0
0.0
t
1.0
0.0
t
0.0
1.0
t
1.0
0.0
t
1.0
0.0
Food
Bell
E-Cell
C-Cell
Gland
Food
Bell
C
E
G
Sensor cells
Actuator
Figure 8: Being in the conditioned state, the food sensor suddenly bec omes deactivated while the bell keeps on ringing.
Thus, the activation of the “C”-cell becomes weaker. This im plies a decrease of activation of the “E”-cell which ﬁnally r esults
in a decline of gland activity.

Sun SPARC 141.7.12.105
Organism, e.g. on
somewhere
an i86-PC
Organism as Applet
in a browser, on
architecture 'foo',
on Sun SPARC
131.215.48.50
Central GA-server, e.g.
Organism, e.g. on
evaluation request for
register request
new genotype or
'go on' reply
genotype reply:ζ
new client central GA-server
Φ(ζ)
Figure 9: Left: genotype evaluation in clients of diﬀerent architect ure, using TCP/IP for communication with the
asynchronous genetic algorithm. Right: communication between th e GA-server and a client hosting an organism.
application and clients, each of which hosting one indi-
vidual of the current GA population.
As Java is supposed to be platform independent,
clients can be started from every computer for which
an accurate Java 1.1 virtual machine or browser exists
(Figure 9). The clients are hybrids, which means that
they can be started as Java Applet by choosing the html
page from our WWW-server, or with the help of a boot-
loader program which dynamically downloads the client
and starts it as a Java application (no browser neces-
sary).
A client automatically sends a request-to-register to
the central GA-server after it was started, and receives
from the server a genotype ζ. The client then starts
up a simulation as described in Section 2.4. After a
certain number of simulation cycles, it sends the geno-
type’s ﬁtness Φ( ζ) (average reinforcement signal during
simulation) to the server. By comparing Φ( ζ) to the
ﬁtness of other genotypes in the database, the server de-
cides if it is worth to keep this genotype or if the client
should be assigned a new one. If the server has to send a
new genotype, it either takes a suspended one out of its
database, or constructs one through the processes of re-
combination and/or mutation from genotypes of known
ﬁtness already present in the population (Figure 9). Fit-
ter genotypes are more likely to be selected for recombi-
nation and/or asexual copying then genotypes of lower
ﬁtness. This leads to an increase of the average ﬁtness
over time.
5 Conclusion
We introduced a developmental and behavioral model
based on artiﬁcial gene expression which shares key prop-
erties with natural neural development. Within this
model we succeeded to construct simple systems with
properties which are believed to be essential [11, 12]
for higher self-organizing information processing sys-
tems, such as deterministic structure development, self-
limiting growth, growth following diﬀusion gradients,
computation of logical functions, pacemaker behavior
and simple adaptation (sensitization, habituation, as-
sociative classical conditioning) [8, 9]. Furthermore, we
showed how an evolutionary search for genomes coding
for information-processing network structures can be dis-
tributed in a platform-independent manner such that
the unused CPU power of the Internet can be tapped
to search for ANNs that reduce the gap between the ab-
stract models and neurophysiology.
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