The Brain in a Box

An Encoding Scheme for Natural Neural Networks

Martin Pyka

, Tilo Kircher

, Sascha Hauke

and Dominik Heider

Section BrainImaging, Department of Psychiatry, University of Marburg, Marburg, Germany

Telecooperation Group, Technische Universit

at Darmstadt, Darmstadt, Germany

Department of Bioinformatics, University of Duisburg-Essen, Duisburg and Essen, Germany

Keywords:

Neural Networks, Artiﬁcial Development, CPPN.

Abstract:

To study the evolution of complex nervous systems through artiﬁcial development, an encoding scheme for

modeling networks is needed that reﬂects intrinsic properties similiar to natural encodings. Like the ge-

netic code, a description language for simulations should indirectly encode networks, be stable but adaptable

through evolution and should encode functions of neural networks through architectural design as well as sin-

gle neuron conﬁgurations. We propose an indirect encoding scheme based on Compositional Pattern Produc-

ing Networks (CPPNs) to fulﬁll these needs. The encoding scheme uses CPPNs to generate multidimensional

patterns that represent the analog to protein distributions in the development of organisms. These patterns

form the template for three-dimensional neural networks, in which dendrite- and axon cones are placed in

space to determine the actual connections in a spiking neural network simulation.

1 INTRODUCTION

If evolutionary development, i.e., the stepwise im-

provement through mutation and crossover, is seen as

a means of understanding natural networks, two open

questions in particular present themselves. First, how

must a description language for neural networks (e.g.

complex nervous systems) look like? And second,

what are appropriate ﬁtness functions to evolve com-

plex networks? In this article, we propose an answer

to the ﬁrst question.

Because such a description language has to be ca-

pable of developing natural neural networks, it is im-

perative to know the intrinsic properties of the net-

works that should be modeled. In neuroscientiﬁc dis-

ciplines, numerous properties of natural neural net-

works are well-accepted that, in our opinion, should

receive considerably more attention in those disci-

plines that try to model networks by evolutionary

means.

For instance, the cortex of the brain consists of

layers of different neuron types interconnected with

each other. The connections are a side-effect of the

position of the neurons within the layer, their anatom-

ical form and their intersection with other neurons

(Wolpert, 2001). Connections between neurons can

be either speciﬁc (a clear and precise determination

of connections, e.g. as it can be found in the thalamus

(Basso et al., 2005)) or driven by side effects (e.g.

adjacent regions are interconnected with each other).

Furthermore, the (genetic) encoding scheme for

natural organisms itself has certain properties that

should be considered in simulations. In biological

organisms, comparatively few genes (e.g., the hu-

man genome consists of circa 22,000 genes) encode

complex structures by using highly indirect mecha-

nisms. The encoding involves local interactions be-

tween cells, diffusion of substrates and gene regu-

latory networks. One purpose of these mechanisms

is to generate global patterns from local interactions

between genes that serve as axes for the organiza-

tion and reﬁnement of (cellular) structures (Mein-

hardt, 2008; Raff, 1996; Curtis et al., 1995). From this

notion, it can be concluded that an encoding scheme

for the evolutionary exploration of neural networks

should work in a highly indirect manner as well and

serve as a pattern generator for subsequent structural

elements. If the network architectures should be im-

provable via evolutionary mechanisms, the encoding

scheme must fulﬁll requirements for generating stable

network architectures across generations, but should

also facilitate alterations in the network that can im-

pact (and improve) their functionality.

In the following, we propose an encoding scheme,

196

Pyka M., Kircher T., Hauke S. and Heider D..

The Brain in a Box - An Encoding Scheme for Natural Neural Networks.

DOI: 10.5220/0004152801960201

In Proceedings of the 4th International Joint Conference on Computational Intelligence (ECTA-2012), pages 196-201

ISBN: 978-989-8565-33-4

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

which we call Brain In a Box (BIB), that aims to gen-

erate networks with the outlined properties via pattern

generation. It comprises three steps which will be de-

scribed in detail in the following sections: i) Generate

global patterns of protein densities, representing po-

tential neuron locations and their properties; ii) Con-

vert these patterns into a 3D-representation of neurons

where axon- and dendrite-cones are used to determine

connections between neurons; iii) Run a neural net-

work simulation using the inferred network conﬁgura-

tion. We think, that this encoding scheme is not only

useful in the evolutionary exploration of network ar-

chitectures, but can also serve as a model to describe

real neural networks.

2 STEP 1: GLOBAL PATTERN

GENERATION

In a BIB, network architectures are inferred from neu-

rons placed in a three-dimensional space. Each neu-

ron can be characterized by several properties im-

pacting their potential connections to other neurons.

These properties include, for instance, its position and

its orientation in space. Potential axons and dendrites

are indicated through cones projecting away from the

neuron (Fig. 4). The length of the cone and its width

(deﬁned by its angle) determine a volume that may

intersect with cones of other neurons. In case of an

intersection, a density parameter of the neuron deter-

mines the chance for a potential connection. In a bi-

ological sense, different types of proteins form – de-

pending on its concentration, location and combina-

tion with other proteins – the structure of a neuron.

Connections between neurons are just a subsequent,

probabilistic side-effect of their properties and place-

ment in space.

BIBs are generated by mapping the three space co-

ordinates, a distance and a bias value to several output

values, representing the above mentioned properties

of the neuron (Fig. 1). The mapping is computed by

Compositional Pattern Producing Networks (CPPNs)

which model how gene regulatory networks (GRNs)

generate global patterns, while exploiting computa-

tional shortcuts in the simulated world (Stanley, 2007)

(Fig. 2). An additional output node encodes whether

a neuron is built at a given location. Intermediate

genes are activated either by the (initial) input values

or by other genes and generate density distributions

that modulate, for instance, axon length and orienta-

tion of the neuron. Thus, a single gene may have a

modulating inﬂuence on either one or several prop-

erties of neurons. This means, the functional role of

neurons is controlled by genes inﬂuencing the struc-

x y z d b

n nx ny nz aa al ad da dl dd

Compositional Pattern Producing Network

normal vector axon parameter dendrite parameterneuron

Figure 1: Structure of a CPPN for a BIB. The axes of the

three-dimensional space (x, y, z) a bias value (b) and a dis-

tance value indicating the distance for a given point to the

center of the volume (d (Stanley, 2007)), are mapped via a

CPPN (see also ﬁg. 2) on various parameters of the neu-

ron. The output values encode, whether a neuron should be

placed (n), the orientation of the neuron (nx, ny, nz), and

the angle, length and density of the axon and dendrite cones

(aa, al, ad, da, dl, dd).

tural properties of the neurons.

x y z

Figure 2: Example of a Compositional Pattern Producing

Network. Global axes serve as input for concatenated func-

tions generating two- or three-dimensional patterns. The

functions represent the analog to density distributions of

genes which activate other genes.

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Figure 3: 3D-representation of a neural network using a res-

olution of 20 × 20 × 20 neurons. For clarity’s sake, cones

for axon and dendrite connections are not displayed. In the

lower left corner, the volume generated by the CPPN is de-

picted in a higher resolution.

3 STEP 2: CONVERSION OF THE

NETWORK INTO 3D

The patterns generated in the previous step are used

to generate a 3D-representation of the neural network

(ﬁg. 3). Only at this stage, it is necessary to spec-

ify the spatial resolution in which the neural network

should be generated. A routine calls the CPPN with

all coordinate values in the given resolution and deter-

mines, whether or not a neuron should be placed at a

given location, and if yes, which properties the axon-

and dendrite cones have (Fig. 4). If axon- and den-

drite cones intersect, density parameters of the cones

provided by the CPPN (ranging from 0.0 to 1.0) are

multiplied and used as probability for a directed con-

nection generated between both neurons.

Since BIBs can also be used to model real (bio-

logical) neural systems, such as cortex layers, the 3D-

representation provides a means to obtain face valid-

ity upon a given anatomical structure. Here, again the

role of genes within the model can be characterized

and validated against the potential role of biological

candidate genes.

4 STEP 3: SPIKING NEURAL

NETWORK SIMULATION

The network information obtained from the pre-

vious step can subsequently be used in any type

of neural network simulations to simulate and test

Figure 4: Three-dimensional representation of a neuron. A

neuron is characterized by its location in space, its orienta-

tion and some parameters describing the cones for potential

axons and dendrites. The green cone represents potential

axon-connections and the blue cone potential dendrite con-

nections.

the network. In our BIB-code, we chose Brian

(www.briansimulator.org, (Goodman and Brette,

2008)) as framework for implementing a spiking neu-

ral network system. For each 3D-neuron generated

in the second step, a neuron was generated in Brian

and connected with other neurons as determined in

the 3D-representation. The time-course of each neu-

ron was deﬁned as

−(V − E

)

, (1)

where V is the activity of a neuron for a certain point

t in time. E

can be regarded as an attractor or resting

state of the neuron and τ is a time-scaling factor. If

V exceeds a threshold V

, the neuron sends a spike

through the axon and V is set to a reset-level V

, from

which it approximates E

as described in Eq. (1).

5 SIMULATION

In the following, we will demonstrate that BIBs might

be a valuable framework for two purposes: On the

one hand, BIBs can be used to develop networks

with evolutionary means. This implies, however, a

well-deﬁned ﬁtness-function that reﬂects somehow

the function of a dedicated region in a brain. On the

other hand, BIBs can be used as a model to imitate bi-

ological relationships between genotype and pheno-

type of brain structures.

Having said this, we argue that, in principle, con-

troller networks can be evolved and BIBs can gener-

ate networks, in which groups of neurons on different

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scales can be altered in a semantical meaningful man-

ner.

5.1 Proof of Principle

In a ﬁrst test, we show that the evolutionary algo-

rithm implemented in NeuroEvolution of Augment-

ing Topologies (NEAT) (Stanley, 2002) also works for

BIBs. This is achieved via a simulation, in which an

organism, controlled by a network of sensors, neurons

and motors, can navigate on a 2D-plane with different

concentrations of a notional substrate called “feed”.

The plane is represented by a 10x10 raster, in which

the center of each ﬁeld has a different concentration

of the substrate. The “feed” concentration of any in-

termediate point in the continuous space is the inter-

polated value of the four adjacent centers (Fig. 5a).

The ﬁtness value of the organism increases by har-

vesting the “feed” of the raster ﬁeld on which it is lo-

cated. At the same time, the “feed” volume decreases

on the ﬁeld when it is harvested. Thus, the organ-

ism has to move to other ﬁelds in order to “consume”

more “feed”. Additionally, the ﬁtness of the organism

is decreased by the number of neurons (divided by

100). Thus, a solution with less neurons is regarded

as more efﬁcient.

To interact with the environment, we amended the

BIB-model by sensors and motors, which represent

special types of neurons. Sensors are able to measure

the “feed”-concentration at their location. Motor neu-

rons can generate an impulse for a movement. The

direction of the movement is determined by the rela-

tive direction between the center of the neural net and

the position of the motor neuron in the network. As

we used spiking neurons for the transition of informa-

tion, the “feed” concentration was converted into a se-

quence of spikes originating from the sensor neurons.

A higher concentration of “feed” leads to a higher ﬁr-

ing rate. Likewise, a higher ﬁring rate of a motor neu-

ron generates a stronger impulse for a movement into

a certain direction. The CPPN of a BIB was extended

by two output nodes encoding the existence of sen-

sor and motor neurons for a given location. A BIB,

converted to a neural network, was simulated for 5

seconds (simulation time) in intervals of 250ms.

The evolutionary algorithm of NEAT was used to

improve the network conﬁguration. 100 individuals

per generation were tested and the four best individu-

als of each species were used to create the next gener-

ation. Crossover was always applied in order to gen-

erate a new genome. New genes were added with a

probability of 0.1, new connections between genes

with a probability of 0.2 and continuous changes

of the connection weights with a probability of 0.2.

Figure 5: Some elements of the simulation. a) The organ-

ism moves on a 2D-plane with different concentrations of

a ﬁctive substrate called “feed”. The concentration of any

point on this plane is the interpolated value of the centers

of each ﬁeld. b) The frequency of a spiking motor neuron

determines the strength of a movement impulse towards its

direction. The cumulated sum of movement impulses is the

direction in which the organism moves.

Three-dimensional networks were created with a res-

olution of ﬁve neurons per axis.

The whole evolutionary search was repeated ten

times. In all runs, networks were found in which sen-

sors forwarded information to motor neurons in or-

der to move the organisms towards higher concentra-

tions of “feed”. In some solutions, normal neurons

were completely avoided to transport sensory infor-

mation to the motor neurons. Figure 6 shows some

of the networks that have been generated. The evo-

lutionary search revealed that those BIBs that have

wider axon cones in order to reach more motor neu-

rons, show increased ﬁtness because they can move

faster towards higher concentrations of “feed”. This

example provides evidence that changes in the BIB

can lead to abstract alterations in the neural model

(such as axon cone width) that can inﬂuence the ﬁt-

ness of the model.

As the network is generated from a CPPN,

changes in the expression intensity of a gene can

have a global inﬂuence on the anatomy of the net-

work, when they are directly or indirectly (via gene-

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Figure 6: Some networks generated by the evolutionary

search for BIBs. Yellow spheres represent sensor neurons,

red spheres represent motor neurons. Cones indicate axon-

and dendrite areas of the neurons. a) One of the ﬁrst net-

works, that successfully controlled the organism towards

more “feed”. b) and c) are more sophisticated versions of a)

with less neurons and wider axon-cones leading to a faster

movement of the organism and less energy consumption. d)

depicts the averaged ﬁtness for one run over 20 generations.

interactions) involved in the activation of certain neu-

ral parameters. However, continuous changes of

gene-values cause continuous changes in the net-

work architecture. Thus, an evolution-driven search

can gradually improve ﬁtness by altering the weights

of the cis-regulatory elements. New genes or cis-

elements can have a stronger inﬂuence of the network

architecture leading to new innovations.

5.2 BIBs as a Model for Complex

Nervous Systems

A BIB does not model a developmental growth pro-

cess in terms of local interactions, cell division or cell

movement but encodes the properties of a given lo-

cation directly. However, this does not mean that a

BIB does not reﬂect the outcome of a growth process.

As already argued for CPPNs (Stanley, 2007), genes

and their corresponding proteins are involved in a cas-

cade of local interactions and diffusion processes that

generate gradients of various protein concentrations

(Meinhardt, 2008; Raff, 1996; Curtis et al., 1995).

These gradients serve as local axes for subsequent lo-

cal reactions with other genes. From this perspective,

genes, not directly involved in building structural ele-

ments, can be regarded as density functions activating

or inhibiting other genes and thereby facilitating coor-

dination on various spatial scales. Therefore, CPPNs

(and BIBs) implicitly model the chronology of devel-

opmental events and cell interactions by concatenat-

ing functions with each other as depicted in Fig. 2.

Figure 7: BIB as model for cortical layers. a) A CPPN

with ﬁve genes generates patterns along the x-axis for cer-

tain properties of the neurons. b) 2D-Slides of the output

values modulated by the genes.

The BIB-scheme amends the properties of CPPNs by

modeling connections between neurons as side-effect

of their location, orientation, anatomical structure and

coincidental intersections with other neurons. This

means that, like in biological systems, genes control

global properties of the anatomy, single connections

between neurons are derived from that as side-effect.

The BIB-approach might be therefore a valuable

tool to combine gene interactions, single neuron prop-

erties and large-scale cortical and subcortical struc-

tures in one model. Rudimentary examples for this

are given in ﬁgure 3 and 7. Figure 7a shows a CPPN

for generating cortical layers. Using one coordinate

axis, eight genes contribute to the formation of neu-

ronal elements and the length and angle of the axon

and dendrite cones. The density distribution of the

output genes controlling the properties of the neurons

are depicted in ﬁg. 7b. Like MRI- or diffusion tensor

imaging (DTI) data, the pictures show the anatomical

properties of the network in one slice (for instance,

neuron locations, projections) in a color encoding.

These data can be sampled in any resolution to gen-

erate the three-dimensional representation of the cor-

tical layers (ﬁg. 8). In future work, such models can

serve as a model to better understand the relation be-

tween certain genes and their phenotypic inﬂuence on

the neural network.

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Figure 8: The neural network generated from the CPPN de-

picted in ﬁg. 7. Neurons with axon and dendrite cones are

placed in space according to the local properties generated

by the CPPN a). These cones are used to detect connections

between neurons, depicted in b).

6 DISCUSSION

In this paper, we proposed a model for generating

complex neural networks based on an indirect encod-

ing scheme for three-dimensional patterns, which we

call Brain in a Box, or BIB. Connections between

neurons are only affected indirectly, as their position,

orientation, axon and dendrite parameters determine

the connection to other neurons. In this regard, sin-

gle genes in a BIB modify abstract properties of the

network, such as the thickness of neuronal layers, the

degree of branching or the proximity of neural groups.

Thus, BIBs reﬂect a way of modeling networks that,

to our mind, corresponds more to biological mech-

anisms for the emergence of complex nervous sys-

tems originating from gene regulatory networks. In

this context, BIBs might also be a valuable tool to de-

scribe structural properties of real biological networks

through gene interactions. By exploiting properties of

CPPNs, BIBs show an inherent stability across gener-

ations and but also the ability to change global prop-

erties of the network architecture to increase ﬁtness.

Future research will focus on deﬁning appropri-

ate ﬁtness-functions to model certain functional prop-

erties of areas in complex nervous systems through

BIBs. These functions may include short- and long-

term plasticity for the storage of information, learning

of patterns / sequences for motor control or the detec-

tion of errors between perceived and expected infor-

mation.

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