**Structure in human consciousness:**

**A fractal approach to the topology of the self
perceiving an outer world in an inner space**

Erhard Bieberich

Department of Biochemistry and Biophysics

Medical College of Virginia Campus of Virginia Commonwealth University

Richmond, VA, 23298-0614, U.S.A.

Correspondence to: Dr. Erhard Bieberich

Department of Biochemistry and Molecular Biophysics

Medical College of Virginia of Virginia Commonwealth University

1101 East Marshall Street Box 614

Richmond, VA, 23298-0614, U.S.A.

Phone: 804-828-9217, fax: 804-828-1473

E-mail: EBIEBERI@HSC.VCU.EDU

Key words: consciousness, self, neural
network, fractals, brain, artificial intelligence

__Abstract__

In human consciousness a world of separated objects is perceived by an inner observer who is experienced as an undivided feeling of one-self. A topological correlation of the self to the world, however, entails a paradoxical situation by either merging all separated objects into one or splitting the self in as many disconnected sub-selves as there are objects perceived. This study introduces a model suggesting that the self is generated in a neural network by algorithmic compression of spatial and temporal information into a fractal structure. A correlation of an inner observer to parts of a fractal structure inevitably entails a correlation to the whole, thereby preserving the undividedness of the self. Molecular mechanisms for the generation of a fractal structure in a neural network and the possibility of experimental investigation will be discussed.

__Introduction__

Consciousness has remained to be an enigmatic property of the human mind since its nature was discussed in the early days of philosophy. The most elusive phenomenon observed with consciousness is its ability to generate an inner imagination or view of the outer world. It is based on a very common sense experience of a "thinking thing" inside our mind though unprovable by critical scientific analysis. A lack of appropriate scientific means to analyze human consciousness results in the view that mind function is reducible to mere electro-physiological mechanics of the nervous system. Reductionistic models are useful in explaining the computation of nerve signals by specialized areas of the brain but are not able to explain the integration of conscious experience emerging from these signals. The simultaneous perception of spatially or temporally distinct information apparently does not destroy the feeling of an undivided self as being the only observer aware of the inner world (Baars J., 1997; Strawson G., 1997). The experience of the self in a one-to-all correspondence can be established by the principle of fractality. In a fractal structure iterative pointing at one element correlates the pointing entity to the entire fractal as being a downscaled part in each of its elements. The underlying geometry has been intensively analyzed by Benoit Mandelbrot and shown to provide a general principle of structurization in nature (Mandelbrot B., 1982). The present study discusses a model for application of fractality to integration of information perceived in consciousness in order to explain the preservation of an undivided inner observer, the self. An underlying algorithm for fractal integration will be developed and correlated with a neurophysiological process in the nerve cell membrane. On basis of this process an experimental approach for the investigation of a putatively fractal structure in the nerve cell membrane will be suggested. The following sections will be composed of discussion parts and more detailed mathematical descriptions. For brief reading, the mathematical analysis can be skipped without loss of crucial information.

__Results and Discussion__

__The self and the world as a topological
space__

In order to install an algorithm for the description of the
self and the topology of consciously perceived spatio-temporal information we
have to develop a symbolic notation which is able to cope with a standard
mathematical form. (The operations __and__ and __or__ will be used for
"union" and "intersection" as in set theory or according to the equivalent
symbols in Boolean logic, respectively).

Suppose that the self can be defined as a set S with only one
member: S itself. S is introduced as a quality in order to have a variable which
can be included into an algorithmic description. The first premise for the
definition of S is that it has to be an irreducible entity which under any
operation cannot be divided into different sub-selves. Nevertheless, S is the
entity in each of us which is able to perceive all objects, sensations and
emotions we are aware of in our consciousness. Now suppose there is a
topological space X in which this information is distributed. We will define a
group of spatially distributed sensations as subset A being disjointed from the
complement subset nonA. Accordingly, the intersection of A and nonA will form an
empty set: A __and__ nonA = 0. As shown in Fig. 1, a one-to-one
correspondence of S to A or nonA leads to two putative solutions each of which
being contradictory to the initial properties of S or A or nonA, respectively.
Solution 1 favors either the combination S __and__ A or S __and__ nonA.
Since A and nonA are complementary and in turn disjointed subsets in X, the
topological space will become disconnected referring to the embedded subset.
Depending on which information S is focused on it will also be embedded in the
respective and in turn distinct subspace or it will be split into two disjointed
subsets s_{1} or s_{2}. This implies the logically contradictory
situation: S unequals S. On the other hand, solution 2 will preserve the
undividedness of S but in consequence abandons the spatial difference between A
and nonA. This also implies a contradictory situation in which a homogeneous
mixture of sensations is experienced: A equals nonA. The logical dilemma arises
from the attempt to apply the distributive law to the expression S __and__ (A
__or__ nonA) which will inevitably reduce S to the correlation S __and__ A
or S __and__ nonA. A solution of this dilemma is possible if a part of a
geometrical object is always composed of a downscaled version of the entire
structure. Pointing at one element entails inevitably correlation to the whole.
Applied to S, a correlation to A or nonA preserves its undividedness since any
attempt to reduce S by correlation to a part iteratively re-establishes a
one-to-all correspondence. The scale-invariance of correlation is known to be
realized by generation of a fractal structure and will be evaluated in the
following mathematical section.

________________________________________________________________

__Mathematical analysis 1__: Scale-invariance of correlation
in fractal structures:

Solutions 1 and 2 in Fig. 1 will be expressed in terms of
operator functions on S in order to resolve the logical contradiction to the
initial conditions by generation of the self. The logical states A and nonA will
be merely characterized by the ability to cover sub-spaces x_{i} of X.
By definition it follows that x_{1}(A) __or__ x_{2}(nonA) =
X, and x_{1}(A) __and__ x_{2}(nonA) = 0. The simultaneous
realization of the two logical states will be described by a linear
superposition with f, g and h acting as operator functions on S:

f{S __and__ X} = g{S __and__ A} + h{S __and__ nonA};
Eq. 1

The undividedness of S will now be preserved by the introduction of a scaling operator z such that S remains invariant under scaling by z:

f{S __and__ X} = f[z {S __and__ x_{1}(A)}] = f[z
{S __and__ [x_{1}(A) __or__ x_{2}(nonA)]}]; Eq.
2

The realization of the scaling invariance of S will be
approached by a construction of X according to self-similar or fractal
structures. A general algorithm for the description of fractals develops from
the idea that in self-similar structures there is always the same correlation
between the size of pieces x and their number N(x) (Peitgen H.-O. et al., 1992).
This correlation can be interpreted in the sense that the entire structure
perceived by the self is a scale-invariant magnification of each part of it. A
possible solution for equation 2 with x = S __and__ X is then given by a the
following power law (Liebovitch L.S. and Toth I., 1990):

f(x) = C x^{(1-D)}; Eq. 3

where, C is a constant and D = lim(x __to__ 0)
[logN(x)/log(1/x)]

The fractal dimension D is a constant for the geometrical description of self-similar structures (Peitgen H.-O. et al., 1992).

In the present context the power law describes the topological
scale-invariance of the dyadic combination S __and__ X.

________________________________________________________________

In Fig. 2A the principle of self-similarity is illustrated by
iterative tiling of a plane giving rise to the construction of a Sierpinski
triangle. A particular feature of fractal tiling is again given by the principle
of scale -invariance: a correlation to one tile by iterative downscaling entails
a scale-invariant correlation to the entire structure. It is just this property
of the fractal concept which provides a solution for the correlation of S to
distinct tiles under preservation of its irreducibility. There is, however, one
crucial pitfall in any iterative downscaling operation. No matter how small the
tiles are, they are still separated. Infinite downscaling operations require an
__infinite__ structurization of space and time which is hard to imagine for a
physical substrate. Instead it is assumed that there is a spatial state reached
by __finite __downscaling when the fractal is composed of separated but yet
"connected" (not independent) states. The connectedness can be realized by
engaging spatially separated states in an intermediate pre-state of not being A
or nonA, but temporally synchronizing a subsequent manifestation of the two
states. This phenomenon of fractal coherence will be described by a combination
of eqs. 1 and 3, with x = S __and__ X and X = x_{1}(A) __or__
x_{2}(nonA), which establishes a scale-invariant and non-distributive
correlation of S to X:

f{S __and__ X} = C {S __and__ [x_{1}(A) __or__
x_{2}(nonA)]}^{(1-D)} ; Eq. 4

It should be emphasized at this point, that the term fractal coherence is introduced as a mere mathematical construct in order to meet the principles of scale-invariance and non-distributiveness. We will now evaluate how this operation for fractal construction might be realized in our brain (or in any appropriate physical environment).

__The tiling of the world in our mind__

In order to describe the construction of a fractal structure in
a neurophysiological substrate it is necessary to define an construction
algorithm compatible with the physical properties of this substrate. It should
be noted, however, that the primary goal of this section is to develop a
figurative but still completely abstract algorithm for the realization of
spatial information perceived in our consciousness. A putative correlation to a
neurophysiological substrate will be discussed in the following sections. A
general directive for the construction of fractal structures is given by a
specific affine linear transformation w_{i}(P_{i}) of n points
P_{i} in a metric system R(N), with N = spatial dimension (Peitgen H.-O.
et al., 1992). A transformation w generating self-similar structures by
iterative downscaling is given by the Hutchinson operator (Peitgen H.-O. et al.,
1992). The application of this operator on different geometrical directives for
spatial distribution of points is able to entail the generation of fractal
structures which are transformable into each other. Figure 2 illustrates this
equivalence for the Sierpinski triangle which can be constructed by iterative
rotation and contraction of a triangle in a plane (Fig. 2A), or by a tree like
network as in Fig. 2C. The fractal is implicitly contained within a modified
version of the "Apollonian gasket" generated by a distribution of hexagons shown
in Fig. 2B (see Fig. 3 for comparison). The equations underneath the figures are
of descriptive nature derived from the algorithmic similarity for the
construction of different fractal structures (Peitgen H.-O. et al., 1992). A
comparison of the Sierpinski triangle (Fig. 2A) with the modified (hexagonal)
Apollonian gasket (Fig. 2B) indicates a construction directive for the spatial
adjustment of signal flow by linear affine transformations in order to generate
a fractal in a neural network.

________________________________________________________________

__Mathematical analysis 2:__ Spatial adjustment of signal
flow for fractal construction

A covariant component of the input vector v_{i}' is
iteratively subtracted until the output vector v_{i} is orthogonal to
u_{i}. This computational process is similar to a model introduced by
Pellionisz and Llinas for signal adjustment of sensory input to executive motor
output vectors in the cerebellum (Pellionisz A. and Llinas R., 1980). By this
adjustment the system of u-vectors is consistent with the construction of the
Sierpinski triangle by a tree like network with the enclosed angles determining
the direction cosines (Fig. 2C). It should be noted for purpose of
generalization that the network construction of the fractal is again directed by
a linear affine transformation, this time derived from a more convenient polar
coordinate notation. Linear affine transformation is driven by rotation (by
angle as indicated in Fig. 2) and translation of a triangular plane in each step
of iteration.

________________________________________________________________

The equivalence of the Sierpinski triangle to a tree like
network establishes the spatial adjustment of the affine transformation as a
process "growing" by time. This property is most remarkable as it is in
accordance to the distribution of information in neural networks with dendritic
structure (Peitgen H.-O. et al., 1992). Once the network splits into
sufficiently small branches coherence occurs as described in the preceding
section. In Fig. 2B, it is illustrated the simultaneous effect of three signal
vectors u_{1-3} on one point element surrounded by a hexagonal area
under coherence. This array will be defined as the smallest fractal d{SX}
representing the elementary tile for the construction of space perceived in
consciousness. The metric underlying the elementary tile d{SX} can be formed by
a coordinate system with the principle axes x_{i} lying in the plane
enclosed by the tile (see Fig. 2B). In the case of the Sierpinski triangle the
vectors u_{i} pointing to the vertices are collinear to x_{i}.
Rotation will superimpose them irrespective of the construction directive
applied (Fig. 2A or C). In order to describe the perception of space by an inner
observer it is suggested that these vectors generate the three components
X_{i},Y_{i}, Z_{i} for one point in a three-dimensional
space perceived by the self. As illustrated in Fig. 2D the respective operation
"lifts up" the spatial information given by the fractal in R2 to a point seen in
R3. The eye in Fig. 2D representing the self as the inner observer of visual
information adopts a dual position in either R2 or R3, thereby indicating a
general consideration for any RN. As shown in Fig. 2D the components of any
point perceived in consciousness are subject to transformation to a coordinate
system opposite to the position of the observer. Adjacent tiles don't simply
fuse in a coherent area of homogeneously mixed information (equivalent to
Solution 2 in Fig. 1) but superimpose their texture without giving up their
individual spatial distribution. Each elementary tile behaves like a "point",
but forms a connected space with other tiles covering the entire coherent
lattice. The size of adjacent fractals may vary depending on the actual
coherence length. This does not violate the principle of self-similarity, but
gives rise to points in R3 with different distance to R2 and may distribute them
in a three-dimensional space as it is experienced in our mind. It should be
emphasized that the space perceived in consciousness is not to be assigned to an
Euclidean space somewhere in brain. In fact, the directive suggested for
construction of this "hyper"space (see Fig. 2D) implies the impossibility to
form a common space with the vectors used for its generation in a
neurophysiological substrate. Instead, it is only possible to investigate the
neuro-physical principles underlying this construction in an experimentally
accessible Euclidean space. The hexagonal version of the Apollonian gasket is
meant as an approximation for a model structure generated in a neuronal
substrate coping with fractality as well as coherence. We will now evaluate how
the transformation from a fractal network to a neuronal substrate under
coherence may develop.

__Principles of fractal coherence__

The distribution of the self to the elements of a fractal structure is attempted by a probabilistic approach derived from the theoretical basis of the generation of fractal images (Peitgen H.-O. et al., 1992). According to this model the spatial distribution of the self (equivalent to the image density) is depending on the distribution density of the preceding image (pre-image). This approach appears to be justified since fractal distribution follows a certain path determined by the self-similarity in each step of construction. The construction directive applied is called Markov process and can be expressed for any type of fractal structure. The advantage of this process is given by a spatial adjustment of signal flow (by Hutchinson operation) reacting to the actual signal distribution in each step of iteration. Iterative distribution of signals or the self onto downscaled space intervals will proceed until the spatial limit for coherence is reached. The distribution density of S derived from the Markov operation will then be transformed to that of the space under coherence. A mathematical analysis will show that a Markov operation can be converted to a quantum computational process relying on the equilibrium density of the coherent state.

________________________________________________________________

__Mathematical analysis 3__: Conversion of a fractal to a
coherent state

Suppose that in a pre-coherent state t<t_{c} the
self S hits the space intervals dx_{1}=x_{c}-x_{1} and
dx_{2}=x_{2}-x_{c} with the probabilities
p_{1}(dx_{1}) and p_{2}(dx_{2}). The
distribution of the self is then given by a Markov operation according to
fractal image construction as described in (Peitgen H.-O. et al.,
1992):

M(v) = p_{1} v(S) w_{-1}(dx_{1}) +
p_{2 }v(S) w_{-1}(dx_{2}); Eq. 4

with w(dx) = Hutchinson operator with w_{-1} =
pre-image on dx

v(S) = integral of {u(dx,t) dx}

u(dx,t) = distribution density of "S" on the pre-image

An iteration of w(dx) leads to a fractal distribution of S on
sequentially contracting intervals of dx. After sufficient downscaling
iterations, S is driven to the attractor x_{c}+/-dx_{l} which is
equivalent to the point of coherence. The probabilities for the distribution of
S onto dx derived from the Markov process are transformed into those for the
description of a coherence equilibrium.

_{}p(eq) = ½ (p_{1} + p_{2}); Eq.
6_{}

This equation resembles an algorithmic description recently
introduced for quantum computation of bulk spin states (Gershenfeld N.A. and
Chuang I.L., 1997). In anlogy to the non-distributive superposition of the
logical states A and nonA, the coherence equilibrium of i eigenstates on each
scale can be calculated in terms of energy levels a_{i}.

_{}p_{i} ~ _{}a_{i} =
hf_{i}/(2kT); Eq. 7

(in terms of Boltzmann factor k), f_{i} = frequency of
the wave-function).

________________________________________________________________

The mathematical analysis eventually resulting in eq. 7
indicates that the recently developed principles in quantum computation may be
useful for a description of the logical superposition S __and__ (A __or__
nonA) in the space experienced in human consciousness. An adequate description
has to take into account all combinations of space intervals with the self
(e.g., with "spin up" equivalent to "hit by the self" and "spin down" equivalent
to "not hit by the "self"). With N space intervals we will find a coherence
equilibrium of 2^{N} combinations (Gershenfeld N.A. and Chuang I.L.,
1997). The discrete probability for each combination will then be derived from
those of the Markov process. Quantum computation as expressed by eq. 7, however,
provides no proof that the energy levels are spatially distributed according to
a fractal structure, that is to say: the coherence equilibrium is scale
invariant. A simple power law according to a_{i} = const. h
f_{i}^{(1-D)}/(2kT) (see eq. 3) appears not to be sufficient to
justify this conclusion. The scale invariance, however, is as discussed earlier
the prerequisite in the present model that a logical superposition of
eigenstates results in a fractal distribution of energy nodes. In other words,
for implication of a conscious experience, it is to be shown that quantum
computation is based on a fractal structure implicitly contained within a space
under coherence.

__A neurophysiological substrate of the self__

On search of a physiological mechanism in our brain which is
able to generate consciousness it is crucial to find a substrate being able to
build up a coherent state with underlying fractal geometry. As depicted in Fig.
2C any dendritic network may then be programmed on directing the flow of spatial
information according to a fractal set-up. The undividedness of the self has
often led to the impression that there is a substance under coherence in our
brain. The physical description for the excitation of coherence in biological
substrates was introduced by Fröhlich about thirty years ago (Fröhlich H.,
1968). Recent applications of his theory to an explanation of consciousness by
Hameroff, Pribram and Penrose favor the idea that it is generated by a network
of neuronal microtubules forming an extended coherent phase (Jibu M. et al.,
1994; Penrose R., 1994). In the present study the generation of physically
extended coherent states is avoided. The connection of the self to a world
perceived by it is suggested to be achieved by algorithmic compression of
spatial information according to a fractal structure in a neural substrate. It
should be noted, that the informational flow for fractal construction is subject
to "pre-conscious" neuronal computation and can be evenly distributed to any
space of the brain. This circumvents the necessity that the realization of the
self is restricted to a microscopic spot somewhere in the brain. However, in
order to avoid a speculation on non-local (remote) connectedness of events
perceived in the human mind, the neuronal computation has to be based on each
distinct information represented in consciousness. A microscopic coherent state
is most likely generated under the influence of neuronal signal transduction.
The nerve cell membrane is permanently subject to electro-physiological
excitation owing to ion fluctuations which are triggered by transduced signals
from different spatial directions. The ion fluctuations are depending on the
conformational changes of ion channel proteins integrated within the nerve cell
membrane and modulated by surrounding membrane lipids. Recently, a Monte-Carlo
simulation of conformational coherence of membrane lipid arrays in a clustered
lattice surrounding integral membrane proteins (e.g., ion channel proteins) has
shown a sharp increase in coherence length at the transition temperature of the
membrane (Sperotto M.M. and Mouritsen O.G., 1991). Interestingly, the phase
transition temperature is extremely close to the body temperature which is often
correlated with the composition of the lipids in the nerve cell membrane
(Perillo M.A. et al., 1994; Becker K. and Rahmann H., 1995). For a hexagonally
shaped lattice a coherence length of more than 60 was found which is consistent
with a cluster area of about 10^{-17} m^{2} and hence
corresponds to approximately 1/10^{8} of the total membrane surface of
the nerve cell body. This model is in good agreement to the construction
directive for a hexagonal Apollonian gasket as shown in Fig. 2B. A number of
10^{8} ion channel proteins (corresponding to 0.2 fmol) plus surrounding
membrane arrays could build up a lattice of coherent areas on the entire
membrane surface of one cell body. It is more likely, however, that a connected
lattice is composed of smaller patches of lipid arrays under coherence. This
assumption is consistent with recent observations on integral membrane proteins
forming clustered patches (Koh et al., 1994) on the cell surface.

Fig. 3 summarizes the vertical signal flow through five distinct but interdependent levels of "coordinate systems" in the brain. Level 1 describes the local information of the outer world projected onto the retina and is followed by its transfer to visual centers in the brain on level 2. Each center houses a network of converging and diverging nerve connections (level 3) with the goal to integrate and compress the information onto the next level. Level 1 to 3 is fully compatible with the current reductionistic view of neuronal computation and can be intensively described by this type of analysis. Level 4 is most remarkable as it comprises the set-up mechanisms for conscious experience which still elude reductionistic description. According to the model presented here the outer world is reconstructed from the inner point of view of the self on level 5. This level represents the "hyper"space which cannot be assigned to an euclidean space in the brain, but which is constructed by neuro-physiological processes within the brain. We will now discuss how this process can be evaluated by experimental investigation.

__Experimental investigation of fractal coherence in neuronal
networks__

The creation of fractal coherence in a neuronal network is
suggested to provide an experimental approach to investigate neurophysiological
processes underlying consciousness in human mind. The iterative generation of a
coherent structure with inherent fractal organization is assumed to control and
to trigger the signal flux in the network. A model for fractal organization is
derived from the modified Apollonian gasket shown in Fig. 2B. In this model the
vectors u_{i} for construction of the tiles direct the flow of control
signals i(u_{i},t) setting up a polygonal or circular shaped array in a
neurophysiological substrate under coherence as shown in Fig. 4. In Fig. 4C,
I(t) stands for the ion flux for the time t through an ion channel in the center
of one polygonal tile. The system flux I(t) is perpendicular to the control
currents i(u_{i},t) which modulate the opening of the ion channel
protein. They propagate along the membrane and induce coherence in an array of
membrane lipids surrounding the ion channel. The conformation of the channel
protein and that of the surrounding membrane lipids is linked to each other in a
complex manner since the majority of lipids constituting the nerve cell membrane
is negatively charged and may react sensitively to alterations in Na/K-ion
fluxes. Integration of ion fluxes in the nerve cell membrane is consistent with
currently developed models for signal integration in the dendritic trunks or the
cell bodies of nerve cells (Hoppensteadt F.C., 1989; Koch C., 1997; Orpwood,
R.D., 1994; Segev I., 1998; Sporns et al., 1989). Fig. 4 C and D illustrate how
this principle may be realized in the nerve cell membrane or in a putative
"conscious bio-electronic chip". The system current I(t) conducted through the
coherent space area with the size A is determined by the control currents
i(u_{i},t) depending on the directional vectors u_{i} (see Fig.
2B) according to:

I(t) = f(A,t) = f[sum of i(u_{i},t) ×
i(u_{i+1},t)] =

f[sum of i(u_{i},t) i(u_{i+1},t) sin
[i(u_{i},t),i(u_{i+1},t)]]; Eq. 8

with u_{i};u_{i+1} = construction vectors of
coherent space with a common angle as in Fig. 2B.

The system current I(t) can be correlated with the self by the
assumption that I(t) is a function of f{S __and__ X}. The function f{S
__and__ X} is then interpreted as a product vector perpendicular on
u_{i}, a model already introduced for the construction of a
three-dimensional space perceived in consciousness as described in Fig. 2D.
Thereby, the self would be composed of a vector field generated by elementary
and coherent tiles of a fractal structure and distributed throughout the brain.
In the present model, the distribution would be triggered by the development of
the system current I(t) which is relying on the conductance or resistence
properties of the coherent area. These properties may be different for a
fractally compared to a non-fractally structurized space. Since an intermediate
decoherence would enter an iterative Markov process of downscaling, the coherent
area may be trapped in a transfinite state between superposition of logical
states by coherence and their particular manifestation on spatially distinct
energy nodes. Iterative reverberation will proceed until eventually sufficient
energy loss induces the final decay of the coherent state. If the system current
is feedback-coupled to the control currents, there will be a characteristic
oscillation frequence or resonance for I(t) distinguishing the fractal from the
non-fractal behavior of the coherent area. It is assumed that there are specific
resonance attractors (or repellors) reacting to harmonic frequency bands. An
experimental verification of these assumptions is expected from the application
of the patch-clamp technique to a nerve cell membrane triggered with control
currents from different directions. A certain area of the membrane can be
excised from the nerve cell by the tip of a micropipette. It will be enclosed by
a polygonal or circular shaped mesh which is spiked by radial electrodes as
shown in Fig. 2B. Fractal behavior can be analyzed by recording the oscillation
characteristics of the action potentials upon tiggering the patch-clamped
membrane with control currents generated between the surrounding electrodes. A
direct structural analysis can be achieved by atomic force microscopy with the
patch-clamped membrane spread on a mechanical support (Lärmer J. et al., 1997).
Recent studies on nerve cell action potentials have shown a strict fractal
dependence of the frequency to the power (amplitude) of the potential signals
(Lowen S.B. et al., 1997). This can be explained by fractal thermal noise or
fractally organized changes in the conformation of ion channel proteins reported
previously (Liebovitch L.S. and Toth T.I., 1990). Another mechanism evoking the
characteristics observed may be the fractal structurization of the nerve cell
membrane. At any time the overall intensity of the potential would depend on the
area size of the "elementary fractals" and the probability for signal release on
their abundance in an extended fractal lattice. According to a model introduced
Beck and Eccles the self is created by a quantum physical process deciding on
the probability of a synaptosomal vesicle to merge with the nerve cell membrane
in order to release its neurotransmitter content (Beck F. and Eccles J.C.,
1992). A combination with a fractal lattice imprinted in the synaptic membrane
would enable to control this process by coherence or decoherence of the
elementary tiles. In this case signal flow along the membrane, fractal onset in
the membrane bilayer and transmitter release would create a unified
computational process for generation of the self in the human
brain.

__Conclusions and Perspectives__

A model for a simultaneous realization of two distinct logical states by an undivided self was taken as a first entry to an explanation of perception in human consciousness. In the present approach the self as the perceiving entity emerges from the distribution of the information perceived to a fractally structurized space in the nerve cell membrane. Each different model at least has to cope with a self behaving like a "pseudoparticular" singularity and yet consciously experiencing a space filled with distinct objects. Previous approaches favor the idea that the self emerges from an operation termed "the holographic paradigm" which is developed from a model of holographic memory storage based on the convolution theorem and introduced by Dennis Gabor about thirty years ago (Gabor D., 1968; Jibu M. et al., 1994; Psaltis D. et al., 1990). This model describes the holographic superposition of information in form of spatially spread wave-functions. Spatial spreading, however, abandons the locality of logical states and in turn undoes the prerequisite for their distinction. When a hologram is imprinted on a grating support then the diffraction pattern is again a distribution of spatially distinct states. In this case, however, the holographic imprint will also face the problem of splitting the integrity of the self.

In a consequent application of a materialistic view it is
inevitable that the self is not an independent entity but a quality inevitably
arising from the proper arrangement of physical processes. That is to say
clearly: the self has always been an immanent property of nature but has to
become conscious just by an appropriately organized physical set-up. According
to a thesis introduced by Liberman it is even possible to go further: each
physical process with energy transformation is accompanied by an emotional
quality if the generation of a self provides a conscious "inner point of view"
(Liberman E.A. et al., 1989). A particular organization of our brain generates
not only a self acting as an observer of objects, but is also observing itself,
it is self-conscious. According to the present model this requires the inclusion
of additional information in a fractal structure since the mere perception of
physical objects does not inevitably entail that the perceiving entity is aware
of its own existence. Fig. 5 illustrates a model describing the development of
self-consciousness from a conscious self. The self related to the space
perceived in consciousness evolves in time from {S __and__ X(t_{1})}
to {S __and__ X(t_{2})}. The evolution of {S __and__ X} is driven
by a computation within the coherent state, and eventually is distributed by the
collapsing fractal. The awareness of the self as continuity emerges from a
confrontation of the present {S __and__ X(t_{2})} with the past {S
__and__ X(t_{1})} in a common fractal "space-time" which is the
benefit of having a memory. The mechanisms of this process are too elusive as to
be an additional part of the present discussion but are likely to involve a
superposition of stored nerve signals consistent with the "holographic paradigm"
(Gabor D., 1968; Jibu M. et al., 1994; Psaltis D. et al., 1990; Stamenow M.I.,
1996).

The self may develop in time by iteration of finite "time singularities" (similar to the those of spatial distribution) without losing its temporal integrity. It should be noted that just this integrity warrants the experience of sensations evolving in time like the beauty felt by listening to music. The continuous experience of one-self, however, does not necessarily entail a feeling of being a unique identity. Birds may have a continuous impression of their singing but one would doubt that they define themselves as personalities. In order to build up this identity the self must be fed with a big deal of one's memory in each moment of its existence. Only this simultaneous integration of past and present information ensures that we wake up in the morning as the same person we used to be when we fell asleep the night before. As shown in Fig. 5D this information maybe included within consciousness by a multi-fractal structure. It is possible that a self-conscious self with a feeling of its own identity is the precious gift only man awarded from nature. This does not exclude that in earlier stages of evolution simpler forms of a self were realized. In this study the application of the fractal concept to the internal structurization of the nerve cell membrane suggests a way for experimental verification by patch-clamping technique. The construction of the respective computational element may even found the technical basis for the artificial creation of conscious events in a machine.

__Acknowledgment:__ The author wishes to thank Dr. Robert
K.Yu and Dr. Matthew Tranduc for fruitful discussions and critically reading the
manuscript.

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__Legends to the Figures__

__Figure 1__:

__Algebra of sets in human consciousness__

Euler-Venn diagram for the description of products formed by the an irreducible "self" with spatially disjoint subsets A and nonA. Two paradoxical situations arise from a distributive intersection of S with A or nonA: Solution 1: S is split into two complementary subsets S(A) and S(nonA) and in turn unequals itself. Solution 2: The integrity of S is preserved but A and nonA are no longer complementary to each other and in turn cannot be distinguished by S. The ability of simultaneous perception of A and nonA by an undivided S is illustrated by solution "?".

__Figure 2__

__Fractals in neural networks__

A fractal downscaling according to the Sierpinski triangle can
proceed by tiling (A) or dendritic networking (C). The two downscaling pathways
merge to the same coherent structure with a fractal imprint indicated by a
hexagonal tiling according to a modification of the Apollonian gasket (B). This
can be interpreted as a model for a fractal distribution of wave-nodes for
spatially spread energy states. The vectors given for the construction of each
structure are subject to linear transformation as described in a general
notation by determinant matrices in the equations underneath. The matrix
coefficients describe a transformation of
P_{i}(x_{i},y_{i}) to
P_{i}'(x_{i}',y_{i}') thereby generating a vector
v_{i}. The operations for (A) and (C) are equivalent using a notation
derived from either cartesian or polar coordinates. The matrix for polar
coordinates is based on the assumption that there is first a rotation by
followed by a translation. Iterative downscaling is achieved by the Hutchinson
operator w. In (C) the contraction after rotation is obtained by the scaling
factor r. The spatial adjustment of the v- and u-vectors is reached by
elimination of the non-diagonal matrix coefficients in (B). Under coherence a
different coordinate system with three principle axis x_{i} is used.
This system forms the basis for the definition of points perceived in the space
of consciousness (D). The self is either aware of one point in a three
dimensional space (eye in the lower half) or of three points in a
two-dimensional space (eye in the upper half). Depending on the spatial limit
for coherence in each fractal, the distinct distribution of points forms a
hyper-plane (e.g., P_{a}-P_{c}) which is experienced as a
three-dimensional structure in the human mind.

__Figure 3__:

__Signal flow in the brain__

The divergent or convergent flow of nerve signals is described by direction vectors for each level of neuronal structures. Level 1, retina projection of the outer space; level 2, operational brain centers; level 3, neuronal network; level 4, generation of a fractal structure in a nerve cell (Apollonian gasket formed by patches of membrane proteins and surrounding lipids); level 5, space and objects perceived by the mind.

__Figure __4:

__Creation of fractal coherence__

Biological, electronic, and hybrid elements for the generation
of microscopic fractals. Control currents i_{i} are superimposed to a
coherent distribution of oscillation nodes. Iterative decoherence and
reverberation to the fractal determines the resonance frequency of the system
current I. This process is suggested to generate conscious experience. A:
"Classical" transistor; B: hybrid element, patch clamped nerve cell membrane
with surrounding electrode spikes; C: nerve cell, Apollonian gasket as fractal
in the membrane; D: electronic element (field effect transistor) for generation
of a fractal (Sierpinski triangle); E-F: three elements combined to clusters
with reentry of I directing i_{i}.

__Figure 5__:

__Different levels of the self__

In A spatial information is perceived by one-self but temporal connection is only given in B. In C the actual information from sensual organs is integrated with memory and provides the feeling of being a unique identity. STM = short term memory; LTM = long term memory.

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