Bose –Einstein Condensation in Complex
networks related to Consciousness
Monendra Grover*1 and Ritu Grover2 1 Amity Institute of Biotechnology
Current Address: National Bureau of Plant Genetic Resources, New Delhi 2. Directorate of Training and Technical Education, New Delhi
*Author for correspondence, email: monendra_grover@yahoo.com
Abstract: Many complex systems such as World Wide Web, business and citation networks have been shown to follow Bose statistics and can undergo Bose-Einstein condensation. In this paper it is shown that neural networks connected by gap junctions may also follow Bose statistics and undergo Bose-Einstein condensation. The neural networks have been shown to be important in maintaining neural synchrony which in turn has been implicated in consciousness.
Introduction
Though the complex systems such as World Wide Web, business and citation networks display irreversible and non-equilibrium nature, these networks follow Bose Statistics and can undergo Bose-Einstein condensation .A common feature of the systems mentioned above is that the nodes self-organize into a complex network, whose topology and evolution reflect typical dynamics. In this paper it is postulated that the networks in the brain related to the function of consciousness can be mapped into an equilibrium Bose gas (Huang, 1987), neurons mapping to energy levels and gap junctions corresponding to particles. It is inferred that such networks may undergo Bose-Einstein (BE) condensation, in which a single set of neurons captures a macroscopic fraction of dendrites/axons/glia through gap junctions. Some of the mathematical equations used in this paper (Equations 1 to 18) have been worked out for World Wide Web, business and citations in previous studies (Bianconi and Barabasi 2001). In this study the application of these equations for networks related to consciousness in the brain is shown. Marshall (1989) has suggested that membrane proteins in brain neurons form Bose-Einstein condensates whose coherence accounts for "binding" is essential to consciousness. However Bose Einstein condensation has not been studied in consciousness at the level of neural networks.
Neural synchrony has been linked to consciousness. Gap junctions or electrical synapses are open windows between adjacent neurons and have been shown to mediate gamma EEG/coherent 40 Hz and other synchronous activity (Dermietzel, 1998, Dicsi, 1989, Hormuzdi et al. 2004, Bennet and Zukin, 2004, LeBeau et al. 2003 Friedmand and Strowbridge, 2003, Buhl et al. 2003, Rozental et al. 2000, ,Perez-Velaquez and Carlen, 2000,Galarreta and Hestrin 1999, Gibson et al. 1999) which has in turn been related to consciousness (Hameroff, 2006). Gap junctions occur between axons and dendrites, between glia, between neurons and glia, between axons and axons and between neural dendrites. The gap junctions bypass chemical synapses (Traub et al. 2002, Froes and Menzes, 2002, Traub et al. 2002, Bezzi and Volterra, 2001). Gap junction connected neurons have both continuous membrane surfaces and continuous cytoplasmic interiors since ions, nutrients and other material pass through the gap junctions. Neurons connected by gap junctions are electrically coupled and depolarize synchronously (Kandel et al. 2000). A single neuron may have many gap junctions and only some of these are open at given time, with openings and closings regulated by cytoskeletal microtubules and/ or phosphorylation via metabotropic receptor activity (Hatton, 1998). The increase in pH in the cell increases gap junctional coupling and decrease in pH results in the decrease of gap junctional coupling. The increase in cytoplasmic concentration of Calcium can result in decrease in gap junction coupling. In the ensuing discussion the neurons with greater number of open junctions are taken as being more fit than those having less number of open junctions. Thus the fitness function ηi is defined as
y iCa
H
z
PO
C
2 2 3 4
Where C is a constant, [(PO4)3
intracellular hydrogen ions and x is the exponent of hydrogen ion concentration and [Ca2+] is the intracellular calcium concentration and y is the exponent of the calcium ion concentration. It is notable that x, y and z may be equal to 1 or any other real number.
The network of the consciousness in the brain grows through the inclusion of new neurons through the transmission of the electrical/chemical signals. The neurons in the network have the different ability to form connections with new neurons. This differential ability is accounted for a fitness parameter which is assigned to each neuron (defined above) chosen from a distribution of ρ (η). This may account for the different values for consciousness vector (Grover, 2011) in different organisms/states of the consciousness. The probability П i that
a new neuron connects one of its m links to a neuron ialready present in the network depends on the number of gap junctions kiand on the fitness ηi of neuron isuch that
l l l i i i
k
k
Equation 1An energy εi is assigned to each neuron, determined by the fitness ηi through the relation
i i
1
log
Equation 2
In this relation β =1/T or inverse temperature. A link between the two neurons i and j with energies εi andεj
corresponds to two noninteracting particles on the energy levels εi andεj . Adding a new neuron to the network
corresponds to a new energy level εi and 2m open gap junctions to the system. Of these open gap junctions some
are deposited on the level εi corresponding to the m outgoing link processed by neuron i while the other open
gap junctions are distributed between the other energy levels. Each neuron added to the system at time ti with
energy εi is characterized by the occupation number ki (εi, t, ti ), denoting the number of gap junctions (particles)
that the neuron (energy level) has at time t. The rate at which level εi acquires new particles (open gap junctions)
is
t i i i i i i iZ
t
t
k
e
m
t
t
t
k
,
,
,
,
Equation 3
Where Z t is the partition function defined as
t j j j j it
e
k
t
t
Z
1,
,
Equation 4It is assumed that each neuron increases its connectivity according to a power law
Equation 5
Where f (ε) is the energy dependent dynamic exponent. The energy levels are chosen from the distribution g (ε) = βρ(e-βε)e-βε. Z can now be determined by averaging over g (ε), i.e.,
Equation 6 Where
f
e
g
d
z
1
1
Equation 7
t it
d
g
dt
e
k
t
t
Z
1 0
,
,
0
,
,
( ) i f i j i it
t
m
t
t
k
is the inverse fugacity and α = min ε[1 – f((ε)] >0. Since z is positive for any β ≠ 0 the term chemical potential µ
has been introduced where z = eβµ (Bianconi and Barabasi. 2001 b) which allows the equations 6 and 7 to be written as
mt
Z
e
tt
lim
Equation 8
By using equation 8 the continuum equation 3 can be solved where the dynamic exponent is
e
f
Equation 9By combining Equations 7 and 9 the chemical potential is the solution of the Equation
1
1
1
,
e
g
d
I
Equation 10
Equation 10 indicates that in the thermodynamic limit (t →∞) the fitness model maps into a Bose gas. For an ideal gas of volume v = 1, we have (Huang, 1987)
1
d
g
n
Equation 11Where n(ε) is the occupation number of a level with energy ε. Equation (10) indicates that for the gas inspired by the fitness model the occupation number can be described by Bose statistics (Huang, 1987)
1
1
e
n
Equation 12
i.e. the evolving network of neurons maps into a Bose gas.
The solutions Equation 5, Equation 6 and Equation 9 exist only when there is a µ that satisfies Equation 10. However I (β, µ) defined in equation 10 takes its maximum at µ =0, thus, when I (β, 0) < 1 for a given β and g (ε). Equation 10 has no solution. This is reminiscent of Bose-Einstein condensation indicating that finite n0 (β)
fraction of particles condense on the lowest energy level. Due to mass conservation at time t, t energy levels are populated by 2mt particles, i.e.,
Equation 13
When I (β, 0) <1, Equation 13 is replaced with
n
o
mtI
mt
mt
,
2
Equation 14Where n0(β) is given by Equation 7
0
,
1
I
mt
n
o
Equation 15The occupancy of the lowest energy level corresponds to the number of links the neuron with the largest fitness has. Thus the emergence of a non zero n0(β) , a characteristic of Bose-Einstein condensation in
quantum gases, represents the phenomenon of fittest neuron acquiring a finite fraction of the gap junctions, independent of the size of the network
,
,
,
2
1
mtI
mt
t
t
k
mt
t to
o to
The mapping to a Bose gas predicts the existence of three distinct phases characterizing the dynamic properties of neural networks: 1. scale –free phase 2. A fit-get rich phase. 3. A Bose-Einstein condensate
1. Scale-Free phase. When all the neurons have the same fitness, i.e.,
the model reduces to the scale free model (Barabasi and Albert, 1999). The model describes a behavior in which the oldest neurons acquire most links. Equation 9 predicts that f(ε) =1/2; i.e., according to Albert et al. 1999 all neurons increase their connectivity as t ½, the older neurons in the network, with smaller ti having largerki..
However the share of links of the oldest neuron decays to zero as t -½, in the thermodynamic limit.
2. Fit get rich phase - This phase emerges in systems where nodes have different fitness such as in the system considered in this paper where neurons have different fitness defined by fitness function (see above). In this case Equation 10 has a solution, i.e., I (β, µ) = 1. Equation 5 indicates that each neuron increases its connectivity in time but the dynamic exponent is larger for nodes with higher fitness (Bianconi and Barabasi, 2001). Thus fitter neurons which join the system surpass the less fit but ‘older’ neurons by acquiring links at higher rates.
3. Bose-Einstein Condensate – Bose-Einstein condensation emerges when equation 10 has no solution, at which point Equations 5, 9 and 10 break down. In the competition for links the neuron with the largest fitness emerges as the winner. Thus a finite fraction of particles (gap junctions) [n0(β)] land on this energy level. Thus
Bose-Einstein condensation predicts a situation in which the fittest neuron despite the presence of neurons that compete for links always acquires a finite fraction of links (Equation 15).
To demonstrate the existence of a phase transition from the Fit-get-rich phase to a Bose-Einstein condensate, the energy distribution is represented as follows
C
g
Equation 16Where is a free parameter. For this class of distributions the condition for a Bose Condensation is
1
1
1
maxmin 1 max
t x
e
x
dx
, Equation 17
Where εmin (t) corresponds to the fittest neuron present in the system at time t. The lower bound for the critical
temperature at which phase shift takes place can be defined as
11
max
1
2
BE
T
Equation 18Conclusions
Various types of complex networks such as World Wide Web, citations and maps can be mapped on to Bose gas. In this paper it is shown that the neural network involved in consciousness can be mapped on to a Bose gas and may undergo Bose-Einstein condensation. The neurons are taken as nodes and open gap junctions as links in this study. The neuron with maximum open gap junctions has been taken as node with maximum fitness.
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