E. C. CASTRO, B. A. R. PELI CANO
Copyright © 2013 SciRes. OJAppS
to the transition probability, the trajectory of least action
for the motion can be described. The optimized path in
free space does not depend on the frequency of the par-
ticle to pass through a certain point, but depends on the
most probable point in space in relation to the particle’s
current position. Even though the transition probability is
independent of the history of the Brownian motion, the
path of least action is indirectly affected by the initial
positions.
Brownian motion in a neural network still has a clas-
sical characteristic, i.e. the trajectory most likely follows
a straight line fro m the starting position to the d esignated
point. Whereas, the installation of the cond ition of a final
point never appears in the transition probability calcula-
tion. It appears to be that it is inherent to the transition
probability itself that the Brownian motion is subject to
follow a horizontal of vertical displacement. The final
position can then be reached if the Brownian motion ap-
proaches the point, which can be done by orienting the
edges of t he directed graph.
REFERENCES
[1] L. Zhao, Y.-C. Lai, K. Park and N. Ye, “Onset of Traffic
Congestion in Complex Networks,” Physical Review E,
Vol. 71, No. 2, 2005.
http://dx.doi.org/10.1103/PhysRevE.71.026125
[2] B. Tadic, S. Thurner and G. J. Rodgers, “Traffic on Com-
plex Networks: Towards Understanding Global Statistical
Properties from Microscopic Density Fluctuations,” Phy -
sical Review E , Vol. 69, No. 3, 2004.
[3] G. Yan, T. Zhou, B. Hu , Z.-Q. Fu and B.-H. Wang , “Effi-
cient Routing on Complex Networks,” Physical Review E,
Vol. 73, No. 4, 2006.
http://dx.doi.org/10.1103/PhysRevE.73.046108
[4] C. Daganzo, “The Cell Transmission Model, Part 2: Net-
work Traffic,” Transportation Research Part B: Metho-
dological, Vol. 29, No. 2, 1995, pp. 79-93.
http://dx.doi.org/10.1016/0191-2615(94)00022-R
[5] S. Allesina and M. Pascual, “Network Structure, Preda-
tor-Prey Modules, and Stability in Large Food Webs,”
Theoretical Ecology, Vol. 1, No. 1, 2008.
http://dx.doi.org/10.1007/s12080-007-0007-8
[6] R. Milo, S. Shen-Orr, S. It zkovitz, N. Kashtan, D. Chklo-
vskii and U. Alon, “Network Motifs: Simple Building
Blocks of Complex Networks,” Scienc e , Vol. 298, No.
5594, 2002, pp. 824-827.
http://dx.doi.org/10.1126/science.298.5594.824
[7] S. Strogatz, “Exploring Complex Networks,” Nature, Vol.
410, 2001, pp. 268-276.
http://dx.doi.org/10.1038/35065725
[8] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.-U.
Hwang, “Complex Networks: Structure and Dynamics,”
Vol. 424, No. 4-5, 2006, pp. 175-308.
http://dx.doi.org/10.1016/j.physrep.2005.10.009
[9] L. S. Schulman, “Brownian Motion and the Wiener Inte-
gral; Kac’s Proof,” Techniques and Applications of Path
Integration, John Wiley & Sons Inc., Hoboken, 1981, pp.
53-64.
[10] M. Chaichian and A. Demichev, “Wiener’s Treatment of
Brownian Motion: Wiener Path Integrals,” Path Integrals
in Physics: Stochastic Processes and Quantum Mechanics,
Vol. 1, IOP Publishing, Bristol, 2001, pp. 22-38.
[11] R. Bernstein and S. Bernstein, “Biology,” Wm. C. Publi-
shers, Dubuque, 1996.
[12] P. Raven and G. Johnson, “Biology,” 6th Edition, Mc-
Graw-Hill Companies, New York, 2002.
[13] N. Campbell, J. Reece and L. Mitchell, “Biology,” 5th Edi-
tion, Benjamin Cummings, San Francisco, 1999.
[14] M. D. Odom and R. Sharda, “A Neural Network Model
for Bankruptcy Prediction,” Neural Network s, 1990 IJCNN
International Joint Conference on, Vol. 2, 1990, pp. 163-
168. http://dx.doi.org/10.1109/IJCNN.1990.137710
[15] A. Grabowski, “Interpersonal Interactiona and Human
Dynamics in a Large Social Network,” Physical A, Vol.
385, pp. 363-369.
[16] D. J. Watts and S. H. Strogatz, “Collective Dynamics of
‘Small-World’ Networks,” Nature, Vol. 393, 1998, pp.
440-442. http://dx.doi.org/10.1038/30918
[17] Z. C. Lou, Y. H. Lai, L. L. Chen, X. Zhou, Z. Dai and X.
Y. Zou, “Identific ation of Human Protein Complexes from
Local Sub-Graphs of Protein-Protein Interaction Network
Based on Random Forest with Topological Structure Fea-
tures,” Analytica Chimica Acta, Vol. 718, 2012, pp.
32-41. http://dx.doi.org/10.1016/j.aca.2011.12.069
[18] A. Sharma, S. Costantini and G. Colonna, “The Protein-
Protein Interaction Network of the Human Sirtuin Family,”
Biochemica et Biophysica Acta. Article in Press.
[19] A. L. Barabasi and R. Albert, “Emergence of Scaling in
Random Networks,” Science, Vol. 286, No. 5439, 1999,
pp. 509-512.
http://dx.doi.org/10.1126/science.286.5439.509
[20] G. J. Ortega, R. G. Sola and J. Pastor, “Complex Network
Analysis of Human ECoG Data,” Neuroscience Letters,
Vol. 447, 2008, pp. 129-133.
http://dx.doi.org/10.1016/j.neulet.2008.09.080
[21] S. Thornton and J. Marion, “Classical Dynamics of Parti-
cles and Systems,” Brooks/Cole, Thomson Learning, Bel -
mont, 2004.
[22] N. Mordant, J. De lour, E. Leveque, A. Arneodo and J. F.
Pinton, “Long Time Correlations in Lagrangian Dynam-
ics: A Key toIntermittency in Turbulence,” Phys. Rev.
Lett., Vol. 89, No. 254502, 2002.
http://dx.doi.org/10.1103/PhysRevLett.89.254502
[23] N. Mordant, E. Leveque and J. F. Pinton, “Experimental
and Numerical Study of the Lagrangian Dynamics of
High Reynolds Turbulence,” New Journal of Physics, Vol.
6, No. 116, 2004.
[24] N. De Marco Garcia and T. Jessell, “Early Motor Neuron
Pool Identity and Muscle Nerve Trajectory Defined by
Postmitotic Restrictions in Nkx6.1 Activity ,” Neuron , Vol.
57, No. 2, 2008.