Least Action Trajectory in Neural Networks

Ellison C. Castro, Bhazel Anne R. Pelicano

Department of Physical Sciences, University of the Philippines, Baguio City, Philippines.

DOI: 10.4236/ojapps.2013.33B002                

Abstract

The study of complex networks had developed over the years to include systems such as traffic, predator-prey interactions, financial market, and even the world wide web. Complex network studies encompass biology, chemistry, physics, and even engineering and economics [1-6]. However, the dynamics of such complex networks are yet to be understood fully [7,8]. In this paper, we will be focusing mostly on the possible learning ability in a complex network. To do this, an optimization process is used via Wiener process [9,10]. It is apparent from the sample lattice shown that the final position was not a basis of the transition probability, or it was never used to calculate the probability, since the transition probability only considers the current position. The final point is reached because of the orientation of the edges, where each edge is facing the final point, an aspect of the nervous system (afferent and efferent nerves) [11-13]. No matter how random the orientation of the neurons is, each directs to the central nervous system for processing and is transmitted away for reaction.

Keywords

Neural Networks; Optimization; Wiener process

Conflicts of Interest

The authors declare no conflicts of interest.

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 Complex Networks: Towards Understanding Global Statistical Properties from Microscopic Density Fluctuations,” Physical Review E, Vol. 69, No. 3, 2004.
[3]	G. Yan, T. Zhou, B. Hu, Z.-Q. Fu and B.-H. Wang, “Efficient 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: Network Traffic,” Transportation Research Part B: Methodological, 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, PredatorPrey 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. Itzkovitz, N. Kashtan, D. Chklovskii and U. Alon, “Network Motifs: Simple Building Blocks of Complex Networks,” Science, 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 Integral; 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. Publishers, 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 Edition, Benjamin Cummings, San Francisco, 1999.
[14]	M. D. Odom and R. Sharda, “A Neural Network Model for Bankruptcy Prediction,” Neural Networks, 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, “Col-lective 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, “Identification of Human Protein Complexes from Local Sub-Graphs of Protein-Protein Interaction Network Based on Random Forest with Topological Structure Features,” 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 Particles and Systems,” Brooks/Cole, Thomson Learning, Belmont, 2004.
[22]	N. Mordant, J. Delour, E. Leveque, A. Arneodo and J. F. Pinton, “Long Time Correlations in Lagrangian Dynamics: 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.