Percolation (from the Latin word percolatio, meaning filtration[1]) is a theoretical model used to understand the way activation and diffusion of neural activity occurs within neural networks.[2] Percolation is a model used to explain how neural activity is transmitted across the various connections within the brain. Percolation theory can be easily understood by explaining its use in epidemiology. Individuals who are infected with a disease can spread the disease through contact with others in their social network. Those who are more social and come into contact with more people will help to propagate the disease quicker than those who are less social. Factors such as occupation and sociability influence the rate of infection. Now, if one were to think of neurons as individuals and synaptic connections as the social bonds between people, then one can determine how easily messages between neurons will spread.[2] When a neuron fires, the message is transmitted along all synaptic connections to other neurons until it can no longer continue. Synaptic connections are considered either open or closed (like a social or unsocial person) and messages will flow along any and all open connections until they can go no further. Just like occupation and sociability play a key role in the spread of disease, so too do the number of neurons, synaptic plasticity and long-term potentiation when talking about neural percolation.

Percolating cluster

A key aspect of percolation is the concept of percolating clusters, which are single large groups of neurons that are all connected by open bonds and take up the majority of the network. Any signals that originate at any point within the percolating cluster will have a greater impact and diffusion across the network than signals that originate outside of the cluster. This is similar to a teacher spreading an infection to a whole community through contact with the students and subsequently with the families than an isolated businessman that works from home.

History and background

Percolation theory was originally purposed by Broadbent and Hammersley as a mathematical theory for determining the flow of fluids through porous material.[3] An example of this is the question originally purposed by Broadbent and Hammersley: "suppose a large porous rock is submerged under water for a long time, will the water reach the center of the stone?".[4] Since its founding, percolation theory has been used in both applied fields and mathematical modeling, areas such as engineering, physics, chemistry, communications, economics, mathematics, medicine and geography. From a mathematical perspective, percolation is uniquely able to exhibit both algebraic and probabilistic relationships graphically.[3] In network and cognitive sciences, percolation theory is often used as a computational model that has the benefit of testing theories on neural activity before any physical testing is necessary.[5] It can also be used as a model to explain experimental observations of neural activity to a certain extent.[6]

Current research

Percolation has been developed outside of the cognitive sciences; however, its application in the field has proven it to be a useful tool for understanding neural processes. Researchers have focused their attention not only studying how neural activity is diffused across networks, but also how percolation and its aspect of phase transition can affect decision making and thought processes.[5] Percolation theory has enabled researchers to better understand many psychological conditions, such as epilepsy, disorganized schizophrenia and divergent thinking.[2] These conditions are often indicative of percolating clusters and their involvement in propagating the excess firing of neurons. Seizures occur when neurons in the brain fire simultaneously, and often these seizures can occur in one part of the brain which may then transfer to other parts. Researchers are able to facilitate a better understanding of these conditions because "the neurons involved in a seizure are analogous to the sites in a percolating cluster".[2] Disorganized schizophrenia is more complex as the activity is indicative activity in a percolating cluster; however, some researchers have suggested that the percolation of information does not occur in a small cluster but on a global functional scale.[7] Attention as well as percolation also plays a key role in disorganized and divergent thinking; however, it is more likely that directed percolation, that is a directionally controlled percolation, is more useful to study divergent thinking and creativity.

Table of recent research

Below is a table of some of the studies and experiments that have involved percolation. The majority of these studies focus on the application of percolation theory to neural network processing from a computational approach.

Researcher(s)ArticleOverview
J.P. Eckmann, E. Moses, O. Stetter, T. Tlusty, & C. ZbindenLeaders of neuronal cultures in a quorum percolation modelResearchers use quorum percolation to describe the impacts of neural activity initiation in neural networks
S. Leleu-MervielOn the relevance of percolation theory to the acquisition of human skillsThis paper discusses the contributions that percolation theory has made to the acquisition of human skills
R. Kozma, M. Puljic & L. PerlovskyModeling goal-oriented decision making through cognitive phase transitionsResearchers use neuropercolation models to outline the implications to decision making
R. Kozma, M. Puljic, P. Balister, B. Bollobas & W. J. FreemanPhase transitions in the neuropercolation model of neural populations with mixed local and non-local interactionsResearchers aim to model phase transitions within the brain using percolation to understand neurphil properties
L. K. Gallos, H. A. Makes & M. SigmanA small world of weak ties provides optimal global integration of self-similar modules in functional brain networksResearchers use a modified percolation theory to attempt show a new way of thinking about the modularity of the brain as opposed to the widely accepted view of small world networks
D. B. Van, P. Gong, M. Breakspear & C. Van LeeuwenFragmentation: Loss of global coherence or breakdown of modularity in functional brain architecture?Researchers purpose that disorganized schizophrenia is not due to small-world organization but in fact a breakdown in local organization that leads to global modularity of functioning

Other applications

Percolation theory has been applied to a wide variety of fields of study, including medicine, economics, physics, as well as other areas of psychology, such as social sciences and industrial and organizational psychology. Below is a table of other areas of study that apply percolation theory as well as recent research information.

Area of studyResearchersArticle
Medicine/EpidemiologyS. B. KhandelwalEcology of infectious diseases with contact networks and percolation theory
EconomicsD. Duffie, G. Manso & S. MalamudInformation Percolation in Segmented Markets
Industrial/Organizational PsychT. GrebelNetwork evolution in basic science
Social PsychR. Flores, M. Koster, I. Lindner, & E. MolinaNetworks and collective action
PhysicsN. Poccia, A. Ansuini & A. BianconiFar from Equilibrium Percolation, Stochastic and Shape Resonances in the Physics of Life
GeographyH. E. Winzeler, P. R. Owens, S. W. Waltman, W. J. Waltman, Z. Libohova & D. BeaudetteA Methodology for Examining Changes in Soil Climate Geography through Time: U.S. Soil Moisture Regimes for the Period 1971–2000
Artificial intelligenceI. Rebollo, M. Grana, & B. CasesEffect of spatial percolation on the convergence of a graph colouring boid swarm
Emergency management and crisis preventionJ.L. WyboPercolation, temporal coherence of information, and crisis prevention

Future research

Percolation theory is widely used and impacts many different fields; however, the research in network science can still be developed further. As a computational model, percolation has its limitations in that it cannot always account for the variability of real-life neural networks. Its limitations do not hinder its functionality in total, just in some cases.[6] In order for one to understand small-world networks better, a closer objective look at percolation in neural networks is needed. The best possible way for this to occur would be to combine the applications of percolation modelling and experimental stimulation of artificial neural networks.[8][9][10][11][12][13][14][15][16][17][18]

References

  1. ↑ Leleu-Merviel, Sylvie (2004). "On the relevance of percolation theory to the acquisition of human skills". EISTA 2004. Volume II: Education and Training Systems, Technologies and Applications. IIIS Editor. pp. 391–395.
  2. 1 2 3 4 Friedenbreg, J., Silverman, G. (2012). Cognitive Science: An Introduction to the Study of Mind (2nd Ed.). Thousand Oaks, CA: SAGE Publications.
  3. 1 2 Beffara, Vincent; Sidoravicius, Vladas (July 11, 2005). "Percolation theory". arXiv:math/0507220. Bibcode:2005math......7220B. {{cite journal}}: Cite journal requires |journal= (help)
  4. ↑ Broadbent, S.R.; Hammersley, J.M. (1957). "Percolation processes, I and II". Proceedings of the Cambridge Philosophical Society. 53 (3): 629–641. Bibcode:1957PCPS...53..629B. doi:10.1017/s0305004100032680. S2CID 84176793.
  5. 1 2 Kozma, Robert; Puljic, Marko; Perlovsky, Leonid (March 2009). "Modeling goal-oriented decision making through cognitive phase transitions". New Mathematics and Natural Computation. 05 (1): 143–157. doi:10.1142/S1793005709001246.
  6. 1 2 Eckmann, J.P.; Moses, E.; Stetter, O.; Tlusty, T.; Zbinden, C. (2010). "Leaders of neuronal cultures in a quorum percolation model". Frontiers in Computational Neuroscience. 4: 132. arXiv:1009.5355. Bibcode:2010arXiv1009.5355E. doi:10.3389/fncom.2010.00132. PMC 2955434. PMID 20953239.
  7. ↑ Van, D. B.; Gong, P.; Breakspear, M.; van Leeuwen, C. (2012). "Fragmentation: Loss of global coherence or breakdown of modularity in functional brain architecture?". Frontiers in Systems Neuroscience. 6: 20. doi:10.3389/fnsys.2012.00020. PMC 3316147. PMID 22479239.
  8. ↑ Kozma, Robert; Puljic, Marko; Balister, Paul; Bollobás, Bela; Freeman, Walter J. (June 2005). "Phase transitions in the neuropercolation model of neural populations with mixed local and non-local interactions". Biological Cybernetics. 92 (6): 367–379. doi:10.1007/s00422-005-0565-z. PMID 15920663. S2CID 912262.
  9. ↑ Haggstrom, O.; Jonasson, J. (2006). "Uniqueness and non-uniqueness in percolation theory". Probability Surveys. 3: 289–344. arXiv:math/0612812. Bibcode:2006math.....12812H. doi:10.1214/154957806000000096. S2CID 7444277.
  10. ↑ Gallos, L. K.; Makse, H. A.; Sigman, M. (2012). "A small world of weak ties provides optimal global integration of self-similar modules in functional brain networks". PNAS. 109 (8): 2825–2830. arXiv:1102.0604. Bibcode:2012PNAS..109.2825G. doi:10.1073/pnas.1106612109. PMC 3286928. PMID 22308319.
  11. ↑ Bansal Khandelwal, Shweta (May 2008). Ecology of infectious diseases with contact networks and percolation theory (Thesis). hdl:2152/3910. ProQuest 622038307.
  12. ↑ Duffie, Darrell; Malamud, Semyon; Manso, Gustavo (September 2014). "Information percolation in segmented markets". Journal of Economic Theory. 153: 1–32. doi:10.1016/j.jet.2014.05.006. S2CID 6162521.
  13. ↑ Grebel, Thomas (July 2012). "Network evolution in basic science". Journal of Evolutionary Economics. 22 (3): 443–457. doi:10.1007/s00191-012-0266-4. S2CID 154217698. ProQuest 1282239642.
  14. ↑ Flores, Ramón; Koster, Maurice; Lindner, Ines; Molina, Elisenda (October 2012). "Networks and collective action" (PDF). Social Networks. 34 (4): 570–584. doi:10.1016/j.socnet.2012.06.003. ProQuest 1027510949.
  15. ↑ Poccia, N.; Ansuini, A.; Bianconi, A. (2011). "Far from Equilibrium Percolation, Stochastic and Shape Resonances in the Physics of Life". International Journal of Molecular Sciences. 12 (10): 6810–6833. doi:10.3390/ijms12106810. PMC 3211012. PMID 22072921.
  16. ↑ Rebello, I.; Grana, M.; Cases, B. (2012). "Effect of spatial percolation on the convergence of a graph colouring boid swarm". International Journal on Artificial Intelligence Tools. 21 (6): 1250015. doi:10.1142/S0218213012500157.
  17. ↑ Wybo, J.L. (2012). "Percolation, temporal coherence of information, and crisis prevention". Safety Science. 57: 60–68. doi:10.1016/j.ssci.2013.01.021.
  18. ↑ Winzeler, H. E.; Owens, P. R.; Waltman, S. W.; Waltman, W. J.; Libohova, Z.; Beaudette, D. (2013). "A Methodology for Examining Changes in Soil Climate Geography through Time: U.S. Soil Moisture Regimes for the Period 1971–2000". Soil Science Society of America Journal. 77 (1): 213–225. Bibcode:2013SSASJ..77..213W. doi:10.2136/sssaj2012.0123.
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