# American Institute of Mathematical Sciences

2014, 11(1): 139-148. doi: 10.3934/mbe.2014.11.139

## Structural phase transitions in neural networks

 1 Mathematical Center, University of Lund, Box 118, Lund S-221 00, Sweden

Received  January 2013 Revised  June 2013 Published  September 2013

A model is considered for a neural network that is a stochastic process on a random graph. The neurons are represented by integrate-and-fire" processes. The structure of the graph is determined by the probabilities of the connections, and it depends on the activity in the network. The dependence between the initial level of sparseness of the connections and the dynamics of activation in the network was investigated. A balanced regime was found between activity, i.e., the level of excitation in the network, and inhibition, that allows formation of synfire chains.
Citation: Tatyana S. Turova. Structural phase transitions in neural networks. Mathematical Biosciences & Engineering, 2014, 11 (1) : 139-148. doi: 10.3934/mbe.2014.11.139
##### References:
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##### References:
 [1] M. Abeles, "Local Cortical Circuits: An Electrophysiological Study,", Studies of Brain Function, (1982). Google Scholar [2] M. Abeles, "Corticonics: Neural Circuits of the Cerebral Cortex,", First edition, (1991). Google Scholar [3] I. Ayzenshtat, E. Meirovithz, H. Edelman, U. Werner-Reiss, E. Bienenstock, M. Abeles and H. Slovin, Precise spatiotemporal patterns among visual cortical areas and their relation to visual stimulus processing., J. Neurosci., 30 (2010), 11232. Google Scholar [4] E. Bienenstock, A model of neocortex,, Network, 6 (1995), 179. Google Scholar [5] J.-P. Eckmann, E. Moses, O. Stetter, T. Tlusty and C. Zbinden, Leaders of neuronal cultures in a quorum percolation model,, Front. Comput. Neurosci., 4 (2010). Google Scholar [6] I. R. Fiete, W. Senn, C. Z. H. Wang and R. H. R. Hahnloser, Spike-time-dependent plasticity and heterosynaptic competition organize networks to produce long scale-free sequences of neural activity,, Neuron., 65 (2010), 563. doi: 10.1016/j.neuron.2010.02.003. Google Scholar [7] W. J. Freeman, "Mass Action in the Nervous System: Examination of the Neurophysiological Basis of Adaptive Behavior through the EEG,", Academic Press, (1975). Google Scholar [8] T. E. Harris, "The Theory of Branching Processes,", Die Grundlehren der Mathematischen Wissenschaften, (1963). Google Scholar [9] R. H. Hahnloser, A. A. Kozhevnikov and M. S. Fee, An ultra-sparse code underlies the generation of neural sequences in a songbird,, Nature, 419 (2002), 65. Google Scholar [10] J. Hertz and A. Prgel-Bennet, Learning synfire-chains by self-organization,, Network, 7 (1996), 357. Google Scholar [11] J. Iglesias and A. E. Villa, Emergence of preferred firing sequences in large spiking neural networks during simulated neuronal development,, Int. J. Neural Syst., 18 (2008), 267. Google Scholar [12] J. Iglesias and A. E. Villa, Effect of stimulus-driven pruning on the detection of spatiotemporal patterns of activity in large neural networks,, Biosystems, 89 (2007), 287. Google Scholar [13] S. Janson, T. Luczak, T. Turova and T. Vallier, Bootstrap percolation on the random graph $G_{n,p}$,, Annals of Applied Probability, 22 (2012), 1989. doi: 10.1214/11-AAP822. Google Scholar [14] R. Kozma, M. Puljic, P. Balister, B. Bollobás and W. Freeman, Phase transitions in the neuropercolation model of neural populations with mixed local and non-local interactions,, Biol. Cybernet., 92 (2005), 367. doi: 10.1007/s00422-005-0565-z. Google Scholar [15] S. Kunkel, M. Diesmann and A. Morrison, Limits to the development of feed-forward structures in large recurrent neuronal networks,, Frontiers in Computational Neuroscience, 4 (2011). doi: 10.3389/fncom.2010.00160. Google Scholar [16] R. Mooney and J. F. Prather, The HVC microcircuit: The synaptic basis for interactions between song motor and vocal plasticity pathways,, J. Neurosci., 25 (2005), 1952. Google Scholar [17] G. Mongillo, O. Barak and M. Tsodyks, Synaptic theory of working memory,, Science, 319 (2008), 1543. Google Scholar [18] J. Montgomery and D. Madison, Discrete synaptic states define a major mechanism of synapse plasticity,, Trends in Neurosciences, 27 (2004), 744. Google Scholar [19] Y. Prut, E. Vaadia, H. Bergman, I. Haalman, S. Hamutal and M. Abeles, Spatiotemporal structure of cortical activity: Properties and behavioral relevance,, J. Neurophysiol., 79 (1998), 2857. Google Scholar [20] M. Puljic and R. Kozma, Activation clustering in neural and social networks,, Complexity, 10 (2005), 42. doi: 10.1002/cplx.20075. Google Scholar [21] E. T. Rolls and A. Treves, The neuronal encoding of information in the brain,, Progress in Neurobiology, 95 (2011), 448. Google Scholar [22] E. T. Rolls, "Memory, Attention, and Decision-Making. A Unifying Computational Neuroscience Approach,", Oxford University Press, (2008). Google Scholar [23] C. Trengove, C. van Leeuwen and M. Diesmann, High-capacity embedding of synfire chains in a cortical network model,, J. Comput. Neurosci., 34 (2012), 185. doi: 10.1007/s10827-012-0413-9. Google Scholar [24] T. S. Turova, The emergence of connectivity in neuronal networks: From bootstrap percolation to auto-associative memory,, Brain Research, 1434 (2012), 277. Google Scholar [25] T. Turova and A. Villa, On a phase diagram for random neural networks with embedded spike timing dependent plasticity,, BioSystems, 89 (2007), 280. Google Scholar [26] A. E. P. Villa, I. V. Tetko, B. Hyland and A. Najem, Spatiotemporal activity patterns of rat cortical neurons predict responses in a conditioned task,, Proc. Natl. Acad. Sci. U.S.A., 96 (1999), 1106. Google Scholar [27] A. Waddington, P. A. Appleby, M. De Kamps and N. Cohen, Triphasic spike-timing-dependent plasticity organizes networks to produce robust sequences of neural activity,, Frontiers in computational Neuroscience, 6 (2012). doi: 10.3389/fncom.2012.00088. Google Scholar
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