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:
[1]

M. Abeles, "Local Cortical Circuits: An Electrophysiological Study,", Studies of Brain Function, (1982).

[2]

M. Abeles, "Corticonics: Neural Circuits of the Cerebral Cortex,", First edition, (1991).

[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.

[4]

E. Bienenstock, A model of neocortex,, Network, 6 (1995), 179.

[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).

[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.

[7]

W. J. Freeman, "Mass Action in the Nervous System: Examination of the Neurophysiological Basis of Adaptive Behavior through the EEG,", Academic Press, (1975).

[8]

T. E. Harris, "The Theory of Branching Processes,", Die Grundlehren der Mathematischen Wissenschaften, (1963).

[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.

[10]

J. Hertz and A. Prgel-Bennet, Learning synfire-chains by self-organization,, Network, 7 (1996), 357.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

G. Mongillo, O. Barak and M. Tsodyks, Synaptic theory of working memory,, Science, 319 (2008), 1543.

[18]

J. Montgomery and D. Madison, Discrete synaptic states define a major mechanism of synapse plasticity,, Trends in Neurosciences, 27 (2004), 744.

[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.

[20]

M. Puljic and R. Kozma, Activation clustering in neural and social networks,, Complexity, 10 (2005), 42. doi: 10.1002/cplx.20075.

[21]

E. T. Rolls and A. Treves, The neuronal encoding of information in the brain,, Progress in Neurobiology, 95 (2011), 448.

[22]

E. T. Rolls, "Memory, Attention, and Decision-Making. A Unifying Computational Neuroscience Approach,", Oxford University Press, (2008).

[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.

[24]

T. S. Turova, The emergence of connectivity in neuronal networks: From bootstrap percolation to auto-associative memory,, Brain Research, 1434 (2012), 277.

[25]

T. Turova and A. Villa, On a phase diagram for random neural networks with embedded spike timing dependent plasticity,, BioSystems, 89 (2007), 280.

[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.

[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.

show all references

References:
[1]

M. Abeles, "Local Cortical Circuits: An Electrophysiological Study,", Studies of Brain Function, (1982).

[2]

M. Abeles, "Corticonics: Neural Circuits of the Cerebral Cortex,", First edition, (1991).

[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.

[4]

E. Bienenstock, A model of neocortex,, Network, 6 (1995), 179.

[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).

[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.

[7]

W. J. Freeman, "Mass Action in the Nervous System: Examination of the Neurophysiological Basis of Adaptive Behavior through the EEG,", Academic Press, (1975).

[8]

T. E. Harris, "The Theory of Branching Processes,", Die Grundlehren der Mathematischen Wissenschaften, (1963).

[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.

[10]

J. Hertz and A. Prgel-Bennet, Learning synfire-chains by self-organization,, Network, 7 (1996), 357.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

G. Mongillo, O. Barak and M. Tsodyks, Synaptic theory of working memory,, Science, 319 (2008), 1543.

[18]

J. Montgomery and D. Madison, Discrete synaptic states define a major mechanism of synapse plasticity,, Trends in Neurosciences, 27 (2004), 744.

[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.

[20]

M. Puljic and R. Kozma, Activation clustering in neural and social networks,, Complexity, 10 (2005), 42. doi: 10.1002/cplx.20075.

[21]

E. T. Rolls and A. Treves, The neuronal encoding of information in the brain,, Progress in Neurobiology, 95 (2011), 448.

[22]

E. T. Rolls, "Memory, Attention, and Decision-Making. A Unifying Computational Neuroscience Approach,", Oxford University Press, (2008).

[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.

[24]

T. S. Turova, The emergence of connectivity in neuronal networks: From bootstrap percolation to auto-associative memory,, Brain Research, 1434 (2012), 277.

[25]

T. Turova and A. Villa, On a phase diagram for random neural networks with embedded spike timing dependent plasticity,, BioSystems, 89 (2007), 280.

[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.

[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.

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