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Wave speed in excitable random networks with spatially constrained connections.

Vladimirov N, Traub RD, Tu Y - PLoS ONE (2011)

Bottom Line: We further show that the maximum length of connection is a much better predictor of the wave speed than the mean length.When tested in networks with various degree distributions, wave speeds are found to strongly depend on the ratio of network moments <k2>/<k> rather than on mean degree <k>, which is explained by general network theory.Our results suggest practical predictions for networks of electrically coupled neurons, and our mean field method can be readily applied for a wide class of similar problems, such as spread of epidemics through spatial networks.

View Article: PubMed Central - PubMed

Affiliation: IBM T. J. Watson Research Center, Yorktown Heights, New York, United States of America. nikita.vladimirov@gmail.com

ABSTRACT
Very fast oscillations (VFO) in neocortex are widely observed before epileptic seizures, and there is growing evidence that they are caused by networks of pyramidal neurons connected by gap junctions between their axons. We are motivated by the spatio-temporal waves of activity recorded using electrocorticography (ECoG), and study the speed of activity propagation through a network of neurons axonally coupled by gap junctions. We simulate wave propagation by excitable cellular automata (CA) on random (Erdös-Rényi) networks of special type, with spatially constrained connections. From the cellular automaton model, we derive a mean field theory to predict wave propagation. The governing equation resolved by the Fisher-Kolmogorov PDE fails to describe wave speed. A new (hyperbolic) PDE is suggested, which provides adequate wave speed v() that saturates with network degree , in agreement with intuitive expectations and CA simulations. We further show that the maximum length of connection is a much better predictor of the wave speed than the mean length. When tested in networks with various degree distributions, wave speeds are found to strongly depend on the ratio of network moments / rather than on mean degree , which is explained by general network theory. The wave speeds are strikingly similar in a diverse set of networks, including regular, Poisson, exponential and power law distributions, supporting our theory for various network topologies. Our results suggest practical predictions for networks of electrically coupled neurons, and our mean field method can be readily applied for a wide class of similar problems, such as spread of epidemics through spatial networks.

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Effect of link lengths (radii) distribution on wave speed.A. The radii distributions between nodes in the random networks: black o - fixed value, cyan x - uniform, red (blue) triangles - exponentially increasing (decreasing), green squares - bell-shaped distribution (see Methods for detailed formulae). B. Wave speeds in the networks with corresponding radii distributions (markers are consistent with panel A). Broken lines are computed mean maxima out of  radii samples from each distribution, used as a plausible estimates of the true wave speeds (solid lines). Networks are Erdös-Rényi SCC, so the degree distribution (links per node) is Poissonian.
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pone-0020536-g007: Effect of link lengths (radii) distribution on wave speed.A. The radii distributions between nodes in the random networks: black o - fixed value, cyan x - uniform, red (blue) triangles - exponentially increasing (decreasing), green squares - bell-shaped distribution (see Methods for detailed formulae). B. Wave speeds in the networks with corresponding radii distributions (markers are consistent with panel A). Broken lines are computed mean maxima out of radii samples from each distribution, used as a plausible estimates of the true wave speeds (solid lines). Networks are Erdös-Rényi SCC, so the degree distribution (links per node) is Poissonian.

Mentions: To study the effects of other possible radii distributions, we simulated networks with five distributions (Figure 7A): the uniform radii distribution, the fixed-value distribution (), a bell-shaped and two exponential distributions (increasing and decreasing, respectively). The speeds of wave propagation in resulting random networks are qualitatively similar and always significantly higher than the average value of the corresponding distribution (Figure 7B). Broken lines in Figure 7B are mean maxima of radii generated from each distribution, which served as a good estimate of wave speed in four out of five distributions. Note that the distribution for which our maximum link hypothesis works the least is the one with an exponentially small probability of reaching the maximum link length (see the purple line in Figure 7A).


Wave speed in excitable random networks with spatially constrained connections.

Vladimirov N, Traub RD, Tu Y - PLoS ONE (2011)

Effect of link lengths (radii) distribution on wave speed.A. The radii distributions between nodes in the random networks: black o - fixed value, cyan x - uniform, red (blue) triangles - exponentially increasing (decreasing), green squares - bell-shaped distribution (see Methods for detailed formulae). B. Wave speeds in the networks with corresponding radii distributions (markers are consistent with panel A). Broken lines are computed mean maxima out of  radii samples from each distribution, used as a plausible estimates of the true wave speeds (solid lines). Networks are Erdös-Rényi SCC, so the degree distribution (links per node) is Poissonian.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3108581&req=5

pone-0020536-g007: Effect of link lengths (radii) distribution on wave speed.A. The radii distributions between nodes in the random networks: black o - fixed value, cyan x - uniform, red (blue) triangles - exponentially increasing (decreasing), green squares - bell-shaped distribution (see Methods for detailed formulae). B. Wave speeds in the networks with corresponding radii distributions (markers are consistent with panel A). Broken lines are computed mean maxima out of radii samples from each distribution, used as a plausible estimates of the true wave speeds (solid lines). Networks are Erdös-Rényi SCC, so the degree distribution (links per node) is Poissonian.
Mentions: To study the effects of other possible radii distributions, we simulated networks with five distributions (Figure 7A): the uniform radii distribution, the fixed-value distribution (), a bell-shaped and two exponential distributions (increasing and decreasing, respectively). The speeds of wave propagation in resulting random networks are qualitatively similar and always significantly higher than the average value of the corresponding distribution (Figure 7B). Broken lines in Figure 7B are mean maxima of radii generated from each distribution, which served as a good estimate of wave speed in four out of five distributions. Note that the distribution for which our maximum link hypothesis works the least is the one with an exponentially small probability of reaching the maximum link length (see the purple line in Figure 7A).

Bottom Line: We further show that the maximum length of connection is a much better predictor of the wave speed than the mean length.When tested in networks with various degree distributions, wave speeds are found to strongly depend on the ratio of network moments <k2>/<k> rather than on mean degree <k>, which is explained by general network theory.Our results suggest practical predictions for networks of electrically coupled neurons, and our mean field method can be readily applied for a wide class of similar problems, such as spread of epidemics through spatial networks.

View Article: PubMed Central - PubMed

Affiliation: IBM T. J. Watson Research Center, Yorktown Heights, New York, United States of America. nikita.vladimirov@gmail.com

ABSTRACT
Very fast oscillations (VFO) in neocortex are widely observed before epileptic seizures, and there is growing evidence that they are caused by networks of pyramidal neurons connected by gap junctions between their axons. We are motivated by the spatio-temporal waves of activity recorded using electrocorticography (ECoG), and study the speed of activity propagation through a network of neurons axonally coupled by gap junctions. We simulate wave propagation by excitable cellular automata (CA) on random (Erdös-Rényi) networks of special type, with spatially constrained connections. From the cellular automaton model, we derive a mean field theory to predict wave propagation. The governing equation resolved by the Fisher-Kolmogorov PDE fails to describe wave speed. A new (hyperbolic) PDE is suggested, which provides adequate wave speed v() that saturates with network degree , in agreement with intuitive expectations and CA simulations. We further show that the maximum length of connection is a much better predictor of the wave speed than the mean length. When tested in networks with various degree distributions, wave speeds are found to strongly depend on the ratio of network moments / rather than on mean degree , which is explained by general network theory. The wave speeds are strikingly similar in a diverse set of networks, including regular, Poisson, exponential and power law distributions, supporting our theory for various network topologies. Our results suggest practical predictions for networks of electrically coupled neurons, and our mean field method can be readily applied for a wide class of similar problems, such as spread of epidemics through spatial networks.

Show MeSH
Related in: MedlinePlus