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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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Wave speeds in networks with different degree distributions.Wave speeds in networks of six different degree distributions (explained in legend) are plotted A. against mean degree ; B. against ratio of network moments . Note the convergence of wave speeds in B. The mean field formulae are shown in both panels by broken lines (Eqn. 7 in A; Eqn. 11 in B). Inset. The  versus  in the simulated networks. Line markers are consistent with legend in panel A. Errorbars are smaller than symbols, due to simulations on many networks of large size (1000×1000).
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pone-0020536-g008: Wave speeds in networks with different degree distributions.Wave speeds in networks of six different degree distributions (explained in legend) are plotted A. against mean degree ; B. against ratio of network moments . Note the convergence of wave speeds in B. The mean field formulae are shown in both panels by broken lines (Eqn. 7 in A; Eqn. 11 in B). Inset. The versus in the simulated networks. Line markers are consistent with legend in panel A. Errorbars are smaller than symbols, due to simulations on many networks of large size (1000×1000).

Mentions: In all networks, the lengths of links between nodes were uniformly distributed in (). As one can see in Figure 8A, the wave speed profiles vary widely when plotted against network mean degree . However, they merge into nearly the same shape when plotted against (Figure 8B, inset). The key role of ratio is evident from general network theory. For a randomly chosen link, the degree of a node on its end follows the nearest-neighbor distribution , where is the original network degree distribution [6]. The mean degree of a connected node is therefore different from a randomly picked node: , so our ratio is merely the nearest-neighbor mean degree. In other words, when activation travels from one node to another, the degree of a node in the end of a link follows the nearest-neighbor distribution , which has a mean of and gives the actual branching ratio of the activation in the new node.


Wave speed in excitable random networks with spatially constrained connections.

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

Wave speeds in networks with different degree distributions.Wave speeds in networks of six different degree distributions (explained in legend) are plotted A. against mean degree ; B. against ratio of network moments . Note the convergence of wave speeds in B. The mean field formulae are shown in both panels by broken lines (Eqn. 7 in A; Eqn. 11 in B). Inset. The  versus  in the simulated networks. Line markers are consistent with legend in panel A. Errorbars are smaller than symbols, due to simulations on many networks of large size (1000×1000).
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3108581&req=5

pone-0020536-g008: Wave speeds in networks with different degree distributions.Wave speeds in networks of six different degree distributions (explained in legend) are plotted A. against mean degree ; B. against ratio of network moments . Note the convergence of wave speeds in B. The mean field formulae are shown in both panels by broken lines (Eqn. 7 in A; Eqn. 11 in B). Inset. The versus in the simulated networks. Line markers are consistent with legend in panel A. Errorbars are smaller than symbols, due to simulations on many networks of large size (1000×1000).
Mentions: In all networks, the lengths of links between nodes were uniformly distributed in (). As one can see in Figure 8A, the wave speed profiles vary widely when plotted against network mean degree . However, they merge into nearly the same shape when plotted against (Figure 8B, inset). The key role of ratio is evident from general network theory. For a randomly chosen link, the degree of a node on its end follows the nearest-neighbor distribution , where is the original network degree distribution [6]. The mean degree of a connected node is therefore different from a randomly picked node: , so our ratio is merely the nearest-neighbor mean degree. In other words, when activation travels from one node to another, the degree of a node in the end of a link follows the nearest-neighbor distribution , which has a mean of and gives the actual branching ratio of the activation in the new node.

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