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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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Initiation of wave in a CA model on a random network.The first 4 time steps of wave initiation are shown for an 11×11 network. A. t = 0; B. t = 1; C. t = 2; D. t = 3. Colorcode: red for active, blue for refractory, black for excitable cells. Lines show links between cells, red square shows the connectivity footprint of the central cell (shown only in A). Parameters:  = 4,  (small for demonstration purposes).
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pone-0020536-g002: Initiation of wave in a CA model on a random network.The first 4 time steps of wave initiation are shown for an 11×11 network. A. t = 0; B. t = 1; C. t = 2; D. t = 3. Colorcode: red for active, blue for refractory, black for excitable cells. Lines show links between cells, red square shows the connectivity footprint of the central cell (shown only in A). Parameters:  = 4, (small for demonstration purposes).

Mentions: Initiation of wave in a small 2D network is shown in Figure 2 (first four time steps). Although directions of links are random, activity propagates outwards from the initial point because it is followed by refractory state, prohibiting backward propagation. Propagation of waves in large 2D networks is shown in Figure 3A–C. Waves in networks with round and square footprint do not differ qualitatively (not shown), because neighbors of each node are chosen as random lattice points from round or square neighborhood (respectively), which differ only in relatively small corner areas.


Wave speed in excitable random networks with spatially constrained connections.

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

Initiation of wave in a CA model on a random network.The first 4 time steps of wave initiation are shown for an 11×11 network. A. t = 0; B. t = 1; C. t = 2; D. t = 3. Colorcode: red for active, blue for refractory, black for excitable cells. Lines show links between cells, red square shows the connectivity footprint of the central cell (shown only in A). Parameters:  = 4,  (small for demonstration purposes).
© Copyright Policy
Related In: Results  -  Collection

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

pone-0020536-g002: Initiation of wave in a CA model on a random network.The first 4 time steps of wave initiation are shown for an 11×11 network. A. t = 0; B. t = 1; C. t = 2; D. t = 3. Colorcode: red for active, blue for refractory, black for excitable cells. Lines show links between cells, red square shows the connectivity footprint of the central cell (shown only in A). Parameters:  = 4, (small for demonstration purposes).
Mentions: Initiation of wave in a small 2D network is shown in Figure 2 (first four time steps). Although directions of links are random, activity propagates outwards from the initial point because it is followed by refractory state, prohibiting backward propagation. Propagation of waves in large 2D networks is shown in Figure 3A–C. Waves in networks with round and square footprint do not differ qualitatively (not shown), because neighbors of each node are chosen as random lattice points from round or square neighborhood (respectively), which differ only in relatively small corner areas.

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