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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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Related in: MedlinePlus

Wave speed predicted by the parabolic and hyperbolic PDEs compared to simulations of CA on random networks.The parabolic (Fisher-Kolmogorov) PDE gives wave speed  that indefinitely grows with network degree (red line and diamonds). In contrast, the suggested hyperbolic PDE (given in text) provides a reaso`nable wave speed  (given in text, shown by green line and diamonds). The  grows moderately and saturates to the maximum possible speed , in agreement with CA simulations (blue circles) and intuitive expectations. The solid lines show analytic formulae, the diamonds show simulations of corresponding full PDE systems.
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pone-0020536-g004: Wave speed predicted by the parabolic and hyperbolic PDEs compared to simulations of CA on random networks.The parabolic (Fisher-Kolmogorov) PDE gives wave speed that indefinitely grows with network degree (red line and diamonds). In contrast, the suggested hyperbolic PDE (given in text) provides a reaso`nable wave speed (given in text, shown by green line and diamonds). The grows moderately and saturates to the maximum possible speed , in agreement with CA simulations (blue circles) and intuitive expectations. The solid lines show analytic formulae, the diamonds show simulations of corresponding full PDE systems.

Mentions: Taking into account only the first time derivative, we arrive at a linearized version of Fisher-Kolmogorov equation(5)where is a diffusion coefficient [22], is the growth rate, and the second-order extinction term is omitted for wave speed analysis. The well-known [23] formula for Fisher-Kolmogorov wave speed gives infinite growth of speed with mean degree (Figure 4, upper line). Taking high-order terms into account in the right hand side of the PDE does not alter the principal behavior of wave speed (simulations not shown), demonstrating that parabolic PDEs are not suitable for wave speed prediction.


Wave speed in excitable random networks with spatially constrained connections.

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

Wave speed predicted by the parabolic and hyperbolic PDEs compared to simulations of CA on random networks.The parabolic (Fisher-Kolmogorov) PDE gives wave speed  that indefinitely grows with network degree (red line and diamonds). In contrast, the suggested hyperbolic PDE (given in text) provides a reaso`nable wave speed  (given in text, shown by green line and diamonds). The  grows moderately and saturates to the maximum possible speed , in agreement with CA simulations (blue circles) and intuitive expectations. The solid lines show analytic formulae, the diamonds show simulations of corresponding full PDE systems.
© Copyright Policy
Related In: Results  -  Collection

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

pone-0020536-g004: Wave speed predicted by the parabolic and hyperbolic PDEs compared to simulations of CA on random networks.The parabolic (Fisher-Kolmogorov) PDE gives wave speed that indefinitely grows with network degree (red line and diamonds). In contrast, the suggested hyperbolic PDE (given in text) provides a reaso`nable wave speed (given in text, shown by green line and diamonds). The grows moderately and saturates to the maximum possible speed , in agreement with CA simulations (blue circles) and intuitive expectations. The solid lines show analytic formulae, the diamonds show simulations of corresponding full PDE systems.
Mentions: Taking into account only the first time derivative, we arrive at a linearized version of Fisher-Kolmogorov equation(5)where is a diffusion coefficient [22], is the growth rate, and the second-order extinction term is omitted for wave speed analysis. The well-known [23] formula for Fisher-Kolmogorov wave speed gives infinite growth of speed with mean degree (Figure 4, upper line). Taking high-order terms into account in the right hand side of the PDE does not alter the principal behavior of wave speed (simulations not shown), demonstrating that parabolic PDEs are not suitable for wave speed prediction.

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