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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 speed derived from the hyperbolic PDE compared to CA simulations.The wave speed  (red line, high) is derived assuming all links have maximum length. CA simulations are shown in two variants, with maximum-length links (red triangles) and generic random-length links (blue circles). The naive speed scaling  (blue line, low) is derived assuming that link lengths are uniformly distributed. This discrepancy is explained in Results, showing that maximum-length link is a better predictor of wave speed. The dashed lines show the high-order analysis, proving that the hyperbolic PDEs capture the wave behavior sufficiently well, and derivatives of order above 2 are not necessary.
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pone-0020536-g005: Wave speed derived from the hyperbolic PDE compared to CA simulations.The wave speed (red line, high) is derived assuming all links have maximum length. CA simulations are shown in two variants, with maximum-length links (red triangles) and generic random-length links (blue circles). The naive speed scaling (blue line, low) is derived assuming that link lengths are uniformly distributed. This discrepancy is explained in Results, showing that maximum-length link is a better predictor of wave speed. The dashed lines show the high-order analysis, proving that the hyperbolic PDEs capture the wave behavior sufficiently well, and derivatives of order above 2 are not necessary.

Mentions: The wave speed is shown in more detail in Figure 5 (upper solid line). One can see that falls near to CA simulations on networks where all links have maximum length (Figure 5, triangles), as expected. Surprisingly, also approximates well the CA simulations on networks where links have random length (Figure 5, circles), which is our primary model. This phenomenon is explained in the next section.


Wave speed in excitable random networks with spatially constrained connections.

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

Wave speed derived from the hyperbolic PDE compared to CA simulations.The wave speed  (red line, high) is derived assuming all links have maximum length. CA simulations are shown in two variants, with maximum-length links (red triangles) and generic random-length links (blue circles). The naive speed scaling  (blue line, low) is derived assuming that link lengths are uniformly distributed. This discrepancy is explained in Results, showing that maximum-length link is a better predictor of wave speed. The dashed lines show the high-order analysis, proving that the hyperbolic PDEs capture the wave behavior sufficiently well, and derivatives of order above 2 are not necessary.
© Copyright Policy
Related In: Results  -  Collection

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

pone-0020536-g005: Wave speed derived from the hyperbolic PDE compared to CA simulations.The wave speed (red line, high) is derived assuming all links have maximum length. CA simulations are shown in two variants, with maximum-length links (red triangles) and generic random-length links (blue circles). The naive speed scaling (blue line, low) is derived assuming that link lengths are uniformly distributed. This discrepancy is explained in Results, showing that maximum-length link is a better predictor of wave speed. The dashed lines show the high-order analysis, proving that the hyperbolic PDEs capture the wave behavior sufficiently well, and derivatives of order above 2 are not necessary.
Mentions: The wave speed is shown in more detail in Figure 5 (upper solid line). One can see that falls near to CA simulations on networks where all links have maximum length (Figure 5, triangles), as expected. Surprisingly, also approximates well the CA simulations on networks where links have random length (Figure 5, circles), which is our primary model. This phenomenon is explained in the next section.

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