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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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Role of maximum link lengths in wave propagation.A. The distribution of link lengths between the cells at the wave front, and the cells which triggered their firing. The front cells (top 1 or 5 %) were selected by their positions in a wave. The mean distances are given in the legend, parameters , . B. Estimate of the wave speed by numerical estimate of mean maxima of  i.i.d radii taken from uniform distribution  (broken line). The formula  is the expected value of the mean maxima (solid line). The CA simulations of wave speed are shown by circles. As seen, the mean maxima give a good wave speed estimate, in contrast to naive scaling  derived earlier from the PDE (dotted line).
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pone-0020536-g006: Role of maximum link lengths in wave propagation.A. The distribution of link lengths between the cells at the wave front, and the cells which triggered their firing. The front cells (top 1 or 5 %) were selected by their positions in a wave. The mean distances are given in the legend, parameters , . B. Estimate of the wave speed by numerical estimate of mean maxima of i.i.d radii taken from uniform distribution (broken line). The formula is the expected value of the mean maxima (solid line). The CA simulations of wave speed are shown by circles. As seen, the mean maxima give a good wave speed estimate, in contrast to naive scaling derived earlier from the PDE (dotted line).

Mentions: However, this naive scaling based on uniform radii distribution underestimates the wave speed of CA roughly by a factor of two (Figure 5, lower solid line, compare to circles). This happens because cells in the wave front have actually non-uniform distribution of links from which they have received their activation (Figure 6A). The distribution is strongly biased towards the longest links. A wave front generates a new wave front at the next time step by sending activity through the longest links out of available . To support this notion, we generated sets of i.i.d. discrete random variables uniformly distributed in and computed their mean maxima as a plausible estimate of wave speed, that is a contribution of activity propagation from each single node to a global propagation of wave per unit time. As one can see in Figure 6B (broken line), the mean maxima of random radii gives a good measure of CA wave speed (circles), especially for high . These numerical calculations are supported by analytic formula for the expected mean of maxima (black solid line), , derived for a uniform continuous distribution . This formula gives a very good prediction of CA wave speed at high , demonstrating that the wave propagation is indeed mainly determined by the mean maximum of radii, which converges in the limit () to the maximum possible radius . In other words, it is not the average, but rather the maximum link length that determines the wave speed in a random network with random radii distribution.


Wave speed in excitable random networks with spatially constrained connections.

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

Role of maximum link lengths in wave propagation.A. The distribution of link lengths between the cells at the wave front, and the cells which triggered their firing. The front cells (top 1 or 5 %) were selected by their positions in a wave. The mean distances are given in the legend, parameters , . B. Estimate of the wave speed by numerical estimate of mean maxima of  i.i.d radii taken from uniform distribution  (broken line). The formula  is the expected value of the mean maxima (solid line). The CA simulations of wave speed are shown by circles. As seen, the mean maxima give a good wave speed estimate, in contrast to naive scaling  derived earlier from the PDE (dotted line).
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3108581&req=5

pone-0020536-g006: Role of maximum link lengths in wave propagation.A. The distribution of link lengths between the cells at the wave front, and the cells which triggered their firing. The front cells (top 1 or 5 %) were selected by their positions in a wave. The mean distances are given in the legend, parameters , . B. Estimate of the wave speed by numerical estimate of mean maxima of i.i.d radii taken from uniform distribution (broken line). The formula is the expected value of the mean maxima (solid line). The CA simulations of wave speed are shown by circles. As seen, the mean maxima give a good wave speed estimate, in contrast to naive scaling derived earlier from the PDE (dotted line).
Mentions: However, this naive scaling based on uniform radii distribution underestimates the wave speed of CA roughly by a factor of two (Figure 5, lower solid line, compare to circles). This happens because cells in the wave front have actually non-uniform distribution of links from which they have received their activation (Figure 6A). The distribution is strongly biased towards the longest links. A wave front generates a new wave front at the next time step by sending activity through the longest links out of available . To support this notion, we generated sets of i.i.d. discrete random variables uniformly distributed in and computed their mean maxima as a plausible estimate of wave speed, that is a contribution of activity propagation from each single node to a global propagation of wave per unit time. As one can see in Figure 6B (broken line), the mean maxima of random radii gives a good measure of CA wave speed (circles), especially for high . These numerical calculations are supported by analytic formula for the expected mean of maxima (black solid line), , derived for a uniform continuous distribution . This formula gives a very good prediction of CA wave speed at high , demonstrating that the wave propagation is indeed mainly determined by the mean maximum of radii, which converges in the limit () to the maximum possible radius . In other words, it is not the average, but rather the maximum link length that determines the wave speed in a random network with random radii distribution.

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