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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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Traveling waves of activity in random networks.Traveling waves emerging in the CA model on random networks with A. square, B. quasi-1D connectivity footprint. The cell which initiates the wave is shown by a red asterisk. Active cells are white, refractory and excitable cells are black. Directions of wave propagation are shown by arrows. C. A snapshot of wave () with spatial profiles of all three states: grey for excitable, bold red for active, light blue for refractory cell density. In the center, the wake of excitable cells (grey) grows by recovering from the refractory state (blue). D. Profiles of the active state at four time steps, showing two traveling waves emerged from a single active cell. Once formed, the speed and width of a wave remain constant. Profiles were calculated by averaging active cell counts over 100 bins along X. Parameters ,  ( is shown in the bottom right corners in A,B).
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pone-0020536-g003: Traveling waves of activity in random networks.Traveling waves emerging in the CA model on random networks with A. square, B. quasi-1D connectivity footprint. The cell which initiates the wave is shown by a red asterisk. Active cells are white, refractory and excitable cells are black. Directions of wave propagation are shown by arrows. C. A snapshot of wave () with spatial profiles of all three states: grey for excitable, bold red for active, light blue for refractory cell density. In the center, the wake of excitable cells (grey) grows by recovering from the refractory state (blue). D. Profiles of the active state at four time steps, showing two traveling waves emerged from a single active cell. Once formed, the speed and width of a wave remain constant. Profiles were calculated by averaging active cell counts over 100 bins along X. Parameters , ( is shown in the bottom right corners in A,B).

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)

Traveling waves of activity in random networks.Traveling waves emerging in the CA model on random networks with A. square, B. quasi-1D connectivity footprint. The cell which initiates the wave is shown by a red asterisk. Active cells are white, refractory and excitable cells are black. Directions of wave propagation are shown by arrows. C. A snapshot of wave () with spatial profiles of all three states: grey for excitable, bold red for active, light blue for refractory cell density. In the center, the wake of excitable cells (grey) grows by recovering from the refractory state (blue). D. Profiles of the active state at four time steps, showing two traveling waves emerged from a single active cell. Once formed, the speed and width of a wave remain constant. Profiles were calculated by averaging active cell counts over 100 bins along X. Parameters ,  ( is shown in the bottom right corners in A,B).
© Copyright Policy
Related In: Results  -  Collection

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

pone-0020536-g003: Traveling waves of activity in random networks.Traveling waves emerging in the CA model on random networks with A. square, B. quasi-1D connectivity footprint. The cell which initiates the wave is shown by a red asterisk. Active cells are white, refractory and excitable cells are black. Directions of wave propagation are shown by arrows. C. A snapshot of wave () with spatial profiles of all three states: grey for excitable, bold red for active, light blue for refractory cell density. In the center, the wake of excitable cells (grey) grows by recovering from the refractory state (blue). D. Profiles of the active state at four time steps, showing two traveling waves emerged from a single active cell. Once formed, the speed and width of a wave remain constant. Profiles were calculated by averaging active cell counts over 100 bins along X. Parameters , ( is shown in the bottom right corners in A,B).
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