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 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.
Highlights
Different types of networks are found across many scales, from metabolic networks in a single cell, to neural networks in brain, up to social and technological global networks
The system we describe the system and its dynamic properties, along with simulations that demonstrate its typical behavior
To treat the system analytically, we reduce it to a quasi one-dimensional network: the length of links is limited only along X but may be unlimited in the Y coordinate
Summary
Different types of networks are found across many scales, from metabolic networks in a single cell, to neural networks in brain, up to social and technological global networks. The maximum length of connection is usually limited by the available resources or other natural restrictions. To address this problem, spatial generalizations of the ER graph were suggested. Two nodes can be connected only if the distance between them is below threshold rc [8] This model was used to simulate spatio-temporal activity in networks of electrically coupled neurons [9]. Another example is the Waxman model, in which the probability that two nodes are connected is a decreasing function of distance between the nodes [10]. The latter model was used to simulate the Internet [11]
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