Abstract

 
 
 We consider a model for an N × N lattice network of weakly coupled neural oscilla- tors with periodic boundary conditions (2D square torus), where the coupling between neurons is assumed to be within a von Neumann neighborhood of size r, denoted as von Neumann r-neighborhood. Using the phase model reduction technique, we study the existence of cluster solutions with constant phase differences (Ψh, Ψv) between adjacent oscillators along the horizontal and vertical directions in our network, where Ψh and Ψv are not necessarily to be identical. Applying the Kronecker production representation and the circulant matrix theory, we develop a novel approach to analyze the stability of cluster solutions with constant phase difference (i.e., Ψh,Ψv are equal). We begin our analysis by deriving the precise conditions for stability of such cluster solutions with von Neumann 1-neighborhood and 2 neighborhood couplings, and then we generalize our result to von Neumann r-neighborhood coupling for arbitrary neighborhood size r ≥ 1. This developed approach for the stability analysis indeed can be extended to an arbitrary coupling in our network. Finally, numerical simulations are used to validate the above analytical results for various values of N and r by considering an inhibitory network of Morris-Lecar neurons.
 
 
Highlights
An average neuron forms and receives one to ten thousand synaptic connections for sending and receiving information
We aim to extend the known analytical results concerning the existence and stability of cluster solutions in a two dimensional network topologies to include two dimensional finite lattices with periodic boundary conditions
We review the phase reduction method by which a general network model of identical weakly coupled oscillators can be reduced to a phase model
Summary
An average neuron forms and receives one to ten thousand synaptic connections for sending and receiving information. Since there are at least 1011 neurons in the human brain, there are 1014 ∼ 1015 synaptic connections that are formed in the brain. The summation of this input at the cellular level combine to allow neurons to perform complicated information processing and when considered as a whole brain, or neural region, allow for the complex cognitive task that we use to live our lives to be performed. Understanding our brains ability to organize and create coherent patterns out of the collection of electrical activity from billions of coupled individual neurons is of much research interest. The synchronization and cluster solutions, as defined below, in networks of large populations of neurons play an important role in various brain functions [43, 9, 41, 22, 16, 6, 27, 48, 14, 47, 44, 40, 45]
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