Analyzing synchronized clusters in neuron networks

Abstract

The presence of synchronized clusters in neuron networks is a hallmark of information transmission and processing. Common approaches to study cluster synchronization in networks of coupled oscillators ground on simplifying assumptions, which often neglect key biological features of neuron networks. Here we propose a general framework to study presence and stability of synchronous clusters in more realistic models of neuron networks, characterized by the presence of delays, different kinds of neurons and synapses. Application of this framework to two examples with different size and features (the directed network of the macaque cerebral cortex and the swim central pattern generator of a mollusc) provides an interpretation key to explain known functional mechanisms emerging from the combination of anatomy and neuron dynamics. The cluster synchronization analysis is carried out also by changing parameters and studying bifurcations. Despite some modeling simplifications in one of the examples, the obtained results are in good agreement with previously reported biological data.

One-way dependent clusters and stability of cluster synchronization in directed networks

“…We now associate a dynamics to each node, by considering a network with N nodes, which are connected through L different kinds of links. We model networks of this kind through the following set of dynamical equations ( i = 1, …, N ) 33 , which provides a rather general description of deterministic systems governed by pairwise interactions where is the n -dimensional state vector of the i th node, is the vector field of the isolated i th node describing the uncoupled dynamics of x i , is the coupling strength of the k th kind of link, A k is the weighted adjacency matrix with respect to the k th kind of link, for which the pairwise interaction between two generic nodes i and j is described by the nonlinear function , and δ k is the transmission delay characteristic of the k th kind of link. We assume that each individual node can be of one out of M different types (with M ≤ N ): f i ( x ) = f j ( x ) if i and j are of the same type, f i ( x ) ≠ f j ( x ) otherwise.…”

“…In particular, we analyze the stability of clusters by linearizing Eq. ( 1 ) about a state corresponding to exact synchronization among all the nodes within each cluster 33 . The Master Stability Function approach 42 , which evaluates the stability of synchronized solutions through linearization, is based on two standard steps: (i) finding the variational equations of the network about the synchronized solutions and (ii) expressing these variational equations in a new system of coordinates, which decouples the perturbation dynamics along the transverse manifold from that along the synchronous manifold.…”

“…In the case of directed networks, the matrix is not a null block, in general. Therefore, each matrix B k can be reported in block-upper triangular form 27 , 33 , but, at the best of our knowledge, there are no methods to find the matrix T that transforms the perturbation modes in an irreducible form. We remark that a key requirement is that the matrix is decomposed so as to separate perturbations along the synchronous manifold from those along the transverse manifold.…”

“…The key variational equation for studying cluster stability has the following structure 33 : where Ψ 1 and are time-dependent matrices that depend on the synchronous solution s ( t ).…”

“…( 4 ). Taking into account that T is an orthogonal matrix (i.e., T − 1 = T T ), each matrix B k has the following structure: where the block contains only 0 entries (the proof of this statement is in the section “Methods”), whereas the block is null for undirected networks and contains nonzero entries for directed networks 33 . Therefore, in general, B k can be written as This implies that the transverse perturbations ( η Q +1 , …, η N ) accounting for cluster stability do not depend on the parallel perturbations ( η 1 , …, η Q ) along the synchronous manifold.…”