Abstract. A coupled cell network describes interacting (coupled) individual systems (cells). As in networks from real applications, coupled cell networks can represent inhomogeneous networks where different types of cells interact with each other in different ways, which can be represented graphically by different symbols, or abstractly by equivalence relations.Various synchronous behaviors, from full synchrony to partial synchrony, can be observed for a given network. Patterns of synchrony, which do not depend on specific dynamics of the network, but only on the network structure, are associated with a special type of partition of cells, termed balanced equivalence relations. Algorithms in Aldis (2008) and Belykh and Hasler (2011) find the unique pattern of synchrony with the least clusters. In this paper, we compute the set of all possible patterns of synchrony and show their hierarchy structure as a complete lattice.We represent the network structure of a given coupled cell network by a symbolic adjacency matrix encoding the different coupling types. We show that balanced equivalence relations can be determined by a matrix computation on the adjacency matrix which forms a block structure for each balanced equivalence relation. This leads to a computer algorithm to search for all possible balanced equivalence relations. Our computer program outputs the balanced equivalence relations, quotient matrices, and a complete lattice for user specified coupled cell networks. Finding the balanced equivalence relations of any network of up to 15 nodes is tractable, but for larger networks this depends on the pattern of synchrony with least clusters.
For regular homogeneous networks with simple eigenvalues (real or complex), all possible explicit forms of lattices of balanced equivalence relations can be constructed by introducing lattice generators and lattice indices [Kamei, 2009]. Balanced equivalence relations in the lattice correspond to clusters of partially synchronized cells in a network. In this paper, we restrict attention to regular homogeneous networks with simple real eigenvalues, and one-dimensional internal dynamics for each cell. We first show that lattice elements with nonzero indices indicate the existence of codimension-one synchrony-breaking steady-state bifurcations, and furthermore, the positions of such lattice elements give the number of partially synchronized clusters. Using four-cell regular homogeneous networks as an example, we then classify a large number of regular homogeneous networks into a small number of lattice structures, in which networks share an equivalent clustering type. Indeed, some of these networks even share the same generic bifurcation structure. This classification leads us to explore how regular homogeneous networks that share synchrony-breaking bifurcation structure are topologically related.
Regular homogeneous networks are a class of coupled cell network, which comprises one type of cell (node) with one type of coupling (arrow), and each cell has the same number of input arrows (called the valency of the network). In coupled cell networks, robust synchrony (a flow-invariant polydiagonal) corresponds to a special kind of partition of cells, called a balanced equivalence relation. Balanced equivalence relations are determined solely by the network structure. It is well known that the set of balanced equivalence relations on a given finite network forms a complete lattice. In this paper, we consider regular homogeneous networks in which the internal dynamics of each cell is one-dimensional, and whose associated adjacency matrices have simple eigenvalues (real or complex). We construct explicit forms of lattices of balanced equivalence relations for such networks by introducing key building blocks, called lattice generators, along with integer numbers called lattice indices. The properties of lattice indices allow construction of all possible lattice structures for balanced equivalence relations of regular homogeneous networks of any number of cells with any valency. As an illustration, we show all 14 possible lattice structures of balanced equivalence relations for four-cell regular homogeneous networks.
The spindle assembly checkpoint is a cell cycle surveillance mechanism that ensures the proper separation of chromosomes prior to cell division at mitosis. Aurora kinases play critical roles in mitotic progression and hence small-molecule inhibitors of Aurora kinases have been developed as a new class of potential anticancer drugs. In this paper we present for the first time an integrated pharmacokinetic-pharmacodynamic model of the functional effects of CYC116 (a known inhibitor of Aurora kinases A and B) on the spindle assembly checkpoint. We use the model to simulate two common experimental systems: cell culture and p.o. dosing of mice and present predictions of the effects of CYC116 for a range of doses and drug scheduling regimes. The model reveals that a critical peak drug concentration is required to cause aberrant kinetochore-microtubule attachments. The model also predicts that provided this threshold concentration is exceeded, a high total oral dose causes a high number of aberrant attachments within any given damaged cell. However, the proportion of cells which enter anaphase with aberrant attachments is associated with the total length of time for which the plasma concentration is maintained above the threshold. Moreover, our model reveals that the length of prometaphase/metaphase is a nonlinear function of drug dose and this time period can be extended or shortened. Finally, a strong saturation effect on CYC116 efficacy is predicted by the model. We discuss how these predictions may have implications for further drug trials using CYC116 and other similar AK inhibitors.
For a regular coupled cell network, synchrony subspaces are the polydiagonal subspaces that are invariant under the network adjacency matrix. The complete lattice of synchrony subspaces of an n-cell regular network can be seen as an intersection of the partition lattice of n elements and a lattice of invariant subspaces of the associated adjacency matrix. We assign integer tuples with synchrony subspaces, and use them for identifying equivalent synchrony subspaces to be merged. Based on this equivalence, the initial lattice of synchrony subspaces can be reduced to a lattice of synchrony subspaces which corresponds to a simple eigenvalue case discussed in our previous work. The result is a reduced lattice of synchrony subspaces, which affords a well-defined non-negative integer index that leads to bifurcation analysis in regular coupled cell networks.
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