Cyclic attractor

The second case auxiliary system has one cyclic attractor... 

For one catalytic cycle, relaxation in the subspace = 0 is approximated by relaxation of a chain that is produced from the cycle by cutting the limiting step (Section 2). For reaction networks under consideration (with one cyclic attractor in auxiliary discrete dynamical system) the direct generalization works for approximation of relaxation in the subspace = 0 it is sufficient to perform the following procedures ... 

The graph of the sequences of states may lead to one or more final states (attractors), each of which may be a stable state (already detected at the level of the state tables), a cyclic attractor, or perhaps a non-periodic attractor (e.g., a chaotic situation). 

A Jacobi or Gauss-Seidel iteration on (6) will provide us with the coordinates of the steady state, or it will cycle indefinitely, depending on the slope of the functions. On the other hand, determining the trajectory by numerical integration of equation (5) will lead to a stable steady state or to a limit cycle depending on the slope of the functions. There is thus an obvious formal similarity between the two situations. However, the steepness corresponding to the transition from a punctual to a cyclic attractor is much smaller in the first case (in which the cyclic attractor is an iteration artifact) as in the second case (in which the cyclic attractor is close to the real trajectory). 

The classification of logical network structures imposed by the hypercube description depends on the signs of the focal point coordinates - associated with each orthant of phase space, which leads to the hypercube representation of the allowed flows. We consider that two different networks are in the same dynamical equivalence class if their directed A-cube representations can be superimiposed under a symmetry of the A-cube. For example, in three dimensions there is only one cyclic attractor (see Fig. 3b), but this can appear in eight different orientations on the 3-cube. From a dynamical perspective, exactly the same qualitative dynamics can be found in any of these networks provided the focal points are chosen in an identical fashion. However, from a biological... 

In the above, we have given a computational scheme that allows us to define the connections and interactions between components in biochemical networks and to determine the dynamics in the resulting networks. For an arbitrary network, it is not possible to give a precise description of the dynamics without carrying out numerical simulations. However, all the networks obey certain dynamic rules that are set by the stmcture as embodied in the directed A-dimensional hypercube. Moreover, for networks that show certain structural feamres, such as cyclic attractors, it is possible to make precise statements about the dynamics even without further mathematical analysis or simulation. In other cases, analytical techniques are available to give insight into the dynamics observed—for example, in the cases in which it is possible to prove limit cycles... 

Our initial studies of dynamics in biochemical networks included spatially localized components [32]. As a consequence, there will be delays involved in the transport between the nuclear and cytoplasmic compartments. Depending on the spatial structure, different dynamical behaviors could be faciliated, but the theoretical methods are useful to help understand the qualitative features. In other (unpublished) work, computations were carried out in feedback loops with cyclic attractors in which a delay was introduced in one of the interactions. Although the delay led to an increase of the period, the patterns of oscillation remained the same. However, delays in differential equations that model neural networks and biological control systems can introduce novel dynamics that are not present without a delay (for example, see Refs. 57 and 58). 

Cyclic Attractors and Limit Cycles in Higher Dimensions... 

In order for Eq. (32) to display cyclic behavior it is necessary that there be a cycle in the state transition diagram. The cycle we have discussed for N = 3 (Fig. 4) has a special property. All transitions between a state on the cycle and a neighboring state not on the cycle are oriented toward the states on the cycle. If this is true, we call the cycle a cyclic attractor For N=3, the number of distinct cyclic attractors is 1. Since this case has been of chemical interest, it is of potential interest to investigate the cyclic attractors in higher dimensions. 

From the definition, each state on a qrclic attractor specifies the orientations of N — 1 edges on the state transition diagram. Therefore, the maximum length Lmax for a cyclic attractor in N dimensions must satisfy the relation ... 

The simple circuits in the graph of the iV-cube defined by the cyclic attractors have been called snakes in the combinatorial literature. ... 

Fig. 10. The state transition diagrams for the three cyclic attractors found in four dimensions. If the orientation of an edge is not specified, that edge can have either orientation without effecting the cyclic attractor.

Fig. 11. The three limit cycles found for the cyclic attractors in Fig. 10, by numerically integrating the piecewise linear equation for each cyclic attractor with all thresholds equal to 0.5. Only the values at the thresholds were computed. The concentration of each chemical at each iteration for one cycle are shown. Tlie cycles appear to be stable so that all points in any of the eight volumes on each of the cycles approach the cycle.

Figs. 3d and 4, since it represents the cyclic attractor found in the state transition diagram of Eq. (27), iV = 4. For any number of dimensions there will always be a cyclic attractor through 2N volumes corresponding to the cyclic attractor found for Eq. (27). Numerical integration of Eq. (27) for n = 8, N = 5,6, 7 has indicated stable limit cycle attractors for each case where both the period of the oscillation and the amplitude increase as N increases. 

These numerical examples and the analysis in Section 4.2 suggest the following conjecture. Cyclic attractors in the state transition diagram of Equation (22) imply stable limit cycle oscillations in Equation (22) provided N>3. 

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