Coupled loops with time delays are common in physiological systems such as neural networks. We study a Hopfield-type network that consists of a pair of one-way loops each with three neurons and two-way coupling (of either excitatory or inhibitory type) between a single neuron of each loop. Time delays are introduced in the connections between loops, and the effects of coupling strengths and delays on the network dynamics are investigated. These effects depend strongly on whether the coupling is symmetric (of the same type in both directions) or asymmetric (inhibitory in one direction and excitatory in the other). The network of six delay differential equations is studied by linear stability analysis and bifurcation theory. Loops having inherently stable zero solutions cannotbe destabilized by weak coupling, regardless of the delay. Asymmetric coupling is weakly stabilizing but easily upset by delays. Symmetric coupling (if not too weak) can destabilize an inherently stable zero solution, leading to nontrivial fixed points if the gain of the neuron response function is not too negative or to oscillation otherwise. In the oscillation case, intermediate delays can restabilize the zero solution. At the borderline of the weak coupling region (symmetric or asymmetric), stability can change with delay ranges. When the coupling strengths are of the same magnitude, the oscillations of corresponding neurons in the two loops can be in phase, antiphase (symmetric coupling), or one quarter period out of phase (asymmetric coupling) depending on the delay.

The explosive growth in knowledge of the genome of humans and other organisms leaves open the question of how the functioning of genes in interacting networks is coordinated for orderly activity. One approach to this problem is to study mathematical properties of abstract network models that capture the logical structures of gene networks. The principal issue is to understand how particular patterns of activity can result from particular network structures, and what types of behavior are possible. We study idealized models in which the logical structure of the network is explicitly represented by Boolean functions that can be represented by directed graphs on n-cubes, but which are continuous in time and described by differential equations, rather than being updated synchronously via a discrete clock. The equations are piecewise linear, which allows significant analysis and facilitates rapid integration along trajectories. We first give a combinatorial solution to the question of how many distinct logical structures exist for n-dimensional networks, showing that the number increases very rapidly with $\mathrm{n.}$ We then outline analytic methods that can be used to establish the existence, stability and periods of periodic orbits corresponding to particular cycles on the n-cube. We use these methods to confirm the existence of limit cycles discovered in a sample of a million randomly generated structures of networks of 4 genes. Even with only 4 genes, at least several hundred different patterns of stable periodic behavior are possible, many of them surprisingly complex. We discuss ways of further classifying these periodic behaviors, showing that small mutations (reversal of one or a few edges on the n-cube) need not destroy the stability of a limit cycle. Although these networks are very simple as models of gene networks, their mathematical transparency reveals relationships between structure and behavior, they suggest that the possibilities for orderly dynamics in such networks are extremely rich and they offer novel ways to think about how mutations can alter dynamics.

We analyze a class of ordinary differential equations representing a simplified model of a genetic network. In this network, the model genes control the production rates of other genes by a logical function. The dynamics in these equations are represented by a directed graph on an n-dimensional hypercube (n-cube) in which each edge is directed in a unique orientation. The vertices of the n-cube correspond to orthants of state space, and the edges correspond to boundaries between adjacent orthants. The dynamics in these equations can be represented symbolically. Starting from a point on the boundary between neighboring orthants, the equation is integrated until the boundary is crossed for a second time. Each different cycle, corresponding to a different sequence of orthants that are traversed during the integration of the equation always starting on a boundary and ending the first time that same boundary is reached, generates a different letter of the alphabet. A word consists of a sequence of letters corresponding to a possible sequence of orthants that arise from integration of the equation starting and ending on the same boundary. The union of the words defines the language. Letters and words correspond to analytically computable Poincaré maps of the equation. This formalism allows us to define bifurcations of chaotic dynamics of the differential equation that correspond to changes in the associated language. Qualitative knowledge about the dynamics found by integrating the equation can be used to help solve the inverse problem of determining the underlying network generating the dynamics. This work places the study of dynamics in genetic networks in a context comprising both nonlinear dynamics and the theory of computation.