Synergetic Analysis of the Haeussler-von der Malsburg Equations for Manifolds of Arbitrary Geometry

We generalize a model of Haeussler and von der Malsburg which describes the self-organized generation of retinotopic projections between two one-dimensional discrete cell arrays on the basis of cooperative and competitive interactions of the individual synaptic contacts. Our generalized model is independent of the special geometry of the cell arrays and describes the temporal evolution of the connection weights between cells on different manifolds. By linearizing the equations of evolution around the stationary uniform state we determine the critical global growth rate for synapses onto the tectum where an instability arises. Within a nonlinear analysis we use then the methods of synergetics to adiabatically eliminate the stable modes near the instability. The resulting order parameter equations describe the emergence of retinotopic projections from initially undifferentiated mappings independent of dimension and geometry.Comment: Dedicated to Hermann Haken on the occasion of his 80th birthda

Optimal linear stability condition for scalar differential equations with distributed delay

Linear scalar differential equations with distributed delays appear in the study of the local stability of nonlinear differential equations with feedback, which are common in biology and physics. Negative feedback loops tend to promote oscillations around steady states, and their stability depends on the particular shape of the delay distribution. Since in applications the mean delay is often the only reliable information available about the distribution, it is desirable to find conditions for stability that are independent from the shape of the distribution. We show here that for a given mean delay, the linear equation with distributed delay is asymptotically stable if the associated differential equation with a discrete delay is asymptotically stable. We illustrate this criterion on a compartment model of hematopoietic cell dynamics to obtain sufficient conditions for stability

Instability of Turing patterns in reaction-diffusion-ODE systems

The aim of this paper is to contribute to the understanding of the pattern formation phenomenon in reaction-diffusion equations coupled with ordinary differential equations. Such systems of equations arise, for example, from modeling of interactions between cellular processes such as cell growth, differentiation or transformation and diffusing signaling factors. We focus on stability analysis of solutions of a prototype model consisting of a single reaction-diffusion equation coupled to an ordinary differential equation. We show that such systems are very different from classical reaction-diffusion models. They exhibit diffusion-driven instability (Turing instability) under a condition of autocatalysis of non-diffusing component. However, the same mechanism which destabilizes constant solutions of such models, destabilizes also all continuous spatially heterogeneous stationary solutions, and consequently, there exist no stable Turing patterns in such reaction-diffusion-ODE systems. We provide a rigorous result on the nonlinear instability, which involves the analysis of a continuous spectrum of a linear operator induced by the lack of diffusion in the destabilizing equation. These results are extended to discontinuous patterns for a class of nonlinearities.Comment: This is a new version of the paper. Presentation of results was essentially revised according to referee suggestion