Advances in neuroscience have been closely linked to mathematical modeling beginning with the integrate-and-fire model of Lapicque and proceeding through the modeling of the action potential by Hodgkin and Huxley to the current era. The fundamental building block of the central nervous system, the neuron, may be thought of as a dynamic element that is “excitable”, and can generate a pulse or sp...

In a recent paper it was shown that, for chemical reaction networks possessing a subtle structural property called concordance, dynamical behavior of a very circumscribed (and largely stable) kind is enforced, so long as the kinetics lies within the very broad and natural weakly monotonic class. In particular, multiple equilibria are precluded, as are degenerate positive equilibria. Moreover, u...

Neural networks are currently implemented on digital Von Neumann machines, which do not fully leverage their intrinsic parallelism. We demonstrate how to use a novel class of reconfigurable dynamical systems for analogue information processing, mitigating this problem. Our generic hardware platform for dynamic, analogue computing consists of a reciprocal linear dynamical system with nonlinear f...

We show that a continuous map or a continuous flow on Rn with a certain recurrence relation must have a fixed point. Specifically, if there is a compact set W with the property that the forward orbit of every point in R n intersects W then there is a fixed point in W . Consequently, if the omega limit set of every point is nonempty and uniformly bounded then there is a fixed point. In this note...

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The process of establishing an acceptable model of an observed dynamic system from measured data is a challenging task that occupies a major portion of the work of the mathematical modeler. In this thesis, we propose a knowledge-based approach to automated modeling of dynamic systems based on equation discovery methods. Most work in equation discovery is concerned with assisting the empirical a...

We study positive shadowing and chain recurrence in the context of linear operators acting on Banach spaces or even normed vector spaces. show that for there is only one recurrent set, this set a closed invariant subspace. prove every transitive dynamical system with property frequently hypercyclic and, as corollary, we obtain hypercyclic.