Temperature-dependent stochastic dynamics of the Huber-Braun neuron model.

The response of a four-dimensional mammalian cold receptor model to different implementations of noise is studied across a wide temperature range. It is observed that for noisy activation kinetics, the parameter range decomposes into two regions in which the system reacts qualitatively completely different to small perturbations through noise, and these regions are separated by a homoclinic bifurcation. Noise implemented as an additional current yields a substantially different system response at low temperature values, while the response at high temperatures is comparable to activation-kinetic noise. We elucidate how this phenomenon can be understood in terms of state space dynamics and gives quantitative results on the statistics of interspike interval distributions across the relevant parameter range.

Parameter and state estimation of experimental chaotic systems using synchronization.

We examine the use of synchronization as a mechanism for extracting parameter and state information from experimental systems. We focus on important aspects of this problem that have received little attention previously and we explore them using experiments and simulations with the chaotic Colpitts oscillator as an example system. We explore the impact of model imperfection on the ability to extract valid information from an experimental system. We compare two optimization methods: an initial value method and a constrained method. Each of these involves coupling the model equations to the experimental data in order to regularize the chaotic motions on the synchronization manifold. We explore both time-dependent and time-independent coupling and discuss the use of periodic impulse coupling. We also examine both optimized and fixed (or manually adjusted) coupling. For the case of an optimized time-dependent coupling function u(t) we find a robust structure which includes sharp peaks and intervals where it is zero. This structure shows a strong correlation with the location in phase space and appears to depend on noise, imperfections of the model, and the Lyapunov direction vectors. For time-independent coupling we find the counterintuitive result that often the optimal rms error in fitting the model to the data initially increases with coupling strength. Comparison of this result with that obtained using simulated data may provide one measure of model imperfection. The constrained method with time-dependent coupling appears to have benefits in synchronizing long data sets with minimal impact, while the initial value method with time-independent coupling tends to be substantially faster, more flexible, and easier to use. We also describe a method of coupling which is useful for sparse experimental data sets. Our use of the Colpitts oscillator allows us to explore in detail the case of a system with one positive Lyapunov exponent. The methods we explored are easily extended to driven systems such as neurons with time-dependent injected current. They are expected to be of value in nonchaotic systems as well. Software is available on request.

Optimized synchronization of chaotic and hyperchaotic systems.

A method of synchronization is presented which, unlike existing methods, can, for generic dynamical systems, force all conditional Lyapunov exponents to go to -∞ . It also has improved noise immunity compared to existing methods, and unlike most of them it can synchronize hyperchaotic systems with almost any single coupling variable from the drive system. Results are presented for the Rossler hyperchaos system and the Lorenz system.

Modeling and detecting localized nonlinearity in continuum systems with a multistage transform.

A general method is presented for modeling spatially extended systems that may contain a localized source of nonlinearity. It has direct applications to structural health monitoring (SHM) where physical damage may cause such nonlinearity and also communications channels which may exhibit localized nonlinearity due to bad electrical contacts or component nonlinearity. The method uses a multistage nonlinear transform in order to model the system dynamics. We discuss the application to SHM and provide a preliminary test of the method with experimental data from a randomly shaken beam with loose bolts. We discuss the application to telecommunications, provide an experimental observation of symmetric nonlinearity in a "bad" electrical contact, and provide a preliminary test of using this method to remove nonlinear echo (and thereby improve data rate) on a telephone line used for data transmission.