We study the unexpected disappearance of stable homoclinic orbits in regions of parameter space in a neural field model with one spatial dimension. The usual approach of using numerical continuation techniques and local bifurcation theory is insufficient to explain the qualitative change in the model's behaviour. The lack of robustness of the model to small perturbations in parameters is surprising and the phenomenon may be of broader significance than just our model. By exploiting the Hamiltonian structure of the timeindependent system, we develop a numerical technique with which we discover that a small, separate solution curve exists for a range of parameter values. As the firing rate function steepens, the small curve causes the main curve to break and stable homoclinic orbits are destroyed in a region of parameter space. Numerically, we use level set analysis to find that a codimension-one heteroclinic bifurcation occurs at the terminating ends of the solution curves.By replacing the firing rate function with a step function, we show analytically that the bifurcation is related to the value of the firing threshold. We also show the existence of heteroclinic orbits at the breakpoints using a travelling front analysis in the time-dependent system.

Although the active properties of airway smooth muscle (ASM) have garnered much modeling attention, the passive mechanical properties are not as well studied. In particular, there are important dynamic effects observed in passive ASM, particularly strain-induced fluidization, which have been observed both experimentally and in models; however, to date these models have left an incomplete picture of the biophysical, mechanistic basis for these behaviors. The well-known Huxley cross-bridge model has for many years successfully described many of the active behaviors of smooth muscle using sliding filament theory; here, we propose to extend this theory to passive biological soft tissue, particularly ASM, using as a basis the attachment and detachment of cross-linker proteins at a continuum of cross-linker binding sites. The resulting mathematical model exhibits strain-induced fluidization, as well as several types of force recovery, at the same time suggesting a new mechanistic basis for the behavior. The model is validated by comparison to new data from experimental preparations of rat tracheal airway smooth muscle. Furthermore, experiments in noncontractile tissue show qualitatively similar behavior, suggesting support for the protein-filament theory as a biomechanical basis for the behavior.

We investigate Turing bifurcations in a neural field model with one spatial dimension. For some parameter values the resulting Turing patterns are stable, while for others the patterns appear transiently. We show that this difference is due to the relative position in parameter space of the saddle-node bifurcation of a spatially periodic pattern and the Turing bifurcation point. By varying parameters we are able to observe transient patterns whose duration scales in the same way as type-I intermittency. Similar behavior occurs in two spatial dimensions.

We study a nonlinear, one-dimensional neural field model based upon a partial integro-differential equation, that is used to model spatial patterns in working memory. Through the application of Fourier transforms to the PIDE, steady states of spatially localised areas of high activity can be represented by solutions of a fourth order ODE. Recent research has shown a high density of gap junctions in areas of the brain that experience epileptic events. We extend the model by including a diffusion-like term to model gap junctions and derive a sixth order ODE which we use to investigate changes in the dynamics of spatially localised solutions. We find that symmetric homoclinic orbits to a zero steady state exist for a wide area of parameter space. Numerical work shows families of solutions are destroyed as the strength of the term modelling gap junctions increases.

Rationale: Airway hyperresponsiveness (AHR) is one of the defining features of asthma. It has been proposed that an increase in airway smooth muscle (ASM) shortening velocity could contribute to the development of AHR by allowing the muscle to shorten further during each expiration. It is well known, however, that muscle behaves differently under dynamic as opposed to static conditions, so it is unclear whether steady-state shortening velocity can predict the response of ASM under the cyclic loading conditions associated with breathing. Methods: We investigated the effect of increased ASM shortening velocity on the response to cyclic loading using both a mathematical model and experiments. We used a two-state cross-bridge model based on that of Huxley (1957). By increasing all rate constants in in vitro proportion, shortening velocity could be changed without altering isometric force. Experiments were conducted using trachealis ASM strips from Fisher and Lewis rats. Fisher rats exhibit innate AHR and faster trachealis shortening velocity compared to Lewis rats. Muscle strips were studied under supramaximal methacholine stimulation after a 1hr equilibration period. In both model and experiments, the muscle was initially held at constant length so that isometric force ( ) could be measured. The force on the muscle was then varied Fmax sinusoidally about a mean value of 0.33 , with an amplitude of ±0.25 and a frequency of 1.2Hz, for 4 min. Results: In both the Fmax Fmax model and experiments, the muscle exhibited periods of stretch and shortening during each cycle of force oscillation but the amount of shortening exceeded the amount of stretch, resulting in a net decrease in length. The model predicted that an increase in maximal steady-state shortening velocity would increase the net muscle shortening per oscillation cycle. Shortening in the model continued indefinitely unless a force-length relationship was included, in which case mean length approached a steady-state value that was attained more quickly by the faster muscle. Preliminary experimental results (2 Fisher, 5 Lewis) indicated that total shortening in 4 min was greater in Fisher than Lewis muscle strips (mean shortening, %: Fisher: 38.0±0.6; Lewis: 32.5±3.5). Measurements of isotonic shortening velocity at 0.2 supported previous observations that Fisher trachealis shortens more quickly under steady-state conditions (velocity 1 min after Fmax activation, lengths/s: Fisher: 0.094±0.006; Lewis: 0.081±0.010). Conclusions: Both the modelling and experimental results support the proposition that an increase in ASM shortening velocity could contribute to AHR by increasing net muscle shortening and therefore bronchoconstriction. This abstract is funded by: NIH HL87791, HL87788, HL87792, HL87401; FRSQ Am J Respir Crit Care Med