A modeling approach on why simple central pattern generators are built of irregular neurons.

The crustacean pyloric Central Pattern Generator (CPG) is a nervous circuit that endogenously provides periodic motor patterns. Even after about 40 years of intensive studies, the rhythm genesis is still not rigorously understood in this CPG, mainly because it is made of neurons with irregular intrinsic activity. Using mathematical models we addressed the question of using a network of irregularly behaving elements to generate periodic oscillations, and we show some advantages of using non-periodic neurons with intrinsic behavior in the transition from bursting to tonic spiking (as found in biological pyloric CPGs) as building components. We studied two- and three-neuron model CPGs built either with Hindmarsh-Rose or with conductance-based Hodgkin-Huxley-like model neurons. By changing a model's parameter we could span the neuron's intrinsic dynamical behavior from slow periodic bursting to fast tonic spiking, passing through a transition where irregular bursting was observed. Two-neuron CPG, half center oscillator (HCO), was obtained for each intrinsic behavior of the neurons by coupling them with mutual symmetric synaptic inhibition. Most of these HCOs presented regular antiphasic bursting activity and the changes of the bursting frequencies was studied as a function of the inhibitory synaptic strength. Among all HCOs, those made of intrinsic irregular neurons presented a wider burst frequency range while keeping a reliable regular oscillatory (bursting) behavior. HCOs of periodic neurons tended to be either hard to change their behavior with synaptic strength variations (slow periodic burster neurons) or unable to perform a physiologically meaningful rhythm (fast tonic spiking neurons). Moreover, 3-neuron CPGs with connectivity and output similar to those of the pyloric CPG presented the same results.

Sound synchronization of bubble trains in a viscous fluid: experiment and modeling.

We investigate the dynamics of formation of air bubbles expelled from a nozzle immersed in a viscous fluid under the influence of sound waves. We have obtained bifurcation diagrams by measuring the time between successive bubbles, having the air flow (Q) as a parameter control for many values of the sound wave amplitude (A), the height (H) of the solution above the top of the nozzle, and three values of the sound frequency (fs). Our parameter spaces (Q,A) revealed a scenario for the onset of synchronization dominated by Arnold tongues (frequency locking) which gives place to chaotic phase synchronization for sufficiently large A. The experimental results were accurately reproduced by numerical simulations of a model combining a simple bubble growth model for the bubble train and a coupling term with the sound wave added to the equilibrium pressure.

Synchronization of two bubble trains in a viscous fluid: experiment and numerical simulation.

We investigate the interactions of two trains of bubbles, ejected by nozzles immersed in a viscous fluid, due only to the solution's circulation. The air fluxes (Q(1),Q(2)) are controlled independently, and we constructed parameter spaces of the periodicity of the attractors. We have observed complex behavior and many modes of phase synchronization that depend on these airflows as well as on the height (H) of the solution above the tops of the nozzles. Such synchronizations are shown in details in the parameter space (Q(1),Q(2)) and also in the (Q(1),H) space. We also observed that the coupling strength between the two trains of bubbles increases when the solution height increases. The experimental results were reasonably explained by numerical simulations of a model combining a simple bubble growth model for each bubble train and a coupling term between them, which was assumed symmetrical and proportional to the growth velocities.

Period adding cascades: experiment and modeling in air bubbling.

Period adding cascades have been observed experimentally/numerically in the dynamics of neurons and pancreatic cells, lasers, electric circuits, chemical reactions, oceanic internal waves, and also in air bubbling. We show that the period adding cascades appearing in bubbling from a nozzle submerged in a viscous liquid can be reproduced by a simple model, based on some hydrodynamical principles, dealing with the time evolution of two variables, bubble position and pressure of the air chamber, through a system of differential equations with a rule of detachment based on force balance. The model further reduces to an iterating one-dimensional map giving the pressures at the detachments, where time between bubbles come out as an observable of the dynamics. The model has not only good agreement with experimental data, but is also able to predict the influence of the main parameters involved, like the length of the hose connecting the air supplier with the needle, the needle radius and the needle length.

Chaotic bubbling and nonstagnant foams.

We present an experimental investigation of the agglomeration of bubbles obtained from a nozzle working in different bubbling regimes. This experiment consists of a continuous production of bubbles from a nozzle at the bottom of a liquid column, and these bubbles create a two-dimensional (2D) foam (or a bubble raft) at the top of this column. The bubbles can assemble in various dynamically stable arrangement, forming different kinds of foams in a liquid mixture of water and glycerol, with the effect that the bubble formation regimes influence the foam obtained from this agglomeration of bubbles. The average number of bubbles in the foam is related to the bubble formation frequency and the bubble mean lifetime. The periodic bubbling can generate regular or irregular foam, while a chaotic bubbling only generates irregular foam.

Champagne experiences various rhythmical bubbling regimes in a flute.

Bubble trains are seen rising gracefully from a few points on the glass wall (called nucleation sites) whenever champagne is poured into a glass. As time passes during the gas-discharging process, the careful observation of some given bubble columns reveals that the interbubble distance may change suddenly, thus revealing different rhythmical bubbling regimes. Here, it is reported that the transitions between the different bubbling regimes of some nucleation sites during gas discharging is a process which may be ruled by a strong interaction between tiny gas pockets trapped inside the nucleation site and/or also by an interaction between the tiny bubbles just blown from the nucleation site.

Period-adding route in sparkling bubbles.

Chains of bubbles are seen rising along the wall whenever champagne is poured into a glass. The careful observation of a given bubble chain often reveals that the interbubble distance suddenly changes during the degassing process, indicating different bubbling regimes in this elusive phenomenon of effervescence. We report the transitions between these different bubbling regimes that present sequences of multiple periods known as the period-adding route.

Period-adding bifurcations and chaos in a bubble column.

We obtained period-adding bifurcations in a bubble formation experiment. Using the air flow rate as the control parameter in this experiment, the bubble emission from the nozzle in a viscous fluid undergoes from single bubbling to a sequence of periodic bifurcations of k to k+1 periods, occasionally interspersed with some chaotic regions. Our main assumption is that this period-adding bifurcation in bubble formation depends on flow rate variations in the chamber under the nozzle. This assumption was experimentally tested by placing a tube between the air reservoir and the chamber under the nozzle in the bubble column experiment. By increasing the tube length, more period-adding bifurcations were observed. We associated two main types of bubble growth to the flow rate fluctuations inside the chamber for different bubbling regimes. We also studied the properties of piecewise nonlinear maps obtained from the experimental reconstructed attractors, and we concluded that this experiment is a spatially extended system.

Bistability in bubble formation.

We obtain experimental data on time intervals of a bubble train generated from a nozzle with the air flow rate as the control parameter. Varying the length of the hose that connects the proportionating solenoid valve to the nozzle, we generate bifurcation diagrams showing period-adding cascades, among other dynamical phenomena. Then we construct a two-parameter family of one-dimensional maps whose bifurcation diagrams qualitatively match the experimental ones. The model indicates the existence of parameters where two attractors coexist, a phenomenon called bistability, and the same behavior is fully confirmed in the experiment.