Enrique Alvarez-Lacalle

Low viscosity contrast fingering in a rotating Hele-Shaw cell

We study the fingering instability of a circular interface between two immiscible liquids in a radial Hele-Shaw cell. The cell rotates around its vertical symmetry axis, and the instability is driven by the density difference between the two fluids. This kind of driving allows studying the interfacial dynamics in the particularly interesting case of an interface separating two liquids of comparable viscosity. An accurate experimental study of the number of fingers emerging from the instability reveals a slight but systematic dependence of the linear dispersion relation on the gap spacing. We show that this result is related to a modification of the interface boundary condition which incorporates stresses originated from normal velocity gradients. The early nonlinear regime shows nearly no competition between the outgrowing fingers, characteristic of low viscosity contrast flows. We perform experiments in a wide range of experimental parameters, under conditions of mass conservation (no injection), and characterize the resulting patterns by data collapses of two characteristic lengths: the radius of gyration of the pattern and the interface stretching. Deep in the nonlinear regime, the fingers which grow radially outwards stretch and become gradually thinner, to a point that the fingers pinch and emit drops. We show that the amount of liquid emitted in the first generation of drops is a constant independent of the experimental parameters. Further on there is a sharp reduction of the amount of liquid centrifugated, punctuated by periods of no observable centrifugation.

Detection, properties, and frequency of local calcium release from the sarcoplasmic reticulum in teleost cardiomyocytes.

Calcium release from the sarcoplasmic reticulum (SR) plays a central role in the regulation of cardiac contraction and rhythm in mammals and humans but its role is controversial in teleosts. Since the zebrafish is an emerging model for studies of cardiovascular function and regeneration we here sought to determine if basic features of SR calcium release are phylogenetically conserved. Confocal calcium imaging was used to detect spontaneous calcium release (calcium sparks and waves) from the SR. Calcium sparks were detected in 16 of 38 trout atrial myocytes and 6 of 15 ventricular cells. The spark amplitude was 1.45±0.03 times the baseline fluorescence and the time to half maximal decay of sparks was 27±3 ms. Spark frequency was 0.88 sparks µm−1 min−1 while calcium waves were 8.5 times less frequent. Inhibition of SR calcium uptake reduced the calcium transient (F/F0) from 1.77±0.17 to 1.12±0.18 (p = 0.002) and abolished calcium sparks and waves. Moreover, elevation of extracellular calcium from 2 to 10 mM promoted early and delayed afterdepolarizations (from 0.6±0.3 min−1 to 8.1±2.0 min−1, p = 0.001), demonstrating the ability of SR calcium release to induce afterdepolarizations in the trout heart. Calcium sparks of similar width and duration were also observed in zebrafish ventricular myocytes. In conclusion, this is the first study to consistently report calcium sparks in teleosts and demonstrate that the basic features of calcium release through the ryanodine receptor are conserved, suggesting that teleost cardiac myocytes is a relevant model to study the functional impact of abnormal SR function.

Characterization of the nonlinear content of the heart rate dynamics during myocardial ischemia

We develop a method to quantify the changes in heart rate dynamics during local myocardial ischemia induced by a percutaneous transluminal coronary angioplasty procedure (PTCA). The method introduces an index measuring the nonlinear content of the beat-to-beat (RR) time series by using nonlinear time series techniques such as surrogate data analysis and average mutual information. The index is applied to RR data from 67 subjects obtained before, during, and after the ischemic period and shows an increase in the nonlinearity of the cardiac control dynamics during ischemic and reperfusion stages. The nonlinear index is also used to characterize the effects of performing the coronary occlusion at different arteries and distances. We observe that the effect of ischemia becomes larger as the occlusion distance is reduced, and that most of the changes in the nonlinear content of the dynamics occur at long time scales typically related to sympathetic modulation of the cardiac rhythm (6-25 s).

Slow and fast pulses in 1-D cultures of excitatory neurons.

We analyze the characteristics of front propagation in activity of 1-D neuronal cultures by numerical simulations, using only excitatory dynamics. Experimental results in 1-D cultures of hippocampal neurons from rats have shown the spontaneous generation of a slow, low amplitude pulse that precedes a high amplitude, fast pulse that propagates through all the system. Notably, this transition appears both with and without the presence of functioning inhibitory synapses. In accordance with previous work, we demonstrate that purely excitatory integrate and fire neurons with depression in the synapses suffice to produce fast and uniform pulses but cannot explain the appearance of slow, weak pulses. We propose to explain the slow pulses by increasing the complexity of the neuron model in a purely excitatory network with connectivity as close to the experiments as possible. This approach allows us to show that spike frequency adaptation is a fundamental ingredient for the initiation process of the pulse. The introduction of a slow variable that mimics the presence of the slow K +  channels in the soma and produces spike frequency adaptation increases strongly the persistence of the transient activity before the emergence of the fast pulse up to temporal and spatial scales comparable with the experiments. Finally, we demonstrate that proper levels of additive white noisy currents generate such pulses spontaneously, fully reproducing the experimental results.

Systematic weakly nonlinear analysis of interfacial instabilities in Hele-Shaw flows

We develop a systematic method to derive all orders of mode couplings in a weakly nonlinear approach to the dynamics of the interface between two immiscible viscous fluids in a Hele-Shaw cell. The method is completely general: it applies to arbitrary geometry and driving. Here we apply it to the channel geometry driven by gravity and pressure. The finite radius of convergence of the mode-coupling expansion is found. Calculation up to third-order couplings is done, which is necessary to account for the time-dependent Saffman-Taylor finger solution and the case of zero viscosity contrast. The explicit results provide relevant analytical information about the role that the viscosity contrast and the surface tension play in the dynamics of the system. We finally check the quantitative validity of different orders of approximation and a resummation scheme against a physically relevant, exact time-dependent solution. The agreement between the low-order approximations and the exact solution is excellent within the radius of convergence, and is even reasonably good beyond this radius.

Minimal model for calcium alternans due to SR release refractoriness

In the heart, rapid pacing rates may induce alternations in the strength of cardiac contraction, termed pulsus alternans. Often, this is due to an instability in the dynamics of the intracellular calcium concentration, whose transients become larger and smaller at consecutive beats. This alternation has been linked experimentally and theoretically to two different mechanisms: an instability due to (1) a strong dependence of calcium release on sarcoplasmic reticulum (SR) load, together with a slow calcium reuptake into the SR or (2) to SR release refractoriness, due to a slow recovery of the ryanodine receptors (RyR2) from inactivation. The relationship between calcium alternans and refractoriness of the RyR2 has been more elusive than the corresponding SR Ca load mechanism. To study the former, we reduce a general calcium model, which mimics the deterministic evolution of a calcium release unit, to its most basic elements. We show that calcium alternans can be understood using a simple nonlinear equation for calcium concentration at the dyadic space, coupled to a relaxation equation for the number of recovered RyR2s. Depending on the number of RyR2s that are recovered at the beginning of a stimulation, the increase in calcium concentration may pass, or not, over an excitability threshold that limits the occurrence of a large calcium transient. When the recovery of the RyR2 is slow, this produces naturally a period doubling bifurcation, resulting in calcium alternans. We then study the effects of inactivation, calcium diffusion, and release conductance for the onset of alternans. We find that the development of alternans requires a well-defined value of diffusion while it is less sensitive to the values of inactivation or release conductance.

Calcium alternans is a global order-disorder phase transition. Robustness on Ryanodine Receptor release dynamics

Electromechanical alternans is a beat-to-beat alternation in the strength of contraction of a cardiac cell which appears often due to an instability of calcium cycling. The global calcium signal in cardiomyocytes is the result of the combined effect of several thousand micron scale domains called Calcium Release Units (CaRU), coupled through diffusion, where the flow of calcium among different cell compartments is regulated by stochastic signaling involving the ryanodine receptor (RyR). Recently, numerical simulations have suggested that the transition from regular Ca cycling to alternans is an order-disorder phase transition consistent with the Ising universality class. Inside the cell, groups of CaRU form transient areas within the cell where alternans appear. However, global alternans appears only as a result of the synchronization of the oscillation phase among different subunits. We show here that this transition is indeed robust and universal upon changes in the behavior of the RyR. Using three different set of parameters for the transition rates among open, closed and inactivated states in the RyR, we show that different RyR behavior leads to the same type of order-disorder transition.

Oscillatory regime in excitatory media with global coupling: Application to cardiac dynamics

The goal of this paper is to describe the effects of simple electro-mechanical coupling in isotropic two-dimensional (2D) cardiac tissue. To this aim, we show that the Nash-Panfilov two variable model [PRL 95, 258104 (2005)] for electrical activation, which couples active stress directly to transmembrane potential, can be reduced in the linearly elastic regime to an excitatory system with global coupling. In the linear limit, numerical simulations of both models give the same dynamic evolution, including the appearance of an ectopic focus with origin at the center. Indeed, after an initial excitation, mechano-electrical coupling can generate sustained oscillations in the form of successive waves originated at the center. These oscillations have a large basin of attraction for different sample lengths and values of the stretching current, specially when the recovery time of the excitatory cells is short. We finally present and discuss the appearance of oscillatory waves whose origin is not the center of the 2D sample but a ring of tissue around it. These waves appear spontaneously under some conditions even when the first excitation is generated at the center.