Asymptotics of Conduction Velocity Restitution in Models of Electrical Excitation in the Heart

Addendum: Action potential propagation and block in a model of atrial tissue with myocyte-fibroblast coupling

The analytical theory of our earlier study (Mortensen et al., 2021, Math. Med. Biol., 38, 106-131) is extended to address the outstanding cases of fibroblast barrier distribution and myocyte strait distribution. In particular, closed-form approximations to the resting membrane potential and to the critical parameter values for propagation are derived for these two non-uniform fibroblast distributions and are in good agreement with numerical estimates.

Action potential propagation and block in a model of atrial tissue with myocyte-fibroblast coupling

The electrical coupling between myocytes and fibroblasts and the spacial distribution of fibroblasts within myocardial tissues are significant factors in triggering and sustaining cardiac arrhythmias, but their roles are poorly understood. This article describes both direct numerical simulations and an asymptotic theory of propagation and block of electrical excitation in a model of atrial tissue with myocyte-fibroblast coupling. In particular, three idealized fibroblast distributions are introduced: uniform distribution, fibroblast barrier and myocyte strait-all believed to be constituent blocks of realistic fibroblast distributions. Primary action potential biomarkers including conduction velocity, peak potential and triangulation index are estimated from direct simulations in all cases. Propagation block is found to occur at certain critical values of the parameters defining each idealized fibroblast distribution, and these critical values are accurately determined. An asymptotic theory proposed earlier is extended and applied to the case of a uniform fibroblast distribution. Biomarker values are obtained from hybrid analytical-numerical solutions of coupled fast-time and slow-time periodic boundary value problems and compare well to direct numerical simulations. The boundary of absolute refractoriness is determined solely by the fast-time problem and is found to depend on the values of the myocyte potential and on the slow inactivation variable of the sodium current ahead of the propagating pulse. In turn, these quantities are estimated from the slow-time problem using a regular perturbation expansion to find the steady state of the coupled myocyte-fibroblast kinetics. The asymptotic theory gives a simple analytical expression that captures with remarkable accuracy the block of propagation in the presence of fibroblasts.

Predicting critical ignition in slow-fast excitable models

Linearization around unstable traveling waves in excitable systems can be used to approximate strength-extent curves in the problem of initiation of excitation waves for a family of spatially confined perturbations to the rest state. This theory relies on the knowledge of the unstable traveling wave solution as well as the leading left and right eigenfunctions of its linearization. We investigate the asymptotics of these ingredients, and utility of the resulting approximations of the strength-extent curves, in the slow-fast limit in two-component excitable systems of FitzHugh-Nagumo type and test those on four illustrative models. Of these, two are with degenerate dependence of the fast kinetic on the slow variable, a feature which is motivated by a particular model found in the literature. In both cases, the unstable traveling wave solution converges to a stationary "critical nucleus" of the corresponding one-component fast subsystem. We observe that in the full system, the asymptotics of the left and right eigenspaces are distinct. In particular, the slow component of the left eigenfunction corresponding to the translational symmetry does not become negligible in the asymptotic limit. This has a significant detrimental effect on the critical curve predictions. The theory as formulated previously uses an heuristic to address a difficulty related to the translational invariance. We describe two alternatives to that heuristic, which do not use the misbehaving eigenfunction component. These new heuristics show much better predictive properties, including in the asymptotic limit, in all four examples.

Fast-slow asymptotics for a Markov chain model of fast sodium current

We explore the feasibility of using fast-slow asymptotics to eliminate the computational stiffness of discrete-state, continuous-time deterministic Markov chain models of ionic channels underlying cardiac excitability. We focus on a Markov chain model of fast sodium current, and investigate its asymptotic behaviour with respect to small parameters identified in different ways.

Atrial Conduction Velocity Correlates with Frequency Content of Bipolar Signal

BACKGROUND

Anisotropy in conduction velocity (CV) is a key substrate abnormality influencing atrial arrhythmias. In skeletal muscle fibers, CV and frequency content of the surface electromyogram signal are directly related. We hypothesized that in human atria the frequency content of the bipolar signal, recorded on the endocardial surface, is directly related to the local CV.

METHODS

In 15 patients submitted to ablation of supraventricular arrhythmias, incremental pacing was performed through an octapolar catheter inserted into the coronary sinus (CS), alternatively from both extremities in two different sequences: CS bipole 1-2 as the pacing site and CS bipole 7-8 as the detection site in the first, and vice versa in the second. The pacing cycle length (PCL) was stepwise decreased from 600 ms to 500 ms, 400 ms, 300 ms, until 250 ms. Estimation of the CV was performed as the ratio between the distance traveled by the propagating pulse and the propagation time. The frequency distribution of the signal energy was estimated using the fast Fourier transform, and the characteristic frequency (CF) was estimated as the barycenter of the frequency spectrum.

RESULTS

A total of 2,496 bipolar signals were analyzed; CV and CF were estimated and compared. The single patient and group data analysis showed a significant direct correlation between CV and CF of the local bipolar signal.

CONCLUSIONS

Comparing the degree of spectral compression among signals registered in different points of the endocardial cardiac surface in response to decreasing PCL enables to map local differences in CV, a useful arrhythmogenic substrate index.