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The authors have declared that no competing interests exist.

Conceived and designed the experiments: NC ABO IT KB. Performed the experiments: NC. Analyzed the data: NC. Contributed reagents/materials/analysis tools: NC. Wrote the paper: ABO IT KB.

Space-fractional operators have been used with success in a variety of practical applications to describe transport processes in media characterised by spatial connectivity properties and high structural heterogeneity altering the classical laws of diffusion. This study provides a systematic investigation of the spatio-temporal effects of a space-fractional model in cardiac electrophysiology. We consider a simplified model of electrical pulse propagation through cardiac tissue, namely the monodomain formulation of the Beeler-Reuter cell model on insulated tissue fibres, and obtain a space-fractional modification of the model by using the spectral definition of the one-dimensional continuous fractional Laplacian. The spectral decomposition of the fractional operator allows us to develop an efficient numerical method for the space-fractional problem. Particular attention is paid to the role played by the fractional operator in determining the solution behaviour and to the identification of crucial differences between the non-fractional and the fractional cases. We find a positive linear dependence of the depolarization peak height and a power law decay of notch and dome peak amplitudes for decreasing orders of the fractional operator. Furthermore, we establish a quadratic relationship in conduction velocity, and quantify the increasingly wider action potential foot and more pronounced dispersion of action potential duration, as the fractional order is decreased. A discussion of the physiological interpretation of the presented findings is made.

Excitable media models are typical mathematical tools used to reproduce

Classical equations describing electrical propagation in space at a macroscopic level are based on modelling strategies that represent the tissue as a continuous medium characterised by space average quantities according to the homogenisation principle [

The spatial complexity and heterogeneity of the structure in which an observed transport phenomenon takes place might lead to significant deviations from the standard laws of diffusion [

To the best of our knowledge, the work by Bueno-Orovio et al. [

Bueno-Orovio et al. [

For these reasons, in this manuscript we propose a systematic study of how the numerical solution of a standard excitable media model is affected by the introduction in the mathematical formulation of the model of a space-fractional operator of non-integer order

We begin by introducing the two main components of this work, that is, the monodomain formulation of the Beeler–Reuter cell model and the fractional differential operator of interest, namely the fractional Laplacian (−Δ)^{α/2} on an insulated finite domain. After formulating the space-fractional modification of the considered excitable media model, we then describe the methodology adopted in order to compute the numerical solution of the modified system of equations. We present the results of a set of numerical simulations designed in order to investigate how the presence of the fractional operator in space and its order

Mathematical models of electrical signal propagation in cardiac electrophysiology consist of suitable spatially dependent formulations of specified cell models reproducing the response of a single excitable cell to an applied electrical stimulus. All cell models describe the temporal evolution of the transmembrane potential

In order to account for pulse propagation, spatial dependence is typically introduced via the bidomain formulation of the considered cell model (for example, see [_{m} is the membrane capacitance per unit area, _{ion} is the sum of all transmembrane ionic currents, _{stim} is the electrical stimulus.

The specification of the vector _{ion} in

Space-fractional differential equations are differential equations in which the classical integer-order differential operators in space are substituted by non-integer counterparts, often referred to as fractional operators. As previously mentioned, differential equations involving non-integer order derivatives have shown to be very powerful tools in the description of transport phenomena whose characteristics substantially deviate from the classical Gaussian and Markovian assumptions and have been adopted as mathematical models for a variety of practical applications.

Among fractional operators in space, the fractional Laplacian (−Δ)^{α/2} [

The restriction of fractional operators to finite domains is by no means trivial and a fundamental role in the definition and implementation of the solution of the problem is played by the particular boundary conditions to which the model is coupled. With fractional operators the mere specification of a local condition at the boundaries is no longer sufficient for the problem to be well-posed, and even though well-established methodologies have been developed for the case of homogeneous Dirichlet boundary conditions (see for example [

For the particular application considered in this paper, typical boundary conditions are defined under the assumption that the finite domain considered is insulated and hence, in the standard case, they are specified as homogeneous Neumann boundary conditions.

On an insulated one-dimensional domain [0, ^{α/2} can be defined via the spectral mapping theorem and the eigenvalue decomposition of the continuous standard Laplacian (−Δ), as originally proposed by Ilić et al. [

Since the theoretical results obtained in Cusimano [

Let us consider the BR monodomain formulation on an insulated finite interval [0, _{ion}, ^{α/2} is the one-dimensional fractional Laplacian of order

Specifically, (−Δ)^{α/2} is the continuous operator with eigenfunctions

Throughout the rest of the paper we assume that the cell model description in system Eqs (_{ion},

In order to compute the numerical solution of system Eqs (_{f}], with _{f} > 0, we adapt the strategy proposed by Whiteley [_{f}] and starting from a given initial condition, to compute the numerical solution at each time iteration in two steps. First, the value of the transmembrane potential

The only difference between the strategy presented here and the one proposed by Whiteley [

In particular, we look for the solution ^{α/2} on the insulated domain, that is, in the form _{k} with uniform time step Δ^{k}: = _{k}), and ^{k}: = _{k}). The solution of _{k+1}, while the ionic term is considered fixed at _{k}, that is, by solving
_{j}, the computation of ^{k+1} can be fully diagonalised and each integral coefficient can be updated independently at each time step as follows:
^{α/2} and ^{k} and ^{k}.

Once the updated value of

Note that for practical reasons the eigenfunction expansion of the solution and the source term must be truncated after a finite number of terms, and the integral coefficients _{i} =

In this section we provide a set of numerical results aimed at studying the effect of reducing the order

Let us consider the spatial domain [0, _{f}] with _{f} = 500 ms, and a uniform time step equal to Δ_{m} = 1^{−2}, ^{−1}, and ^{−1}. The transmembrane potential is assumed to be initially equal to ^{−1} everywhere.

Each simulation proposed here was run for a certain temporal interval [0, _{stim}] without the application of any external stimulus so that the steady state of each variable could be reached before any AP was triggered. The electrical stimulus was applied at _{stim} to a small region _{stim} = 40 ^{−3} on the sub-interval _{stim} = 10 ms for two consecutive milliseconds, we are able to trigger the propagation across the domain of an AP.

In order to assess convergence of the numerical solution produced by our solution method and to identify acceptable values of the key parameters _{i} ∈ [0, _{i} (APD_{50} and APD_{90}, respectively), and the relative change in the conduction velocity (CV) measured as the ratio of the distance between two nodes of [0, _{50} and APD_{90} were defined as the time intervals between activation and the time when the voltage (at the considered point) reached 50% and 90% of repolarization to its resting value, respectively. In both cases, linear interpolation was used to obtain better resolved time values.

In order to compute the quantities of interest for our metrics, we focus on three equally spaced nodes, namely _{1}(_{2}(_{3}(

The solution is computed for all values of _{stim} = 40 ^{−3} was applied on [0,0.05 cm] at _{stim} = 10 ms for two consecutive milliseconds and then removed.

_{50} and APD_{90} only at the mid-point _{2} but very similar qualitative results were obtained for these metrics also at _{1} and _{3} (results not shown). In order to compute the relative velocity of _{1} and _{3}.

As is clearly visible in

In all simulations performed to compute the errors displayed in

_{2} increases as _{1} and _{3} obtaining very similar results (not shown).

The solution is computed for all values of ^{−4} ms. In all simulations, the stimulus _{stim} = 40 ^{−3} was applied on [0,0.05 cm] at _{stim} = 10 ms for two consecutive milliseconds and then removed.

In light of the proposed convergence results, we decided to set ^{−2} for all the considered values of _{f}]. Similarly, from _{50} and APD_{90} is smaller than 0.01% and hence, ensures that the value of these two biomarkers is computed with an even higher precision.

In this section we investigate the effect of reducing the value of the fractional order

We begin by studying how changes in _{1}, _{2}, and _{3} generated by the same applied stimulus _{stim} = 40 ^{−3} for

(A) Solution obtained when _{stim} = 40 ^{−3} was applied on [0,0.05 cm] at _{stim} = 10 ms for two consecutive milliseconds and then removed.

In the standard case depicted in

In order to better visualise these differences we focus on a single node in space and compare the AP shape generated at this node by _{stim} = 40 ^{−3} for ten different values of _{2} are plotted in _{1} and _{3} (results not shown).

In all cases the stimulus _{stim} = 40 ^{−3} was applied on [0,0.05 cm] at _{stim} = 10 ms for two consecutive milliseconds and then removed.

As

a decrease in both the depolarization peak height and the dome peak;

a visibly more pronounced early repolarization phase;

an increase in the AP foot width (the AP foot is defined as the first portion of the quick AP depolarization);

a shift towards the right of the AP, corresponding to a later activation time for the considered node and indicating a reduction in the CV of the propagation pulse.

In order to quantify the effects observed qualitatively in ^{b} +

(A) AP depolarization peak height, (B) dome peak height, (C) early repolarization minimum.

A better visualisation of the effects produced by varying the fractional order _{2} is given in

To aid the visualisation of the differences produced by varying the fractional order, we align the AP foot of the ten solution profiles considered so that the activation time (AT) of the node _{2} coincides for all values of

The differences in the activation time of _{2} observed for different values of

The CV corresponding to the reciprocal of the gradient of each of the ten straight lines in (A) was considered as data in (B) and a quadratic dependence of these data points from the fractional order

In particular, we find that the relationship between the CV and

In this section we make some considerations on the action potential duration (APD), and as a measure of APD we consider the APD_{90} previously introduced in the text. One important characteristic observed in standard simulations of excitable media models is the dependence of the repolarization front advance on the domain geometry and on the proximity to the boundaries [

Differences in the APD across the spatial interval are present for all values of

For a fixed

The shortening in APD in proximity to the boundary is essentially linked to the finiteness of the spatial domain and to the increase in the repolarization electrotonic current due to the loss of the more depolarized neighbouring cells, as observed by Cherry and Fenton [

We stress that the presented results on APD dispersion are not a simple effect due to the small domain size chosen for our simulations. In fact, when longer cables are considered dispersion of APD is still present and becomes increasingly pronounced as the fractional parameter

The introduction of a fractional operator in space in the BR monodomain formulation resulted in the solution of the model being characterised by a reduction of the CV and various changes in a number of other features of the AP shape. Clearly, a reduction in the CV could be obtained also by considering the standard BR monodomain formulation (^{−1}, generating an excitation wave with CV = 12.2903 cm⋅s^{−1}, and compare the AP generated at the usual three equidistant nodes _{1}, _{2}, _{3}, to the solution obtained at the same nodes when ^{−1}.

The continuous blue line corresponds to the standard solution obtained with ^{−1}. The dashed red line represents the fractional solution obtained by setting ^{−1}. In both cases the stimulus _{stim} = 40 ^{−3} was applied on [0,0.05 cm] at _{stim} = 10 ms for two consecutive milliseconds and then removed.

Even though the choice of these two combinations of parameters _{3} the distinction between the repolarization phase of the AP generated in the standard and the fractional cases becomes visible, identifying also the more pronounced shortening of APD as another distinctive feature of the fractional modification of the model. Achieving a larger CV in the fractional case can be simply obtained by increasing the conductivity parameter

The results presented in this manuscript show that the introduction of a space-fractional operator in the monodomain formulation of the BR cell model affects both the temporal profile of the solution and the spatial propagation of the applied stimulus. Moreover, we identify well defined patterns in the value of a set of solution features corresponding to quantities of physiological interest as functions of the fractional order

Although a direct comparison with experimental data is not provided in this work, the characteristics observed in the solution behaviour for

Similar conclusions can be made by examining the AP foot shape. Previous studies on the AP foot morphology [

Another important aspect of our study concerned the identification of crucial differences between the solution of the standard formulation and of the space-fractional modification of the model. We highlighted the fact that the modification in the solution profile produced by the simple reduction of

The study of the considered features of electrical wave propagation in two-dimensional and three-dimensional extensions of the space-fractional model proposed in this paper will be investigated in our future research.

In this manuscript, we chose a simplified model of cardiac electrophysiology, namely the BR monodomain model, and considered its space-fractional modification on an insulated interval [0, ^{α/2} of order ^{α/2} on the insulated finite domain and incorporate in its definition the correct representation of insulating boundary conditions in both the standard and the purely fractional cases.

By computing the solution of the considered space-fractional modification of the BR monodomain model for multiple values of the fractional parameter in its domain of definition, we were able to establish specific patterns in the solution behaviour and in the value of a set of solution features of interest as functions of

The extension of this work to more elaborated excitable media models does not pose a challenge from the theoretical perspective and would only require additional effort in the implementation of a more complicated system of differential equations and the computation of its solution. The real limitation of the presented analysis is the fact that we only considered a one-dimensional finite domain. Experimental studies showing direction-dependent characteristics of the AP shape in two spatial dimensions, such as [

Complete mathematical description of the equations for the BR cell model as provided in the original work by Beeler and Reuter [

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Here we report the analytic expression of the fitting curves introduced in the Results section, provide specific values for the parameters involved (with 95% confidence bounds), and use a set of statistics to evaluate how well the proposed curves approximate the data.

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We thank David Kay from the Department of Computer Science of the University of Oxford (UK) for many useful discussions on various aspects of the study proposed in this paper and for providing helpful comments on the manuscript draft before submission.