# Cardiac Electrophysiology Meshfree Modeling through the Mixed Collocation Method
## 1 Introduction  
In the last decades, computational modeling and simulation has taken a growing role as a method to deepen the understanding of cardiac function in health and disease [1, 2]. Novel in silico models of increasing complexity are continuously being developed to simulate the electrophysiology [3] and mechanics of the heart [4] from the cell [5] to the whole-organ level [6]. At the tissue and organ levels, electrophysiology is simulated by using the well-known bidomain [7] and monodomain [8] models. The latter is a simplified version of the former under the assumption of equal anisotropy ratios for the intracellular and extracellular spaces. The monodomain model is more computationally efficient than its bidomain counterpart and is able to produce accurate transmembrane potential values in the absence of extracellularly applied currents [9]. Commonly, state-of-the-art simulators [10, 11] employ the finite element method (FEM) to solve either the bidomain or the monodomain model for the simulation of cardiac electrophysiology. FEM is a mature and robust numerical method, but its requirement for a good-quality mesh may pose challenges to generate realistic heart models with reasonable computational cost [12]. Alternative meshfree methods can alleviate the mesh requirement and have been proposed for both cardiac electrophysiological [13, 14, 15] and mechanical [12, 16] simulations, as well as brain biomechanics [17]. Among the different proposed meshfree solutions, models based on the element-free Galerkin (EFG) method offer high convergence rate and high resolution of localized steep gradients [18]. Nevertheless, special treatment for the imposition of essential boundary conditions is required since the approximation functions do not possess the Kronecker delta property. Recently, cell-based maximum entropy (CME) approximants were used in the EFG method to alleviate this limitation [19]. CME possesses the weak Kronecker delta property where approximation functions of internal nodes vanish on the boundaries. Therefore, essential boundary conditions can be imposed directly, as in the FEM. However, the CME approximants give rise to complex integrals requiring a large number of quadrature points for accurate integration that may lead to increased computational cost. Similarly, the computational cost of methods based on smoothed particle hydrodynamics (SPH) may be significantly higher than that of mesh-based methods. Furthermore, the standard SPH formulation may imply inaccurate computation of gradients of constant and linear fields (first-order incomplete approximation) [20]. To overcome these problems, the total Lagrangian formulation of SPH [21] and gradient normalization [20] were applied in [12; 13] to accurately simulate the propagation of the electrical impulse in the heart and cope with large deformations in the context of cardiac mechanics. In the present study, we propose the mixed collocation method (MCM) as an alternative to mesh-based and previously mentioned meshfree methods for cardiac electrophysiology simulation. Our main motivation is the purely meshfree property of MCM that allows us to completely alleviate the time-consuming mesh-generation process of mesh-based methods like the FEM. The MCM is based on the meshfree local Petrov-Galerkin (MLPG) method [22; 23]. MLPG implies quadrature over locally-defined domains, providing the flexibility to select the trial and test functions from different spaces. In the mixed collocation variant of the MLPG method, the Dirac function is used as test function, and both the field function and its gradient are interpolated by the trial function. As a result, the computational cost is decreased since the local integration is reduced to nodal summation. Moreover, while collocation methods suffer from inaccurate imposition of Neumann boundary conditions [24], the accuracy is ameliorated in the MCM due to the interpolation of the field function's gradient.