POD-DEIM MODEL ORDER REDUCTION FOR THE MONODOMAIN REACTION-DIFFUSION EQUATION IN NEURO-MUSCULAR SYSTEM

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1 6th European Conference on Coputational Mechanics (ECCM 6) 7th European Conference on Coputational Fluid Dynaics (ECFD 7) 1115 June 2018, Glasgow, UK POD-DEIM MODEL ORDER REDUCTION FOR THE MONODOMAIN REACTION-DIFFUSION EQUATION IN NEURO-MUSCULAR SYSTEM N. EMAMY,1, P. LITTY 1, T. KLOTZ 2, M. MEHL 1 and O. RÖHRLE 2 1 Institute for Parallel and Distributed Systes, University of Stuttgart, Universitätstr. 38, Stuttgart, Gerany nehzat.eay,iria.ehl@ipvs.uni-stuttgart.de, 2 Institute of Applied Mechanics (CE), SiTech Research Group on Continuu Bioechanics and Mechanobiology, University of Stuttgart Pfaffenwaldring 5a, Stuttgart, Gerany thoas.klotz@echbau.uni-stuttgart.de, roehrle@sitech.uni-stuttgart.de, Key words: Model order reduction, Neuro-uscular syste, Coupled probles, Multiscale, Reaction-diffusion, POD, DEIM Abstract. We apply the POD-DEIM odel order reduction to the propagation of the transebrane potential along 1D uscle fibers. This propagation is represented using the onodoain partial differential equation. The onodoain equation, which is a reaction-diffusion equation, is coupled through its reaction ter with a set of ordinary differential equations, which provide the ionic current across the cell ebrane. Due to the strong coupling of the transebrane potential and ionic state variables, we reduce the all together proposing a total reduction strategy. We copare the current strategy with the conventional strategy of reducing the transebrane potential. Considering the current approach, the discrete syste atrix is slightly odified to adjust for the size. However, size of the precoputed reduced syste atrix reains the sae, which eans the sae coputational cost. The current approach appears to be four orders of agnitude ore accurate considering the equivalent nuber of odes on the sae grid in coparison to the conventional approach. Moreover, it shows a faster convergence in the nuber of POD odes with respect to the grid refineent. Using the DEIM approxiation of nonlinear functions in cobination with the total reduction, the nonlinear functions corresponding to the ionic state variables are also approxiated besides the nonlinear ionic current in the onodoain equation. For the current POD-DEIM approach, it appears that the sae nuber of DEIM interpolation points as the nuber of POD odes is the optial choice regarding stability, accuracy and runtie. 1 Introduction The neuro-uscular syste is a coplex ultiscale coupled syste, for which cheoelectroechanical odels are proposed as e.g. in 1, 2, 3. Realistic siulations of such 1

2 odels are coputationally extreely deanding. One challenging part of the odel is the partial differential equation (PDE) describing the propagation of the transebrane potentials along uscle fibers. The PDE is coupled with a set of nonlinear ordinary differential equations (ODEs), which reseble ionic currents at the cell ebrane. To reduce the coputational costs, one could apply odel order reduction techniques. For spiking neurons, the odel order reduction is applied in 4, where the proper orthogonal decoposition (POD) is used together with the proposed discrete epirical interpolation ethod (DEIM) to approxiate the nonlinear ionic current resulting fro the Hodgkin and Huxley (1952) odel 5. For paraeter identification in cardiac electrophysiology, the POD approach is used to build a reduced basis for the extracellular and transebrane potentials 6. The POD-DEIM approach is used in 7 to estiate the cardiac conductivities and in 8 to study electroyographic signals for skeletal uscles. The usual reduction approach as used in the above entioned studies, is to consider snapshots of the transebrane potential and nonlinear ionic current to build the POD bases. However, as the transebrane potential and ionic state variables are strongly coupled, we focus on a reduction strategy to reduce the all together. We call this strategy and the usual approach as total and partial reduction strategies, respectively, which we introduce in section 4. In sections 2 and 3, we show the onodoain equation and its discrete version, respectively. The introduced partial and total reduction strategies are copared in section 5 for the POD approach. The POD-DEIM approach using the total reduction is studied in section 6. 2 Propagation of the transebrane potentials along uscle fibers Considering skeletal uscles, the propagation of transebrane potentials along uscle fibers are well approxiated as 1D probles using the onodoain equation. The onodoain equation coprises the diffusion of the transebrane potentials along 1D uscle fibers in doain Γ R and the icroscopic reactions existing at the cell ebrane, V t y t = 1 A C ( ( x σ eff V x ) ( ) ) A I ion y, V, I sti in Γ, (1a) = G (y, V ), (1b) where V is the transebrane (action) potential, C is the capacitance of the uscle fiber transebrane (sarcolea), A is the fiber s surface to volue ratio and σ eff is the effective conductivity. x denotes the spatial coordinate along the fiber. I ion is the ionic current flowing over the ion-channels and -pups. Further state variables are suarized in y, e. g. the states of different ion-channels, which depend on V, as well. I sti is an externally applied stiulation current due to a stiulus fro the nervous syste. G suarizes the right-hand-side of all nonlinear ODEs associated with the state variables y. Here, we consider the biophysically otivated odel of Hodgkin and Huxley (1952) 5, which is the base odel to describe the ionic echanis in the cell ebrane. For this odel, y R 3 represent the three gating variables related to activation/inactivation 2

3 of potassiu and sodiu ion-channels. We apply hoogeneous Neuann boundary conditions for V at both ends of the 1 c long fiber. The initial values for V and y are set according to the Hodgkin and Huxley odel. The stiulation current is applied at the iddle of the fiber. 3 Discrete syste For the tie integration of the onodoain equation (1a), we apply the Godunov splitting schee, where we integrate the reaction and diffusion ters in two steps. The reaction ter is integrated together with (1b) using the explicit Euler schee and the diffusion ters is integrated using the iplicit Euler schee. V y (n+1) = V (n) y (n) V (n+1) = V + dt A C 1 ( (n) I ion V + dt C x G ( V (n), y (n)) V (σ (n+1) ) eff x, y (n), I (n) ) sti V denotes an interediate transebrane potential. After the spatial discretization using the FEM ethod, we have the following discrete syste of equations, which define our full order odel, v (n) (n) v F(v, y (n) ) + Bu (n) = +, (3a) y (n+1) y (n) G(v (n), y (n) ) v (n+1) in Γ, (2a) (2b) = v + Av (n+1), (3b) where v, Bu R n and A R n n. y R 3n, F : I R n and G : I R 3n for I R 4n. Bu ipleents the stiulation current as an input ter, which is adjustable linearly with the nuber of nodes in the grid. 4 Reduction strategy To study the potential for odel order reduction, the singular value decoposition (SVD) is applied on the snapshot (saple of trajectory) atrix S = s 1,..., s ns, S = VΣW T. Here we follow the notation in 9, where Σ = diag(σ 1,..., σ r ) R r r, with σ 1 σ 2... σ r > 0. The rank of S is r in(n, n s ). n s is the nuber of snapshots. The snapshots are gathered every tie step s for 10 s (n s = 20000) for grids of n = 10, 20, 40, 80, 160 and 320 nodes. We consider two strategies with respect to the snapshots. First, for a partial reduction, we choose the snapshots of the transebrane 3

4 potential to have the snapshot atrix, v 1 v 1... v s 1 S n ns = v n v n... v s n Second, for a total reduction, we consider the snapshots of the transebrane potential and the three state variables, v 1 v 1... v s 1 y 11 y y s 1 1 y 2 1 y y s 2 1 y 31 y y s 3 1 S 4n ns = v n v n... v s n y 1 n y 1n... y s 1 n y 2n y 2n... y s 2 n y 3n y 3n... y s 3 n Note that V, which contains the left singular vectors of S, has the diensions n r and 4n r considering the partial and total reduction strategies, respectively. The singular values are shown in Figures 1 and 2 for the two strategies. There are jups in the singular values, at which one can set a threshold for the reduction. A convergence with respect to grid refineent could be observed for both strategies. We consider a threshold of for the singular values and show the nuber of singular values, which are greater than this threshold in Table 1. Nuber of the odes k shows to converge by grid refineent for both strategies. The k/r ratio shows the ratio of the odes corresponding to the transebrane potential for both strategies. The sall ratios show that the reduction is ore effective using fine grids. Using the total reduction, a faster convergence in k by the grid refineent and uch saller ratios k/r on the sae grid are achieved in coparison to the partial reduction. Table 1: Coparison of the singular values considering the partial and total reduction strategies for different grids. k/r shows the ratio of the nuber of singular values greater than to the rank of snapshot atrix. Partial reduction Total reduction n r k k/r

5 Singular values Singular values nodes 20 nodes 40 nodes 80 nodes 160 nodes 320 nodes Nuber of singular values Relative nuber of singular values Figure 1: Singular value decoposition of the snapshot atrix containing the transebrane potentials using the partial reduction strategy. Singular values Singular values nodes 20 nodes 40 nodes 80 nodes 160 nodes 320 nodes Nuber of singular values Relative nuber of singular values Figure 2: Singular value decoposition of the snapshot atrix containing the transebrane potential and three ion-channel state variables using the total reduction strategy. 5 POD-Galerkin approach The set of left singular vectors of the snapshot atrix provides the basis for the POD approxiation of the snapshots, where the iniu L2-nor error of the approxiation is given by n s k 2 r s j (s T j v i )v i = σi 2. (4) j=1 i=1 2 i=k+1 5

6 We use the KerMor 1 10 library and apply the Galerkin projection of the syste (3) above. Using the partial reduction, we have the following reduced syste to solve, ṽ y (n+1) ṽ (n+1) = ṽ(n) y (n) + V T k F(v (n), y (n) ) + V T k Bu(n) G(v (n), y (n) ), (5a) = ṽ + V T k AV k ṽ (n+1), (5b) where v = V k ṽ. The reduced transebrane potential ṽ R k and V k is the atrix of k left singular vectors of the snapshot atrix with k < r. Considering the total reduction, we have where v y ṽ ỹ (n+1) ṽ (n+1) ṽ = V k ỹ = ṽ(n) ỹ (n) + V T k F(v (n), y (n) ) G(v (n), y (n) ) + V T k Bu (n), (6a) = ṽ + Vk T AV k ṽ (n+1), (6b). As entioned before, by the total reduction, V k has the diension 4n k. Therefore, atrix A is odified by adding extra lines of zeros to adjust its size in the following for, a 11 a a 1n a 21 a a 2n A 4n 4n = a n1 a n2... a nn However, the reduced syste atrix à = VT k AV k has the sae diension k k for both partial and total reduction strategies and could be precoputed. Bu is also odified for the total reduction, accordingly. In Figure 3, we copare the two strategies for the errors of the transebrane potential vs. runtie using a grid of 80 nodes. The relative L2-nor errors are coputed with respect to the full order odel (3) and averaged over the whole 1 6

7 coputational tie. Using the total reduction rather than the partial reduction, ore accurate solutions are achieved in saller runties. Considering 30 odes for projecting v, the error is four orders of agnitude saller with a runtie of approxiately 24 s. Therefore, we continue with the total reduction strategy for the following studies using the sae grid and intervals of 5 odes for v. In Figure 4, we show the errors of v vs. runtie and speedup for different grids. As expected the speedups becoe considerable for finer grids. Mean relative L2-nor error Runtie (s) Partial reduction Full reduction Figure 3: Coparison of the partial and total reduction strategies. Errors of the transebrane potential with respect to the full order odel are shown. A grid of 80 nodes is eployed. Each point on the plot resebles 5 odes for the partial and 20 odes for total reduction (5 relevant odes for the transebrane potential), respectively. 7

8 Mean relative L2-nor error Mean relative L2-nor error nodes 40 nodes 80 nodes 160 nodes 320 nodes Runtie (s) Speedup Figure 4: Errors of the transebrane potential vs. runtie and speedup with respect to the full order odel. The POD approach with the total reduction strategy is applied. A grid of 80 nodes is eployed. Each point on the plot resebles 5 odes of the transebrane potential. 6 POD-DEIM approach Besides the total reduction strategy in equation (6a), we apply the discrete epirical interpolation ethod (DEIM) 9 to the nonlinear functions F and G, ṽ ỹ (n+1) = ṽ(n) ỹ (n) + Vk T U (P T U ) 1 P T F(v (n), y (n) ) G(v (n) + V, y (n) k T Bu (n), (7a) ) ṽ (n+1) = ṽ + V T k AV k ṽ (n+1) (7b) where U, P R 4n contain the projection basis and interpolation points, respectively. The basis U = u 1,..., u is found by applying POD, where the SVD is perfored on the snapshots, F1 1 F F ns 1 G 11 G G s 1 1 G 2 1 G G s 2 1 G 31 G G s 3 1 H = Fn 1 Fn 2... F n ns 1 2 n G 1 n G 1 n... G s 1 n G 2n G 2n... G s 2 n G 3n G 3n... G s 3 n The atrix P = e 1,..., e contains coluns of the identity atrix, where indices { 1,..., } are found through the DEIM algorith with respect to the basis {u 1,..., u }. Applying the above, the coputational coplexity of the nonlinear functions are independent of n. Because the functions ust be evaluated only at the interpolation 8

9 points, P T F R G 1. Therefore, the atrix vector ultiplication of size (k 4n) ties (4n 1) in equation (6a) is replaced by (k ) ties ( 1). The atrix Vk TU (P T U ) 1 R k is precoputed. We show the errors of the transebrane potential vs. runtie for the POD-DEIM approach in Figure 6. A grid of 80 nodes is used as before. Each point on the plot resebles 20 POD odes (5 relevant odes for the transebrane potential). /4n = 0.4,..., 1 is the ratio of the interpolation points. Using the DEIM approxiation, the runtie could be reduced considerably for the sae level of accuracy as expected. As one ay also find out fro the figure, the nuber of the DEIM interpolation points should be chosen equal or greater than nuber of the POD odes, k. The optial nuber would be = k for the iniu runtie. A saller nuber of interpolation points, would ean issing parts of the basis for the snapshots of the nonlinear functions, which are already included in the snapshots of the solution. According to our siulations, if < k while having less than 60 percent of the interpolation points, the solution is not stable when adding ore POD odes by increasing k. If ore than 60 percent of the points are used, the solution is stable but the error is not reduced further and is doinated by the error of the DEIM approxiation. Our finding of the optial choice is in agreeent with 4, where they also found epirically the equal nuber of POD and DEIM odes as the best choice. Mean relative L2-nor error POD-DEIM 0.4 POD-DEIM 0.45 POD-DEIM 0.5 POD-DEIM 0.55 POD-DEIM 0.6 POD-DEIM 0.65 POD-DEIM 0.7 POD-DEIM 0.75 POD-DEIM 0.8 POD-DEIM 0.85 POD-DEIM 0.9 POD-DEIM 0.95 POD-DEIM Runtie (s) Figure 5: POD-DEIM approach for odel order reduction. Errors of the transebrane potential with respect to the full order odel are shown. Ratios /4n for DEIM show the ratio of interpolation points used for evaluating the nonlinear functions. A grid of 80 nodes is eployed. Each point on the plot resebles 20 odes (5 relevant odes for the transebrane potential). 9

10 7 Conclusion and outlook Due to the strong coupling between the transebrane potential and ionic state variables, we propose a total reduction of the all together to build the POD basis. Coparing to the conventional partial reduction, where only the transebrane potential is reduced, three ionic state variables resulting fro the Hodgkin and Huxley (1952) odel are additionally considered to build the snapshot atrix. The SVD decopositions of snapshot atrices using both reduction strategies show jups in the singular values on different grids. However, using finer grids, the jups disappear, which could be used as a sign for grid convergence. Considering a grid of 80 nodes and a threshold of for the singular values, only 42 percent of the odes are above the threshold using the total reduction, while that would be 98 percent of the odes using the partial reduction. The effectiveness of the total reduction is approved by coparing the ean relative L2-nor errors of transebrane potential for both strategies. Using 30 odes, the errors are four orders of agnitude saller for the total reduction. To efficiently evaluate the nonlinear ionic current, we use the DEIM approxiation. To adjust for the total reduction, the nonlinear functions corresponding to the state variables on the right-hand side of the nonlinear ODEs are approxiated as well. Varying the nuber of DEIM interpolation points for a fixed nuber of POD odes could reduce the runtie for the sae accuracy as expected. However, if the nuber of points is saller than the odes, the solution is either unstable (using less than 60 percent of the points) or the errors are doinated by the DEIM approxiation error. This could be understood as issing DEIM points because the snapshots of the nonlinear functions are present in and influence the snapshots of the solution. The equal nuber of DEIM points and POD odes appears to be the optial choice, which is stable and accurate with considerably reduced runtie. Our overall conclusion is that the potential for odel order reduction lies ainly in reduction of the reaction ter of the onodoain equation. This should be verified by considering ore coplicated odels for cheical reactions such as the Shorten odel 11, which contains ore than 50 state variables. 8 Acknowledgents This research was funded by the Baden-Württeberg Stiftung as part of the DiHu project of the High Perforance Coputing II progra and the Cluster of Excellence for Siulation Technology (EXC 310/1). REFERENCES 1 O. Röhrle, J. B. Davidson, and A. J. Pullan. A physiologically based, ulti-scale odel of skeletal uscle structure and function. Frontiers in Physiology, 3, T. Heidlauf and O. Röhrle. Modeling the Cheoelectroechanical Behavior of Skeletal Muscle Using the Parallel Open-Source Software Library OpenCMISS. Coputational and Matheatical Methods in Medicine, 2013:1 14,

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