State Estimation Problem for the Action Potential Modeling in Purkinje Fibers

Transcription

1 APCOM & ISCM -4 th Deceber, 203, Singapore State Estiation Proble for the Action Potential Modeling in Purinje Fibers *D. C. Estuano¹, H. R. B.Orlande and M. J.Colaço Federal University of Rio de Janeiro UFRJ, Departent of Mechanical Engineering PEM/COPPE, Brazil *Corresponding author: Abstract In this wor we use the Particle Filter Method to solve a state estiation proble resulting fro the application of Hodgin-Huxley's odel to Purinje fibers, by applying Liu and West's Auxiliary Sapling Iportance Resapling (ASIR) algorith. This algorith allows the siultaneous estiation of state variables and paraeters. The estiation of the action potential in Purinje fibers can be related to the identification of heart anoalies. The use of Bayesian particle filters is of great interest for such specific application, since they tae into account uncertainties in the atheatical odels for the evolution of the state variables and the easureents. Siulated easureents are used in this wor to exaine the accuracy of the Particle Filter Method under analysis. Keywords: Particle Filters, Bayesian estiate, Hodgin-Huxley's odel, Purinje fibers. Introduction Hodgin and Huxley (952) proposed a odel for the action potential in an axon, in ters of an electric circuit with capacitance and ionic currents. Sodiu and potassiu ions are the ost iportant in the action potential and are distinguished in ters of their own proper currents, in coparison to the other ions. The odel involves a non-linear syste of four ordinary differential equations, whose coefficients are given in ters of functions of the applied potential. Although Hodgin-Huxley's odel has been originally proposed for the experiental data involving an axon, it has also been used to odel the action potential in heart cells, lie Purinje fibers (Noble, 962). State estiation probles are dynaically solved within the Bayesian fraewor (Kaipio and Soersalo, 2004; Arulapala et al., 200). In this fraewor, an attept is ade to utilize all available inforation in order to reduce the aount of uncertainty present in inferential or decisionaing probles. As new inforation is obtained, it is cobined with previous inforation to for the basis for statistical procedures. The foral echanis that cobines the new inforation with the previously available inforation is nown Bayes theore (Kaipio and Soersalo, 2004). Monte Carlo ethods have been developed in order to represent the posterior density in ters of rando saples and associated weights and can be applied to non-linear odels with non-gaussian errors (Kaipio and Soersalo, 2004; Arulapala et al., 200; Ristic et al., 2004; Doucet et al., 200; Orlande et al., 202), such as the one under analysis in this wor. In this paper we extend our previous wor (Estuano et al., 203) in order to copare the results obtained with the Sapling Iportance Resapling (SIR) algorith and the Auxiliary Sapling Iportance Resapling (ASIR) algorith (Kaipio and Soersalo, 2004; Arulapala et al., 200; Ristic et al., 2004; Doucet et al., 200), to results obtained with the algorith proposed by Liu and West (200). This paper is focused on the use of Hodgin-Huxley s odels for the action potential in Purinje Fibers (Noble, 962). The three algoriths are copared in ters of their coputational ties and RMS errors. We note, beforehand, that Liu and West's algorith is the ost general of the three algoriths listed above, since uncertainties in the odel paraeters are taen into account through Gaussian ernel soothing (Liu and West, 200). Other recent applications of inverse

2 probles to Hodgin-Huxley's odel include the wors of Doos and Lovell (2004) and Meng et al. (20). Hodgin-Huxley s Model Hodgin and Huxley, in their classical paper of 952, exained the behavior of an axon under the effects of an iposed electric current across the cell ebrane. The cell electric potential was assued to be independent of the position within the cell, that is, the intracellular electric resistance was neglected. In their experients, Hodgin and Huxley (952) observed that the conductance of soe ions across the cell ebrane, lie sodiu and potassiu, varied with changes in the axon's potential. The iposed electric current across the cell ebrane was then odeled in ters of capacitive and ionic currents. Being the sodiu and potassiu ions recognized as the ost iportant ones in this process, their currents were treated separately fro those corresponding to the other ions, which were quantified in a global anner and referred to as leaage current. For the odel, an inflow of ions was assued as positive. A basic difference between axons and Purinje fibers results fro the fact that in the last ones the potassiu flow is governed by both a fast and a slow channel dynaics. In addition, the flow of ions other than sodiu and potassiu through the cell ebrane can be neglected, so that the analogous electric circuit for the proble is presented in Figure. The iposed electric current is null for the case involving Purinje fibers because these cells are auto-excitable (Noble, 962). Therefore, the equation for the action potential in Purinje fibers is given by (Noble, 962): dv C + GNa( V VNa) + GK ( V VK) = 0 () dt Figure. Electric circuit for a Purinje fibers (Noble, 962) The equations given by Noble (962) for sodiu and potassiu conductances of a Purinje fiber are given, respectively, by: G K G = G h+ G (2) ax 3 Na Na Na, l V + 90 V + 90 =.2exp exp +.2n (3) where and n represent the open fraction, or probability of the channels being open, for sodiu and potassiu, respectively, while h is the probability of the channel being closed for the sodiu ions. The variables and n are also referred to as the activations of the ion transfer through the 2

3 cell ebrane, while h is referred to as the inactivation for the sodiu ion transfer. In equation (2), G refers to the axiu sodiu conductance. ax Na Hodgin and Huxley (952) proposed the following ordinary differential equations to describe the ion channels opening/closing dynaics: d = α( ) + β dt dh = αh( h) + βhh dt dn = αn( n) + βnn dt (4-6) The paraeters for the coputation of the activations and n and inactivation h in Eq. (4)-(6) are given as (Noble, 962): V + 48 α = 0. V + 48 exp 5 V + 42 βh = exp + 0 ( V + ) β = V + 8 exp 5 ( V + ) αn = V + 50 exp 0 V αh = 0.7 exp V βn = exp (7-9) (0-2) Other paraeters for the application of Hodgin-Huxley's odel for a Purinje fiber are presented in Table (Estuano et al., 203). Table. Paraeters for Hodgin-Huxley's odel for Purinje fiber Paraeter Values Paraeter Values Paraeter Values ( µ F c 2 ) 2 VNa ( V ) 40 GNa, l ( S ) 0.4 C G ( S ) 400 ( ) ax Na State Estiation Proble V V -00 In order to define the state estiation proble, consider a odel for the evolution of the vector x in the for (Kaipio and Soersalo, 2004; Arulapala et al., 200; Ristic et al., 2004; Doucet et al., 200; Orlande et al., 202): x = f ( x, v ) (3) nx where the subscript =, 2,, denotes a tie instant t in a dynaic proble. The vector x R is called the state vector and contains the variables to be dynaically estiated. This vector advances in accordance with the state evolution odel given by Eq. (3), as a non-linear function of nv the state variables x and of the state noise vector v R. Consider also that easureents n z z R are available at t, =, 2,. The easureents are related to the state variables x through the general function h in the for z = h ( x, n ) (4) 3

4 nn where n R is the easureent noise. Equation (4) is referred to as the observation (easureent) odel. The state estiation proble ais at obtaining inforation about x based on the state evolution odel (3) and on the easureents z: = { zi, i=, K, } given by the observation odel (4). The state estiation proble addressed in this wor deals with Hodgin-Huxley's odel applied to Purinje fibers. Therefore, the state variables are given by T x = [ V,, h, n] (5) with state evolution odels given by Eqs. () to (2). Measureents of the cell potential, V, are supposed available for the estiation of the state variables. Due to its nonlinear character, the Particle Filter Method was used for the solution of the present state estiation proble (Kaipio and Soersalo, 2004; Arulapala et al., 200; Ristic et al., 2004; Doucet et al., 200; Orlande et al., 202). In this ethod, the required posterior density function is represented by a set of rando saples (particles) with associated weights, which are then used for the sequential coputation of its associated statistics. The particle filter algoriths generally ae use of an iportance density, which is proposed to represent another density that cannot be exactly coputed, that is, the sought posterior density in the present case. Then, saples are drawn fro the iportance density instead of the actual density (Kaipio and Soersalo, 2004; Arulapala et al., 200; Ristic et al, 2004; Doucet et al., 200; Orlande et al., 202). i The set of particles fro tie t 0 to tie t is denoted as { x0:, i = 0, K, N } and their associated weights as { w i, i= 0, K, N}, where N is the nuber of particles. The weights are noralized, so that N i= w i 4 =. The sequential application of the particle filter ight result in the degeneracy phenoenon, where after a few states very few particles have negligible weight (Kaipio and Soersalo, 2004; Arulapala et al., 200; Ristic et al., 2004; Doucet et al., 200; Orlande et al., 202). An attept to overcoe this difficulty is to use a resapling step in the application of the particle filter, where particles with sall weights are discarded and particles with large weights are replicated. In the Sapling Iportance Resapling (SIR) algorith, resapling is applied every tie step (Arulapala et al., 200; Ristic et al., 2004). Although the resapling step reduces the effects of the degeneracy proble, it ay lead to a loss of diversity and the resultant saple ay contain any repeated particles, which is ore liely to occur in the case of sall state evolution noise (Kaipio and Soersalo, 2004; Arulapala et al., 200; Ristic et al., 2004; Doucet et al., 200; Orlande et al., 202). In addition, in the SIR algorith the state space is explored without the inforation conveyed by the easureents at the specific instant that the state variables are sought. With the Auxiliary Sapling Iportance Resapling (ASIR) algorith an attept is ade to overcoe these liitations, by perforing the resapling step at tie t -, with the available easureent at tie t (Arulapala et al., 200; Ristic et al., 2004). The resapling is based on soe point estiate µ i that characterizes π(x x i -), which can be the ean of π(x x i -) or siply a saple of π(x x i -). If the state evolution odel noise is sall, π(x x i -) is generally well characterized by µ i i, so that the weights w are ore even and the ASIR algorith is less sensitive to outliers than the SIR algorith. On the other hand, if the state evolution odel noise is large, the single point estiate µ i in the state space ay not characterize well π(x x i -) and the ASIR algorith ay not be as effective as the SIR algorith.

5 We note that the functions f (.) and h (.), in the evolution and observation odels, respectively, contain several constant paraeters, here denoted as the vector θ. However, in general, such paraeters are not deterinistic or ight not be deterinistically nown. Therefore, the saples i i need to be extended to { x, θ : i = 0, K, N} with associated weights { w i : i= 0, K, N}. In this wor, the algorith developed by Liu and West (200), and based on the ASIR algorith, is used for the solution of the state estiation proble, with the vector of state variables given by equation (5) and the vector of paraeters given by: Τ θ = [ C, ax GNa, GNa, l] (7) Such paraeters were selected for the present analysis because they are the ones with larger variabilities in the open literature. The algoriths of the SIR and ASIR filters, as well as the one due to Liu and West, can be found in Kaipio and Soersalo (2004), Arulapala et al. (200), Ristic et al. (2004), Liu and West (200), Orlande et al. (202), Colaço et al. (202) and are not repeated here for the sae of brevity. Results and Discussion In this paper, the three algoriths described above are copared in ters of their RMS errors and coputational ties, for cases dealing with axons and Purinje fibers. The CPU ties correspond to a coputational code running under the Matlab platfor, in an Intel i5 CPU with 6 Gb of RAM eory. The RMS error is coputed as M M = ( ) ( ) 2 i est i exa i (8) RMS = x t x t where the subscripts est and exa denote the estiated and exact values of the state variable xt ( i ) at tie t i, while M is the nuber of tie steps. A siilar definition was used to copute the RMS errors for each paraeter θ. The RMS errors were copared in ters of the eans of 00 repetitions of the particle filter estiates, in order to avoid any bias resulting fro different sets of siulated easureents used in the analysis (Hailton et al., 203). Siulated easureents of the cell potential, V, were utilized in the present wor. Such easureents were generated fro a nuerical siulation of the deterinistic dynaic proble for the Purinje fibers. Errors in the siulated easureents were additive, uncorrelated, with a Gaussian distribution, zero ean and a constant standard deviationσ, so that the lielihood function at tie t is given by π ( ) exp [ ( ) ( )] 2πσ 2σ od 2 z x = V 2 2 t V t (9) where the superscript od refers to the easureent variable coputed with the observation odel given by Eq. (4). Siulated easureents were considered available in tie intervals of 0 s. For the results presented below, the paraeters given by Eqs. () to (2) and Table were used in the analysis. The initial conditions for these cases were taen V 0 70V h 0 = (Estuano et al., 203). The as ( ) =, ( ) =, n ( ) = and ( ) 5

6 errors in the evolution odel were additive, uncorrelated, with Gaussian distribution, zero ean and a constant standard deviation of 5% of the absolute value of the state variables at the initial tie. The standard deviations of the easureents were taen as 5% of the axiu absolute value of the easured variable, that is, σ = 0.05 V,ax. The results obtained for the estiation of the state variables and the odel paraeters with Liu and West's algorith are presented in figures 2 and 3, respectively. Such results were obtained with 500 particles. Figure 2 shows an excellent agreeent between exact and estiated state variables, even for those for which easureents are not available, such as, n and h. Siilarly, uncertainties in the odel paraeters are appropriately taen into account as depicted fro figure 3. Note in this figure that the exact values of the paraeters fall within the confidence intervals of the estiates. Figures 2 and 3 reveals the robustness of Liu and West's algorith as applied to the present proble, which was capable of accurately estiating state variables and odel paraeters, despite the large uncertainties in the evolution and easureent odels, as well as in the odel paraeters Exact Measureent Estiate 99% Conf. Interv..4.2 Exact Estiate 99% Conf. Interv. V (V) Activation Exact Estiate 99% Conf. Interv..2 Exact Estiate 99% Conf. Interv. 0.7 Activation n Inactivation h Tepo s 0 Figure 2: Estiation of the state variables Tables 2-4 present the RMS errors and the coputational ties for the three Particle filter algoriths exained in this wor, for various nubers of particles (N p ). Let us first exaine tables 2 and 3, which present the results obtained with the SIR and the ASIR algoriths, respectively. Such algoriths deal only with the estiation of the state variables. These tables show, as expected, a reduction on the RMS errors, followed by an increase in the coputational tie, as the nuber of particles is increased. In addition, we notice in tables 2 and 3 that the RMS errors tend to approach a constant value as the nuber of particles is increased. According to tables 2 and 3, siilar RMS errors were obtained with the ASIR algorith by using less particles than those of the SIR algorith. Although the ASIR algorith is ore expensive than the SIR algorith in ters of coputational tie for the sae nuber of particles, the ASIR algorith is capable of providing accurate results with a uch saller nuber of particles. As a result, the coputational ties are saller with the ASIR algorith than with the SIR algorith, for results of coparable accuracy. 6

7 C (µf.c -2 ) G Na S G Nal S Figure 3: Estiation of the odel paraeters Table 2: RMS errors and coputational ties for the SIR algorith N p CPU Tie (s) V (V) n h Table 3: RMS errors and coputational ties for the ASIR algorith N p CPU Tie (s) V (V) n h The results obtained with Liu and West's algorith (200) are presented in table 4. A coparison of tables 3 and 4 reveal an increase on the coputational tie, for the sae nuber of particles, when Liu and West's algorith was used, caused by the siultaneous estiation of paraeters and state variables. Anyhow, both state variables and paraeters can be accurately estiated (see also figures 2 and 3) with this algorith. In addition, the RMS error was reduced when the uncertainties on the paraeters was taen into account, as copared to the original ASIR algorith. 7

8 Table 4: RMS errors and coputational ties for Liu and West's algorith N p CPU Tie (s) V (V) n h C µ F c 2 ( ) G (S) G Na Na, l (S) Conclusions Particle filter ethods are the ost general techniques for the solution of state estiation probles involving nonlinear and non-gaussian odels. In this paper, three different particle filter algoriths were applied to the estiation of state variables of the odel proposed by Hodgin and Huxley to describe the action potential in excitable cells. Cases involving Purinje fiber are exained in the paper, by using siulated easureents of the action potential. Although the SIR and the ASIR algoriths are capable of accurately estiating the state variables, we notice that the ore general algorith by Liu and West allows the siultaneous estiation of the state variables and odel paraeters. Furtherore, such quantities can be estiated with better accuracy than those related to the estiation of only the state variables. Acnowledgeents The support provided CNPq, ANP/PRH37, CAPES and FAPERJ, Brazilian agencies for the fostering of science, is greatly appreciated. References A. Hodgin and A. Huxley (952), A Quantitative Description of Mebrane Current and It's Application to Conduction and Excitation in Nerve, Journal of Physiology, 7, pp B. Ristic, S. Arulapala and N. Gordon (2004), Beyond the Kalan Filter, Artech House, Boston. D. Noble (962), Modification of Hodgin- Huxley Equations Applicable to Purinje Fiber Action and Pace Maer Potentials, Journal of Physiology, 60, pp D. Estuano, H. Orlande and M. Colaço (203), State Estiation Proble For Hodgin-Huxley S Model: A Coparison of Particle Filter Algoriths, 4th Inverse Probles, Design and Optiization Syposiu (IPDO- 203), Albi, France, June F. Hailton, M. Colaço, R. Carvalho and A. Leiroz (203), Heat Transfer Coefficient Estiation of an Internal Cobustion Engine Using Particle Filters, Inverse Probles in Science and Engineering, DOI:0.080/ H. Orlande, M. Colaço, G. Duliravich, F. Vianna, W. Silva, H. Fonseca and O. Fudy (202), State Estiation Probles in Heat transfer, International Journal for Uncertainty Quantification, 2, pp J. Kaipio and E. Soersalo (2004), Statistical and Coputational Inverse Probles, Applied Matheatical Sciences 60, Springer-Verlag. J. Liu and M. West (200), Cobined Paraeter and State Estiation in Siulation-Based Filtering, Chapter 0 in Sequential Monte Carlo Methods in Practice, A. Doucet, N. Freitas, N. Gordon (editors), Springer, New Yor. L. Meng, M. A. Kraer and U. T. Eden (20), A sequential Monte Carlo approach to estiate biophysical neural odels fro spies, Journal of Neural Engineering, 8(6), DOI: 0.088/ /8/6/ M. Colaço, H. Orlande, W. Silva and G. Duliravich (202), Application of Two Bayesian Filters to Estiate Unnown Heat Fluxes in a Natural Convection Proble, Journal of Heat Transfer, 34, pp S. Arulapala, S. Masell, N. Gordon and T. Clapp (200), A Tutorial on Particle Filters for On-Line Nonlinear/Non-Gaussian Bayesian Tracing, IEEE Trans. Signal Processing, 50, pp S. Doos and N. H. Lovell (2004), Paraeter estiation in cardiac ionic odels, Progress in Biophysics & Molecular Biology, 85, pp