Estimating Electrical Conductivity Tensors of Biological Tissues Using

Microelectrode Arrays

ELAD GILBOA,1 PATRICIO S. LA ROSA,2 and ARYE NEHORAI1

1Preston M. Green Department of Electrical and Systems Engineering, Washington University in St. Louis, St. Louis,MO 63130, USA; and 2Division of General Medical Sciences, Department of Medicine, School of Medicine, Washington

University in St. Louis, St. Louis, MO 63110, USA

(Received 1 December 2011; accepted 21 April 2012)

Associate Editor Zahra Moussavi oversaw the review of this article.

Abstract—Finding the electrical conductivity of tissue ishighly important for understanding the tissue’s structure andfunctioning. However, the inverse problem of inferringspatial conductivity from data is highly ill-posed andcomputationally intensive. In this paper, we propose a novelmethod to solve the inverse problem of inferring tissueconductivity from a set of transmembrane potential andstimuli measurements made by microelectrode arrays(MEA). We first formalize the discrete forward model oftransmembrane potential propagation, based on a reaction–diffusion model with an anisotropic inhomogeneous electri-cal conductivity-tensor field. Then, we propose a novelparallel optimization algorithm for solving the complexinverse problem of estimating the electrical conductivity-tensor field. Specifically, we propose a single-step approxi-mation with a parallel block-relaxation optimization routinethat simplifies the joint tensor field estimation problem into aset of computationally tractable subproblems, allowing theuse of efficient standard optimization tools. Finally, usingnumerical examples of several electrical conductivity fieldtopologies and noise levels, we analyze the performance ofour algorithm, and discuss its application to real measure-ments obtained from smooth-muscle cardiac tissue, usingdata collected with a high-resolution MEA system.

Keywords—Inverse solution, Electrical conductivity, Bido-

main model, Tensor field, Parallel optimization, Alternating

optimization, Microelectrode array, Biological tissues.

INTRODUCTION

Transmembrane potential propagation in biologicaltissue results when the ionic concentrations change in

either the intracellular or extracellular domains. Po-tential propagation is correlated to the medium’sconductivity, and as a mechanism of intercellularcommunication it plays an important role in tissue andorgan functioning, e.g., exocytosis and muscle con-tractions.20 However, in order to relate transmembranepotential measurements to the electrophysiologicalstates of cells in the tissue, we depend on mathematicaland computational models to capture the interaction.A classical approach to modeling spatiotemporaltransmembrane potential propagation is based on thegeneralized cable theory, combined with dynamicmodels of ionic concentration gradients. This para-metric model relates changes in the transmembranepotential to changes in ionic currents through themembrane, taking into account the effective electricalconductivity and geometry of the tissue. The bidomainmodel treats the tissue as two continuous domains, andis a macroscale model of the electrical behavior aver-aged over many cells, taking into account both theintracellular and extracellular current flows. Althoughthis model has been used extensively in numericalsimulations of the electrical behavior of anisotropicmyocardiac tissues,31 in neuroscience it has recentlybeen used for analyzing the non-homogeneity of theextracellular domain in nerve tissues.5,6

In the last few years, much progress has been madein developing high-resolution microelectrode arrays(MEA) that allow electrophysiological measurementsof biological tissues with high spatiotemporal resolu-tion.7,16 Using this technology, it is possible to effec-tively and directly measure transmembrane potentialpropagation by parallel measurements of the tissue atdifferent locations. By employing the biodomainmodel to analyze the MEA data, we gain a deeperunderstanding of the underlying biophysical nature of

Address correspondence to Elad Gilboa, Preston M. Green

Department of Electrical and Systems Engineering, Washington

University in St. Louis, St. Louis, MO 63130, USA. Electronic mail:

egilboa@wustl.edu, gilboae@ese.wustl.edu, plarosa@dom.wustl.

edu, nehorai@ese.wustl.edu

Annals of Biomedical Engineering (� 2012)

DOI: 10.1007/s10439-012-0581-9

� 2012 Biomedical Engineering Society

tissue. However, several model parameters must firstbe estimated, including ones related to the ionic cur-rent dynamics, cell geometry, and electrical tissueconductivities. Several approaches are available tomodel and estimate cellular ionic currents, and theinterested reader is referred to the technical literaturefor details.12,15,45 However, inferring the conductivityparameters from the data by inverse techniques is ahighly challenging problem.13 The estimation problemsbecomes even more significant when the tissue isassumed to be composed of multiple anisotropicinhomogeneous regions.

In this work, we develop a mathematical frameworkfor solving the inverse problem of estimating theeffective electrical tissue conductivities from a set ofelectric potentials and stimulus measurements. Inparticular, we formulate the problem in a systemidentification framework, using a parametric modelbased on the generalized cable theory. In this frame-work, experiments are performed by exciting the sys-tem and observing its input/output over a timeinterval.39 Unfortunately, solving this ill-posed inverseproblem is highly complex.41 Specifically, it suffersfrom high dimensionality (as one must estimate thetensor matrix for each point in space), nonlinearity(due to the nonlinear extracellular field potentialdynamics), and stochasticity (as the observations arecorrupted by noise). The application of sophisticatedmethods, such as nonlinear filters (e.g., particle andunscented Kalman filters36) or traditional constraintoptimizations (e.g., augmented Lagrangian meth-ods27), becomes computationally prohibitive due to thecomplexity of the estimation problem, especially whena high resolution grid is considered.

The contributions of this paper are twofold. First,we introduce a discrete forward model of transmem-brane potential based on a diffusion–reaction modelwith an anisotropic inhomogeneous electrical conduc-tivity tensor field. Second, we propose a novel paralleloptimization algorithm for solving the complex inverseproblem of estimating the conductivity tensor field.Specifically, we propose a single-step approximationwith a parallel block-relaxation optimization method.This combination simplifies the joint tensor field esti-mation problem into a set of computationally tractableproblems, allowing the use of efficient standard opti-mization algorithms. We analyze the performance ofour algorithm using numerical examples of severalelectrical conductivity field topologies and noise levels,and discuss its application to real measurementsobtained from cardiac tissue, using a high resolutionMEA system.

The notational conventions adopted in this paperare as follows: italic font indicates a scalar quantity,e.g., a; lowercase boldface indicates a vector quantity,

e.g., a; upper case italic bold indicates a matrix quan-tity, e.g., A. The matrix transpose is indicated by asuperscript ‘‘T’’ as in AT, and the identity matrix of sizen 9 n is denoted as In. A lowercase roman font indi-cates a function, e.g., g(t) and a lowercase bold romanfont indicates a vector function, e.g., gt. The set Sn

denotes the vector space of symmetric n 9 n matrices,and the subsets of nonnegative definite matrices andpositive definite (PD) matrices are denoted by Snþ andSnþþ; respectively. jj � jjF is the Frobenius norm,

defined as jjAjjF ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Pmi¼1Pn

j¼1 jaijj2

q

: Writing the time

index as a subscript indicates the vector at the nthtimepoint, e.g., an � a½nDt�; or indicates vectors from aset of timepoints, e.g., a0:n � fa0; a1; . . . ; ang:

The remaining part of this paper is organized asfollows: in ‘‘Materials and Methods’’ section we pres-ent the numerical scheme used to discretize the forwardproblem. Then we formulate the inverse problem in adiscrete setting and propose a method to solve thecomplex inverse problem. Finally, in ‘‘Results’’ sectionwe present results of the proposed method for bothsimulated and real data.

MATERIALS AND METHODS

Forward Model

To model electrical propagation in biological tissuecomposed of elements of different conductivities, weuse generalized cable theory, namely, the monodomainapproach.19 The monodomain is a specific case of thebidomain models which have been successfully usedfor modeling extracellular potentials in several tis-sues.23,24,29,42 In the monodomain model, biologicaltissue is reduced to a two- or three-dimensional cellgrid, where the electrical behavior is governed by a setof reaction–diffusion equations.31 The diffusion part ofthis model represents the spatial evolution of thetransmembrane potential in a domain with changingconductivities. The reaction part models the voltage-dependent dynamics of the tissue as a function of threelocal currents: (i) a capacitative current through thecells’ membranes, (ii) a cell-ionic current with a volt-age-dependent dynamic jionðvðr; tÞ;w;/; tÞ; and (iii)external or spontaneous stimulations jstimðr; tÞ: Oursystem can be written as follows:

r �DðrÞrvðr; tÞ|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

Diffusion

¼ am cm@vðr; tÞ@t

þ jionðvðr; tÞ;w;/; tÞ � jstimðr; tÞ� �

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Reaction

;

ð1Þ

GILBOA et al.

vðr; 0Þ ¼ v0; ð2Þ

rvðr; tÞ � nðrÞ ¼ 0; r 2 @C; ð3Þ

where t 2 ½0;T�; the spatial vector r belongs to C � Rp;the domain C is a bounded Euclidean subset, n denotesthe normal to the boundary, and ¶C is the boundary ofdomain C. In this work we consider the 2D case, wherep = 2. Furthermore, jstimðr; tÞ is the stimulus volumecurrent density (A/m3); cm is the membrane capaci-tance per unit area (F/m2); am is the surface-to-volumeratio of the membrane (1/m); and DðrÞ 2 S2þþ is thePD conductivity tensor.4 jionðvðr; tÞ;w;/; tÞ is the ionicvolume current density (A/m3) of a biological cell, andit can be chosen to fit a specific dynamic, with w cor-responding to the internal state vector, and / tothe model parameters. For simplicity, we consider ahomogenous cell dynamic in the tissue; namely, / isconsistent in all the cells (in ‘‘Results’’ section, we usedthe extended FitzHugh–Nagumo (FHN) equations31,34

as a fairly general and simple representation of a cell’sionic currents; however, more extensive models can beused). Equations (2) and (3) present the initial tem-poral and boundary conditions, respectively. In par-ticular, we use the homogenous Neumann boundarycondition since we assume that there will be no currentthrough the borders of the domain, and we use the zerostate response33 for the initial values since we assumethat the system is initially relaxed at its restingpotential v0 and cannot initiate a spontaneousresponse.

Modeling Tissue Anisotropy

In order to infer the underlying conductivity struc-ture of the tissue, we represent biological tissue asa continuous field of conductivity tensors, D(r) inEq. (1), which models local extracellular conductivitieswithin the tissue.19 To simplify the problem, we con-sider working with only a thin slice of tissue that can berepresented as a 2D plane. The conductivity tensor isgiven by

Dðx; yÞ ¼ rxðx; yÞ rxyðx; yÞrxyðx; yÞ ryðx; yÞ

� �

; ð4Þ

where rx(x, y), rxy(x, y), and ry(x, y) are the conduc-tivity values in the horizontal, diagonal, and verticaldirections, respectively. The tensor field is an indexed setof tensors in space, and is referred to as isotropic if all theconductivity tensors are directionally independent(symmetric), that is, rx(x, y) = ry(x, y) = r0 and rxy

(x, y) = 0 for all x, y. Otherwise, if some conductivitytensors in the field are directionally dependent, it isreferred to as anisotropic. If the conductivity tensors are

constant throughout the field, then the field is referred toas homogeneous; otherwise, it is inhomogeneous.

A diffusion tensor can be expressed in terms ofits eigenvalues k ¼ ðk1; k2; k3Þ and eigenvectors E ¼ðe1; e2; e3Þ as D ¼ ETdiagðkÞE: The tensor can be rep-resented as an ellipsoid (Fig. 1), whose radii (eigen-values) represent the amount of diffusion (flow) in eachof the main directions (eigenvectors).1

In Fig. 2, we can see the general representation of atensor and a tensor field in 3D, and examples ofhomogeneous and inhomogeneous anisotropic 2Dtensor fields. The anisotropy inhomogeneity propertyof the conductivity tensors plays a major role in thespatial evolution of the wave propagation in the tis-sue.9 In this paper, we will focus on developing amathematical formulation for finding the best tensorfield representation to fit the electrical measurements.

Discretization

To transfer our model from a continuous domaininto a discrete vector space formulation, we will firstproceed to discretize the continuous diffusion term ofEq. (1), which is given as

r �Dðx; yÞrvðx; y; tÞ ¼@@x

@@y

" #Trxðx; yÞ rxyðx; yÞrxyðx; yÞ ryðx; yÞ

� �

�@vðx;y;tÞ

@x

@vðx;y;tÞ@y

" #

; ð5Þ

to a discrete term over the regular lattice domainC¢ = [x1, xK] 9 [y1, yK].

In particular, we apply the Finite-DifferenceMethod(FDM) with an forward-time central-space scheme40 toapproximate the derivatives, taking into account thespatially varying tensor field of Eq. (5). In the follow-ing, we present the two main steps for discretization ofEq. (5), but for brevity we will ignore the temporalcomponent of the extracellular voltage v(x, y, t). First,we expand the matrix multiplication of rvðx; yÞ withD(x, y), and are left with three second derivatives,

FIGURE 1. Diffusion tensors shown as ellipsoids with theircorresponding eigenvectors.14

Estimating Electrical Conductivity Tensors of Biological Tissues

which correspond to current densities: jx, jxy, and jy.Therefore, Eq. (5) can be written as

r �Dðx; yÞrvðx; yÞ

¼@ rxðx; yÞ @vðx;yÞ@x

�

@x|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

jx

þ@ ryðx; yÞ @vðx;yÞ@y

�

@y|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

jy

þ@ rxyðx; yÞ @vðx;yÞ@x

�

@yþ@ rxyðx; yÞ @vðx;yÞ@y

�

@x|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

jxy

: ð6Þ

Second, each term of Eq. (6) is further expandedusing a central-space difference FDM scheme. Forexample, jx can be expanded as

jxðrx; rxy; ryÞ

¼

rxðxþ Dx; yÞ � rxðx� Dx; yÞ2Dx

� �

� vðxþ Dx; yÞ � vðx� Dx; yÞ2Dx

� �

!

þ rxðx; yÞvðxþ Dx; yÞ þ vðx� Dx; yÞy

Dx2

� �

: ð7Þ

The term jx(rx, rxy, ry) represents the current densityalong the x direction at a single point on the grid (x, y).Concatenating all points into a column stack vector,we can write Eq. (7) as

jxðHÞ ¼ GxdiagðrxÞGxvþ diagðrxÞGxxv; ð8Þ

where H ¼ fH1;H2; . . . ;HKg is a joint set of K tensors

of the form Hk ¼rxk rxyk

rxyk ryk

� �

at location k. We will

refer to this set H as the discrete conductivity-tensor

field. The matrices Gx 2 RK�K and Gxx 2 RK�K (whereK is the number of nodes) are the linear operators ofthe first and second discrete spatial derivative opera-tors in Eq. (7).40 The vectors v and rx are column stack

representations of the two dimensional potentialfunction v(x, y) and tensor function rx(x, y), respec-tively, at time t.10 We derive jxy and jy similarly. Noting

the dependence in t, we can write the three currentdensities as

jxtðHÞ ¼GxdiagðrxÞGxvt þ diagðrxÞGxxvt;

jxytðHÞ ¼ 2GxydiagðrxyÞvt þ GxdiagðrxyÞGyvt

þ GydiagðrxyÞGxvt;

jytðHÞ ¼GydiagðryÞGyvt þ diagðryÞGyyvt:

ð9Þ

Next, using the forward-time central-space FDMscheme, we discretize the right side of Eq. (1). Weobtain the following discrete form of Eq. (1):

jxnðHÞ þ jxynðHÞ þ jynðHÞ

¼ am cmvnþ1 � vn

Dtþ G v0:n;w0:n;/ð Þ þ un

�

: ð10Þ

Finally, separating vn+1 to left side of the equation anddiscretizing the boundary and initial value conditions,we arrive at the following discrete representation of thesystem:

vnþ1 ¼Dt

amcmjxnðHÞ þ jxynðHÞ þ jynðHÞ�

� Dtcm

g v0:n;w0:n;/ð Þ þ unð Þ þ vn; ð11Þ

where the external stimulus input jstimðr; nÞ is written asthe vector un. Further, g v0:n;w0:n;/ð Þ is a nonlinearfunction that depends on the cell dynamic model,model parameters /; previous states v0:n; and previousinternal states w0:n: For full derivation on state dis-cretization in for the FHN model, please refer to theSupplemental materials. To completely define the dis-crete system, we add the following discretized initialand Neumann boundary conditions: initially inhomogenous resting potential v0 ¼ v01; no currentthrough the boundaries Gxvn ¼ 0;Gyvn ¼ 0 for hori-zontal, vertical boundaries, respectively.

FIGURE 2. (a) Depicts a general tensor in 3D space, while (b) illustrates a 3D spatially-discrete tensor field, where each tensor isrepresented by an ellipsoid. (c) and (d) Illustrate 2D constant anisotropic-homogeneous and anisotropic-inhomogeneous 2D fields,respectively.

GILBOA et al.

Note that we can write our discrete system as

vnþ1 ¼AðDt;am;cm;Dx;Dy;HÞvnþ g v0:n;w0:n;/ð Þþ bun;

ð12Þ

where the A matrix depends on the model parameters.As can be observed from Eq. (12), even when we ignorethe nonlinear part, the discrete system’s stability issensitive to the model parameters (am and cm), theFDM parameters (Dt, Dx, and Dy), the conductive-tensor field estimation H; and the input values un.Finding an analytical formulation for the stabilitycriterion is highly challenging and will be considered infuture work. However, consideration is needed whenchoosing parameter values, and in some cases ofinstability, more complex numerical approximationmethods should be considered.40 Furthermore, it isbeneficial to use a numerical optimization method thatcan recover from instability, e.g., the standard Matlabimplementation of the sequential quadratic program-ming method described in Ref. 28.

Measurement Model

In a highly dense sensor system such as an MEA,the electrodes capture the extracellular field potential,which is the product of a highly complex network ofneighboring cells, making interpretation very difficult.The complexity of the system is greatly simplified whenusing the monodomain model, as it is a macroscopicmodel that describes the average behavior of the bio-electric fields on a larger scale than the size of a singlecell.35 Hence, the state vn in our model represents theaverage local extracellular field, which is measuredthrough a noisy sensor to give spatiotemporal mea-surements yn 2 RQ: Here we will assume that yn arecorrupted with an additive white Gaussian noise ofmean zero and variance c2, and that the wavelengths ofthe dynamics are large compared to the spatial densityof the MEA. Therefore, the discretized evolution andmeasurement models are given by

vnþ1 ¼ fðv0:n;w0:n;/;HÞ þ bun; ð13Þ

yn ¼ vn þ m: ð14Þ

Note that in this work we set Q = K as we assume thatthe MEA is dense enough to allow complete coverageof the region in interest (ROI); however, in cases wherethe system exhibits subsampling, Q will be less than K.

Solving the Inverse Problem

To restate the inverse problem, the goal is to esti-mate H; the discretized conductivity tensor field, fromthe set of measurements y0:T; the set of inputs u0:T; andthe discrete system model of Eqs. (13) and (14). We use

an indirect method to estimate H; by transforming theproblem into an optimization problem.41 One indirectapproach to solve this problem is using a constraintoptimization scheme, specifically, the augmentedLagrangian method. This method has been used suc-cessfully in the field of geoscience for solving problemsof permeability identification in convection-diffusionmodels.26,27 In the augmented Lagrangian method, theproblem is written as a joint optimization routine,where the error between the model and observations isminimized (usually in a least squares sense), with themodel parameters acting as constraints. However, thismethod can easily become computationally prohibi-tive. Even for efficient algorithms, one must alternatebetween optimization steps with high dimensionality:the number of nodes in the grid O(K), and the numberof grid points times the number of time-steps O(KT).These optimizations will quickly become intractable aseither the number of grid points or number of samplesincreases.

Another optimization approach that can be consid-ered is the initial value approach. This method uses thefact that the model evolution function in Eq. (13) isdeterministic, depending entirely on the initial values ofthe states and the known control sequence. An intuitiveapproach is to minimize the least-squares (LS) differ-ence between the observations and the model trajectoryby altering only the initial values v0 and w0. However,the recurrence structure of the model makes it difficultto determine the stability of the solution, and increasesthe complexity in calculating the gradients for themodelparameters estimation.38 Furthermore, the solution isvery susceptible to noise and depends sensitively oninitial conditions and parameters, which makes theinitial value approach inapplicable unless chosen in veryclose proximity to the unknown true values.37

Rather than using such a recurrence approach, an-other method that shows more promising results is arecursive state space technique that evaluates the costfunction in a sequential way, using the observations.37

Filtering is used to approximate the system statevariables from the noisy observations. However, tra-ditional filtering methods such as Kalman or ExtendedKalman will not be sufficient to deal with the complexnonlinear dynamics of the system.17 A particle filterwill also prove problematic since even for a verymodest size grid, the system will be in a high dimen-sional space, which makes the particle filter ineffective.Earlier work has also been done with the unscentedKalman filter (UKF) for nonlinear reaction–diffusionmodels.37 However, a problem with using the UKF isapproximation errors in cases of strong nonlinearity,such as in a polynomial of high degree (>2). Anotherdifficulty is that this approach will offer computa-tionally tractable solutions for only low dimensional

Estimating Electrical Conductivity Tensors of Biological Tissues

systems. For systems with even a modest grid size (e.g.,9 9 9), the problem becomes too complex to be solveddirectly, and some works suggest using a patch filter asa dimension reduction scheme.36 However, using thedimension reduction scheme in our problem mightadversely reduce the spatial resolution. As the inverseproblem is too complex to allow filtering, we propose asimpler, nonrecurrent method based on the ideas ofone-step prediction and block-relaxation optimization.

One-Step-Ahead Prediction

When dealing with invasive electrophysiologicalrecordings, it is acceptable to assume low sensormeasurement noise and use the observed data as adirect measurement of the average local field potentialstates, yn � vn:

16,43 A high signal to noise ratio (SNR)is further supported when different methods of noisereduction are used, e.g., applying a smoothing filterbetween consecutive observations or experiments.18

This assumption simplifies the model by substitutingthe observed measurements for the hidden states, andthe simplified discrete system model can be written as

ynþ1 ¼ fðyn;wn;/;HÞ þ bun: ð15Þ

We notice from Eq. (15) that this method is a feed-forward predictive model, and is a member of theclassical nonlinear autoregressive exogenous (NARX)family, where the nonlinear model depends on pastmeasurements (filtered), and current and past values ofthe input (exogenous) series.25 However, feeding noisymeasurements to the nonlinear model will affect thestability of the system, and hence robustness to noise isan important consideration in the algorithm design. Arobust design can be found by using a suitable per-formance bound for estimated noise, and is animportant subject of future research. In this work, wewill discuss the effects of noise on the algorithm, andexamine possible ways to improve robustness in‘‘Sensitivity Measurement Noise’’ section.

The one-step-ahead prediction (OSP) score functioncan be formulated as the LS difference between theone-step predictions at each point in time and themeasurements,38 and is given by1

VOSP y0:N;w0; u0:N;/;Hð Þ

¼X

N

n¼1

X

K

k¼1yn;k � f yn�1;w0;/;Hð Þn;k� bun;k

� 2

: ð16Þ

However, the estimation of the unknown tensorsH ¼ fH1;H2; . . . ;HKg using Eq. (16) is still highlycomplex due to its high dimensionality and thedependencies between the components of each tensor,which must be PD, and between different tensors in thefield. The interdependencies between the tensors in thefield occur because every tensor can affect the overallscore function, and a change in one could result indifferent optimal values for the others. Solving thejoint complex interdependent estimation problemrequires a highly complicated semi-definite program-ming routine, which will often be intractable, timeconsuming, or converge to a local optimum. In orderto simplify the problem, we propose a novel methodthat uses parallel relaxation optimization to separatethe highly complex semi-definite optimization into aset of smaller, less complex optimizations but stillmaintains mutual influence in order to converge to abetter overall solution.

Parallel Block Relaxation Optimization

To develop a computationally efficient optimizationalgorithm to solve the complex joint nonlinear semi-definite problem, we use a variance on the commonsequential block-relaxation method,11 commonlyknown as alternating optimization (AO).8 To restateour problem, the unknown parameters, on which we aretrying to perform optimization, are positively definedtensor matrices with inner dependencies in their com-ponents. Performing optimization in the tensor space isa highly nontrivial task. In recent years, a few methodshave been developed to perform gradient descent intensor space using an intrinsic geodesic marchingscheme with the Riemannian framework.30 However,these methods come with a high computational burdenin the form of intensive use of matrix inverses, squareroots, logarithms; moreover, exponentials are usuallyinvolved.2 A simpler approach will be to break down thetensors to their components and use constraints toguarantee PD; however, this will dramatically increasethe dimensionality of the problem.

In order to solve this complex problem, we turnedto the block-relaxation optimization class of algo-rithms. In this class, a complex optimization problemis solved by iteratively solving a series of easily handledsubproblems. The optimization algorithm for eachsubproblem is simpler since only a subset of theparameters are considered, keeping the rest of theparameters constant at their current value. The algo-rithm then iteratively cycles through the differentsubproblems and updates the parameters in each sub-set until convergence is reached. However, even for amodest size grid, the high-dimensionality of the

1To simplify our inference algorithm, for certain dynamic models

such as the FitzHugh–Nagumo, we can use the method of variation

of parameters in order to have the evolution function dependence on

w0:n brought down to depend only on w0 and v0:n (see Supplemental

materials Eq. (4)).

GILBOA et al.

problem will make the alternating optimization algo-rithm computationally expensive because it will haveto sequentially alternate between many parametersubspaces. To improve computational costs, we varythe sequential block-relaxation optimization algorithm(i.e., AO) so that instead of simply using a sequentialscheme for solving the subproblems, we parallelize thealgorithm, solving them all at once and joining themafter each iteration.

To use parallel block-relaxation optimization, westart with the complex joint optimization problemgiven by

H½j� argminH¼fH1;H2;...;HKg8k;Hk2S2þþ

� VOSP y0:N;w0; u0:N;/; H½j�1�;H

�

; ð17Þ

and separate it into K optimization problems of theform

H½j�k argminHk¼frxk;rxyk

;rykgHk2S2þþ

� VOSP y0:N;w0; u0:N;/; H½j�1�i6¼k ;Hk

� h i

; ð18Þ

where H½j�1� are known values from the previous iter-ation. The arg min function of Eq. (18), evaluates thenew value of the tensor Hk that will optimally decrease(or increase) the objective function when only thatsingle tensor is allowed to change.

Using a parallel scheme, however, can introduceconvergence problems because the separate subprob-lems are greedily optimized within each iteration. Thiscan often lead to a poor convergence rate and con-vergence to a local optimum. To improve the conver-gence properties of the algorithm, we introduce anadditional term to the arg min in Eq. (18), penalizinglarge changes of the parameters between consecutiveiteration steps. The step-size penalty feature is added tothe algorithm to reduce the probability of the algo-rithm becoming stuck at a local minimum that is worsethan the global one. This feature is very similar to theidea behind the acceptance probability in Monte-Carlomethods, such as simulated annealing, where theacceptance probability function is usually chosen sothat the probability of accepting a move decreaseswhen it causes a large change, thus making small uphillmoves more likely than large ones.32,2 Further, havingthis penalty term in the optimization score functionguarantees that the tensors will remain PD. In thiswork, we use a standard Frobenius norm as the dis-tance function between tensors; however, there are

other distance measures, e.g., Riemannian Metrics orlog-Euclidean metrics, that offer a more rigorous andgeneral framework for handling tensor operations, butthey are more computationally intensive.2,4

The K optimization routines can then be written as

H½j�k argminHk¼frxk;rxyk

;rykg

Hk2S2þþ

� VOSP y0:N;w0; u0:N;/; H½j�1�i 6¼k ;Hk

� h

þkd Hk; H½j�1�k

� i

; ð19Þ

where k is the Lagrangian hyper-parameter. The jointparameter set at iteration j; H½j�; is simply the union of thesolutions to theK optimization problems, and is given by

H½j� ¼ H½j�1 ; H½j�2 ; . . . ; H½j�K

n o

: ð20Þ

Finally, we use these ideas to write a parallel OSPalgorithm (pOSP) which iterates between parallelestimation of conductivity tensors and estimation ofthe initial internal states w0. In order to simplify theoptimization of w0; we assume that initially the systemis close to the resting state and is homogeneous. Thisassumption lowers the dimensions of w0 since we cantreat it as a scalar w0 multiplied by a vector of ones, 1.However, in situations where this assumption does notapply, standard vector optimization methods can beused. The full iterative pOSP algorithm is presented inAlgorithm 1.3

The last term of the arg min is an optional spatial

penalization function h Hk; H½j�1�

�

; which penalizes

large spatial deviations from a priori information of

ALGORITHM 1. Parallel OSP.

procedure pOSP(H0;w00) . H0 2 S2þþ;w

00 2 R

H08k2K H0

w00 w0

0 1

j 0

repeat

j j þ 1

ParFor k 2 K

H½j �k arg minHk2S2þþVOSP Hk ; H

½j�1�i 6¼k ;w

½j�1�0 ; y

�

þ kd Hk ; H½j�1�k

�

þ h Hk ; H½j�1�

�

EndParFor

w½j �0 arg minw02RV ðH½j �;w0; yÞ

w½j �0 w

½j �0 1

until convergence

return

end procedure

2This parameter can also be viewed, in a game theoretic framework,

as quantifying the amount of cooperation or ‘‘trust’’ the players

(tensors) have in one another.

3Minimization is performed using a standard Matlab implementation

of the sequential quadratic programming method described in

Ref. 28.

Estimating Electrical Conductivity Tensors of Biological Tissues

the tissue. Prior information can be provided by othersources, such as anatomical data, or from biologicalknowledge of the tissue properties, such as smoothnessin smooth muscle tissues. We will further explore theeffects of adding a spatial penalization in ‘‘Results’’.

RESULTS

To analyze the performance of the pOSP algorithm,we compiled a number of test simulations to examinethe algorithm’s ability to estimate the tensor field ofdifferent field topologies under varying noise levels. Fora fairly general and simple representation of a cell’sionic current dynamics, we use the extended FHNequations.34 Then, the continuous model can be writtenas follows:

jionðvðr; tÞ; tÞ ¼ �1

�1ðkðvðr; tÞ � v1Þðv2 � vðr; tÞÞðvðr; tÞ

� v3Þ � wðr; tÞÞ; ð21Þ

@wðr; tÞ@t

¼ �2ðbvðr; tÞ � cwðr; tÞ þ dÞ; ð22Þ

where jionðr; tÞ is the ionic volume current density(A/m3) of a biological cell. Here, jstimðr; tÞ is the stim-ulus volume current density (A/m3) and is a knowndeterministic input series, and wðr; tÞ is the internalstate of the system. The rest of the model parametersand their simulation values are presented in Table 1.After discretizing the FHN system, we can write thenonlinear evolution function as follows:

fðv0:n;w0:n;/;HÞ ¼ c jxnðHÞ þ jxynðHÞ þ jynðHÞ�

� dnw0 � g v0:n;/ð Þ; ð23Þ

where w0 is the vector of the initial state w anddn ¼ c amj

�1e��2cnDt is a known scalar function.

Step-Size Penalty k

The hyper-parameter k, which was introduced in theprevious section, is the weight of the penalty term weadded in order to control the step size of the pOSPalgorithm. To better understand the influence of thisparameter on the convergence properties of the algo-rithm, we varied k and recorded the resulting squarederror, as shown in Fig. 3b. Theoretically, low k results ingreedy optimizations of each tensor and the ‘‘strongestsurvive’’ phenomenon (convergence to a local opti-mum). This phenomenon can be observed in Fig. 3c,where a few dominating tensors appear while others arediminished. On the other hand, high k corresponds to aslow convergence rate, as illustrated in Fig. 3d, whereafter ten iteration the estimate is still close to the initialfield. For this problem, the optimal k value can beobserved from Fig. 3b at around 1024, and the corre-sponding estimate is illustrated in Fig. 3e. As can beobserved from Fig. 3e, the estimated conductivity-ten-sor field, although much better then the other two esti-mations, is still only partially accurate, namely, only onthe main diagonal. This is a typical example whereincreasing the number of stimuli will greatly improvethe estimated results. We will further discuss this topicin the following section. In the following simulationsand analysis, we used a constant k = 1024 for all thesubproblems. The effect of noise, inhomogeneity, ortime-dependence on the optimal ks, and an efficient wayfor finding the optimal ks, are not tackled in this workand are left open for future research.

To test the significance of the step-size penalty onthe optimization algorithm, we compared the standardsequential AO to the parallel block-relaxation opti-mization, with and without a step-size penalty (k =1024), for various noise levels. We compared both theaccuracy and the computational time for the algo-rithms, using ten iterations on a toy 8 9 8 Gaussianmixture as illustrated in Table 2. From Fig. 4, we canobserve that parallel block-relaxation optimizationalgorithm with step-size penalty (PBROSP) achievesconsiderably better estimation results than the onewithout the step-size penalty. Furthermore, for alltested noise levels, the PBROSP estimation offerscomparable estimation accuracy to the standard AOalgorithm at a significantly lower computation time.

Richness of Inputs

The choice of input signals used to excite the systemis a fundamental factor in the algorithm’s ability tocorrectly estimate the spatially varying conductivity-tensor field.22,39 The input must possess sufficientrichness in both its spatial and temporal excitationsto provide enough information to fully identify the

TABLE 1. FitzHugh–Nagumo reaction parameters (/) cate-gorized by their corresponding control, and the values used

for the simulations.

am Surface-to-volume ratio

of the membrane

5.758710 9 105 m21

cm Membrane capacitance

per unit area

0.01 Fm22

e1 Sharpness of the edges 100 Xm2

e2 Excitability duration 1 s21

c Control the excitability

threshold of the cell

0.1

b 1

d 0.0520 V

v1 Control the range of v(r,t) 20.02 V

v2 20.04 V

v3 20.065 V

j 1e4 1/V2

The values of the reaction parameters for the simulations were

fitted to a slow wave from a pregnant human’s uterine myocyte, as

presented in Ref. 19.

GILBOA et al.

system. In this paper, we do not attempt to analyze theoptimal experimental design but rather present a fewexamples of the improvement that can be achieved byusing a rich input set.

For our simulations, we considered three types ofspatially varying tensor fields: constant, Gaussian mix,and circular (Table 2). We chose these spatial struc-tures because they correspond respectively to a con-stant field, piecewise constant field with discontinuous

edges, and a completely varying smooth field. Fur-thermore, in all fields the size of the tensors werechosen at random.

We tested the improvement of the pOSP conduc-tivity-tensor estimation with varying numbers ofstimuli (Table 2). The stimuli were sequential, with 150time-steps between consecutive stimuli. We used a8 9 8 grid and ran the simulation for 400 time-stepsDt; for a total of 20 s. The evoked stimuli were half a

FIGURE 3. Estimation results of the pOSP algorithm on the constant 9 3 9 tensor field (a) for different values of k. (b) Squareerror sensitivity of the algorithm to the parameter k. All the pOSP estimations were calculated with ten iterations and same initialvalues. (c)–(e) Illustrate the results of using the pOSP estimation using a low, high, and optimal (1024) k, respectively. The plots ofthe tensor fields were made using the matlab tool as described in Ref. 3.

TABLE 2. The results of the pOSP algorithm for three 8 3 8 conductivity-tensor fields, constant, Gaussian mix, and circular.

With each sequential experiment, we introduced an additional stimulus to the simulation and observed the resultant conductivity-tensor field

estimation of the pOSP. In the figures, the red dots illustrate the locations of the stimuli, and there was a period of 150 time-steps between

consecutive stimuli.

Estimating Electrical Conductivity Tensors of Biological Tissues

second each, with amplitudes of 1 mV. In Fig. 5, wepresent the resulting log mean squared error (MSE)between the estimated conductivity-tensor field and theoriginal simulation field, for the three fields considered,as a function of a number of stimuli. For both theconstant and circular fields, the MSE (using theFrobenius norm) is monotonically decreasing, meaningthat the additive inputs improved the overall estimate.Furthermore, we can observe that the greatestimprovement in estimation occurred when the spatiallocation of a new stimulus created a significantly dif-ferent wave propagation compared to the waves thatwere created by previous stimuli. This can be seen in

the constant field in Table 2, where the first threestimuli were along the same main direction based onthe topology of the field, and as a result, they allproduced a similar wave propagation. However, thefourth stimulus location was different enough to pro-duce a new wave that allowed for more information onthe structure of the top left corner of the field. Theresults were not as conclusive in the Gaussian mix case,where not enough time was given between sequentialstimuli, causing interference in the waves, whicheventually lowered the accuracy of estimation.

Sensitivity to Modeling Error

Although the extended FHN equations model theextracellular field dynamics for a wide variety ofexcitable systems, they also present a challenge: apotentially small deviation in the parameters couldcause a large modeling error. In fact, trying to infer theconductivity tensor field without properly fitting theseparameters will almost certainly result in inaccurateresults. We study the robustness of the algorithm byassuming an error of one parameter at a time (Fig. 6).The purpose of this study is to illustrate how much ofan error we will have in the tensor field estimation(Fig. 6a), and in the OSP estimation (Fig. 6b) if thereaction parameters are not properly fitted. Studyingthe parameters v1; v2; v3; �1; and �2; we can see that thealgorithm is especially sensitivity to the polynomialroot v3, which controls the minimum range of the FHNmodel. However, v3 is easy to estimate since the min-imum range of the extracellular potential can beinferred from both data and prior knowledge. FromFig. 6, we can also observe that the OSP predictionerror and conductivity tensor errors each have a min-imum when the parameters used for the estimation

20 25 30 35 40 45 5010

0

101

102

103

dB SNR

MS

E

PAOAOAOSPPAOSP

(a)

20 25 30 35 40 45 500

10

20

30

40

50

dB SNR

Com

puta

tiona

l Tim

e (m

in)

PAO

AO

AOSP

PAOSP

(b)

FIGURE 4. A comparison between the standard sequential alternating optimization (AO), alternating optimization with step-sizepenalization (AOSP), parallel block-relaxation optimization (PBRO), and parallel block-relaxation optimization with step-sizepenalization (PBROSP) algorithms. As can be observed from (a), addition of a step-size penalty improves the accuracy of both theparallel and sequential optimizations schemes. Both the PBROSP and AOSP produced the best estimations, and their error wasalmost identical for all noise levels. However, as (b) shows, the PBROSP had considerably faster execution time than any of theother optimization schemes, at all noise levels. (All runs were executed on a Windows7 system with parallel Matlab package.)

1 2 3 4 51

1.5

2

2.5

3

3.5

4

Number of Stimulations

log

MS

E

ConstantGaussian MixCircular

FIGURE 5. The effect of an increasing number of stimuli onthe log mean squared error (using the Frobenius norm),between the original conductivity-tensor field and its esti-mate.

GILBOA et al.

were equal to the true parameters that were used tosimulate the wave phenomena. Further, both errorsdecrease smoothly and monotonically as the parame-ters get closer to the true value. These results suggestthat even though the estimation is sensitive to thereaction parameters, a good approximation can still beachieved by inferring the reaction parameters from thedata, using the OSP error and prior knowledge.

Sensitivity Measurement Noise

In this section, we test the sensitivity of the pOSP(using the PBROSP optimization method) to mea-surement noise. We used a constant 9 9 9 tensor fieldseparated into two sections by a zero tensor column in

the middle (Fig. 7a). By stimulating only the left sec-tion (Fig. 7c) at different noise levels, we were able toexamine noise effects on both the areas which the wavepassed through (explored) and the unexplored areaswhere the wave did not reach (which hence contain noinformation). As can be seen from Fig. 7f, the pOSPestimation gives accurate results for low noise levels,with high anisotropy in the explored area, and lowconductivity tensors in the middle column. In theunexplored area, due to lack of information needed toimprove the estimation of the conductivity tensors, thetensors remained close to their initial values. As weincrease the noise, moving our system away from theassumption of high SNR (Fig. 7e), the results ofthe pOSP estimation begin to degrade. In Fig. 7g, the

FIGURE 6. Sensitivity of the pOSP estimation to changes in the reaction parameters, using estimations over the Gaussian mixtensor field and varying one parameter each time. The reaction parameters used for estimation varied from 0.5 its true value to 3/2.Hence, if the ratio variable r is the ratio of the value chosen for estimation to the true value, then r ‰ [0:5,1:5]. (a) Illustrates the OSPprediction error from Eq. (16) and (b) illustrates the actual tensor field error, calculated using the Frobenius norm between the trueand the estimated tensor fields.

FIGURE 7. The effect of noise on the pOSP algorithm in both the explored and unexplored areas. The wave propagation wassimulated using the true conductivity tensor field (a), and an isotropic tensor field as the initial field for the pOSP (b). The stimulusin all simulations was at location (x, y) 5 (2, 3) (c), and the observations were corrupted with additive noise of 50 dB (d) and 35 dBSNR (e). (f), (g) Illustrate pOSP estimations for cases of low noise (50 dB SNR) and high noise (35 dB SNR), respectively. (h)–(j)Show the effects of adding a spatial penalization (l 5 10213), temporal filter (three time-steps), and multiple experiments averaging(five trials), respectively.

Estimating Electrical Conductivity Tensors of Biological Tissues

estimation of the tensors in the explored areas worsen,with many tensors losing their anisotropic informationand becoming isotropic. Problems also occur in theunexplored area, where the tensors exhibit a falsedirectional anisotropic property, as can be seen in theright side of Fig. 7g. This false directionality can easilymislead experimentalists into arriving at wrong con-clusions about the conductivities in this area. Toimprove the algorithm’s robustness to noise, severalfiltering and constraining schemes can be employed.For clarity, we will divide the methods into three cat-egories: (i) preprocessing, (ii) processing within pOSP,and (iii) postprocessing. In this subsection we analyzethree approaches in the preprocessing step and withinthe pOSP algorithm, leaving discussion on postpro-cessing, such as fusion of conductivity-tensor fieldestimation, to future work.

(1) Temporal Filter: In this method, a slidingtemporal window filter is applied to the noisymeasurements in the preprocessing step, prioron using them in the pOSP algorithm. Themethod exploits the fact that often in experi-ments where the wave propagation is mea-sured, the sampling frequency is highcompared to the propagation speed of thewave. Thus, consecutive measurements wouldbe fairly similar, and averaging them willachieve higher SNR. However, yields poorresults using a three time-steps window in thisexample, as can be seen in Fig. 7i, is becausethe wave propagation dynamics were too ra-pid and were significantly degraded by theaveraging.

(2) Spatial Penalization: In this method, robust-ness to noise is achieved by using a constraintwithin the pOSP algorithm. Spatial penaliza-tion can be added to bring in prior knowledgeof the tissue, such as anatomical structure,from other modalities. In this work, we willconsider the less constraining assumption ofsmoothness as our spatial penalty. The idea isto penalize large variations between neigh-boring tensors, thereby directing the algo-rithm to choose a smoother solution for thetensor field. For the spatial penalizationfunctions, we choose a function in the fol-lowing form,

h Hk;H½j�1�

�

¼X

K

k¼1l L2

xyrxþL2xyrxyþL2

xyry

�

;

ð24Þ

where L2xy is the 2D discrete Laplace operator,10

and l is the penalty weight. The penalization

gives particularly good results in cases, such asour example, where the data was taken from atissue that exhibits a high degree of smoothness(see Fig. 7h).

(3) Multiple Post-Stimulus Averaging: In caseswhere conditions are stationary enough toconduct multiple trials of the same post-stimulus experiment, averaging the dataacross trials can efficiently reduce the noisewithout degrading the dynamics. As we cansee from Fig. 7j, the tensor field estimated bythis method is similar to the one the in lownoise case (Fig. 7f), and the method was veryeffective in reducing the noise and improvingthe results.

Finally, we tested the pOSP estimation using fivestimuli (an ‘‘all stimuli’’ experiment) under differentnoise levels. We considered three cases: 25 dB SNRwith no temporal filtering, 25 dB SNR with a low passsliding window of three time-steps, and 50 dB SNRwith no temporal filtering. As we can observe from theresults in Table 3, even with high noise the pOSP canproduce good estimates of the original conductive-tensor fields. Ten runs of the pOSP with differing ini-tial fields were performed to check for sensitivity to theinitial guess, all produced similar results. The estimatedtensor fields from the first run are illustrated inTable 3.

Testing on Real Data

We applied the pOSP method to a normalized dataset from cardiomyocyte tissue of a newborn mouse,recorded at the Italian Institute of Technology usingthe high-resolution 4096-channel MEA platform of3Brain GmbH, Switzerland.

Since the data is already normalized, and to simplifyour computational effort, we scaled the data to a par-ticular range of values between [20.07, 20.01], whilethe reaction parameters / were chosen to fit thedynamics of distinct waves of the normalized measure-ments. However, when possible, the method should beapplied directly to the real transmembrane potentialmeasurements. To simplify the computations, we low-ered the resolution of the data into a 20 9 20 grid byperforming block averaging. In Fig. 8a, we present bothsnapshots of the transformed measurements and anactivity measure of the tissue. The activity measure wascalculated as the average potential of the five mostactive measurements. We can divide the observed wavephenomenon into three phases: In the first 15 ms, thewave propagates from the center toward the periphery.In the second phase, between 15 and 23 ms, we canobserve a much faster wave propagation speed. The

GILBOA et al.

wave propagates to the right in a funnel shape, thendisperses as it reaches the boundary. In the followingthird phase, no propagation is observed and the mea-surements are of noise only. For purposes of illustra-tion, we estimated the conductivity tensor fieldcorresponding to the first phase of propagation.

To use the pOSP algorithm, first the optimalreaction parameters must be inferred from the data.Finding the optimal reaction parameters is a complextask, and ideally should be estimated as an added stepto the iterative pOSP algorithm, using prior biologicalknowledge of the tissue as constraints. There aremany important questions, such as what is the opti-mal fitting strategy, how to handle spatial and tem-poral varying systems, and how to best integrate thereaction parameters’ inference with the conductivityinference. However, these questions are not tackled inthis paper and are left for future research. In thiswork we limit ourself to the simplifying assumptionof spatially homogeneous dynamics with time invari-ant parameters. Since the difference in propagationbetween the first and second phase suggests that thereis a difference between the reaction parameters of thecells involved in these phases, we infer the reactionparameters by fitting the FHM model to the wavedynamics of single sensor measurements, whichpresent wave propagation in the first phase only(Fig. 8b).

To define the boundary conditions of the tissue,locations where the sensors had a total cumulativeactivity less than the threshold were considered to benot conductive, and hence the conductivity tensors inthese locations were set to zero. The initial state values

were set to the theoretical resting state potential of amembrane potential v0 = v01, and the first set ofmeasurements was used as input to the system u0 = y0.We used tensor-volume constraints of [0.3, 2], a spatialpenalization penalty of l = 10212 (see ‘‘SensitivityMeasurement Noise’’ section), and a step-size penaltyk = 1024. The conductivity tensor field was estimatedusing the pOSP algorithm and the measurements in thefirst phase, and is shown in Fig. 8c.

Next we checked that our estimated tensor fieldindeed significantly improves the model predictionresults compared to using a noninformative tensorfield. For the null hypothesis, we calculated the OSPerrors for 30 random tensor fields, and used theseresults to find the mean OSP error and confidenceinterval. In Fig. 8d, we noted the mean OSP error ofrandom fields as ‘‘Mean Random Tensor Field’’, andthe confidence interval, corresponding to 98%, is pre-sented in gray. Further, we also compared the OSPerrors for other noninformative tensors: an isotropichomogenous field (circles), a zero tensor field (all ten-sors set to zero), and a constant tensor field (Table 2).While all of the noninformative tensor fields tested werewithin the confidence interval of the random fields, wecan observe that for the first 13 ms the estimated tensorfield resulted in an improvement to the OSP error whichis statistically significant (Fig. 8d). After 13 ms, theOSP errors of the estimated and noninformative fieldsare relatively equal, and are within the confidenceinterval. This effect can be explained by the changes inboth the spatial location of the wave and the change indynamics, as explained above (Fig. 8a). These changesresulted in the estimated tensor field from the first phase

TABLE 3. Figures illustrate pOSP estimations for ‘‘all stimuli’’ experiments under different noise levels.

We considered three cases: 25 dB SNR with no temporal filtering, 25 dB SNR with a low pass sliding window of three time-steps, and 50 dB

SNR with no temporal filtering.

Estimating Electrical Conductivity Tensors of Biological Tissues

of the wave becoming noninformative for the rest of thepropagation.

DISCUSSION

Estimating the electrical conductivity from thereaction diffusion model and a set of measurementscan be thought of as estimating of the functionalelectrical conductivity of the tissue, in the sense that weestimate the electrical conductivity tensors only fromareas in the domain where the transmembranepotential wave travels (flow of information). Thebenefits of our approach are: (i) It provides anisotropicheterogeneous values of conductivity, and (ii) theconductivity values represent the ability of the media

to allow the passage of information, and thus, aretightly related to the underlying dynamics and func-tional connections of the tissue (for example, in nervetissue, cells can be connected anatomically but notfunctionally).

Our proposed method could also potentially beapplied for extracellular analysis of nerve tissues. Apopular macroscale approach of modeling corticalneural tissue, with its immense number of intertwinedconnections, is a continuous two-dimensional mediumwith neural field equations describing the tissuedynamics.9 Furthermore, this approach can representthe propagation of traveling wavefronts, which play animportant role in cortical information processing.7,21

Although the reaction–diffusion model is a simplifi-cation of the general neural field equations as it is

FIGURE 8. Conductivity tensor field estimation of propagating waves in a slice of cardiomyocyte tissue from a newborn mouse.(a) illustrates the instantaneous tissue activity, calculated as the average of the five most active measurements at each point intime, and depicts the instantaneous measurement grid (yk) of certain time points. The FHN reaction parameters were fitted to awave from a single measurement sensor (b), and the conductivity tensor field was estimated from the first 15 ms of data (c), whichcorresponds to the expanding wave phenomena. (d) illustrates the mean (‘‘Mean Random Tensor Field’’) and confidence interval(98%, in gray) of one-step-ahead prediction (OSP) errors for 30 random tensor fields. Further, comparison of the OSP errors isshown for the estimated conductivity tensor field, an isotropic homogenous field (circles), a zero tensor field (all tensors set tozero), and a constant tensor field (Table 2). While all of the noninformative tensor fields tested were within the confidence intervalof the random fields, it can be observed that for the first 13 ms the estimated tensor field resulted in a statistically significantimprovement (d). After 13 ms, the estimated tensor field OSP error converged with the rest of the noninformative fields. Thisconvergence occurred because the wave’s dynamics and location has changed, resulting in the estimated tensor becoming anoninformative tensor field.

GILBOA et al.

limited to only local interactions,21 the two modelsshare many dynamic characteristics44 suitable formodeling wave propagation in nerve tissue.

To achieve better results with the pOSP algorithm, alearning scheme (e.g., cross-validation) can be used inorder to infer the model parameters from the data.These parameters consist of the penalty term k, thereaction parameters (ionic current dynamics), cellgeometry parameters, boundary conditions, bounds onexpected conductivity tensors (how much current isreasonable), and the spatial penalty function. Settingthe spatial penalty should be based on prior biologicalknowledge of the tissue considered. In this work weused a simple smoothness penalty as we assumed thetissue conductivity should present a smooth tensorfield. However, the actual penalty value does not havea direct biological meaning (10212). For our real dataanalysis, we chose the penalty value that gave the bestresults for the simulated field that seem to fit our priorassumption of partwise smooth tissue (Gaussian Mixin Table 2). We wish to emphasize that this is a heu-ristic approach, based on experiments from simulateddata. How to optimally decide on this penalty term, orperhaps how to learn it from the data, is a subject offuture research.

Another important part of the pOSP algorithm isthe knowledge of the evoked stimuli; however, inspontaneous stimuli experiments, this knowledge israrely available. There are several methods for dealingwith the lack of information. We can either try toestimate the stimuli by using Laplacian methods thatincorporate high-pass spatial filters to enhance focalactivities and reduce widely distributed activity, or wecan use a latent approach and infer it from the data.

Finally, we showed that for the tested simulationsand initial fields used in this study, the parallel block-relaxation optimization with step-size penalty achievedbetter convergence behavior than the common AOalgorithm (whose convergence properties are provenand well understood8). However, a more rigorousstudy must be made in order to fully understand theconvergence properties of our algorithm.

CONCLUSION

We formulated a novelmethod for solving the inverseproblem of inferring the conductivity structure of abiological tissue from a set of spatiotemporal measure-ments. We lowered the complexity of the optimizationby using a single-step approximation employing a par-allel block-relaxation optimization method. This meth-od breaks the original joint problem into a set of smallersubproblems that are solved in parallel, and avoidsconverging to local minima by forcing cooperation.

Cooperation was achieved using a step-size penalty, andthe score function was formulated using a OSP. Weanalyzed the performance of our method using numer-ical examples of several electrical conductivity fieldtopologies and noise levels, and discussed its applicationto real measurements obtained from a smooth cardiacmouse tissue slice, using data collected with the high-resolution 4096-channel MEA platform. In the future,we will consider optimizing model parameter fittingfrom the data by employing more advanced learningschemes and better utilizing prior biological informa-tion. Further, we will extend the model to nonhomo-geneous reaction dynamics and establish amethodologyfor fusing conductivity tensor field information fromdifferent post-stimulus experiments.

ELECTRONIC SUPPLEMENTARY MATERIAL

The online version of this article (doi:10.1007/s10439-012-0581-9) contains supplementary material,which is available to authorized users.

ACKNOWLEDGMENTS

MEA data set recorded at Italian Institute ofTechnology by using the high-resolution 4096-channelMEA platform of 3Brain GmbH, Switzerland. Thiswork was supported in part by the McDonnell Inter-national Scholars Academy Fellowship, and also inpart by a National Science found CCF-0963742. Anyopinions, findings, conclusions, or recommendationsexpressed in this publication are those of the authorsand do not necessarily reflect the views of the NSF.

REFERENCES

1Aja-Fernandez, S., R. de Luis Garcia, D. Tao, and X. Li.Tensors in Image Processing and Computer Vision. Lon-don: Springer, 2009.2Arsigny, V., P. Fillard, X. Pennec, and N. Ayache. Log-euclidean metrics for fast and simple calculus on diffusiontensors. Magn. Reson. Med. 56:411–421, 2006.3Barmpoutis, A., B. C. Vemuri, T. M. Shepherd, and J. R.Forder. Tensor splines for interpolation and approxima-tion of DT-MRI with applications to segmentation ofisolated rat hippocampi. IEEE Trans. Med. Imaging26(11):1537–1546, 2007.4Batchelor, P. G., M. Moakher, D. Atkinson, F. Cala-mante, and A. Connelly. A rigorous framework for diffu-sion tensor calculus. Magn. Reson. Med. 53:221–225, 2005.5Bazhenov, M., P. Lonjers, S. Skorheim, C. Bedard, andA. Destexhe. Non-homogeneous extracellular resistivityaffects the current-source density profiles of updown state

Estimating Electrical Conductivity Tensors of Biological Tissues

oscillations. Philos. Trans. R. Soc. 369(1952):3802–3819,2011.6Bedard, C., and A. Destexhe. Generalized theory forcurrent-source-density analysis in brain tissue. Phys. Rev. E84(4):041909, 2011.7Berdondini, L., K. Imfeld, A. Maccione, M. Tedesco, S.Neukom, M. Koudelka-Hep, and S. Martinoia. Activepixel sensor array for high spatio-temporal resolutionelectrophysiological recordings from single cell to largescale neuronal networks. Lab Chip 9:2644–2651, 2009.8Bezdek, J. C., and R. J. Hathaway. Convergence of alternat-ing optimization. Neural Parallel Sci. Comput. 11:351–368,2003. http://dl.acm.org/citation.cfm?id=964885.964886.9Bressloff, P. C. Traveling fronts and wave propagationfailure in an inhomogeneous neural network. Phys. D155:83–100, 2001.

10Castleman, K. R. Diginal Image Processing. EnglewoodCliffs: Prentice Hall, 1996.

11de Leeuw, J., and G. Michailides. Block RelaxationMethods in Statistics. Technical Report, Department ofStatistics, University of California at Los Angeles, 1993.

12Dokos, S., and N. H. Lovell. Parameter estimation in cardiacionic models. Prog. Biophys. Mol. Biol. 85(2–3):407–431,2004. Modelling Cellular and Tissue Function. http://www.sciencedirect.com/science/article/pii/S0079610704000306.

13Graham, L. S., and D. Kilpatrick. Estimation of the bi-domain conductivity parameters of cardiac tissue fromextracellular potential distributions initiated by pointstimulation. Ann. Biomed. Eng. 38(12):3630–3648, 2010.

14Hagmann, P., L. Jonasson, P. Maeder, J. Thiran, V. Wedeen,and R. Meuli. Understanding diffusion MRI imaging tech-niques: from scalar diffusion-weighted imaging to diffusiontensor imagingandbeyond.Radiographics26:S205–S223, 2006.

15He, Y., and D. Keyes. Reconstructing parameters of theFitzHugh-Nagumo system from boundary potential mea-surements. J. Comput. Neurosci. 23:251–264, 2007. doi:10.1007/s10827-007-0035-9.

16Imfeld, K., S. Neukom, A. Maccione, Y. Bornat, S. Mar-tinoia, P. Farine, M. Koudelka-Hep, and L. Berdondini.Large-scale, high-resolution data acquisition system forextracellular recording of electrophysiological activity.IEEE Trans. Biomed. Eng. 55(8):2064 –2073, 2008.

17Julier, S., and J. Uhlmann. Unscented filtering and non-linear estimation. Proc. IEEE 92(3):401–422, 2004.

18Kaipio, J., and E. Somersalo. Nonstationary InverseProblems, chapter 4. In: Statistical and ComputationalInverse Problems, Vol. 60, edited by S. Antman, J. Mars-den, and L. Sirovich. Berlin: Springer, 2005.

19La Rosa, P. S., H. Eswaran, H. Preissl, and A. Nehorai.Multiscale forward electromagnetic model of uterine con-tractions during pregnancy. In revision forBMCMed. Phys.

20Latash, M. L. Neurophysiological Basis of Movement.Urbana: Human Kinetics Publishers, 2008.

21Lu, Y., Y. Sato, and S. Amari. Traveling bumps and theircollisions in a two-dimensional neural field. Neural Comput.23(5):1248–60, 2011.

22Mehra, R. K. Optimal input signals for parameter esti-mation in dynamic systems-survey and new results. IEEETrans. Autom. Control 19:753–768, 1974.

23Miller, W., and D. Geselowitz. Simulation studies of theelectrocardiogram. I. The normal heart. Circ. Res. 43(2):301–315, 1978.

24Miller, W., and C. Henriquez. Finite element analysis ofbioelectric phenomena. CRC Crit. Rev. Biomed. Eng.(USA) 18(3):207–233, 1990.

25Nelles, O. Nonlinear System Identification: From ClassicalApproaches to Neural Networks and Fuzzy Models.Berlin, New York: Springer, 2001.

26Nilssen, T., K. Karlsen, T. Mannseth, and X.-C. Tai.Identification of diffusion parameters in a nonlinear con-vection-diffusion equation using the augmented lagrangianmethod. Comput. Geosci. 13:317–329, 2009.

27Nilssen, T. K., and X. C. Tai. Parameter estimation withthe augmented Lagrangian method for a parabolic equa-tion. J. Optim. Theory Appl. 124(2):435–453, 2005.

28Nocedal, J., and S. J. Wright Numerical Optimization.Springer Series in Operations Research. New York:Springer, 2006.

29Penland, R. C., D. M. Harrild, and C. S. Henriquez.Modeling impulse propagation and extracellular potentialdistributions in anisotropic cardiac tissue using a finitevolume element discretization. Comput. Vis. Sci. 4:215–226,2002. doi:10.1007/s00791-002-0078-4.

30Pennec, X., P. Fillard, and N. Ayache. Riemannianframework for tensor computing. Int. J. Comput. Vis.66:41–66, 2006.

31Pilkington, T.C. High Performance Computing in Bio-medical Research. Boca Raton: CRC Press, 1993.

32Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P.Flannery. Numerical Recipes: The Art of Scientific Comput-ing (3rd ed.). New York: Cambridge University Press,2007.

33Proakis, J. G., and D. K. Manolakis. Digital Signal Pro-cessing (4th ed.). Upper Saddle River: Pearson PrenticeHall, 2007.

34Rauch, J., and J. Smoller. Qualitative theory of the Fitz-Hugh-Nagumo equations. Adv. Math. 27:12–44, 1978.

35Sepulveda, N., B. Roth, and J. Wikswo. Finite element bi-domain calculations. In: IEEE Engineering in Medicine andBiology Society 10th Annual International Conference, 1988.

36Sitz, A., J. Kurths, and H. U. Voss. Identification of non-linear spatiotemporal systems via partitioned filtering.Phys. Rev. 68:016202, 2003.

37Sitz, A., U. Schwarz, J. Kurths, and H. U. Voss. Estima-tion of parameters and unobserved components for non-linear systems from noisy time series. Phys. Rev. E66:016210, 2002.

38Sjoberg, J., Q. Zhang, L. Ljung, A. Benveniste, B. Deylon,P. Glorennec, H. Hjalmarsson, and A. Juditsky. Nonlinearblack-box modeling in system identification: a unifiedoverview. Automatica 31:1691–1724, 1995.

39Soderstrom, T., and P. Stoica. System Identification. UpperSaddle River: Prentice Hall, 1989.

40Strikwerda, J. C. Finite Difference Schemes and PartialDifferential Equations. Pacific Grove: Wadsworth andBrooks, 1989.

41Sun, N. Inverse Problems in Groundwater Modeling.Boston: Kluwer Academic Publishers, 1994.

42Tung, L. A bi-domain model for describing ischemicmyocardial D-C potentials. Ph.D. dissertation. Cambridge,MA: MIT, 1978.

43Valdes-Sosa, P. A., A. Roebroek, J. Daunizeau, andK. Friston. Effective connectivity: influence, causality andbiophysical modeling. NeuroImage 58:339–361, 2011.

44Vanag, V. K., and I. R. Epstein. Localised patterns inreaction-diffusion systems. Chaos 17:037110, 2007.

45Willms, A. R., D. J. Baro, R. M. Harris-Warrick, and J.Guckenheimer. An improved parameter estimation methodfor Hodgkin-Huxley models. J. Comput. Neurosci. 6:145–168, 1999. doi:10.1023/A:1008880518515.

GILBOA et al.