From Cardiac Cells to Genetic Regulatory NetworksR. Grosu1, G. Batt2, F. Fenton3, J. Glimm4, C. Le Guernic5, S.A. Smolka1 and E. Bartocci1,4

1Department of Computer Science, Stony Brook University, Stony Brook, NY2Institut National de Recherche en Informatique and en Automatique, Le Cesnay Cedex, France

3Department of Biomedical Sciences, Cornell University, Ithaca, NY4Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY

5Department of Computer Science, New York University, New York, NY

Abstract—One of the fundamental questions in the treatmentof ventricular cardiac disorders, such as tachycardia and fibril-lation, is under what circumstances does such a disorder arise?To answer to this question, we develop a multiaffine hybridautomaton (MHA) cardiac-cell model, and restate the originalquestion as one of identification of the parameter ranges underwhich the MHA model accurately reproduces the disorder. TheMHA model is obtained from the minimal cardiac model of oneof the authors (Fenton) by first bringing it into the form of acanonical, genetic regulatory network, and then linearizing itssigmoidal switches, in an optimal and global way. By leveragingthe Rovergene tool for genetic regulatory networks, we are thenable to successfully identify parameter ranges of interest.

I. INTRODUCTION

A fundamental question in the treatment of cardiac abnor-malities, such as ventricular tachycardia and fibrillation [1](see Figure 1), is under what conditions does such a disorderarise? To answer this question, in vitro and in vivo experi-mentation is nowadays complemented with the mathematicalmodeling, analysis and simulation of (networks of) cardiaccells. Among the myriad existing models, (partial) differential-equations models (DEMs) are arguably the most popular.

The past two decades have witnessed the development ofincreasingly sophisticated DEMs, which unravel in great detailthe underlying genomic and proteomic processes [2], [3],[4], [5]. Such models are essential in the understanding ofthe molecular interactions, and in the development of noveltreatment strategies. However, they also have two significantdrawbacks: 1) They often contain too many parameters tobe reliably and robustly identified from experimental data.2) They are often too complex to render their formal analysisor even simulation tractable. We refer to such models asdetailed molecular models (DMMs).

Approximation (or abstraction) is a well-established tech-nique in science and engineering for dealing with complexity.In dynamical systems possessing very fast transient regimes,compared to the rest of the system dynamics, one mayuse approximation to systematically eliminate these regimesand compensate for their elimination [6]. For example, inenzymatic reactions, a substrate reacts very quickly with anenzyme to produce a compound, which subsequently, andmuch more slowly, breaks down into a product of the reactionand the enzyme itself. In this case, one can use the so-calledquasi-steady-state assumption to eliminate the fast reaction

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Fig. 1. Emergent behavior in cardiac-cell networks. Top: Electrocardiogram.Middle and bottom: Simulation and experimental mappings of voltage wavesoccurring in a small rectangular area on the surface of the heart.

and derive a sigmoidal dependence of the product on the logof the substrate, called the Michaelis-Menten equation [7], [8].

Similar to the rectangular (step or Heaviside) switches usedin digital-computer models, sigmoidal switches (dependencies)occur everywhere in biological models: from molecular tocellular models, and from organ to population models [9], [10].In most cases, they result from the same kind of abstractionsdiscussed above: eliminate fast, transitory components. Unlikein digital-computer models, however, the switching speed ofsigmoids plays an important role. Biology is sophisticated!

DEMs with state variables whose rate of change is con-trolled with sigmoidal switches are still intractable from ananalysis point of view. Research in genetic regulatory net-works overcomes this problem by approximating sigmoidswith either steps or with ramps [11], [12], [13], [14]. Thisleads to a piecewise-affine (multiaffine) model, respectively. Insuch models, the dynamics within a hyper-rectangular region,is completely determined by the dynamics of its corners,enabling model analysis through the use of powerful discreteabstraction and model-checking techniques [13].

In prior work, one of the authors (Fenton) co-developed anextremely versatile electrical model for cardiac cells involvingjust 4 state variables and 26 parameters [15]. For reasons tobe made clear, we refer to this model as the minimal resistormodel (MRM). After its parameters are identified from eitherexperimental data or DMM-based simulation results, the MRMis able to accurately reproduce the desired behavior. In fact,

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Fig. 2. Overview of our approachthe MRM identified from experimental data reproduces theexperimentally observed behavior with greater accuracy thanany of the DMMs. Moreover, its simulation speed is orders ofmagnitude faster than that of the DMMs.

The success of the MRM intuitively relies on the same kindof abstraction as discussed above: the large variety of currentstraversing the cell membrane are lumped together in only threecurrents: the fast input current, the slow input current, and theslow output current. These currents are regulated by three gatevariables, which together with the voltage, define the MRM’sstate variables [15]. The lumping process is akin to removingand compensating for fast components [8], [6].

In the MRM context, one may restate the cardiac disorderquestion as follows: what are the parameter ranges for whichthe MRM accurately reproduces cardiac abnormalities? Oncethese ranges are identified, one may exploit the correspondencebetween the MRM and DMMs to infer the correspondingparameter ranges in the DMMs. The molecular relevance ofthe DMMs then allows to target treatment strategies to themolecular components responsible for the disorder.

Despite its simplicity compared to DMMs, the MRM isstill intractable from an analysis point of view. Its electricalformulation, not only uses sigmoidal switches to control thegating variables, but also uses them to model gated resistors.As such, sigmoids occur both as numerators and denominatorsin the state equations. As part of our effort to simplify theMRM, we prove that sigmoids are closed under reciprocaloperation. This allows us to bring the MRM to a canonicalform, which we call the minimal conductance model (MCM).Intriguingly, this is the form of a genetic regulatory networkmodel (GRM). Hence, this transformation not only exposes theunity of biology, but also allows us to leverage tools developedfor GRMs for the analysis of cardiac models.

While in a qualitative GRM sigmoidal switches are approxi-mated with steps or ramps, this is impossible to do in theMCM without considerably distorting its original behavior. Wetherefore approximate the shallow sigmoids of the MCM witha succession of ramps, the number of which depends on the de-sired accuracy. For analysis purposes, it is critical to minimizethe number of ramps used and to avoid arbitrary choices. Wetherefore adapt a dynamic programming technique [16], to findthe optimal number of segments, typically of different length,minimizing for all sigmoids at the same time, a sigmoidal-linearization error. This results in a hybrid-automaton modelwith multiaffine behavior in each mode (MHA).

By recasting the intractable parameter-range identifica-tion problem for MCMs in terms of MHAs, we now havea tractable problem. Moreover, certain MHA parameter-identification problems can be seen as a special case of theGRM parameter-identification problem: find the uncertain (due

S+ (u,!,k) = 11+ e"2k (u"! )

R+ (u,!1,!2 ) = (u < !1)?0 : (u < !2 )? u "!1

!2 "!1

:1

k = 1 / (!2 "!1)

H + (u,!) = (u < ! )?0 :1

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to experimental barriers or measurement errors) parameterranges that lead to a robust behavior satisfying a giventemporal logic property [13]. Hence, for these disorders, wecan leverage tools already developed for GRMs to address theMHA problem. Conversely, qualitative GRM can exploit ourlinearization technique in order to obtain accurate piecewise-multiaffine models from the experimental data.

The particular cardiac-disorder question addressed by thispaper is, under what circumstances may cardiac-cell excitabil-ity be lost? A region of unexcitable cells within cardiac tissue,can be responsible for tachycardia or fibrillation: 1) The regionbecomes an obstacle, in the way of the propagating electricalwave; 2) This triggers a spiral rotation of the wave, a conditionknown as tachycardia; 3) The spiral may then eventuallybreak up into many spirals, a condition known as fibrilla-tion. Studying the parameter ranges for which cardiac cellsloose excitability, and identifying the responsible molecularprocesses, is therefore an important question in the treatmentof cardiac disorders. Loss of excitability can be formulated asan LTL formula, for which the Rovergene tool [13] automat-ically infers nontrivial parameter ranges. To the best of ourknowledge, this is the first result of this kind.

Our overall approach is summarized in Figure 2. The restof this paper is organized accordingly, as follows.. Section IIintroduces biological switches and their formal description.Section III reviews the MRM. In Section IV, we transformthe MRM to an MCM, which is linearized in Section V. Sec-tion VI considers the parameter-range-identification problem.Section VII concludes and discusses future work.

II. BIOLOGICAL SWITCHING

As discussed in Section I, biological switching is sigmoidal.In this paper, we are interested in a particular class of on (+)and off (-) sigmoidal switches, namely the logistic functions.The sigmoidal on-switch is shown in Figure 3. The off-switchis defined as: S

−(u,k,θ) = 1−S+(u,k,θ).

We typically scale S so that it varies between a minimumvalue um and a maximum value uM , both positive:

S+(u,k,θ,um,uM ) = um +(uM − um)S+(u,k,θ)

S−(u,k,θ,um,uM ) = S

+(u,k,θ,uM ,um)

Note that S+(u,k,θ) = (1 + tanh(k(u− θ)))/2, which is an-

other way of expressing the logistic function.

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#

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+#'(

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-.)/&+/01#

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Fig. 4. APD, DI, and restitution at 10% of the maximum value of the AP.

If an on-sigmoid is very steep, then it can be approximatedwith a Heaviside (or step) switch, as shown in Figure 3.The off-step is defined as: H

−(u,θ) = 1−H+(u,θ). As with

sigmoids, step-switches can be scaled between um and uM .If an on-sigmoid is steep but not very steep, it can be

approximated with a ramp, as shown in Figure 3. The off-rampis defined as: R

−(u,θ1,θ2) = 1−R+(u,θ1,θ2) Ramps can also

be scaled between between um and uM .If the sigmoid is slow, then, as shown in Section V, it can

be approximated with a sequence of ramps.

III. THE MINIMAL RESISTOR MODEL

Based on previously published data [17], Fenton co-developed a minimal (resistor) model (MRM), of the actionpotential produced by human ventricular myocytes [15]. Anaction potential (AP) is a change in the cell’s transmembranepotential u, as a response to an external stimulus (current).If the stimulus e comes from the neighboring cells, then itsvalue is ∇(D∇u), where D is the diffusion coefficient and∇ is the gradient operator. The shape of an AP, its duration(APD), the diastolic interval (DI), and the AP restitution curve(dependence of the APD on the DI) are depicted in Figure 4.Intuitively, the membrane acts like a capacitor, requiring sometime to recharge after it discharges. The more time it has torecharge, the greater (and longer) the AP. Note that the APvalue u is scaled between 0 and 1.5 in the MRM model.

The MRM differs from more complex ionic models (DMM)in that instead of reproducing a wide range of ion channelcurrents, it considers the sum of these currents partitionedinto three main categories: fast inward Jfi (Na-like), slowinward Jsi (Ca-like), and slow outward Jso (K-like). Theflow of these total currents is controlled by a fast channelgate v and two slow gates w and s. Together, they retainenough structure such that with parameters fitted from eitherexperimental data or from DMM simulation results, the MRMaccurately reproduces the behavior in question.

Among fitted parameters are the voltage-controlled resis-tances τv , τw, and τs, and the equilibrium values v∞ and w∞.The differential equations for the state variables are as follows:

u = e− (Jfi(u, v) + Jsi(u, w, s) + Jso(u))v =H

−(u, θv) (v∞ − v)/τ−v (u) −H

+(u, θv)v/τ+v

w =H−(u, θw)(w∞ − w)/τ

−w (u)−H

+(u, θw)w/τ+w

s =(S+(u, ks, us)− s)/τs(u)

TABLE IPARAMETER VALUES FOR MRM AND MCM

MRM MCM MRM MCMPar Val Par Val Par Val Par Val

θo 0.006 θo 0.006 τ−v1 60 g−v1 0.01666

u−w 0.03 u�−w 0.0406 τ−v2 1150 g−v2 0.00086

θw 0.13 θw 0.13 τ−w1 60 g−w1 0.01666θv 0.3 θv 0.3 τ−w2 15 g−w2 0.06666uso 0.65 u�so 1.4824 τo1 400 go1 0.0025us 0.9087 us 0.9087 τo2 6 go2 0.16666uu 1.55 uu 1.55 τso1 30.0181 gso1 0.03331w∗∞ 0.94 w∗∞ 0.94 τso2 0.9957 gso2 1.00431k−w 65 k−w 65 τs1 2.7342 gs1 0.36453kso 2.0458 kso 2.0458 τs2 16 gs2 0.0625ks 2.0994 ks 2.0994 τfi 0.11 gfi 9.0909τ+v 1.4506 g+

v 0.68936 τsi 1.8875 gsi 0.5298τ+w 200 g+

w 0.005 τw∞ 0.07 gw∞ 142.8571

where the three currents are given by the following equations:Jfi(u, v) =−H

+(u, θv)v(u− θv)(uu − u)/τfi

Jsi(u, w, s) =−H+(u, θw)ws/τsi

Jso(u) =+H−(u, θw)u/τo(u) + H

+(u, θw)/τso(u)The voltage-controlled resistances are defined as follows:

τ−v (u) =H

+(u, θo, τ−v1

, τ−v2

)τo(u) =H

−(u, θo, τo2 , τo1)τs(u) = H

+(u, θw, τs1 , τs2)τ−w (u) =S

−(u, k−w , u

−w , τ

−w2

, τ−w1

)τso(u) =S

−(u, kso, uso, τso2 , τso1)

Finally, the steady state values for gates v and w are:v∞(u) =H

−(u, θo)w∞(u) =H

−(u, θo)(1− u/τv∞) + H+(u, θ

−v )w∗

∞

The values of the parameters for the epicardial (surface)myocytes, as fitted in [15], are given in Table I.

IV. THE MINIMAL CONDUCTANCE MODEL

While much simpler than DMMs, the MRM model isstill intractable from an analysis perspective. Its electricalformulation not only uses sigmoidal (and step) switches tocontrol the state variables, but also uses them to control thevalue of the resistances. As a consequence, sigmoids occurboth as numerators and denominators in the state equations.

In order to simplify the MRM model, we prove that scaledsigmoids (or steps) are closed under division; that is, thereciprocal of a scaled sigmoid is also a sigmoid. This resultallows us to bring the MRM model to a canonical form, whichwe call the minimal conductance model (MCM).

Theorem 1 (Sigmoid closure). For a, b > 0, scaled sigmoidsare closed under multiplicative inverses (division):

S+(u, k, θ, a, b)−1 = S

−(u, k, θ + ln(a/b)/2k, b−1

, a−1)

Proof: The proof proceeds by successively transformingthe inverse of a scaled sigmoid to a scaled sigmoid:

S+(u,k,θ,a,b)−1 =

1a+ b− a

1 + e−2k(u− θ)

=1 + e

−2k(u− θ)

b + ae−2k(u− θ)=

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Fig. 5. Response of MCM HA state variables to a stimulus.

1a× a− b + b + ae

−2k(u− θ)

b + ae−2k(u− θ)=

1a−

1a −

1b

1 + ab e−2k(u− θ)

=

1a−

1a −

1b

1 + e−2k(u− (θ + ln a−ln b

2k ))= S

−(u,k,θ +ln

ab

2k,1b,1a)

Obviously, H+(u, θ, a, b)−1 =H

−(u, θ, b−1

, a−1). Table I

gives the values of the conductances gi =1/τi of the MCMfor each resistance τi. For each threshold ui in the MRMmodel, we also provide the associated MCM threshold u

�i.

Revising the MRM model by replacing each factor 1/τi withgi and each threshold ui with the associated MCM thresholdu�i results in the differential equations for the MCM model.

Note that in the MCM, sigmoids and steps only appear atthe numerator. For further clarity and future reference, weprovide here the complete definition of the MCM model. Thedifferential equations for the state variables are:

u = e− (Jfi(u, v) + Jsi(u, w, s) + Jso(u))v =H

−(u, θv) (v∞ − v)g−v (u) −H+(u, θv)v g

+v

w =H−(u, θw)(w∞ − w)g−w (u)−H

+(u, θw)w g+w

s =(S+(u, ks, us)− s) gs(u)

where the three currents are given by the following equations:

Jfi(u, v) =−H+(u, θv)(u− θv)(uu − u)v gfi

Jsi(u, w, s) =−H+(u, θw)ws gsi

Jso(u) =+H−(u, θw)u go(u) + H

+(u, θw) gso(u)

The voltage-controlled conductances are defined as follows:

g−v (u) =H

−(u, θo, g−v2

, g−v1

)go(u) =H

+(u, θo, go1 , go2)gs(u) = H

−(u, θw, gs2 , gs1)g−w (u) =S

+(u, k−w , u

�−w , g

−w1

, g−w2

)gso(u) =S

+(u, kso, u�so, gso1 , gso2)

Finally, the steady state values for the gates v and w are:

v∞(u) =H−(u, θo)

w∞(u) =H−(u, θo)(1− u gv∞) + H

+(u, θo)w∗∞

(!v " u < uu )!u = #(D#u) + (u $!v )(uu $ u)v gfi + ws gsi $ gso(u)!v = $v gv

+

!w = $w gw+

!s = S+ (u,ks ,us ) gs2$ s gs2

u !"v

!w " u < !v

!u = #(D#u) + ws gsi $ gso(u)!v = $v gv2

$

!w = $w gw+

!s = S+ (u,ks ,us ) gs2$ s gs2

!o " u < !w

!u = #(D#u) $ u go2

!v = $v gv2

$

!w = (w%* $ w) gw

$ (u)!s = S+ (u,ks ,us ) gs1 $ s gs1

0 ! u < "o !u = #(D#u) $ u go1

!v = (1$ v) gv1$

!w = (1$ u gw%$ w) gw

$ (u)!s = S+ (u,ks ,us ) gs1 $ s gs1

u < !v

u < !w

u !"w

u !"o

u < !o

Fig. 6. Hybrid automaton for the MCM model.

An interesting feature of the MCM is that it has thecanonical form of a genetic regulatory network (GRN) model.

Definition 1 (GRM). The sigmoidal form of a genetic regu-latory network model (GRM) consists of a set of differentialequations in which the i-th equation has the following form:

ui =mi�

j=1

aij

nj�

k=1

S±(uk, kk, θk)−

m�i�

j=1

bij

n�j�

k=1

S±(uk, kk, θk)

where S± are either on- or off-sigmoidal switches, and ai and

bi are the expression and inhibition coefficients, respectively.

Approximating sigmoids with steps (or ramps) in the GRM,that is, assuming that switching is very steep (or steep),results in a set of piecewise-affine (-multiaffine) differentialequations [11], [12] (or [13]). The second summand is oftena simple decay term. We believe that that further advanceswill uncover genetic regulatory networks that contain slowsigmoids as well.

Since each (free) variable on the right-hand side of thedifferential equations of the MCM corresponds to a ramp,the MCM has the form of a GRM, where the very steep (orsteep) sigmoids have been approximated with steps (or ramps).The u

2 term in Jfi can be understood as the product of twocomplementary on- and off-ramps. This is not allowed in [13]but, in general, is a valid term of a GRM [9], [10].

The step-switches occurring in the differential equations ofthe MCM, indicate that the MCM specifies a mixed discreteand continuous behavior. In fact, the MCM is equivalent to a

hybrid automaton (HA). Consider the partition of the u-axisby the thresholds occurring in step-switches. Each mode of theHA corresponds to the u-interval defined by two successivethresholds, and each transition corresponds to the discretejump of one of the step-switches. The MCM HA is givenin Figure 6. Nonlinear terms are shown in red color andexponential degradation terms in blue. The currents have beenexpanded and broken according to the modes.

The behavior of the MCM HA state variables in time andas a response to an super-threshold external stimulus is shownin Figure 5. The voltage intervals corresponding to the modesof the HA are highlighted using the same color as in the HA.

Mode [0 θo) (orange) is a recovering resting mode. In thismode, gates v and w open to their maximum value, and gate s

remains closed. Slow sigmoids S+(u, ks, us) and g

−w (u) have

essentially their minimum value. The only transmembranecurrent is the slow output current Jso(u), whose overallbehavior mimics the ionic (potassium) K-current. This currentcauses an exponential decay of u. Conductances g

−v1

and g−w1

control the recovery speed of v and w. Hence, their values areimportant in properly reproducing AP restitution.

Mode [θv, uu) (green) is a successful AP mode, initiated bya super-threshold stimulus. Factor (u− θv)(uu−u) in the fast-input current Jfi mimics the fast opening of the (sodium) Na-channel. This leads to a dramatic membrane depolarization,during which u reaches its peak value uu. At the same time,but with a slight delay, gate v, which mimics the closing of theNa-channel, closes, thus blocking the Jfi current. The closing-time of v is solely controlled by the rate constant g

+v and the

initial value of v. The slow-input (calcium) Ca-like current,Jsi, is still flowing, which prolongs the duration of the AP,providing the cardiac muscle enough time to contract. Thevalue of Jsi is essentially controlled by gate s, which mimics,through its slow sigmoid, the behavior of the Ca-channelopening-gates. Gate w, which mimics the Ca-channel closing-gate, eventually blocks Jsi, at rate g

+w . The slow-output, K-like

current, Jso, reaches its peak value when the slow sigmoid gso

switches on towards its maximum value gso2 (u > u�so).

In mode [θo , θw) (blue), gate v starts closing at rate g−v2

,while gate w is still opening. The closing/opening of thesegates does not affect the value of u, as this still decays at ratego2 . It does, however, affect the initial values of v and w forthe next AP, which in turn affects the length of this AP. It alsoaffects the AP propagation speed, the so-called AP conductionvelocity (CV) in a myocyte network.

In mode [θw , θv) (pink), gate v closes at the same rate asbefore, but now gate w is also closing, at rate g

+w . Current Jso

changes from an exponential decay to a sigmoid, and the slow-input current starts flowing proportional to ws. Gate s adjuststhe “expression” coefficient of its slow sigmoid to gs2 .

V. THE MULTIAFFINE MODEL

Although the MCM is simpler than the MRM and consid-erably simpler than the DMMs, its analysis is still intractabledue to the presence of sigmoidal switches. GRN modelsovercome this problem by assuming that every sigmoid is steep

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!!!!1#2!%!6!<8=!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!:(>'*%#./'*/&$;(2!'#!0#/).?(2!

!!!!!!!!$./@22!6!0#)'A%*3B)*3CD!$./E/?(>!6!%*3D!@22#2!#1!A%*3B%C!6!F

!!!!!!!!1#2!.!6!)8%*7!!!!!!!!!!!!!!!!!!!!!!!!!:(>'*./'(2$(?.4'(*%#./'!'#!0#/).?(2

!!!!!!!!!!!!.1!A(22#2A.B%C!66!E/1C!!!!!!!@22#2!#1!,./(!)(+$(/'!A.B/C!/#'!04)5(?

!!!!!!!!!!!!!!!!1#2!0!6!38"!!!!!!!!!!!!!!!!!!!:(>'!0&2G(*/&$;(2!'#!0#/).?(2

!!!!!!!!!!!!!!!!!!!!)(A0C!6!)(+$(/'@22#2A>A.8%CB!-AHB.8%CCD!A.B%C*)(+$(/'!(22#2

!!!!!!!!!!!!!!!!(/?D!1#2!H!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!(22#2A.B%C!6!$4>A)(CD!!I4>.$&$!,./(!)(+$(/'!(22#2!

!!!!!!!!!!!!(/?D!.1!

!!!!!!!!!!!!0&22@22!6!0#)'A.B)*3C!J!(22#2A.B%CD!!!)*)(+$(/'*%#,-,./(!(22#2

!!!!!!!!!!!!.1!A0&22@22!K!$./@22C!!!!!!!!!!!!!!!!!!!!!!!!!9$4,,(2!(22#2L

!!!!!!!!!!!!!!!!$./@22!6!0&22@22D!$./E/?(>!6!.D!!M%?4'(!(22#2!4/?!%42(/'

!!!!!!!!!!!!(/?D!.1

!!!!!!!!(/?D!1#2!.

!!!!!!!!0#)'A%B)C!6!$./@22D!!!!!!!!!!!!!)*)(+$(/'*%#,-,./(!$./.$4,!0#)'

!!!!!!!!14'5(2A%B)C!6!$./E/?(>D!!!!N4)'!%#./'O)!14'5(2!#/!'5(!%#,-,./(

!!!!(/?D!1#2!%

(/?D!1#2!)

P(B4B;B>;Q!6!@>'240'R/)S(2D!

(/?

enough to be accurately approximated with either one step orone ramp. While this assumption is debatable for GRMs, itdoes not hold for MCMs. Its sigmoids are too slow to beapproximated by either a single step or a single ramp, withoutseriously distorting the original MCM behavior.

Slow sigmoids can be accurately approximated with se-quences of ramps. This, however, raises a new question: howto choose as few ramps as possible, while still maintaininga desired approximation error? Additionally, since each suc-cessive pair of ramp thresholds introduces a new mode in the

!

"#$%&'($!)*+,+-.!/!0*12*$&344(456+78!9$:#&;

!!!!6+7;!!<'1'&'=*>!%#4?*@0*12*$&!,0!,$!6@?*%&(4!,$>!,$!7@?*%&(4!

A#&:#&;!!!!!!!*;!!344(4!("!&B*!C'$*!0*12*$&!-*&D**$!&B*!"'40&!,$>!C,0&!:('$&!!!!,+-;!!EB*!%(*"'%'*$&0!>*"'$'$1!&B'0!0*12*$&

9$'&',C'=,&'($;z!/!0'=*568F!G!/!05H8F!!!I'$>!(#&!&B*!$#2-*4!("!:('$&0!("!6+7

J(2:#&*!K@0*12*$&!C'$*,4@'$&*4:(C,&'($!("!56+78!%(*""'%'*$&0!!!!!,!/!575$8!@!75K88!L!565$8!@!65K88F-!/!575K8!M!65$8!@!75$8!M!65K88!L!565$8!@!65K88F

J(2:#&*!:*4:*$>'%#C,4@>'0&,$%*!*44(4!"(4!,-(?*!C'$*!0*12*$&*!/!NF!!!!!!!!!!!!!!!9$'&',C'=*!344(4"(4!:!/!K;G!!!!!!J(2:#&*!*44(4!"(4!&B*!*,%B!:('$&!($!&B*!%#4?*!!!!*!/!*!O!575:8!@!,!M!65:8!@!-8H !L!5,HOK8F!!P%%#2#C,&*!C*,0&!0Q#,4*!*$>F*$>

!

"#$%&'($!)*+,+-+.-/!0!1.&2,%&3$45*2!!6#&7#&8!!!!!!!*+,+-+.-8!!34!'$!&9*!(#&7#&!("!(7&':,;<'$*,23772(.':,&'($=$'&',;'>,&'($8'-!0!>*2(4?@+@ABCD!.-!0!>*2(4?@+@ABCD!!E-!0!>*2(4?F+!@+@ABCD!!G('$&4!:,&2'%*4,!0!>*2(4?F+!@+@CD!!!!-!0!>*2(4?F+!@+@CD!!!*2!0!>*2(4?F+!@+@CD!F(*""'%'*$&4H*22(21.&2,%&!*22(2!,$I!%(*""'%'*$&!:,&2'%*4"(2!4!0!@8JB8B!!!!!!!!K2,L*24*!7(;E;'$*!4*M:*$&4!'$!'$L*24*!(2I*2!!!!'-?4+4ABC!!0!GD!!!!!!!!!!!!!!!!!!!!!!N*&!;,4&!7('$&!$#:-*2!!!!.-?4+4ABC!0!.?'-?4+4ABCCD!!!!N*&!.JL,;#*!"(2!&9'4!7('$&!!!!!"(2!%!0!B8F!!!!!!!!K2,L*24*!,;;!%#2L*4!!!!!!!!E-?%+4+4ABC!0!E?%+'-?4+4ABCCD!!N*&!EJL,;#*!"(2!&9'4!7('$&!!!!*$ID!!!!!!!!"(2!'!0!48JB8B!!!!!!!K2,L*24*!72*I*%*44(2!7('$&4!'$!'$L*24*!(2I*2!!!!!!!!'-?4+'C!0!",&9*2?'-?O+'ABC+'CD!!!!N*&!72*I*%*44(2!7('$&!$#:-*2!!!!!!!!.-?4+'C!0!.?'-?4+'CCD!!!!!!!!!!!!!!!!!!!N*&!.JL,;#*!!"(2!&9'4!7('$&!!!!!!!!!"(2!%!0!B8F!!!!!!K2,L*24*!,;;!%#2L*4!!!!!!!!!!!!E-?%+4+'C!0!E?%+'-?4+'CCD!!!!!!!!!N*&!EJL,;#*!"(2!&9'4!7('$&!!!!!!!!!!!!)*2?%+4+'C+!,?%+4+'C+!-?%+4+'C/!0!F(:7#&*!*22+!,!,$I!-!"(2!4*M:!?.+EC!!!!!!!!!!!!!!!!4*M:*$&122(2?!.?'-?4+'C8'-?4+'ABCC+!E?%+'-?4+'C8'-?4+'ABCC!CD!!!!!!!!*$I!!!!!!!!*$ID*$ID!*$I

HA, how to choose the same thresholds for all sigmoids?In the following, we show that all of these goals are achievable;i.e., there is an optimal solution, to the slow-sigmoid approxi-mation problem, which minimizes a given approximation errorin a global way. Our approach is based on and extends adynamic programming algorithm developed in the computergraphics community for approximating digitized polygonalcurves [16] with minimal error. The MATLAB code for themain function, optimalLinearApproximation, is shown on previouspage, where for readability, comments are displayed in green.

The function’s input is a set of curves (digitized with thesame number of points), and a number S of segments to beused by the polylines, optimally interpolating the curves. Thecurves are given as a vector x of P x-coordinates, and a matrixy of C rows, each consisting of P y-coordinates.

The function’s output consists of matrices e, a, b and ofvector xb. Each entry e(c,s,i), a(c,s,i) and b(c,s,i) gives the error,

!"#"

!1 !2 = !o

S+ (u,us ,ks )

(1! ugw" )gw! (u)

gw! (u)

q + m(!2 "!1)R+(u,!1,!2)

$"

m(!2 "!1)

Fig. 7. Linearization of the orange mode of the MCM HA.

slope, and y-intercept, respectively, of the i-th segment, in theoptimal interpolation polyline of curve c, using s segments.Each entry xb(s,i) is the x-coordinate of breaking point i, ofthe optimal interpolation polylines, using s segments.

The function first determines the number of points P ineach curve, and the number of curves C. It then initializes thedynamic programming storage tables cost(P,S) and father(P,S).Each entry cost(p,s) stores the cost from point 1 to point p,of the optimal interpolation polyline consisting of s segments.Each entry father(p,s), stores the predecessor of point p on theoptimal-cost polyline consisting of s segments. To speed upthe search, we use an error matrix error(P,P), such that eachentry error(p,q) caches the maximum error of the segment (p,q)with respect to all of the given curves.

Then, in a classic dynamic programming fashion, optimalLi-nearApproximation fills its solution tables bottom up. First, for allpoints in the curve (except 1), it computes the cost and father,of the 1-segment polyline starting from point 1. Then, knowingthe optimal cost of all s-segment polylines from point 1 to anypoint i that is less than or equal to p, it computes the optimalcost of an s+1-segment polyline from point 1 to point p+1, bychoosing the s-segment polyline, whose cost is minimal whenincreased with error(i,p+1).

The value stored in error(p,q) is computed with the (nested)function segmentError. Its input consists of vectors x and y,defining a curve segment. Its output consists of error e, andcoefficients a and b of the line y(x) = ax+ b passing throughthe first and the last points of the curve segment. Error e iscomputed by summing up, for each point p on the curve, thesquare of the perpendicular distance from (x(p),y(p)) to y.

Once the solution tables are completely filled, optimalLine-arApproximation calls nested function extractAnswer to traversetable father in reverse order, and produce matrices e, a, b, andxb. These matrices have the same format as the output of thecaller function, optimalLinearApproximation.

Our implementation of segmentError also allows the use oflinear regression instead of linear interpolation. This leadsto an optimal approximation that, for the same error, hasfewer segments. However, linear regression also introduces

!2 = !o

!3

!4

!5

!6

!7

!8 = !w

!"

S+ (u,us ,ks )

gw! (u)

Fig. 8. Linearization of the blue mode of the MCM HA.

!9

!11

!10

!11 = ! v

!"

!8 = !w

S+ (u,us ,ks )

gso(u)

Fig. 9. Linearization of the pink mode of the MCM HA.

discontinuities at the breaking points of the optimal polylines,as the line segment resulting from regression does not typicallystart and end on the curve. When approximating a singlecurve, one can choose as new breaking points the points wherethe polyline segments intersect. Unfortunately, it is not clearhow to generalize this approach to a set of curves, withoutintroducing unnecessary breaking points.

As in GRMs, we assume that the thresholds and slopes ofsteps and sigmoidal switches are known and fixed. This is areasonable assumption, as these parameters can be accuratelyidentified from simulation or experimental data. We can thuslinearize the MCM HA one mode at a time. The result is amultiaffine hybrid automaton (MHA).

Figure 7 presents our linearization of the orange mode ofthe MHA. There are three nonlinear functions: sigmoids g

−w (u)

and S+(u, us, ks), and product (1−ugw∞)g−w . The last is

treated separately, as the linearization of g−w (u) multiplied by

(1−ugw∞) results in a u2 term. A two-segment linearization

(two modes in the MHA) results in a very small error.Figure 8 presents our linearization of the blue mode of the

MHA. There are two nonlinear functions: sigmoids g−w (u)

and S+(u, us, ks). In this case, we needed a six-segment

linearization (six modes in the MHA) to achieve a smallapproximation error. Note that the sensitivity of the MCM

!12 = ! v

!13

!14

!15

!16

!17

!18

!19

!20

!21

!22

!23

!24

!25

!26 = uu!"

(u !"v )(uu ! u)gfi

S+ (u,us ,ks )

gso(u)

Fig. 10. Linearization of the green mode.

!"#

$%!#$%!

#!"#

&#'()*#

+#'(

,*#

-./.(01#23).)&45#-4631#-+17#$8/3#-4631#'9:#50(;)*#

Fig. 11. Comparison of the AP shape and restitution in cable.

behavior to the linearization error is also very important.Figure 9 presents our linearization of the pink mode of the

MHA. There are two nonlinear functions: sigmoids gso(u) andS

+(u, us, ks). A four-segment linearization (four modes in theMHA) achieves a small enough approximation error.

Finally, Figure 10 presents our linearization of the greenmode of the MHA. This is the most sophisticated mode.It contains three nonlinear functions: sigmoids gso(u) andS

+(u, us, ks) and the parabolic term (u− θv)(uu−u). Al-though gate v closes very rapidly, nullifying the parabolicterm in Jfi , voltage u traverses in the meantime the entireinterval [θv, uu). Hence, one needs to linearize the parabolicterm over this entire interval. This leads to a quite costly,but inevitable linearization via 14 segments (14 modes in theMHA). In our experiments, fewer segments have lead to anunacceptable approximation of the MCM behavior.

To asses the accuracy of the MHA, we performed extensive1D and 2D simulations, in a cable of 100 cells and a gridof 800× 800 cells, respectively. Although the 1D simulationwas used to determine the behavior of a single cell only—for example, cell number 50—the use of a cable is necessary,as it is known that cells behave differently when interactingwith neighboring cells. Many cardiac models, for example [2],accurately reproduce the AP when simulated in isolation, butfail to reproduce the desired behavior in a cable.

Figure 11 shows the restitution curves of the MCM andMHA models. Each point APD(d) on these curves, wasobtained by first pacing the MCM and the MHA models at

Fig. 12. Spiral wave and tip movement comparison in an 800× 800 grid.

the largest DI value, and then abruptly changing the pacing toDI d. For each value of d, we also compared the AP shapesAP(d). One such comparison is given in Figure 11. In bothcases (restitution and AP shape), the MHA approximated theMCM with sufficient accuracy.

In Figure 12, we compare the behavior of the MCM andMHA models on a 2D grid of 800× 800 cells. The comparisonuses a well-established protocol for the initiation of a spiralwave in cardiac ventricular tissue. We have also tracked themovement of the tip of the spiral over time, which is shownas a dark-blue curve in Figure 12. The 2D simulation alsoconfirms the very good accuracy of the MHA model.

The simulations where performed on a CUDA PC equippedwith four GPU processors. *****EZIO SHOULD COM-PLETE THE SETUP DESCRIPTION***** In this setting, weobserved a 1.43 speedup in MHA simulation times comparedto MCM. Note that it is possible to table sigmoid and parabolavalues to speedup MCM simulation times [18]. This strategy,however, considerably increases the memory demand for thesame accuracy and speed, and renders analysis intractable. Ourlinearization approach can be viewed as an optimal tabling.

!12 = !v < u " uu = !26

!u = #(D#u) + R(u,!i ,!i+1,u fii,u fii+1

)i=12

25$ v gfi + ws gsi % R(u,!i ,!i+1,usoi ,usoi+1) gsoi=12

25$!v = %v gv

+

!w = %w gw+

!s = ( R(u,!i ,!i+1,usi ,usi+1)

i=12

25$ % s) gs2

u !"v

!8 = !w " u < !v = !12

!u = #(D#u) + ws gsi $ R(u,!i ,!i+1,usoi ,usoi+1) gsoi=8

11%!v = $v gv2

$

!w = $w gw+

!s = ( R(u,!i ,!i+1,usi ,usi+1)

i=8

11% $ s) gs2

!2 = !o " u < !w = !8

!u = #(D#u) $ u go2

!v = $v gv2

$

!w = (w%* $ w) R(u,!i ,!i+1,uwi ,uwi+1

)i=2

7& gwb!s = ( R(u,!i ,!i+1,usi ,usi+1

)i=2

7& $ s) gs1

!0 = 0 " u < !o = !2 !u = #(D#u) $ u go1

!v = (1$ v) gv1$

!w = ( (R(u,!i ,!i+1,uwi+ ,uwi+1

+ )i=0

1% $ wR(u,!i ,!i+1,uwi$ ,uwi+1

$ )) gwa!s = ( R(u,!i ,!i+1,usi ,usi+1

)i=0

1% $ s) gs1

u < !v

u < !w

u !"w

u !"o

u < !o

Fig. 13. The multiaffine hybrid automaton.

VI. PARAMETER-RANGE IDENTIFICATION

The linearization algorithm presented in Section V, returnsfor each mode [θ1, θ2), the parameter sequences ai and bi, andthe threshold sequence xi. Subscript i ranges over the numberof segments chosen, in order to fulfil a desired approximationerror e. For each i, the returned values define a line segmenty(x) = ax+ b within the interval [xi, xi+1).

For the first segment, x1 = θ1, and for the last segment,xn = θ2. Now consider segment [xi, xi+1). The minimumvalue of y(x) is yi = aixi + bi and the maximum value of y(x)is yi+1 = aixi+1 + bi. Together with the threshold values, theydefine the scaled ramp R

±(x, xi, xi+1, yi, yi+1). This is an on(+) ramp, if yi≤ yi+1 and an off (-) ramp if yi≥ yi+1.

Since the ramps have to be summed up, for each i> 1, wehave to adjust the y coordinate, by subtracting the maximumvalue of the previous ramp. Hence, these ramps becomeR±(x, xi, xi+1, yi− yi−1, yi+1− yi−1).Once the scaled ramps are computed and summed up, for

each mode of the MCM HA, one obtains a multiaffine hybridautomaton (MHA), as shown in Figure 13. The remainingparameters of the MHA are now highlighted in red color. TheMHA modes have become super-modes, each consisting of asmany sub-modes as there are distinct indices in the sums.

The MHA however, is not suitable as input to Rovergenefor two reasons: 1) Rovergene expression terms have to bescaled ramps; 2) Rovergene does not support steps. The firstproblem is overcome by replacing variables with ramps. Forexample, variable v occurring on the right-hand side of u in thegreen mode, is replaced with the ramp R

+(v, 0, 1). The secondproblem is overcome by replacing steps with very-steep ramps.This amounts to introducing for each threshold θi, separatingmodes [θi−1, θi) and [θi, θi+1), a just-before θi threshold

θ−i . The equations in mode [θi−1, θi) are now multiplied

with R−(u, θ

−i , θi) and the ones in mode [θi , θi+1) are now

multiplied with R+(u, θ

−i , θi). The MHA now becomes:

u = e−R−(u, θ

−o , θo)R+(u,0,θ

−o ,0,θ

−o ) go1 −

R+(u,θ

−o ,θo)R−(u,θ

−w ,θw)R+(u,θo,θ

−w ,θo,θ

−w ) go2 +

R+(u,θ

−w ,θw)R+(s,0,1)R+(w,0,1) gsi−

R+(u,θ

−w ,θw)

�25i=8 R(u,θi,θi+1,usoi ,usoi+1) +

R+(u,θo,θv)R+(v, 0, 1)�25

i=12 R(u,θi,θi+1,ufii ,ufii+1) gfi

v =R−(u,θ

−o ,θo)R−(v,0,1) g

−v1−

R+(u, θ

−o ,θo)R−(u,θo,θv)R+(v,0,1) g

−v2−

R+(u,θo,θv)

�25i=12 R

+(v,0,1) g+v

w =R−(u,θ

−o ,θo)

�1i=0(R(u,θi,θi+1,u

+wi

,u+wi+1

)−R(w,0,1)R(u,θi,θi+1,u

−wi

,u−wi+1

)) gwa +R

+(u,θ−o ,θo)R−(u,θ

−w ,θw)

(w∗∞−R(w,0,1))

�7i=2 R(u,θi,θi+1,u

+wi

,u+wi+1

) gwb −R

+(u,θ−w ,θw)R(w,0,1) g

+w

s =(R−(u,θ−w ,θw) gs1 +R

+(u,θ−w ,θw) gs2)�25

i=0 R+(u,θi,θi+1,usi ,usi+1)−R

+(s,0,1)

where the thresholds (except for the new ones), voltages, andconductances match the ones in the MHA. We call this systema piecewise-multiaffine differential equations model (MEM).

As discussed in Section I, a biologically relevant questionthat we would like to answer is, under what circumstances acell may loose excitability? At molecular level, this is due toan improper functioning of the cardiac-cell ionic channels.

To identify the molecular processes responsible for this dis-order, we first reformulate the above question, as a parameter-range identification question: What are the parameter rangesfor which the above MEM fails to generate an AP?

This property may be specified in linear temporal logic(LTL) as follows: G (u < θv), where G is the LTL globally(always) temporal operator. The property states that in allexecutions of the DEM (an implicit universal quantification ofLTL over executions) and in all moments of time (an explicitquantification in LTL with operator G, over all states of asingle execution), the voltage value is below θv . We wouldlike this property to hold for all stimulus durations. In termsof the MCM HA in Figure 13, this property is true due to theinterplay of the ranges of conductances go1 , go2 , gsi and gso.

To identify these ranges in an automatic fashion we use theRovergene tool [13] co-developed by one of the co-authors(Batt) with input the above MEM and LTL formula, and initialregion: u∈ [0, θ1], 0∈ [0.95, 1], w∈ [0.95, 1] and s∈ [0, 0.01].

The u-thresholds and the initial region impose the followingpartition on the ranges of state variables: [0,. . .,θ29] for u (wehave added the just-before thresholds), [0, 0.95, 1] for v andw and [0, 0.01, 1] for s. The parameter ranges with biological

!"!!#$

!"%&$

!"&$

go1 ! 166.9496

go2 ! 7.6982

0.9688 gso -0.24784 gsi ! 26.0888

!$ '$

($

)*+,$-+./$%!$ %'$ 0!$ 0'$ &!$

Fig. 14. Simulation results for a sample in each of the above ranges.

significance were taken to be: [1, 180] for go1 , [0, 10] for go2 ,[0.1, 100] for gsi and [0.9, 50] for gso.

The behavior of the MEM in each hypercube of the state-space partition is completely determined by its corners, so theexistence of transitions from one hypercube to its neighbours,can be computed, by evaluating the MEM in the corners. Ineach corner, the MEM becomes an affine system in the MEMparameters. Solving these systems one obtains the separatinghyperplanes of positive and negative sign of the derivativesin the MEM. Finally, taking into account the desired LTLproperty, one obtains the parameter ranges for which theproperty is satisfied. In our case these ranges returned are:

166.9494≤ go1 ≤ 180, 7.6982≤ go1 ≤ 10

−0.24784 gsi + 0.9688 gso ≤ 26.0888

They have the following meaning. If go1 ≥ 166.9494 thenno matter how long a stimulus of magnitude 1 is applied, thevoltage u never leaves the orange interval (mode) [0, θo]. If onthe other hand go1 < 166.9494, then u reaches the blue interval(mode) [θo, θw). Since we are considering stimuli of any width(note that time is abstracted away by Rovergene), once u

enters the blue range, its behavior is completely determined bythis mode. If go2 ≥ 7.6982 then u can never leave the mode.If go2 < 7.6982 then u will enter the pink interval (mode)[θw, θv). In this mode the behavior of u is determined by theinterplay between gso and gsi. If the above linear combinationis satisfied, one can never leave this mode.

The corresponding simulation, for a sample of values inthe above parameter ranges is shown in Figure 14. To makesure that we run the same model as Rovergene, we alsodeveloped a Rovergene simulation tool that, given a Rovergenemodel as input, simulates its dynamic behavior in Matlab. Thissimulation tool proved to be an invaluable debugging tool,during the model encoding in Rovergene.

VII. CONCLUSIONS AND FUTURE WORK

Although formal techniques were used before to analyzecardiac-cell properties (see e.g. our work in [19], [20]), thispaper presents to the best of our knowledge, for the first time,an approach to automatically identify the parameter rangesof a biologically-relevant cardiac model, guaranteeing that themodel accurately reproduces a particular cardiac disorder.

The approach takes the nonlinear cardiac-model in [15],brings it first into a genetic regulatory network sigmoidalform, then linearizes and formats it to a piecewise-multiaffineset of differential equations, and finally leverages the toolspreviously developed for the automatic parameter-range iden-tification in genetic regulatory networks [13], to automaticallycheck a disorder expressed in linear temporal logic.

In particular, the property we studied in this paper is lack ofcardiac-cell excitability. This is an invariant property, wheretime is abstracted away. In future work, we plan to investigatemore sophisticated LTL properties of singles cells, as well asreachability properties of cell networks (e.g. spirals).

Many abnormalities responsible for cardiac disorders aretime or rate dependent properties, that cannot be checkedwith the Rovergene tool, due to its underlying finite-automataabstraction. For example, action potential duration or spiralbreakup (fibrillation) are such properties. In future work wetherefore plan to investigate novel parameter-range identifica-tion approaches, that use timed-automata[21], [22] or linear-automata[23], [24] abstractions instead.

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[2] C. H. Luo and Y. Rudy, “A dynamic model of the cardiac ventricularaction potential. I. simulations of ionic currents and concentrationchanges,” Circulation Research, vol. 74, no. 6, pp. 1071–1096, Jun.1994.

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