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ventricular arrhythmia is the leading cause of cardiac arrest and sudden death . clinical observations and implantable cardioverter defibrillators have accumulated a substantial amount of data on the occurrences of ventricular arrhythmia in patients [ 1 , 36 , 37 , 38 , 52 , 59 - 61 ] . temporal patterns of initiations and terminations of ventricular arrhythmia tend to exhibit substantial variations across different time scales , and their occurrences do not correlate decisively with medication , exertion , stress , lifestyles and similar factors . arrhythmia events are not random ; they show circadian patterns [ 36 , 60 ] and also tend to occur in clusters . however , the detection times between consecutive events and clusters are spread out over time [ 37 , 52 , 59 ] , making it difficult to understand their causes and make predictions about their occurrences . unlike the circadian patterns , these clusterings or their patterns of occurrences are not affected by the long - term administration of antiarrhythmic drugs [ 59 ] . in spite of the abundant data in existence , the basis for non - circadian patterns is not well - understood . many factors , ranging from internal cardiac mechanisms to external chance events play significant roles in shaping the electrocardiogram ( ecg )  recordings and the implantable defibrillator data . separating all of the possible contributions is a formidable task , but understanding the influences of various factors and the extent to which each plays a role may have important consequences for the diagnosis and treatment of tachyarrhythmias . a number of prior studies using special preparations of animal cell cultures establish that complex , spontaneously generated patterns may occur without some of the features peculiar to the heart ( e.g. the 3-dimensional geometry or biological features such as valves , different tissue layers and types , etc ) [ 3 , 25 , 35 , 44 ] . these studies also indicate ( both theoretically and experimentally ) that causes external to the heart , whether random or deterministic , are not always necessary for the occurrence of irregular behavior . a different class of studies that involve interactive self - oscillatory sources also establish the capability of basic cardiac mechanisms to generate complex rhythms without considering the whole heart [ 23 , 28 , 43 ] . from these and similar studies it may be inferred that changes in the basic internal cardiac mechanisms can play a significant role in generating and sustaining arrhythmias . to clarify the role of these basic internal mechanisms at a fundamental level , in this paper we focus on the unidirectional reentrant circulation of action potentials in a loop of ventricular tissue . because of their pervasiveness , relative simplicity and importance in clinical arrhythmogenesis reentrant loops have received a great deal of attention [ 4 , 8 , 11 , 14 - 16 , 20 , 26 , 30 , 41 , 44 , 48 - 50 , 55 ] . studying spontaneous initiation and termination of reentry ( sitr ) patterns in the loop context is considerably simpler than in the whole heart and can offer potentially useful insights into the clinically observed patterns of arrhythmia . in the construction of a workable model , a certain level of abstraction bridges over physiological complexities that do not play a central role in the long - term evolution of temporal patterns .    in this paper we consider a model that combines the traditional discrete loop with a pacer to form an interactive system . fundamental thresholds are dynamically integrated so they can either trigger reentry or inhibit it . although a relatively simple mechanism , this composite system has two important features : it is an essential component of a well - known cardiac anomaly ; and it is capable of generating complex initiation and termination patterns .    in a series of case - studies we show that the loop - pacer mechanism may be responsible for irregularities in onset and termination of reentry beyond such pervasive factors as tissue heterogeneity , the geometry of the heart and other physiological and non - physiological factors . even without such complexities , the sitr  patterns generated by the loop - pacer system can be complex ; i.e. they contain many regular features yet their evolution over time is difficult to predict . we trace this complexity to the interaction of a threshold with bistability of the reentrant circulation and in some cases , also to the inherent discontinuity of phase resetting interactions between the loop and the pacer . bistability is an inherent physiological characteristic that may be attributed to non - uniformities in the  restitution of action potential duration ( apd ) ; this feature emerges prominently when conduction velocity ( cv ) is properly matched with apd . when this happens ( e.g. when the length of the loop is within a certain range ) the sitr patterns become considerably more complex and unpredictable . in this section we describe the main features of the loop - pacer model , leaving a few additional details to section 4 . the pacer is a self - oscillatory mechanism that is tied to the reentrant circuit ( the loop in this paper ) via a time lag or delay parameter and a special threshold . consider a loop of cardiac tissue consisting of cells that conduct action potentials and assume that a single conducting pathway connects the loop to the rest of the heart . let the physical length of the loop be denoted by @xmath4 measured in centimeters . divide the loop into @xmath5 sets or aggregates of cells , each of which may be called a _ cell aggregate _ or for brevity , just a cell . the cell aggregate that is connected to the only pathway out of the loop is also the gateway through which pulses enter the loop or exit it ; we label it cell 1 ; see figure 1 . this cell is adjacent not only to cell 2 , but also to cell @xmath5 at its other end . [ schematicloop ]        for each integer @xmath6 between @xmath7 and @xmath8 label the length of the @xmath6-th cell @xmath9 these cell aggregates lengths are not necessarily equal but of course , they add up to @xmath10 since each cell aggregate must have at least one cardiac cell in it , @xmath11 has a natural lower bound , namely the nominal length of a single cardiac cell ( about 0.01 cm or 100 microns ) . the closer all @xmath11 are to this lower bound , the greater is the number @xmath5 of cell aggregates in a loop of fixed length @xmath4 ; for a 12 cm loop with cells of length 0.01 cm , the number of cells is 1200 . while for discrete modeling it is not necessary to pick @xmath5 that large , it needs to be large enough ( hence each cell aggregate small enough ) that the conduction velocity from one end of an aggregate to the other can be taken to be approximately constant . this assumption is essential if the restitution of conduction time ( ct ) is defined in terms of the restitution of cv , as we do later on . we assume that all the cardiac cells within a given aggregate are identical . if all cells in the _ entire loop _ have identical physiological characteristics then we may also let cell aggregates all have equal lengths . in such a case , the loop is said to be homogeneous . otherwise the loop is heterogeneous .    to initiate unidirectional circulation , it is necessary that a unidirectional block ( ub ) exist somewhere in the loop ; for simplicity , we place it in cell 1 , the gateway to the loop ; see figure 1 . thus cell 1 is able to activate cell 2 but not cell @xmath12 for our simulations , we use a simplified version of the ub time window that is defined by a threshold value for the diastolic interval ( di ) . thus conduction is blocked in any cell whose di is not greater than the threshold @xmath13 ; see section 4 below for more details . such a minimum value is used in [ 26 ] as a termination mechanism for explaining the experimental results of [ 20 ] ; also see [ 19 ] . we use the term restitution  generally for relations that give a particular quantity , e.g. action potential duration or conduction time , as a function of the diastolic interval ( di ) . the di ( usually measured in milliseconds , ms ) is the rest or recovery period for the cell which essentially starts with the end of the refractory period and ends when the cell is activated again .      in its most basic form , the action potential duration or apd is the length of time ( usually measured in milliseconds , ms ) that a cell is active after excitation . for our purposes , we may think of apd as a cell s effective refractory period ( erp ) during which no excitations , even strong ones , can elicit new action potentials . the restitution of apd , which can be experimentally measured or derived from ionic models , is the most extensively studied of restitution relations [ 2 , 4 - 8 , 10 , 11 , 21 , 24 , 26 , 29 , 32 , 33 , 42 , 45 , 48 , 50 , 53 - 57 ] . any apd restitution function , whether experimentally measured or analytically derived can be used in this model .  for the numerical simulations in this paper , we use an apd  restitution function that is fitted to the experimental data by koller , et al [ 33 ] where apd values were recorded in two types of patients , those with and those without structural heart disease ( shd ) . the following is a possible fit to their averaged data displayed in figure 1 in [ 33 ] for shd  patients : @xmath14 with parameter values    [ c]|c|c|c|c|c|c|c|c|c|c|c|c|@xmath15 & @xmath16 & @xmath17 & @xmath18 & @xmath19 & @xmath20 & @xmath21 & @xmath22 & @xmath23 & @xmath24 & @xmath25 & @xmath26 + 350 & 157 & 8 & 20 & 1700 & 1200 & 0.0021 & 0.025 & 0.0004 & 80 & 136 & 82 +    a graph of this function is shown in figure 2 . expressing the apd as a function of di in the above manner represents only a first order of approximation ( see section 4 below ) . [ apdrc ]          each cell conducts an action potential through it in a finite amount of time . this time interval , usually measured in milliseconds , is referred to as the conduction time ( ct ) [ 4 , 26 , 50 ] . experimentally measured ct restitution functions are not as readily available for human hearts as the apd restitution functions ; however , they may be readily derived from cv restitution functions through the relation@xmath27 where @xmath28 and @xmath29 are , respectively , the ct and the cv restitution functions . this derivation is valid if as mentioned earlier , @xmath29 is essentially constant over the length of each cell ( or cell aggregate ) and thus no spatial dependence is required in @xmath30    for discussions of cv restitution functions see [ 2 , 7 , 8 , 11 , 12 , 14 - 16 , 20 , 22 , 53 , 54 , 56 ] . for our simulations we use the following expression @xmath31,\quad di>0 \label{ct}%\ ] ] with parameter values :    [ c]|c|c|c|c|@xmath32 & @xmath33 & @xmath34 & @xmath35 + 0.1 cm & 0.07 cm / ms & 1 & 0.02 ms@xmath36 +    these numbers are obtained using ( [ cv2ct ] ) and a reasonable fit to the human - rescaled , guinea pig data for cv from [ 22 , 54 ] . the value of @xmath32 corresponds to 10 nominal cells . the number @xmath33 in ( [ ct])is the maximum cv ; in more common ( though less uniform ) units , the fitted number in the above table would be 0.7 m / s . by changing the parameters in ( [ ct ] ) we may achieve conduction slow - downs in different ways , as shown in figure 3 . [ ctrc ]     and @xmath37 . ( a ) @xmath38 and @xmath39 ; ( b ) @xmath40 and @xmath39 ; and ( c ) @xmath41 and @xmath42.,width=288 ]      in the absence of reentry , a pacer ( the sinus node or an ectopic focus ) drives the ventricular contraction cycle . if reentry is initiated and interrupted then the pacer and the loop fall out of phase and it is necessary to carefully model their interaction . for this purpose we define two rhythms or clocks : the pacer s rhythm which has a prescribed beat pattern ( fixed or variable ) and the system s rhythm which is reset when reentry is initiated or terminated . these two rhythms are related via a pacer threshold ( seen the next section ) and a delay term that we now define . the _ time of occurrence _ for the @xmath43-th beat ( following a fixed reference beat ) can be tracked and calculated using the following quantity , namely , the sum of usually variable cycle lengths ( after a di adjustment ) :    @xmath44    let the time of occurrence of the @xmath45-th pacer beat be denoted by @xmath46 ( starting from the same reference point in time as that for @xmath47 ) . the difference @xmath48 is the duration of each such beat . in our simulations @xmath49 is defined arbitrarily in order to facilitate our study of the sitr patterns . if the pacer is the sinus node then @xmath49 is influenced by a variety of deterministic and stochastic factors , such as circadian , weekly and seasonal variations , lifestyles , drugs , the autonomic system and so on . thus a single , all purpose formula for @xmath49 does not exist . the two rhythms @xmath47 and @xmath46 fall out of phase if reentry occurs as noted earlier . to model their interaction , we define the _ first _ pacer pulse that reaches cell 1 in beat @xmath43 as the _ least _ integer @xmath50 such that @xmath51    the variable delay term @xmath52 here represents a _ time lag _ that results from the blocking of the pacer by the retrograde reentrant wave propagating toward the pacer . a number of things may affect @xmath52 ; these include the different types of tissue in which pulses propagate ( retrograde and antegrade ) , the electrotonic spread of current in conducting tissue between the pacer and the loop , and the effect of reentrant waves on the pacer s intrinsic beat rate . for simplicity we do not consider the electrotonic currents and tissue heterogeneity . we assume also that the pacer itself is protected in the sense that it s intrinsic beat rate is not affected by the reentrant activity . therefore , we use a simple definition for @xmath52 as follows:@xmath53{ll}% \delta , & \text{if pulse } n-1\text { was reentrant}\\ di^{\ast } , & \text{otherwise } % \end{array } \right .   , \ \delta > di^{\ast}.\ ] ]    if the loop is not in  reentry , then @xmath52 may take the minimum value @xmath13 since cell 1 must have at least that much rest time after its erp before it can be reactivated . high frequency reentrant waves typically disassociate a slow pacer such as the sinus node . this situation is modeled here by choosing a sufficiently large value of the delay parameter @xmath54 so as to inhibit the pacer from interfering with the reentrant circulation . small values for @xmath54 , which promote pacer interference , may be feasible in certain circumstances , e.g. when there are pathways from the pacer s site to the site of the loop that are protected from the fast reentrant waves by pathologies .    to determine the index value @xmath50 we may generate the sequence @xmath46 independently of @xmath47 and for each @xmath43 mark the values @xmath50 that satisfy ( [ kn ] ) . this process is carried out internally during each simulation run ; an analytical relationship is generally not easy to find . however , if @xmath55 is constant then such a relationship is easy to find ; in this case we have @xmath56 with @xmath57 i.e. , the least integer that is greater than or equal to the ratio @xmath58 this simple relationship makes explicit the discontinuity that results from phase resetting in our model . we distinguish between two primary dynamic modes for the loop - pacer system : the _ reentry mode _ where an action potential circulates in the loop by itself and the _ paced mode _ where the action potential in the loop comes from the pacer . the system s mode in each beat changes on the basis of threshold relations to be defined below . the modes are defined by systems of partial difference equations that track the circulating pulse in both space and time . introductory material on ordinary and partial difference equations can be found in [ 9 , 17 , 31 , 40 , 51 ] . technically , the dynamical system consisting of the loop , the pacer and the associated thresholds is a polymodal structure in the sense of [ 51 ] . let @xmath59 be the di of cell @xmath6 in beat @xmath60 define the _ firing indicator _ function as @xmath61{ll}% 1 , & \text{if } di_{i , n-1}>di^{\ast}\text { and } \phi_{i-1,n}=1\\ 0 , & \text{if } di_{i , n-1}\leq di^{\ast}\text { or } \phi_{i-1,n}=0\text { } % \end{array } \right .   , \ i=2,\ldots , m.\end{aligned}\ ] ]    this quantity specifies conditions under which the @xmath6-th cell fires an action potential in beat @xmath43 ; these conditions simply require that ( a ) the preceding cell @xmath62 fired and ( b ) the di of cell @xmath6 in the preceding beat @xmath63 exceeded the minimum value @xmath13 as is required for propagation . we assume that cell 1 fires in every beat whether by a reentrant pulse or by a pulse from the pacer . let @xmath64 denote the action potential duration of cell @xmath6 in beat @xmath43 so that if the @xmath6-th cell does not fire in beat @xmath43 then @xmath65 ( otherwise , @xmath66 ) . the dynamic _ activation duration _ is defined as @xmath67 where @xmath68 is the apd restitution function for cell @xmath6 . note that @xmath69 for cell 1 in every beat . in the following sections ( except in case - study 4 ) we assume that the loop is apd homogeneous so that @xmath70 for all @xmath71 where @xmath72 is given by ( [ apd - shd ] ) .      in this mode the pacer determines the di values @xmath59 rather than a reentrant pulse in the primary loop ; the circulation is assumed unidirectional owing to an active ub . we use the following system of @xmath5 equations :    [ udc11]@xmath73    here @xmath74 is the ct restitution function for cell @xmath75 the above system of equations determines the di values using the first pacer beat after time instance @xmath47 ( beat @xmath43 in progress ) . figure 4 illustrates equations ( [ udc4 ] ) and ( [ re2 ] ) . the first five terms in ( [ udc4 ] ) simply express in mathematical terms what we might state less precisely using the english language . they are the same for tissue fibers as they are for loops ; see , e.g. [ 68 ] . the last two terms in the definition of @xmath59 in ( [ udc4 ] ) allow for proper adjustment of the di value ( without double counting ) when cell @xmath6 does not fire in beat @xmath43 ( see comments on the reentry mode below ) . in typical loop models in prior literature all cells fire so these two terms would drop out .     cells ; a is apd , d is di and ct is the conduction time . the sizes of a , di and ct are selected arbitrarily for clarity of illustration.,width=336 ]      in this mode the reentrant loop drives the ventricles ectopically . thus @xmath76 is the difference between the apd of cell @xmath7 and the total conduction time for unidirectional circulation once around the ring , i.e.    [ re0]@xmath77    it is not hard to show that the system of @xmath5 equations ( [ re0 ] ) is equivalent to the coupled map lattice equations in [ 51 ] when @xmath78 for all @xmath6 and @xmath60    equations ( [ re0 ] ) are not valid if some of the di  values fall below the threshold @xmath13 or if a stray pacer pulse gets into the loop ( i.e. if the right hand side of ( [ re1 ] ) exceeds the right hand side of ( [ udc1 ] ) ) . the following threshold relations ensure a consistent scenario . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ tc : the _ circulation threshold _ ( or _ head - tail _ or _ conduction block _ _ threshold _ ) _ _ is based on the following set of inequalities : @xmath79    they ensure that the di values @xmath59 in equations ( [ re0 ] ) exceed @xmath80 as is necessary for the generation of action potentials . if any of the above inequalities fails in a cell @xmath6 then a wavefront reaches its own tail in that particular cell and circulation stops until the arrival of the next pacer pulse . in this case we say that tc fails in cell @xmath75 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ tp : the _ pacer _ _ threshold _ ( or _ phase resetting _ _ threshold _ ) is defined by the following inequality:@xmath81    this inequality ensures that the reentrant pulse ( rather than a stray one from the pacer ) reactivates cell 1 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    * mode switching criteria*. _ equations ( [ re0 ] ) are used if tp holds and also tc holds in cell 1 . if for some cell _ @xmath82 _ tc fails , then _ ( [ re2 ] ) _ can still be used to find the di values for beat n , but we switch to equations ( [ udc11 ] ) for beat _ @xmath83__. if tc fails for _ _ @xmath84 _ or if tp fails then we use ( [ udc11 ] ) for beat n. _      in the absence of unidirectional block the pacer sends two pulses in the loop with cell 1 activating both cell 2 and cell @xmath5 . these pulses propagate in opposite directions to eventually annihilate each other in some cell in the loop . transitions into and out of this bidirectional circulation mode are governed by tc , tp and a ub threshold ( tub ) ; see section 4 below . we assume for simplicity that tub never fails so that the bdc mode need not be discussed in this paper . in this section we present the results of numerical simulations of our threshold model along with some analytical observations . due to the large number of different types of behavior that can occur by changing one or more of the many variables involved , providing a comprehensive list of essentially different patterns is not feasible here . we therefore present our main results in the form of a few model case - studies and for additional results we refer the interested reader to our web site : _ www.people.vcu.edu / hsedagha / sitr_.    each case - study below consists of one or more runs ,  i.e. sets of iterations of the propagation equations . each run is based on a fixed set of parameter values and we call each iteration in a given run a beat . for each run we need to specify @xmath5 initial di values @xmath85 . this specification of di  values is a technical representation of the occurrence of a premature stimulation in our model . computational accuracy is not a major issue in the qualitative studies in this paper . however , calculating the values of the various nonlinear functions ( e.g. the exponentials in the apd and cv  restitution functions ) generate round - off errors that can accumulate in certain cases ( e.g. runs in case - study 4 ) . therefore , results obtained by different programs , codes or levels of precision ( single or double ) may be different from some of those that we have presented , although the qualitative features are usually retained . the following conventions facilitate the presentation of results in this section :    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ for a loop consisting of @xmath5 cells if a reentrant pulse is blocked at cell @xmath86 in a particular beat , then we define the _ reentry value _ of that beat as the ratio @xmath87 this ratio will be abbreviated as @xmath88.$ ]    if a reentrant pulse completes a turn around the loop then the reentry value is 1 . if a reentrant pulse is blocked in cell @xmath86 after @xmath89 complete turns around the loop then the reentry value is @xmath90.$ ]    if a beat occurs in the paced mode ( i.e. one of the thresholds tc or tp fails ) then we assign it the value @xmath91 thus @xmath89 consecutive paced beats appear as @xmath92 . if the pattern is locked in the reentry mode , we use the symbol @xmath93 if it is locked in the paced mode , we use @xmath94 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    adding up consecutive reentry mode beats and consecutive paced mode beats gives a sequence of numbers consisting of integers whose sign tell us the mode of system in various sets of beats . examples of mode sequences and more details on them are found in the case - studies below . we call the loop - pacer system _ bistable _ if there are two or more coexisting stable di configurations or state vectors , each of which can be reached from a particular region of the @xmath5-dimensional state space . if there are more than two distinct , coexisting stable states then the system is _ multistable_.    bistability may be attributed to the non - concavity of the apd restitution function [ 50 ] . the apd function ( [ apd - shd ] ) whose graph has bumps and twists is also non - concave . proper ct restitution parameters are required to realize bistability ; i.e. the ct and apd restitution parameters must be properly matched . because the length @xmath4 of the loop affects the conduction time through it , bistability may emerge or fade as @xmath4 is changed with other ct parameters fixed [ 50 ] . a bistable regime can affect sitr  patterns by causing the violation of the circulation threshold tc without any changes in the apd or ct parameters , or any changes in the pacing rate . this situation occurs if the bistable regime satisfies the following _ threshold bistability _ condition :    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ t - bistability _ : there are two distinct stable states in the reentry mode : in one state the di values cross @xmath13 and cause the failure of the circulation threshold tc at some cell within the loop , but in the other state the di values are always greater than @xmath13 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    an example of t - bistability is shown in figure 7 below ; other examples of t - bistability occur in the next section ( case - study 1 ) . the @xmath5-dimensional state space of a t - bistable system is partitioned into two regions or basins of attraction . as a sitr pattern evolves with the motion of the di state vector , mode changes occur if the state vector crosses over into the basin of attraction of a different stable regime .      in this section we establish that t - bistability lends a measure of unpredictability to sitr  patterns that is not due to phase resetting disruptions by the pacer ( i.e. tp  failures ) . for instance , comparing the mode sequences in runs 1.2 , 1.4 and 1.6 below we see that the initial few bursts of fast , reentrant beats provide no obvious clues about the eventual modes that the patterns lock into . consider a homogeneous loop with the apd  and ct restitutions given by ( [ apd - shd ] ) and ( [ ct ] ) . we also set the length @xmath4 and other parameters as :    [ c]|c|c|c|c|c|@xmath4 & @xmath5 & @xmath13 & @xmath54 & @xmath95 + 12.5 cm & 125 & 15.3 ms & 120 ms & 320 ms +    where @xmath95 is the fixed cycle length of the pacer ; it is chosen small in this case - study to facilitate re - initiations of reentry . the value of @xmath54 is set high enough to inhibit pacer interference ( tp never fails ) . the following table summarizes the results of simulations ; it is followed by a series of observations and elaborations . [ c]|c|c|c|run no . & @xmath96 & sitr pattern mode sequence + 1.1 & 200 ms & @xmath97 + 1.2 & 100 ms & @xmath98,-2,\infty\right\ }   $ ] + 1.3 & 80 ms & @xmath99 + 1.4 & 70 ms & @xmath100,-\infty\right\ }   $ ] + 1.5 & 60 ms & @xmath101 + 1.6 & 50 ms & @xmath102,-2,\infty\right\ }   $ ] +    \(i ) in run 1.1 the sitr pattern immediately locks into a regular form of 2 paced beats followed by 2 reentrant beats . in runs 1.2 , 1.3 , 1.5 and 1.6 the sitr  pattern locks into the reentry mode .  in run 1.4 the paced mode is locked into . note that the only thing that is changing from one run to the next is the initial di values . \(ii ) in runs 1.2 , 1.3 , 1.5 and 1.6 the di values in the sustained or locked reentry modes converge to a fixed number and the eventual values of the fundamental parameters are    [ c]|c|c|c|di & apd & cycle length cl = apd+di + 59.4 ms & 173.6 ms & 233 ms +    the fixed , limiting di value 59.4 represents a stable _ convergent state_. this equilibrium di is locally stable ( attracting ) because using the restitution functions ( [ apd - shd ] ) and ( [ ct ] ) we compute [ 4 , 26]@xmath103    \(iii ) in runs 1.2 , 1.4 and 1.6 all the spontaneously terminated reentry bursts have oscillating ( quasiperiodic ) di . thus apd  and cl also oscillate similarly . the oscillatory state is stable and eventually crosses @xmath104 ms . hence there is t - bistability ( the oscillatory one and the convergent one ) which is responsible for mode changes in each run . for example , in run 1.2 the 2nd initiation of reentry ( the 32-beat burst ) starts with the di  state - vector in the basin of attraction of an oscillatory state which expands enough for the di  values to cross @xmath13 ( tc fails ) ; see figure 5 . this terminates the second reentry burst at cell 73 in the 41st beat . this is the same type of termination mechanism as that in [ 19 , 20 , 26 ] ; this type of oscillations in di are also seen in [ 56 ] where they are related to the t wave alternans in ecg recordings ; also see [ 35 ] in this regard . with the 3rd initiation of reentry , the di state - vector falls in the basin of attraction of the convergent state which then effectively traps  the di trajectory and locks the sitr pattern into the reentry mode ( here then , simplicity of behavior is not a good thing ) . [ run1 - 2 ]    . amplification of oscillations in di leads to termination of reentry in beat 41 , cell 73 because @xmath105 . with beat 44 reentry is re - initiated and the system locks into the stable convergent state ( after some transient oscillations).,width=432 ]    \(iv ) examination of the simulation data for run 1.4 shows a recurrent failing of tc in cell 1 after the last burst of 23 reentrant beats . for all sufficiently large values of @xmath43 ,    [ c]|c|c|c|@xmath76 & @xmath106 & @xmath107 + 96.6 & 223.4 & 320 +    the number 96.6 is the equilibrium di for the paced mode with pacing period @xmath108 . the paced - mode equilibrium happens to be stable because in the absence of reentry the paced mode is governed by the one - dimensional difference equation@xmath109 with @xmath110 i.e. @xmath111 why the transition to the stable paced equilibrium occurs in this run and not others is unclear . figure 6 shows the changes in di values in this run . [ run1 - 4 ]    . the four interruptions of reentry due to conduction block ( tc fails ) are seen as brief jumps to high di . the last interruption terminates reentry and locks the system in the paced mode.,width=432 ]    \(iv ) in run 1.5 , the given initial di values put the state vector in the basin of attraction of the convergent state of the system . this is evidently not the case in run 1.6 with even shorter di . in run 1.6 , as in run 1.4 , the initial mode is oscillating - di reentry . we now consider some longer loops than in case - study 1 for comparison . in these loops bistability is present but does not satisfy the t - bistability condition . a smaller variety of different sitr  patterns are obtained ( with all other system parameters having the same values as in case - study 1 ) . we consider two different lengths :    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ @xmath112 cm ( @xmath113 ) where stable oscillating states exist all of which fail tc , but the convergent mode is absent since the di fixed point of about @xmath114 ms is unstable [ 4 , 26]@xmath115    @xmath116 cm ( @xmath117 ) where all of the stable oscillating states ( there is at least one ) satisfy tc . the convergent mode is again absent for the same reason as above:@xmath118 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    to save space let us abbreviate repetitions or locked mode patterns with bars:@xmath119    the following table summarizes the results of simulations ; it is followed by a series of observations and elaborations . [ c]|c|c|c|c|run no . & @xmath85 & @xmath112 cm & @xmath116 cm + 2.1 & 200 ms & @xmath120 & @xmath121 + 2.2 & 100 ms & @xmath122,\overline{-2,2}\right\ }   $ ] & @xmath123 + 2.3 & 80 ms & @xmath124 & @xmath125 + 2.4 & 70 ms & @xmath126,-2,17[79],\overline{-2,5[79]}\right\ }   $ ] & @xmath127 + 2.5 & 60 ms & @xmath128 & @xmath129 + 2.6 & 50 ms & @xmath130,\overline{-2,2}\right\ }   $ ] & @xmath131 +    \(i ) the sitr patterns for the 14 cm loop are expected to lock into the reentry mode since the stable states do not fail tc to cause terminations . the exception is run 2.6 where possibly owing to the greater length of the loop , a pulse from the pacer has time to activate cell 1 and block the premature simulation from reentering the loop ( threshold tp fails ) . it is not clear why reentry does not re - initiate as in run 2.5 . \(ii ) in the case of the 13 cm loop the repeated terminations can be attributed to the fact that the stable states all fail tc ; thus the patterns can not lock into the reentry mode . \(iii ) for both of these loops , the eventual form of the sitr  pattern is more predictable than the t - bistable loop of case - study 1 . on the other hand , the transient bursts of fast beats still seem difficult to predict . * run 3.1 * ( _ more t - bistability _ ) . this run presents a complex sitr pattern that locks into a long cycle of several initiations and terminations . make the following changes to parameters in run 1.2 :    [ c]|c|c|@xmath34 & @xmath95 + 1.3 & 315 ms +    note that the increase in @xmath34 from 1 to 1.3 in this run causes a slight elevation of the ct restitution curve ( i.e. , a conduction slow - down ) at low di values ; see figure 3 . \(i ) in this run , after a transient period of spontaneous initiations and terminations , a long 62-beat pattern emerges that is repeated ; i.e. the sitr  pattern locks into a 62-beat cycle containing several bursts of fast reentrant beats . the mode sequence for this run is as follows , with the pattern between starred numbers repeating : @xmath132,-1,[35],-2,4,-2,7[24],-1,[18],-2,8[18],^{\ast}% -1,[12],-2,7[12],-1,[12],-2,\\ &   10[49],-2,5[49],-2,4,-2,7[22],-1,[19],-2,7[19],^{\ast}% -1,[12],-2,7[12],-1,[12],-2,\ldots\}\end{aligned}\ ] ]    \(ii ) the di equilibrium of approximately @xmath133 ms is unstable in this run since@xmath134    this instability is caused by the greater value of @xmath34 and results in the absence of the convergent state . nevertheless , there is t - bistability where one of the stable oscillating states does not fail tc but another oscillating state does . in the first oscillatory state , di values come very close to @xmath13 ; see figure 7 . this proximity increases the sensitivity to transient effects of mode changes and makes locking into the reentry mode an improbable event . [ run3 - 1 ]     for all @xmath6 , the graph crosses the dashed horizontal line @xmath135 ; for the sake of illustrating the quasiperiodic nature of oscillations in reentry mode , the graph ignores tc since reentry would terminate with the first such crossing . ( b ) @xmath136 for all @xmath6 . reentry is sustained with quasi - periodic variation of di.,width=528 ]    * run 3.2 * ( _ pacer action _ ) . in case - studies 1 and 2 we see that the convergent reentry pattern is sticky  i.e. once this pattern is attained , reentry does not self - terminate by the circulation mechanism in the loop . however , pacer interaction with the loop can lead to termination in some cases by changing the convergent pattern to a terminating oscillatory one in a t - bistable case . all parameters are as in run 1.5 ( hence there is t - bistability ) except that now :    [ c]|c|c|@xmath54 & @xmath95 + 45 ms & 800 ms +    we have set the pacing period at the nominal sinus length to eliminate the effects of fast pacing . the smaller value of @xmath54 promotes interference by the pacer . \(i ) the mode sequence for this run , which may be compared with run 1.5 , is:@xmath137 \(ii ) reentry is initiated with beat 1 by a premature stimulation , and the di values begin to approach the limiting value 59.4 ms as in run 1.5 . however , tp fails in beat 41 ; a stray pulse from the pacer enters the loop and changes the reentry di pattern from the convergent type to an oscillating one ; see figure 8 . the quantities @xmath138 , @xmath139 in figure 8 are the right hand sides of equations ( [ udc1 ] ) and ( [ re1 ] ) , respectively . [ run4 - 1 ]     ( see text ) . the negative value in beat 41 ( indicated by the arrow ) corresponds to termination of reentry due to failure of tp . ( b ) di values in beats 30 through 60 ; note the phase resetting that occurs in beat 41 and changes the reentrant circulation pattern from convergent to oscillatory in this t - bistable case.,width=528 ]    the oscillating di pattern crosses the @xmath13 threshold ( tc fails ) in beat 55 and reentry terminates in this beat .      when the pacing period is long ( say , within the nominal sinus range of 800@xmath140 ms ) and the loop is homogeneous , _ re - initiations _ of reentry do not occur with the above ct and apd parameters in the absence of premature stimulations or ectopic sources . the low ct ( or high cv ) in the loop blocks reentry when pacing is slow . nevertheless , the rather common occurrences of ventricular tachycardia at low pacing rates in patients [ 58 ] motivate us to study this important special case in the context of a heterogeneous loop with a variable - rate pacer . suppose that the loop consists of two patches of cells . patch 1 consists of cells 1 through @xmath86 and patch 2 contains cells @xmath141 and beyond where of course , @xmath86 is a positive integer less than @xmath8 the total number of cells . the cells in patch 1 may have different apd and/or ct restitution parameters than those in patch 2 . for simulations , we also assume that the pacer has a variable - rate within the aforementioned nominal sinus range . assume that the pacer s oscillation period varies in a sinusoidal fashion:@xmath142    this beat pattern may represent an idealization ( without random effects ) of a single cycle of a complex , multi - pattern stretch of pacer beats . more complex variations may be considered in later studies if needed . the number @xmath143 is the period of a full cycle of variable - rate oscillations , which we refer to as a _ full pacing cycle _ ( fpc ) . the halfway point of the fpc , or the bottom of the fpc well  occurs at @xmath144 at this point , the pacer oscillates with minimum period ( fastest beat rate ) . on the other hand , when @xmath145 or @xmath146 the pacer has its largest oscillation period ( slowest beat rate ) . the number @xmath147 is the fpc amplitude  since it modulates the pacer s oscillation period within the fpc . if @xmath148 then the period is a fixed @xmath95 as in the preceding case - studies . * run 4.1*. we fix the apd and ct parameters for patch 2 to be the same as those used in case - study 1 . patch 1 apd and ct have the same parameters as patch 2 except for the following :    [ c]|c|c|c|@xmath15 & @xmath16 & @xmath33 + 300 ms & 170 ms & 0.05 cm / ms +    thus apd is reduced ( by a shift in the restitution function ) in patch 1 ; also the cells in this patch conduct more slowly with a lower value of @xmath33 ( the maximum cv ) . additional loop and pacer parameters are set as follows :    [ c]|c|c|c|c|c|c|c|@xmath4 & @xmath95 & @xmath147 & @xmath143 & @xmath86 & @xmath54 & @xmath149 + 13.5 cm & 800 ms & 350 ms & 1000 & 50 & 140 ms & 600 ms +    the integer @xmath86 gives the number of cells in patch 1 and the three pacing parameters @xmath150 and @xmath147 define a variable pacing protocol with fpc period of 1000 beats and amplitude 350 about a fixed , nominal sinus trend of 800 ms . \(i ) the mode sequence of the sitr pattern obtained is@xmath151,-1,[51],-1,[51],-1,[51],-1,[51],-2,\infty\}.\ ] ]    \(ii ) the mode sequence exhibits a pattern of 5 incomplete reentry initiations followed by a pacer beat ( or 2 in one case ) . each incomplete reentry start which is blocked at the patch junction ( cell 51 ) corresponds to a short cycle length ( about 224 ms ) compared to the pacer s period of approximately 460 ms near the middle of the fpc . hence the pattern of 5 alternating reentry / pacer beats resembles a series of premature ventricular contractions ( pvc ) in a bigeminal form before reentry finally takes hold for good . \(iii ) we note that if @xmath152 or fewer , then no reentry is initiated in this run . on the other hand , larger @xmath86 than 50 sustain reentry more quickly by increasing the percentage of slow cells in the loop . * run 4.2*. in the preceding run the pacer did not interfere with the reentrant circulation ( tp did not fail ) because @xmath153 ms was large enough to block the pacer . we now show that decreasing @xmath54 to promote pacer interactions may cause the eventual termination of reentry in the preceding run . we make only one change in this run relative to run 4.1 by setting @xmath154 ms . \(i ) the mode sequence of the resulting sitr pattern is as follows:@xmath155,-1,\ldots,[51],-1}% } , [ 51],-2,1,-1,11,-3,1,-1,4,-1,6,-3,1,-1,\\ &   4,-1,6,-3,1,-1,11,-1,6,-3,\underset{\text{38 times}}{\underbrace { [ 51],-1,\ldots,[51],-1}},[51],-374,\ldots\}\end{aligned}\ ] ]    after several pvc s and reentrant bursts of different durations , the pattern locks into the paced mode until reentry is initiated in the next fpc . this mode sequence is sensitive to computer round - off errors ; however , the varied nature of mode changes persists . figure 9 gives a graphical representation of mode changes for this run . \(ii ) the first deviation from the mode sequence of run 4.1 occurs with a tp failure that interrupts reentry after just one full cycle . some of the @xmath156 entries in the mode sequence above ( mostly the ones _ not _ associated with conduction blocks in cell 51 ) indicate failures of tp also . in this section we present additional supporting details about the model . experimental measurements and other studies reveal the difficulty of fitting a single curve to the data , particularly at the lower end of the di scale . to address this and related issues , refinements have been considered in the forms of memory and latency . beat to beat memory has been studied in [ 13 , 19 , 21 , 23 , 28 , 46 , 56 ] and has been modeled in a number of ways . in many cases , modeling memory can be accomplished by using di  values from several earlier stimulations . latency ( roughly a brief period between the arrival of a pulse and the firing of the cell ) has also been expressed as a restitution function of di [ 12 , 13 ] . if this effect is added to apd we obtain a composite restitution function that may no longer be monotonic for small di values . for simplicity , we ignore both memory and latency in this paper ; however , they can ( and need to ) be included in the model for greater refinement in future studies .      the apd restitution function may be measured experimentally using two different pacing protocols : dynamic or standard ( s1-s2 ) [ 29 , 32 ] . these protocols generally produce apd curves that are shifted or have different slopes . for a discussion of this issue regarding the apd restitution that we use in this paper see [ 33 ] . the fit in ( [ apd - shd ] ) is to the dynamic restitution data in [ 33 ] . the apd values ( and thus the restitution function ) may be affected by the steady fast pacing of the ventricles [ 34 ] or of the atria [ 39 ] . the implied shift in the apd restitution can have a significant effect on the sitr  patterns in certain borderline cases . for simplicity we do not consider such effects in this paper ; however , a suitable modification of the apd restitution function that incorporates explicit time dependence can accommodate dynamically - induced shifts .      experimental evidence is offered in [ 53 ] that the apd restitution curve may be non - monotonic for small di values . working with a apd restitution curve having a single local minimum at low di , the same study demonstrates the occurrence of spatial bifurcations along the length of a periodically paced fiber of cardiac tissue . although not involving a loop structure , the work in [ 53 ] uses restitution functions in a discrete model . a localized form of non - monotonicity ( like a bump or dent ) in apd may occur at any di value and has interesting consequences for additional complexity in reentry mode . in [ 48 ] where a continuous model of the loop as a cable is used , this type of non - monotonicity is shown to be responsible for the appearance of spatio - temporal chaos in apd and other key variables over the length of the loop . the occurrence of spatio - temporal chaos on the loop may raise the level of unpredictability in sitr if the amplitudes of oscillations in di are large enough to let the wavefront reach its tail ( thus causing a threshold failure ) . various anatomical and functional mechanisms for ub in a loop are discussed in the model study [ 49 ] . these mechanisms include a properly timed premature stimulus delivered to a suitable location in the loop , inhomogeneities in the degree of cellular uncoupling and gap junction resistance , in membrane excitability and in fiber cross sectional area . the discussion in [ 49 ] points to time , space and voltage windows of vulnerability  for the occurrence of unidirectional block and initiation of reentry . of these , the time window is seen to be the easiest to measure and its value is related to the space and voltage windows via standard mathematical formulas . each cell requires a certain amount of depolarizing current to fire an action potential . this _ activation threshold _ can be represented by a restitution function [ 27 ] or equivalently , the strength - interval curve [ 13 ] shifted to the left so that the origin represents the end of the erp . in figure 10 this curve is shown as a decreasing restitution function @xmath157 .    in principle , at any time following the end of the erp a sufficiently large current will be able to elicit an action potential . however , in the absence of external electrical shocks , there is a limited amount of current @xmath158 available to depolarize the next cell . if a restitution of @xmath158 is given then the intersection point of @xmath157 and @xmath158 is the @xmath159 threshold or cut - off value @xmath13 in the sense that if @xmath160 then action potential is fired and propagation occurs . the time interval @xmath13 is similar in nature to the chronaxie  [ 47 ] although the definition of @xmath13 requires another restitution curve in addition to @xmath161 a monotonically increasing restitution function for @xmath158 might be appropriate ; see figure 10 . [ currents ]     and @xmath162 . in this figure , @xmath163 ( see text).,width=288 ]    for our purposes here we do not need explicit formulas for either @xmath157 or @xmath164 the value @xmath13 may be chosen arbitrarily or estimated experimentally without reference to its potential sources . values as high as 160 ms were used in [ 26 ] . in human cases the value of @xmath13 in vivo is likely to be in a much smaller range . we use the conservatively small value of about 15 ms in this study . an analog of the time window in [ 49 ] can be defined here using the restitution functions @xmath157 and @xmath158 . we define two sets of currents restitution functions for cell 1 corresponding to the two possible directions in the loop : @xmath165 , @xmath166 and @xmath167 , @xmath168 for the left  and the right  directions . the asymmetry creates a _ ub window _ in cell 1 as follows : let @xmath169 and @xmath170 with at least one of these inequalities strict . let @xmath13 be the ( unique ) intersection point of @xmath165 and @xmath167 and let @xmath162 to be the intersection point of @xmath166 and @xmath171 then @xmath172 and the interval between these two values is where the ub occurs . for if @xmath76 represents the di of cell 1 in a cycle or beat @xmath173 then the conditions @xmath174 imply that propagation is possible in one direction because the threshold marked by @xmath13 is crossed but it is inhibited in the other direction where the @xmath162 threshold is not crossed . we emphasize that the unidirectional block as defined here is functional and may not occur even if the ub window exists ( for anatomical or functional reasons ) . if the number @xmath76 does not enter the the ub window ( i.e. the interval from @xmath13 to @xmath162 ) in a particular beat @xmath43 then the ub  does not materialize . this option adds another mode to the loop - pacer system , namely the bdc mode mentioned previously . in addition to tc and tp , the following threshold is associated wtih the bdc mode :    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ tub : @xmath175 with beat @xmath63 in the paced  mode . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    this inequality defines the _ ub threshold _ to ensure that cell 1 conducts unidirectionally . if beat @xmath63 is in reentry mode then conduction in beat @xmath43 is automatically unidirectional whether tub holds or not , because the pulse is transmitted from a still active cell @xmath12 to limit the variety of cases to consider , we assume in this paper that @xmath162 is so large that the unidirectional block is always present in cell 1 ( permanent ub ) so tub always holds . in the preceding sections , we developed an interactive loop - pacer threshold model that is a part of a familiar cardiac anomaly . using experimental data on restitution parameters from the existing literature for action potential duration and conduction time , we obtained various conditions and parameter values that would cause irregular patterns of spontaneous initiations and terminations of reentry . a major cause of unpredictable behavior is threshold bistability where one stable regime causes a failure of the circulation threshold tc and another does not . in this situation the initiation and termination of reentry depends on the location of the di state vector in the @xmath5-dimensional state space , where the nonlinearity in propagation equations makes it difficult to track the state vector . thus , it is quite difficult to forecast the long - term behavior of an evolving sitr  pattern for a t - bistable loop . another potential source of unpredictability is the interference by the pacer . if reentry occurs , then the pacer and the reentrant circuit are thrown out of phase so a phase resetting pacer pulse may unexpectedly end the reentrant circulation . a number of prior studies of a different nature from ours also find complex sitr patterns in simple settings , such as in sheets [ 3 , 25 ] , loops [ 44 ] and aggregates [ 35 ] of animal cardiac tissue . in [ 3 ] and [ 25 ] complex patterns in monolayers of growing cell cultures are linked to changing densities and to local inhomogeneities , respectively . in [ 35 ] bursting beat patterns are studied in cell aggregates under external pacing . closer to the subject matter of this paper , in [ 44 ] a ring - shaped lab preparation is considered with two localized pacemakers . threshold relations for handling the pacers in this loop appear in the mathematical discussion . in a different class of studies , namely , the modulated parasystole , a permanent ectopic rhythm coexists with the sinus or another competing rhythm . in such relatively simple systems , complex beat patterns arise that are highly sensitive to parameter changes [ 23 , 28 , 43 ] . like the tp threshold in our model , a mathematical relation with a discontinuity is responsible for the complexity of temporal patterns in parasystole ; see [ 23 , 43 ] . we omitted many features from our discussion here that one might include in an enhanced version of this model . for example , ub may be treated functionally by refining the ub time window and considering bidirectional propagation in the loop , as noted above . other enhancements to this model might include adding memory and latency to the apd restitution function . the inclusion of memory is especially important in clarifying the dynamic significance of delayed responses ( not to mention improved fits to the apd restitution data ) . additional directions for exploring the dynamics of the loop - pacer system include consideration of multiple exit / entry points into the loop that can affect reentry , a systematic approach to heterogeneity and also some theory to better model the time lag @xmath52 that influences the loop - pacer interactions . a somewhat different direction that may be explored would be to incorporate a programmed shock routine to artificially end long bursts of fast , reentrant beats .    with some of the above enhancements the sitr patterns will likely become more complex and less predictable , though it is also possible that some of these enhancements ( e.g. memory ) can have a moderating effect . understanding the precise mechanisms that generate such complex sitr patterns may in turn offer new insights into the nature and causes of long - term temporal patterns of tachyarrhythmia occurrences .      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a two - component model is developed consisting of a discrete loop of cardiac cells that circulates action potentials as well as a pacing mechanism . 
 physiological properties of cells such as restitutions of refractoriness and of conduction velocity are given via experimentally measured functions . 
 the dynamics of circulating pulses and their interactions with the pacer are regulated by two threshold relations . 
 patterns of spontaneous initiations and terminations of reentry ( sitr ) generated by this system are studied through numerical simulations and analytical observations . 
 these patterns can be regular or irregular ; causes of irregularities are identified as the threshold bistability of reentrant circulation ( t - bistability ) and in some cases , also phase - resetting interactions with the pacer .     * complex patterns of spontaneous initiation and termination *    * of reentrant circulation in a loop of cardiac tissue *    h. sedaghat@xmath0 , m.a . 
 wood@xmath1 , j.w . 
 cain@xmath0 , c.k . 
 cheng@xmath2    c.m . 
 baumgarten@xmath3 , d.m . 
 chan@xmath0    @xmath0department of mathematics and the center for the study of biological complexity ;    @xmath1department of internal medicine - cardiology and the pauley heart center ;    @xmath2department of computer science and the center for the study of biological complexity ;    @xmath3department of physiology and the pauley heart center ;    virginia commonwealth university , richmond , virginia , 23284 - 2014 , usa    * keywords *  reentry ; loop ; pacer ; thresholds ; bistability ; difference equations