sudden cardiac death is a major cause of mortality in the industrialized world , and ventricular fibrillation ( vf ) is the leading cause of sudden cardiac death .
an electrical wavebreak in the heart tissue may lead to vf . according to the so - called restitution hypothesis , rate - dependent alteration of action potential duration ( apd ) and refractoriness ( r )
the changes in apd that accompany the changes in rate , reflect the dependence of apd on the preceding diastolic interval ( di ) , a relationship characterized by the electrical restitution curve .
all of these mean that the modeling and simulating of restitution property in the cardiac cell is very important .
cellular automata ( ca ) may be considered as a method for modeling discrete dynamic systems .
a ca consists of a discrete system of lattice sites ( cells ) having various initial values .
these cells evolve in discrete time steps , as each cell assumes a new state based on the rules , that is , the states of its local neighborhood and a finite number of previous time steps . the neighborhood is described by specifying the set of cells that are the neighbors of a given cell .
( a ) von neumann ; ( b ) moore a ca lattice may be 1-d or multidimensional . for 2-d ca ,
if a cell which is still in the refractory state is re - stimulated by an adequate large stimulus ( s2 in figure 2 ) , the generated apd ; apdn+1 will be shorter than that in the normal case , apdn , that is , when the cell is stimulated after complete refractory .
illustration of the restitution property , apd and di the action potential duration restitution property refers to the relationship between apdn+1 and the previous diastolic interval din , as shown in figure 2 .
di is defined by the duration between the end of the cell 's re - polarization and the next applied stimulation [ figure 2 ] .
the plot of apdn+1 versus din is called the restitution curve [ figure 3 ] .
a sample restitution curve restitution curve may be explained as a function f : apdn+1 = f(din )   ( 1 ) experimental measurements and observations confirmed that the restitution curve may be fitted by a sigmoid function like : there are different methods for experimental plotting of the restitution curve , like the s1-s2 standard and the dynamical restitution method .
action potential duration is usually explained as apd95 or apd90 , apd95 is defined as the time distance between the s2 stimulus ( beginning of ap ) and 95% of the cell 's full repolarization . in this
regarding figure 2 , di is defined as follows : di=(s1-s2 interval)-(apd95 caused by last s1 )   ( 3 )
figure 1 illustrates two types of neighborhoods that are generally used in literature . in this article a model based on ca
this article is organized as follows : section 2 reviews the restitution property . in section 3
the proposed automata model is introduced , as also the method of implementation of the restitution property in the proposed model .
two state variables are assigned to each cell ; one for the action potential variable ( apmn ) and the other for the refractory variable ( rmn ) , where m , n , and t denote the column number , row number , and the time step , respectively .
the variable smn stands for the stimulus applied to the cell , which is in the coordinate ( m , n ) in time t. table 1 summarizes the description of the other parameters .
definition of parameters used in the proposed model the transition rule ( automata rule ) in the proposed model is defined as follows : 
 i.if smn is greater than the excitation threshold ;  and apmn = rmn = 0 , the cell in position mn ; cellmn will be excited in the next time step ; t+1 and apmn = apup and rmn=0ii.if smn
<  and rmn = apmn = 0 , the cell will not be stimulated and stays in its previous ( rest ) stateiii.if rmn , + apmn  0 , the cellmn will take the path in the state space , as shown in figure 4 .
notice that rmn starts decreasing when the apmn reaches its maximum value after a complete evolution in the state space the rmn and apmn will return to their rest states again 
  if smn is greater than the excitation threshold ;  and apmn = rmn = 0 , the cell in position mn ; cellmn will be excited in the next time step ; t+1 and apmn = apup and rmn=0 if smn <  and rmn = apmn = 0 , the cell will not be stimulated and stays in its previous ( rest ) state if rmn , + apmn  0 , the cellmn will take the path in the state space , as shown in figure 4 . notice
that rmn starts decreasing when the apmn reaches its maximum value after a complete evolution in the state space the rmn and apmn will return to their rest states again evolution of the cellmn state variables ; apmn and rmn in the state space .
see the text for transition rules the parameter values in the model are set so that the time course of ap and r in a single cell are as depicted in figure 5 . here
the continued and dashed lines show ap and r , respectively . referring to figure 5 ,
it is clear that the general morphology of action potential and refractory in a heart cell are preserved .
( a ) time course of ap and r as implemented in the model ; ( b ) a real ap time course for the r state variable , five phases ( 0 , 1 , 2 , 3 , 4 ) are assumed . in phase 0
phase 1 stands for r when the cell has just entered the rest state . in this phase
, r continues decreasing and a strong enough stimulus can re - excite the cell . in phase 4 , r reaches its minimum value ; 0 . here
, the cell is in rest and normal stimulus can excite it and generate an ap with normal duration , that is , a normal apd .
if a cell is excited during phases 2 or 3 , the restitution property will require that the apd is shorter than normal .
the main aim of this research was the simulation of restitution in a single cell , so a lattice of a single cell without any neighbor was considered .
the parameter rdown was considered as a time - dependent parameter , as : in eq .
4 , t denotes time and  is an adjustable coefficient . the model was implemented by matlab
the model parameters were adjusted so that by stimulating a single cell , the outcome of the ap and r morphology was generally similar to a ventricular cell .
the outcome is depicted on the left side of figure 6 . referring to figure 6 ,
ap and r morphology are reserved as well as the restitution property in figure 6 , three stimuli are applied to a cell , at times 0 , 40 , and 70 milliseconds .
the time interval between the first and second stimulus is long enough . in other words ; di1 is normal and the cell has enough time for full recovery , and this leads to a normal apd2 .
the right hand side of figure 6 indicates that the proposed model can simulate the restitution property , that is , the third apd ( apd3 ) is shorter than the first and second one .
after assurance about the installation of the restitution property in the model , its validity should be justified .
if the proposed model is precise , it should simulate each of these curves by the adjustment of its parameters .
three experimental restitution curves in this research the adjustment of the model was done via the regulation of n ,  , and apdmax .
the normalized mean square error ( nmse ) criterion was used for fitness comparison of the restitution curves . at first , the guinea pig restitution curve of figure 7 was tested .
comparison of the guinea pig restitution curve with the simulated curve created by the model .
the proposed model can simulate the guinea pig restitution curve second , as an attempt for the model validity justification , the rabbit 1 restitution curve of figure 7 was examined .
comparison of the rabbit 1 restitution curve with the simulated curve created by the model .
it is clear that the proposed model can simulate the rabbit 1 restitution curve and finally , the rabbit 2 restitution curve of figure 7 was investigated .
the outcome is illustrated in figure 10 . here also , it is clear that the proposed model can simulate the restitution curve of rabbit 2 , when its parameters are adjusted as shown in the table of figure 10 .
comparison of the rabbit 2 restitution curve with the simulated curve created by the model .
a cellular automata - based model was proposed for simulating restitution . for the validation test of the model ,
the restitution curves that were simulated by the model were evaluated against the experimental results of the validated sources . in this research , the parameters  , n , and apdmax were used for model regulation .
the results showed that the proposed model could not only simulate restitution properly , but could also be regulated for simulating restitution curves of different kinds of cardiac cells . on account of the low computational load of the automata - based cell models , the model would be very useful for simulating propagation of electrical waves in the ventricular tissue and investigation of the wave break and fibrillation , which would form the matter of the authors future researches .
he is with the department of biomedical engineering and the research laboratory of biomedical signals and sensors , iran university of science and technology ( i.u.s.t . ) , tehran , iran .
fateme pourhasanzade is a phd student in biomedical engineering department ; iran university of science and technology ( i.u.s.t . ) , tehran , iran .
 ventricular fibrillation is the cause of the most sudden mortalities . 
 restitution is one of the specific properties of ventricular cell . 
 the recent findings have clearly proved the correlation between the slope of restitution curve with ventricular fibrillation . 
 this ; therefore , mandates the modeling of cellular restitution to gain high importance . 
 a cellular automaton is a powerful tool for simulating complex phenomena in a simple language . 
 a cellular automaton is a lattice of cells where the behavior of each cell is determined by the behavior of its neighboring cells as well as the automata rule . in this paper , 
 a simple model is depicted for the simulation of the property of restitution in a single cardiac cell using cellular automata . at first , two state variables ; action potential and recovery are introduced in the automata model . in second , automata rule is determined and then recovery variable is defined in such a way so that the restitution is developed . in order to evaluate the proposed model , the generated restitution curve in our study is compared with the restitution curves from the experimental findings of valid sources . 
 our findings indicate that the presented model is not only capable of simulating restitution in cardiac cell , but also possesses the capability of regulating the restitution curve . 
INTRODUCTION
AIMS
MATERIALS AND METHODS
RESULTS
CONCLUSIONS
BIOGRAPHIES