Locating Order-Disorder Phase Transition in a Cardiac System

To prevent sudden cardiac death, predicting where in the cardiac system an order-disorder phase transition into ventricular fibrillation begins is as important as when it begins. We present a computationally efficient, information-theoretic approach to predicting the locations of the wavebreaks. Such wavebreaks initiate fibrillation in a cardiac system where the order-disorder behavior is controlled by a single driving component, mimicking electrical misfiring from the pulmonary veins or from the Purkinje fibers. Communication analysis between the driving component and each component of the system reveals that channel capacity, mutual information and transfer entropy can locate the wavebreaks. This approach is applicable to interventional therapies to prevent sudden death, and to a wide range of systems to mitigate or prevent imminent phase transitions.

The APD response to the initial stimulus led to DI shorter than the minimal value in the APD restitution curve. This resulted in a 2:1 response.

SI Appendix 2. APD alternans and conduction block in 1-D cable
At BCL=300 msec, one stimulus generated one action potential, and APD was uniform spatially and temporally (APD=185 msec, Figure 2A, SI Movie 1). Shannon entropy was consistently high throughout the cable length ( Figure 2E), which indicates that the cardiac macrostate is homogeneous across the cable. Channel capacity and mutual information were also consistently high. Mutual information was the lowest of the three information metrics but was close to channel capacity, which indicates that the channel is operating close to the maximum capacity. Transfer entropy was small and variable with fluctuations across the cable length ( Figure 2F).
At BCL=220 msec, one stimulus generated one action potential, but APD at the stimulus site was alternating with long (APD=188 msec) and short APDs (APD=101 msec) ( Figure 2B, top row, SI Movie 2). At a short distance away from the stimulus site on the cable ( Figure 2B, second row), the APD alternans improved and APD became uniform temporally. This is called a node behavior since this site serves as a node to link the regions of concordant and discordant alternans 2 . At midway of the cable ( Figure 2B, third row), APD alternans reappeared but π out of phase from the stimulus site (discordant alternans) 3 . At a further distance away from the stimulus site ( Figure 2B, fourth row), the APD alternans improved and APD became uniform temporally again, showing another node behavior. At furthest distance away from the stimulus site ( Figure 2B, bottom row), the APD alternans reappeared in-phase with the stimulus site (concordant alternans) 3 . Shannon entropy remained high and constant throughout the cable length ( Figure 2G). Channel capacity was high near the stimulus site (cable length=0 cm) but steeply declined to near zero between cable length=10 cm and 25 cm, representing the region of discordant APD alternans. Beyond cable length=30 cm, channel capacity improved at >0.8 bits, reflecting concordant APD alternans. Mutual information followed the trend of the channel capacity, with only slightly lower values. Transfer entropy still fluctuated but declined steadily from the stimulus site and reached the minimum value at the beginning of discordant alternans (cable length around 10 cm), indicating little information transfer from the source to this region ( Figure 2H).
At BCL=208 msec, one stimulus generated one action potential with APD alternans at the stimulus site, alternating with long (APD=185 msec) and short APDs (APD=90 msec) ( Figure 2C, top row, SI Movie 3). At a short distance away from the stimulus site on the cable ( Figure 2C, second row), the APD alternans became discordant, and eventually conduction block occurred ( Figure 2C, third row). At midway and further, only one APD out of two stimuli was conducted ( Figure 2C, fourth and bottom rows). There was a tendency for Shannon entropy to decline toward the site of conduction block, where it increased again and remained high throughout the cable length ( Figure 2I). Channel capacity was high near the stimulus site (cable length=0 cm) but steeply decreased to near zero at around cable length=10 cm, representing the region of discordant APD alternans and conduction block. Beyond the site of block, the channel capacity improved only slightly but continued to be low (<0.16 bits) throughout the cable. Mutual information followed the trend of the channel capacity, with only slightly lower values. Transfer entropy declined steadily but with fluctuations from the stimulus site and reached the minimum value at the beginning of discordant alternans and conduction block (cable length around 10 cm)( Figure 2J). Beyond the site of conduction block, transfer entropy increased slightly and remained constant throughout the remaining cable length, indicating that the information of stimuli from the source continues to be transmitted to this region despite a 2:1 conduction.
At BCL=200 msec, BCL was shorter than APD, and thus hit the refractory period of the previous APD ( Figure 2D, SI Movie 4). As a result, only every other stimulus generated one action potential at the stimulus site (2:1 response). Shannon entropy remained high throughout the cable length, indicating that the 2:1 response does not impact the cardiac macrostate ( Figure 2K). Channel capacity dropped to approximately a half of that of the origin (around 0.5 bits) and maintained the same level throughout the cable, reflecting the 2:1 response. Mutual information followed the trend of the channel capacity, with only slightly lower values. Transfer entropy remained close to zero throughout the cable length ( Figure 2L), indicating little information transfer from the source when a 2:1 response is present.  0.025 cm). B. BCL=220 msec (APD alternans). C. BCL=208 msec (conduction block). D. BCL=200 msec (refractory response). Shannon entropy (bits, green line), channel capacity (bits, red line) and mutual information (bits, blue line) are shown as a function of cable length (cm) in E (BCL=300 msec), G (BCL=220 msec), I (BCL=208 msec), and K (BCL=200 msec). The stimulus site is at cable length = 0 cm. Transfer entropy (bits, red line) is shown as a function of cable length (cm) in F (BCL=300 msec), H (BCL=220 msec), J (BCL=208 msec), and L (BCL=200 msec). The stimulus site is at cable length = 0 cm.

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SI Appendix 3. Information dynamics in 1-D cable Figure 3A-D shows the dynamics of Shannon entropy, channel capacity, mutual information, and transfer entropy as a function of BCL (x-axis) and cable length (y-axis). Shannon entropy was robust against changes in BCL and the location ( Figure 3A). This indicates that BCL has little impact on the cardiac macrostate. We also found that a period-doubling bifurcation was variable depending on the location of the cable (black dashed line). There were three local minima of bifurcation at x=7.1 cm (BCL=236 msec), x=21.5 cm (BCL=236 msec), and x=35.3 cm (BCL=237 msec). Channel capacity showed that those minima of bifurcation were located at the node between concordant and discordant APD alternans, indicating that the variability of bifurcation derives from the node behavior ( Figure 3B). To better understand the dynamics of cardiac macrostate as a function of BCL, we conducted an error-space analysis of each component where the x-axis is the probability ε 0 that an input 0 will be flipped into a 1 (type 0 error), and the y-axis is the probability ε 1 for a flip from 1 to 0 (type 1 error). As BCL declined from 300 msec to 200 msec, the cardiac component at x=5 cm started at the origin of the error space and traveled counterclockwise ( Figure 3E). Both ε 0 and ε 1 reached a peak where discordant alternans was maximum before BCL hit the refractory period and showed a refractory response. At x=20 cm discordant alternans persisted over many BCLs before a conduction block occurred, and eventually a refractory response ensued ( Figure 3F). In contrast, at x=35 cm the trajectory was initially limited to the vicinity of the origin because concordant alternans occurred more frequently than discordant alternans ( Figure 3G). As BCL progressively declined, a conduction block and a refractory response eventually occurred. In the error space, the x-axis is the probability ε 0 that an input 0 will be flipped into a 1 (type 0 error), and the y-axis is the probability ε 1 for a flip from 1 to 0 (type 1 error). Red dots, white lines and black arrows in the error space indicate the trajectory of each component as BCL goes down from 300 msec to 200 msec.

SI Appendix 4. Fenton-Karma model of cardiac excitation
We used a deterministic, simplified ionic model of the cardiac action potential described by Fenton and Karma 4 . We chose this model because it accurately reproduces the critical properties of the cardiac action potential to test our hypothesis, such as restitution properties, APD alternans, conduction block, and spiral wave initiation 2 . The model consists of three variables: the transmembrane potential V , a fast ionic gate u, and a slow ionic gate w.
Here C m is the membrane capacitance (= 1 µF/cm 2 ), and D is the diffusion tensor, which is a diagonal matrix whose diagonal and off-diagonal elements are 0.001 cm 2 /msec and 0 cm 2 /msec, respectively, to represent a 2-D isotropic system 2 .
The current I f i is a fast inward inactivation current used to depolarize the membrane when an excitation above threshold is induced. The current I so is a slow, time-independent rectifying outward current used to repolarize the membrane back to the resting potential. The current I si is a slow inward inactivation current used to balance I so and to produce the observed plateau in the action potential. I ex is the external current 5 . The two gate variables of the model follow first order equations in time. where and The transmembrane potential V and the two gate variables u and w vary from 0 to 1. The three currents are given by the following: We chose the following model parameters to produce action potential dynamics of interest for the study, including APD alternans, conduction block and wavebreak generation: SI Appendix 5. Channel capacity for binary asymmetric channels The channel capacity C between the input X and the output Y is given by where I(X;Y ) is mutual information that quantifies the information content shared between the input X and the output Y .
where p(x, y) denotes the joint probability density function of X and Y . We consider these channels to be a binary asymmetric channel, which is the most general form of binary discrete memoryless channel. The channel has a probability ε 0 that an input 0 will be flipped into a 1 (type 0 error) and a probability ε 1 for a flip from 1 to 0 (type 1 error).
The conditional entropy of Y given X is where h is the binary entropy function defined as h(p) = −p log 2 p − (1 − p) log 2 (1 − p) The mutual information as a function of x is The derivative of I(x) is and thus Therefore, the channel capacity of the binary asymmetric channel is because h 1 1 + z = log 2 (1 + z) − z log 2 z z + 1 (31) C can be rewritten as C = log 2 (1 + z) − log 2 z + ε 1 log 2 z − h(ε 1 ) (32) The input distribution p(x) that maximizes the mutual information and thus achieves the channel capacity is given by 6 p(X = 0) = 1 − p(X = 1) = 1 − ε 1 (1 + z) (1 − ε 0 − ε 1 )(1 + z)