Modified Kuramoto Phase Model for Simulating Cardiac Pacemaker Cell Synchronization

It has been suggested that sick sinus syndrome, which is due to the dysfunction of the sinus node, may result from the sparser gap junctions and/or lower intrinsic frequencies of pacemaker cells that occur with aging. Hence, in this paper, the synchronization mechanism of pacemaker cells that lie in the sinus node of the heart is examined using a modified Kuramoto phase model. Although each element always interacts with all the others in the Kuramoto phase model, in the proposed model, each element interacts only with the neighbors over a certain time (called the interaction time) during Phase 4 of the action potential. The pacemaker cell elements are arranged on a square lattice, and each element connects with the elements surrounding it. The results indicate that the diversity of intrinsic frequencies of pacemaker cells may be necessary for synchronization. Moreover, increasing the proportion of invalid connections causes the elements to take more time to synchronize until eventually they do not synchronize at all, and decreasing the intrinsic frequencies of the elements prevents them from synchronizing. Probably these elucidate the cause of sick sinus syndrome.

• Synchronization
• Sick Sinus Syndrome
• Oscillator
• Overdrive Suppression Test
• Gap Junction

1. Introduction

Synchronization is a universal phenomenon associated with oscillations. The Moon revolves around the Earth with the same period as that of its rotation. The tide-producing force from the Earth to the Moon causes the synchronization of the revolution and rotation of the Moon. Moreover, the spontaneous synchronization of firefly lights is one of several extremely fascinating exhibitions that occur in nature. Some mathematical models have been proposed to simulate this phenomenon [1] . Each firefly is assumed to be a light-emitting oscillator that generates a limit cycle. Another example is the synchronization of a series of Josephson junctions, each element of which interacts only with its close neighbors [2] . Similarly, each pacemaker cell of the sinoatrial node (a group of pacemaker cells in the wall of the right atrium of the heart), which is regarded as an electrical oscillator generating a limit cycle, is loosely coupled with its neighboring pacemaker cells so that those cells synchronize. The electrical impulse current of a single pacemaker cell is not sufficient to travel through the impulse-conducting system, so the ventricles of the heart do not contract. In contrast, synchronized pacemaker cells generate an impulse with sufficiently high current to travel through the system so that the ventricles contract and pump out blood normally. Hence, the failure of this synchronization is presumed to cause sinoatrial arrest [3] . This interruption of the cardiac cycle generally lasts a few seconds before the atrioventricular junction lying in the middle of the impulse-conducting system begins pacing and restores slower ventricular contractions. Sinoatrial arrest is a part of sick sinus syndrome, the symptoms of which include fainting, vertigo, and weakness. Gap junctions in the sinoatrial node have recently been demonstrated to be a key player in the electrical coupling underlying synchronization [4] . Because gap junctions have low resistance, local circuit currents propagate over short distances. Jalife proposed a very probable hypothesis called the democratic consensus hypothesis, which is based on an experiment using rabbit sinoatrial pacemaker cells [5] . This hypothesis states that although the individual pacemaker cells in the sinoatrial node beat at slightly different intrinsic frequencies, they interact mutually by electrical coupling to fire at a “consensus” rate. It has been suggested that when two independent groups of fast and slow pacemaker cells are connected through low-resistance junctions, the period resulting from their mutual entrainment should be a function of their respective intrinsic frequencies, their phase relations, and the degree of electrical coupling. Moreover, Jalife et al. elucidated the mechanisms of sinoatrial pacemaker synchronization using a computer simulation of 81 to 225 coupled cells [6] . Pacemaker activity has been simulated using differential equations that describe transmembrane ionic currents. These results support the hypothesis that sinoatrial node synchronization occurs through a “democratic” process resulting from the phase-dependent interactions of thousands of pacemakers.

The Kuramoto phase model has been used to simulate the synchronization of firefly signals [7] . Although this firefly signal model is simple, the respective frequency, phase relation, and degree of coupling are included as variables in the model. The responsiveness of pacemaker cells is rather different from that of fireflies in that the action potential of each pacemaker cell has a refractory period, during which it does not respond to external stimuli at all. However, a pacemaker cell can respond to external stimuli during Phase 4 of the action potential (described in detail below). Jalife reported various patterns of interactions between fast and slow pacemaker cells, that is, simple harmonic (e.g., 1:1, 2:1, and 1:2) and more complex (e.g., 3:2 and 5:4) ratios [5] . Because such complex phenomena have not been reported in the observation of firefly signals, we speculate that these various ratios may be due to the refractory period. Specific ionic currents, over time, slowly depolarize Phase 4 for pacemaking. Hence, it is assumed that each pacemaker cell is influenced by its neighbors, the phases of which are within a certain range of its phase. This range is called the interaction time. We modified the Kuramoto phase model by incorporating a variable that represents the period of Phase 4. In the modified Kuramoto phase model, for each pacemaker cell, the intrinsic frequency, Phase 4 period, phase relation, and degree of coupling between it and the neighboring pacemaker cells can be modulated independently as variables to observe various patterns. The differences between the Kuramoto phase model and the proposed model are that although each element always interacts with all the others in the former, each element interacts only with its neighbors during the Phase 4 period in the latter. We examine how the synchronization of pacemaker cells depends on those variables. Such an examination is presumed to be rather difficult using differential equations describing transmembrane ionic currents.

The repetitive high-frequency stimulation test of the sinoatrial node (called the overdrive suppression test) is used to examine its function clinically. The sinoatrial node comes to a standstill immediately after the repetitive stimulation, then resumes a regular rhythm after a certain pause (called the sinus node recovery time). The length of this pause is presumed to reflect the degree of dysfunction of the sinoatrial node [8] [9] . Although some mechanisms of this dysfunction have been described [10] , the factor that determines the length of the sinus node recovery time is unknown. One of the purposes of the present study is to infer that factor using the modified Kuramoto phase model. Additionally, an unexpected finding is that slightly different intrinsic frequencies of the pacemaker cell elements promote their synchronization, although it was expected that the same frequencies would make those elements synchronize more easily.

2. Model

The simplest model consists of two connected elements, as illustrated in Figure 1(a). Figure 1(b) shows the minimum square lattice in which there is at least one element that is unconnected to each element. For example, the first element does not connect with the fourth element. Figure 1(c) shows a square lattice consisting of nine elements, which is the minimum lattice in which each element has four connections. For example, the elements are numbered in Figure 1(c): the element on the left corner of the first row is element 1, the element of the right corner of the first row is element 3, and the element of the right corner of the last row is element 9. Every element interacts with four elements: every element inside the lattice connects with its immediate neighbors, and everyone on the lattice sides connects with its immediate neighbors and elements on the op-

posite side. Because every element is equivalent to the others with respect to position, no boundary condition is necessary. Hence, this two-dimensional lattice is assumed to be the surface of a three-dimensional torus made by connecting the top side with the bottom and the left side with the right. The elements directly connected in this manner are regarded as neighbors. In the modified Kuramoto phase model, the dynamics of the i-th cell ( i = 1 , 2 , 3 , ⋯ , n ) is represented as follows:

d θ i d t = f i − 1 N ∑ m = 1 N     K i i m sin ( θ i − θ i m )     × 1 − sign ( | rem ( θ i − θ i m , 2 π ) | − π × G i ) 2     × 1 − sign ( rem ( θ i , 2 π ) + π ( 1 − sign ( rem ( θ i , 2 π ) ) ) − π × H i ) 2 ; (1)

In this equation, n is the total number of the elements of the lattice, q_i is the phase of the i-th element, f_i is the intrinsic frequency of the i-th element, N is the number of couplings with the i-th element (N = 1 in Figure 1(a), 2 in Figure 1(b), and 4 in Figure 1(c)), and K i i m is the degree of interaction between the i-th and i_m-th elements. For example, i_m for the first element of Figure 1(c) are 2, 3, 4, and 7, and i_m for the fifth element is 2, 4, 6, and 8. The rem function is the remainder operation: r = rem ( a , b ) returns the remainder after the division of a by b and the result has the same sign as dividend a. For example, rem ( − 5 π , 2π ) = − π . Function y = sign(x) returns 1 if x > 0, 0 if x = 0, and −1 if x < 0. Moreover, G_i and F_i are coefficients between 0 and 2. The phase of every cycle is 2p.

The first line of the equation is just the Kuramoto phase model for the interactions between each element and its neighbors. The second line returns 1 if the difference between q_i and θ i m , for any integer multiple of 2p, is less than p × G_i, 0 if it is larger than p × G_i, and 1/2 if it is equal to p × G_i. However, it is assumed never to equal to p × G_i exactly. Hence, the second line means that only if the difference between the phase of the i-th element and that of any neighbor is within p × G_i is the i-th element influenced by that neighbor (Figure 2). When

rem ( θ i , 2π ) ≥ 0 , the third line returns 1 if rem ( θ i , 2π ) is less than p × H_i, 0 if it is larger than p × H_i, and 1/2 if it is equal to p × H_i. When rem ( θ i , 2π ) < 0 , the third line returns 1 if rem ( θ i , 2π ) + 2 π is less than p × H_i, 0 if it is larger than p × H_i, and 1/2 if it is equal to p × H_i. It is also assumed neither to be exactly equal to p × H_i nor to be < 0. Thus, the third line means that only if the phase of the i-th element, for any integer multiple of 2p, is between 0 and p × H_i is the i-th element influenced by the neighbors. Figure 2 shows trains of action potentials for two pacemaker cells. The action potential mainly consists of Phases 0, 3, and 4, because Phases 1 and 2 are small. Phase 0 is the period of rapid depolarization caused by a fast inflow of calcium ions and Phase 3 consists of repolarization caused by a fast outflow of potassium ions. These periods make up the refractory period. Phase 4 is the period during which the inflow of sodium ions begins, and thereafter, the inflow of calcium ion continues before firing beyond the threshold [11] . Because the i-th element interacts with the neighbors during Phase 4 and the ionic current through the gap junctions depends on the phase, it is presumed that elements interact with each other when the neighbors are in or near Phase 4. Using p × G_i, any element influencing the i-th element is restricted to neighbors in or near Phase 4 correctly from rem ( θ i , 2π ) − π × G i to rem ( θ i , 2π ) + π × G i . The interaction time is defined as π × G i × 2 . This is the interval between A1 and A2 in Figure 2.

The assumptions are summarized as follows:

Assumption 1: The elements are independent oscillators. In the Kuramoto phase model, each element interacts with all the others. In contrast, in the proposed model, each element interacts only with the connecting elements (neighbors).

Assumption 2: The frequency of each element varies marginally around a certain common frequency. Hence, F_i, the intrinsic frequency of the i-th element ( i = 1 , 2 , 3 , ⋯ , n ), is the sum of a common frequency F_c and random frequency F r i . Common frequency F_c is fixed as 0.4 arbitrarily. The random frequency is given individually and is generated from a uniform distribution of random numbers between 0 and 0.1. Then, F r i is denoted as (0, 0.1) (Table 1, Table 2). Hence, F i = 0.4 + F r i .

Assumption 3: The degree of interaction between two neighbors varies marginally around a certain common degree. Hence, K i i m , the degree of interaction between the i-th element and its i_m-th neighbor, is the sum of a common degree K_c and random degree K r i m . The random degree is given individually and is generated from a uniform distribution of random numbers between 0 and 1. Then, K r i m is denoted as (0, 1) (all Tables). Hence, K i i m = K c + K i r i m .

Assumption 4: The duration of Phase 4 is represented as p×H_i, during which the i-th element interacts with the neighbors. The value of H_i is from 0 to 1. For the sake of model simplicity, it is the same for all elements. Hence, all H_i are expressed as H. The phase of one cycle is 2p. Because the duration of Phase 4 (=p × H) is approximately one half of one cycle [11] , H is assumed to be 1. The start

Table 1

Effects of F r , G c , and lattice size on synchronization (SYNC)

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