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Properties of a multicomponent excitable medium
J. fheor. Biol. (1972) 36,61-80
Properties of a Multicomponent Excitable Medium V. S. MARKIN AND Yu. A. CTUSMAD~ Institute of Hecfrochemisfry, Academy of Sciences, Moscow, U.S.S.R. (Received 12 January 1971, and in revisedform
19 November 1971)
The present paper is concerned with the problem of excitation propagation in a multicomponent excitable medium, which is considered as a model of neuron nets of the syncytial type. The best example is the cardiac tissues, which form a heterogeneous system of different units with common axoplasma. The sundry cardiac tissues differ quantitatively in their properties such as excitation threshold and refractory time. If units of each type are uniformly distributed in space and sufficiently “mixed” with one another, the mediumaproves to be macroscopically homogeneous. In the first part of the paper a method is described which is suitable for consideration of an excitable medium. In the case of a one-component system the velocity of excitation and profile of nerve impulse is obtained by this method. It is interesting to note that the impulse can have two velocities, but only the larger velocity proves to be stable. In the second part of the paper, complicated types of excitation in a two-component medium are considered. It is generally assumed that the complex behaviour of excitation of the echo or rhythm transformation type may arise only in macroscopically inhomogeneous media. In this paper is shown that similar phenomena can be observed also in macroscopically homogeneous media consisting of units of different types. In a two-component medium a moving “ectopic centre” arises, which is a source of nerve impulses. This theory explains some experimental results on cardiac arrhythmia.
The problem of excitation propagation in neuron networks is of great importance for modern neurophysiology. Various authors have made a study of the networks of both kinds: of synaptic transmissions and of the syncytial type in which all fibres have a common axoplasm (Wiener & Rosenblueth, 1946; Bellman, 1962; Rerkinblit, Kovalev, Smoljaninov & Chailakhjan, 1966). The present paper is concerned with networks of the second type. The method of describing excitation propagation over the network depends on the number of structure units which are exoited simultaneously. Ifat each moment of time only a few units are excited, the discrete structure of the network proves to be of importance. The propagation of impulses 61
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over such a network was considered by Pastushenko, Markin dz Chismadjev (1969a). The other limiting case is a network in which many units are excited at once. The detailed structure of this network is not important: it can be described in terms of a continuum with certain averaged characteristics (Wiener & Rosenbluth, 1946). For such a medium in the one-dimensional case the change in membrane potential cpin time and space is described by a well-known equation (Hodgkin & Huxley, 1952):
This can be readily generalized for the three-dimensional case if both internal and external media of the syncytium are macro-isotropic:
Here C is the capacitance of membranes confined within unit volume of medium, R is the sum of resistances of axoplasm and external electrolyte (also confined in unit volume of medium), and Jis the ionic current flowing through the membrane of fibres confined within unit volume. The direction outwards is considered as being the positive direction of the membrane current, and the membrane potential is determined as the difference of the internal and external potentials. If the internal or external space of the syncytium is macroscopically unisotropic or inhomogeneous the problem becomes more complicated. In this case it is impossible to derive a single equation for membrane potential. Instead, one obtains two coupled equations for internal and external potentials. They may be reduced to one equation only if the resistance of internal or external spaces (for example, external) is negligible :
c!2=$. at
‘P-J
dxi rkax,
'
(3)
Here bik is the conductivity tensor of the internal space (Landau & Lifshitz, 1957). The current J generated by the membrane depends on the potential and time-including the time elapsed since the last excitation. Since the potential is generally varying in space, the membrane current J ultimately depends also on the co-ordinates. Given a correct dependence of the generated current on potential and time, it is possible by means of equations (1) to (3) to describe a variety of widely different excitation conditions of the medium. Apart from macroscopic inhomogeneities, the medium may contain excitable units of different types. This property is quite common, cardiac syncytium being among networks of this kind (Hoffman dc
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Cranefield, 1960). These various units we shall mark with subscript k, which if necessary can be both discrete and continuous. Units of different types may have different excitation thresholds cp: and generate different currents Jk. If units of each type are uniformly distributed in space and sufficiently “mixed” with one another, a medium of this kind proves to be homogeneous or, to be more precise, macrohomogeneous. It is generally assumed (Berkinblit et al., 1966; Smoljaninov, 1969; Krynsky, 1968) that the complex behaviour of excitation propagation of the echo or rhythm transformation type may arise in macroscopically inhomogeneous excitable media. It will be shown below that similar or some other phenomena can be observed also in macroscopically homogeneous media consisting of units of different types. 2. Excitation Propagation in a One-component Medium Let us begin our treatment with a medium in which all units have the same properties. The ionic current generated by this medium during excitation can be described by a set of Hodgkin-Huxley equations. Since these equations can be solved only numerically, for analysis of complex cases it is expedient to use simple models of membrane current generation. One such simplified model was developed by Kompaneetz & Gurovich, 1966; Markin & Chismadjev, 1967; Markin 8c Pastushenko, 1969; Pastushenko & Markin, 1969; and Markin, 1970. It is well known experimentally that the ionic current flowing through the membrane during the passage of the nerve impulse should be alternating. In other words, while on reaching the threshold the current immediately flows inside the fibre, after some time it changes its direction and flows outwards. This statement is confhmed by the fact that after the impulse has passed the membrane the potential reverts to its initial value. Let us approximate the true ionic current shown by a dashed line on the plot by two “tables” (Fig. 1). In other words, let us assume that at a certain moment corresponding to the beginning of excitation, the current, directed inside the fibre and equal in modulus to J’, is switched on. After the time z’ the current changes to an opposite one and is equal to J”. This phase lasts for the time 7’. Let us seek only one-dimensional automodel solutions of equation (1) or, in other words, the solutions in the form of a travelling wave. For this purpose let us introduce a new variable < = x- Vi, where V is the impulse propagation velocity. Equation (1) transforms to the form d*dO -+VRCd$-RJ(C)=O. dt’
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1. Current generated by the membrane upon excitation.
Considering cp to be the deviation from the resting membrane potential, let us set the boundary condition &co) = 0. At the opposite end of the axis at C-W- co let us impose the requirement for finiteness of function q(c). One may demand cp(- co) = 0. But in the general case this is not so. The exact value of cp(- co) is determined by the relation between J'7' and J”r”. It will be shown below that if J'7' = Jn7!!, (5) then cp(- co) = 0. By this reason we shall use the relation (5) in the following way. An analytical solution of equation (4) can be readily obtained for the potential (Markin & Chismadjev, 1967). If the current generation is switched on in the point 5 = 0, then we have in the region 0 < 5 &) = (~/Y~R@)[J’+J” ,-V2RC(*‘+rX)_(~‘+~n) e-V’RCr’] e-VRCC; (f9 in the region - V7' < Lj < 0 &) = (~/~ZRC~)[J” e-V’RC(r’++‘) -(J’ +J”> e-V2RCz’] e-VRCC - (J’/ VC)( + (J’/ V2RC2) ; in the region - V(7'+2") < 4: < - V7' cp(() = (J”/v~RC~) ,-V’RC(~‘+~“)-VRCC+(J’~/VC)~+ [z’(J’ +J”)/c] and in the region < < V(7' + z")
-(J”/V2RC2);
(7)
(8)
q(C) = (J'Z'--J"7y/c. (9) The last equation with the condition (5) gives (p(t) = 0. So far in the solution of equation (4) the velocity figured as a parameter, but it is yet to be found. To determine the velocity let us now make use of the condition that the potential at the point where the membrane current is
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switched on should be equal to the excitation threshold q(0) = fp*. (10) Substituting this condition into equation (6), we fmd the equation for determining the velocity : (~/Y~R@)[J:J’+J”
e-ylRC(r'+r")_(~+~")
e--f']
= @.
(11)
It is very convenient to solve this equation graphically. In Fig. 2 we have plotted the left-hand side of equation (11) i.e. q(O) vs. velocity for the
V (m/s4
FIG. 2. Determination
of the impulse rate.
giant axon of squid. The parameters were chosen so: R = 235 x lo4 Q/cm, C = O-157 pF/cm, J’ = 63 pA/cm, J” = 40 PA/cm, r’ = 0.35 msec, r” = o-55 msec, cp* = 18 -5 mV, d = O-05 cm. The solution is found as the intersection of this curve with the straight line ~(0) = rp*. There appear to be two such intersections and thus the impulse can have two velocities. Such a possibility has been pointed out in the literature (Huxley, 1959; Cooley & Dodge, 1966). Let us see whether both the solutions are stable. Let us begin with the impulse propagating with greater velocity. Let the velocity increase somewhat due to fluctuation. Then, as is clear from Fig. 2, the potential at the front point of the impulse will decrease a little and thus prove to be below the excitation threshold. As a result, such an impulse should slow down. If, however, fluctuation decreases the impulse velocity, the potential at the front point of the impulse will exceed the threshold and excitation will propagate faster. Thus, the impulse corresponding to a large velocity proves to be stable. Such an analysis for an impulse with lesser velocity shows the fluctuation increase of velocity to lead to a potential rise at the front point and hence to further impulse acceleration. And vice versa, with decreasing velocity, the impulse slows down further. Thus an impulse with lesser velocity is unstable. The presence of two propagation velocities found above is not associated with any particular form of ionic current. It reflects only the fact of the change in T.B. 5
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the direction of this current. The problem just considered is to a certain extent similar to the problem of laminar flame propagation with heat leakage. In that case there exist also two steady-state solutions, one of which is unstable (Spalding, 1957). For a larger velocity we can obtain from equation (11) an approximate analytical formula v = JJ~/RC$*. (12) It is clear from this formula that the impulse velocity is determined only by the region where the potential increases and does not depend on the reverse current region. To obtain this formula we neglected exponential terms in equation (11). This is possible only if the condition (1 +J”/J’) e-r’J’IQ*C & 1 (13) is fulfilled. The sense of inequality, (6), is that the maximum potential of the impulse should be much larger than the excitation threshold [cf. equation (7)]. This seems to be the case in practice. Calculation with formula (12) of the conduction velocity in giant axon of the squid with parameters listed above gave the value 23.4 m/set. The experimental value is 21.2 m/set (Hodgkin & Huxley, 1952). Now we can plot on the Fig. 3(a) (curve 2) the action potential of giant axon of the squid calculated with the formulae (6) to (9). On the same figure we plotted (curve 1) the real action potential (Hodgkin 8c Huxley, 1952). It is evident that our simple model permits us to calculate action potentials rather well. But there is also an obvious discrepancy : the calculated action potential does not have a region of hyperpolarization. However this point is not critical. This discrepancy may be easily avoided if one takes into consideration leakage of current through the resting membrane. Such a calculation was performed by Undrovinas, Pastushenko & Markin (1972). It was shown that consideration of the resting membrane conductivity changes only slightly the form and the velocity of impulse. The reason is the following. A resting nerve fibre has length constant and time constant A =mR:, T,,, = r,,,C, (14) where r,,, is the resting membrane resistance of unit length of nerve fibre. So the membrane potential in front of a nerve impulse calculated by formulae (6) to (9) must be changed due to r,,,. But in the preceding calculation we supposed that rm = co. Now we evaluate the error introduced by this supposition, Apart from the length constant of nerve fibre Iz, the nerve impulse has its own length constant-it is the distance at which the membrane potential rises electrotonically in front of the impulse, We shall call it “the nose of nerve
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!-is”l -1.5
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Fb. 3. Curreut and potential of the impulse.
impulse”. At this distance the nerve impulse “feels the obstacles”, for example the inhomogenities of the fibre. As follows from equation (a), the length of “impulse nose” is equal to I = l/VRC. (15) If the “impulse nose” is much shorter than the length constant of the nerve fibre 14 A, (la) then resting membrane conductivity cannot change significantly the potential distribution in front of the impulse. Indeed, an electrotonic rise of the potential in front of the impulse takes @lace during the time t*be = l/V= l/PRC. (17) The membrane charge may leak through its conductance during the time r,,,. So if 4irc Q %n~ (18) the resting membrane conductance cannot contribute significantly to the potential distribution in front of nerve impulse. The inequality (18) with (14), (15) and (17) gives I2 4 A2. (19)
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and consequently gives the less strict inequality (16). This seems to be the case in neurophysiology. In our example of a giant fibre of squid of 0.05 cm diameter, the length constant A = 0.5 cm, and “the impulse nose” 1 = O-116 cm. So the inequalities (16) and (19) are fulfilled very well. For this reason the resting membrane conductance makes only a small contribution to a nerve impulse velocity, at least for the considered example. But this is also correct for nerve fibres of any diameter with a similar membrane. Indeed the parameters in formula (12) depends on fibre diameter d in the following way: J’ - d, C - d and R - l/d’ (if the internal resistance is much greater than the external). So nerve conduction velocity is proportional to the square root of fibre diameter, Y - &, as is usually found in experiment. Because r,,, - l/d, both A and 1 are proportional to the square root of the fibre diameter. It means that the inequalities (16) and (19) do not depend of fibre diameter. So we may conclude that the resting membrane conductivity has little significance for calculation of the nerve conduction velocity in unmyelinated fibres. Such a calculation performed by Undrovinas et al. (1972) for the squid giant axon gives a conduction velocity equal to 21-5 m/set. The resting membrane conductivity can also intluence slightly the form of the action potential and, for example, the region of hyperpolarization may be obtained. This pattern is presented in Fig. 3(a), curve 3 [Undrovinas et al., 19721. In a normal spike, the amplitude of the action potential is much greater than the threshold potential. The ratio of these two quantities is called a safety factor lc = %L&*.
(20)
On the base of present theory we can calculate the safety factor. Using the formulae (7) and (12) one obtains approximately Ic = J’T’/@*
- In (1 +J”/J’).
(21)
For the squid giant axon this formula gives K = 7-l. However, in the case of an unstable impulse propagating with a lesser velocity, the action potential is only a little higher than the threshold. Such a difference between the impulse amplitudes was pointed out also by Huxley (1959) and Cooley & Dodge (1966). Using the described model, Markin & Pastushenko (1969), Pastushenko & Markin (1969), Pastushenko, Markin & Chismadjev (1969) studied the passage of nerve impulses along fibres of varying size and along branching fibres, and investigated the electric interaction of adjacent nerve fibres (Markin, 1970).
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3.ChwdbadoaalExcitatlea~lna Mmltiuxu~ Me&m Let us now assume that the medium consists of two types of units. In a recent paper Kukushkin 8z Sakson (1971) showed that Purkinje fibres and muscle fibres in the heart may be taken as an example of such two-types of units. Let us denote the current generated by them by Ji and J2, the membrane operation times by zl and ?2 and the excitation thresholds by cp: and &, respectively. Solving the problem by the method described above, we can obtain the shape of the running impulse (spike) and its velocity. The qualitative results are as follows. If the excitation thresholds do not differ too much, a collective excitation arises which propagates with a certain velocity. In such an impulse it is units with a lower threshold that are excited first, to be followed shortly by the excitation of units with higher thresholds, If the thresholds differ greatly, there can exist an impulse propagating only through the low threshold units. This possibility will be explored later. Here we shall consider collective excitation. Let us assume for simplicity that the excitation thresholds of d.iEerent units coincide, cp: = qf = (p*, and it is only the currents generated by them (Ji and 2J) that differ. In this case the equation for the impulse velocity will assume the form (1/V2&)[J;+JJ;+J;
e-V’RC(r~‘+r~“)+J; -(J;+J’;)
e-Y2RCW+n”)
e-Y2RC?~‘+;+~;)
e-Y3RCd]
=
@.
(22)
If the operation times of different units z; and z; are markedly different, the plot of the left-hand side of this equation has the shape of a doublepeaked curve (Fig. 4). The impulse velocity is determined by the intersection of this curve with a horizontal line drawn at the height cp*. It can easily be seen that in some cases four intersections are possible and hence four steady-state solutions. It was shown earlier that the intersection of the straight line with the descending section of the curve corresponds to a stable impulse, and the
I
(4)
Fro. 4. Determination
(3)
(2)
(I)
4
of the cdkctive impulse rate in a two-component
medium.
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intersection with the ascending section to an unstable one. Thus in the model under consideration, the two different stable impulses can propagate with velocities indicated as (1) and (3) in Fig. 4. In the general case these velocities can be determined from equation (22) only numerically. A large velocity, however, can be found analytically with a good accuracy: v, = J(J; +J;)/RC2cp* (23) potential of the impulse is much higher than the threshold
if the maximum value. It is of interest to consider further the case where units of the same type, e.g. type (2), are rendered unexcitable; for example, they may be refractory. Then excitation runs only along units of type (l), units of type (2) remaining passive. Only one stable impulse with a velocity V, can exist in this case. If condition (13) is satisfied, this velocity is approximately equal to
v, = ~J;/RC~~*.
(24)
If condition (13) is not satisfied, it is necessary to solve equation (22) numerically. If it is units of type (1) which are unexcitable, the impulse propagates only through units of type (2). We shall denote its velocity by V,. It is evident that the velocities of “single” impulses running along fibres of one type are smaller than the velocity of a collective impulse due to the shunting effect of refractory units. 4. Reverberator Let us consider a medium with the properties described above and let us assume, in addition, that the units of type (1) possess an absolute refractory period rl and the units of type (2), r2. Let us ignore the relative refractory period, i.e. let us assume that when these absolute periods elapse, the excitabilities of the respective fibres will be completely restored at once. For definiteness let us assume the refractory period of fibres of type (2) to be greater than that of the fibres of type (I), r2 > rl. Let us suppose that at the origin of the medium (at the point with the coordinate x = 0), stimulation with frequency v = l/T is applied. The first stimulation results in a collective excitation propagating with the velocity V,. For illustration, let us plot the path travelled by excitation, showing the excitation of the units of the fist type by a solid line and that of the units of the second type by a dashed line (Fig. 5). In the plot the first collective excitation is shown by a double line-solid and dashed-starting from the origin. If the stimulation frequency is chosen in such a way that l/r, > v > l/r . the fihres of the second type will not have the time to come
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5. Arising of a reverberator in a two-component medium.
out of the refractory state when the second stimulation is applied at the same point x = 0. Therefore a collective impulse does not arise upon second stimulation, but only an excitation impulse of the fibres of the tit type which propagates with the velocity V,. In the plot it is shown by a single solid line with a smaller slope than the double line. For this reason, the distance between the collective impulse and the excitation impulsk following it increases. Thus, after some time the impulse running along the fibres of the first type comes into a region in which the fibres of the second type are also no longer in the refractory state. As a result, the fibres of the second type become excited and a new collective impulse arises which follows the initial collective impulse travelling with the same velocity or with the velocity V,. In Fig. 5 this is shown by a double line into which the single solid line passes. It is at the intersection point that the collective impulse arises. In addition, there also arises the excitation impulse of the fibres of the second type, travelling in the opposite direction with the velocity V,. In the plot it is shown by a dashed line running downward until it intersects with the time axis. The third stimulation is then applied. Again it excites only the first fibres, and at first the impulse runs over the region in which the second fibres are refractory. In the figure this is a single solid line. After some time the excitation impulse of the first fibres comes into .a region in which the second fibres are also excitable. Here again a collective excitation impulse arises, travelling in the forward direction, and also an excitation impulse of the second fibres, travelling in the reverse direction. In the case shown in Fig. 5 no further stimulation of the medium is produced, but nevertheless the activity of the medium does not cease. Before reaching the edge of the medium
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the excitation impulse of the second fibres, travelling in the reverse direction, comes into a region in which the fibres of the first type have had time “to rest” and pass out of the refractory state. As a result there arises a collective impulse also travelling in the reverse direction, and an excitation impulse of the fibres of the first type, travelling in the forward direction. The collective impulse travelling in the reverse direction reaches the edge of the medium (X = 0) and here fades away as if adsorbed by the edge of the medium. But the excitation impulse of the first fibres, travelling in the forward direction, after some time excites the collective impulse travelling in the forward direction and the impulse of the second fibres travelling in the reverse direction. Subsequently, this pattern will be repeated a number of times, gradually shifting in the forward direction. In other words, a reverberator arises in the system, moving through the medium with a certain velocity and emitting forward and backward collective excitation impulses, this process occurring in a homogeneous medium. If stimulation on the medium is discontinued, the activity in it will persist until the reverberator reaches the opposite edge of the medium, where it will become extinct. Inside the reverberator a periodical switching of excitation from one type of unit to the other takes place. Excitation runs forward along the units of the first type and backward along the units of the second type. For this reason, we can define the reverberator frequency v,. A simple kinematic calculation, which for lack of space we shall omit here, leads to the following result (25) vr = vc- w/(Kr, - V,r,). In a similar manner we can calculate the velocity of motion of the reverberator V,. It is equal to K = KW2 - rdllKr2 - WJ. (26) It is interesting to note that both v, and V, do not depend on the velocity V, ! The velocity of motion of the reverberator can be very small if the refractory periods of different fibres are similar. In each cycle the reverberator emits one collective impulse forward and one backward. The emission frequency is equal to the reverberator frequency v,. However; owing to the motion of the reverberator itself a Doppler effect should arise, i.e. the frequency of the forward impulses vr, should be higher than the frequency of the backward impulses vb. By means of the usual Doppler effect formulae, we find 1 V, v’=~vI/vc=~; V, K-V, vb=l+c=
(V,+V&,-2V,<’
(27)
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Thus, the frequency of the forward impulses is equal to the maximum possible excitation frequency of the entire medium, l/rz. This also can be seen in Fig. 5. In the case illustrated in Fig. 5, the reverberator arose as the result of a threefold stimulation of the medium. In other cases some other number of stimulations may be necessary to excite the reverberator. Analysis shows that in the general case with the frequency of stimulation v, the number of stimulations necessary to excite the reverberator is determined by the relation
n/>1+vc-VA v, (V, + v-2) x v, x(&if This formula is applicable if the frequency of stimulation lies within the range r2 > l/v > rl. It is clear that if the stimulation period l/v approaches the greatest refractory period r2, the number of stimulations necessary for a reverberator to arise tends to infinity. T’his means that infrequent stimulation produces only the usual collective impulses. To excite a reverberator it is necessary to stimulate the medium with a frequency exceeding its least natural frequency. This fact has a close analogy in electrophysiology. It is known (Hoffinan & Cranefield, 1960) that complex heart excitations such a fibrillation can be brought about by means of several extrasystoles. If the excitations that follow the first excitation, with a frequency greater than the least natural frequency of the medium, are considered as extrasystoles, we may draw an analogy with complex excitation behavior of cardiac syncytium. There is no doubt that many important properties of a real network are not reflected in this scheme; we have considered only the one-dimensional case, and we suppose that in two- and three-dimensional cases one can obtain new effects. It is interesting to note that the reverberator properties do not depend on the mode of excitation. A reverberator always has the same frequency and velocity of motion, and it always emits collective impulses with the same frequency. In this respect, we can say that rhythm transformation is realized in the suggested model. If the stimulation frequency is less than the least natural frequency of the medium (in this case l/r,), this rhythm does not vary and the medium responds with the usual collective impulses. If, however, the stimulation frequency proves to be higher, this stimulation rhythm undergoes transformation and ultimately the medium responds with collective impulses with the frequency l/r2 which is the maximum possible one for it. It is important to note that rhythm transformation occurs not at the stimulation point, but rather at a distance from it (compare Berkinblit et al., 1%6). The impulse going in the retrograde direction may be considered as an echo phenomenon. The first echo can be registered after the
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second stimulation of the medium (Fig. 5). The time of the echo return elapsed since the second stimulation, 6,, depends on the period of stimulation T. A simple calculation gives this function: (2% W? = V,K + V&r2 - TWAKVI). This is plotted in Fig. 6. If T < rl or T > rz, no echo is evoked, because, in the first case, the units of both types are refractory and no impulse can be initiated at all; in the second case, all units are excitable and a collective impulse arises.
FIG. 6. Dependence of latency of reciprocal excitation 0. on time between stimulations T.
After starting the reverberator, if stimulation of the medium is not stopped, one more reverberator will arise after some time, then again one more, etc. In this connection, there arises a question as to how the reverberators will interact with one another and with single impulses. It is well known that when two nerve impulses collide, they extinguish each other. Therefore, in order to destroy a nerve impulse it is sufficient to let another impulse travel toward it. This is not so with a reverberator. It is impossible to destroy a reverberator by a counter impulse. The reason is quite clear: each reverberator has in front of it a protective “coat” of impulses emitted by it. A counter-impulse will destroy only one of these protective impulses, leaving the rest of them untouched. Since the velocity of motion of a reverberator is small, one might attempt to destroy it by sending an impulse in pursuit. However this is also impossible since a reverberator emits protective impulses not only in front of it but also behind it. We should like to note especially that the reverberator can not be destroyed by a pursuing impulse even immediately after its “birth”. So if the stimulation of the medium is not stopped the second reverberator will arise, then the third and so on. Thus in order to destroy a reverberator it is necessary to “bombard” it with a series of a sufficiently large number of impulses. Another reverberator would be most suitable for this purpose. Let us consider the interaction of two reverberators moving toward each other (Fig. 7). For simplicity, we shall not draw a detailed structure of a reverberator, as in Fig. 5. Instead, we shall represent a reverberator by a wavy line and the impulses emitted by it by simple straight lines. The two reverberators move toward each other, but the one moving in the positive direction
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arose somewhat later. Its “coat” of protective impulses will then be thinner, and as a result it will be the one to be destroyed. A case is possible when both reverberators will perish as a result of interaction. This may occur under symmetrical conditions (Fig. 8). In conclusion, let us consider “the pursuit” of reverberators. Let two reverberators follow each other (Fig. 9). In this case neither is destroyed. They both “bombard” each other and thus protect themselves successfully. Colliding, the protective impulses perish where simple straight lines intersect in the figure.
Fro. 7. Collision of reverberators involving destruction of one of them.
FIQ. 8. Collision and destruction of two reverberators.
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\ 9. Pursuit
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\ reverberators.
5. Decay of Group Excitation Let us consider a medium consisting of units of both types whose excitation thresholds differ markedly. For definiteness let the excitation threshold of units of the first type be higher, and the velocity, V,, of the impulse running along these first type units be lower than in the case of units of the second type (velocity V2). In addition, let us assume that the impulse running along the fibres of the second type cannot excite fibres of the first type on account of their high threshold. Back excitation, however, is possible. There need not be a great difference between the thresholds for such a phenomenon to be observed. It is necessary only that the maximum potential of the impulse running along the second fibres should be confined between thresholds cp: and qz. Using the formula (7), one can easily find the maximum potential and hence the condition of the breakup of the group excitation: fpf < (XJV,ZRC”) In J;/[(J; +Jg) e-Y22RCr2’-J; e-Y2ZRC(*2’+*21)]< qy. (30) In such a medium a complex excitation behavior can be observed even with a single stimulation. Let stimulation be applied at the point x = 0, so that fibres of both types should be excited at the same time. Owing to a considerable difference in the thresholds, no collective impulse arises in this case. Instead, excitation breaks up into two single impulses, each of them running along its own units. As before, we shall represent the impulse running along the units of the first type by a solid line and the impulse running along the units of the second type by a dashed line (Fig. 10). The slope of the dashed lines is steeper since the velocity of the respective impulses is greater. After application of stimulation, the excitation impulse of the second fibres is the first to move forward. It does not excite the fibres of the first
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10. Decay
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type owing to their high threshold. It is followed by an excitation impulse of the fibres of the first type moving with a lesser velocity. In principle, this impulse can excite the fibres of the second type, but this does not take place since they appear to be in the refractory state. However, owing to the velocity difference, the distance between the impulses gradually increases and after some time the impulse running along the first fibres enters the region in which the second fibres have already come out of the refractory state. Here the second fibres are again excited and a fast impulse with the velocity V, will pass through them forward and backward. All this will be repeated periodically. Thus, once produced, the slow excitation wave in the first fibres generates periodically fast excitation waves in the second fibres, which become detached from it and move away forward and backward. Let us find the parameters of such excitation behavior. An analysis shows that the emission frequency of fast waves is equal to v,, = (V’- V,)/V,r,. Due to the Doppler effect, the frequency of forward and backward waves will be different:
Thus, the fast waves run forward with the maximum possible frequency l/r, and back with a somewhat lesser frequency. It can be readily seen that there is a close analogy between the properties of the excitation behavior described here and, the reverberator properties described earlier. For this reason, the consideration regarding the pursuit and collision of reverberators discussed above are applicable here as well.
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6. Discussion
We suppose that this theory can describe some phenomena in heart excitation. Indeed it is usually supposed that cardiac tissues form a heterogeneous system of different units with a common axoplasma (Brooks, Hoffman, Suckling & Orias, 1955). The sundry cardiac tissues differ quantitatively in their properties. Auricular and ventricular muscles differ in their electrical excitability, conduction velocity and limiting frequency for the propagation of impulses, and the magnitude, pattern and duration of their action potential. The atrioventricular (AV) node, and His and Purkinje fibres differ from the auricle and ventricle, and also differ from each other. So the macrohomogeneous multicomponent excitable model formulated above seems to be an adequate one. Here we studied only a one-dimensional problem of the two-component excitable medium. But even in this rather simple case some interesting
-T FIG. 11. Dependence of time separation two subsequent registrations 0. on time separating stimulations T.
results can be obtained. We demonstrated the existance of the rhythm transformation phenomenon, which was observed in a heart long ago (Lucas, 1910). The impulses initiated at the auricle or ventricle with significantly different delays after a first impulse, may reach the distal muscle with a constant time interval after the arrival of the first. The same phenomenon in a single tissue was demonstrated in a number of papers (Moe, Preston & Burlington, 1956; Rosenblueth, 1958). This phenomenon was explained by the existence of the relative refractory period (Moe et al., 1956) and by a temporary stop of the impulse (Rosenbluth, 1958). It can be explained on the basis of the present theory. Indeed, let us consider two subsequent stimulations of the medium with period T (Fig. 5), and register the time separation 8, of the impulses at the distal part of the medium. It is then easy to realize that we shall have the curve given in Fig. 11. If T > r2, we have a straight line with a slope equal to one. In the interval between r1 and r2 there is a horizontal line. The explanation is very simple. Only the units of type 1 are excitable and the impulse travels through them until it excites a collective impulse. This collective impulse will arrive at the
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distal part with time separation rz. Such dependences were observed by Moe et al. (1956) and Rosenblueth (1958) in the atrioventricular node. It should also be noted that in the literature up to now there is discussion of the problem of the existence of two conducting paths in the atrioventricular node (Moe et al., 1956; Vermeulen & Wellens, 1971). Now we should like to discuss briefly the manner in which this theory can be of value in elucidating the genesis of cardiac arrhythmias. It is well known that disorders of conduction are a possible cause of arrhythmia (Hoffman, 1966). This mechanism implies two different pathways for nerve impulses, separated by a region of non-conducting tissue. Making such morphological assumptions it is not a difficult problem to obtain re-entry phenomena or circulation of impulses around an obstacle (Wiener & Rosenblueth, 1946 ; Krynsky, 1968). But the question is whether we can really find such nonconducting regions in the heart. It is well known that it is possible to initiate flutter or fibrillation in purely homogeneous regions of the heart (Scherf, 1966). In this paper we showed that arrhythmia can be obtained in a macroscopically homogeneous two-component medium. We can say that in a two-component medium a moving ectopic centre arises which is a source of nerve impulses. The parameters of this centre are not functions of morphological inhomogeneities of the medium. This theory explains some new experimental results about arrhythmia (Kukushkin t Sakson, 1971). As an example of a two-component excitable medium Kukushkin & Sakson (1971) recently studied the muscle fibres and Purkinje fibres in a frog heart. Histologically, ordinary cardiac muscle fibres and Purkinje fibres are continuous, one merging into the other without morphological interruption (Kugler & Parkins, 1956). In spite of some differences in electrophysiological properties, excitation can cross the junctional region between the two types of fibres in either direction (Hoffman & Suckling, 1953). The shorter action potential of the muscle fibres allows it to be re-excited earlier than the Purkinje fibres (Kao & Hofhnan, 1958). Kukushkin & Sakson (1971) studied the excitation of the frog heart ventricle under two subsequent stimulations, and interpreted the second excitation as a reciprocal one, or as an echo phenomenon. Present theory explains this phenomenon and gives an equation (29) for a latency 0, (see also Fig. 6). In preparations of heart tissue with high latency a series of impulses arise after the second electrical stimulus, while in preparations with small latency only a local response was measured. From equations (28) and (29) of this paper we can obtain the critical condition for initiation of arrhythmia : O,jrl > 1. (32)
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From this relation it is clear that arrhythmia in agreement withexperimental results must appear in preparations with high latency. To obtain more obvious evidence of the reality of this theory it is necessary to carry out measurements with multi-electrode registration, which permits one to obtain the electrical pattern of the preparation in the space. New available methods in electrophysiology of heart allow the hope that such measurements can be done in the immediate future. REFERENCES BELLMAN, R. (1962) ed. Mathematical Problems in the Biological Sciences. Providence, RI. : American Mathematical Society. BERKINBLIT, M. B., KOVAL-EV, S. A., SMOUANINOV, V. V. & CHAILAKHJAN, L. M. (1966). In MoaWy struct~~-f~tsio~~noy organizatsii nekotorykh biologichesk~h system. (I. hi. Gelfand, ed.) p. 71. Moscow, “Nauka”. BROOKS, C. McC., HOFFMAN, B. F., SUCKLING, E. E. & ORIAS, 0. (1955). Excitability of the Heart. New York: Grune. COOLEY, J. W. & DODGE, F. A. (1966). Biophys. J. 6, 582. HOD~KXN, A. L. & HUXLEY, A. F. (1952). J. Physiof., Land. 117, 500. HOFFMAN, B. F. (1966). In Mechanisms and Therapy of Cardiac Arrhythmias, (L. S. Dreifus & W. Likoff, eds.) p. 27. New York, London: Grune and Stratton. HOFFMAN, B. F. & CRANFIELD, P. F. (1960). Electrophysiofogy of the Heart. New York: McGraw-Hill. HOFFMAN, B. F. & SUCKLING, E. E. (1953). Am. J. Physiol. 173, 312. HUXLEY, A. F. (1959). J. Physiol., Lund. 148, 80. KAO, C. Y. & HOFFMAN, B. F. (1958). Am. J. Physiol. 194, 187. KOMPANEETZ, A. S. & GUROVICH, V. Ts. (1966). Biojizika 11, 30t. KRyNsKY, V. J. (1968). Problem? Kibern. 20, 59. KUGLER, J. H. & PARKINS, J. B. (1956). Anat. Res. 126, 335. KIJKUSHKIN, N. I. & SAKSON, M. E. (1971). Biofizika 16, 135. LANDAU, L. D. & LIFSHITZ, E. M. (1957). Electrodinamika sploshnykh sred. Moskva: G.I.T.T.L. LUCAS, K. (1910). J. Physiol., Land. 41, 368. MARKIN, V. S. & CHI~MADJEV, Yu. A. (1967). Biofizika 12, 900. MARKIN, V. S., PASTUSHENKO, V. F. (1969). Biofizika 14, 326. MARKIN, V. S. (1970). Biofzika 15, 120. MOE, G. K., PRESTON, J. B. & BURLINGTON, H. (1956). Circulation Res. 4, 357. PASTUSHENKO, V. F. & MARKIN, V. S. (1969). Biofizika 14, 517. PASTIJS~NKO, V. F., MARKIN, V. S. & CHISMADJEV, Yu. A. (1969). Biojizika 14,883,1072. ROSENBLUHTH, A. (1958). Am. J. Physiol. 194, 171. SCHERF, D. (1966). In Mechanisms and Therapy of Cardiac Arrhythmias, (L. S. Dreifus & W. Likoff, eds) p. 129. New York, London: Grune & Stratton. SMOLJANINOV, V. V. (1969). Biofiika 14, 336. SPALDING, D. B. (1957). Proc. R. Sot. A 240, 83. UNDRO~INAS, A. I., PASTUSHENKO, V. F., MARKIN, V. S. (1972). Dokl. Akad. Nauk SSSR 204, NI. VERMEULEN, A. & WELLENS, W~NER, N. & ROSENIILUETH,
t English translation Biophysics.
H. J. (1971). A. (1946).
Br. Heart J. 33, 320. Arch. Inst. Cardiologia Mexico 16, 205.
of articles published in Biofizika
are available in the journal
Properties of a Multicomponent Excitable Medium V. S. MARKIN AND Yu. A. CTUSMAD~ Institute of Hecfrochemisfry, Academy of Sciences, Moscow, U.S.S.R. (Received 12 January 1971, and in revisedform
19 November 1971)
The present paper is concerned with the problem of excitation propagation in a multicomponent excitable medium, which is considered as a model of neuron nets of the syncytial type. The best example is the cardiac tissues, which form a heterogeneous system of different units with common axoplasma. The sundry cardiac tissues differ quantitatively in their properties such as excitation threshold and refractory time. If units of each type are uniformly distributed in space and sufficiently “mixed” with one another, the mediumaproves to be macroscopically homogeneous. In the first part of the paper a method is described which is suitable for consideration of an excitable medium. In the case of a one-component system the velocity of excitation and profile of nerve impulse is obtained by this method. It is interesting to note that the impulse can have two velocities, but only the larger velocity proves to be stable. In the second part of the paper, complicated types of excitation in a two-component medium are considered. It is generally assumed that the complex behaviour of excitation of the echo or rhythm transformation type may arise only in macroscopically inhomogeneous media. In this paper is shown that similar phenomena can be observed also in macroscopically homogeneous media consisting of units of different types. In a two-component medium a moving “ectopic centre” arises, which is a source of nerve impulses. This theory explains some experimental results on cardiac arrhythmia.
The problem of excitation propagation in neuron networks is of great importance for modern neurophysiology. Various authors have made a study of the networks of both kinds: of synaptic transmissions and of the syncytial type in which all fibres have a common axoplasm (Wiener & Rosenblueth, 1946; Bellman, 1962; Rerkinblit, Kovalev, Smoljaninov & Chailakhjan, 1966). The present paper is concerned with networks of the second type. The method of describing excitation propagation over the network depends on the number of structure units which are exoited simultaneously. Ifat each moment of time only a few units are excited, the discrete structure of the network proves to be of importance. The propagation of impulses 61
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over such a network was considered by Pastushenko, Markin dz Chismadjev (1969a). The other limiting case is a network in which many units are excited at once. The detailed structure of this network is not important: it can be described in terms of a continuum with certain averaged characteristics (Wiener & Rosenbluth, 1946). For such a medium in the one-dimensional case the change in membrane potential cpin time and space is described by a well-known equation (Hodgkin & Huxley, 1952):
This can be readily generalized for the three-dimensional case if both internal and external media of the syncytium are macro-isotropic:
Here C is the capacitance of membranes confined within unit volume of medium, R is the sum of resistances of axoplasm and external electrolyte (also confined in unit volume of medium), and Jis the ionic current flowing through the membrane of fibres confined within unit volume. The direction outwards is considered as being the positive direction of the membrane current, and the membrane potential is determined as the difference of the internal and external potentials. If the internal or external space of the syncytium is macroscopically unisotropic or inhomogeneous the problem becomes more complicated. In this case it is impossible to derive a single equation for membrane potential. Instead, one obtains two coupled equations for internal and external potentials. They may be reduced to one equation only if the resistance of internal or external spaces (for example, external) is negligible :
c!2=$. at
‘P-J
dxi rkax,
'
(3)
Here bik is the conductivity tensor of the internal space (Landau & Lifshitz, 1957). The current J generated by the membrane depends on the potential and time-including the time elapsed since the last excitation. Since the potential is generally varying in space, the membrane current J ultimately depends also on the co-ordinates. Given a correct dependence of the generated current on potential and time, it is possible by means of equations (1) to (3) to describe a variety of widely different excitation conditions of the medium. Apart from macroscopic inhomogeneities, the medium may contain excitable units of different types. This property is quite common, cardiac syncytium being among networks of this kind (Hoffman dc
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Cranefield, 1960). These various units we shall mark with subscript k, which if necessary can be both discrete and continuous. Units of different types may have different excitation thresholds cp: and generate different currents Jk. If units of each type are uniformly distributed in space and sufficiently “mixed” with one another, a medium of this kind proves to be homogeneous or, to be more precise, macrohomogeneous. It is generally assumed (Berkinblit et al., 1966; Smoljaninov, 1969; Krynsky, 1968) that the complex behaviour of excitation propagation of the echo or rhythm transformation type may arise in macroscopically inhomogeneous excitable media. It will be shown below that similar or some other phenomena can be observed also in macroscopically homogeneous media consisting of units of different types. 2. Excitation Propagation in a One-component Medium Let us begin our treatment with a medium in which all units have the same properties. The ionic current generated by this medium during excitation can be described by a set of Hodgkin-Huxley equations. Since these equations can be solved only numerically, for analysis of complex cases it is expedient to use simple models of membrane current generation. One such simplified model was developed by Kompaneetz & Gurovich, 1966; Markin & Chismadjev, 1967; Markin 8c Pastushenko, 1969; Pastushenko & Markin, 1969; and Markin, 1970. It is well known experimentally that the ionic current flowing through the membrane during the passage of the nerve impulse should be alternating. In other words, while on reaching the threshold the current immediately flows inside the fibre, after some time it changes its direction and flows outwards. This statement is confhmed by the fact that after the impulse has passed the membrane the potential reverts to its initial value. Let us approximate the true ionic current shown by a dashed line on the plot by two “tables” (Fig. 1). In other words, let us assume that at a certain moment corresponding to the beginning of excitation, the current, directed inside the fibre and equal in modulus to J’, is switched on. After the time z’ the current changes to an opposite one and is equal to J”. This phase lasts for the time 7’. Let us seek only one-dimensional automodel solutions of equation (1) or, in other words, the solutions in the form of a travelling wave. For this purpose let us introduce a new variable < = x- Vi, where V is the impulse propagation velocity. Equation (1) transforms to the form d*dO -+VRCd$-RJ(C)=O. dt’
64
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4
1. Current generated by the membrane upon excitation.
Considering cp to be the deviation from the resting membrane potential, let us set the boundary condition &co) = 0. At the opposite end of the axis at C-W- co let us impose the requirement for finiteness of function q(c). One may demand cp(- co) = 0. But in the general case this is not so. The exact value of cp(- co) is determined by the relation between J'7' and J”r”. It will be shown below that if J'7' = Jn7!!, (5) then cp(- co) = 0. By this reason we shall use the relation (5) in the following way. An analytical solution of equation (4) can be readily obtained for the potential (Markin & Chismadjev, 1967). If the current generation is switched on in the point 5 = 0, then we have in the region 0 < 5 &) = (~/Y~R@)[J’+J” ,-V2RC(*‘+rX)_(~‘+~n) e-V’RCr’] e-VRCC; (f9 in the region - V7' < Lj < 0 &) = (~/~ZRC~)[J” e-V’RC(r’++‘) -(J’ +J”> e-V2RCz’] e-VRCC - (J’/ VC)( + (J’/ V2RC2) ; in the region - V(7'+2") < 4: < - V7' cp(() = (J”/v~RC~) ,-V’RC(~‘+~“)-VRCC+(J’~/VC)~+ [z’(J’ +J”)/c] and in the region < < V(7' + z")
-(J”/V2RC2);
(7)
(8)
q(C) = (J'Z'--J"7y/c. (9) The last equation with the condition (5) gives (p(t) = 0. So far in the solution of equation (4) the velocity figured as a parameter, but it is yet to be found. To determine the velocity let us now make use of the condition that the potential at the point where the membrane current is
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switched on should be equal to the excitation threshold q(0) = fp*. (10) Substituting this condition into equation (6), we fmd the equation for determining the velocity : (~/Y~R@)[J:J’+J”
e-ylRC(r'+r")_(~+~")
e--f']
= @.
(11)
It is very convenient to solve this equation graphically. In Fig. 2 we have plotted the left-hand side of equation (11) i.e. q(O) vs. velocity for the
V (m/s4
FIG. 2. Determination
of the impulse rate.
giant axon of squid. The parameters were chosen so: R = 235 x lo4 Q/cm, C = O-157 pF/cm, J’ = 63 pA/cm, J” = 40 PA/cm, r’ = 0.35 msec, r” = o-55 msec, cp* = 18 -5 mV, d = O-05 cm. The solution is found as the intersection of this curve with the straight line ~(0) = rp*. There appear to be two such intersections and thus the impulse can have two velocities. Such a possibility has been pointed out in the literature (Huxley, 1959; Cooley & Dodge, 1966). Let us see whether both the solutions are stable. Let us begin with the impulse propagating with greater velocity. Let the velocity increase somewhat due to fluctuation. Then, as is clear from Fig. 2, the potential at the front point of the impulse will decrease a little and thus prove to be below the excitation threshold. As a result, such an impulse should slow down. If, however, fluctuation decreases the impulse velocity, the potential at the front point of the impulse will exceed the threshold and excitation will propagate faster. Thus, the impulse corresponding to a large velocity proves to be stable. Such an analysis for an impulse with lesser velocity shows the fluctuation increase of velocity to lead to a potential rise at the front point and hence to further impulse acceleration. And vice versa, with decreasing velocity, the impulse slows down further. Thus an impulse with lesser velocity is unstable. The presence of two propagation velocities found above is not associated with any particular form of ionic current. It reflects only the fact of the change in T.B. 5
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the direction of this current. The problem just considered is to a certain extent similar to the problem of laminar flame propagation with heat leakage. In that case there exist also two steady-state solutions, one of which is unstable (Spalding, 1957). For a larger velocity we can obtain from equation (11) an approximate analytical formula v = JJ~/RC$*. (12) It is clear from this formula that the impulse velocity is determined only by the region where the potential increases and does not depend on the reverse current region. To obtain this formula we neglected exponential terms in equation (11). This is possible only if the condition (1 +J”/J’) e-r’J’IQ*C & 1 (13) is fulfilled. The sense of inequality, (6), is that the maximum potential of the impulse should be much larger than the excitation threshold [cf. equation (7)]. This seems to be the case in practice. Calculation with formula (12) of the conduction velocity in giant axon of the squid with parameters listed above gave the value 23.4 m/set. The experimental value is 21.2 m/set (Hodgkin & Huxley, 1952). Now we can plot on the Fig. 3(a) (curve 2) the action potential of giant axon of the squid calculated with the formulae (6) to (9). On the same figure we plotted (curve 1) the real action potential (Hodgkin 8c Huxley, 1952). It is evident that our simple model permits us to calculate action potentials rather well. But there is also an obvious discrepancy : the calculated action potential does not have a region of hyperpolarization. However this point is not critical. This discrepancy may be easily avoided if one takes into consideration leakage of current through the resting membrane. Such a calculation was performed by Undrovinas, Pastushenko & Markin (1972). It was shown that consideration of the resting membrane conductivity changes only slightly the form and the velocity of impulse. The reason is the following. A resting nerve fibre has length constant and time constant A =mR:, T,,, = r,,,C, (14) where r,,, is the resting membrane resistance of unit length of nerve fibre. So the membrane potential in front of a nerve impulse calculated by formulae (6) to (9) must be changed due to r,,,. But in the preceding calculation we supposed that rm = co. Now we evaluate the error introduced by this supposition, Apart from the length constant of nerve fibre Iz, the nerve impulse has its own length constant-it is the distance at which the membrane potential rises electrotonically in front of the impulse, We shall call it “the nose of nerve
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Fb. 3. Curreut and potential of the impulse.
impulse”. At this distance the nerve impulse “feels the obstacles”, for example the inhomogenities of the fibre. As follows from equation (a), the length of “impulse nose” is equal to I = l/VRC. (15) If the “impulse nose” is much shorter than the length constant of the nerve fibre 14 A, (la) then resting membrane conductivity cannot change significantly the potential distribution in front of the impulse. Indeed, an electrotonic rise of the potential in front of the impulse takes @lace during the time t*be = l/V= l/PRC. (17) The membrane charge may leak through its conductance during the time r,,,. So if 4irc Q %n~ (18) the resting membrane conductance cannot contribute significantly to the potential distribution in front of nerve impulse. The inequality (18) with (14), (15) and (17) gives I2 4 A2. (19)
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and consequently gives the less strict inequality (16). This seems to be the case in neurophysiology. In our example of a giant fibre of squid of 0.05 cm diameter, the length constant A = 0.5 cm, and “the impulse nose” 1 = O-116 cm. So the inequalities (16) and (19) are fulfilled very well. For this reason the resting membrane conductance makes only a small contribution to a nerve impulse velocity, at least for the considered example. But this is also correct for nerve fibres of any diameter with a similar membrane. Indeed the parameters in formula (12) depends on fibre diameter d in the following way: J’ - d, C - d and R - l/d’ (if the internal resistance is much greater than the external). So nerve conduction velocity is proportional to the square root of fibre diameter, Y - &, as is usually found in experiment. Because r,,, - l/d, both A and 1 are proportional to the square root of the fibre diameter. It means that the inequalities (16) and (19) do not depend of fibre diameter. So we may conclude that the resting membrane conductivity has little significance for calculation of the nerve conduction velocity in unmyelinated fibres. Such a calculation performed by Undrovinas et al. (1972) for the squid giant axon gives a conduction velocity equal to 21-5 m/set. The resting membrane conductivity can also intluence slightly the form of the action potential and, for example, the region of hyperpolarization may be obtained. This pattern is presented in Fig. 3(a), curve 3 [Undrovinas et al., 19721. In a normal spike, the amplitude of the action potential is much greater than the threshold potential. The ratio of these two quantities is called a safety factor lc = %L&*.
(20)
On the base of present theory we can calculate the safety factor. Using the formulae (7) and (12) one obtains approximately Ic = J’T’/@*
- In (1 +J”/J’).
(21)
For the squid giant axon this formula gives K = 7-l. However, in the case of an unstable impulse propagating with a lesser velocity, the action potential is only a little higher than the threshold. Such a difference between the impulse amplitudes was pointed out also by Huxley (1959) and Cooley & Dodge (1966). Using the described model, Markin & Pastushenko (1969), Pastushenko & Markin (1969), Pastushenko, Markin & Chismadjev (1969) studied the passage of nerve impulses along fibres of varying size and along branching fibres, and investigated the electric interaction of adjacent nerve fibres (Markin, 1970).
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3.ChwdbadoaalExcitatlea~lna Mmltiuxu~ Me&m Let us now assume that the medium consists of two types of units. In a recent paper Kukushkin 8z Sakson (1971) showed that Purkinje fibres and muscle fibres in the heart may be taken as an example of such two-types of units. Let us denote the current generated by them by Ji and J2, the membrane operation times by zl and ?2 and the excitation thresholds by cp: and &, respectively. Solving the problem by the method described above, we can obtain the shape of the running impulse (spike) and its velocity. The qualitative results are as follows. If the excitation thresholds do not differ too much, a collective excitation arises which propagates with a certain velocity. In such an impulse it is units with a lower threshold that are excited first, to be followed shortly by the excitation of units with higher thresholds, If the thresholds differ greatly, there can exist an impulse propagating only through the low threshold units. This possibility will be explored later. Here we shall consider collective excitation. Let us assume for simplicity that the excitation thresholds of d.iEerent units coincide, cp: = qf = (p*, and it is only the currents generated by them (Ji and 2J) that differ. In this case the equation for the impulse velocity will assume the form (1/V2&)[J;+JJ;+J;
e-V’RC(r~‘+r~“)+J; -(J;+J’;)
e-Y2RCW+n”)
e-Y2RC?~‘+;+~;)
e-Y3RCd]
=
@.
(22)
If the operation times of different units z; and z; are markedly different, the plot of the left-hand side of this equation has the shape of a doublepeaked curve (Fig. 4). The impulse velocity is determined by the intersection of this curve with a horizontal line drawn at the height cp*. It can easily be seen that in some cases four intersections are possible and hence four steady-state solutions. It was shown earlier that the intersection of the straight line with the descending section of the curve corresponds to a stable impulse, and the
I
(4)
Fro. 4. Determination
(3)
(2)
(I)
4
of the cdkctive impulse rate in a two-component
medium.
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intersection with the ascending section to an unstable one. Thus in the model under consideration, the two different stable impulses can propagate with velocities indicated as (1) and (3) in Fig. 4. In the general case these velocities can be determined from equation (22) only numerically. A large velocity, however, can be found analytically with a good accuracy: v, = J(J; +J;)/RC2cp* (23) potential of the impulse is much higher than the threshold
if the maximum value. It is of interest to consider further the case where units of the same type, e.g. type (2), are rendered unexcitable; for example, they may be refractory. Then excitation runs only along units of type (l), units of type (2) remaining passive. Only one stable impulse with a velocity V, can exist in this case. If condition (13) is satisfied, this velocity is approximately equal to
v, = ~J;/RC~~*.
(24)
If condition (13) is not satisfied, it is necessary to solve equation (22) numerically. If it is units of type (1) which are unexcitable, the impulse propagates only through units of type (2). We shall denote its velocity by V,. It is evident that the velocities of “single” impulses running along fibres of one type are smaller than the velocity of a collective impulse due to the shunting effect of refractory units. 4. Reverberator Let us consider a medium with the properties described above and let us assume, in addition, that the units of type (1) possess an absolute refractory period rl and the units of type (2), r2. Let us ignore the relative refractory period, i.e. let us assume that when these absolute periods elapse, the excitabilities of the respective fibres will be completely restored at once. For definiteness let us assume the refractory period of fibres of type (2) to be greater than that of the fibres of type (I), r2 > rl. Let us suppose that at the origin of the medium (at the point with the coordinate x = 0), stimulation with frequency v = l/T is applied. The first stimulation results in a collective excitation propagating with the velocity V,. For illustration, let us plot the path travelled by excitation, showing the excitation of the units of the fist type by a solid line and that of the units of the second type by a dashed line (Fig. 5). In the plot the first collective excitation is shown by a double line-solid and dashed-starting from the origin. If the stimulation frequency is chosen in such a way that l/r, > v > l/r . the fihres of the second type will not have the time to come
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5. Arising of a reverberator in a two-component medium.
out of the refractory state when the second stimulation is applied at the same point x = 0. Therefore a collective impulse does not arise upon second stimulation, but only an excitation impulse of the fibres of the tit type which propagates with the velocity V,. In the plot it is shown by a single solid line with a smaller slope than the double line. For this reason, the distance between the collective impulse and the excitation impulsk following it increases. Thus, after some time the impulse running along the fibres of the first type comes into a region in which the fibres of the second type are also no longer in the refractory state. As a result, the fibres of the second type become excited and a new collective impulse arises which follows the initial collective impulse travelling with the same velocity or with the velocity V,. In Fig. 5 this is shown by a double line into which the single solid line passes. It is at the intersection point that the collective impulse arises. In addition, there also arises the excitation impulse of the fibres of the second type, travelling in the opposite direction with the velocity V,. In the plot it is shown by a dashed line running downward until it intersects with the time axis. The third stimulation is then applied. Again it excites only the first fibres, and at first the impulse runs over the region in which the second fibres are refractory. In the figure this is a single solid line. After some time the excitation impulse of the first fibres comes into .a region in which the second fibres are also excitable. Here again a collective excitation impulse arises, travelling in the forward direction, and also an excitation impulse of the second fibres, travelling in the reverse direction. In the case shown in Fig. 5 no further stimulation of the medium is produced, but nevertheless the activity of the medium does not cease. Before reaching the edge of the medium
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the excitation impulse of the second fibres, travelling in the reverse direction, comes into a region in which the fibres of the first type have had time “to rest” and pass out of the refractory state. As a result there arises a collective impulse also travelling in the reverse direction, and an excitation impulse of the fibres of the first type, travelling in the forward direction. The collective impulse travelling in the reverse direction reaches the edge of the medium (X = 0) and here fades away as if adsorbed by the edge of the medium. But the excitation impulse of the first fibres, travelling in the forward direction, after some time excites the collective impulse travelling in the forward direction and the impulse of the second fibres travelling in the reverse direction. Subsequently, this pattern will be repeated a number of times, gradually shifting in the forward direction. In other words, a reverberator arises in the system, moving through the medium with a certain velocity and emitting forward and backward collective excitation impulses, this process occurring in a homogeneous medium. If stimulation on the medium is discontinued, the activity in it will persist until the reverberator reaches the opposite edge of the medium, where it will become extinct. Inside the reverberator a periodical switching of excitation from one type of unit to the other takes place. Excitation runs forward along the units of the first type and backward along the units of the second type. For this reason, we can define the reverberator frequency v,. A simple kinematic calculation, which for lack of space we shall omit here, leads to the following result (25) vr = vc- w/(Kr, - V,r,). In a similar manner we can calculate the velocity of motion of the reverberator V,. It is equal to K = KW2 - rdllKr2 - WJ. (26) It is interesting to note that both v, and V, do not depend on the velocity V, ! The velocity of motion of the reverberator can be very small if the refractory periods of different fibres are similar. In each cycle the reverberator emits one collective impulse forward and one backward. The emission frequency is equal to the reverberator frequency v,. However; owing to the motion of the reverberator itself a Doppler effect should arise, i.e. the frequency of the forward impulses vr, should be higher than the frequency of the backward impulses vb. By means of the usual Doppler effect formulae, we find 1 V, v’=~vI/vc=~; V, K-V, vb=l+c=
(V,+V&,-2V,<’
(27)
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Thus, the frequency of the forward impulses is equal to the maximum possible excitation frequency of the entire medium, l/rz. This also can be seen in Fig. 5. In the case illustrated in Fig. 5, the reverberator arose as the result of a threefold stimulation of the medium. In other cases some other number of stimulations may be necessary to excite the reverberator. Analysis shows that in the general case with the frequency of stimulation v, the number of stimulations necessary to excite the reverberator is determined by the relation
n/>1+vc-VA v, (V, + v-2) x v, x(&if This formula is applicable if the frequency of stimulation lies within the range r2 > l/v > rl. It is clear that if the stimulation period l/v approaches the greatest refractory period r2, the number of stimulations necessary for a reverberator to arise tends to infinity. T’his means that infrequent stimulation produces only the usual collective impulses. To excite a reverberator it is necessary to stimulate the medium with a frequency exceeding its least natural frequency. This fact has a close analogy in electrophysiology. It is known (Hoffinan & Cranefield, 1960) that complex heart excitations such a fibrillation can be brought about by means of several extrasystoles. If the excitations that follow the first excitation, with a frequency greater than the least natural frequency of the medium, are considered as extrasystoles, we may draw an analogy with complex excitation behavior of cardiac syncytium. There is no doubt that many important properties of a real network are not reflected in this scheme; we have considered only the one-dimensional case, and we suppose that in two- and three-dimensional cases one can obtain new effects. It is interesting to note that the reverberator properties do not depend on the mode of excitation. A reverberator always has the same frequency and velocity of motion, and it always emits collective impulses with the same frequency. In this respect, we can say that rhythm transformation is realized in the suggested model. If the stimulation frequency is less than the least natural frequency of the medium (in this case l/r,), this rhythm does not vary and the medium responds with the usual collective impulses. If, however, the stimulation frequency proves to be higher, this stimulation rhythm undergoes transformation and ultimately the medium responds with collective impulses with the frequency l/r2 which is the maximum possible one for it. It is important to note that rhythm transformation occurs not at the stimulation point, but rather at a distance from it (compare Berkinblit et al., 1%6). The impulse going in the retrograde direction may be considered as an echo phenomenon. The first echo can be registered after the
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second stimulation of the medium (Fig. 5). The time of the echo return elapsed since the second stimulation, 6,, depends on the period of stimulation T. A simple calculation gives this function: (2% W? = V,K + V&r2 - TWAKVI). This is plotted in Fig. 6. If T < rl or T > rz, no echo is evoked, because, in the first case, the units of both types are refractory and no impulse can be initiated at all; in the second case, all units are excitable and a collective impulse arises.
FIG. 6. Dependence of latency of reciprocal excitation 0. on time between stimulations T.
After starting the reverberator, if stimulation of the medium is not stopped, one more reverberator will arise after some time, then again one more, etc. In this connection, there arises a question as to how the reverberators will interact with one another and with single impulses. It is well known that when two nerve impulses collide, they extinguish each other. Therefore, in order to destroy a nerve impulse it is sufficient to let another impulse travel toward it. This is not so with a reverberator. It is impossible to destroy a reverberator by a counter impulse. The reason is quite clear: each reverberator has in front of it a protective “coat” of impulses emitted by it. A counter-impulse will destroy only one of these protective impulses, leaving the rest of them untouched. Since the velocity of motion of a reverberator is small, one might attempt to destroy it by sending an impulse in pursuit. However this is also impossible since a reverberator emits protective impulses not only in front of it but also behind it. We should like to note especially that the reverberator can not be destroyed by a pursuing impulse even immediately after its “birth”. So if the stimulation of the medium is not stopped the second reverberator will arise, then the third and so on. Thus in order to destroy a reverberator it is necessary to “bombard” it with a series of a sufficiently large number of impulses. Another reverberator would be most suitable for this purpose. Let us consider the interaction of two reverberators moving toward each other (Fig. 7). For simplicity, we shall not draw a detailed structure of a reverberator, as in Fig. 5. Instead, we shall represent a reverberator by a wavy line and the impulses emitted by it by simple straight lines. The two reverberators move toward each other, but the one moving in the positive direction
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arose somewhat later. Its “coat” of protective impulses will then be thinner, and as a result it will be the one to be destroyed. A case is possible when both reverberators will perish as a result of interaction. This may occur under symmetrical conditions (Fig. 8). In conclusion, let us consider “the pursuit” of reverberators. Let two reverberators follow each other (Fig. 9). In this case neither is destroyed. They both “bombard” each other and thus protect themselves successfully. Colliding, the protective impulses perish where simple straight lines intersect in the figure.
Fro. 7. Collision of reverberators involving destruction of one of them.
FIQ. 8. Collision and destruction of two reverberators.
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\ reverberators.
5. Decay of Group Excitation Let us consider a medium consisting of units of both types whose excitation thresholds differ markedly. For definiteness let the excitation threshold of units of the first type be higher, and the velocity, V,, of the impulse running along these first type units be lower than in the case of units of the second type (velocity V2). In addition, let us assume that the impulse running along the fibres of the second type cannot excite fibres of the first type on account of their high threshold. Back excitation, however, is possible. There need not be a great difference between the thresholds for such a phenomenon to be observed. It is necessary only that the maximum potential of the impulse running along the second fibres should be confined between thresholds cp: and qz. Using the formula (7), one can easily find the maximum potential and hence the condition of the breakup of the group excitation: fpf < (XJV,ZRC”) In J;/[(J; +Jg) e-Y22RCr2’-J; e-Y2ZRC(*2’+*21)]< qy. (30) In such a medium a complex excitation behavior can be observed even with a single stimulation. Let stimulation be applied at the point x = 0, so that fibres of both types should be excited at the same time. Owing to a considerable difference in the thresholds, no collective impulse arises in this case. Instead, excitation breaks up into two single impulses, each of them running along its own units. As before, we shall represent the impulse running along the units of the first type by a solid line and the impulse running along the units of the second type by a dashed line (Fig. 10). The slope of the dashed lines is steeper since the velocity of the respective impulses is greater. After application of stimulation, the excitation impulse of the second fibres is the first to move forward. It does not excite the fibres of the first
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excitation.
type owing to their high threshold. It is followed by an excitation impulse of the fibres of the first type moving with a lesser velocity. In principle, this impulse can excite the fibres of the second type, but this does not take place since they appear to be in the refractory state. However, owing to the velocity difference, the distance between the impulses gradually increases and after some time the impulse running along the first fibres enters the region in which the second fibres have already come out of the refractory state. Here the second fibres are again excited and a fast impulse with the velocity V, will pass through them forward and backward. All this will be repeated periodically. Thus, once produced, the slow excitation wave in the first fibres generates periodically fast excitation waves in the second fibres, which become detached from it and move away forward and backward. Let us find the parameters of such excitation behavior. An analysis shows that the emission frequency of fast waves is equal to v,, = (V’- V,)/V,r,. Due to the Doppler effect, the frequency of forward and backward waves will be different:
Thus, the fast waves run forward with the maximum possible frequency l/r, and back with a somewhat lesser frequency. It can be readily seen that there is a close analogy between the properties of the excitation behavior described here and, the reverberator properties described earlier. For this reason, the consideration regarding the pursuit and collision of reverberators discussed above are applicable here as well.
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6. Discussion
We suppose that this theory can describe some phenomena in heart excitation. Indeed it is usually supposed that cardiac tissues form a heterogeneous system of different units with a common axoplasma (Brooks, Hoffman, Suckling & Orias, 1955). The sundry cardiac tissues differ quantitatively in their properties. Auricular and ventricular muscles differ in their electrical excitability, conduction velocity and limiting frequency for the propagation of impulses, and the magnitude, pattern and duration of their action potential. The atrioventricular (AV) node, and His and Purkinje fibres differ from the auricle and ventricle, and also differ from each other. So the macrohomogeneous multicomponent excitable model formulated above seems to be an adequate one. Here we studied only a one-dimensional problem of the two-component excitable medium. But even in this rather simple case some interesting
-T FIG. 11. Dependence of time separation two subsequent registrations 0. on time separating stimulations T.
results can be obtained. We demonstrated the existance of the rhythm transformation phenomenon, which was observed in a heart long ago (Lucas, 1910). The impulses initiated at the auricle or ventricle with significantly different delays after a first impulse, may reach the distal muscle with a constant time interval after the arrival of the first. The same phenomenon in a single tissue was demonstrated in a number of papers (Moe, Preston & Burlington, 1956; Rosenblueth, 1958). This phenomenon was explained by the existence of the relative refractory period (Moe et al., 1956) and by a temporary stop of the impulse (Rosenbluth, 1958). It can be explained on the basis of the present theory. Indeed, let us consider two subsequent stimulations of the medium with period T (Fig. 5), and register the time separation 8, of the impulses at the distal part of the medium. It is then easy to realize that we shall have the curve given in Fig. 11. If T > r2, we have a straight line with a slope equal to one. In the interval between r1 and r2 there is a horizontal line. The explanation is very simple. Only the units of type 1 are excitable and the impulse travels through them until it excites a collective impulse. This collective impulse will arrive at the
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distal part with time separation rz. Such dependences were observed by Moe et al. (1956) and Rosenblueth (1958) in the atrioventricular node. It should also be noted that in the literature up to now there is discussion of the problem of the existence of two conducting paths in the atrioventricular node (Moe et al., 1956; Vermeulen & Wellens, 1971). Now we should like to discuss briefly the manner in which this theory can be of value in elucidating the genesis of cardiac arrhythmias. It is well known that disorders of conduction are a possible cause of arrhythmia (Hoffman, 1966). This mechanism implies two different pathways for nerve impulses, separated by a region of non-conducting tissue. Making such morphological assumptions it is not a difficult problem to obtain re-entry phenomena or circulation of impulses around an obstacle (Wiener & Rosenblueth, 1946 ; Krynsky, 1968). But the question is whether we can really find such nonconducting regions in the heart. It is well known that it is possible to initiate flutter or fibrillation in purely homogeneous regions of the heart (Scherf, 1966). In this paper we showed that arrhythmia can be obtained in a macroscopically homogeneous two-component medium. We can say that in a two-component medium a moving ectopic centre arises which is a source of nerve impulses. The parameters of this centre are not functions of morphological inhomogeneities of the medium. This theory explains some new experimental results about arrhythmia (Kukushkin t Sakson, 1971). As an example of a two-component excitable medium Kukushkin & Sakson (1971) recently studied the muscle fibres and Purkinje fibres in a frog heart. Histologically, ordinary cardiac muscle fibres and Purkinje fibres are continuous, one merging into the other without morphological interruption (Kugler & Parkins, 1956). In spite of some differences in electrophysiological properties, excitation can cross the junctional region between the two types of fibres in either direction (Hoffman & Suckling, 1953). The shorter action potential of the muscle fibres allows it to be re-excited earlier than the Purkinje fibres (Kao & Hofhnan, 1958). Kukushkin & Sakson (1971) studied the excitation of the frog heart ventricle under two subsequent stimulations, and interpreted the second excitation as a reciprocal one, or as an echo phenomenon. Present theory explains this phenomenon and gives an equation (29) for a latency 0, (see also Fig. 6). In preparations of heart tissue with high latency a series of impulses arise after the second electrical stimulus, while in preparations with small latency only a local response was measured. From equations (28) and (29) of this paper we can obtain the critical condition for initiation of arrhythmia : O,jrl > 1. (32)
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From this relation it is clear that arrhythmia in agreement withexperimental results must appear in preparations with high latency. To obtain more obvious evidence of the reality of this theory it is necessary to carry out measurements with multi-electrode registration, which permits one to obtain the electrical pattern of the preparation in the space. New available methods in electrophysiology of heart allow the hope that such measurements can be done in the immediate future. REFERENCES BELLMAN, R. (1962) ed. Mathematical Problems in the Biological Sciences. Providence, RI. : American Mathematical Society. BERKINBLIT, M. B., KOVAL-EV, S. A., SMOUANINOV, V. V. & CHAILAKHJAN, L. M. (1966). In MoaWy struct~~-f~tsio~~noy organizatsii nekotorykh biologichesk~h system. (I. hi. Gelfand, ed.) p. 71. Moscow, “Nauka”. BROOKS, C. McC., HOFFMAN, B. F., SUCKLING, E. E. & ORIAS, 0. (1955). Excitability of the Heart. New York: Grune. COOLEY, J. W. & DODGE, F. A. (1966). Biophys. J. 6, 582. HOD~KXN, A. L. & HUXLEY, A. F. (1952). J. Physiof., Land. 117, 500. HOFFMAN, B. F. (1966). In Mechanisms and Therapy of Cardiac Arrhythmias, (L. S. Dreifus & W. Likoff, eds.) p. 27. New York, London: Grune and Stratton. HOFFMAN, B. F. & CRANFIELD, P. F. (1960). Electrophysiofogy of the Heart. New York: McGraw-Hill. HOFFMAN, B. F. & SUCKLING, E. E. (1953). Am. J. Physiol. 173, 312. HUXLEY, A. F. (1959). J. Physiol., Lund. 148, 80. KAO, C. Y. & HOFFMAN, B. F. (1958). Am. J. Physiol. 194, 187. KOMPANEETZ, A. S. & GUROVICH, V. Ts. (1966). Biojizika 11, 30t. KRyNsKY, V. J. (1968). Problem? Kibern. 20, 59. KUGLER, J. H. & PARKINS, J. B. (1956). Anat. Res. 126, 335. KIJKUSHKIN, N. I. & SAKSON, M. E. (1971). Biofizika 16, 135. LANDAU, L. D. & LIFSHITZ, E. M. (1957). Electrodinamika sploshnykh sred. Moskva: G.I.T.T.L. LUCAS, K. (1910). J. Physiol., Land. 41, 368. MARKIN, V. S. & CHI~MADJEV, Yu. A. (1967). Biofizika 12, 900. MARKIN, V. S., PASTUSHENKO, V. F. (1969). Biofizika 14, 326. MARKIN, V. S. (1970). Biofzika 15, 120. MOE, G. K., PRESTON, J. B. & BURLINGTON, H. (1956). Circulation Res. 4, 357. PASTUSHENKO, V. F. & MARKIN, V. S. (1969). Biofizika 14, 517. PASTIJS~NKO, V. F., MARKIN, V. S. & CHISMADJEV, Yu. A. (1969). Biojizika 14,883,1072. ROSENBLUHTH, A. (1958). Am. J. Physiol. 194, 171. SCHERF, D. (1966). In Mechanisms and Therapy of Cardiac Arrhythmias, (L. S. Dreifus & W. Likoff, eds) p. 129. New York, London: Grune & Stratton. SMOLJANINOV, V. V. (1969). Biofiika 14, 336. SPALDING, D. B. (1957). Proc. R. Sot. A 240, 83. UNDRO~INAS, A. I., PASTUSHENKO, V. F., MARKIN, V. S. (1972). Dokl. Akad. Nauk SSSR 204, NI. VERMEULEN, A. & WELLENS, W~NER, N. & ROSENIILUETH,
t English translation Biophysics.
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