The essence of minimal models of spark-mediated Ca2+ waves is to model Ca2+ release sites as idealized point sources, that is, an array of Dirac delta functions, denoted by ¿(x — xi), where is the spatial position of the ith release site. While the Dirac delta function is actually not a function at all (rather, it is a distribution, the limit of a sequence of functions), it is often thought of as a sharply peaked function that is zero everywhere except x = x^ In addition, the delta function is normalized so that

and it has the so-called sifting property f (x)«(x - x^dx = f (x4). (8.23)

Armed with delta functions, we can model spark-mediated Ca2+ waves by modifying the propagating front model (8.15) so that Ca2+ release occurs only at regularly spaced release sites, that is,

i where xi is the location of the ith release site with kinetics given by f (c) [Keener, 2000].

We can further idealize this model for spark-mediated waves by removing the Ca2+-dependence of the release rate f (c). Instead, assume that the source strength fi(t) of each release site is a square pulse function of time, fi(t) = — H(t - ti)H(ti + rR - t).

In this expression H(t) is the Heaviside step function, so that H(t) = 0 for t < 0, H(t) = 1 for t > 0. The product of these Heaviside functions represents the source turning "on" at time t = ti and remaining on for duration tr, i.e., it turns "off" at time t = ti + tr. The constant a represents the source amplitude and has units of ^M • ^m because from (8.22) we see that the delta function has units of 1/^m. The normalization factor (1/tr) is chosen so that fi (t) dt = a. (8.25)

Substituting this form for the Ca2+ release rate into (8.24) gives the "fire-diffuse-fire" model of spark-mediated Ca2+ wave propagation, dc d2c a

— = D— + — V «(x - xi)H(t - ti)H(ti + tr - t). (8.26)

To finish our presentation of the fire-diffuse-fire model, we must specify the location of each release site (xi) and the times (ti) at which Ca2+ release begins. If we further assume a regular array of release sites, then xi = id where d is the site spacing. Because we have Ca2+-induced Ca2+ release in mind, we assume that Ca2+ release at site i begins when the local Ca2+ concentration c(xi,t) achieves a fixed threshold Ca2+ concentration cg (here 0.1 ^M). In Exercise 12 the reader can implement such a fire-diffuse-fire simulation.

Interestingly, Figure 8.12 shows that the fire-diffuse-fire model supports continuous as well as propagating Ca2+ signals. In Figure 8.12A, the time constant for Ca2+ release, tr, is 1 s and the propagating signal is similar to the traveling front solutions presented earlier in this chapter. Conversely, Figure 8.12B presents a simulation using tr = 10 ms. Here spark—like Ca2+ releases lead to a propagating signal that is distinctly saltatory. Note that the continuous wave is traveling at 11.3 ^m/s. while the saltatory wave is traveling at 67 ^m/s. The long duration of Ca2+ release in the continuous case appears to slow the velocity of the propagating signal.

The fire-diffuse-fire model can be analyzed to give insight into the continuous and saltatory limits of Ca2+ wave propagation. As the reader may expect, it is not tr alone but rather a dimensionless parameter that determines the existence and form of propagating signals. Indeed, it can be shown that the relevant dimensionless quantity is DTR/d2. The continuous limit corresponds to DTR/d2 ^ 1 and the saltatory limit to DTR/d2 1. Below we study these two limits separately to determine (you guessed it) how diffusion influences wave velocity in both limits.

The continuous limit pertains when DTR/d2 ^ 1, that is, when diffusion is fast and the release time long compared to the intersite separation. This limit could be achieved in a simulation by increasing the density of release sites (d ^ 0) while simultaneously decreasing the release rate (ct) so that the release per unit length (^/d) is constant.

Because the thresholding that determines Ca2+ release times (t4) is the only nonlinear part of our model (8.26), we can convolve the release rates /¡(t) with the diffusion kernel to write an explicit expression for c(x, t):

where we have assumed a symmetric profile c(x, t) and the firing of N — 1 sites on either side of the origin. Equation (8.27) is not yet a solution because we have the unknown parameters ti to determine. To find the time at which site N fires we evaluate (8.27) at (t = tN,x = xN) and set c(xN, tN) = cg. This yields ce =

an expression that can be solved for tN. At this point we seek traveling-wave like solutions with regular firing times. This implies that ti = ¿A, where A is the time interval between adjacent site firings and a velocity of propagation given by v = d/A. By substituting A = tN/N and solving for A it can be shown [Keizer et al., 1998] that the velocity of such a propagating wave has the following dependence on model parameters:

Figure 8.13 Relationship between a = cgd/a and the dimensionless firing interval (A = 1/v) for the fire-diffuse-fire model in the saltatory limit. Since g(A) < 1, waves do not propagate for a > 1. From [Keizer et al., 1998].

Figure 8.13 Relationship between a = cgd/a and the dimensionless firing interval (A = 1/v) for the fire-diffuse-fire model in the saltatory limit. Since g(A) < 1, waves do not propagate for a > 1. From [Keizer et al., 1998].

Notice that the factor of -\Jd/tr is similar to what we have seen before for traveling fronts and pulses. The existence of traveling wave solutions depends on \Jir/dc9.

The saltatory limit of the fire-diffuse-fire model corresponds to DTR/d2 ¿Ï 1 in (8.26), that is, when diffusion is slow and release time short compared to the intersite separation. This limit could be achieved in a simulation by decreasing the time constant for Ca2+ release (tr) while maintaining a fixed density of release sites (d constant). Because the normalization factor (1/tr) is chosen to satisfy (8.25) regardless of tr, the release rate per unit length (a/d) remains constant as tr ^ 0, and (8.26) becomes ddc d2 c dt = ^ + - xi) ¿(t - ii)- (8.29)

We now nondimensionalize space (x* = x/d, x** = xi/d = i), time (t* = tD/d2, t** =

tiD/d2), and concentration (c* = c/c9) and drop asterisks to write c 2c 1

i where a = cgd/a. Using the diffusion kernel we obtain an implicit expression for c(x, t),

where the presence of the Heaviside function indicates no contribution from sites that have not yet fired.

Assuming that the sites for which — N < i < N have fired at times ti through tN (left and right pairs simultaneously) we want to determine tN+1. The dimensionless

0.54

Figure 8.14 For a range of values for a, the saltatory limit of the fire-diffuse-fire model is simulated by successively calculating the dimensionless firing interval (A) for n = 1, 2, ••• using (8.32) and the criterion c(n,U) = 1. The wave was initiated by simultaneously firing all sites for -15 < n < 15. A period doubling cascade begins at a ~ 0.512 and terminates near a ~ 0.535, beyond which waves do not propagate. From [Keizer et al., 1998].

0.54

Figure 8.14 For a range of values for a, the saltatory limit of the fire-diffuse-fire model is simulated by successively calculating the dimensionless firing interval (A) for n = 1, 2, ••• using (8.32) and the criterion c(n,U) = 1. The wave was initiated by simultaneously firing all sites for -15 < n < 15. A period doubling cascade begins at a ~ 0.512 and terminates near a ~ 0.535, beyond which waves do not propagate. From [Keizer et al., 1998].

threshold for firing is now c = 1, so from (8.31) we have c(xN+1 ,tN+1) — 1 —

Because we are interested only in long time solutions, we consider the large N limit and neglect all the terms in the above sum with i < 0, that is, we are following a rightward traveling wave that eventually is not influenced by the sites to the left of the origin. Under this approximation (8.32) simplifies to

This expression is an implicit map for tN+1 as a function of all the previous firing times, (tN, tN-1, ...,t1) ^ tN+1. traveling-wave-like solutions correspond to fixed points of this map with regular firings, that is, tN+1 — tN + A with constant A (giving a dimensionless velocity of v — 1/A). Substituting A — tN+1 - tN in (8.33) we obtain a —

Defining n — N + 1 - i and taking the limit N ^ro we obtain

4nnA

It can be shown that 0 < g(A) < 1. The first equality holds in the high—velocity limit (A ^ 0) and the second for low velocity (A ^ ro). Since g(A) is monotonic, we can numerically calculate a unique solution A — g-1(a) when 0 < a < 1, i.e., the range of g(A). At this point it is worth remembering that a — cgd/u. Thus, a necessary condition for the existence of a velocity is that a < 1. If the sites are too far apart or

too weak or the threshold too high, there can be no propagating waves. Interestingly, it can be shown that propagation failure occurs through a sequence of instabilities as a is increased well before the condition a = 1 is reached (see further reading). When a is small (a/dcg ^ 1), propagating wave solutions exist and are stable. It can be shown that the dimensional velocity v in this saltatory propagation limit is given by

that is, it scales linearly with D, quite unlike propagating waves in the continuous limit, which have velocity that scales with \/~D as in (8.28). If fact, whenever tr is sufficiently small (even if a is not particularly small), the saltatory wave velocity predicted by the fire—diffuse—fire model scales linearly with D.

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