Initiation of excitation waves

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C. Frank Starmer (2007), Scholarpedia, 2(2):1848. doi:10.4249/scholarpedia.1848 revision #129751 [link to/cite this article]
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Curator and Contributors

1.00 - C. Frank Starmer

Figure 1: Sustained propagation or collapse depends on the relationship between a disturbed region and the critical nucleus. Black is the single cell threshold. The red, high amplitude pulse, excites a region < critical nucleus and collapses while the blue pulse excites a region > critical nucleus and initiates a propagating front.

Single excitable cells (often described by reaction-diffusion models) at equilibrium have a threshold of excitation (FitzHugh-Nagumo Model). Small transient disturbances rapidly decay while larger disturbances initiate significant responses over a longer period of time. When cells are coupled, a disturbance can propagate. Here we explore the nature of the critical (liminal) disturbed region required to initiate sustained propagation.

Contents

Liminal Excitation Region: Requirements for initiating a sustainable propagating disturbance

The chemistry of matches provides a ready model for exploring requirements for success and failure of a alteration in the match head equilibrium (i.e. flame) to propagate. The pressure associated with dragging the match across an abrasive surface plays a role in the excited region (see Figure 2, Figure 3 and Figure 4 below). Gentle pressure results in little friction and occasionally the interaction of the match-head and heat from the friction produces small spark that self-extinguished. Greater pressure increases the friction which increases the temperature of match-head and produces a propagating flame. Creating a sustainable flame (propagating disturbance in equilibrium) is quite similar to creating a reentrant cardiac rhythm. Cardiac arrhythmias are often the result unexpected excitation of a region of the heart. Starting an excitation wave in the heart is similar to striking a match and is sensitive to the nature of the ignition process: a small region of excitation creates a wave that decays while a larger region of excitation creates a wave that propagates through the heart tissue. This article explains the nature of initiating a propagating disturbance, specifically the requirements for an initial disturbance to propagate or collapse.

Figure 2: Preparing to strike a match
Figure 3: Ignition after rubbing the match against the abrasive surface
Figure 4: Stable propagation following flame size > liminal size


Model of Excitable Switch and Cable: Background

Figure 5: Phase plot of the FHN Switch

In the mid 1930s, Rushton explored the dependence of successful wave formation and propagation on the length of an excited region. Rushton used the term liminal to characterize the threshold excited region necessary for a wave to form and propagate. The FitzHugh-Nagumo model simplified the description of an excitable cell such that Rushton's approach to understanding excitation and propagation could be refined. As a further simplification, let us use a FitzHugh-Nagumo switch that retains the dynamic excitable component but uses a constant inhibitory current.

We start with the FitzHugh-Nagumo (FHN) characterization of a switch: \[\frac{dU}{dt} = f(U) - W \] where \(U\) is the membrane potential, \(f(U) = U(1-U^2)\) is the excitatory current/voltage relationship, \(W\) is a constant inhibitory current. The utility of the FHN model is based on the cubic excitation current/voltage relationship which is the least complex function available for characterizing excitable processes and approximates the current-voltage relationship of cardiac and nerve cells. Here we assume that the time scales of the activation variable \(U\) and the recovery variable \(W\) are so different we can consider \(W\) nearly constant during the whole initiation stage. The behavior of the FHN switch is determined by the phase plane plot (Figure 5) and displays 2 stable states similar to a switch. Of the three zeros, the left (\(P\)) and right (\(S\)) are stable while the middle zero (\(Q\)) is unstable. When the initial phase point has a potential \(U < Q\ ,\) it migrates toward \(P\) while when \(U > Q\ ,\) it migrates toward \(S\ .\)

Numerical experiments with a switch

Here we demonstrate the switch transitions. With \(W = -8/27\) the three equilibria are at \(P\) (\(U = -0.7908\ ,\) stable), \(Q\) (\(U = -0.3333\ ,\) unstable) or \(S\) (\(U = 1.1241\ ,\) stable). We switch the potential from one equilibrium to the other by setting the \(U(0) < Q\) or \(U(0) > Q\ .\)

Figure 6: Phase portrait of a FitzHugh-Nagumo switch transition from with \(U(t = 0) = -0.330\) to \(U(t=\infty) = 1.1241\)
Figure 7: Dynamics of a FitzHugh-Nagumo switch transition from \(U(t=0) = -0.330\) to \(U(t=\infty) = 1.1241\ .\)

If we set \(U(0) > Q\) the phase point migrates from \(Q\) to \(S\ ,\) a stable equilibrium which we shall label excited.

Figure 8: Phase portrait of a FitzHugh-Nagumo switch transition from with \(U(t = 0) = -0.335\) to \(U(t=\infty) = -0.7908\)
Figure 9: Dynamics of a FitzHugh-Nagumo switch transition from \(U(t=0) = -0.335\) to \(U(t=\infty) = -0.7908\)

If we set \(U(0) < Q\ ,\) the phase point migrates from \(Q\) to \(P\ ,\) a stable equilibrium which we shall label resting.

Exploring propagation along an excitable cable (trigger waves)

What happens when switches are joined in a linear array? What are the conditions for sustained switch transitions propagating to adjacent switches? We build an excitable cable by coupling individual switches via a diffusive process: \[\tag{1} \frac{\partial U}{\partial t} = f(U) - W + D \nabla^2 U\]

where \(D\) is the diffusion coefficient. (this is also known as the Zeldovich-Frank-Kamenetskii model, excitation waves are seen here as trigger waves). With this model, we explore the fate of an excited region by altering the initial conditions for \(U(x, t=0)\) and watch how \(U(x,t)\) evolves. Specifically, we shall explore

  • initial conditions that form a wave that subsequently propagates
  • initial conditions that form a wave that subsequently collapses.

Rushton determined that a minimum length of excitable cable must be initially excited in order for a wave to form and propagate. For his explorations, he used a short (5 mm) region that approximates point stimulation. Beyond the excited region, the potential dropped exponentially, a result of passive diffusion along the cable. To simplify analysis, we use a rectangular pulse centered at the middle of the cable for exploring the fate of initial conditions of different amplitudes or spatial extents.

Here we simplify the analysis by using uniform excitation of a short length of cable and demonstrate that the threshold potential, \(U\ ,\) depends on the length of the excited region. Neu's analysis indicated that for broad pulses, the threshold amplitude required for excitation was that of the excitation potential (U(x) > Q) while for narrow pulses, the amplitude of the excited region increases as the pulse width is decreased in a manner that delivers a fixed charge to the cable.

Figure 10: U(36.25 < x < 43.75 t = 0) = -.271 (red, collapsing); -.270 (blue, expanding) Dynamics of an expanding and collapsing pulse

Figure 10, displays the expanding and collapsing dynamics of the transition of the square wave initial condition (width = 7.5 arbitrary units). During the early moments the spatial extent of perturbed region slightly expands at the base and while the peak initially increases a small amount, momentarily hesitates before either expanding or collapsing. The initial condition pulse evolves to a pair of propagating trigger waves if the amplitude, U(peak,0) = -0.270), is greater than some threshold (green curve while the pulse collapses if the initial amplitude, U(peak,0 = -0.271), is less than this threshold. Below, we will illustrate that U(x) during the hesitation interval appears invariant to the width of the perturbed region when the amplitude is near the threshold of propagation.

Figure 11: U(30 < x < 50, t = 0) = -.331 (red, collapsing); -.3305 (blue, expanding) Dynamics of an expanding and collapsing pulse

Increasing the spatial extent of the rectangular initial condition reduces the threshold for propagating trigger waves and alters the initial evolution of the potential, U. Figure 11 displays the initial condition that either initiates a pair of propagating trigger waves (blue) when the potential \(U(30 < x < 50,0) = -0.3305\ ,\) or collapses, \(U(30 < x < 50,0) = -0.331\ ,\) when the amplitude is slightly below threshold for propagation. Similar to the pulse evolution above, Figure 11 shows the square wave initial condition expanding at the base and then contracting while the peak increases, hesitates and then either collapses or continues to grow to become a pair of propagating trigger wave. Again, the fate of the initial pulse is determined by the amplitude.



Results of Neu et al. (1997) for initiating a propagating disturbance with pulse stimulation are an important theoretical characterization of required relationship between stimulus width and strength, narrower stimuli requiring larger amplitude and vice versa (see Figure 20 below). If talking about biological excitability, however, there is a certain upper limit of the stimulus amplitude that an excitable membrane can realistically endure. Correspondingly, this put a certain lower limit to the liminal region.

An analytical characterization of the critical nucleus

Figure 12: Potential function
Figure 13: Phase trajectory

Figure 10 and Figure 11 illustrate the evolution of a pulse initial condition to an intermediate function,\(U_{crit}(x)\ ,\) before either collapsing or expanding into a pair propagating trigger waves. Here we explore a possibility of \(U_{crit}(x)\) as the solution for the stationary pulse, e.g. the critical nucleus, that satisfies: \[U_{xx} + U(1 - U^2) + W = 0 \] with the boundary condition, \(U(x) \rightarrow P \) as \(\mid X \mid \rightarrow \infty\) and \(P\) is the smallest (real) root of the cubic equation \(U - U^3 + W = 0\) (see Figure 5).

To find the stationary solution, set \(U_t=0\) and rewrite the above equation as \[U_{xx} = U(U^2 - 1) - W = -\frac{dV(U)}{dU}\] where \(U_x = dU/dx\) and introduce the potential function, \(V(U)\) as shown in Figure 12 which exhibits two stable eigenvalues and 1 unstable eigenvalue. \[ V(U) = WU + \frac{U^2}{2} - \frac{U^4}{4} \] Integrating \(U_{xx}\) yields \[ \frac{U^{'2}}{2} + V(U) = E = constant \] a family of trajectories that depend on the value of the integration constant, \(E\) (Figure 13). Integrating again \[ \int{\frac{dU}{\sqrt{2(E - V(U)}}} = x - x_0 \] Rewriting this in terms of the expanded potential function results in \[ \int{\frac{\sqrt{2}dU}{\sqrt{(U-\alpha)(U-\beta)(U-\gamma)(U-\delta)}}} = x - x_0 \] where \(\alpha > \beta > \gamma > \delta \) are roots of the quartic \(E_0 + \frac{U^4}{4} - \frac{U^2}{2} - Wu\ .\) When \(W = -8/27\) then \(c_{min} = -0.79076\) and \(E_0 = V(P) = -0.0194\ .\) The roots of \(E_0 - V(U)\) are: \[ \gamma = \delta = P = -0.79076, \beta = -0.0749225, \alpha = 1.656437. \] Using the theory of Jacobi elliptic functions, integration yields the stationary wave as shown in Figure 14.

Figure 14: Stationary U(x)

\[ \begin{matrix} \tanh^2[\mu(x - x_0)] = d \frac {U - \beta}{U - \alpha} \\ d = \frac {\alpha - \gamma}{\beta - \gamma} = 3.418637 \end{matrix} \] with \( \mu = \frac{1}{2\sqrt{2}} \sqrt{\beta - \delta} \sqrt{\alpha - \gamma} = 0.467948 \) Let \(\mu^{-1}\) be a measure of the liminal width of the stationary wave (the stable manifold of the solution) that separates collapsing pulses from expanding pulses. Then solving for \(U_{crit}(x)\) \[ U_{crit}(x,0)= \frac{\alpha \tanh^2(\mu x) - \beta d}{\tanh^2(\mu x) - d} , -\infty < x < \infty \] where \( \alpha, \mu, \beta \) and \(d\) are functions of the roots of the potential function.

The critical nucleus as an unstable stationary wave

The nature of the critical nucleus can be readily demonstrated by using initial conditions that are scaled larger than, equal to or smaller than the critical nucleus. These cases are shown below: A pulse slightly smaller than the critical nucleus\[0.99 U_{crit}(x)\ ;\] a pulse equal to the critical nucleus\[U(x,0) < U_{crit}(x)\ ;\] and a pulse slightly greater than the critical nucleus\[ 1.01 U_{crit}(x)\ .\] Figure 15, Figure 16 and the Figure 17 illustrate the time dependent dynamics of a collapsing pulse, a stationary pulse and an expanding pulse.

Figure 15: \(U(x) < U_{crit}(x)\ :\) collapsing pulse
Figure 16: \(U(x) = U_{crit}(x)\ :\) stable pulse
Figure 17: \(U(x) > U_{crit}(x)\ :\) expanding pulse


The stimulus strength - spatial extent relationship

Typically experiments with excitable tissue, small external electrodes are used to inject charge into the coupled cells where the stimulus duration is changed in order to alter the amount of injected charge. These strength-duration experiments revealed a minimal current (rheobase) required to initiate propagation as well as a minimum pulse width. To avoid the complexity of injecting current at a single point (or over a small region) as with experiments, here we alter the spatial extent of the initial condition and then determine the threshold for propagation as a function of spatial extent of the excited region. Pulse widths are selected so that part falls within the critical nucleus while other parts fall outside. Shown below are two examples: Case 1 (left), the spatial extent is 1.25 cells, less than the spatial extent of the critical nucleus and the amplitude exceeds that of the peak amplitude of the critical nucleus. Case 2 (middle), the spatial extent is 10 cells and exceeds that of the critical nucleus while the amplitude is less than the peak of the critical nucleus.

The pulse evolution following the initial condition demonstrates that for narrow initial conduction pulses, the subsequent spatial distribution of potential, the maximum U(x,t) either collapses to that of the peak potential of the critical nucleus while the spatial extent increases to that comparable to the spatial extent of the critical nucleus (Case 1). For wide initial conditions, the maximum U(x,t) increases to that of the peak of the critical nucleus while the spatial extent collapses to approximate the spatial extent of the critical nucleus. In both cases, the evolution in potential from the initial condition to the critical nucleus is the result diffusion. As shown above, the development of U(x,t) hesitates while approximating the profile of the critical nucleus before either collapsing or expanding to become a pair of propagating trigger waves.

Figure 18: Initial Condition: U(39.375 < x < 40.625, t = 0). Dynamics of propagation and collapse of a narrow pulse with an amplitude near the threshold for propagation. U(peak ic) = +0.8180 (green, propagation) and U(peak ic) = +0.8179 (red, collapse). The critical nucleus is blue
Figure 19: Initial Condition: U(30 < x < 50, t = 0). Dynamics of propagation and collapse of a wide pulse with an amplitude near the threshold for propagation. U(peak ic) = -0.3020 (green, propagation) and U(peak ic) = -0.3030 (red, collapse). The critical nucleus is blue
Figure 20: The propagation threshold initial condition amplitude as a function of initial condition pulse width for an FHN cable. This relationship parallels the strength-duration curves experimentally determined with point stimulation of cardiac muscle or nerve fibers


Summary

From a simple FitzHugh-Nagumo switch we have identified conditions that define whether an initial disturbance of an excitable cable will collapse or evolve to propagating trigger waves. We found an analytical solution for a standing wave in a medium characterized by reaction and diffusion that depends only on model parameters. These results are quite useful when exploring how spiral waves and reentrant cardiac arrhythmias arise.

Spiral waves and reentrant cardiac arrhythmias are often initiated by forming a discontinuous propagating wave. Discontinuities can result from collisions of a continuous wave with an obstacle or can arise from disturbances in a medium with spatially non-uniform excitability, a vulnerable medium (see Vulnerability of Cardiac_Dynamics). The analytical results above provide a basis for identifying conditions that support either wave formation or wave fractionation by testing whether liminal requirements are met. For example as a wave collides with an obstacle, will the outcome produce fragmentation of the front or restore front continuity? Similarly, these results are of use in studies of wave diffraction. When a wave propagates in a two dimensional medium and collides with an obstacle containing a slit, propagation beyond the slit depends on the relationship between the width of the slit and the liminal region. When the slit dimensions are greater than the liminal region, then the wave fragment passing through the slit will continue to propagate. In addition, the ends of the newly formed wave fragment will curl and form counter-rotating spirals.

References

  • Rushton, W.A.H. Initiation of the propagated disturbance. Proc. R. Soc. Lond. (Biol.) 124:210-243, 1937.
  • FitzHugh, R. Impulses and physiologic states in theoretical models of nerve membrane. Biophysical J. 1:445-466, 1961
  • Nagumo, J., Arimoto, S. and Yoshizawa, S. An active pulse transmission line simulating nerve axon. Proceedings of the IRE 50:2061-2070, 1962.
  • Noble, D. and Stein, R.B. The threshold conditions for initiation of action potentials by excitable cells. J. Physiol (London) 187:129-162, 1966.
  • Flores, G. The Stable manifold of the standing wave of the Nagumo equation, J. Differential Equations, 80: 306-314, 1989.
  • Flores, G. Stability analysis for the slow traveling pulse of the FitzHugh-Nagumo system, SIAM J. Math. Anal. 22: 392-399, 1991.
  • Fozzard, H.A. and Schoenberg, M. Strength-duration curves in cardiac Purkinje fibers: Effects of liminal length and charge distribution. J. Physiol. (London) 226:593-618, 1972.
  • Neu, J. C., Preissig, R. S. and Krassowska, W. Initiation of propagation in a one-dimensional excitable medium, Physica D. 102:285-299, 1997.
  • Bountis, T., Bezerianos, T. and Starmer, C.F. Wavefront formation in an excitable medium by perturbation of solitary pulse solutions. Progress in Theoretical Physics, 139:12-33 (supplement), 2000
  • Ostrovskii, L.A. and Yakhno, V.G. Formation of pulses in an excitable medium. Biofizika 20:489-493, 1975
  • Starobin, J., Zilberter, Y.I. and Starmer, C.F. Vulnerability in one-dimensional excitable media. Physica D. 70:321-341, 1994
  • Starobin, J. M. and Starmer, C.F. Boundary Layer Analysis of Waves Propagating in an Excitable Medium Phys Rev E 54:430-437, 1996

Internal references

External Links

See Also

Alternans, Barkley Model, Cardiac Arrhythmia, Drift of Spiral Waves, FitzHugh-Nagumo Model, Noble Model, Models of Cardiac Cell, Models of Heart, Reaction-Diffusion Systems, Restitution, Scroll Wave, Scroll Wave Turbulence, Spiral Breakup, Spiral Waves, Symmetry Breaking in Reaction-Diffusion Systems, Traveling Wave Vulnerability of Cardiac Dynamics

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