Initiation of excitation waves
| C. Frank Starmer (2007), Scholarpedia, 2(2):1848. | doi:10.4249/scholarpedia.1848 | revision #129751 [link to/cite this article] |
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.
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 Fig <ref>F1</ref>, <ref>F2</ref> and <ref>F3</ref> 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.
Model of Excitable Switch and Cable: Background
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: \[\frac{\partial U}{\partial t} = f(U) + W \] 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 (Fig.<ref>F4</ref>) 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 \(U(0) > Q\).
If we set \(U(0) > Q\) the phase point migrates from \(Q\) to \(S\), a stable equilibrium which we shall label excited.
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:
- <math ZFK>\frac{\partial U}{\partial t} = f(U) + W + D \nabla^2 U</math>
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. <review> Is this applicable to FHN? Is it true that there is minimal length that should be initially excited, regardless of the perturbation amplitude? If not, the above statement must be amended as now it leaves a false impression. (Reviewer A)
The minimal length criteria applies to FHN. Excited regions below some spatial extent will not propagate, independent of the perturbation amplitude. The initial condition for the potential, U, can be viewed as the result of an increase the charge carrier within the perturbed region. For propagation to succeed, these charges must diffuse into adjacent regions thus increasing the potential in these regions until threshold is reached. The mobility of these charge carriers is determined by the diffusion constant and this relationship (at least as I understand it) is the basis of the liminal region.
My problem is I do not believe in this qualitative argument. I think that if the stimulus amplitude is high enough it can compensate for the lack of spatial extent. It seems to me that the cited paper by Neu et al 1997 contains rather convincing evidence that stimulus width can be arbitrary small provided that the amplitude is large enough.
Clearly, my qualitative argument was not a clean mathematical argument, rather a qualitative physical argument. I have altered the text to refer to Neu's analytical result, indicating that the equivalent of a delta-like function with zero width is able to excite a cable with the amplitude based on transferring the appropriate charge. This is obviously not possible with real physiologic systems and the wording clarifies this issue. Thanks for pushing me to dissociate the mathematical argument from the physical argument (physical meaning with realistic physiologic electrical parameters). </review>
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.
Fig.<ref>F10</ref>, 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, momemtarily 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.
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. Fig.<ref>F12</ref> displays the initial condition that either initiates a pair of propagating trigger waves (green) 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, Fig.<ref>F12</ref> 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.
Neu's results for initiating a propagating disturbance with pulse stimulation are an important theoretical characterization of requirements to form an excited region > the critical nucleus. In a physical sense, though, the membrane properties limit the amount of current that can be transferred from an excited region to adjacent cells. Consequently the physical equivalent of an infinitely thin pulse is not achievable in a physiciologic system. From a physiological perspective, the liminal region depends on conditions local to the region of excitation as these properties limit the available current required to diffuse into adjoining cells.
An analytical characterization of the critical nucleus
Figs <ref>F10</ref> and <ref>F12</ref> 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 Fig.<ref>F4</ref>).
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 Fig.<ref>F17</ref> 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, \(E\) (Fig.<ref>F18</ref>). Integrating again \[ \int{\frac{dU}{\sqrt{2(E - V(U)}}} = x - x_0 \] Rewriting this in terms of the expanded potential fuction 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 Fig.<ref>F19</ref>.
\[ \begin{matrix} \tanh^2[\mu(x - x_0)] = d \frac {U - \beta}{U - \alpha} \\ d = \frac {\alpha - \gamma}{\beta - \gamma} = 3.418637 \end{matrix} \] 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) of 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\[ 1.01 U_{crit}(x)\]. Fig.<ref>F20</ref> shows the dynamics of a collapsing pulse where \(U(x) < U_{crit}(x)\), while Fig.<ref>F21</ref>shows a stationary pulse where \(U(x) = U_{crit}(x)\) and the Fig.<ref>F22</ref> shows the dynamics of an expanding pulse where \(U(x) > U_{crit}(x)\)
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.
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 requirments 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
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- 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. (1989), "The Stable manifold of the standing wave of the Nagumo equation", J. Differential Equations, 80: 306-314.
- Flores, G. (1991), "Stability analysis for the slow traveling pulse of the FitzHugh-Nagumo system", SIAM J. Math. Anal. 22: 392-399.
- 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. (1997) "Initiation of propagation in a one-dimensional excitable medium", Physica D. 102:285-299.
- 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
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
