Why Do Oscillations Occur
If the following three conditions on the Morris—Lecar equations hold, then oscillations will occur:
• the V nullcline has the inverted "N" shape like that in Figure 2.9B;
• a single intersection of the V and wnullclines occurs between the maximum and minimum of the "N;"
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Figure 2.10 (A) The K+, Ca2+, and total current (1Ca + 1K + ieak — Iapp) when w = 0.35. States 1 and 3 are stable steady states, and state 2 is unstable as indicated by the velocity vectors. (B) The phase plane for the MorrisLecar model for Iapp = 150 pA, except that t(V) has been increased by a factor of ten. The values w = 0.468 and 0.251 correspond to the maximum and minimum of the Vnullcline. The points 1, 2, and 3 at w = 0.350 are the steady states in (A). Inset shows the voltage record for a single spike.
• the rate of change of V is much greater than w.
All three conditions are met for the parameter values in Table 2.4, giving rise to oscillations in Figure 2.9A. The importance of the slow change in w, i.e., the "delay" of the delayed rectifier, can be seen by examining the trajectories in Figure 2.9B. If the rate of change of w were large with respect to V, then the trajectories would not depolarize and hyperpolarize rapidly as they do in Figure 2.9A. This is tested by decreasing the value of the parameter thereby increasing the value of the characteristic time for relaxation of w.
By altering the time scale we see why oscillations occur when the rate of change of V is very much faster than w and the nullclines have the shape in Figure 2.9. In this case, we can treat changes in V under the assumption that w is constant, and we need only to consider the voltage equation with w fixed:
Because only the voltage is changing on this time scale, we can examine its dynamical behavior using the one—dimensional phase portrait, rather than a phase plane. This is shown in Figure 2.10A, where the total current, which is proportional to the rate of change of V, is plotted along with the Ca2+ and K+ currents for w = 0.35 and Iapp = 150 pA. The total current vanishes at three points, which are the steady states for the voltage when w is fixed. States 1 and 3 are stable, and state 2 is unstable, as
C— = gcam<»(V  Vca)  gKw(V  VK)  gL(V  VL) + Japp. (2.38)
indicated by the velocity vectors in the figure. Note that at state 1 the membrane is polarized and the outward K+ current dominates the Ca2+ inward current, whereas the opposite holds true at state 3. As we shall see, the oscillations can be thought of as transitions back and forth between the polarized and depolarized states, driven by slow changes in activation of the delayed rectifier current.
Figure 2.10A also explains why the Vnullcline in the Morris—Lecar model has the inverted "N" shape: The K+ and leak currents exceed the inward Ca2+ current at polarized voltages between states 1 and 2, whereas the Ca2+ current exceeds the other currents between states 2 and 3. This voltagedependent competition between inward and outward currents leads to a maximum and minimum in total current and therefore the inverted "N" shape for the nullcline.
The three steady states of the voltage also can be found graphically in the phase plane of Figure 2.10B by locating the intersection of the line w = 0.35 with the Vnullcline. It is clear from the figure that if w exceeds 0.468 (the maximum on the right branch of the Vnullcline), then the voltage has only a single polarized steady state (intersection with w) on the far left branch of the Vnullcline. Similarly, if w is smaller than 0.251 (the minimum on the left branch of the Vnullcline), then the voltage has only a single depolarized steady state on the right branch. For 0.251 < w < 0.468 two stable steady states and one unstable state occur, and the voltage is said to be bistable.
To understand how bistability on the fast time scale (neglecting w) leads to oscillations in the complete system, we need to understand how w changes on the longer time scale. Assume that initially, the membrane is polarized at state 1 in Figure 2.10B, with w = 0.35. Because w = 0.35 is above the wnullcline at point 1, w will decrease. As w decreases, V will stay close to the Vnullcline because it relaxes rapidly to the closest steady—state value. The trajectory thus follows the polarized branch, as indicated by the heavy line, until the minimum at w = 0.251 is reached. Beyond the minimum (near a), stable polarized states no longer exist, and V rapidly relaxes to the only remaining steady state (near b), which is on the depolarized, far right branch of the Vnullcline. During the depolarization, however, the wnullcline is crossed. Thus on the depolarized branch, w increases and tracks the Vnullcline upward until the maximum at w = 0.468 is reached (near c) and the membrane rapidly repolarizes to the polarized branch (near d).
The abrupt transitions from the polarized to depolarized branch and back again have led to the name relaxation oscillator for systems of equations that have wellseparated time scales. For the parameters in Figure 2.9, the Morris—Lecar model is not a relaxation oscillator. However, when the characteristic time for w is increased by a factor of 10 (by decreasing the limit cycle (heavy line in Figure 2.10B) closely approximates that for a relaxtion oscillator. If the characteristic time were increased sufficiently, or the characteristic time for V were decreased sufficiently by altering the capacitance, then the trajectory would coincide with the bistable portions of the Vnullclines, and the rapid excursions of the voltage would occur precisely at w = 0.251 and 0.468. The inset in Figure 2.10B shows a single voltage spike that illustrates the rapid upstroke and downstroke for the limit cycle. The inset also illustrates that the
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Figure 2.11 Excitability in the MorrisLecar model for Iapp = 60 pA. (A) An initial deviation of the voltage to 22 mV relaxes rapidly to the steadystate voltage, whereas a deviation to 17 mV produces an action potential. (B) The trajectories in (A) represented in the phase plane. When V changes much faster than w, the location of the Vnullcline (longdashed line) sets the threshold for action potential spikes.
shape of the depolarized and polarized portions of the spike reflects the shape of the two branches of the Vnullcline.
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torin stevenson23 days ago
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