The Radial Isochron Clock

Section 10.8 of Intermediate Physics for Medicine and Biology describes the radial isochron clock, a toy model for electrical stimulation of nerve or muscle. Russ Hobbie and I write:

Many of the important features of nonlinear systems do not occur with one degree of freedom. We can make a very simple model system that displays the properties of systems with two degrees of freedom by combining the logistic equation for variable r with an angle variable θ that increases at a constant rate:

We can interpret (r,θ) as the polar coordinates of a point in the xy plane. When [time] t has increased from 0 to 1 the angle has increased from 0 to 2π, which is equivalent to starting again with θ = 0. This model system has been used by many authors. Glass and Mackey (1988) have proposed that it be called the radial isochron clock.

Fig. 10.19 of IPMB.

We use this model to analyze phase resetting. Let the clock run until it settles into a stable limit cycle, in which case the signal x(t) is a sinusoidal oscillation. Then apply a stimulus that suddenly increases x by an amount b (see Fig. 10.19 in IPMB) and observe the resulting dynamics. The system returns to its limit cycle, but with a different phase. The first plot in the figure below shows the period T/T₀ of the oscillator just after a stimulus is applied at T_S/T₀; it's the same illustration as in Fig 10.20b of IPMB. Something dramatic appears to be happening at T_S/T₀ = 0.5. What's going on?

The radial isochron clock for different stimulus times and strengths.

The problem with a plot of T/T₀ versus T_S/T₀ is that I have difficulty relating it to the behavior of the signal as a function of time, x(t). Above I plot x versus t for four cases:

• T_S/T₀ = 0.25, b = 0.95 (blue dot). In this case, the stimulus is applied soon after the peak when the signal is decreasing. The sudden jump in x increases the signal so it has further to fall (it must recover lost ground), delaying its descent. As a result is the signal is a behind (is shifted to the right of) the signal that would have been produced had there been no stimulus (red dashed). The figure for b = 1.05 is almost indistinguishable from b = 0.95, so I won’t show it.
• T_S/T₀ = 0.75, b = 0.95 (red dot). The stimulus is applied after the trough when the signal is increasing. The stimulus helps it rise, so it reaches its peak earlier (is shifted to the left) compared to the signal with no stimulus. Again, the figure for b = 1.05 is similar.
• T_S/T₀ = 0.50, b = 0.95 (green dot). When we apply the stimulus near the bottom of the trough, the behavior depends sensitively on stimulus strength b. If b were exactly one and it was applied precisely at the minimum, the result would be x = 0 forever. This would be an unstable equilibrium, like balancing a pencil on its tip. If b was not exactly one, then the key issue is if the signal starts slightly negative (in phase with the unperturbed signal) or slightly positive (out of phase). For b = 0.95, the signal moves to a slightly negative value that corresponds to a trough, meaning that the resulting signal is in phase with the unperturbed signal.
• T_S/T₀ = 0.50, b = 1.05 (yellow dot). If b was a little stronger, then the stimulus moves x to a slightly positive value corresponding to a peak, meaning that the resulting signal is out of phase with the unperturbed signal. Because T/T₀ = 1.5 is equivalent to T/T₀ = 0.5 (the phase just wraps around), the jump of T/T₀ in the top frame does not correspond to a discontinuous physical change.

The drama at T_S/T₀ = 0.5 and b = 1 arises because the stimulus nearly zeros out the signal. The phase of the signal changes from zero to 180 degrees as b changes from less than one to greater than one, but the amplitude of the signal r goes to zero, so the variables x and y change in a continuous way. Some of the homework problems for Section 10.8 in IPMB ask you to explore this on your own. Try them.

The moral of the story is that an abstract illustration—such as Fig. 10.20b in Intermediate Physics for Medicine and Biology—summarizes the behavior of a nonlinear system, but it can’t replace intuition about how the system behaves as a function of time. You need to understand your system “in your gut.” This isn’t true just for the radial isochron clock; it's true for any system. Forget this lesson at your peril!