the impulsive stimuli term, consider the example of a population of many fireflies, and imagine that the stimuli at $t=0$ and at $t=\tau $ are external flashes of light at those times, where the constants $\hat d_0$ and $\hat d_1$ in Eq. (2) represent the intensity of these flashes. We hypothesize that a firefly will be induced by a stimulus flash to move its flashing phase toward synchronism with the stimulus flash. Thus a firefly that has just recently flashed will reset its phase by retarding it, while a firefly that was close to flashing will advance its phase. The amount of advance or retardation is determined by the ‘reset function’, $h(\theta )$. Since the reset function $h(\theta
)$ depends on properties of the fireflies, we do not specify it further. Let $\theta ^+_i$ and $\theta ^-_i$ represent the phases of oscillator $i$ just after and just before a stimulus flash at $t=0$ or $t=\tau$. Then we have from Eq. (1) that $$\int ^{\theta ^{+}_{i}}_{\theta _i^-} \frac{d\theta }{h(\theta
)}=\hat d_p ; \ \ p=0,1 \ .$$ Letting $F(\theta )=\int ^\theta d\theta /h(\theta )$, we obtain $$\theta ^+_i=F^{-1}(\hat d_p+F(\theta ^-_i)) \ .$$ In our subsequent analysis in Sec. III, we will 