However, time constants do not always capture the effective time scales.
This notion of effective time scales for the underlying autonomous system is connected with its eigenvalues.
The initial time interval of transient behavior that determines the effective time scales depends on both the eigenvalues' real part and Ω nat, and not only on Ω nat.
Figure 4 also illustrates the roles of α and ϵ in determining both the effective time scales for system (29)–(30) and the amplitude of the voltage response (maximum value of v for any given trajectory) to constant perturbations ( A t ).
This effect is more pronounced for smaller values of α (compare top panels a and b) because of the product αϵ in (30), which affects the effective time scales.
For the purpose of our study, we have qualitatively characterized the effective time scales of the isolated neurons by looking at their response to instantaneous DC (tonic) inputs and the shapes of the corresponding trajectories.
We have identified the effective time scales of the isolated neurons that are relevant for their interaction with oscillatory inputs with the time scales that govern the behavior of the initial portion of the trajectory from its initial point until it reaches its highest voltage value.
The effective time scales defined in this way may or may not represent the dynamic behavior of the autonomous trajectories for all subsequent times t, depending on the parameter values, but they are appropriate to describe the interaction between neurons and oscillatory inputs.
In effect, our approach assumes a fixed effective time-scale of admixture, and ignores the structured transition matrices that are expected due to a fixed pedigree.
Although an increase in α increases the horizontal component of the vector field (through the product αϵ) and hence reduces the effective time-scale separation, this is not enough to prevent limit cycle trajectories from moving beyond v max ( 0 ), and the system is able to exhibit resonance.
A cell's firing frequency can be described in terms of an effective time-scale operating in the subthreshold regime and resulting from current balances occuring there.
