Table 1.
Parameters of the model.
Figure 1.
Illustration of the concept that the faster process contributes more to rhythm generation.
A. Time courses of activity (a, black), the synaptic recovery variable (s, red), and the adaptation variable (θ, blue) for τθ/τs = 10. The range of variation of s (Δs) is about 10 times larger than the range of variation of θ (Δθ). Thus, according to the correlative measure s contributes more to the rhythmicity. B. Similar time courses for τθ/τs = 0.1. The cellular adaptation now appears to contribute more than the synaptic recovery variable. C. Plot of the variations of C = (R−1)/(R+1) with the τθ/τs ratio. Closed circles obtained from simulations with θ0 = 0; open circles for θ0 = 0.18. When θ is much faster than s (τθ/τs is small), C is close to -1 indicating that θ is the dominant process. When τθ ≈ τs, C≈0 indicating that both processes have equal contribution to the rhythm. At large τθ/τs, C approaches 1 and s is the dominant process. Points labeled A and B refer to the cases illustrated in panels A and B. Dashed curve, variations of c = (r−1)/(r+1) with τθ/τs; r = (w/g)(τθ/τs) and w = g = 1. Results are similar if we keep τθ/τs = 1 and vary w/g instead.
Figure 2.
Illustration of the blockade approach.
A. time course of network activity before (“control,” g = 1) and after (“θ block,” g = 0) blocking the adaptation process θ. These simulations were obtained for τθ/τs = 1 and for the three values of θ0 indicated. Vertical dashed line indicates the time when the process was blocked. B. Effects of blocking θ on the lengths of the active and silent phases (AP, SP), represented as percentage of “control”. No bars are shown when rhythmic activity was abolished. There are more cases where θ block results in decrease of SP duration (ii, iii, v, vi, viii) than increase in AP duration (vi, viii, ix). The interpretation is that θ contributes more to delay episode onset than to provoke episode termination. The results of the blockade experiment depend on the value of θ0, the activity threshold in the absence of adaptation, unlike the predictions from the correlative approach (Figure 1C). The blockade experiments also produces similar results in pairs of cases (iii, v) and (vi, viii) that have different τθ/τs ratio; this is also in opposition to the correlative approach.
Figure 3.
Variations of the time constants τs and τθ have different effects on the activity pattern.
A. Relative change in AP (red, diamonds) and SP (blue, stars) as τs (i) or τθ (ii) is varied. For comparison, the linear change in both AP and SP when τθ and τs are varied together by the same factor is shown (iii, τs & τθ). B. Variations of the duty cycle with τs (i), τθ (ii) and τs & τθ (iii).
Figure 4.
Construction of a measure of the contribution of s to episode termination.
Increasing τs by δτs at the beginning of an episode slows down s slightly (thick red curve), so the active phase is lengthened by δAP (thick black curve).
Figure 5.
Variations of the contributions of s and θ with τθ/τs.
A. Contributions of s to episode termination (CsAP, red, diamonds) and initiation (CsSP, blue, stars). B. Contributions of θ to episode termination (CθAP, red, diamonds) and initiation (CθSP, blue, stars). C. Both sums CsAP + CθAP (red, diamonds) and CsSP + CθSP (blue, stars) are close to 1, demonstrating the consistency of the measures. D. Combined measures CAP (red, diamonds) and CSP (blue, stars), as defined in Eq 11–12, superimposed with the prediction from the correlative measure c (dashed curve, as in Figure 1C). CAP≈−1: θ controls the active phase; CAP≈0: both θ and s have equal contributions to setting the duration of the active phase; CAP≈1: s controls the active phase (and similarly for CSP and the silent phase). Variations of CAP show that the relative contribution of s to the termination of the active phase increases with τθ/τs, in agreement with the correlative approach. On the other hand, CSP remains close to -1, showing that the subtractive process (θ) controls episode onset over the whole range.
Figure 6.
Variations of CAP and CSP with network connectivity and cell excitability (for τθ/τs = 1, g = 1).
CAP increases with increased synaptic connectivity, as would be expected from the correlative measure (Eq 4), with equal contributions from both processes (CAP = 0) when (w/g) (τθ/τs) = 1. In contrast, CSP is always close to -1, the subtractive feedback process sets the length of the silent phase regardless of the value of w. Finally, both CAP and CSP are unaffected by changes in θ0, showing that cell excitability does not influence which process controls episodic activity.
Figure 7.
The blockade experiment does not inform on the relative contributions of each process to rhythmic activity.
Panels A, B, C, D correspond to cases shown in panels v, vi, viii, ix in Figure 2. For each case, the change in AP and SP durations following θ block is shown next to the variations of CAP (red, diamonds) and CSP (blue, stars) with g (the maximum amplitude of cellular adaptation). Blocking θ means changing g from 1 to 0. As g reaches 0, both CAP and CSP reach 1 since s becomes the only variable controlling episodic activity. In A and B, the contributions measures are also the same before the block (g = 1, rectangle highlights), nevertheless the blockade leads to different changes in AP and SP durations. Thus, these changes in durations after the block cannot be used to predict the respective contributions of each process before the block. Panels C and D illustrate the same points, with similar contributions measures before the block (oval highlights) but different effects of the block on AP and SP durations. Finally, panels B and C show that despite different contributions measures (oval vs. rectangle highlight) before the block, the resulting effect of the block on AP and SP durations are the same. Again, results from the block do not provide much information about the respective contributions of each process before the block.
Figure 8.
Alternate estimation of the relative contributions of each process using phase plane analysis.
A. Representation of the system in the a,s-plane. The system trajectory is shown as a thick black curve with arrows at the transitions between activity phases. The trajectory follows the dynamic nullcline (thin black S-shaped curve) which moves left during the silent phase and reaches the thick gray nullcline on the left at episode onset. At onset, the trajectory reaches the low knee (s(t) = sk(t)). During the active phase, the dynamic nullcline moves to the right toward the thick gray nullcline on the right. It reaches it at episode termination, as the trajectory reaches the high knee. Note that the lower portion of the nullcline is much more sensitive to θ than the higher portion. LK, low knee; HK, high knee of the a-nullcline. B. Variations of CAP and CSP (calculated using the phase plane approximation illustrated in A) with w (for τθ/τs = 1 and θ0 = 0). There is good agreement with the computational method based on small perturbations in the time constants (compare with Figure 6B). C. For large values of θ0, such that high activity episodes require that θ be close to 0, the computational calculation of the relative contributions (left panel) and the phase plane estimation (right panel) can disagree. In the case shown, τθ/τs = 0.1 (w = g = 1), so the phase plane method estimates that θ should control both active and silent phase (right panel). The disagreement with the computed CAP and CSP (left panel) is a result of the geometric argument used to estimate the contributions neglecting the fact that the speed of variation of s and θ can slow down dramatically when approaching their asymptotic values (see text).