Correlative approach: comparing the variations of the feedback processes

The rationale for this first approach is that if a process varies greatly during the high-activity episodes (active phases) and the inter-episode intervals (silent phases), then it is likely that it contributes significantly to episode termination and onset. On the other hand, if the variations are small, it is likely that the contribution of the process is small. This approach thus relies on observing a relationship between the time course of a process and the system's behavior. Its pitfall, that correlation does not imply causation, is well known.

Experimentally, one can record spontaneous or evoked postsynaptic potentials or currents in target neurons [11], [12], [13], [14]. The variations of this postsynaptic response during the interval of time between two episodes of activity would represent the variations of the effective connectivity, or available synaptic strength, w^.s. Similarly, one may record the degree of adaptation or the current responsible for this adaptation at various times during the silent phase [11], [14]. The variations of the current with time would be equivalent to the variations of g^.θ. Here we assume that there are only two slow feedback processes, represented by s and θ, which can be measured unequivocally and with sufficient precision. This is an ideal situation that will not often be encountered experimentally; we show that even with such ideal conditions we may not be able to determine the contributions of the two slow processes using the correlative approach.

If s varies by Δs and θ by Δθ over one phase of the oscillation, then according to the correlative approach the ratio (4)measures the contribution of s relative to that of θ. We have shown previously [8] that if s and θ vary exponentially with time constants τ_s and τ_θ, then Δs/Δθ ≈ τ_θ/τ_s . Thus, (5)

Assuming that w and g are similar – we set w and g to 1 unless noted otherwise – the ratio of the contributions of the two processes to the rhythmic activity is inversely proportional to the ratio of their time constants, so the faster process contributes more than the slower process. This is illustrated in Figure 1AB where we plot the variations of a (network activity), s and θ for the cases r = τ_θ/τ_s = 0.1 (A) and r = τ_θ/τ_s = 10 (B). In the case shown in Figure 1A, we expect s (red curve) to contribute more to episode onset/termination because it is the faster process, while in the case shown in Figure 1B θ (blue curve) is faster and thus expected to have the major contribution.

We define a quantitative measure of the contribution of the two processes by (6)(or, using the approximation given by Eq. 5, c = (r−1)/(r+1)). C varies between −1 and 1. If C is near 1 then s determines the episode onset and termination (i.e., θ has no role). If C≈−1 then θ controls episode onset and termination. Intermediate values of C indicate that both processes contribute. This measure is plotted as a function of r in Figure 1C, and clearly demonstrates the shift of control (according to the correlative definition) from θ to s as the s dynamics are made progressively faster relative to θ. The filled circles result from simulations with the cell excitability parameter θ₀ set to 0 (relatively high cell excitability). The open circles were obtained using θ₀ = 0.18 (low cell excitability). The differences are very small, showing that, according to this measure, the respective contributions of the two processes depend very weakly on θ₀. The dashed curve in Figure 1C is obtained by plotting c  =  (r−1)/(r+1). Since the points obtained from plotting C lie almost on this curve, one concludes that, according to the correlative approach, the contributions of the two slow processes depend only on the ratio r  =  (w/g) (τ_θ/τ_s). Thus, the faster that one process is relative to the other the greater its contribution will be to rhythm generation. Similarly, the greater the relative weight of a process, the greater its contribution. Finally, since each process covers the same range during the active and silent phase, these results do not distinguish between episode initiation and termination. That is, the correlative approach predicts that the contribution of each process is the same for episode initiation and termination.

Blocking approach: deleting one feedback process

The rationale for this second approach is that if a process is important to a system's behavior, then removing it will have a large effect. This type of experiment is widely used in biology and includes pharmacological block, surgical ablation, and gene knockout. If, for example, θ represents the activation of a potassium current responsible for cellular adaptation, then one could block this current pharmacologically or genetically and measure the effect on network activity. We block the θ process by setting g = 0 and observe the effect on the length of both the active and silent phases after transient effects have died down. If we see a large increase in the active phase duration, then we conclude that this process is important in terminating the active phase. Similarly, if after the block we see a decrease in silent phase duration then we conclude that recovery of this process is important for episode initiation. The pitfall of this approach is that after blocking a process we obtain a different system.

Figure 2 illustrates the results obtained with this approach, for different values of the parameter θ₀. Figure 2A shows the time course of network activity before and after blocking θ in the case τ_θ = τ_s. When cell excitability is too high (e.g., θ₀ = 0.06), synaptic depression alone cannot bring the network to a low activity state and rhythmicity is lost after the block. For lower cell excitability (higher θ₀, middle and right columns), blocking θ leads to changes in the lengths of both the active and silent phases, to various degrees. These changes in active and silent phase durations (AP and SP), after transient effects have died out, are represented on Figure 2B for different values of the ratio τ_θ/τ_s. Can we infer the importance of θ variations on rhythm generation from these changes?

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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 τ_θ/τ_s = 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 τ_θ/τ_s ratio; this is also in opposition to the correlative approach.

https://doi.org/10.1371/journal.pcbi.1001124.g002

We first note that for low θ₀ rhythmic activity is lost after blocking θ, for all values of the ratio τ_θ/τ_s. Thus, variations in θ are required for rhythm generation in these cases. In the other cases shown, blocking θ has large effects on the active and silent phase durations, but these effects are difficult to interpret. For instance, we expect the block to increase the active phase in proportion to θ's contribution to episode termination. Thus, it seems that θ contributes significantly to episode termination in cases vi, viii and ix (where there is a large increase in AP after the block), but does not contribute much to episode termination in case iii (where there is no change in AP after the block). In cases ii and v the active phase duration actually decreases after the block, which is hard to interpret. Similarly, we expect the decrease in silent phase duration following θ block to be in accordance with θ's contribution to episode initiation, since residual adaptation delays episode onset. Thus, we would say that θ contributes significantly to episode onset in cases ii, iii, v, vi and viii. But again, we have an unexpected case (ix) where SP increases after the block.

The blockade experiment illustrated in Figure 2 suggests that there are more cases where θ has a significant contribution on episode initiation (ii, iii, v, vi, viii) than on episode termination (vi, viii, ix). This is in contradiction with the correlative approach, which suggested that θ had a similar contribution to both episode termination and initiation. There are also cases, such as vi and viii, where the effects of the block are similar, suggesting that θ's contribution to episodic behavior is similar in those cases. But cases vi and viii correspond to different values of the ratio τ_θ/τ_s. According to the correlative approach, the contribution of each process should vary with τ_θ/τ_s (Figure 1C), so again the blockage approach and correlative approach disagree. Finally, on each row of Figure 2B the effect of the blockage varies with the value of the parameter θ₀. This again contradicts the correlative analysis, which showed little dependence on θ₀.

The strong perturbation to the system effected by the block is responsible for the counterintuitive decrease in AP observed in cases ii, v and increase in SP observed for case ix. These changes reflect system compensation; after the block and after transients have died out, the unblocked process, s, covers a different range of values, so AP and SP are modified. This compensation could be avoided by measuring AP and SP just after the block instead of letting it equilibrate. This is illustrated in Figure 2Aii, where the block initially increases AP, then decreases it as SP is decreased by the absence of θ. Interpretation of the block experiment would therefore be facilitated by considering only transient behavior, but this would be difficult to do experimentally in most cases. For instance if we block a K⁺ channel pharmacologically then the kinetics of drug application and binding to the channels will interfere with the transient effects.

In summary, we find that the correlative and blockage approaches suggest different interpretations about the contributions of the negative feedback processes to rhythm generation. In the following, we show that neither approach gives a satisfactory description of the contributions of the slow processes. This is because each approach suffers its own pitfall. The first approach is purely correlative, i.e., it links variations in one process to the behavior of the system, but cannot establish causation. To obtain causation it is necessary to determine how the system responds to a perturbation to one of these processes, as in the blocking approach. Unfortunately, by perturbing the system, we change it. The loss of periodic activity after blocking θ (as in cases i, iv, vii in Figure 2) shows that this process may be necessary for maintaining rhythmic activity, but it does not indicate what was the contribution of θ before the block.

A new measure of the relative contributions of the negative feedback processes

The goal here is to derive a measure that allows one to draw a causal link between each slow process and the activity pattern that does not involve a strong perturbation to the system. Suppose that s is the only negative feedback process regulating episodic activity, so it contributes 100% to both episode termination and initiation. Then doubling τ_s will (approximately) double both AP and SP. If s is not the only negative feedback process and therefore has only a partial contribution to episode termination and initiation, then doubling τ_s will still increase AP and SP but by a smaller factor. Thus, the contribution of s to the episodic activity can be determined by the fractional change in AP and SP durations following a change in τ_s. To illustrate this idea, we plot both AP and SP durations as either τ_s or τ_θ is varied in Figure 3A.

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Figure 3. Variations of the time constants τ_s and τ_θ have different effects on the activity pattern.

A. Relative change in AP (red, diamonds) and SP (blue, stars) as τ_s (i) or τ_θ (ii) is varied. For comparison, the linear change in both AP and SP when τ_θ and τ_s are varied together by the same factor is shown (iii, τ_s & τ_θ). B. Variations of the duty cycle with τ_s (i), τ_θ (ii) and τ_s & τ_θ (iii).

https://doi.org/10.1371/journal.pcbi.1001124.g003

Figure 3Ai shows that AP varies more with τ_s than does SP. This suggests that s has more influence on episode termination than on episode initiation. The variations of AP and SP with τ_θ (Figure 3Aii) show the opposite trend, suggesting that θ has more influence on episode initiation than on episode termination. These trends are also illustrated by the variations of the duty cycle ( =  AP/(AP+SP)) with τ_s and τ_θ (Figure 3B). The duty cycle increases with τ_s, but decreases with τ_θ. Finally, comparing Figures 3A i and ii, we observe that the variations of AP with τ_s and τ_θ are similar, suggesting that s and θ have comparable contributions on episode termination. On the other hand, SP varies more with τ_θ than with τ_s, suggesting that θ has a stronger influence on episode initiation than does s. This example suggests that the contributions made by the slow processes to the episodic activity can be determined by varying the time constants of the processes and observing the effects on AP and SP durations. We now use this idea to construct a quantitative measure of these contributions.

We first construct a measure of the contribution of s to episode termination, as illustrated in Figure 4. At the beginning of an episode, τ_s is increased by δτ_s. If s contributes to episode termination, slowing down s increases AP by δAP. We can quantify the contribution of s to episode termination by evaluating the ratio of the relative change in AP, δAP/AP, divided by the relative change in τ_s, δτ_s/τ_s. We thus define the normalized contribution of s to episode termination as (7)

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Figure 4. Construction of a measure of the contribution of s to episode termination.

Increasing τ_s by δτ_s at the beginning of an episode slows down s slightly (thick red curve), so the active phase is lengthened by δAP (thick black curve).

https://doi.org/10.1371/journal.pcbi.1001124.g004

If s has no influence on episode termination, slowing it down has no effect and δAP  =  0. If s is the only process contributing to episode termination, then the active phase duration is the time it takes for s to decrease from its value at the beginning of an episode to its value at the transition between AP and SP. Since we consider relaxation oscillations, the transition time between active and silent states is negligible. Thus, a fractional change in τ_s leads to the same fractional change in AP (δAP/AP  =  δτ_s/τ_s) so that C^s_AP = 1. Therefore, C^s_AP has a value between 0 (s does not contribute to episode termination) and 1 (s is the only process contributing to episode termination). We quantify the contribution of s to episode initiation similarly using (8)

We define the contributions of θ to episode termination and initiation in a similar way:(9)(10)

These measures have the same motivation as the blockage experiment, but can be computed with small perturbations to the system. We use δτ/τ = 4% so the perturbation is small but nevertheless has a measurable effect. In addition, we look at the acute effect of the perturbation, i.e., we do not wait until the system equilibrates.

Figure 5A shows the contributions of s to episode termination (C^s_AP) and initiation (C^s_SP) as the ratio τ_θ/τ_s is varied, determined through numerical simulations as shown in Figure 4. C^s_AP increases as this ratio is increased, that is, s contributes more to episode termination as it becomes faster relative to θ. When s is much slower than θ, C^s_AP is close to 0. For s much faster than θ, C^s_AP is close to 1. When s and θ have similar speed C^s_AP is close to 0.5, suggesting that the divisive and subtractive feedback processes contribute equally to episode termination when their time constants are similar. This relationship between the contribution of feedback processes to episode termination and the ratio of their time constants is in agreement with the prediction from the correlative approach (Figure 1C). However, the contribution of s to the silent phase, C^s_SP, varies differently with τ_θ/τ_s. Although it increases with τ_θ/τ_s, this increase is so weak that C^s_SP is below 0.1 even if s is 10 times faster than θ. This consistently low C^s_SP suggests that regardless of the relative time constants of the two negative feedback processes, s never contributes significantly to episode onset, in sharp contrast with the prediction from the correlative approach.

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Figure 5. Variations of the contributions of s and θ with τ_θ/τ_s.

A. Contributions of s to episode termination (C^s_AP, red, diamonds) and initiation (C^s_SP, blue, stars). B. Contributions of θ to episode termination (C^θ_AP, red, diamonds) and initiation (C^θ_SP, blue, stars). C. Both sums C^s_AP + C^θ_AP (red, diamonds) and C^s_SP + C^θ_SP (blue, stars) are close to 1, demonstrating the consistency of the measures. D. Combined measures C_AP (red, diamonds) and C_SP (blue, stars), as defined in Eq 11–12, superimposed with the prediction from the correlative measure c (dashed curve, as in Figure 1C). C_AP≈−1: θ controls the active phase; C_AP≈0: both θ and s have equal contributions to setting the duration of the active phase; C_AP≈1: s controls the active phase (and similarly for C_SP and the silent phase). Variations of C_AP show that the relative contribution of s to the termination of the active phase increases with τ_θ/τ_s, in agreement with the correlative approach. On the other hand, C_SP remains close to -1, showing that the subtractive process (θ) controls episode onset over the whole range.

https://doi.org/10.1371/journal.pcbi.1001124.g005

Figure 5B shows that the contributions of θ to episode termination (C^θ_AP) and initiation (C^θ_SP) vary in the opposite way to C^s_AP and C^s_SP. If τ_s is much larger than τ_θ then s does not affect AP while θ strongly affects AP. As the ratio τ_θ/τ_s increases, the contribution of s to episode termination increases while the contribution of θ decreases, in such a way that the sum of the contributions of s and θ stays around 1 (C^s_AP + C^θ_AP ≈ 1) as shown in Figure 5C. The effect of θ on SP is always strong, while the effect of s is weak, regardless of τ_θ/τ_s. The sum of the contributions of s and θ to episode initiation also stays around 1 (C^s_SP + C^θ_SP≈1). Thus s and θ have complementary contributions to the episodic activity and our measure is self-consistent. The relationship C^s_xP + C^θ_xP≈1 is a consequence of the fact that s and θ are the only processes controlling AP and SP. That is, if we increase both of their time constants by a factor k, then AP and SP both increase by the same factor k (Figure 3Aiii). This can be written, in the case of the active phase, as: AP(k τ_s, k τ_θ)  =  k AP(τ_s, τ_θ). Application of Euler's theorem for homogeneous functions yields: and, after dividing each side by AP, results in C^s_AP + C^θ_AP = 1.

Since we are dealing with only two slow processes, we can combine the measures defined for s and θ (Figures 5A and 5B) into single measures by defining(11)(12)

With this definition, C_AP and C_SP vary between −1 to 1. A value close to −1 signifies that θ is the dominant process; a value close to 1 signifies that s is the dominant process; a value near 0 means that s and θ have similar contributions. These are plotted in Figure 5D as a function of τ_θ/τ_s. We see that C_AP rises from −1 to near 1 as τ_θ/τ_s increases, indicating that θ dominates the AP when it varies more rapidly than s, and s dominates when it varies more rapidly than θ. This agrees with the result obtained with the correlative approach (dashed curve, c =  (r−1)/(r+1)). In contrast, the SP is controlled by θ for the full range of τ_θ/τ_s; this was not predicted by the correlative approach.

Conditions of applicability of the measure

The contribution measures defined above are meaningful only if specific conditions are satisfied. The most important condition is that each variable or process contributes to the same aspect of system behavior. For instance we cannot compare the contribution of a slow negative feedback process, such as our s or θ, which terminates an episode of activity, to the contribution of a fast negative feedback variable that could be responsible for fast cycling during the high activity phase. Second, the variables must vary monotonically during each phase of the activity. If not, then increasing their time constant may not increase the duration of a phase in a predictable way and the sum of the contributions of the variables to that phase may not equal 1.

We use a relaxation oscillator with a clear distinction between active and silent phases. The measure can be applied to other types of oscillations, as long as active and silent phases can be clearly distinguished. In more complex cases, it may be necessary to divide a period of activity into more than two phases. More generally, the method could be applied to non-oscillatory systems, for example to determine the contribution that different variables make to return the system to an equilibrium following a perturbation. Also, the measure is not limited to two negative feedback processes. We have chosen feedback processes of different types, subtractive and divisive, because we find the problem of disentangling their relative contributions to be quite challenging. This measure can be applied with feedback processes of the same type, as long as they contribute to the same behavior. We have used the method to compute the respective contributions of two subtractive feedback processes to burst generation and shown that the results can be used to predict the occurrence of phase-independent resetting [15]. Finally, we use a deterministic model. Noise would not qualitatively affect our measure, as long as it does not affect the mechanisms for the transitions between phases. If noise is part of the transition mechanism [16] our method cannot be applied as it is, since noise would also contribute to the transitions.

Since the measure requires a model of the system, the validity of its results depends on the validity of the model. Models may incorporate various degrees of realism, so it is important that the measure be robust to model details. For instance, if we add a fast variable to the relaxation oscillator model, so that fast oscillations (spikes) are produced during each active phase, the two slow negative feedback processes may still terminate episodes (bursts) like in the relaxation case. Thus, the relative contributions of each slow variable to burst onset and termination should not change qualitatively. We have demonstrated such robustness with a model of bursting in pancreatic islets [15].

The effect of network excitability on the contributions of s and θ

We now evaluate how the parameters that control network excitability, w (network connectivity) and θ₀ (average cellular threshold), affect the contributions of s and θ to rhythm generation. Variations of C_AP and C_SP with w are represented in Figure 6, for three different values of θ₀ (and for τ_θ/τ_s = 1). Clearly, C_AP increases with w, i.e., synaptic depression contributes more to episode termination when network connectivity is high. However, this is not true for episode initiation, as C_SP is almost unaffected by w. There is in fact a slight tendency for C_SP to increase at the lowest values of w, which is more visible if s is faster than θ (not shown). Changes in θ₀ do not affect either C_AP or C_SP significantly. This is in agreement with the correlative approach, but in contrast to the results of the blockade experiment (Figure 2).

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Figure 6. Variations of C_AP and C_SP with network connectivity and cell excitability (for τ_θ/τ_s = 1, g = 1).

C_AP increases with increased synaptic connectivity, as would be expected from the correlative measure (Eq 4), with equal contributions from both processes (C_AP = 0) when (w/g) (τ_θ/τ_s)  = 1. In contrast, C_SP is always close to -1, the subtractive feedback process sets the length of the silent phase regardless of the value of w. Finally, both C_AP and C_SP are unaffected by changes in θ₀, showing that cell excitability does not influence which process controls episodic activity.

https://doi.org/10.1371/journal.pcbi.1001124.g006

In summary, the ratio τ_θ/τ_s and connectivity w – but not θ₀ – strongly affect C_AP, while none of these have a significant effect on C_SP. In general, both feedback processes s and θ play roles in the episode termination, but only θ controls episode initiation. The relative influence of s and θ to episode termination varies with parameter values. The correlative approach is roughly correct for predicting the contributions of the two processes to episode termination, but not to episode initiation. This approach makes a direct comparison between the time scales of the two processes, scaled by their relative strength (w and g), evaluating r = (w/g) (τ_θ/τ_s). But this ratio is not the ratio of the contributions of the two processes to episode initiation. In fact, we show below that the weighted time scales cannot be compared directly but must be rescaled, the correct ratio beingwhere the scaling factor a_k is the activity level at the transitions between active and silent phases. At episode termination, a_k≈1 so the correlative approach is approximately right. However, at episode onset a_k≈0, so r_rescaled ≈ 0, meaning that s does not contribute significantly unless r >> 1. Looking back at Eq. 1, it is evident that s generally has little effect when activity is low. Such a simple fact was not revealed using the correlative and blockade approaches, stressing again that these standard experimental approaches are not always useful for determining the contributions of different variables to rhythmic activity.

Effects of g and problems of the blockade experiment

The analysis above suggests that the correlative approach can reasonably estimate the contribution of each process to episode termination, but misses the fact that s contributes little to episode onset (Figure 5D). Results from both the blockade simulations and the analysis above suggest that θ is more important for episode initiation than episode termination. However, we have seen that the blockade approach does not typically provide a good indication of the contribution of θ to the AP and SP durations (Figure 2B). To further demonstrate this, we plot in Figure 7 the variations of C_AP and C_SP with g (curves), the maximal “conductance” of the adaptation process θ, in four of the cases illustrated in Figure 2B (v, vi, viii, ix). The values of both C_AP and C_SP decrease as g is increased, indicating that the influence of θ in the control of the rhythm increases with g. As g decreases towards 0, both C_AP and C_SP increase toward 1 since s is the only slow process when g = 0. This is true for all four cases. However, C_SP only increases noticeably when g approaches 0, illustrating again that the subtractive feedback process controls the silent phase in most cases.

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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 C_AP (red, diamonds) and C_SP (blue, stars) with g (the maximum amplitude of cellular adaptation). Blocking θ means changing g from 1 to 0. As g reaches 0, both C_AP and C_SP 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.

https://doi.org/10.1371/journal.pcbi.1001124.g007

Comparing Figure 7A–B, we see that the C_AP curve is similar in both panels, as is the C_SP curve. The bar plots show the effects of a blockade simulation, where g = 1 before the blockade and g = 0 afterwards. In Figure 7A the blockade results in a 50% reduction in the AP duration, while in Figure 7B there is a very large increase in the AP duration following blockade. Yet, according to the C_AP curves the contribution of θ to the AP duration is nearly the same in both cases when g = 1 (green and yellow boxes). Similarly, C_SP is similar in panels C and D for g = 1, yet the blockade results in decreased SP duration in C, but increased SP duration in D. Thus, the effects of the blockade on AP and SD durations do not provide much information on the respective contributions of the two processes before the blockade.

Next, we compare cases shown in Figure 7B and 7C. We notice that C_AP differs between the two cases, showing that when g = 1 the s variable contributes significantly to episode termination in one case (Figure 7B) but not the other (Figure 7C). Yet, after blockade the changes in AP/SP (bar plots) are similar in both cases. Again, results from the blockade approach do not indicate what was the contribution of each process before the blockade.

A geometric measure of the relative contributions of the negative feedback processes

For the mathematically simple system used in this work, we can use a geometrical argument to derive approximate formulas for C_SP and C_AP. If the system is two-dimensional with one slow process, s, the trajectory could be drawn in the a,s-phase plane and would follow the a-nullcline (except for fast jumps at the transitions between active and silent phase). For the three-dimensional system presented here, the trajectory in the three-dimensional a,s,θ-phase space follows the surface defined by da/dt = 0 [8]. We can project the three-dimensional trajectory and surface into the a,s-plane. This results in a two-dimensional trajectory that follows a dynamic a-nullcline (Figure 8A). The effect of the third variable (θ) in this two-dimensional representation is to move and deform the dynamic a-nullcline (the thin, black S-shaped curve in Figure 8A). Increasing θ moves the nullcline rightward.

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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) = s_k(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 C_AP and C_SP (calculated using the phase plane approximation illustrated in A) with w (for τ_θ/τ_s = 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, τ_θ/τ_s = 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 C_AP and C_SP (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).

https://doi.org/10.1371/journal.pcbi.1001124.g008

At the end of the active phase, the trajectory falls from the high- to the low-activity state and the dynamic nullcline is at its rightmost position (thick, discontinuous, grey S-shaped curve on the right of the diagram). During the silent phase, s increases so the system's trajectory moves to the right while θ decreases so the a-nullcline is transformed leftward. When the trajectory passes the low knee (LK) of the nullcline, the trajectory jumps to the upper branch. At this point the nullcline has reached its leftmost position (the thick grey S-shaped curve on the left), since θ will now again begin to increase and the a-nullcline will be transformed rightward.

To compare the contributions of s and θ to the termination of the silent phase, we can therefore compare the length traveled by the trajectory (controlled by s) with the length traveled by the low knee (controlled by θ). Assuming that their speeds are nearly uniform, we can compare the instantaneous variation of the trajectory's position ds to the instantaneous variation of the knee ds_k due to the variation of θ, dθ. We can show [8] that ds ≈ dθ (τ_θ/τ_s) and that ds_k ≈ (g/w) (dθ/a_k) where a_k is the activity level at the knee (its value varies little with θ). Thus, the ratio of the contributions of s and θ is (13)

This formula applies to both active and silent phases, however the activity level at the knee, a_k, differs between the two phases. During the silent phase, a_k is close to 0 so ds/ds_k is very small, i.e., s generally contributes little to the termination of the silent phase. On the other hand, during the active phase a_k is close to 1, so ds/ds_k ≈ (τ_θ/τ_s) (w/g). If (τ_θ/τ_s) (w/g) ≈1 then the two slow processes contribute similarly to active phase termination. This shows that the relative contributions of s and θ are qualitatively different for the different phases of activity. It also explains why the intuitive approach illustrated in Figure 1 is correct for the active phase (where a_k≈1), since from Eq. 5 and Eq. 13 r ≈ ds/ds_k. If (τ_θ/τ_s) (w/g) ≈ 1, then r ≈ 1 and the correlative approach predicts equal contributions of the feedback variables (Figure 1C). On the other hand, during the silent phase a_k≈0 so r is not a good approximation to ds/ds_k and the correlative approach is invalid.

To compute ds/ds_k for both phases, we must compute a_k (Eq. 13) for both knees of the dynamic a-nullcline shown Figure 8A. For this we note that the nullcline is defined by da/dt = 0. Solving for s, we obtain(14)

For each value of θ, the knees are defined by and differentiating Eq. 14 gives:(15)which has two solutions a_k, each corresponding to a knee, provided the right hand side is greater than 2. The values of θ at onset and termination of the episodes, to be used in Eq 15, were obtained from the durations of the active and silent phases obtained from simulations [8]. Finally, when θ₀ is changed there is a similar but opposite change in the range of variation of θ, so θ + θ₀ is not affected much by a change in θ₀. Thus the solutions of Eq. 15 are not very sensitive to θ₀. This explains why the relative contributions of s and θ are little affected by θ₀, as seen in Figure 6.

Since we identify ds/ds_k to the ratio of the contribution of the two slow variables for each phase, C^s_xP/C^θ_xP, the combined measures C_xP defined in Eq 11–12 correspond to the ratios (ds/ds_k − 1)/(ds/ds_k + 1). These ratios are computed for both active and silent phases as a function of w and shown on Figure 8B. Comparison with Figure 6 (middle panel) shows that this geometric measure of the contributions of the slow processes is in good agreement with the empirical measure constructed above using sensitivities to the slow variables' time constants.

Finally, we point out that there are rare situations when the two measures (Eqs. 7–12 vs. Eq. 13) do not give similar results. Such a case is shown in Figure 8C, for which the parameter θ₀ is large (average cell excitability is low) and τ_s is 10 times greater than τ_θ. Because θ₀ is large, even when θ decreases to its minimum during the silent phase, s may not be sufficiently large for an episode to start, particularly if the connectivity is low. In that case, an episode is not started until s reaches the value corresponding to the low knee. Even if this is a small distance, it can take a long time since s is so slow. Thus, changing τ_s can have a strong effect on the silent phase and C_SP determined from Eq. 12 becomes positive (Figure 8C, left panel) instead of close to -1 as computed using Eq. 13 (Figure 8C, right panel). In other words, using a measure based on time indicates a strong contribution of s in that particular situation, while a measure based on geometry indicates a marginal contribution of s to episode initiation. This discrepancy between the two measures appears because θ does not vary uniformly. It slows down considerably as it approaches its asymptotic value, “waiting” for s to reach the low knee. Thus the dynamics of s now play a major role in terminating the silent phase. Note that θ still has a strong effect on the s dynamics during the silent phase (it determines the location of the low knee of the a-nullcline in Figure 8A), but θ's dynamics do not affect the silent phase duration much, so the measure that relies on perturbing the time constants finds it has little contribution.

Experiments alone might not determine the relative contributions of the slow processes

We have first attempted to use approaches inspired from experimental methodology to determine the relative contributions of the two feedback processes to rhythm generation. These included comparison of the time course of each process (the correlative approach) and blocking one of the processes.

The correlative approach simply compares the amount of variation of each process, scaled by each process' strength or conductance. Since the two processes vary by the same amount during the active and silent phase, this approach does not distinguish between active and silent phase. According to this approach, the relative contribution depends only on the ratio of their time constants (τ_θ/τ_s) and on the ratio of their strength (w/g). It predicts that if these two ratios are close to 1 then both feedback processes contribute similarly to the rhythm. In the example shown here this is a good approximation for the active phase. However, for the silent phase, this intuitive rule fails, because an additional scaling factor must be introduced to compare the contributions of the two different negative feedback types. This scaling factor is significantly different from unity for the silent phase; it reflects the fact that the divisive feedback process, being a multiplicative factor to the activity, has very little effect at low activity (i.e., during the silent phase).

The blockade approach suggests that the subtractive process might be more important in setting the silent phase duration, since blocking this process affected the silent phase duration more often than the active phase duration. In this way it provides a piece of information that is missed by the correlative approach. However, similar effects of the blockade on AP and SP durations were found in cases where the ratio of time constants was different (and different effects when that ratio was identical), contradicting the correlative approach and, as shown in Figure 7, contradicting our measure of the relative contributions of each process. Furthermore, unlike the correlative approach, the blockade experiment suggests a strong effect of θ₀ (which biases the input/output relationship of the system). In general, however, this parameter has little effect on the contribution of each process (cf. Figure 6).

These disappointing results from the two experimental approaches are due to their well known pitfalls: passive observation only establishes an association without proving a causal relationship, while perturbations to the system, such as blockade experiments, can qualitatively change the system being studied. The use of total blockade may be considered extreme. A partial block can potentially be more informative than a complete block because a small enough perturbation may indicate a trend in a component's influence and preclude switching the system to a different mode of operation (see e.g., [17], [18], [19]). In other words, if the perturbation is small enough the effect on the activity may be close to linear so the effect of the partial block can be quantified and provide information on the role of the process that is partially blocked. However, partial blockade cannot provide a quantitative measure with the properties (summation to 1) of the C values developed here.

Our approach, instead, is to use small perturbations to the time constants of the feedback processes and look at the effect immediately following the perturbations. This minimizes the perturbation to the system, while quantifying the relative contribution of the two slow processes to the rhythmic behavior. This method could be applied to many oscillatory systems that rely on the interplay between positive feedback and several negative feedback processes. However, for most known experimental conditions, this method seems impossible to implement. To apply the method requires 1) the ability to change the time constants of the variables of interest one by one, 2) these changes must remain small but have measurable effects and 3) the system's behavior immediately after the changes must be measured, without waiting for transients to die out. For example, in the context of a neural network, there is currently no technique available to change the time constant of synaptic depression by a small amount, quickly and without affecting other network parameters. Thus, in many cases, the question of determining the contributions of different negative feedback processes in rhythm generation (using our approach) may only be addressed with computational models.

One example in which our approach could be used in an experimental setting is the electrical oscillatory activity of single cells. The mathematical formalism used to describe the mean activity of an excitatory network is similar to the Hodgkin-Huxley formalism commonly used to describe the electrical activity of excitable cells [8], [20], [21], [22]. In excitable cells, the sodium or calcium channels generate voltage-dependent inward current, providing fast positive feedback that increases membrane potential, while the delayed activation of outward potassium currents and inactivation of the inward currents provide negative feedback. An outward current has an opposite influence to the excitatory inward current and therefore provides subtractive feedback; on the other hand the inactivation of an inward current is a multiplicative term reducing the amount of positive feedback and therefore is a divisive feedback process. Preliminary results with the Hodgkin-Huxley model of nerve excitability [20] in a repetitive spiking mode suggest that while both sodium current inactivation and potassium (K⁺) current activation contribute to terminating an action potential, it is mostly the de-activation of the K⁺ current that initiates the next spike (J. Tabak, unpublished results). This could be verified experimentally for electrically compact cells using the dynamic clamp technique, which allows one to introduce a model-generated ionic current into a cell [23], [24]. For example, one could pharmacologically block the Na⁺ current, then re-introduce it into the cell using the dynamic clamp. Because the added current is computed from a model, it would be possible to change its inactivation time constant by a desired amount and measure the effect of this perturbation on the duration of the spike or interspike interval. To our knowledge, a similar experiment has been done only once, to show that increasing the inactivation time constant of a low-voltage-activated calcium current would result in longer bursts in invertebrate neurons [25]. While both divisive and subtractive feedback can in principle terminate bursts in neurons [26] it is usually the latter that is considered to regulate bursting, in the form of slow, calcium-activated K⁺ currents. The experiment described in [25] provided strong support for a role of low-voltage-activated calcium current inactivation (divisive feedback) in burst termination.

Other analysis techniques

Modeling is being established as an essential tool for understanding complex biological systems [27], complementing experimental approaches. But more than mere simulations of systems of differential equations, which are akin to experiments, it is the qualitative analysis of the models that provides new insights into a system's dynamics. Qualitative model analysis techniques include phase plane and bifurcation analysis, but these techniques become more difficult to apply as the number of variables increases. The commonly used fast-slow analysis, which simplifies model analysis by formally separating the equations into fast and slow subsystems, may have limited usefulness when many variables operate on the same time scale.

An extension of fast-slow analysis that can deal with many variables operating on the same time scale is the Dominant Scale Method (DMS) [28]. This method follows one variable of interest along an oscillatory trajectory (for instance, voltage in a cellular oscillator model) and determines the sensitivity of this variable at each point on its trajectory to other variables that are present in its differential equation. During different epochs of time, only a few variables may significantly affect the primary variable, so the model can be reduced to a few variables during each epoch. Thus, a complex model is transformed into a sequence of simpler models using only the dominant variables, and qualitative analysis of the dynamics is possible for each successive epoch [29]. The DMS can evaluate the relative contributions of variables that have different roles, unlike the measure presented here. However, our approach uses the sensitivity of observable features of the system behavior (AP and SP), not the sensitivity of a variable to other variables. For this reason, one may use our approach to identify cases where a variable has very little effect on the primary variable but nevertheless controls the duration of a given phase of the activity (as discussed in last section of Results).

Our approach to measure the contribution of feedback processes to rhythmic behavior is to compute the sensitivity of the AP and SP to the time constants for these processes. Other techniques that use sensitivities of observables of a system to control parameters are Metabolic Control Analysis and Biochemical Systems Theory [30], [31], which have been used to analyze metabolic and gene regulatory networks. Important features of these approaches include summation theorems, for instance the sum of the sensitivities of the level of a metabolite to control coefficients is equal to 1. A similar summation theorem holds in our analysis, where the contributions of the two slow variables to the AP or SP duration sum to 1. These techniques are usually applied to the control of steady states, but they have also been used to describe how observables such as the period and amplitude of an oscillatory system are regulated by control parameters [32], [33]. The control of these observables is usually distributed across control parameters [33]. Here, we found that the control of the active phase is distributed across the divisive and subtractive feedback processes, but control of the silent phase is mostly operated by the subtractive process, θ. That is, θ is the “rate limiting factor” in the termination of the silent phase.

Finally we mention parameter search techniques, which are usually developed to find parameter sets that lead to a target behavior. These techniques can also be used to determine what parameter changes must be done to qualitatively affect a system's activity and provide information about the robustness of such activity [34]. Furthermore, by finding different parameter sets that produce similar system behavior, it is possible to determine the relationships between parameters that allow a behavior to be maintained [35] or to evaluate how each model parameter influence a given characteristic of the behavior using nonlinear regression [36]. This “database approach” indirectly provides information about the role played by some variables of the system and how a variable can take over when another variable is eliminated. It can be used to explore the behavior of a model in different regions of parameter space [37]. An intriguing observation is that different parameter combinations in a wide area of parameter space may produce similar oscillatory patterns [38]. If two distinct parameter sets produce the same system behavior, does this mean that a variable might have different roles in different networks that produce similar activity? This question could be answered with a combination of the database approach and the analysis technique developed here.

Conclusion

We have developed a computational method to quantify the relative contributions of feedback processes to active and silent phases of episodic activity. We have considered a case involving both subtractive and divisive processes. If both processes have similar strength and time scales, they contribute equally to terminate the active phase. This is consistent with our intuition and predicted by the correlative approach. Interestingly, it is the recovery from the subtractive process that sets the duration of the silent phase. This is because the divisive feedback is a multiplicative factor to the system's activity and therefore plays little role during the silent phase. Thus, different phases of the activity are controlled differently by the negative feedback processes. Experimental methodologies do not in general provide this type of information, so the determination of the relative contributions of different variables to a biological system's activity will usually require the development of a computational model. The method presented here can be applied to a wide array of oscillatory systems.