Excitable Neurons, Firing Threshold Manifolds and Canards
© J. Mitry et al.; licensee Springer 2013
Received: 30 October 2012
Accepted: 20 March 2013
Published: 14 August 2013
We investigate firing threshold manifolds in a mathematical model of an excitable neuron. The model analyzed investigates the phenomenon of post-inhibitory rebound spiking due to propofol anesthesia and is adapted from McCarthy et al. (SIAM J. Appl. Dyn. Syst. 11(4):1674–1697, ). Propofol modulates the decay time-scale of an inhibitory GABAa synaptic current. Interestingly, this system gives rise to rebound spiking within a specific range of propofol doses. Using techniques from geometric singular perturbation theory, we identify geometric structures, known as canards of folded saddle-type, which form the firing threshold manifolds. We find that the position and orientation of the canard separatrix is propofol dependent. Thus, the speeds of relevant slow synaptic processes are encoded within this geometric structure. We show that this behavior cannot be understood using a static, inhibitory current step protocol, which can provide a single threshold for rebound spiking but cannot explain the observed cessation of spiking for higher propofol doses. We then compare the analyses of dynamic and static synaptic inhibition, showing how the firing threshold manifolds of each relate, and why a current step approach is unable to fully capture the behavior of this model.
Excitable neurons are typically at rest, but can fire action potentials in response to certain forms of stimulation. Many textbooks refer to excitability as an all-or-none response, i.e. a “subthreshold” synaptic input leads to a small graded postsynaptic potential while a “superthreshold” input evokes an action potential. One then seeks to find an action potential threshold, i.e., a particular voltage value that demarcates the all-or-none response. However, this is a misconception. Using geometric analysis of neural models, it was FitzHugh  who first noticed that a firing threshold, if it exists, is never a number but a manifold.
Another characteristic feature of neurons is the existence of processes that evolve on multiple time-scales. The interaction of ionic currents acting on different time-scales is responsible for the creation of action potentials in neurons. This time-scale feature leads to mathematical models of neurons often referred to as singular perturbation problems. These models are particularly amenable to analysis using geometric singular perturbation theory (GSPT) [9, 14] with the specific aim of giving predictions of model dynamics based on singular limit observations. Threshold phenomena are closely related to folded critical manifolds of such singularly perturbed neural problems. In the famous 2-dimensional singularly perturbed FitzHugh–Nagumo model, it is the repelling middle branch of the folded (cubic-shaped) critical manifold that forms the firing threshold manifold in the singular limit. In the full system, this firing threshold manifold perturbs to a nearby repelling slow manifold.
In higher dimensional models with more than one slow variable, the geometry of such singularly perturbed problems becomes quite intricate and folded singularities play a prominent role. Folded singularities lie on a fold of the critical manifold where stable and unstable branches of this (higher dimensional) manifold meet. Canards [2, 3, 8, 15, 21, 23, 24] are trajectories associated with folded singularities and connect the stable and unstable branches of the critical manifold. These special solutions have been identified as important objects in explaining complex oscillatory patterns known as mixed-mode oscillations (MMOs) . There now exists a substantial amount of literature on applications of canard theory, and we refer the interested reader to detailed tables on relevant literature provided in a review on MMOs .
Canards form boundaries of different dynamic behavior—they are separatrices by nature. More importantly, they encode slow time-scales, i.e., temporal information is reflected in the geometry of a canard. In the two-dimensional case, Izhikevich  clearly highlights the fact that canard trajectories and repelling slow manifolds provide the best approximation to the firing threshold manifold, hence giving a mathematical “structure” to the famous No Man’s Land by R. FitzHugh . More recently, Desroches et al.  discuss the relationship between canards and excitability thresholds in planar slow-fast systems by identifying inflection lines of the flow.
In the present study, we focus on a higher dimensional neural model and the specific role of folded saddle canards as firing threshold manifolds. We observe that varying one of the slow time-scales changes the boundaries of different dynamic behaviors. The slow time-scale is thus encoded in the position of the saddle canard separatrix, and so, remarkably, a change in a slow time-scale can be “seen” in a geometric object. Folded saddle canards have been shown to form firing threshold manifolds in general Morris–Lecar/FitzHugh–Nagumo type neural models with dynamic current input  and even in a climate model of the so-called “compost-bomb instability” by Wieczorek et al. . In both, under variation of a particular parameter, the position of a folded saddle canard varies explaining the excitability properties of the model.
This geometric observation becomes important when we try to understand changes in neural dynamics. Neurons constantly sense their environment, e.g., temperature, acidity or glucose, and they encode fluctuations of these parameters by changing their ion channel dynamics. The famous Hodgkin–Huxley model of the squid giant axon  incorporates temperature changes through a -temperature factor that increases the speed of the ion channel gates with increasing temperature. If such environmental changes are encoded in different speeds of slow ion channels, then we expect this to be reflected in a change of the firing threshold manifold of the neuron. Thus, identifying the cause of different neural dynamics through the specific position of a canard in a singularly perturbed system becomes a valuable diagnostic tool to understand this phenomenon, and it is the focus of this work.
2 Propofol and Rebound Spiking
As a case study, we investigate the role of the general anesthetic propofol on rebound spiking in the central nervous system . Many general anesthetics, including propofol, prolong the duration of GABAergic inhibitory postsynaptic currents (IPSCs), and this action contributes to the behavioral properties of these drugs . Mathematically speaking, propofol changes the slow time-scale of the deactivation of the GABAa receptor channel. Paradoxically, low doses of propofol causes excitation rather than sedation. This behavior can already be observed in an isolated single cell model that receives GABAergic IPSCs . We adapt this propofol neuron model formulated in  slightly in order to more clearly emphasize the role of folded saddle canards in the observed dynamic behavior. We note that only minimal adjustments have been made in order to preserve the qualitative behavior of the model, namely the observation of post-inhibitory rebound spiking for a window of GABAa synaptic time-scale values. The modification consists of two parameter changes, the details of which are given below. The essential difference is that this modification shifts the resting membrane potential to a lower, more hyperpolarized, voltage value, allowing a more uniform separation of time-scales over a range of GABAa time-scale values. This modification enables us to make full use of the machinery of GSPT. More details of the relation between the two models can be found in the last section of the paper.
The membrane currents consist of a fast sodium current, , a potassium current, , and a leak current, , collectively referred to as the spiking currents, and a slow muscarinic potassium M-current, . This model neuron receives GABAa IPSCs, here modeled by the current . The applied current, , models tonic external drive.
Propofol neuron and network model system parameters
Current balance equation constants
Maximal ion channel conductances
Voltage-gated Na+ channels
Voltage-gated K+ channels
Slow acting voltage-gated K+ channels
Synaptic GABAa receptor channels
Ionic current reversal potentials
Voltage-gated Na+ channels
Voltage-gated K+ channels
Synaptic GABAa receptor channels
2.1 Dynamic Inhibition, but not the Current Step Protocol, Leads to Cessation of Spiking with Increased Inhibition
Using a traditional approach, the spiking behavior is studied with a step protocol. Here, an applied current is switched on, kept at a constant level, then removed. By holding constant, the dynamics in s are lost, thus rendering the step protocol system five-dimensional as opposed to the six-dimensional dynamic inhibition system (1). This step protocol is usually able to reproduce spiking patterns, while simplifying input dynamics, and thus give insight into the associated spiking behavior. However, applying a step protocol to the present model, we find a single transition from inactivity to isolated spiking as is increased. As is further increased a transition from single spiking to a doublet, and then to a triplet of spikes is observed (Fig. 3). A maximum of three spikes is observed despite further increases in duration of inhibition. The step protocol is unable to reproduce a cessation of spiking for increased synaptic inhibition. It thus becomes apparent that there is a necessary dynamic mechanism required to yield a specific range of spiking under variation of synaptic inhibition time-scale.
In the present study, our aim is to identify firing threshold manifolds of the propofol model, both with dynamic and with static inhibition, thus explaining the spiking behavior under variation of propofol dosage and duration of current step inhibition, respectively. Key to this aim is the use of GSPT for which a detailed analysis of time-scales is necessary. This time-scale analysis is presented in Sect. 3. Identifying multiple time-scales in system (1) implies a splitting of solution trajectories of (1) into segments of fast and slow dynamics. These fast and slow dynamics are captured by lower dimensional subsystems, termed the layer and reduced problems, respectively. GSPT uses these lower dimensional subsystems, studied in Sect. 4, to provide insight into the geometric structures which govern the behavior of the model, and thus to predict the dynamics of the full (higher dimensional) system (1). Results are given in Sect. 5 for the case of dynamic inhibition. In particular, we identify a singular canard of folded saddle type as the separatrix that forms the firing threshold manifold. In Sect. 6, a similar analysis is carried out for the case of static inhibition; again a singular limit prediction of the spiking threshold manifold is identified. In both the dynamic and static inhibition cases, a numerical confirmation of singular limit predictions is made by calculating the true firing threshold manifolds. This emphasizes the predictive power of GSPT. We also point out that the firing threshold manifold in the case of static inhibition is in fact the structure which the firing threshold manifold in the case of dynamic inhibition approaches in the limit as , i.e., in the transition from slow to fast synaptic GABAa inhibition decay rates. In Sect. 7, we discuss the original and modified propofol models and their respective analyses. Finally, in Sect. 8, we make some concluding remarks about canards and excitability in neural models.
3 Time-Scales and Dimensional Analysis
By observation of the time traces in Figs. 2 and 3, it can be argued that there exists a multiple time-scales structure within the typical solution trajectories of the modified propofol model (1). In order to investigate this behavior further, a dimensional analysis of the system is performed so as to roughly determine the time-scales on which each variable evolves.
3.1 The Modified Propofol Model Has Three Distinct Time-Scales
Note that the , values given in parentheses refer to the time-scale functions for subthreshold . The time-scale on which s evolves is given directly by the value of .
It is now possible, based on the above numbers, to propose a hierarchy of variables according to the time-scales on which they evolve. We classify the variables v and m superfast, h and n fast, and w and s slow. Recall the value of directly determines the time-scale on which s evolves. Accordingly, care should be taken in the following singular perturbation analysis since it may no longer be valid to consider s slow as becomes sufficiently small. (In , s is considered a fast variable when is small.)
Another way to check time-scales separation is to consider the maximum magnitude of the time derivative of each state variable over the course of a full spiking trajectory; see Fig. 16. For , we observe that the for is roughly given by ; compare these values with the inverse of the time-scales given in (5). This suggests that our proposed time-scale hierarchy with w and s slow is reasonable. As alluded to earlier, the specific insight of this model is the interplay between the slow M-current and the inhibitory synaptic current contributing to rebound spiking.
where measures the time-scale separation between the fast variables and the superfast variables while measures the time-scale separation between the fast variables and the slow variables . Thus, the dimensionless system (6) is a singularly perturbed problem with singular perturbation parameters and . This suggests an inherent three time-scales problem. The specific insight of the model and, therefore, the corresponding analysis should focus on the interplay of the slow M-current and the inhibitory synaptic current contributing to rebound spiking. We therefore group the fast and superfast variables together into one “fast” pool and consider system (6) a 4-fast/2-slow / problem. In doing so, we choose ε as the main singular perturbation parameter and keep δ fixed.
4 Geometric Singular Perturbation Theory
The identification of distinct fast and slow time-scales in the modified propofol model allows us to utilize the methods of GSPT to identify the key differences in geometric structure that account for the differences in spiking dynamics between static and dynamic inhibition. Thus, we proceed to identify two lower dimensional subsystems that govern the fast and slow dynamics in order to give us insight into the behavior of the full higher dimensional system.
4.1 Layer Problem
Moving further up on the uppermost surface there exists a line of Hopf bifurcations, , indicating the boundary between and a second attracting surface of equilibria, . Stable limit cycles emanate from the Hopf line through supercritical Hopf bifurcations and terminate in homoclinic cycles, which are homoclinic to the lower fold curve, (this bifurcation structure is known as a saddle node on invariant cycles, or SNIC, bifurcation). The Hopf line lies outside the physiological range of interest (i.e., ); however, the associated limit cycles emanate from outside this region of interest and terminate just within. Thus, the layer problem contains two stable attractors; the lower branch of the critical manifold and the set of stable limit cycles.
4.2 Reduced Problem
The reduced problem is a two-dimensional differential algebraic equation. Within (10), the first four equations dictate that the dynamics occur on the critical manifold , and the last two equations describe the dynamics of the slow variables w and s thereon. Note the algebraic equation defining , , can be solved for respectively , but not for v reflecting the folded geometry of the manifold; see Fig. 4. Hence, it suffices to study the flow on the critical manifold in one single coordinate chart, either the -chart or the -chart where is defined as a graph.
The flow described by (15), respectively (16), is equivalent to that of (13), respectively (14), on the attracting surface, , and the repelling surface, , while reversed on the repelling surface, . This is due to the rescaling of time by , which is positive on and and negative on .
4.2.1 Canards Form a Separatrix for Solutions of the Reduced Problem
Here, we aim to use the geometry of the reduced system in order to identify a manifold that separates the trajectories of the reduced problem into two distinct behaviors: those that return to an equilibrium or rest state and those that proceed to the fold curve where the fast dynamics again become important in establishing a spiking solution. Important to the identification of this manifold is finding folded singularities. We find a folded saddle equilibrium; the canard solution of which is the relevant manifold that separates the behavior of the reduced trajectories.
Ordinary singularities are equilibria of the desingularized flow (15), respectively (16), of the reduced flow, (10), and of the original system, (9). Folded singularities, on the other hand are generally not equilibria of the reduced flow or the original system.
Within (15), respectively (16), we find three stable nodes, , and , which constitute the set of ordinary singularities, . Note that lies on , lies on and lies on . The position and stability of these equilibria are independent of the value of the parameter .
The folded saddle canard is a separatrix and effectively organizes the solution trajectories of the reduced problem. Depending on which side of the trajectory is initially, solution trajectories of (15) travel along and either meet the fold curve for or move toward, later traveling along, . In the former case, the trajectory is no longer accurately approximated by the reduced problem at the fold due to a finite time blow up of system (15) and subsequent dynamics are dictated by the layer problem. In the latter case, the solution trajectory terminates at , prevented by from approaching . Compare Figs. 5 and 6.
5 Firing Threshold Manifolds and Dynamic Inhibition
Using information from the reduced and layer flows, we are able to give a singular limit prediction of post-inhibitory rebound spiking as observed in system (1). Recall that the critical manifold is given by the set of equilibria of the layer problem. Trajectories of the layer problem approach this set along so-called fast fibres. In this context, the critical manifold forms the set of base points of these fast fibers. Hence, base points allow for a connection between the flows of the layer and reduced problems. In particular, the relationship between the base point of the fast fibre through the initial condition due to inhibition and the singular canard solution determines whether a trajectory goes on to spike or not; see Fig. 9.
5.1 Nonsingular Canards: The True Threshold Manifolds for Spiking Activity
An important result from GSPT is that a singular canard of folded saddle type perturbs to a nearby canard for the full model problem . Consequently, the perturbed canard forms the true firing threshold manifold of the full system (1). In the following, we confirm numerically this firing threshold manifold by calculating the canard. We note here that due to numerical challenges in calculating canards for systems in , with more than one fast variable, we consider a reduction of the six-dimensional modified propofol model. Resolving this issue is left as a point of focus for future work. By setting each of m, h and n to their respective steady-state values, we derive a three-dimensional reduction of our original system. This reduction is valid as we have determined that m, h and n evolve on a fast time-scale; this system simply approximating this fact by having these processes act instantaneously. Within this reduced system the range of values for which spiking occurs is similar to the original with ; however, the spiking mechanism is partially disabled and the trajectory is unable to fully reset. At least 2 fast variables are necessary to provide the necessary repolarization mechanism. We note that this does not affect sub or perithreshold dynamics, and consequently does not affect the onset of spiking. Hence, a model reduction to 1 fast variable locally near the fold is justified.
Using the continuation package AUTO , and closely following the techniques outlined in , the nonsingular canard, , is found for and is then continued in . The nonsingular canards of the modified propofol model lie at the intersection of the attracting and repelling slow manifolds, and , respectively. These manifolds correspond to and , respectively, for . The attracting and repelling manifolds of the perturbed system are calculated using the homotopy continuation of solution trajectories to a suitable boundary value problem. Here, we make use of the normal hyperbolicity of the critical manifold . Namely, for small , the slow manifolds and are smooth perturbations of the critical manifold , away from the fold curve F where normal hyperbolicity is lost. A detailed description of slow invariant manifold calculations and canard detection and continuation is provided for the self-coupled FitzHugh–Nagumo system within the AUTO manual, demo fnc.
6 Firing Threshold Manifolds and the Classical Step Protocol
6.1 Singular Perturbation Analysis
In a singular perturbation analysis similar to that above, we consider four fast variables, , and a single slow variable, w. Note that s dynamics are no longer considered in a current step protocol; , and thus dependence on s, is here replaced by a constant value. In the singular limit, the layer problem defines a one-dimensional cubic-shaped critical manifold in -space. As per the above analysis, only the lower branch of the critical manifold is an attracting branch; the upper two branches being unstable. While the shape and stability properties of the critical manifold are independent of the value of , the position of the critical manifold is shifted in the direction of negative w as the synaptic current is set to during inhibition (otherwise ; see Fig. 11). Note that here the critical manifold for is precisely the section defined by through the critical manifold derived above, i.e., the critical manifold for is here given by . Once on the critical manifold, dynamics are described by the reduced problem; here formulated as (10) with and adjusted as per the step protocol. The reduced problem reveals, as before, the existence of three stable node equilibria, one on each branch of the critical manifold. We label the equilibria as before for , whereas the equilibria on the shifted critical manifold, i.e., for , are denoted , .
Initially, the state point is held at rest at with . At the onset of synaptic inhibition the critical manifold shifts, and so defines a new stable equilibrium, . In the singular limit (Fig. 11), the trajectory falls instantaneously, traveling along a vertical fast fibre, onto the stable branch of the shifted critical manifold. This base point is thus a projection of onto the shifted critical manifold. The singular trajectory is then described by the reduced problem, slowly moving along this lower branch toward . Once synaptic inhibition is removed, the critical manifold shifts back to its original position. At this point, the layer problem dictates a rapid vertical ascent; the trajectory no longer remains on the critical manifold. The subsequent dynamics are thus dependent on the length of the applied constant synaptic inhibition. If the duration of inhibition is relatively short, at the removal of inhibition, the trajectory simply returns to the now overhead lower branch of the critical manifold. This corresponds to an unsuccessful post-inhibitory rebound spike. If; however, the duration of inhibition allows the trajectory time enough to sufficiently approach then, at the removal of inhibition, the trajectory undergoes a spike in v. This behavior corresponds to a successful post-inhibitory rebound spike. The associated threshold manifold is given as the concatenation of the middle branch of the critical manifold and the fast fiber through the lower fold point, shown in Fig. 11.
6.1.1 The Singular Limit Predicts Only One Spiking Transition with Increasing Duration of Inhibition
We here determine that the singular limit predicts the minimum duration of inhibition such that a trajectory is still able to spike is 6 ms. Since the layer problem acts instantaneously, this value is determined solely according to the dynamics within the reduced problem. We simulate the singular reduced problem along the lower branch of the critical manifold, noting the time at which the singular trajectory passes the w-value of the lower fold. Note here we use the singular problem to avoid the distortion of time within the desingularized problem due to a position dependent rescaling of time in the process of desingularization.
6.2 Nonsingular Firing Threshold Manifold
6.2.1 A Canard Solution Forms the Nonsingular Separatrix Between Spiking and Nonspiking Solutions
As per the analysis of dynamic inhibition above, we again find a geometric object which demarcates the boundaries of trajectories with different spiking behavior. Within the nonsingular system the perturbed separatrix is located using a shooting method, in backward time, from the stable node . As before, the propofol model used to calculate this separatrix requires the reduction , for , in order to allow calculation along an unstable manifold. Here, we see that the nonsingular separatrix forms the boundary of spiking and nonspiking behavior (compare Figs. 12a and 12b). We note here that the threshold manifold is a canard, as per the dynamic inhibition analysis. Accordingly, we find that trajectories, which begin exponentially close to this structure follow it for a significant amount of time, even onto a repelling portion of the associated critical manifold, such as that in Fig. 12b near the fold point. The deviation of the perturbed threshold manifold compared with that of the singular analysis near the critical manifold lower fold is explained by a fold analysis within singular perturbation theory (compare Figs. 11 and 12). Here, we expect, and indeed find, that the nonsingular manifold perturbs a distance which is from the fold [16, 22]. Hence, we have identified the non-singular threshold manifold for the current step protocol.
6.2.2 Explanation of a Finite Number of Rebound Spikes as Inhibition Time Is Increased
As the duration of the step protocol increases, the number of rebound spikes increases steadily to a maximum of three spikes (compare Figs. 3d and 12b). This observation is explained by a limited reset in the direction of w during each rebound spike. After a single spike, as seen in Fig. 12b, the corresponding trajectory remains to the right of the threshold manifold. Three consecutive rebound spikes are required before the trajectory returns to an excitable resting state at , having moved past the threshold manifold. At this point, we see that further prolonging the duration of synaptic inhibition has no additional effect. Once the trajectory is sufficiently close to during synaptic inhibition, regardless of the actual duration of inhibition, the subsequent dynamics remain unchanged. This results in a maximum number of rebound spikes. This maximum spike number is encoded within the time-scale separation between the fast and slow dynamics, i.e., within ε. For fixed ε, the maximum spike number is set according to the average shift in the slow dynamics per (fast) spike event. As , we observe that the maximum spike number increases as the reset in w reduces per spike (work not shown). The singular limit picture confirms this finding, here showing no net shift in w per spike for .
6.3 The Folded Saddle Canards of the Dynamic Protocol Converge Toward the Spiking Threshold Manifold of the Current Step Protocol as
7 Comparison of the Original and Modified Propofol Models
The propofol network model presented here is a modified form of the propofol neuron network model in . The propofol model is the same 6–dimensional system of equations (Eqs. (1) and (2)) as the modified propofol model but with two parameter changes. The modified propofol model is phenomenologically similar to the original propofol model, which shows rebound spiking only for values of between 8 and 48 ms (compared with the modified model: 8–21 ms) and was also examined using geometric singular perturbation theory . In particular, a canard of folded saddle type was identified as a firing threshold manifold. In Sects. 7.1–7.3, we detail our reasons for modifying the original propofol model.
7.1 Modifying the Propofol Model: Increasing the Time-Scale Separation Allows for a more Accurate Singular Limit Trajectory Approximation
Both models, the original and the modified propofol model, have a global stable equilibrium that plays the role of the resting membrane potential. We note that this stable equilibrium (v-value of −63.6 mV) of the original propofol model lies quite close to the fold curve, . Additionally, we observe that the fold-curve has an almost constant v-value ( along ). Given the fairly uniform structure of the critical manifold in the direction of s near , the position of the stable equilibrium ensures a proximity of the post-inhibitory initial condition and thus of the post-inhibitory dynamics to .
Thus, we propose the modified propofol model of post-inhibitory rebound spiking. This modified propofol model retains the basic phenomenological and geometric features of the propofol model while the position of the stable equilibrium has been moved away from the fold curves. This modification allows for a more marked separation of time-scales during the early dynamics, and thus describes a system more accurately approximated by a singular limit prediction (Fig. 14b).
To formulate the modified propofol model the steady-state function is shifted by 3 mV in the direction of negative v. This results in a lower global stable equilibrium point (v-value of −65.8 mV), and thus a lower resting membrane potential. However, this modification also makes it more difficult to generate a rebound spike. In order to counter this effect, the value of the maximal synaptic conductance is increased so as to effectively increase the strength of synaptic inhibition. Here, we set to 4 mS, i.e., .
As previously noted, the behavior of the modified propofol model is similar to the original propofol model in that post-inhibitory rebound spiking occurs only for intermediate values of (Fig. 2). Thus, by our modifications, we have not lost the rebound-spiking “window” and we have gained a more accurate geometrical representation of nonsingular trajectories by their singular limit. However, even more importantly, the modified model allows us to use the same fast/slow decomposition to predict the spiking transition at both interval boundaries ( small and large). This was not possible in the original propofol model. We describe this in more detail next.
7.2 Discrepancies Between the Singular and Nonsingular Trajectories in the Propofol Model
Although the singular limit of the original propofol model provides a good approximation for cessation of spiking when is large (52 ms for the singular limit versus 48 ms for nonsingular trajectories), when we compare singular global trajectories with the nonsingular trajectories of the propofol model, we find that the geometry of the singular limit trajectories does not accurately predict the geometry of the nonsingular trajectories (Fig. 14). In particular, the early dynamics of the singular limit trajectories do not accurately mimic those of the nonsingular trajectories using the proposed slow/fast splitting. As mentioned before, this poor predictive power of geometric singular perturbation theory can be explained by the proximity of the trajectory near the fold , throughout its evolution. The time-scale splitting of the “fast” and “slow” variables away from the fold does not hold anymore in a neighborhood of the fold . Without this time-scale splitting, and thus the identification of a singularly perturbed problem, regular GSPT analysis does not yield a reliable prediction of system dynamics.
We thus conclude that while the singular limit analysis still provides a good prediction for the large spiking transition in the original propofol model, the particular geometry of this system distorts the time-scale separation, and thus requires an alternative approach of geometric singular perturbation theory using a blow-up analysis along the fold-curve  followed by a blow-up of the folded saddle singularity [21, 24]. This two-step approach is left for future work.
7.3 In the Original Propofol Model, Considering s Slow or Fast Does not Explain the Spiking Transition when Is Small
In the original propofol model, considering s as a slow variable cannot predict the spiking transition when is small; see Fig. 10 in . Similarly, considering s as a fast variable cannot predict the spiking transition when is small either; see explanation on p. 13 in . Hence, using geometric singular perturbation theory in the context of either a 4-fast/2-slow time-scales or a 5-fast/1-slow time-scales separation does not predict the transition between spiking and nonspiking when is small in the original propofol model. The answer to this “riddle” about the appropriate dynamics of s for small in the original propofol model lies in an intermediate time-scale of order , . Such an intermediate time-scale reveals itself when using a (cylindrical) blow-up analysis of the fold, [22, 24]. As mentioned above, we leave this blow-up analysis of the original propofol model for future work.
7.4 Comparison of Geometric Structures in the Original and Modified Propofol Model when Is Small
In , the authors use a 5-fast/1-slow time-scale separation to examine the geometry of the spiking transition when is small. Interestingly, this results in the same resting state () singular limit geometry as the 4-fast/1-slow time-scale problem used here to analyze the current step protocol (Fig. 12). This makes sense because considering s fast allows the manifold to be reached on a fast time-scale, and thus the system reduces to the same 4-fast/1-slow time-scale problem used here. As shown in , when s is considered one of the fast variables, as in the original propofol model, the stable manifold of in Fig. 12 approximates well the threshold for spiking. Recall that this is shown in Fig. 13, i.e., as the canard separatrix is well approximated by the firing threshold manifold of the current step protocol. Hence, by changing parameters to allow the construction via geometric singular perturbation theory of canards, this paper clarifies the underlying geometry of the original propofol model.
8 Concluding Remarks
An important feature of most physiological systems is that they evolve on multiple time-scales. The theory of differentiable dynamical systems for two time-scales (slow/fast) has a successful history in explaining a wide range of physiological behavior such as electrical spiking and bursting in neurons . Complex pattern generation in such slow/fast systems is almost exclusively related to loss of normal hyperbolicity of invariant critical manifolds, which is associated with bifurcation sets in the fast subsystem. In particular, folded critical manifolds are ubiquitous in such systems (see, e.g., Fig. 4). Canard theory deals exactly with these slow/fast time-scales systems where loss of normal hyperbolicity occurs, and its theory is applicable to problems with arbitrary dimensions [2, 3, 8, 15, 16, 21, 23, 24]. A recent success story of canard theory is that it provides an explanation for mixed-mode oscillations (MMOs), a frequently observed mix of small and large amplitude oscillation patterns in slow/fast time-scale physiological models. Canards of folded node and folded saddle-node type play a key role in explaining these patterns. The interested reader is referred to the current review on MMOs , and the extensive reference list to applications therein.
Canard theory provides also a new direction for understanding transient dynamics of biological systems that have multiple time-scales. The propofol model studied here is a prime example. We demonstrate here the use of canard theory to explain the dynamics of rebound spiking for a specific range of propofol doses. In the context of neuronal excitability, we identify canards of folded saddle type as firing threshold manifolds. It is remarkable that dynamic information such as the temporal evolution of an external drive (GABAergic inhibition in this study) is encoded in the location of an invariant manifold—the canard. It is the variable positioning of the canard separatrix that explains the observed rebound spiking for a specific range of propofol doses (Fig. 9). The same role of folded saddle canards as firing threshold manifolds was recently identified in a class of Morris–Lecar/FitzHugh–Nagumo type models with dynamic external drive . Since mathematical models of physiological phenomena (both neuronal and nonneuronal) frequently show abrupt transitions in behavior and have dynamics which are encoded by multiple time-scales, the methods of GPST and canard theory are likely applicable to a much broader range of problems within the biological sciences.
Appendix: Steady State and Time-Scale Functions
Electronic Supplementary Material
This work was supported by NSF Grant DMS-0602204 (MM), NSF Grant DMS-071670 (NK and MM) and ARC Grant FT-120100309 (MW).
- Bai D, Pennefather PS, MacDonald JF, Orser BA: The general anesthetic propofol slows deactivation and desensitization of GABA A receptors. J Neurosci 1999, 19(24):10635–10646.Google Scholar
- Benoît E:Systèmes lents-rapides dans et leur canards. Astérisque 1983, 109–110: 159–191.Google Scholar
- Benoît E, Callot J, Diener F, Diener M: Chasse au canard. Collect Math 1981, 31–32: 37–119.Google Scholar
- Brøns M, Kaper T, Rotstein H: Introduction to focus issue: mixed mode oscillations: experiment, computation, and analysis. Chaos 2008., 18: Article ID 015101 Article ID 015101Google Scholar
- Desroches M, Guckenheimer J, Krauskopf B, Kuehn C, Osinga H, Wechselberger M: Mixed-mode oscillations with multiple time-scales. SIAM Rev 2012, 54: 211–288. 10.1137/100791233MathSciNetView ArticleGoogle Scholar
- Desroches M, Krupa M, Rodrigues S: Inflection, canards and excitability threshold in neuronal models. J Math Biol 2013, 67(4):989–1017. 10.1007/s00285-012-0576-zMathSciNetView ArticleGoogle Scholar
- Doedel EJ: AUTO: a program for the automatic bifurcation analysis of autonomous systems. Congr Numer 1981, 30: 265–284.MathSciNetGoogle Scholar
- Dumortier F, Roussarie R: Canard cycles and center manifolds. Mem Am Math Soc 1996., 121: Article ID 577 Article ID 577Google Scholar
- Fenichel N: Geometric singular perturbation theory. J Differ Equ 1979, 31: 53–98. 10.1016/0022-0396(79)90152-9MathSciNetView ArticleGoogle Scholar
- FitzHugh R: Mathematical models of threshold phenomena in the nerve membrane. Bull Math Biophys 1955, 7: 252–278.Google Scholar
- FitzHugh R: Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1961, 1(6):445–466. 10.1016/S0006-3495(61)86902-6View ArticleGoogle Scholar
- Hodgkin AL, Huxley AF: A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 1952, 117: 500–544.View ArticleGoogle Scholar
- Izhikevich EM: Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. MIT Press, Cambridge; 2007.Google Scholar
- Jones CKRT: Geometric singular perturbation theory. Springer Lecture Notes Math. 1609. Dynamical Systems 1995, 44–120.View ArticleGoogle Scholar
- Krupa M, Szmolyan P: Relaxation oscillations and canard explosion. J Differ Equ 2001, 174: 312–368. 10.1006/jdeq.2000.3929MathSciNetView ArticleGoogle Scholar
- Krupa M, Szmolyan P: Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions. SIAM J Math Anal 2001, 33(2):286–314. 10.1137/S0036141099360919MathSciNetView ArticleGoogle Scholar
- McCarthy MM, Brown EN, Kopell N: Potential network mechanisms mediating electroencephalographic beta rhythm changes during propofol-induced paradoxical excitation. J Neurosci 2008, 28(50):13488–13504. 10.1523/JNEUROSCI.3536-08.2008View ArticleGoogle Scholar
- McCarthy MM, Kopell N: The effect of propofol anaesthesia on rebound spiking. SIAM J Appl Dyn Syst 2012, 11(4):1674–1697. 10.1137/100817450View ArticleGoogle Scholar
- Rush ME, Rinzel J: The potassium A-current, low firing rates and rebound excitation in Hodgkin–Huxley models. Bull Math Biol 1995, 57: 899–929.View ArticleGoogle Scholar
- Schwartz RS, Brown EN, Lydic R, Schiff ND: General anesthesia, sleep, and coma. N Engl J Med 2010, 363(27):2638–2650. 10.1056/NEJMra0808281View ArticleGoogle Scholar
- Szmolyan P, Wechselberger M:Canards in . J Differ Equ 2001, 177: 419–453. 10.1006/jdeq.2001.4001MathSciNetView ArticleGoogle Scholar
- Szmolyan P, Wechselberger M: Relaxation oscillations in . J Differ Equ 2004, 200: 69–104. 10.1016/j.jde.2003.09.010MathSciNetView ArticleGoogle Scholar
- Wechselberger M: Existence and bifurcation of canards in in the case of a folded node. SIAM J Appl Dyn Syst 2005, 4: 101–139. 10.1137/030601995MathSciNetView ArticleGoogle Scholar
- Wechselberger M: À propos de canards (Apropos canards). Trans Am Math Soc 2012, 364: 3289–3309. 10.1090/S0002-9947-2012-05575-9MathSciNetView ArticleGoogle Scholar
- Wechselberger M, Mitry J, Rinzel J: Canard theory and excitability. Random and Nonautonomous Dynamical Systems in the Life Sciences 2013. in press in pressGoogle Scholar
- Wieczorek S, Ashwin P, Luke CM, Cox PM: Excitability in ramped systems: the compost-bomb instability. Philos Trans R Soc A, Math Phys Eng Sci 2011, 467(2129):1243–1269. 10.1098/rspa.2010.0485MathSciNetView ArticleGoogle Scholar
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