Dynamical systems analysis of spike-adding mechanisms in transient bursts
© Nowacki et al.; licensee Springer 2012
Received: 24 August 2011
Accepted: 13 February 2012
Published: 24 April 2012
Transient bursting behaviour of excitable cells, such as neurons, is a common feature observed experimentally, but theoretically, it is not well understood. We analyse a five-dimensional simplified model of after-depolarisation that exhibits transient bursting behaviour when perturbed with a short current injection. Using one-parameter continuation of the perturbed orbit segment formulated as a well-posed boundary value problem, we show that the spike-adding mechanism is a canard-like transition that has a different character from known mechanisms for periodic burst solutions. The biophysical basis of the model gives a natural time-scale separation, which allows us to explain the spike-adding mechanism using geometric singular perturbation theory, but it does not involve actual bifurcations as for periodic bursts. We show that unstable sheets of the critical manifold, formed by saddle equilibria of the system that only exist in a singular limit, are responsible for the spike-adding transition; the transition is organised by the slow flow on the critical manifold near folds of this manifold. Our analysis shows that the orbit segment during the spike-adding transition includes a fast transition between two unstable sheets of the slow manifold that are of saddle type. We also discuss a different parameter regime where the presence of additional saddle equilibria of the full system alters the spike-adding mechanism.
How a single spike or a burst of spikes is generated and regulated for neuron cells is one of the most fundamental questions in neuroscience . Spike generation is closely related to neuronal excitability, which is the ability of the cell’s membrane potential to undergo a large excursion, called an action potential or a spike, when subjected to a sufficiently strong stimulus [1–3]. An excitable cell either responds in full to such a stimulus or not at all, which allows for a reliable transmission of information. Therefore, the different mechanisms for excitability and bursting have been widely studied; we refer to Izhikevich  for a comprehensive overview of mechanisms for neurons. Excitable behaviour has also been reported to occur in many other types of cells [1–3] as well as in physical systems, such as lasers [5, 6], electronic circuits [7–10] and chemical reactions . Neuronal excitability can be very sensitive to even relatively small changes in, for example, the biophysical properties of a neuron [12–14] or its morphology . This parameter sensitivity indicates that dynamical systems theory is particularly suited for explaining the rich dynamics found in excitable systems.
The classification of different bursting mechanisms was pioneered by Rinzel , who used a system decomposition into slow and fast subsystems. He showed that the burst can be divided into active (spiking) and silent phases, which follow different types of attractors of the fast subsystem. Hence, a classification of the bursting oscillators is provided by the structure of the bifurcation diagram of the fast subsystem. Rinzel’s classification was extended in Izhikevich’s studies [1, 4]. As an alternative approach, Golubitsky et al.  used singularity theory to classify the bifurcation diagrams of the fast subsystem and, hence, the different bursting mechanisms; see also . These studies are primarily for systems with one slow variable; see Smolen et al.  for an extension to two slow variables and Ermentrout and Terman  for a summary of these ideas along with new results.
A classification of bursting mechanisms, however, does not answer questions about the number of spikes in a particular burst of the same type nor does it explain possible transitions between spiking and bursting. While the latter has received considerable attention over the years, the former has hardly been addressed. Terman  analysed transitions between bursting and tonic (continuous) spiking in a pancreatic β-cell model. He recognised the importance of connecting classical slow-fast analysis with full system bifurcation analysis and identified bifurcations of the periodic bursting solutions that organise the transitions between different parameter regimes. He further studied chaotic spiking that can arise in between such transitions . Recently, Benes et al.  and Kramer et al.  reported that a new type of torus canard can play a role in the transition from spiking to bursting in a model of Purkinje cells. Other recent studies particularly focus on spike-adding in a periodic bursting oscillator; for example, see Govaerts and Dhooge , Guckenheimer and Kuehn , Tsaneva-Atanasova et al.  and Linaro et al. . These studies show that the spike-adding mechanism is formed by a pair of saddle-node bifurcations of periodic orbits of the full system; bursts with different numbers of spikes are, in fact, different periodic attractors of the full system that may coexist only if the number of spikes differs by one [20, 26]. The recent study by Teka et al.  explains how one may predict the precise number of spikes in a burst; here, the use of two slow variables is essential. Methods to regulate the number of spikes are reported by Ghigliazza and Holmes , where a minimal Hodgkin-Huxley model of bursting is proposed to analyse spike-adding and transitions between bursting and tonic spiking in a more general context.
As discussed by Izhikevich , the mechanisms for generating spikes do not depend on whether the neuron exhibits spiking only as a transient phenomenon when subjected to a strong enough stimulus or whether it is spiking or bursting continuously. This is indeed the case if transient bursting is organised by the applied stimulus. Applying a stimulus has the effect of changing the right-hand side of the underlying system of ODEs. The bifurcation diagram of the corresponding fast subsystem typically no longer has a stable equilibrium that corresponds to the resting potential, and a spike or burst arises from new attractors that exist only when the stimulus is on. As soon as the stimulus is switched off, the system relaxes back to the resting potential. Therefore, the mechanism is due to a change in the structure of the bifurcation diagram, which depends on the strength of the applied stimulus. Studies of this nature, where the type of burst is studied in dependence on the strength or duration of the stimulus, have been done, for example, by Tran et al. , Kim et al.  and Stern et al. .
In this paper, we investigate spike-adding in a transient burst in the model of hippocampal pyramidal neurons from Nowacki et al. . In contrast to the above studies, the spike-adding occurs after the applied stimulus has been switched off. Hence, the bursting behaviour is governed by the underlying bifurcation structure of the original system, for which a stable equilibrium exists. The strength of the applied stimulus must be such that a first action potential is generated, but the stimulus is not responsible for generating any additional spikes. Our investigation can be compared to excitability in laser systems [5, 6] where the response after an applied stimulus is explained by the existence of a nearby homoclinic bifurcation with respect to a parameter. This mechanism is due to the presence of a saddle equilibrium that coexists with the stable equilibrium (i.e., the resting potential). The interesting aspect about our model is that there exists only a stable equilibrium, and we have been unable to identify any nearby saddle-type invariant object in the parameter region of interest that could organise homoclinic bifurcations. We explain the transient behaviour following the ideas of Geometric Singular Perturbation Theory (GSPT) [33–36].
Another important difference with existing studies is that we do not rely on simulations to study the effects of an applied stimulus in numerical experiments. Instead, we use a continuation-based approach and show the process of spike-adding transitions during a transient burst in unprecedented detail, which is not possible using brute-force simulation. Our numerical method is based on the continuation of orbit segments as solutions of a two-point boundary value problem; this approach has already been applied to the bifurcation analysis of periodic orbits, including homoclinic or heteroclinic bifurcations , and more recently for the computation of invariant manifolds  and so-called slow manifolds in systems with multiple time scales [39, 40]. We divide the system into two separate orbit segments, with and without current injection, which are coupled only by the boundary conditions. This allows us to continue the orbit segments in a chosen parameter and analyse the precise nature of their continuous deformation even over exponentially small parameter variations. We complement this continuation with the computation of the two-dimensional critical manifold of the fast subsystem, which comprises all the equilibria of the fast subsystem parametrised by two slow variables. The critical manifold is folded and consists of a number of stable and unstable sheets. In our setting, the unstable sheets are of saddle type, and we will refer to them as saddle-unstable sheets. The fast subsystem also has families of periodic orbits that emanate from Hopf bifurcations on the critical manifold, which give rise to the spiking behaviour during a burst. Our analysis shows that, during the spike-adding transition, orbit segments trace saddle-unstable slow manifolds that lie very close to corresponding saddle-unstable sheets of the critical manifold; the distance between these two manifolds is of the same order as the ratio between the contraction/expansion rates towards and on the manifold, which is organised by the difference between the slow and fast time scales [33, 41]. This canard-like behaviour is very similar to behaviour during a spike-adding transition of a periodic burst , but it does not involve bifurcations, and coexistence of bursts with different numbers of spikes is not possible here. Thus, our analysis indicates that it is the presence of canard-like behaviour that organises the spike adding.
As for periodic bursting, the spike-adding transition in our model occurs over an exponentially small parameter interval [42–44]. Within this exponentially small parameter interval, we find an even smaller parameter interval during which the canard-like orbit segment includes a fast transition from a saddle-unstable to another saddle-unstable slow manifold. This phenomenon is similar to the so-called fold-initiated canards that have been observed for periodic orbits . To understand this behaviour, we study the associated slow flow on the critical manifold and identify the effect of folds and folded singularities on the behaviour of the orbit segment. Our findings complement the study for planar systems by Guckenheimer et al.  and confirm that such fold-initiated canard-like behaviour occurs robustly during a spike-adding transition due to continuity of the vector field with respect to the parameter.
Our study concerns the analysis of a spike-adding mechanism where the full system has a unique stable equilibrium that does not undergo any bifurcations. We find that, for different values of model parameters, the system can have additional unstable equilibria that alter the nature of the spike-adding mechanism. More precisely, the appearance of two saddle equilibria on the critical manifold suppresses the fold-initiated transition between saddle-unstable sheets and changes the behaviour of the orbit segment. The presence of the saddle equilibria give rise to a mechanism that is reminiscent of the excitability studies by Wieczorek et al.  and Krauskopf et al. ; a new spike is added via a heteroclinic connection of the perturbed state to a saddle equilibrium of the full system. However, this saddle equilibrium lies outside a neighbourhood of the resting potential, and we observe a canard-like transition before and after the heteroclinic connection that is similar to the canard-like transition that occurs when no additional equilibria are present. We compute the critical manifold for this situation and study the associated slow flow to explain this phenomenon; we find that the saddle equilibrium point lies in the middle of a saddle-unstable sheet of the critical manifold, and it is an attractor with respect to the slow flow on the critical manifold.
This paper is organised as follows: In the next section we present our model of the study. In Section 3, we numerically identify the mechanism of spike-adding via continuation of the orbit segment. Next, in Section 4, we calculate the critical manifold of the fast subsystem along with the slow flow to explain the transition to bursting. Here, we also investigate the transition between two unstable slow manifolds of the saddle type and the changes in the spike-adding mechanism when additional equilibria of the full system are present. We end with a discussion in Section 5.
2 The model
We apply our analysis of transient bursts to pyramidal neuron cells from the CA1 and CA3 regions of the hippocampus. A detailed model of such neurons in Hodgkin-Huxley formalism was presented by Nowacki et al. , but for the purpose of this paper, we study a reduced version of this model. The simplified model consists of four ionic currents, namely, fast and slow inward currents, denoted as and , respectively, and fast and slow outward currents, denoted as and , respectively. Inward currents are responsible for the depolarisation or increase of the membrane potential, whereas outward currents hyperpolarise or decrease the membrane potential and return the cell back to its resting state (a stable equilibrium) [2, 46]. The fast inward current represents the fastest class of spike-generating - and -currents. The rates of change of these currents are usually similar to that of the membrane potential. Therefore, we assume that the gating of is instantaneous [13, 46–48]. The slow inward current mainly corresponds to the transient T-type -current [13, 49–51] and represents the low-voltage activated currents responsible for shaping the subthreshold behaviour of the model. The fast outward current represents high-voltage activated fast -currents that we base on the delayed rectifier -current [13, 46, 47]. Finally, represents muscarinic-sensitive -current [50, 52, 53], which has an activation rate of the same order as that of .
System (1) defined by Equations (2)-(4) evolves on multiple time scales because (as an approximation of the time scale for V) and with have different orders of magnitude. As indicated in Table 1 and are slow variables that vary on a time scale that is (roughly) 10 times slower than and and 100 times slower than V. In particular, this means that our model is capable of firing an arbitrarily large number of spikes during the ADP. More precisely, an increase in , as in Figure 1 and throughout this paper, has the net effect that the slow variable becomes even slower so that more spikes can be fired during the time it takes for to relax back to its equilibrium value. In this paper, we are not interested in the exact nature of this process, but we mention here that a large number of spikes will also be accompanied by a noticeable reduction in oscillation amplitudes. As slows down, the dynamics will resemble more and more the behaviour organised by slow passage through a Hopf bifurcation [26, 54]. We focus on the process of spike adding and take advantage of the difference in time scales in Section 3, where it suffices to consider the time-scale separation between the three fast variables V and , and the two slow variables and .
The gating variables express the fractions of channels in a given state and naturally range over the interval . The natural range of the membrane potential V is bounded by the two Nernst potentials [2, 46], i.e., , where and . It is beneficial for the numerical analysis if all variables vary over a similar range. Therefore, the computations are done using the scaled membrane potential , where . For our numerical investigations, we used the continuation package Auto[55, 56] for solving the boundary value problems. All numerical simulations were done with XPP  using the front-end package XPPy  in Python , and visualisations were done in Python using Matplotlib  and Mayavi .
3 Identifying the spike-adding mechanism
Spike adding happens after a current injection, that is, in the regime where . Hence, any numerical investigation of the transient behaviour must take into account a discontinuous jump from to on the right-hand side of Equation (2). We view the orbit as a concatenation of two orbit segments that are the solution of two boundary value problems and define appropriate boundary conditions to account for the discontinuity in .
Figure 3b shows our starting solution, i.e., a single spike followed by ADP. Along the first plateau of the solution branch up to the first downward peak, all orbit segments are qualitatively like Figure 3b; in particular, the ADP is a small hump. As we follow the solutions into the downward peak, the hump of the ADP for the orbit segments stretches out as shown in Figure 3c, which lies at the bottom of the downward peak. Interestingly, as we follow the solution back up along the downward peak, the orbit segment generates a double step in the ADP, as shown in Figure 3d; we selected the orbit segment with the longest double step (with respect to time). As we continue to trace the solution up along the downward peak, the small spike at the end of the orbit segment grows into a fully developed spike, while the stretched double step retracts; the orbit segment shown in Figure 3e is representative of such a solution, and all orbit segments along the second plateau in Figure 3a are qualitatively like Figure 3e. Figure 3f represents a solution along the next plateau, which exhibits three spikes that are created via the same process as explained above for the two-spike burst. In fact, the same spike-adding process takes place for all spike-adding transitions via the downward peaks in Figure 3a. We emphasise here that the stretched single- and double-step ADPs, as shown in Figure 3c,d, only exist along the downward peaks in Figure 3a, that is, in the exponentially small parameter interval during which a spike-adding transition occurs. Hence, such solutions are unlikely to be observed in actual experiments, and they are also very difficult to find in numerical experiments that use initial-value integration methods. The fact that a spike-adding transition happens over an exponentially small parameter interval, during which the solution measure changes rapidly, suggests that it is organised by the difference in time scales present in system (1). Therefore, in order to obtain a better understanding, we proceed by using GSPT [34–36, 41, 62, 63].
4 Spike-adding organised by the critical manifold
Furthermore, because we are interested in spike-adding phenomena after the brief current injection, we set .
The important objects of study in the singular limit are equilibria and periodic orbits. Since and are parameters, these invariant objects occur in two-parameter families. The -dependent families of equilibria are known as the critical manifold, which we denote by S. The equilibria on S can be stable or unstable, determined with respect to the three-dimensional fast subsystem, and are typically separated by curves of fold or Hopf bifurcations. Similarly, we can expect the existence of -dependent families of periodic orbits that emanate from a curve of Hopf bifurcations on the critical manifold; these periodic orbits can again be stable or unstable with respect to the three-dimensional fast subsystem. Typically, the stable periodic orbits of the fast subsystem organise the spiking phase of the bursting oscillators [16, 19, 26].
The critical manifold S, when considered in the full five-dimensional phase space of system (1), is a two-dimensional surface, or collection of surfaces, and the associated families of periodic orbits form a three-dimensional manifold, or collection of manifolds, that we denote by P. Together, these collections of manifolds organise the behaviour of solutions of (1). If and vary slowly enough, then GSPT guarantees that a solution of (1) (with ) will trace attracting sheets of S or P that correspond to the -dependent families of attractors of the fast subsystem . For example, the transient spikes of system (1) trace the manifold that corresponds to the family of attracting periodic orbits of the fast subsystem, while and are slowly varying [16, 64]. More precisely, solutions of (1) lie on so-called slow manifolds that are perturbations of the different sheets of S and P from the singular limit . Solutions of (1) are characterised by fast transitions between, followed by exponential contraction onto the slow manifolds. The essential difference in behaviour during a spike-adding transition is the fact that the solution of (1) contains a segment that traces a slow manifold associated with a sheet of S that is unstable (of saddle type) rather than attracting; see also [25, 34, 36, 63]. While technically the solutions of (1) trace slow manifolds, we will abuse notation and write ‘sheet of S’ where we mean ‘slow manifold corresponding to the sheet of S.’
Stability properties of the critical manifold S and the manifold P of periodic orbits
Two-dimensional critical manifold S
Three-dimensional manifold P
Dimension stable manifold
Dimension unstable manifold
The maxima and minima of the families of periodic orbits originating from H are shown in Figure 4c,d, using the same two viewpoints as in panels (a) and (b), respectively. The Hopf bifurcation is subcritical along the entire curve so that the emanating family of periodic orbits is unstable (of saddle type), with four-dimensional stable and unstable manifolds; we coloured this family magenta and labelled it . The family of periodic orbits becomes stable via a fold of periodic orbits, after which it is coloured blue and labelled , and ends in a homoclinic bifurcation involving equilibria on the sheet . We refer again to Table 2 for an overview of the different families and their stability properties.
The spike-adding process occurs along the downward peak in Figure 3a, during which the parameter remains almost fixed, but the orbit segments of system (5)-(8) change dramatically. We observe the formation of a stretched ADP, which initially gets increasingly longer and shortens again as we follow the orbit segments along the downward peak in Figure 3a. This transition is initiated by the fact that, at the special value , the injected current perturbs the orbit segment such that lies almost on the four-dimensional stable manifold of the saddle-unstable sheet ; more precisely, has a corresponding slow manifold with a corresponding finite-time stable manifold that comes close to. The closer lies to this four-dimensional manifold, the closer the corresponding orbit segment comes to the slow manifold associated with and the longer it will trace this slow manifold. We approximate the slow manifold by the critical manifold , and Figure 5a shows the orbit segment from Figure 3c, which traces almost up to the fold before it drops down to and converges to the resting potential.
Figure 5b,c,d,e,f illustrates orbit segments for the second upward part of the downward peak in Figure 3a. Interestingly, a double-step ADP is created via a transition from to as shown. The orbit that was previously shown in Figure 3d traces both saddle-unstable sheets to of S all the way up to the fold . This transition from to is, in fact, a robust part of the spike-adding process that is a continuous (parameter-dependent) variation of the case where the folds and are absent, that is, where connects directly to ; see Section 4.2 for more details. After reaching the top fold , the membrane potential V initially increases instead of immediately decreasing down to the stable sheet , and a small spike is created. As we continue to follow the solution up along the downward peak, the spike part of the orbit segment grows and moves back towards the attracting periodic orbit family , as illustrated in Figure 5d,e. Finally, as shown in Figure 5f, the orbit segment traces for only a very short time before the second spike occurs; this orbit segment is selected almost at the end of the downward peak, after which orbit segments stop tracing altogether, and the transition from a one- to two-spike transient burst ends. We refer to Govaerts and Dhooge , Guckenheimer and Kuehn , and Osinga et al.  for similar transitions of periodic bursting oscillators.
We remark here that the manner of eventual convergence to the resting potential depends on the nature of the lift-off from the slow manifolds that correspond to the sheet or . Recall that both sheets and of the critical manifold S have three-dimensional stable manifolds; see Table 2. This means that the associated slow manifolds have a one-dimensional repelling fast component , and orbit segments that trace these saddle-unstable slow manifolds can leave it only along a single fast direction. We can see this in Figure 5 as a lift-off from ‘down’ in V, shown in Figure 5a, or a lift-off from ‘up’ in V, shown in Figure 5f; this uniquely defined change in direction along the one-dimensional repelling fast component is real and not just due to the projection onto -space. The same holds for the sheet , for which Figure 5b,e is a good example that also shows the required lift-off up from in order to reach . In what follows, the notions up and down are with respect to this uniquely-defined change in direction.
The behaviour of the orbit segment of system (1) in relation to the critical manifold S of the fast subsystem that corresponds to the first downward peak in Figure 3a is representative for what happens along the other downward peaks in Figure 3a. Each time reaches a special value such that the orbit segment comes close enough to the four-dimensional stable manifold of , it gets trapped onto (or, more precisely, the corresponding saddle-unstable slow manifold) for increasingly longer times, and the next spike-adding transition begins. For the parameters of Table 1, we found that this process always includes a transition between two saddle-unstable sheets, which organises the double-step ADP solutions. As mentioned before, the two sheets and together are perturbations of a single sheet connected to that can be obtained continuously via a small parameter variation (using a suitable parameter from Table 1), such that the fold curves and disappear in a curve of cusp bifurcations. Therefore, in the next section, we first explain the jump at the end of the canard-like behaviour, that is, the behaviour near the fold that separates the two saddle-unstable sheets and . We then discuss the transition between and in Section 4.2. Section 4.3 illustrates how the spike-adding mechanism can change when additional equilibria are present.
4.1 Slow flow on the critical manifold near
that is, the slow flow on the two-dimensional critical manifold S is defined by two ordinary differential equations for and and a single algebraic constraint . Unfortunately, S is folded with respect to V so that and do not uniquely define V; however, the algebraic constraint does uniquely define or from given pair or , respectively; compare also Figure 4a,b. Hence, it is advantageous to express the slow flow in terms of only one of the slow variables, or , together with the fast variable V.
Figure 7a,b shows a projection onto -coordinates of how the orbit segment follows the slow flow on as it approaches . The first part of the orbit segment lies slightly above (relative to this projection) , and after a jump, another part of this orbit segment lies slightly below . In a neighbourhood of the folded focus on the fold curve , the slow flow has the form of large semi-cycles that cause the orbit segment to trace laterally and, at the same time, push it toward . Since the flow on the top sheet also points towards , as shown in Figure 7c,d, the orbit segment cannot pass and reaches a so-called jump point; compare also with Figure 6b. At the jump point, the fast directions of the flow take over, which causes the formation of a small spike as the orbit segment leaves S; see also Figure 5c. Let us emphasise here that the behaviour of the orbit segments near does not involve interactions with the slow flow on ; the small spike and subsequent drop down to do not intersect the surfaces and . As mentioned earlier in this section, the actual spike formation develops as soon as an orbit segment has reached . After reaching , the orbit segment will lie slightly above (relative to this projection) and experience a lift-off up from (or later only from as shown in Figure 5f). The spike-formation takes place on the fast time scale, and any perceived intersections with and are due to the projection onto -coordinates.
4.2 Slow flow of the critical manifold near the folds and
The formation of a new well-developed spike occurs over an exponentially small parameter interval for which the effect of the injected current is precisely such that the orbit segment comes close to the four-dimensional stable manifold of . The behaviour of the orbit segment near the top fold corresponds to the onset of such a new spike, but the process of reaching , as illustrated in Figure 5a,b, as well as the further development of the spike, as illustrated in Figure 5d,e,f, involves the creation of a double-step ADP; this behaviour is organised by a (fast) jump from to another saddle-unstable sheet . Such a jump, which is actually a jump between the two corresponding saddle-unstable slow manifolds, is a phenomenon that occurs robustly as part of the spike-adding mechanism and has previously been observed for periodic orbits in planar systems; it was reported as a new type of canard called fold-initiated canards in a study by Guckenheimer et al. , and a slightly different version termed ducks with relaxation is discussed by Arnol’d , Ch.4, Sec.5.4]. In fact, the behaviour we observe in our model is essentially planar and very similar to the example discussed by Guckenheimer et al. . Indeed, there are no folded singularities on the folds and , which means that each fold point has the same effect on the dynamics, and the slow flow is essentially one dimensional. Furthermore, the repelling fast component of the slow manifolds associated with and is one dimensional as well.
Let us consider this continuous one-parameter family of orbit segments as identified by the moment of lift-off (first down and then up) from . At the start of the spike-adding process, orbit segments trace only the saddle-unstable sheet before a lift-off down to returns the system to its resting potential; the example in Figure 8a shows an orbit segment that almost reaches . As increases continuously (but only exponentially small), the orbit segments come increasingly closer to until one actually reaches ; these orbit segments grow increasingly longer stretched ADPs.
Using the analysis via the desingularised slow flow (11) as derived in Section 4.1, we can decide what happens when an orbit segment reaches . We find that the desingularised slow flow (11) does not have any equilibria in the neighbourhood of the two folds and , which means that there are no folded singularities on either or ; the fold curve is uniformly attracting, which we indicated by a dark-green colour, and is uniformly repelling, indicated by a light-green colour. Hence, upon reaching , the orbit segment simply jumps down toward the resting potential, and subsequent orbit segments exhibit a lift-off up from . Since the sheet on the other side of is attracting, the fast directions will push these orbit segments toward , provided that the lift-off up from occurs not too far away from . Note that the slow flow on points back to , so orbit segments that reach will flow to and again drop down to ; an example is shown in Figure 8b.
As we continue to increase ever so slightly, orbit segments will converge to closer and closer near , until the lift-off up from happens earlier, far enough from , such that they may reach . In order to determine what happens when an orbit segment reaches , we must remember that system (1) and, hence, the slow flow on S depend continuously on . This means that the family of orbit segments is also continuous, and the orbit segment that reaches is a continuous variation of an orbit segment that experiences a lift-off up from later so that it still converges to and flows back to as well as an orbit segment that experiences a lift-off up from earlier so that it misses altogether and forms a well-developed spike as illustrated in Figure 5f. Therefore, the family of orbit segments must include a subfamily of orbit segments that experience a lift-off from . Just as for , the lift-off will first be down as a continuous variation from the orbit segments that flow along and then up to transform into an orbit segment that misses altogether. This subfamily exists in a (doubly) exponentially small parameter regime of fold-initiated canard behaviour  and consists of orbit segments that exhibit a double-step ADP.
Another way to understand this phenomenon is in terms of singularity theory, which suggests that and are part of the same surface that unfolds a cusp singularity. Recall that the critical manifold S is a collection of families of equilibria of the fast subsystem of (1); the folds and are curves of saddle-node bifurcations that exist robustly in the -parameter plane organised by the two slow variables. Since and are typically not parallel, they will meet; note that and may need to be extended into an unphysical parameter regime of the -plane. Singularity theory tells us that the two fold curves and typically meet at a cusp point and end there, which means that the two sheets and merge into one in this region of the -plane. The existence of a double-step ADP then merely depends on the location relative to the cusp point of the interaction between the orbit segments and the critical manifold S. We remark that a small change in one or more of the parameters given in Table 1 may move the cusp point into the physical regime or such that and no longer exist for physiologically realistic values of and ; in the latter case, the spike-adding transition will not feature a double-step ADP.
4.3 Spike-adding when additional equilibria are present
It turns out that the spike-adding mechanism organised by canard-like behaviour during the downward peaks of Figure 3a always features a double-step ADP stage involving a jump between and . Hence, each downward peak in Figure 3a corresponds to a qualitatively similar transition as discussed for the first one at . If we increase from the fixed value that was used in Figure 3 to the new value , then the nature of spike adding changes due to the presence of additional unstable equilibria of system (1). If we again continue the two-point boundary value problem (5)-(8) as before with but set to its new value, we get a bifurcation diagram similar to the one for shown in Figure 3. In fact, the spike-adding mechanism for the first four additional spikes involves a double-step ADP stage as we have seen in the previous section. However, for , that is, just before the transition from five to six spikes, a saddle-node bifurcation occurs on the saddle-unstable sheet . This creation of two new (unstable) equilibria prevents a double-step ADP stage; the spike-adding mechanism only involves orbit segments exhibiting a stretched ADP with a singe step, and there is no longer a jump between saddle-unstable slow manifolds.
The value for this case with is again special because at the end of the oscillations, when the orbit segment reaches the family of homoclinic orbits where ends, it lies extremely close to the four-dimensional stable manifold of so that it drops down and traces the saddle-unstable sheet of S. The difference with the spike-adding mechanism illustrated in Figure 5 is that the behaviour of the orbit segment on is affected by the presence of the equilibria and . With respect to the two-dimensional slow flow on , the equilibrium is an attractor, and all orbit segments on converge to provided that they lie in its basin of attraction, which is bounded by the one-dimensional stable manifold of the saddle . In terms of the full five-dimensional flow, is obviously unstable, and orbit segments that come close enough to will behave as dictated by the slow flow for only a finite amount of time; this means that convergence to will eventually be followed by a fast repulsion away from . The orbit segment in Figure 9a,c enters a close enough neighbourhood of in the region of the basin of attraction of ; hence, during the time that it is following the slow flow, it converges to , but we can clearly see in Figure 9c that the fast directions take over before it reaches . Since this orbit segment was selected from the falling slope of the downward peak of the spike-adding mechanism, the orbit segment jumps straight down toward , where it converges to the resting potential. Orbit segments on this slope that lie closer to the minimum of the downward peak would come closer to but still jump down toward when the fast directions take over. On the other hand, orbit segments from the rising slope of the downward peak eventually experience a lift-off ‘up’ from so that a large action potential occurs before converging back to the resting potential; this change of direction corresponds to the onset of a new spike, which is more dramatic and abrupt than the gradual increase in V followed by a small-amplitude spike as illustrated in Figure 5.
Continuity of the vector field (1) implies that there exists an orbit segment that actually converges to the saddle and never relaxes back to the resting potential. This happens when is exactly at the value where the perturbed trajectory lies on the four-dimensional stable manifold of the equilibrium . In contrast to the four-dimensional stable manifold of , the manifold is invariant under the flow of (1), and this heteroclinic connection is a well-defined bifurcation for system (1).
We remark here that the presence of additional equilibria, such as and in the example discussed, only affects the spike-adding mechanism if the orbit segments that trace enter the basin of attraction of . If such orbit segments trace on the other side of the stable manifold of , then a double-step ADP stage would occur. We know from our further model analysis (not shown) that the unstable equilibria persists for higher values of as well as , and in all cases that we investigated, these additional equilibria on affect the spike generation in the way described above.
In this paper, we performed a detailed analysis of the mechanisms of spike generation and spike-adding in a transient burst. Based on a reduction of our previous model , we identify these mechanisms using numerical continuation of orbit segments that are solutions to a well-posed boundary value problem. In our analysis, we utilised the separation of time scales in system (1). We calculated the two-dimensional critical manifold S of the fast subsystem, which organises the behaviour of the system. The spike-generation process is characterised by the fact that orbit segments trace saddle-unstable slow manifolds that correspond to saddle-unstable sheets of S. More precisely, there are two saddle-unstable sheets, and , with four-dimensional stable and three-dimensional unstable manifolds; this means that the lift-off from the associated slow manifolds is characterised by a uniquely defined direction. The changes in sign of this direction mark the different phases of the spike-adding transition.
By considering the slow flow on S, we were able to explain the onset of a spike as well as the double-step stretched ADP that leads up to it. For the value of that we considered, the onset of a spike is organised by the top fold of S. This fold contains a folded-focus singularity, but it is not accessible and the spikes are due to (regular) jump points. The folds and that are involved in the double-step ADP do not contain any folded singularities, and they are uniformly attracting and repelling, respectively. Therefore, the first step in the stretched ADP ends at a regular jump point. The second step occurs due to a type of fold-initiated canard because the slow flow points away from ; the fold-initiated canard-like behaviour forms a robust part of the spike-adding mechanism that has also been observed for periodic orbits . The actual spike-adding transition occurs in an exponentially small parameter regime that is very difficult, if not impossible, to find with brute-force integration routines; therefore, it is also highly unlikely to observe anything like a stretched ADP or double-step ADP in experiments.
We found that the nature of the spike-adding mechanism may change if increases slightly. For higher values of , an increase in causes the appearance of two equilibria, and , on that form a saddle-node pair with respect to the slow flow. As it turns out, the presence of these equilibria prevents the double-step ADP. Instead, orbit segments that come close to during the spike-generation process flow towards the attracting equilibrium before a lift-off in the fast direction. This means that the onset of a spike is now organised by rather than the fold , and the stretched ADP involves only a single step. A spike generated by is dramatically different from one generated by . While both spike generations happen in an exponentially small parameter interval, the increase in amplitude of a spike generated by is gradual and should be viewed as a variation of the orbit segment that depends continuously on . On the other hand, a spike generated by is not a continuous variation, and a large-amplitude spike appears abruptly as is increased (on an exponentially small scale). The two families, one with and one without the additional spike, are separated by a heteroclinic connection (via a current injection) from the resting potential to the saddle ; our numerical method for continuation of the family only gets past this discontinuity because we do not impose relaxation back to the resting potential but keep fixed instead. Similar spike-adding mechanisms can be organised by the presence of a saddle periodic orbit or other saddle-type invariant object, but we did not observe this in our model.
In theory, it should be possible to have a double-step ADP as part of the spike-adding transition even when additional equilibria are present. The occurrence of a double-step ADP in this case only depends on whether tracing of commences in the basin of attraction of (restricted to ) or not, which is determined by the stable manifold of the saddle equilibrium . In our numerical explorations, the orbit segments always commence tracing in the basin of attraction of . Hence, we may conclude that and do not have a profound influence on the relative location where orbit segments begin to trace during the spike-adding process. However, other parameters of the system may alter this relative location and provoke a double-step ADP even in the presence of additional equilibria. While this observation indicates a challenge for a precise definition of spike-onset in our context, the two different mathematical notions seem to have the same biological effect.
We believe that the canard-like transition involving saddle-unstable sheets of the critical manifold lies at the heart of any spike-adding mechanism when no invariant saddle-type objects are present. The different phases during the transition, however, could be organised by features other than regular jump points and fold-initiated canards. For example, it should be expected that other folded singularities may appear due to variations in the slow flow on S. An investigation of all possibilities remains an interesting and challenging project for future work.
The authors thank Pablo Aguirre, Mathieu Desroches, Bernd Krauskopf, John Guckenheimer and Peter De Maesschalck for their helpful discussions. JN was supported by the grant EP/E032249/1 from the Engineering and Physical Sciences Research Council (EPSRC). HMO was supported by an EPSRC Advanced Research Fellowship grant, and KT-A was supported by the EPSRC grant EP/I018638/1.
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