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Phase-Amplitude Response Functions for Transient-State Stimuli
© O. Castejón et al.; licensee Springer 2013
Received: 22 December 2012
Accepted: 18 March 2013
Published: 14 August 2013
The phase response curve (PRC) is a powerful tool to study the effect of a perturbation on the phase of an oscillator, assuming that all the dynamics can be explained by the phase variable. However, factors like the rate of convergence to the oscillator, strong forcing or high stimulation frequency may invalidate the above assumption and raise the question of how is the phase variation away from an attractor. The concept of isochrons turns out to be crucial to answer this question; from it, we have built up Phase Response Functions (PRF) and, in the present paper, we complete the extension of advancement functions to the transient states by defining the Amplitude Response Function (ARF) to control changes in the transversal variables. Based on the knowledge of both the PRF and the ARF, we study the case of a pulse-train stimulus, and compare the predictions given by the PRC-approach (a 1D map) to those given by the PRF-ARF-approach (a 2D map); we observe differences up to two orders of magnitude in favor of the 2D predictions, especially when the stimulation frequency is high or the strength of the stimulus is large. We also explore the role of hyperbolicity of the limit cycle as well as geometric aspects of the isochrons. Summing up, we aim at enlightening the contribution of transient effects in predicting the phase response and showing the limits of the phase reduction approach to prevent from falling into wrong predictions in synchronization problems.
List of Abbreviations
PRC phase response curve, phase resetting curve.
PRF phase response function.
ARF amplitude response function.
The phase response (or resetting) curve (PRC) is frequently used in neuroscience to study the effect of a perturbation on the phase of a neuron with oscillatory dynamics (see surveys in [1–3]). For it to be applied, several conditions are required (weak perturbations, long time between stimuli, strong convergence to the limit cycle, etc.) so that the system relaxes back to the limit cycle before the next perturbation/kick is received. In this case, one can reduce the study to the phase dynamics on the oscillatory solution (namely, a limit cycle). However, in realistic situations, we may not be able to determine whether the system is on an attractor (limit cycle); moreover, the system may not show regular spiking, especially because of noise; see for instance [4, 5]. In addition, even in the absence of noise, strong forcing may send the dynamics away from the asymptotic state, eventually close to other nearby invariant manifolds ; thus, both the rate of convergence to the attractor and the stimulation frequency (which can be relatively high; take for instance the case of bursting-like stimuli) play an important role in controlling the time spent in the transient state (away from the limit cycle). All these factors may prevent the trajectories from relaxing back to the limit cycle before the next stimulus arrives and raise the question of the nature of the phase variation away from an attractor (that is, in transient states) and how much can we rely on the phase reduction (PRC).
Recently, tools to study the phase variation away from a limit cycle attractor have been developed. They rely on the concept of isochrons (manifolds transversal to the limit cycle and invariant under time maps given by the flow), introduced by Winfree (see ) in biological problems, from which one can extend the definition of phase in a neighborhood of the limit cycle. In a previous paper [8, 9], we showed how to compute a parameterization of the isochrons (see also [10–12] for other computational methods) as well as the change in phase due to the kicks received when the system is approaching the limit cycle but not yet on it. This approach allowed us to control the phase advancement away from the limit cycle (that is, in the transient states) and build up the Phase Response Functions (PRF), a generalization of the PRCs. In , examples of neuron oscillators were shown in which the phase advancement was clearly different for states sharing the same phase. A review of these tools is presented in Sect. 2.
In Sect. 3, we complete the extension of advancement functions to the transient states by defining the Amplitude Response Function (ARF), and we provide methods to compute it by controlling the changes induced by perturbations in a transversal variable, which represents some distance to the limit cycle. One of the methods presented here to compute the ARFs is an extension of the well-known adjoint method for the computation of PRCs; see, for instance, [13, 14] or Chap. 10 in .
Indeed, the knowledge of both the PRF and the ARF allows us to consider special problems in which these functions can forecast the asymptotic phase of an oscillator under pulsatile repetitive stimuli. In the case of a pulse-train stimulus, the variations of the extended phase and the amplitude can be controlled by means of a 2D map; this 2D map extends the classical 1D map used when the dynamics is restricted to the limit cycle or phase-reduction is assumed; see, for instance Chap. 10 in . Another successful strategy to deal with kicks that send the dynamics away from the limit cycle is the so-called second-order PRC (see [15–17]), which measures the effects of the kick on the next cycle period, taking into account that synaptic input can span two cycles.
As an illustration of the method, in Sect. 4, we then consider a canonical model for which we compute the PRFs and ARFs thanks to the exact knowledge of the isochrons. In this “canonical” example, we apply a two parametric periodic forcing (varying the stimulus strength and frequency) and make predictions both with our 2D map and the classical 1D map; we use rotation numbers to illustrate the differences between the two predictions and we observe differences up to two orders of magnitude in favor of the 2D predictions, especially when the stimulation frequency is high or the strength of the stimulus is large. We also use this example to shed light on the role of hyperbolicity of the limit cycle as well as geometric aspects of the isochrons (see also  for a related study of the effect of isochrons’ shear). Finally, we also present the comparison of the two approaches in a conductance-based neuron model, where we do not know the isochrons analytically.
Summing up, we aim at enlightening the contribution of transient effects in predicting the phase response, focusing on the importance of the “degree” of hyperbolicity of the limit cycle, but also on the relative positions of the isochrons with respect to the limit cycle. Since PRCs are used for predicting synchronization properties, see , Chap. 10 in  or Chap. 8 in , our final goal is to show the limits of the phase reduction approach to prevent wrong predictions in synchronization problems.
2 Set-up of the Problem: Isochrons and Phase Response Functions (PRF)
In this section, we set up the problem and we review some of the results in  that serve as a starting point of the study that we present in this paper.
where , so that .
if the limit cycle is attracting. If the limit cycle is repelling, the same is true with .
The value is the asymptotic phase of x and the isochrons are defined as the sets with constant asymptotic phase, that is, the level sets of the function Θ. The same construction can be extended to limit cycles in higher dimensional spaces, but since the applications in this paper will be restricted to planar systems, we give the definitions in .
where T is the period and λ is the characteristic exponent of the periodic orbit.
Remark 2.1 In , we presented a computational method to compute the parameterization K defined in (6) numerically.
We will use this notation for the rest of the paper.
where denotes partial derivative with respect to the variable V.
3 The Amplitude Response Function (ARF)
A pulse stimulation displaces the trajectory away from the limit cycle, producing a change both in the phase θ and the transversal variable σ, that we will refer to as the amplitude variable. In our notation, the amplitude variable is a distance measure defined by the time from the limit cycle along the orbits of the auxiliary vector field Y, transversal to X, defined in (8). In fact, the orbits of the vector field Y are the isochrons; see, for instance, the blue curve in Fig. 1. The phase-reduction approach assumes that the amplitude decreases to zero before the next pulse arrives and, therefore, the amplitude is always zero at the stimulation time. However, if one wants to consider pulses that arrive before the trajectory relaxes back to the limit cycle, one needs to compute also the amplitude displacement in order to predict the coordinates of the point at the next stimulation time.
In this section, we introduce the amplitude function and Amplitude Response Function (ARF), the analogues of the phase function (3) and the PRF (12) for the variable σ. Finally, we provide a formula to compute them given the diffeomorphism K introduced in (5).
where is the flow associated to the vector field X. The level curves of Σ are closed curves that we will call amplitude level curves or, in short, A-curves.
Expressed in the variables introduced in (5), the motion generated by Z is given by .
where denotes partial derivative with respect to the variable V.
3.2 Computation of the PRFs and the ARFs
In this section, we provide a formula to compute the functions ∇Θ and ∇Σ given the diffeomorphism K introduced in (5).
3.3 The Adjoint Method for the ARF
where is the transpose of the real matrix DX. In this case, the method just requires an initial condition, so that it can be solved uniquely. The initial condition is provided by formula (15).
The same result can be extended to compute the change in the transversal variable σ due to a pulse stimulation. In the following proposition, we provide the differential equation satisfied by where is a point in a neighborhood Ω of the limit cycle γ evolving under the flow of X.
as we wanted to prove. □
4 Periodic Pulse-Train Stimuli
The purpose of this section is to show the convenience of using the response functions away from the limit cycle to obtain accurate predictions of the ultimate phase advancement. To this end, we force a system with pulse-trains of period for trajectories near a limit cycle Γ of period and characteristic exponent λ.
where , and δ is the Dirac delta function. This system can represent, for example, a neuron which receives an idealized synaptic input from other neurons.
Remark 4.1 In the sequel, we will also use , the frequency of the stimulus, and , the frequency of the limit cycle Γ. Then the quotient indicates how many inputs the oscillator receives in one period.
In the latter case, we are assuming that the perturbation happens always on the limit cycle and, therefore, for all n. The possibility that this might not be a realistic assumption (for example, if the stimulus period is too small, the limit cycle is weakly hyperbolic or the strength of the stimulus ϵ is too large) has been already pointed out in the literature; see, for instance,  or Chap. 10 in . However, other factors could play a role, for example, the geometry of the isochrons (curvature, transversality to the limit cycle, etc.). Our aim is to consider some examples and see in which cases the 1D map (23) gives a correct prediction or, on the contrary, one requires the 2D map (22) to correctly assess the true dynamics of the phase variable.
for the 1-dimensional map (23).
These approximate rotation numbers will be our main indicator to compare the dynamics predicted by the 1D map with that of the 2D map. In order to dissect the causes that create the eventual differences between the two maps and highlight the shortcomings of the phase-reduction approach, we have first considered a “canonical” example in which the isochrons can be computed analytically. Next, we consider a conductance-based model, in which the isochrons have to be computed numerically and we obtain similar comparative results.
4.1.1 A Simple Canonical Model
The limit cycle corresponds to and the dynamics on it are given by ; therefore, . The period of the limit cycle Γ is . A parameterization of the limit cycle in terms of the phase , for is .
with and .
Therefore, we find that , and, by the parameterization γ of the limit cycle, . Similarly, , and .
4.1.2 Numerical Simulations
and compare it with the iterations obtained using the maps (22) and (23). In the following, we will call the approximation of the rotation number obtained by this method simply ρ, to distinguish it from and defined previously in (25) and (26), respectively. The following lemma gives a description of the dynamics expected in the 1-dimensional map.
Then, the fixed points of the 1-dimensional map (23) can be computed analytically and:
Iffor all, the map (23) has no fixed points.
If there exists such that and
However, as we have taken squares in Eq. (33), we still have to check whether and are solutions of (33). It is easy to check that always solves (33), while is a solution only when . □
From now on, we will take the initial condition to be , that is, . In order to explore the effect of both the hyperbolicity and the transversality of the isochrons to the limit cycle, we will plot the different approximations of the rotation numbers ρ, , and for different values of the parameters a and α.
Observe also the agreement with the result in Lemma 4.2, which predicts the appearance of the fixed point of the 1D map when , that is, when or, equivalently after substituting , . The fixed point appears at for (panel (a) in Fig. 5) and for (panel (b) in Fig. 5); both values coincide with the downstroke of the corresponding values of .
On the other hand, in Fig. 10, where the contraction to the limit cycle is slow but the isochrons are almost orthogonal to the limit cycle, one can see that the 1D approach diverges from the 2D approach and the analytic one. However, for the range of ϵ and the two different stimulation periods (panels (a) and (b)) considered in Fig. 10, the 1D prediction still gives a fairly good approximation. Moreover, unlike the case where and (see Fig. 5), the 1D approach predicts a similar qualitative behavior as the other two approaches. The results for in Fig. 10(a) raise another interesting question since the analytic rotation number ρ suddenly diverges from the 1D and the 2D rotation numbers. This is due to the fact that the iterates of the analytic map suddenly fail to encircle the critical point of the continuous system (located inside the limit cycle) while the iterates of the 1D and the 2D maps still do it. Thus, the rotation number for the analytic case may not give accurate information.
In conclusion, it seems that for the 2D map to represent a qualitative improvement with respect to the 1D it is necessary to have the combination of weak hyperbolicity of the limit cycle and “weak transversality” of isochrons to it. However, the role of hyperbolicity seems to be much more important, since in the presence of strong hyperbolicity the use of the 2D approach seems completely unnecessary, but for weak hyperbolicity the differences between the 1D and the 2D maps are present also when the isochrons are orthogonal to the limit cycle.
Remark 4.4 Of course, considering a stimulus strength ϵ large enough, both maps (22) and (23) will not give correct predictions, since they are based on first-order approximations. In this case, one should consider PRFs of second (or higher) order to obtain a correct result; see, for instance, [23, 24] for higher-order PRCs. One has to distinguish between these higher order response functions in terms of the stimulus strength from the second-order PRCs above mentioned (see  for instance) that relate to the second cycle after the stimulus.
In the next example, we apply the same methodology to a more biologically inspired case: a conductance-based model for a point-neuron with two types of ionic channels.
4.1.3 A Conductance-Based Model
The parameters of the system are , , , , , , , , , , .
Remark 4.5 We have chosen a value of the parameter for which the system presents weak attraction to the limit cycle. However, for this value of , system (35) is not a model of a spiking neuron, but one with high voltage oscillations. Thus, this example is not intended to deal with a realistic setting of spike synchronization, but to illustrate how to deal with the tools introduced in this paper in the case where one does not explicitly have the parameterization K.
Remark 4.6 In order to compute the parameterization K and the PRFs we have used the methods proposed in . The same ideas can be applied to compute the ARFs. Briefly, the method consists of two steps. First, to compute the value of a given ARF near the limit cycle, where the numeric approximation of the parameterization K is valid, expression (16) is used. Second, to compute the value of some ARF far from the limit cycle, we just integrate the adjoint system (18) backwards in time using an initial condition for ARF close to the limit cycle.
We have introduced general tools (the PRF and the ARF) to study the advance of both the phase and the amplitude variables for dynamical systems having a limit cycle attractor. These tools allow us to study variations of these variables under general perturbation hypotheses and extend the concept of infinitesimal PRCs which assumes the validity of the phase-reduction and is only true under strong hyperbolicity of the limit cycle or under weak perturbations. In fact, the PRFs and ARFs are first-order approximations of the actual variation of the phase and the amplitude, respectively, and so they are supposed to work mainly for weak perturbations; however, being an extension away from the limit cycle makes them more accurate than the PRCs even under strong perturbations. We thus claim that the phase-reduction has to be used with caution since assuming it by default may lead to completely wrong predictions in synchronization problems. We are not dismissing phase-reduction but trying to show the limits beyond which an extended scenario is required.
We have presented a computational analysis to understand the contribution of transient effects in first-order predictions of the phase response, focusing on the importance of the hyperbolicity of the limit cycle, but also on the relative positions of the isochrons with respect to the limit cycle.
In the examples studied, subject to pulse-train stimuli, we have compared the predictions obtained both with the new 2D map defined from the PRF and ARF and the 1D map defined from the classical PRC. Using rotation numbers, we have shown differences up to two orders of magnitude in favor of the 2D predictions, especially when the stimulation frequency is high or the stimulus is too strong. These results confirm previous numerical experiments with specific oscillators; see . On the other hand, we have found that both weak hyperbolicity of the limit cycle and “weak transversality” of isochrons to it are important factors, although the role of hyperbolicity seems to be more crucial. In this paper, these achievements have been tested in a canonical model allowing comparisons with the exact solutions, and other numerical tests have been applied in a conductance-based model. The technique can be applied to other neuron models, and not necessarily for planar systems; n-dimensional systems would only require an additional computational difficulty in computing the associated ARFs.
We would like to emphasize the importance of having good methods to compute isochrons (see [8–12]) since they are the cornerstone to study these transient phenomena that we have observed. They can be useful, not only for the problem illustrated here, but for other purposes like testing how far the experimentally recorded phase variations are from the theoretically predicted ones. In fact, they are the key concept to be able to predict the exact phase variation since, theoretically, if we know the parameterization K that gives the isochrons, the problem reduces to solving, at each step, and , where is the point in the phase space where the pulse perturbation, ϵ w, is applied and . Indeed, the PRFs and ARFs can be computed knowing only the first order in K; in principle, then they are valid only for weak perturbations, but easier to compute. Other refinements could be obtained by computing second order PRFs and ARFs by using the second-order approximations of the isochrons. Further extensions include also the possibility of computing response curves for long (in time) stimulus rather than pulsatile stimuli.
Appendix: The Vector Field for the A-Curves
We prove here that given an analytic local diffeomorphism K, as in (5), satisfying (6), the A-curves are the orbits of a vector field Z, satisfying .
This is equivalent to proving that .
as we wanted to prove.
Partially supported by the MCyT/FEDER grant MTM2009-06973 (DACOBIAN), MTM2012-31714 (DACOBIANO), and Generalitat de Catalunya grant number 2009SGR-859. GH has been also supported by the i-Math contrato flechado and the Swartz Foundation. OC has been also supported by the grant FI-DGR 2011, cofunded by the Secretaria d’Universitats i Recerca (SUR) of the ECO of the Autonomous Government of Catalonia and the European Social Fund (ESF). AG and GH would like to thank the facilities of the Centre de Recerca Matemàtica. Part of this work was done while GH was holding a post-doctoral grant at the Centre de Recerca Matemàtica and AG was visiting it.
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