Phase-Amplitude Descriptions of Neural Oscillator Models
© K.C.A. Wedgwood et al.; licensee Springer 2013
Received: 31 July 2012
Accepted: 18 January 2013
Published: 24 January 2013
Phase oscillators are a common starting point for the reduced description of many single neuron models that exhibit a strongly attracting limit cycle. The framework for analysing such models in response to weak perturbations is now particularly well advanced, and has allowed for the development of a theory of weakly connected neural networks. However, the strong-attraction assumption may well not be the natural one for many neural oscillator models. For example, the popular conductance based Morris–Lecar model is known to respond to periodic pulsatile stimulation in a chaotic fashion that cannot be adequately described with a phase reduction. In this paper, we generalise the phase description that allows one to track the evolution of distance from the cycle as well as phase on cycle. We use a classical technique from the theory of ordinary differential equations that makes use of a moving coordinate system to analyse periodic orbits. The subsequent phase-amplitude description is shown to be very well suited to understanding the response of the oscillator to external stimuli (which are not necessarily weak). We consider a number of examples of neural oscillator models, ranging from planar through to high dimensional models, to illustrate the effectiveness of this approach in providing an improvement over the standard phase-reduction technique. As an explicit application of this phase-amplitude framework, we consider in some detail the response of a generic planar model where the strong-attraction assumption does not hold, and examine the response of the system to periodic pulsatile forcing. In addition, we explore how the presence of dynamical shear can lead to a chaotic response.
KeywordsPhase-amplitude Oscillator Chaos Non-weak coupling
One only has to look at the plethora of papers and books on the topic of phase oscillators in mathematical neuroscience to see the enormous impact that this tool from dynamical systems theory has had on the way we think about describing neurons and neural networks. Much of this work has its roots in the theory of ordinary differential equations (ODEs) and has been promoted for many years in the work of Winfree , Guckenheimer , Holmes , Kopell , Ermentrout  and Izhikevich  to name but a few. For a recent survey, we refer the reader to the book by Ermentrout and Terman . At heart, the classic phase reduction approach recognises that if a high dimensional non-linear dynamical system (as a model of a neuron) exhibits a stable limit cycle attractor then trajectories near the cycle can be projected onto the cycle.
The assumption that phase alone is enough to capture the essentials of neural response is one made more for mathematical convenience than being physiologically motivated. Indeed, for the popular type I Morris–Lecar (ML) firing model with standard parameters, direct numerical simulations with pulsatile forcing show responses that cannot be explained solely with a phase model . The failure of a phase description is in itself no surprise and underlies why the community emphasises the use of the word weakly in the phrase “weakly connected neural networks”. Indeed, there are a number of potential pitfalls when applying phase reduction techniques to a system that is not in a weakly forced regime. The typical construction of the phase response curve uses only linear information about the isochrons and non-linear effects will come into play the further we move away from the limit cycle. This problem can be diminished by taking higher order approximations to the isochrons and using this information in the construction of a higher order PRC . Even using perfect information about isochrons, the phase reduction still assumes persistence of the limit-cycle and instantaneous relaxation back to cycle. However, the presence of nearby invariant phase-space structures such as (unstable) fixed points and invariant manifolds may result in trajectories spending long periods of time away from the limit cycle. Moreover, strong forcing will necessarily take one away from the neighbourhood of a cycle where a phase description is expected to hold. Thus, developing a reduced description, which captures some notion of distance from cycle is a key component of any theory of forced limit cycle oscillators. The development of phase-amplitude models that better characterise the response of popular high dimensional single neuron models is precisely the topic of this paper. Given that it is a major challenge to construct an isochronal foliation we use non-isochronal phase-amplitude coordinates as a practical method for obtaining a more accurate description of neural systems. Recently, Medvedev  has used this approach to understand in more detail the synchronisation of linearly coupled stochastic limit cycle oscillators.
In Sect. 2, we consider a general coordinate transformation, which recasts the dynamics of a system in terms of phase-amplitude coordinates. This approach is directly taken from the classical theory for analysing periodic orbits of ODEs, originally considered for planar systems in , and for general systems in . We advocate it here as one way to move beyond a purely phase-centric perspective. We illustrate the transformation by applying it to a range of popular neuron models. In Sect. 3, we consider how inputs to the neuron are transformed under these coordinate transformations and derive the evolution equations for the forced phase-amplitude system. This reduces to the standard phase description in the appropriate limit. Importantly, we show that the behaviour of the phase-amplitude system is much more able to capture that of the original single neuron model from which it is derived. Focusing on pulsatile forcing, we explore the conditions for neural oscillator models to exhibit shear induced chaos . Finally in Sect. 4, we discuss the relevance of this work to developing a theory of network dynamics that can improve upon the standard weak coupling approach.
2 Phase-Amplitude Coordinates
and Df is the Jacobian of the vector field f evaluated along the periodic orbit u. The derivation of this system may be found in Appendix A. It is straightforward to show that as , and that . In the above, captures the shear present in the system, that is, whether the speed of θ increases or decreases dependent on the distance from cycle. A precise definition for shear is given in . Additionally, describes the θ-dependent rate of attraction or repulsion from cycle.
and is the identity matrix. Here, h and B describe the effect in terms of θ and ρ that the perturbations have. Details of the derivation are given in Appendix A. For planar models, . To demonstrate the application of the above coordinate transformation, we now consider some popular single neuron models.
2.1 A 2D Conductance Based Model
Here, v is the membrane voltage, whilst w is a gating variable, describing the fraction of membrane ion channels that are open at any time. The first equation expresses Kirchoff’s current law across the cell membrane, with representing a stimulus in the form of an injected current. The detailed form of the model is completed in Appendix B.1. The ML model has a very rich bifurcation structure. Roughly speaking, by varying a constant current , one observes, in different parameter regions, dynamical regimes corresponding to sinks, limit cycles, and Hopf, saddle-node and homoclinic bifurcations, as well as combinations of the above. These scenarios are discussed in detail in  and .
In the next example, we show how the same ideas go across to higher dimensional models.
2.2 A 4D Conductance Based Model
3 Pulsatile Forcing of Phase-Amplitude Oscillators
where δ is the Dirac δ-function. This describes T-periodic kicks to the voltage variable. Even such a simple forcing paradigm can give rise to rich dynamics . For the periodically kicked ML model, shear forces can lead to chaotic dynamics as folds and horseshoes accumulate under the forcing. This means that the response of the neuron is extremely sensitive to the initial phase when the kicks occur. In terms of neural response, this means that the neuron is unreliable .
The behaviour of oscillators under such periodic pulsatile forcing is the subject of a number of studies; see, e.g. [27–30]. Of particular relevance here is , in which a qualitative reasoning of the mechanisms that bring about shear in such models is supplemented by direct numerical simulations to detect the presence of chaotic solutions. For the ML model in a parameter region close to the homoclinic regime, kicks can cause trajectories to pass near the saddle-node, and folds may occur as a result .
This system exhibits dynamical shear, which under certain conditions, can lead to chaotic dynamics. The shear parameter σ dictates how much trajectories are ‘sped up’ or ‘slowed down’ dependent on their distance from the limit cycle, whilst λ is the rate of attraction back to the limit cycle, which is independent of θ. Supposing that the function P is smooth but non-constant, trajectories will be taken a variable distance from the cycle upon the application of the kick. When kicks are repeated, this geometric mechanism can lead to repeated stretching and folding of phase space. It is clear that the larger σ is in (15), the more shear is present, and the more likely we are to observe the folding effect. In a similar way, smaller values of λ mean that the shear has longer to act upon trajectories and again result in a greater likelihood of chaos. Finally, to observe chaotic response, we must ensure that the shear forces have sufficient time to act, meaning that T, the time between kicks must not be too small.
This stretching and folding action can clearly lead to the formation of Smale horseshoes, which are well known to lead to a type of chaotic behaviour. However, horseshoes may co-exist with sinks, meaning the resulting chaotic dynamics would be transient. Wang and Young proved that under appropriate conditions, there is a set of T of positive Lebesgue measure for which the system experiences a stronger form of sustained, chaotic behaviour, characterised by the existence of a positive Lyapunov exponent for almost all initial conditions and the existence of a ‘strange attractor’; see, e.g. [28–30].
By comparing with the phase-amplitude dynamics described by Eqs. (8)–(9), we see that the model of shear considered in (15) is a proxy for a more general system, with , and , and .
This simple model with a harmonic form for provides insight into how strange attractors can be formed. Kicks along the isochrons or ones that map isochrons to one another will not produce strange attractors, but merely phase-shifts. What causes the stretching and folding is the variation in how far points are moved as measured in the direction transverse to the isochrons. For the linear system (15) variation in this sense is generated by any non-constant ; the larger the ratio , the larger the variation (see  for a recent discussion).
Although this explicit map is convenient for numerical simulations, we prefer to work with the full stroboscopic map (22)–(23), which is particularly useful for comparing and contrasting the behaviour of different planar single neuron models with arbitrary kick strength. As an indication of the presence of chaos in the dynamics resulting from this system, we evaluate the largest Lyapunov exponent of the map (22)–(23) by numerically evolving a tangent vector and computing its rate of growth (or contraction); see e.g.  for details.
Now that we know the relative contribution of kicks in v to kicks in , it is also useful to know where kicks actually occur in terms of θ as this will determine the contribution of a train of kicks to the dynamics. In Figs. 11c and d, we plot the distribution of kicks as a function of θ. For the ML model, we observe that the kicks are distributed over all phases, while for FHN model there is a grouping of kicks around the region where is roughly zero. This means that kicks will not be felt as much in the ρ variable, and so trajectories here do not get kicked far from cycle. This helps explain why it is more difficult to generate chaotic responses in the FHN model.
After transients, we observe a phase-locked state for the FHN model. For a phase-locked state, small perturbations will ultimately decay as the perturbed trajectories also end up at the phase-locked state after some transient behaviour. This results in a negative largest Lyapunov exponent of −0.0515. We note the sharply peaked distribution of kick phases, which is to be expected for discrete-time systems possessing a negative largest Lyapunov exponent, since such systems tend to have sinks in this case. The phase-locked state here occurs where is small, suggesting that trajectories stay close to the limit cycle. Since kicks do not move trajectories away from cycle, there is no possibility of folding, and hence no chaotic behaviour. For the ML model, we observe chaotic dynamics around a strange attractor, where small perturbations can grow, leading to a positive largest Lyapunov exponent of 0.6738. This time, the kicks are distributed fairly uniformly across θ, and so, some kicks will take trajectories away from the limit cycle, thus leading to shear-induced folding and chaotic behaviour.
In this paper, we have used the notion of a moving orthonormal coordinate system around a limit cycle to study dynamics in a neighbourhood around it. This phase-amplitude coordinate system can be constructed for any given ODE system supporting a limit cycle. A clear advantage of the transformed description over the original one is that it allows us to gain insight into the effect of time dependent perturbations, using the notion of shear, as we have illustrated by performing case studies of popular neural models, in two and higher dimensions. Whilst this coordinate transformation does not result in any reduction in dimensionality in the system, as is the case with classical phase reduction techniques, it opens up avenues for moving away from the weak coupling limit, where . Importantly, it emphasises the role of the two functions and that provide more information about inputs to the system than the iPRC alone. It has been demonstrated that moderately small perturbations can exert remarkable influence on dynamics in the presence of other invariant structures , which cannot be captured by a phase only description. In addition, small perturbations can accumulate if the timescale of the perturbation is shorter than the timescale of attraction back to the limit cycle. This should be given particular consideration in the analysis of neural systems, where oscillators may be connected to thousands of other units, so that small inputs can quickly accumulate.
with an appropriate identification of the interaction functions in terms of the biological interaction between neurons and the single neuron functions . Such phase-amplitude network models are ideally suited to describing the behaviour of the mean-field signal in networks of strongly gap junction coupled ML neurons [36, 37], which is known to vary because individual neurons make transitions between cycles of different amplitudes. Moreover, in the same network weakly coupled oscillator theory fails to explain how the synchronous state can stabilise with increasing coupling strength (predicting that it is always unstable), as observed numerically. All of the above are topics of ongoing research and will be reported upon elsewhere.
Appendix A: Derivation of the Transformed Dynamical System
Appendix B: Gallery of Models
The function models the action of fast voltage-gated calcium ion channels; is the reversal (bias) potential for the calcium current and the corresponding conductance. The functions and similarly describe the dynamics of slower-acting potassium channels, with its own reversal potential and conductance . The constants and characterise the leakage current that is present even when the neuron is in a quiescent state. Parameter values are , , , , , , , , , , , , and .
B.2 Reduced Connor–Stevens Model
Parameter values are , , , , , , , , , and .
B.3 FitzHugh–Nagumo Model
where we use the following parameter values: , , , and .
Electronic Supplementary Material
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