Travelling waves in models of neural tissue: from localised structures to periodic waves
© Meijer and Coombes; licensee Springer. 2014
Received: 21 November 2013
Accepted: 18 February 2014
Published: 6 March 2014
We consider travelling waves (fronts, pulses and periodics) in spatially extended one dimensional neural field models. We demonstrate for an excitatory field with linear adaptation that, in addition to an expected stable pulse solution, a stable anti-pulse can exist. Varying the adaptation strength we unravel the organizing centers of the bifurcation diagram for fronts and pulses, with a mixture of exact analysis for a Heaviside firing rate function and novel numerical schemes otherwise. These schemes, for non-local models with space-dependent delays, further allow for the construction and continuation of periodic waves. We use them to construct the dispersion curve – wave speed as a function of period – and find that they can be oscillatory and multi-valued, suggesting bistability of periodic waves. A kinematic theory predicts the onset of wave instabilities at stationary points in the dispersion curve, leading to period doubling behaviour, and is confirmed with direct numerical simulations. We end with a discussion of how the construction of dispersion curves may allow a useful classification scheme of neural field models for epileptic waves.
Primary 87.19.lj; 87.19.le; 87.19.lq; 87.19.lf
KeywordsNeural field theory Brain wave equation Numerical continuation Anti-pulse Dispersion curve
The analysis of waves in models arising from the study of the nervous system has a long tradition. The seminal example is the development and numerical analysis of the model for action potential propagation in an excitable axonal fibre by Hodgkin and Huxley , and see  for an excellent review. This has been followed by more rigorous mathematical analysis using tools from geometric singular perturbation theory for the existence of pulses (homoclinics) and wave trains (periodics) , as well as the development of a stability theorem . However, the detailed properties of waves in detailed biophysical models is often best pursued with a mixture of both mathematical and numerical analysis. This is nicely exemplified by the work of Miller and Rinzel , who numerically construct the dispersion relation (between speed and period) for steadily propagating periodic wave trains, and then use a mathematical kinematic theory, which applies the dispersion relation, instantaneously, to individual pulses to predict how interspike time intervals can be distorted during propagation. Going beyond the single neuron scenario traveling waves in neurobiology are often studied at the tissue level, using neuroimaging methods such as electroencephalography (EEG). For example these waves can take the form of spindle waves seen at the onset of sleep , the propagation of synchronous discharge during an epileptic seizure  and waves of excitation associated with sensory processing . Such waves are a consequence of synaptic interactions and the intrinsic behaviour of local neuronal circuitry. The propagation speed of these waves is of the order cm s−1, an order of magnitude slower than that of action potential propagation along axons. The class of computational models that are believed to support synaptic waves differ radically from classic models of waves in excitable systems. Most importantly, synaptic interactions are non-local (in space), involve communication (space-dependent) delays (arising from the finite propagation velocity of an action potential) and distributed delays (arising from neurotransmitter release and dendritic processing). In contrast models of axonal fibres are typically based on local partial differential equation models (that track the opening and closing of ion channels and model voltage transport along a fibre with a diffusive process). A common way to model the spatiotemporal evolution of coarse grained variables such as synaptic or firing rate activity in populations of neurons is through the use of a neural field model (in which space is continuous). Despite a growing body of analysis on synaptic waves in neural field models, focusing mainly on fronts and pulses (and see [9, 10] for further discussion) surprisingly relatively little analysis has been performed on periodic waves. Rather in most neural field models of EEG only spatially homogeneous solutions and their instabilities to pattern states have been analysed e.g. [11–13]. Interestingly in some of these models such instabilities have been shown to give rise to periodic travelling waves, e.g. in the work of Curtu and Ermentrout on neural field models with adaptation  and that of Venkov et al.  in models with short-range inhibition, long range excitation and axonal delays, though not subsequently analysed in the full nonlinear regime (far from bifurcation). This begs the question as to whether, like many models for action potential wave trains in axons, it is useful to analyse and classify neural field models in terms of their dispersion curves and the large period limit where one recovers the description of a travelling pulse. This is precisely the question we address in this paper, by considering a minimal one dimensional single population neural field model. Importantly the techniques we develop are readily applicable to more general neural field models, including those with multiple populations in higher dimensions and with various forms of feedback, such as spike frequency adaptation.
In Section “Neural tissue models” we discuss some simple neural field models with a focus on those with purely excitatory connectivity and linear adaptation as these are perhaps the simplest ones that support travelling waves. Direct simulations are used to show that not only do they manifest travelling pulses and periodic wave trains, but also a localised wave of decreased activity that we term an anti-pulse. The existence and stability of both pulses and anti-pulses is determined in Section “Travelling fronts and pulses” for the case of an idealised Heaviside firing rate function. Novel numerical techniques for handling more general firing rate functions are developed in Section “Numerical techniques for analysing neural fields with sigmoidal firing rates”. Not only do these allow us to show that the analytical results obtained for Heaviside firing rates are similar to those for steep sigmoidal functions, they also open the door for the numerical study of periodic waves and their dispersion curves. These are computed in Section “Dispersion curves of wavetrains” and discussed in the context of a kinematic theory that has previously been used so effectively for excitable models of axons to predict and organise irregular patterns of wave propagation. Similarly we find that, when combined with a kinematic theory, dispersion curves for neural field models are a powerful tool for understanding exotic patterns of travelling wave activity (beyond periodic waves). Finally in Section “Discussion and conclusions” we review the results obtained and discuss natural extensions of the work in this paper.
Neural tissue models
Neural field models describe the coarse-grained activity of neuronal populations, and make no attempt to model the detailed behaviour of single neurons. Rather they model the statistical properties of tissue activity that is more directly relevant to EEG. Due to the long-range axonal connectivities in cortex neural field models are typically formulated as non-local integral or integro-differential equations. For a recent review see . Compared to the analysis of local models (of ordinary or partial differential equation type) their analysis (mathematical and numerical) is not as thoroughly developed. Fortunately, in certain physiologically realistic regimes equivalent partial differential equation (PDE) models can be formulated. In particular this then allows the use of powerful techniques from nonlinear PDE theory to be brought to bear, especially the numerical analysis of travelling waves. To illustrate this possibility we now formulate a simple continuum neural field model in one spatial dimension and present some simulations of travelling wave behaviour.
Here τ and κ are constants describing the timescale separation between activity and adaptation and how strong the (linear) adaptation becomes, respectively. More sophisticated models of metabolic processes whose combined effect is to modulate neuronal response could also be considered, though for the purposes of this paper we shall work with (5). Moreover, for illustration we shall work with the explicit choice w(x)= exp(−|x|)/2 and Q=1+∂/∂ t, i.e. α1=1,α2→∞, namely an exponentially decaying synaptic footprint and a synapse with an exponentially decaying response.
Travelling fronts and pulses
The simulation in Figure 1 indicates that a stable travelling anti-pulse solution exists. We will consider the existence and stability for a Heaviside firing rate function, making use of an Evans function approach. This is a powerful tool for the stability analysis of nonlinear waves on unbounded domains. It was originally formulated by Evans  in the context of a stability theorem about excitable nerve axon equations of Hodgkin-Huxley type. The extension to integral models is far more recent, see  for a discussion. For neural field models with axonal delays it has previously been noted these do not typically induce any change of wave stability [22, 23] (though it will affect the shape and speed of a wave). Since our simulations indicate similar effects for the anti-pulse, we will initially consider the case 1/v=0 for simplicity. The discussion below is adapted from that presented in [22, 23], to which we refer the reader for further details.
Existence of the anti-pulse
These are determined analogously as for the anti-pulse but with the requirement that the profile is above threshold only for ξ∈(−Δ,0) . The speed of activating fronts (moving to the right) can be obtained from (18) in the limit Δ→∞. The above equations apply to right-moving waves, i.e. c>0.
Stability of the anti-pulse
A combined view on fronts and pulses
Numerical techniques for analysing neural fields with sigmoidal firing rates
in a computationally efficient and accurate fashion.
Numerical continuation I: Equivalent PDE
It is easily checked that the model has either one or three equilibria with , where is a solution of . Orbits connecting one or two equilibria are homoclinic and heteroclinic solutions corresponding to travelling pulses and fronts, respectively. These are readily analysed using a numerical toolbox such as MATCONT . However, the equivalent PDE approach is limited to certain special choices of synaptic connectivity possessing a rational Fourier transform structure .
Numerical continuation II: Integral model
and in the last line we see that each term separately has a convolution structure. We conclude that ψ(ξ) (with finite v) can be cast into a suitable form by an asymmetric scaling of the connectivity function. In the limit when the wavespeed equals the transmission speed, the scaled connectivity function w(z/a−) approaches a delta distribution. Initial data is taken from a stable wave found with simulations. We then compute periodic solutions of the integral model with standard pseudo-arclength continuation with free parameters the speed c and the size of the spatial domain T.
The diagram contains an additional saddle-node curve and the Hopf curve no longer starts from c=0, but turns. First we note that the heteroclinic cycle still persists. Also another Zero-Hopf point at the left saddle-node bifurcation curve is present together with a similar structure of homoclinic curves. One has a high speed and is associated to an unstable wave as explained above. The other homoclinic curve has several turns until at κ≈0.5425 it approaches the heteroclinic cycle tangentially to a heteroclinic curve. These are the anti-pulse and inactivating front, respectively. The heteroclinic curve is very similar to the one computed for β=42, but the branch terminates when it meets the Hopf curve. The other heteroclinic curve representing the activating front is even split into two branches. Emanating from c=0 it meets the right saddle-node curve tangentially and then exists for 0.2848<κ<0.7647 between two Hopf curves for higher values of c. The upper branch is tangent to the homoclinic curve representing the travelling pulse. This homoclinic curve exists to the right of the saddle-node bifurcation where it turns and then undulates towards the left Zero-Hopf point. There is another homoclinic curve emanating from the left Zero-Hopf point with high speeds but similar to the other purple branches, it corresponds to unstable solutions. The stability of the front and pulse solutions is similar as for higher values of β. It is remarkable that many of the features obtained from the Heaviside analysis persist for not so steep sigmoidal firing rate functions (though the overall bifurcation structure can be more complex).
Travelling waves can be found by starting from a Hopf bifurcation and then to continue periodic orbits in two parameters, e.g. the period T plus the speed c or the strength κ. When the period goes to infinity, the wave approaches one of the homoclinic curves discussed above. In between, in the (c,T)-diagram the curve of periodic orbits makes several turns. These limit point of cycles (LPC) mark the increasing complexity of travelling wave solutions, as we will discuss below. Here the primary LPC’s are indicated by a thick blue solid line.
Dispersion curves of wavetrains
As briefly discussed in the Background, dispersion curves for periodic wavetrains in axonal models have proved very useful for understanding the behaviour of more irregular wavetrains using a kinematic theory. Thus it is first useful to set the scene for dispersion curves in neural field models by reviewing this approach for a simple excitable fibre model with FitzHugh-Nagumo (FHN) dynamics.
Dispersion curves for a FHN model
where (u,v)=(u(x,t),v(x,t)), 0<a<1, b>0, and t>0. Despite its simplicity this model has a rich dynamical structure with a variety of travelling waves and pulses, as discussed and analysed in [33–36]. The dispersion curve for periodic orbits (stationary in a co-moving frame) relates the speed of a wave to its period, giving c=c(T). In the limit of a large period one recovers the homoclinic orbit describing a solitary travelling pulse. For the FHN model the dispersion curve has a slow and fast branch, and a minimum period largely determined by the refractory time-scale. If the shape of the periodic wave (over one period) is pulse-like then one may invoke a kinematic theory to determine the stability of solution branches, as well as describe the evolution of inter-pulse intervals in more exotic wavetrain patterns (that can bifurcate from periodic orbits). For a review of this approach we refer the reader to . Importantly periodic waves are predicted to be stable if c′(T)>0, with wave bifurcations (found to be period-doubling bifurcations for the FHN model) occurring at stationary points in the dispersion curve where c′(T)=0.
Dispersion curves for neural field models
We turn now to the construction of dispersion curves in neural field models for pulse-like periodic waves. Although we may do this explicitly for the case of a Heaviside firing rate function (extending the technique for fronts and pulses used in Section “Travelling fronts and pulses”) we prefer instead to focus on the numerical construction of dispersion curves using the approaches developed in Section “Numerical techniques for analysing neural fields with sigmoidal firing rates”. Moreover, as for the FHN model, we pinpoint the saddle-focus boundary as it predicts where one finds complex dispersion curves. Based on this we show several bifurcation diagrams where we collect the primary backbone of waves and pulses. From these diagrams it is straightforward to understand the shape of dispersion curves for various values of the system parameters.
Discussion and conclusions
In this paper we have developed the numerical tools to explore and continue travelling wave solutions for non-local neural field models with space-dependent axonal delays. Moreover, we have validated our approach against the analytically tractable case of a Heaviside firing rate and shown how bifurcation diagrams of this special case are modified as one moves toward more physiologically realistic shallower sigmoidal firing rate shapes. Interestingly we have shown that as well as pulses and fronts expected for excitatory networks with inhibitory feedback, also anti-pulses are a robust travelling wave solution. Moreover, the bifurcation diagram for travelling localised states, i.e. fronts, pulses, and anti-pulses, is organised around a co-dimension 2 heteroclinic cycle bifurcation. Our main result, however, has been the numerical construction of dispersion curves. We have shown that they offer similar insight into the behaviour and instability of periodic travelling waves as originally found in their application to excitable reaction diffusion systems. Namely, that the eigen-spectrum of fixed points for a homoclinic orbit corresponding to a pulse can strongly affect the shape of the dispersion curve by complex leading eigenvalues arising through a saddle to saddle-focus transition leading to oscillations in the dispersion curve), as the LPC curve in Figure 6 indicates. Not only can this lead to wave instabilities at stationary points which predicted by a kinematic theory, are period doubling bifurcations, it also leads to multi-stability and waves of super-normal speed, as seen in Figures 5, 6 and 11. Such super-normal speeds are greater than that of the isolated pulse and can be read off from the dispersion curve for large values of the period. Furthermore it is possible to see gaps in dispersion curve for the case when β is small.
Importantly our work opens up a novel way to classify the behaviours (and the consequences for spatio-temporal wave propagation) for a broad spectrum of neural field models that have been used in the modelling of EEG, such as the Liley model , and in particular those for epilepsy, such as in the work of Marten et al.  and Goodfellow et al. . Namely, we expect the main similarities, or differences, between these models to be captured through a comparison of their dispersion curves. We conjecture that this may clarify why coherent oscillations, as opposed to travelling waves, are found to be favoured in the model presented in . This, and related work on how to sculpt the shape of dispersion curves through the detailed form of the neural field model using multiple populations, higher-order synapse models, distributions of axonal speeds and so on, will be reported upon elsewhere.
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