Numerical integration can also be used to determine solution stability. In Chapter 4, we constructed piecewise steady state solutions of the neural field models using a step firing rate function. The method of eigenvalue analysis presented in Section 7.5 cannot be used with a piecewise firing rate function. Instead, we add a small perturbation to a steady state and numerically integrate. If the solution tends to the steady state ast→ ∞, then it is stable. Otherwise the steady state is unstable. Suitable numerical integration methods are discussed in the Appendix.
7.7
Conclusion
In this chapter we have discussed five methods that can be used to determine the stability of solutions of the Amari model and the gap junction model. The first three of these have been presented previously in the literature and require a step firing rate function. We have applied them to both the models, with a Mexican hat function, then with the decaying oscillatory coupling function. The last two methods require numerical analysis to determine stability and can be used with the smooth firing rate function.
Amari’s [1] linearised stability analysis finds the stability of single-bump solutions. The stability condition is the same for both coupling and firing rate function combinations in both models. If two single-bump steady states exist for particular system parameters, then there is a stable wider, taller one and an unstable narrower, shorter one. These stability results agree with the numerical analysis in the thesis. We also used Amari’s analysis in
Chapter 2 where steady state solutions of both models with lateral inhibition and a step firing rate function were found analytically.
Pinto and Ermentrout’s stability analysis method [82] uses a perturbation analysis of single-bump steady states. For the Amari model, the stability results agree with Amari’s linearised stability analysis. The analysis could not be extended to the gap junction model as the second spatial derivatives of both the solution and the perturbation at the boundaries of the region of excitation are not known. Numerical simulation, however, showed that the gap junction model has similar stability results to the Amari model.
An Evans function analysis has been previously used to find the stability of travelling fronts in an Amari-type model with finite propagation velocity [16]. The technique relies upon writing the model as a double integral. We used a Green’s function to write the partial integro-differential equation model as an integral equation model. For the Amari model, no travelling fronts were found for the model with lateral inhibition at the specific parameter values we have used throughout the thesis. For the decaying oscillatory coupling function, travelling fronts were found over a wide range of parameters. Evans function analysis showed that the travelling fronts are stable. The results were confirmed by numerical simulation. We could not extend the Evans function analysis to the gap junction model as we were unable to reduce the eventual integral model from a triple integral to a double integral. Evans function analysis was used in Chapter 5 where stable stationary fronts (heteroclinic orbits) were found in the Amari model with decaying oscillatory coupling function.
We presented a method of eigenvalue analysis we developed for both the models with a smooth firing rate function. An eigenvalue equation was solved for a steady state for particular system parameters. The translational invariance of the models means a zero eigenvalue was always found. If the eigenvalues all had a nonpositive real part, then the steady state was stable, otherwise the steady state was unstable. We used eigenvalue analysis to determine the stability of the solutions found with continuation methods in Chapter 3. Finally, we discussed numerical integration as a way to determine solution stability. This method can be used with any coupling and firing rate function combination. The method numerically integrates with a perturbed steady state as the initial condition. If the solution tends to the steady state as time increases then it is a stable steady state.
Turing structures
8.1
Introduction
We now investigate spatio-temporal pattern formation out of equilibrium in the neural field models. We refer to the spatio-temporal patterns as Turing structures in recognition of Alan Turing’s work in 1952 [97]. Turing proposed that pattern formation in biological systems is caused by chemical morphogenesis. It is well known that spatio-temporal patterns can form as a result of spatially uniform equilibrium states losing stability to a spatially heterogeneous perturbation. Both Turing (stationary) and wave (oscillatory Turing) bifurcations lead to a large variety of spatio-temporal patterns. A Turing bi- furcation breaks spatial symmetry, generating a pattern that is stationary in time but oscillatory in space. A wave bifurcation breaks both spatial and temporal symmetries, forming patterns that are oscillatory in both time and space [109]. Examples are found in fluid dynamics, solid-state physics, nonlinear optics, reaction-diffusion systems, chemistry and biology [20, 75]. An example is the Turing structures observed in the Belousov- Zhabotinsky (BZ) reaction [100]. Rotating spirals and target waves have been observed in a variety of two-dimensional physical, chemical and biological reaction-diffusion systems [99]. Once initiated, rotating spiral waves are self-sustaining.
We now briefly discuss several relevant papers in the literature. Hutt et al. [53] applied an analytical stability method to a one-dimensional neural field model with lateral inhibition and axonal delay to find instabilities where spatio-temporal patterns can form. These are referred to as Turing instabilities. The method uses a linearised stability analysis to identify the unstable wavenumbers in a spatial perturbation that can cause a periodic pattern to form. A Turing instability was identified, as seen in activator-inhibitor systems [97]. Hutt et al. observed oscillatory or wave bifurcations in their neural field
model with local inhibition and lateral excitation when the delay exceeded a threshold. The Turing instabilities were independent of delay effects. Recently, Turing structures have also been analysed in several two-dimensional models. Steyn-Rosset al.[94] studied a continuum model of a noise-driven cortex with gap junction connections between inhibitory neurons. Using a linear stability analysis and numerical simulation, they found that Turing structures form when the diffusive coupling between gap junctions is large. Coombes et al. [18] studied another continuum model with space-dependent axonal delays. In this chapter we apply the linear stability analysis of [53] to both the Amari model and the gap junction model. Instead of lateral inhibition, we use the decaying oscillatory coupling function and smooth firing rate function introduced in Chapter 3. We look for Turing structures in both one and two spatial dimensions.
The plan of the chapter is as follows. Section 8.2 looks at the Amari model and the gap junction model in one spatial dimension. For the Amari model, spatially uniform steady states of the system are found as a function of system parameters. Regions of parameter space where Turing instabilities can occur are found using a linear stability analysis. Numerical simulations are undertaken to find Turing structures for particular parameter values. This involves numerically integrating with an initial condition of a spatially uniform steady state plus a very small random spatial perturbation. To explain the results, spatially periodic patterns are represented by Fourier series. Bifurcation analysis finds regions of parameter space where both stable and unstable periodic patterns exist. We see that the transiency of some solutions is related to a type-I intermittency. The same method is then applied to the gap junction model and numerical simulations are presented. In Section 8.3, the analysis is extended to two spatial dimensions for the Amari model only. Numerical simulations are presented.