No abstract

We establish center manifold theorems that allow one to study the bifurcation of small solutions from a trivial state in systems of functional equations posed on the real line. The class of equations includes most importantly nonlinear equations with nonlocal coupling through convolution operators as they arise in the description of spatially extended dynamics in neuroscience. These systems possess a natural spatial translation symmetry but local existence or uniqueness theorems for a spatial evolution associated with this spatial shift or even a well motivated choice of phase space for the induced dynamics do not seem to be available, due to the infinite range forward-and backward-coupling through nonlocal convolution operators. We perform a reduction relying entirely on functional analytic methods. Despite the nonlocal nature of the problem, we do recover a local differential equation describing the dynamics on the set of small bounded solutions, exploiting that the translation invariance of the original problem induces a flow action on the center manifold. We apply our reduction procedure to problems in mathematical neuroscience, illustrating in particular the new type of algebra necessary for the computation of Taylor jets of reduced vector fields.Center-manifold reductions have become a central tool to the analysis of dynamical systems. The very first results on center manifolds go back to the pioneering works of Pliss [21] and Kelley [16] in the finitedimensional setting. In the simplest context, one studies differential equations in the vicinity of a nonhyperbolic equilibrium,The basic reduction establishes that the set of small bounded solutions u(t), t ∈ R, sup |u(t)| < δ ≪ 1, is pointwise contained in a manifold, that is, u(t) ∈ W c for all t. This manifold is a subset of phase space, W c ⊂ R n , contains the origin, 0 ∈ W c , and is tangent to E c , the generalized eigenspace associated with purely imaginary eigenvalues of f ′ (0). As a consequence, the flow on W c can be projected onto E c , to yield a reduced vector field. The reduction to this lower-dimensional ODE then allows one to describe solutions qualitatively, even explicitly in some cases. Of course, the method applies to higher-order differential equation, which one simply writes as first-order equation in a canonical fashion. Extensions to infinite-dimensional dynamical systems were pursued soon after; see for instance [12].Starting with the work of Kirchgässner [17], such reductions have been extended to systems with u ∈ X , a Banach space, where the initial value problem is not well-posed: For most initial conditions u 0 , there does not exist a local solution u(t), 0 ≤ t < δ, say. Local solutions do exist however for all initial conditions on a finite-dimensional center-manifold, and much of the theory is quite analogous to the finite-dimensional case; see [25]. In these theories, one can typically split the phase space in infinite-dimensional linear spaces where solutions to the linearized equation either decay or grow, and a finite-d...

Motivated by a model for the perception of textures by the visual cortex in primates, we analyze the bifurcation of periodic patterns for nonlinear equations describing the state of a system defined on the space of structure tensors, when these equations are further invariant with respect to the isometries of this space. We show that the problem reduces to a bifurcation problem in the hyperbolic plane D (Poincaré disc). We make use of the concept of a periodic lattice in D to further reduce the problem to one on a compact Riemann surface D/Γ , where Γ is a cocompact, torsion-free Fuchsian group. The knowledge of the symmetry group of this surface allows us to use the machinery of equivariant bifurcation theory. Solutions which generically bifurcate are called "H-planforms", by analogy with the "planforms" introduced for pattern formation in Euclidean space. This concept is applied to the case of an octagonal periodic pattern, where we are able to classify all possible H-planforms satisfying the hypotheses of the Equivariant Branching Lemma. These patterns are, however, not straightforward to compute, even numerically, and in the last section we describe a method for computation illustrated with a selection of images of octagonal H-planforms.Communicated by M. Golubitsky.

We study localised activity patterns in neural field equations posed on the Euclidean plane; such models are commonly used to describe the coarse-grained activity of large ensembles of cortical neurons in a spatially continuous way. We employ matrix-free Newton-Krylov solvers and perform numerical continuation of localised patterns directly on the integral form of the equation. This opens up the possibility to study systems whose synaptic kernel does not lead to an equivalent PDE formulation. We present a numerical bifurcation study of localised states and show that the proposed models support patterns of activity with varying spatial extent through the mechanism of homoclinic snaking. The regular organisation of these patterns is due to spatial interactions at a specific scale associated with the separation of excitation peaks in the chosen connectivity function. The results presented form a basis for the general study of localised cortical activity with inputs and, more specifically, for investigating the localised spread of orientation selective activity that has been observed in the primary visual cortex with local visual input.

We establish Fredholm properties for a class of nonlocal differential operators. Using mild convergence and localization conditions on the nonlocal terms, we also show how to compute Fredholm indices via a generalized spectral flow, using crossing numbers of generalized spatial eigenvalues. We illustrate possible applications of the results in a nonlinear and a linear setting. We first prove the existence of small viscous shock waves in nonlocal conservation laws with small spatially localized source terms. We also show how our results can be used to study edge bifurcations in eigenvalue problems using Lyapunov-Schmidt reduction instead of a Gap Lemma.

We examine the existence and stability of traveling pulse solutions in a continuum neural network with synaptic depression and smooth firing rate function. The existence proof relies on geometric singular perturbation theory and blow-up techniques as one needs to track the solution near a point on the slow manifold that is not normally hyperbolic. The stability of the pulse is then investigated by computing the zeros of the corresponding Evans function. This study predicts that synaptic depression leads to the formation of stable traveling pulses with algebraic decay along their back. This characteristic feature differs from the exponential decay of traveling pulses of neural field models with linear adaptation.

We identify a new mechanism for propagation into unstable states in spatially extended systems, that is based on resonant interaction in the leading edge of invasion fronts. Such resonant invasion speeds can be determined solely based on the complex linear dispersion relation at the unstable equilibrium, but rely on the presence of a nonlinear term that facilitates the resonant coupling. We prove that these resonant speeds give the correct invasion speed in a simple example, we show that fronts with speeds slower than the resonant speed are unstable, and corroborate our speed criterion numerically in a variety of model equations, including a nonlocal scalar neural field model.

We consider a nonlocal generalization of the Fisher-KPP equation in one spatial dimension. As a parameter is varied the system undergoes a Turing bifurcation. We study the dynamics near this Turing bifurcation. Our results are two-fold. First, we prove the existence of a two-parameter family of bifurcating stationary periodic solutions and derive a rigorous asymptotic approximation of these solutions. We also study the spectral stability of the bifurcating stationary periodic solutions with respect to almost co-periodic perturbations. Secondly, we restrict to a specific class of exponential kernels for which the nonlocal problem is transformed into a higher order partial differential equation. In this context, we prove the existence of modulated traveling fronts near the Turing bifurcation that describe the invasion of the Turing unstable homogeneous state by the periodic pattern established in the first part. Both results rely on a center manifold reduction to a finite dimensional ordinary differential equation.