On local bifurcations in neural field models with transmission delays

Abstract

Neural field models with transmission delays may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results.

Pitchfork–Hopf bifurcations in 1D neural field models with transmission delays

“…Indeed, since E : X → X ⊙⋆ as defined in (15) maps into Y × {0}, φ ⊙⋆ is of the form (y, 0) for some y ∈ Y . We can therefore find ν by applying Lemma 36 in [21], which we now restate for completeness.…”

“…We can derive expressions for the critical normal form coefficients as in [21], namely, by substituting the expansions (60), (69), and normal form (62) into the homological equation (68) and equating coefficients of the corresponding powers of u, z,z. This leads to linear operator equations of the form…”

“…A more detailed description is given in [21]. Let Y := C (Ω) denote the Banach space of real-valued continuous functions on Ω equipped with the supremum norm ∥y∥ Y := sup r∈Ω |y(r)| (1) V (t) ∈ Y denotes the spatial distribution of the membrane potential at time t, and we sometimes abuse notation and write V (t, r) to denote V (t)(r).…”

“…In [21], local bifurcation theory for systems of the type (NFE) is developed. In the present paper, we show how symmetry arguments and residue calculus can be used to simplify the computations of the spectral properties and the evaluation of the normal form coefficients, respectively.…”

Pulsatile Localized Dynamics in Delayed Neural Field Equations in Arbitrary Dimension

“…In contrast with [19], the presence of delays necessitates to work in complex spaces with a specific quadratic form for the delays [12]. Methods for the reduction to normal form in neural-field equations have recently been developed [26,24]. While the first reference uses a L 2 approach, the second uses a semi-group approach (sun star formalism).…”