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Geometric perturbation theory is universal. A typical example is provided by the 3D wave equation, widely used in acoustics. We face vibrating eardrums as a binaural auditory input stemming from an external sound source. In the setup of internally coupled ears (ICE), which are present in more than half of the land-living vertebrates, the two tympana are coupled by an internal air-filled cavity, whose geometry determines the acoustic properties of the ICE system. The eardrums themselves are described by a 2-dimensional, damped, wave equation and are part of the spatial boundary conditions of the three-dimensional Laplacian belonging to the wave equation in the internal cavity that couples and internally drives the eardrums. In animals with ICE the resulting signal is the superposition of external sound arriving at both eardrums and the internal pressure coupling them. This is also the typical setup for geometric perturbation theory. In the context of ICE it boils down to acoustic boundary-condition dynamics (ABCD) for the coupled dynamical system of eardrums and internal cavity. In acoustics the deviations from equilibrium are extremely small (nm). Perturbation theory is therefore natural and shown to be appropriate. In doing so, we use a time-dependent perturbation theory \`a la Dirac in the context of Duhamel's principle. The relaxation dynamics of the tympanic-membrane system, which neuronal information processing stems from, is explicitly obtained in first order. Furthermore, both the initial and the quasi-stationary asymptotic state are derived and analyzed. Finally, we set the general stage for geometric perturbation theory where (d-1)-dimensional manifolds as subsets of the boundary of a d-dimensional domain are driven by their own dynamics with the domain pressure $p$ and an external source term as input, at the same time constituting time-dependent boundary conditions for $p$.

We generalize the F\"{o}ppl-von K\'arm\'an equations to an initially precurved sheet and present the underlying derivation. A geometrically computed moment of strain replaces the notion of bending moment and results in a geometric formulation of the theory of shells. As the curvature approaches zero, i.e., the sheet becomes flat, the new equations reduce to the classic F\"{o}ppl-von K\'arm\'an ones. The present theory solves the long-standing problem of formulating these equations for an a priori curved shell and applies, for instance, both to shell theory and to strongly curved biomembranes of cells as closed surfaces, exhibiting crumpling as the membrane thickness goes to zero.

The thalamus is the major gate to the cortex and its contribution to cortical receptive field properties is well established. Cortical feedback to the thalamus is, in turn, the anatomically dominant input to relay cells, yet its influence on thalamic processing has been difficult to interpret. For an understanding of complex sensory processing, detailed concepts of the corticothalamic interplay need yet to be established. To study corticogeniculate processing in a model, we draw on various physiological and anatomical data concerning the intrinsic dynamics of geniculate relay neurons, the cortical influence on relay modes, lagged and nonlagged neurons, and the structure of visual cortical receptive fields. In extensive computer simulations we elaborate the novel hypothesis that the visual cortex controls via feedback the temporal response properties of geniculate relay cells in a way that alters the tuning of cortical cells for speed., Comment: 31 pages, 7 figures

An unresolved paradox exists in auditory and electrosensory neural systems {Carr93,Heiligenberg91}: they encode behaviourally relevant signals in the range of a few microseconds with neurons that are at least one order of magnitude slower. We take the barn owl's auditory system as an example and present a modeling study based on computer simulations of a neuron in the laminar nucleus. Three observations resolve the paradox. First, spiking of an integrate-and-fire neuron driven by excitatory postsynaptic potentials (EPSPs) with a width at half maximum of 250 microseconds has an accuracy of 25 microseconds if the presynaptic signals arrive coherently. Second, the necessary degree of coherence in the signal arrival times can be attained during ontogenetic development by virtue of an unsupervised Hebbian learning rule. Learning selects connections with matching delays from a broad distribution of axons with random delays. Third, the learning rule also selects the correct delays from two independent groups of inputs, for example, from the left and right ear.

This chapter begins by summarizing psychophysical and neurophysiological evidence for shift and size invariance and its limitations. It provides a list of constraints from neuroanatomy and physiology that need to be taken into account when developing biologically plausible models. From a computational point of view, there are two basic approaches by which shift and size invariance can be achieved: normalizing the image or extracting invariant features. Models following these two computational approaches are presented and the extent to which they can account for different aspects of the visual system are discussed. Open questions in modeling shift and size invariance in the visual system are addressed.