Fatou’s Associates

Suppose that f is a transcendental entire function, V⊊C is a simply connected domain, and U is a connected component of f-1(V). Using Riemann maps, we associate the map f : U→V to an inner function g : D→D. It is straightforward to see that g is either a finite Blaschke product, or, with an appropriate normalisation, can be taken to be an infinite Blaschke product. We show that when the singular values of f in V lie away from the boundary, there is a strong relationship between singularities of g and accesses to infinity in U. In the case where U is a forward-invariant Fatou component of f, this leads to a very significant generalisation of earlier results on the number of singularities of the map g. If U is a forward-invariant Fatou component of f there are currently very few examples where the relationship between the pair (f, U) and the function g has been calculated. We study this relationship for several well-known families of transcendental entire functions. It is also natural to ask which finite Blaschke products can arise in this manner, and we show the following: for every finite Blaschke product g whose Julia set coincides with the unit circle, there exists a transcendental entire function f with an invariant Fatou component such that g is associated with f in the above sense. Furthermore, there exists a single transcendental entire function f with the property that any finite Blaschke product can be arbitrarily closely approximated by an inner function associated with the restriction of f to a wandering domain

Very Early Development of Nucleus Taeniae of the Amygdala

Ikebuchi M, Nanbu S, Okanoya K, Suzuki R, Bischof H-J. Very Early Development of Nucleus Taeniae of the Amygdala. Brain Behavior And Evolution. 2013;81(1):12-26.The avian nucleus taeniae of the amygdala (TnA) corresponds to part of the mammalian medial amygdala. Like its mammalian counterpart, it has been shown to be involved in the control of social function. According to behavioral observations, such control is already necessary early in the ontogenetic development of a bird. If so, TnA should be one of the earliest differentiating brain structures of the telencephalon. Our anatomical study shows that TnA can already be delineated at posthatching day one. The volume of TnA exhibits a growth spurt between days 1 and 8 posthatch, developing at a faster rate than the entire telencephalon. Our results suggest that between days 1 and 8 the growth of neuropil exceeds the enhancement of neuron number (leading to a decrease of cell density), and an addition at the same pace of new neurons and neuropil thereafter. A plateau is reached at posthatch day 30. The development of TnA precedes that of the song control nuclei and is similar to the early growth of thalamic and telencephalic sensory areas. This adds to the idea that this structure may already be involved in social control at the time of hatching. A proximate cause of the early development of TnA might be the direct afference from the olfactory bulb. Copyright (C) 2012 S. Karger AG, Base

Optical Imaging of Retinotopic Maps in a Small Songbird, the Zebra Finch

Keary N, Voss J, Lehmann K, Bischof H-J, Loewel S. Optical Imaging of Retinotopic Maps in a Small Songbird, the Zebra Finch. PLOS ONE. 2010;5(8): e11912.Background: The primary visual cortex of mammals is characterised by a retinotopic representation of the visual field. It has therefore been speculated that the visual wulst, the avian homologue of the visual cortex, also contains such a retinotopic map. We examined this for the first time by optical imaging of intrinsic signals in zebra finches, a small songbird with laterally placed eyes. In addition to the visual wulst, we visualised the retinotopic map of the optic tectum which is homologue to the superior colliculus in mammals. Methodology/Principal Findings: For the optic tectum, our results confirmed previous accounts of topography based on anatomical studies and conventional electrophysiology. Within the visual wulst, the retinotopy revealed by our experiments has not been illustrated convincingly before. The frontal part of the visual field (0 degrees +/- 30 degrees azimuth) was not represented in the retinotopic map. The visual field from 30 degrees-60 degrees azimuth showed stronger magnification compared with more lateral regions. Only stimuli within elevations between about 20 degrees and 40 degrees above the horizon elicited neuronal activation. Activation from other elevations was masked by activation of the preferred region. Most interestingly, we observed more than one retinotopic representation of visual space within the visual wulst, which indicates that the avian wulst, like the visual cortex in mammals, may show some compartmentation parallel to the surface in addition to its layered structure. Conclusion/Significance: Our results show the applicability of the optical imaging method also for small songbirds. We obtained a more detailed picture of retinotopic maps in birds, especially on the functional neuronal organisation of the visual wulst. Our findings support the notion of homology of visual wulst and visual cortex by showing that there is a functional correspondence between the two areas but also raise questions based on considerable differences between avian and mammalian retinotopic representations