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The aim of this paper is to investigate the oscillatory and asymptotic behavior of solutions of a third-order delay difference equation. By using comparison theorems, we deduce oscillation of the difference equation from its relation to certain associated first-order delay difference equations or inequalities. Examples are given to illustrate the main results.

In this doctoral thesis, several aspects of neuronal activity in the rat superior olivary complex (SOC), an auditory brainstem structure, were analyzed using optical imaging with voltage-sensitive dyes (VSD). The thesis is divided into 5 Chapters. Chapter 1 is a general introduction, which gives an overview of the auditory brainstem and VSD imaging. In Chapter 2, an optical imaging method for the SOC was standardized, using the VSD RH795. To do so, the following factors were optimized: (1) An extracellular potassium concentration of 5 mM is necessary during the incubation and recording to observe synaptically evoked responses in the SOC. (2) Employing different power supplies reduced the noise. (3) Averaging of 10 subsequent trials yielded a better signal-to-noise ratio. (4) RH795 of 100 µM with 50 min prewash was optimal to image SOC slices for more than one hour. (5) Stimulus-evoked optical signals were TTX sensitive, revealing action potential-driven input. (6) Synaptically evoked optical signals were characterized to be composed of pre- and postsynaptic components. (7) Optical signals were well correlated with anatomical structures. Overall, this method allows the comparative measurement of electrical activity of cell ensembles with high spatio-temporal resolution. In Chapter 3, the nature of functional inputs to the lateral superior olive (LSO), the medial superior olive (MSO), and the superior paraolivary nucleus (SPN) were analyzed using the glycine receptor blocker strychnine and the AMPA/kainate receptor blocker CNQX. In the LSO, the known glutamatergic inputs from the ipsilateral, and the glycinergic inputs from the ipsilateral and contralateral sides, were confirmed. Furthermore, a CNQX-sensitive input from the contralateral was identified. In the MSO, the glutamatergic and glycinergic inputs from the ipsilateral and contralateral sides were corroborated. In the SPN, besides the known glycinergic input from the contralateral, I found a glycinergic input from the ipsilateral and I also identified CNQX-sensitive inputs from the contralateral and ipsilateral sides. Together, my results thus corroborate findings obtained with different preparations and methods, and provide additional information on the pharmacological nature of the inputs. In Chapter 4, the development of glycinergic inhibition for the LSO, the MSO, the SPN, and the medial nucleus of the trapezoid body (MNTB) was studied by characterizing the polarity of strychnine-sensitive responses. In the LSO, the high frequency region displayed a shift in the polarity at P4, whereas the low frequency region displayed at P6. In the MSO, both the regions displayed the shift at P5. The SPN displayed a shift in the polarity at E18-20 without any regional differences. The MNTB lacked a shift between P3-10. Together, these results demonstrate a differential timing in the development of glycinergic inhibition in these nuclei. In Chapter 5, the role of the MSO in processing bilateral time differences (t) was investigated. This was done by stimulating ipsilateral and contralateral inputs to the MSO with different t values. In preliminary experiments, the postsynaptic responses showed a differential pattern in the spread of activity upon different t values. This data demonstrates a possible presence of delay lines as proposed by Jeffress in the interaural time difference model of sound localization. In conclusion, this study demonstrates the usage of VSD imaging to analyze the neuronal activity in auditory brainstem slices. Moreover, this study expands the knowledge of the inputs to the SOC, and has identified one glycinergic and three AMPA/kainate glutamatergic novel inputs to the SOC nuclei. Charakterisierung neuronaler Aktivität im auditorischen Hirnstamm von Ratten: Untersuchungen mit einen bildgebenden Verfahren