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We deal with the periodic boundary value problem for a second-order nonlinear ODE which includes the case of the Nagumo type equation $v_{xx} - g v + n(x) F(v) = 0,$ previously considered by Grindrod and Sleeman and by Chen and Bell in the study of the model of a nerve fiber with excitable spines. In a recent work we proved a result of nonexistence of nontrivial solutions as well as a result of existence of two positive solutions, the different situations depending by a threshold parameter related to the integral of the weight function $n(x).$ Here we show that the number of positive periodic solutions may be very large for some special choices of a (large) weight $n.$ We also obtain the existence of subharmonic solutions of any order. The proofs are based on the Poincar\'{e} - Bikhoff fixed point theorem., Comment: 18 pages, 2 figures

In Drosophila, naturally the circadian clock is entrained by environmental light-dark cycles. Photoreceptors like rhodopsins and cryptochrome perceive light signals for light entrainment of the circadian clock. It is known that there is a light-dependent CRY:TIM interaction, leading to the degradation of TIM coupled with molecular re-setting of the clock-gene cyclings. However, functional double mutants (norpAP41;cryb), blocking simultaneously the known rhodopsin mediated transduction cascade and the largely unknown cryptochrome mediated transduction cascade do not confer the circadian clock absolutely blind to light signals. The double mutants� circadian clock still shows residual light entrainment ability, indicating that a novel photoreceptor entrains the circadian clock. On the other hand, in gl60j cryb double mutants the circadian clock is absolutely blind to light signals. The gl60j mutation removes morphological structures like the compound eyes and the ocelli (signaling from them is blocked by norpAP41); gl60j in addition removes the H-B eyelet and a subset of DNs (DN1s). The H-B eyelet projects axons toward the s-LNvs pacemaker neurons. Both in the s-LNvs and in the DN1s the molecular PER and TIM cyclings can still be synchronized by light in norpAP41;cryb double mutant flies. Therefore, we have investigated a possible role of the H-B eyelet in residual light entrainment of the circadian clock. We have shown that the H-B eyelet functions to sense dim white light. Blocking light signaling from it affects the synchronization of molecular TIM cycling in the pacemaker neurons and in a subset of the DNs. However, at higher intensities of light no effect was seen. Therefore, we further investigated a possible role of the DNs, a subset of which also maintained light-synchronized molecular circadian oscillation in norpAP41;cryb double mutant flies. We have shown that the DN2+3s host a self-sustained molecular oscillator, which is entrainable by light dark cycles. Further, it was demonstrated that a novel norpA and cry independent photoreceptor in the dorsal brain could entrain the DN oscillator. As a putative candidate, we have investigated the Rh 7 gene and shown that it is indeed expressed in proximity to the DNs and LNs. Possibly, Rh 7 bypasses the classical visual norpA dependent phototransduction cascade. The data presented in this thesis support the existence of a novel circadian photoreceptor and--pigment and set the stage for analysis of the signaling pathways involved. Finally, isolation of the novel mutant Veela presents a potential missing link in the CRY:TIM signaling and interaction mechanism, which is the central part of the Drosophila light-entrainment pathway., Normalerweise wird die circadiane Uhr der Fruchtfliege Drosophila melanogaster durch Licht/Dunkel Zyklen ihrer Umgebung synchronisiert. Photorezeptoren wie die Rhodopsine und Cryptochrome erkennen Lichtsignale für diese lichtinduzierte Anpassung der inneren Uhr. Man weiß, dass eine lichtabhängige Interaktion der Protein CRY:TIM zu einem Abbau von TIM führt. Dieser Abbau bewirkt ein Zurücksetzten und einen Neustart der inneren Uhr auf molekularer Ebene. Die doppelt mutante Fliege (norpAP41;cryb), in der gleichzeitig die bekannte Rhodopsin Signalkaskade und die weitgehend unbekannte durch Cryptochrome vermittelte Signalkaskade ausgeschaltet ist, ist jedoch nicht, zumindest auf die innere Uhr bezogen, absolut blind. Die innere Uhr der Doppelmutanten kann immer noch durch Licht beeinflusst werden. Dies zeigt, dass es noch einen weiteren, neuen Photorezeptor in der Fliege geben muss. Andererseits sind gl60j cryb Doppelmutanten absolut blind gegenüber Lichtsignalen .Die gl60j Mutation entfernt morphologische Strukturen wie die Komplexaugen und die Ocellen (Signale aus den Ocellen werden in den norpAp41 Fliegen blockiert); zusätzlich sind in den gl60j Fliegen das H-B Äuglein und ein Teil der DNs (DN1s) entfernt. Das H-B Äuglein projiziert Axone in Richtung der s-LNvs Schrittmacher Neurone. Sowohl in den s-LNvs als auch in den DN1s kann die molekulare Rhythmik der Proteine PER und TIM auch in den norpAP41;cryb Doppelmutanten noch durch Licht synchronisiert werden. Darum wurde eine mögliche Rolle des H-B Äugleins in der Lichtrezeption bei der inneren Uhr untersucht. Wir haben gezeigt, dass die Funktion des H-B Äuglein in der Wahrnehmung von schwachen Licht liegt. Eine Blockierung der Lichtweiterleitung aus dem H-B Äuglein beeinflusst die Synchronisation der molekularen TIM Rhythmik in den Schrittmacherneuronen und in einem Teil der DNs. Bei höheren Lichtintensitäten konnte jedoch kein Effekt festgestellt werden. Darum wurden eine weitere Rolle der DNs untersucht und zwar der Teil der DNs welcher in norpAP41;cryb Doppelmutanten auch eine Lichtsynchronisierte molekulare Oszillation zeigt. Hier konnten wir zeigen, das die DN2+3s einen selbst erhaltenden molekularen Oszillator besitzen, welcher durch Licht/Dunkel Zyklen synchronisiert werden kann. Ebenso wurde gezeigt, dass ein weiterer, norpA und cry unabhängiger Photorezeptor im dorsalen Gehirn den DN Oszillator synchronisieren kann. Als ein möglicher Kandidat haben wir das Rh 7 Gen untersucht und zeigen können, dass es in der Tat in der Nähe der DNs und LNs exprimiert wird. Möglicherweise umgeht Rh 7 die klassische norpA abhängige Phototranduktions-Kaskade. Die Ergebnisse die in dieser Doktorarbeit präsentiert werden sprechen für die Existenz eines neuen circadianen Photorezeptors und Pigments und schaffen die Vorraussetzungen für die Analyse dieses weiteren Signalwegs. Zum Schluss wurde eine neue Mutante, Veela, isoliert, welche ein mögliches Bindeglied im CRY:TIM Signal- und Interaktionsmechanismus darstellt.

The work introduces a model for reciprocal connections in neural fields by a nonlocal feedback mechanism, while the neural field exhibits nonlocal interactions and intra-areal transmission delays. We study the speed of traveling fronts with respect to the transmission delay, the spatial feedback range and the feedback delay for general axonal and feedback connectivity kernels. In addition, we find a novel shape of traveling fronts due to the applied feedback and criteria for its occurence are derived.

This paper focuses on the outline of some computational methods for the approximate solution of the integral equations for the neuronal firing probability density and an algorithm for the generation of sample-paths in order to construct histograms estimating the firing densities. Our results originate from the study of non-Markov stationary Gaussian neuronal models with the aim to determine the neuron's firing probability density function. A parallel algorithm has been implemented in order to simulate large numbers of sample paths of Gaussian processes characterized by damped oscillatory covariances in the presence of time dependent boundaries. The analysis based on the simulation procedure provides an alternative research tool when closed-form results or analytic evaluation of the neuronal firing densities are not available., 10 pages, 3 figures, to be published in Scientiae Mathematicae Japonicae

A stability analysis is presented for neural field equations in the presence of axonal delays and for a general class of connectivity kernels and synaptic properties. Sufficient conditions are given for the stability of equilibrium solutions. It is shown that the delays play a crucial role in non-stationary bifurcations of equilibria, whereas the stationary bifurcations depend only on the kernel. Bounds are determined for the frequencies of bifurcating periodic solutions. A perturbative scheme is used to calculate the types of bifurcations leading to spatial patterns, oscillatory solutions, and traveling waves. For high transmission speeds a simple method is derived that allows the determination of the bifurcation type by visual inspection of the Fourier transforms of the connectivity kernel and its first moment. Results are numerically illustrated on a class of neurologically plausible second order systems with combinations of Gaussian excitatory and inhibitory connections.

The author is concerned with the asymptotic stability of traveling wave solutions of integral differential equations arising from synaptically coupled neuronal networks. By using complex analytic functions, he proves that there is no nonzero spectrum of some linear operator $\mathcal L$ in the region Re $\lambda \geq 0$, and $\lambda =0$ is a simple eigenvalue. By applying linearized stability criterion, he shows that the traveling wave solutions are asymptotically stable. Additionally, some explicit analytic functions are found for a scalar integral differential equation.

The present article introduces a neuronal field model for both excitatory and inhibitory connections. A single integro-differential equation with delay is derived and studied at a critical point by stability analysis, which yields conditions for static periodic patterns and wave instabilities. It turns out that waves only occur below a certain threshold of the activity propagation velocity. An additional brief study exhibits increasing phase velocities of waves with decreasing slope subject to increasing activity propagation velocities, which are in accordance to experimental results. Numerical studies near and far from instability onset supplement the work.

In this article, we study the network of two neurons with delay. Using the discrete Lyapunov functional of Mallet-Paret and Sell and the techniques developed recently by Krisztin, Walther and Wu (for the scalar case), we obtain a two-dimensional closed disk bordered by a phase-locked periodic orbit and we have a complete description about the structure of various heteroclinic connections in the global forward extension of a three-dimensional $C^1$-submanifold contained in the unstable set of the trivial solution.

Linear reaction-hyperbolic systems of partial differential equations in one space dimension arise in the study of the physiological process by which materials are transported in nerve cell axons. Probabilistic methods are developed to derive a closed form approximate solution for an initial-boundary value problem of such a system. The approximate solution obtained is a translating solution of a heat equation. An estimate is proved giving the deviation of this approximate traveling wave solution from the exact solution.

The cortical magnification matrix M is introduced founded on a notion similar to that of the scalar cortical magnification factor M. Unlike M, this matrix is suitable to describe anisotropy in cortical magnification, which is of particular interest in the highly gyrified human cerebral cortex. The advantage of our tensor method over other surface-based 3D methods to explore cortical morphometry is that M expresses cortical quantities in the corresponding sensory space. It allows us to investigate the spatial relation between sensory function and anatomical structure. To this end, we consider the calcarine sulcus (CS) as an anatomical landmark for the primary visual cortex (V1). We found that a stereotypically formed 3D model of V1 compared to a flat model explains an excess of cortical tissue for the representation of visual information coming from the horizon of the visual field. This suggests that the intrinsic geometry of this sulcus is adapted to encephalize a particular function along the horizon. Since visual functions are assumed to be M-scaled, cortical folding can serve as an anatomical basis for increased functionality on the horizon similar to a retinal specialization known as visual streak, which is found in animals with lower encephalization. Thus, the gain of surface area by cortical folding links anatomical structure to cortical function in a previously unrecognized way, which may guide sulci development., 22 pages, 6 figures