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The dynamics of biological neural networks are of great interest to neuroscientists and are frequently studied using conductance-based compartmental neuron models. For speed and ease of use, neuron models are often reduced in morphological complexity. This reduction may affect input processing and prevent the accurate reproduction of neural dynamics. However, such effects are not yet well understood. Therefore, for my first aim I analyzed the processing capabilities of 'branched' or 'unbranched' reduced models by collapsing the dendritic tree of a morphologically realistic 'full' globus pallidus neuron model while maintaining all other model parameters. Branched models maintained the original detailed branching structure of the full model while the unbranched models did not. I found that full model responses to somatic inputs were generally preserved by both types of reduced model but that branched reduced models were better able to maintain responses to dendritic inputs. However, inputs that caused dendritic sodium spikes, for instance, could not be accurately reproduced by any reduced model. Based on my analyses, I provide recommendations on how to construct reduced models and indicate suitable applications for different levels of reduction. In particular, I recommend that unbranched reduced models be used for fast searches of parameter space given somatic input output data. The intrinsic electrical properties of neurons depend on the modifiable behavior of their ion channels. Obtaining a quality match between recorded voltage traces and the output of a conductance based compartmental neuron model depends on accurate estimates of the kinetic parameters of the channels in the biological neuron. Indeed, mismatches in channel kinetics may be detectable as failures to match somatic neural recordings when tuning model conductance densities. In my first aim, I showed that this is a task for which unbranched reduced models are ideally suited. Therefore, for my second aim I optimized unbranched reduced model parameters to match three experimentally characterized globus pallidus neurons by performing two stages of automated searches. In the first stage, I set conductance densities free and found that even the best matches to experimental data exhibited unavoidable problems. I hypothesized that these mismatches were due to limitations in channel model kinetics. To test this hypothesis, I performed a second stage of searches with free channel kinetics and observed decreases in the mismatches from the first stage. Additionally, some kinetic parameters consistently shifted to new values in multiple cells, suggesting the possibility for tailored improvements to channel models. Given my results and the potential for cell specific modulation of channel kinetics, I recommend that experimental kinetic data be considered as a starting point rather than as a gold standard for the development of neuron models.

Die vorliegende Arbeit befasst sich mit einer Reihe von Fragestellungen, die Forschungsfeldern wie rauschinduziertem Verhalten, Strukturbildung in aktiven Medien und Synchronisation nichlinearer Oszillatoren erwachsen. Die verwendeten nichtlinearen Modelle verfügen über erregbare, oszillatorische und bistabile Eigenschaften. Zusätzliche stochastische Fluktuationen tragen wesentlich zur Entstehung komplexer Dynamik bei. Modelliert wird, auf welche Weise sich extrazelluläre Kaliumkonzentration, gespeist von umliegenden Neuronen, auf die Aktivität dieser Neuronen auswirkt. Neben lokaler Dynamik wird die Ausbildung ausgedehnter Strukturen in einem heterogenem Medium analysiert. Die raum-zeitlichen Muster umfassen sowohl Wellenfronten und Spiralen als auch ungewöhnliche Strukturen, wie wandernde Cluster oder invertierte Wellen. Eine wesentliche Rolle bei der Ausprägung solcher Strukturen spielen die Randbedingungen des Systems. Sowohl für diskret gekoppelte bistabile Elemente als auch für kontinuierliche Fronten werden Methoden zur Berechnung von Frontgeschwindigkeiten bei fixierten Rändern vorgestellt. Typische Bifurkationen werden quantifiziert und diskutiert. Der Rückkopplungsmechanismus aus dem Modell neuronaler Einheiten und deren passiver Umgebung kann weiter abstrahiert werden. Ein Zweizustandsmodell wird über zwei Wartezeitverteilungen definiert, welche erregbares Verhalten widerspiegeln. Untersucht wird die instantane und die zeitverzögerte Antwort des Ensembles auf die Rückkopplung. Im Fall von Zeitverzögerung tritt eine Hopf-Bifurkation auf, die zu Oszillationen der mittleren Gesamtaktivität führt. Das letzte Kapitel befasst sich mit Diffusion und Transport von Brownschen Teilchen in einem raum-zeiltich periodischen Potential. Wieder sind es Synchronisationsmechanismen, die nahezu streuungsfreien Transport ermöglichen können. Für eine erhöhte effektiven Diffusion gelangen wir zu einer Abschätzung der maximierenden Parameter. / In this work problems are investigated that arises from resarch fields of noise induced dynamics, pattern formation in active media and synchronisation of self-sustained oscillators. The applied model systems exhibit excitable, oscillatory and bistable behavior as basic modes of nonlinear dynamics. Addition of stochastic fluctuations contribute to the appearance of complex behavior. The extracellular potassium concentration fed by surrounding activated neurons and the feeback to these neurons is modelled. Beside considering the local behavior, nucleation of spatially extended structures is studied. We find typical fronts and spirales as well as unusal patterns such as moving clusters and inverted waves. The boundary conditions of the considered system play an essential role in the formation process of such structures. We present methods to find expressions of the front velocity for discretely coupled bistable units as well as for the countinus front interacting with boundary values. Canonical bifurcation scenarios can be quantified. The feedback mechanism from the model for neuronal units can be generalized further. A two-state model is defined by two waiting time distributions representing excitable dynamics. We analyse the instantaneous and delayed response of the ensemble. In the case of delayed feedback a Hopf-bifurcation occur which lead to oscillations of the mean activity. In the last chapter the transport and diffusion of Brownian particles in a spatio-temporal oscillating potential is discussed. As a cause of nearly dispersionless transport synchronisations mechanisms can be identified. We find an estimation for parameter values which maximizes the effective diiffusion.

Neurons can respond to input by firing isolated action potentials or spikes. Sequences of spikes have been linked to the encoding of neuron input. However, many neurons also fire bursts; mechanistically distinct responses consisting of brief high-frequency spike firing. Bursts form separate response symbols but historically have not been thought to encode input. However, recent experimental evidence suggests that bursts can encode input in parallel with tonic spikes. The recognition of bursts as distinct encoding symbols raises important questions; these form the basic aims of this thesis: (1) What inputs do bursts encode? (2) Does burst structure provide extra information about different inputs. (3) Is burst coding robust against the presence of noise; an inherent property of all neural systems? (4) What mechanisms are responsible for burst input encoding? (5) How does burst coding manifest in in-vivo neurons. To answer these questions, bursting is studied using a combination of neuron models and in-vivo hippocampal neuron recordings. Models ranged from neuron-specific cell models to models belonging to three fundamentally different burst dynamic classes (unspecific to any neural region). These classes are defined using concepts from non-linear system theory. Together, analysing these model types with in-vivo recordings provides a specific and general analysis of burst encoding. For neuron-specific and unspecific models, a number of model types expressing different levels of biological realism are analysed. For the study of thalamic encoding, two models containing either a single simplified burst-generating current or multiple currents are used. For models simulating three burst dynamic classes, three further models of different biological complexity are used. The bursts generated by models and real neurons were analysed by assessing the input they encode using methods such as information theory, and reverse correlation. Modelled bursts were also analysed for their resilience to simulated neural noise. In all cases, inputs evoking bursts and tonic spikes were distinct. The structure of burst-evoking input depended on burst dynamic class rather than the biological complexity of models. Different n-spike bursts encoded different inputs that, if read by downstream cells, could discriminate complex input structure. In the thalamus, this n-spike burst code explains informative responses that were not due to tonic spikes. In-vivo hippocampal neurons and a pyramidal cell model both use the n-spike code to mark different LFP features. This n-spike burst may therefore be a general feature of bursting relevant to both model and in-vivo neurons. Bursts can also encode input corrupted by neural noise, often outperforming the encoding of single spikes. Both burst timing and internal structure are informative even when driven by strongly noise-corrupted input. Also, bursts induce input-dependent spike correlations that remain informative despite strong added noise. As a result, bursts endow their constituent spikes with extra information that would be lost if tonic spikes were considered the only informative responses.

Εκτός από τα Τεχνητά Νευρωνικά Δίκτυα, ένα άλλο παρεμφερές πρόβλημα είναι αυτό της μοντελοποίησης των δομικών και λειτουργικών χαρακτηριστικών διαφόρων τμημάτων του Κεντρικού Νευρικού Συστήματος καθώς και των διαφόρων εγκεφαλικών λειτουργιών. Στόχος αυτής της διπλωματικής είναι η δημιουργία ενός μοντέλου του φυσιολογικού νευρώνα και της συγκρότησης νευρωνικών δικτύων που εμπλέκονται σε κάποια εγκεφαλική λειτουργία. Στην ανάπτυξη του μοντέλου λήφθηκαν υπόψη τα ιδιαίτερα νευροανατομικά χαρακτηριστικά και νευροφυσιολογικά χαρακτηριαστικά και οι ιδιότητες που σχετίζονται με τις υπό μελέτη εγκεφαλικές καταστάσεις. Επίσης διερευνήθηκε η αλληλεπίδραση και η αναπτυσσόμενη δυναμική, τόσο σε κυτταρικό επίπεδο όσο και σε συστημικό επίπεδο, καθώς και η δυναμική αλληλεπίδραση νευρωνικών δικτύων. Πραγματοποιήθηκε μακροσκοπική προσέγγιση με τη χρήση μαθηματικών μοντέλων και αναπτύχθηκε ένα GUI περιβάλλον για τη διαχείριση του προγράμματος από το χρήστη. / Apart from the Artificial Neural Networks, another similar problem is the modeling of structural and functional characteristics of different parts of the Central Nervous System and the various brain functions. The aim of this diploma is to create a model of normal neuron and the establishment of neural networks involved in some brain function. In developing the model were taken into account the specific neuroanatomical and neurophysiological characteristics and properties related to the studied brain states. We also investigated the interaction and the growing momentum, both at the cellular level and system level, and the dynamic interaction of neural networks. An macroscopic approach using mathematical models and developed a GUI environment for the management of the program by the user.

Classical value estimation reinforcement learning algorithms do not perform very well in dynamic environments. On the other hand, the reinforcement learning of animals is quite flexible: they can adapt to dynamic environments very quickly and deal with noisy inputs very effectively. One feature that may contribute to animals' good performance in dynamic environments is that they learn and perceive the time to reward. In this research, we attempt to learn and perceive the time to reward and explore situations where the learned time information can be used to improve the performance of the learning agent in dynamic environments. The type of dynamic environments that we are interested in is that type of switching environment which stays the same for a long time, then changes abruptly, and then holds for a long time before another change. The type of dynamics that we mainly focus on is the time to reward, though we also extend the ideas to learning and perceiving other criteria of optimality, e.g. the discounted return, so that they can still work even when the amount of reward may also change. Specifically, both the mean and variance of the time to reward are learned and then used to detect changes in the environment and to decide whether the agent should give up a suboptimal action. When a change in the environment is detected, the learning agent responds specifically to the change in order to recover quickly from it. When it is found that the current action is still worse than the optimal one, the agent gives up this time's exploration of the action and then remakes its decision in order to avoid longer than necessary exploration. The results of our experiments using two real-world problems show that they have effectively sped up learning, reduced the time taken to recover from environmental changes, and improved the performance of the agent after the learning converges in most of the test cases compared with classical value estimation reinforcement learning algorithms. In addition, we have successfully used spiking neurons to implement various phenomena of classical conditioning, the simplest form of animal reinforcement learning in dynamic environments, and also pointed out a possible implementation of instrumental conditioning and general reinforcement learning using similar models.

Η παρούσα Διπλωματική Εργασία εστίασε το ενδιαφέρον της στην διερεύνηση των μοντέλων για νευρώνες (neuron models) ικανών να μιμηθούν την φυσική λειτουργία των βιολογικών νευρώνων. Συγκεκριμένα, έγινε μελέτη κάποιων μοντέλων για νευρώνες που παρουσιαστήκαν τα τελευταία χρόνια και στην συνέχεια, σχεδιάστηκε και υλοποιήθηκε ένα από τα μοντέλα αυτά κάνοντας χρήση κυκλωμάτων χαμηλής τάσης τροφοδοσίας στο πεδίο του υπερβολικού ημιτόνου (Sinh-Domain). Η γρήγορη ανάπτυξη της μικροηλεκτρονικής στην υλοποίηση συστημάτων υψηλής αξιοπιστίας και απόδοσης μικρού βάρους και όγκου όπως φορητών ηλεκτρονικών πολυμέσων, επικοινωνιών, βιοϊατρικών συσκευών, ωθεί στην σχεδίαση των ολοκληρωμένων κυκλωμάτων που τα απαρτίζουν με μειωμένη κατανάλωση ισχύος και κατ’ επέκταση χαμηλής τάσης τροφοδοσίας. Αρχικά, γίνεται μια εισαγωγή για τη σχεδίαση ολοκληρωμένων κυκλωμάτων για λειτουργία σε περιβάλλον χαμηλής τάσης τροφοδοσίας. Ακολούθως, δίνεται η περιγραφή της δομής και της λειτουργίας ενός βιολογικού νευρώνα και στην συνέχεια η περιγραφή των δύο βασικών μοντέλων νευρώνα (neuron models) που μελετήθηκαν στα πλαίσια της συγκεκριμένης εργασίας. Επίσης, παρουσιάζεται μία πρόσφατη υλοποίηση του βασικού μοντέλου νευρώνα των Mihalas και Niebur που ερευνάται στα πλαίσια της παρούσας εργασίας, στο πεδίο του λογαρίθμου καθώς και τα μοτίβα αιχμών τα οποία είναι ικανό να παράγει το συγκεκριμένο μοντέλο στο πεδίο του λογαρίθμου. Τον πυρήνα στην σχεδίαση των συγκεκριμένων μοντέλων για νευρώνες που μελετώνται, αποτελεί η τοπολογία του Tau-Cell. Η συγκεκριμένη βαθμίδα χρησιμοποιείται για την συστηματική σχεδίαση φίλτρων στο πεδίο του λογαρίθμου (Log-Domain filters). Έπειτα, αναλύεται η μέθοδος σχεδίασης κυκλωμάτων, και συγκεκριμένα φίλτρων, χαμηλής τάσης τροφοδοσίας στο πεδίο του υπερβολικού ημιτόνου (Sinh-Domain). Παρουσιάζονται οι βασικούς τελεστές καθώς και τα βασικά cells, για την σχεδίαση κυκλωμάτων στο πεδίο του υπερβολικού ημιτόνου (Sinh-Domain). Στην συνέχεια, περιγράφεται η σχεδίαση της τοπολογίας του Tau-Cell η οποία όπως αναφέραμε, αποτελεί τον πυρήνα στην υλοποίηση μοντέλων για νευρώνες, στο πεδίο του υπερβολικού ημιτόνου και επιβεβαιώνεται η ορθή λειτουργία της συγκεκριμένης βαθμίδας, με την σχεδίαση και εξομοίωση φίλτρων στο πεδίο του υπερβολικού ημιτόνου, με βασικό στοιχείο το Tau-Cell στο Analog Design Environment του λογισμικού της Cadence σε τεχνολογία της AMS CMOS 0.35μm. Αφότου έχει ολοκληρωθεί η σχεδίαση του Tau-Cell στο πεδίο του υπερβολικού ημιτόνου, περιγράφεται στην συνέχεια η υλοποίηση του μοντέλου νευρώνα των Mihalas και Niebur στο πεδίο του υπερβολικού ημιτόνου. Κάνοντας χρήση της βαθμίδας του Tau-Cell στο πεδίο του υπερβολικού ημιτόνου, γίνεται η υλοποίηση και στην συνέχεια η εξομοίωση, δύο βασικών κυκλωμάτων του μοντέλου, με βάση την ήδη υπάρχουσα υλοποίηση τους στο πεδίο του λογαρίθμου, ικανών να παράγουν διάφορα μοτίβα αιχμών (spiking patterns) με βάση το συγκεκριμένο μοντέλο του νευρώνα. Η ορθή λειτουργία των δύο αυτών κυκλωμάτων του μοντέλου με βάση τα μοτίβα αιχμών (spiking patterns) που είναι ικανά να παράγουν, επιβεβαιώνεται από τις εξομοιώσεις στο περιβάλλον του Analog Design Environment του λογισμικού της Cadence σε τεχνολογία της AMS CMOS 0.35μm. Τέλος, παρουσιάζεται η φυσική σχεδίαση (layout) των δύο βασικών κυκλωμάτων του μοντέλου νευρώνα καθώς και τα αποτελέσματα από την post-layout εξομοίωση του μοντέλου. Η φυσική σχεδίαση πραγματοποιήθηκε μέσω του λογισμικού Cadence το οποίο και περιέχει το περιβάλλον φυσικής σχεδίασης αναλογικών ηλεκτρονικών κυκλωμάτων Virtuoso Layout Editor. Η τεχνολογία που χρησιμοποιήθηκε αναφέρεται ως AMS C35D4 CMOS διαστάσεων 0.35μm. / This present M.Sc. Thesis is focused its interest in the study of neuron models that emulate the physical behavior of biological neurons. More specifically, we present a study of some neuron models that have been presented the last years and we proceed with the design and the implementation one of them using low voltage circuits in the Sinh-Domain. Τhe radical technological developments of microelectronics in the systems implementation with high reliability and performance, such as portable electronic devices for multimedia, communications and biomedical systems, demand the design of integrated circuits with reduced power consumption and thus low voltage supply. Initially, an introduction for the design of integrated circuits in low voltage environment is given and, also, the description of the structure and behavior of a biological neuron. Next, an analysis of two recently introduced neuron models realized in the Log-Domain, from which the Mihalas and Niebur neuron model constitutes the basic model studied in the context of this work and, also, the basic spiking patterns, that this implementation of the Mihalas-Niebur neuron model is capable of producing. The core in the implementation of this neuron models, is the topology of Tau-Cell. The topology of Tau-Cell is used for the systematic design of filters in the Log-Domain. Thereafter, is given an analysis of the method of designing low voltage circuits and more specifically filters, in the Sinh-Domain. The basic operators and the principal cells, for designing circuits in the Sinh-Domain are presented. Then, the design and implementation of the Tau-Cell topology which as mentioned is the core for the implementation of neuron models, is realized in the Sinh-Domain. The proper operation of this topology is confirmed through the design and simulation of filters in the Sinh-Domain, in the Analog Design Environment of Cadence using the AMS CMOS 0.35μm technology. After the design of the Tau-Cell in the Sinh-Domain, we continue with the implementation of the Mihalas-Niebur neuron model. Using the topology of Tau-Cell in the Sinh-Domain, we proceed with the implementation and the simulation of the basic two topologies of the neuron model based on the existing implementation in the Log-Domain. The implemented topologies of the neuron are capable of producing spiking patterns based to the Mihalas-Niebur neuron model. The proper operation of these topologies based on the spiking patterns that are capable of producing, is confirmed through the design and simulation in the Sinh-Domain, in the Analog Design Environment of Cadence using the AMS CMOS 0.35μm technology. Finally, is presented the layout design of the main two topologies of the neuron model and also the results of the post-layout simulations. The layout was conducted via the Cadence software through Virtuoso Layout Editor. The technology used is referred as AMS C35D4 CMOS in 0.35μm dimensions.

In vielen Kommunikationssytemen wird Information durch die zeitliche Struktur von Signalen kodiert. Ein neuronales System, welches rhythmische Signale verarbeitet, sollte davon profitieren, seine inhärenten Filtereigenschaften den Frequenzcharakteristika dieser Signale anzupassen. Die Grille Gryllus bimaculatus stellt ein einfaches biologisches System dar, für welches nur wenige, spezifische Modulationsfrequenzen verhaltensrelevant sind. Ich habe einzelne Neuronen im peripheren und höheren auditorischen System der Grille hinsichtlich einer möglichen Anpassung auf diese Frequenzen untersucht. Hierfür habe ich extrazelluläre Elektrophysiologie mit verschiedenen Stimulationsparadigmen kombiniert, welche auf amplitudenmodulierten Tönen basierten. Die Analyse der experimentellen Daten ergab, dass bereits in der auditorischen Peripherie einige der untersuchten Neurone Bandpasseigenschaften aufwiesen, da sie verhaltensrelevante Modulationsfrequenzen mit den höchsten Feuerraten beantworteten. Anhand einfacher mathematischer Modelle demonstriere ich, wie weitverbreitete, zellintrinsische und netzwerkbasierte Mechanismen die beobachteten Feuerratenresonanzen erklären könnten. Diese Mechanismen umfassen unterschwellige Resonanz von Membranströmen, aktivitätsabhängige Adaptation, sowie das Zusammenwirken von Exzitation und Inhibition. Ich zeige, wie eine serielle Kombination solcher elementarer Filter die deutliche Selektivität im Verhalten der Grille erklären könnte, ohne dabei auf ein dediziertes Filterelement zurückzugreifen. Allgegenwärtige neuronale Mechanismen könnten demnach eine dezentralisierte Filterkaskade in einem hochspezialisierten und größenbeschränkten neuronalen System begründen. / In many communication systems, information is encoded in the temporal pattern of signals. For rhythmic signals that carry information in specific frequency bands, a neuronal system may profit from tuning its inherent filtering properties towards a peak sensitivity in the respective frequency range. The cricket Gryllus bimaculatus is a simple biological system for which only a narrow range of modulation frequencies is behaviorally relevant. I examined individual neurons in the peripheral and higher auditory system for tuning towards specific temporal parameters and modulation frequencies. To this end, I combined extracellular electrophysiology with different stimulation paradigms involving amplitude-modulated sounds. Analysis of the experimental data revealed that—even in the auditory periphery—some of the examined neurons acted as tuned band-pass filters, yielding highest firing-rates for behaviorally relevant modulation frequencies. Using simple computational models, I demonstrate how common, cell-intrinsic or network-based mechanisms could account for the experimentally observed firing-rate resonances. These mechanisms include subthreshold resonances, spike-triggered adaptation, as well as the interplay of excitation and inhibition. I present how a serial combination of such elementary filters could explain the strong selectivity evident in the cricket’s behavior—without the need for a dedicated filter element. Pervasive neuronal mechanisms could therefore constitute an efficient, distributed frequency filter in a highly specialized, size-constrained neuronal system.

In dieser Arbeit werden mithilfe der Fokker-Planck-Gleichung die Statistiken, vor allem die Leistungsspektren, von Punktprozessen berechnet, die von mehrdimensionalen Integratorneuronen [Engl. integrate-and-fire (IF) neuron], Netzwerken von IF Neuronen und Entscheidungsfindungsmodellen erzeugt werden. Im Gehirn werden Informationen durch Pulszüge von Aktionspotentialen kodiert. IF Neurone mit radikal vereinfachter Erzeugung von Aktionspotentialen haben sich in Studien die auf Pulszeiten fokussiert sind als Standardmodelle etabliert. Eindimensionale IF Modelle können jedoch beobachtetes Pulsverhalten oft nicht beschreiben und müssen dazu erweitert werden. Im erste Teil dieser Arbeit wird eine Theorie zur Berechnung der Pulszugleistungsspektren von stochastischen, multidimensionalen IF Neuronen entwickelt. Ausgehend von der zugehörigen Fokker-Planck-Gleichung werden partiellen Differentialgleichung abgeleitet, deren Lösung sowohl die stationäre Wahrscheinlichkeitsverteilung und Feuerrate, als auch das Pulszugleistungsspektrum beschreibt. Im zweiten Teil wird eine Theorie für große, spärlich verbundene und homogene Netzwerke aus IF Neuronen entwickelt, in der berücksichtigt wird, dass die zeitlichen Korrelationen von Pulszügen selbstkonsistent sind. Neuronale Eingangströme werden durch farbiges Gaußsches Rauschen modelliert, das von einem mehrdimensionalen Ornstein-Uhlenbeck Prozess (OUP) erzeugt wird. Die Koeffizienten des OUP sind vorerst unbekannt und sind als Lösung der Theorie definiert. Um heterogene Netzwerke zu untersuchen, wird eine iterative Methode erweitert. Im dritten Teil wird die Fokker-Planck-Gleichung auf Binärentscheidungen von Diffusionsentscheidungsmodellen [Engl. diffusion-decision models (DDM)] angewendet. Explizite Gleichungen für die Entscheidungszugstatistiken werden für den einfachsten und analytisch lösbaren Fall von der Fokker-Planck-Gleichung hergeleitet. Für nichtliniear Modelle wird die Schwellwertintegrationsmethode erweitert. / This thesis is concerned with the calculation of statistics, in particular the power spectra, of point processes generated by stochastic multidimensional integrate-and-fire (IF) neurons, networks of IF neurons and decision-making models from the corresponding Fokker-Planck equations. In the brain, information is encoded by sequences of action potentials. In studies that focus on spike timing, IF neurons that drastically simplify the spike generation have become the standard model. One-dimensional IF neurons do not suffice to accurately model neural dynamics, however, the extension towards multiple dimensions yields realistic behavior at the price of growing complexity. The first part of this work develops a theory of spike-train power spectra for stochastic, multidimensional IF neurons. From the Fokker-Planck equation, a set of partial differential equations is derived that describes the stationary probability density, the firing rate and the spike-train power spectrum. In the second part of this work, a mean-field theory of large and sparsely connected homogeneous networks of spiking neurons is developed that takes into account the self-consistent temporal correlations of spike trains. Neural input is approximated by colored Gaussian noise generated by a multidimensional Ornstein-Uhlenbeck process of which the coefficients are initially unknown but determined by the self-consistency condition and define the solution of the theory. To explore heterogeneous networks, an iterative scheme is extended to determine the distribution of spectra. In the third part, the Fokker-Planck equation is applied to calculate the statistics of sequences of binary decisions from diffusion-decision models (DDM). For the analytically tractable DDM, the statistics are calculated from the corresponding Fokker-Planck equation. To determine the statistics for nonlinear models, the threshold-integration method is generalized.

Die vorliegende Arbeit beschäftigt sich mit Mechanismen, die in Einzelzellmodellen zu einer frequenzabhängigen Informationsübertragung führen können. Um dies zu untersuchen, werden Methoden aus der theoretischen Physik (Statistische Physik) und der Informationstheorie angewandt. Die Informationsfilterung in mehreren stochastischen Neuronmodellen, in denen unterschiedliche Mechanismen zur Informationsfilterung führen können, werden numerisch und, falls möglich, analytisch untersucht. Die Bandbreite der betrachteten Modelle erstreckt sich von reduzierten strombasierten ’Integrate-and-Fire’ (IF) Modellen bis zu biophysikalisch realistischeren leitfähigkeitsbasierten Modellen. Anhand numerischer Untersuchungen wird aufgezeigt, dass viele Varianten der IF-Neuronenmodelle vorzugsweise Information über langsame Anteile eines zeitabhängigen Eingangssignals übertragen. Der einfachste Vertreter der oben genannten Klasse der IF-Neuronmodelle wird dahingehend erweitert, dass ein Konzept von neuronalem ’Gedächtnis’, vermittelst positiver Korrelationen zwischen benachbarten Intervallen aufeinander- folgender Spikes, integriert wird. Dieses Model erlaubt eine analytische störungstheoretische Untersuchung der Auswirkungen positiver Korrelationen auf die Informationsfilterung. Um zu untersuchen, wie sich sogenannte ’unterschwelligen Resonanzen’ auf die Signalübertragung auswirken, werden Neuronenmodelle mit verschiedenen Nichtlinearitäten anhand numerischer Computersimulationen analysiert. Abschließend wird die Signalübertragung in einem neuronalen Kaskadensystem, bestehend aus linearen und nichtlinearen Elementen, betrachtet. Neuronale Nichtlinearitäten bewirken eine gegenläufige Abhängigkeit (engl. "trade-off") zwischen qualitativer, d.h. frequenzselektiver, und quantitativer Informations-übertragung, welche in allen von mir untersuchten Modellen diskutiert wird. Diese Arbeit hebt die Gewichtigkeit von Nichtlinearitäten in der neuronalen Informationsfilterung hervor. / Neurons transmit information about time-dependent input signals via highly non-linear responses, so-called action potentials or spikes. This type of information transmission can be frequency-dependent and allows for preferences for certain stimulus components. A single neuron can transmit either slow components (low pass filter), fast components (high pass filter), or intermediate components (band pass filter) of a time-dependent input signal. Using methods developed in theoretical physics (statistical physics) within the framework of information theory, in this thesis, cell-intrinsic mechanisms are being investigated that can lead to frequency selectivity on the level of information transmission. Various stochastic single neuron models are examined numerically and, if tractable analytically. Ranging from simple spiking models to complex conductance-based models with and without nonlinearities, these models include integrator as well as resonator dynamics. First, spectral information filtering characteristics of different types of stochastic current-based integrator neuron models are being studied. Subsequently, the simple deterministic PIF model is being extended with a stochastic spiking rule, leading to positive correlations between successive interspike intervals (ISIs). Thereafter, models are being examined which show subthreshold resonances (so-called resonator models) and their effects on the spectral information filtering characteristics are being investigated. Finally, the spectral information filtering properties of stochastic linearnonlinear cascade neuron models are being researched by employing different static nonlinearities (SNLs). The trade-off between frequency-dependent signal transmission and the total amount of transmitted information will be demonstrated in all models and constitutes a direct consequence of the nonlinear formulation of the models.