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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.

Physical systems far from thermo dynamic equilibrium present excitability and irreversibility. The excitability is responsible for the great sensitivity of these systems to external stimuli while the irreversibility is associated with energy dissipation. The thermal fluctuations, inevitable in any real system, arise due to the interaction between many particles of the system. For such systems one of the best approaches is given by the non-equilibrium Statistical Mechanics, since it is virtually impossible an individualized approach of the motion equations. Many works in the current literature use a Gaussian stochastic modeling (without correlations) to represent the fluctuations. However, there is a growing number of studies reporting the occurrence of correlated fluctuations, mainly related to biological systems. In this thesis we investigate the influence of non-Gaussian stochastic distribution on the properties for two representative excitable models. In the first model we study the influence of distribution on the neural dynamics through the stochastic resonance (SR) mechanism. In the second model we approach the ratchet effect (RE) on directed transport of particles. In both systems we use a non-Gaussian power-law distributed noise obtained through a random multiplicative process (RMP). This process allows a fine tuning of the asymptotic power-law decay exponent. The optimization conditions are reported. In particular, we show that the optimization conditions for resonance and directed transport in Brownian ratchets are reached for a finit decay exponent of the stochastic distribution that represents a Strong non-Gaussian character. As non-Gaussian fluctuations occur with great frequency in natural systems, we believe that the non-Gaussian character can optimize the efficiency on the stochastic transport mechanisms in micro and nanoscale. / Fundação de Amparo a Pesquisa do Estado de Alagoas / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Sistemas físicos fora do equilíbrio termo dinâmico apresentam excitabilidade e irreversibilidade. A excitabilidade é responsável pela grande sensibilidade desses sistemas a estímulos externos enquanto a irreversibilidade está asso ciada à dissipação de energia. As flutuações térmicas, inevitáveis em qualquer sistema real, surgem devido à interação entre as inúmeras partículas do meio. Para tais sistemas uma das melhores abordagens é dada pela Mecânica Estatística de não-equilíbrio, uma vez que é praticamente impossível uma abordagem individualizada das equações de movimento. Muitos trabalhos na literatura atual utilizam uma modelagem estocástica gaussiana (sem correlação) para representar as flutuações. No entanto, há um número crescente de trabalhos que relatam a ocorrência de flutuações correlacionadas, principalmente em sistemas biológicos. Nesta tese nós investigamos a influência da distribuição estocástica não-gaussiana sobre as propriedades de dois modelos excitáveis representativos. No primeiro, estudamos a influência da distribuição sobre a dinâmica neural através do mecanismo de ressonância estocástica (RE). No segundo, abordamos o mecanismo do efeito catraca (EC) sobre o transporte direcionado de partículas. Nos dois sistemas utilizamos um ruído colorido não-gaussiano com distribuição tipo lei de potência obtido através de um processo multiplicativo aleatório (PMA). Esse processo permite o ajuste no do expoente de decaimento assintótico da lei de potência. As condições de otimização são relatadas. Em particular, obtivemos que as condições de otimização para a ressonância e para o transporte direcionado em catracas brownianas são atingidas para um valor finito do expoente da distribuição estocástica que representa um caráter fortemente não-gaussiano. Como flutuações não-gaussianas o correm com muita frequência nos sistemas naturais, acreditamos que o caráter não-gaussiano pode otimizar a eficiência dos mecanismos estocásticos de transporte em micro e nanoescala.

Wir untersuchen globale dynamische Phänomene, die sich von dem Zusammenspiel zwischen Netzwerktopologie und Dynamik der einzelnen Elementen ergeben. Im ersten Teil untersuchen wir relativ kleine strukturierte Netzwerke mit überschaubarer Komplexität. Als geeigneter theoretischer Rahmen für erregbare Systeme verwenden wir das Kuramoto und Shinomoto Modell der sinusförmig-gekoppelten "aktiven Rotatoren" und studieren das Kollektivverhalten des Systems in Bezug auf Synchronisation. Wir besprechen die Einschränkungen, die durch die Netzwerktopologie auf dem Fluss im Phasenraum des Systems gestellt werden. Insbesondere interessieren wir uns für die Stabilitätseigenschaften von Fluss-invarianten Polydiagonalen und die Entwicklungen von Attraktoren in den Parameterräume solcher Systeme. Wir untersuchen zweidimensionale hexagonale Gitter mit periodischen Randbedingungen. Wir untersuchen allgemeine Bedingungen auf der Adjazenzmatrix von Netzwerken, die die Watanabe-Strogatz Reduktion ermöglichen, und diskutieren verschiedene Beispiele. Schließlich präsentieren wir eine generische Analyse der Bifurkationen, die auf der Untermannigfaltigkeit des Watanabe-Strogatz reduzierten Systems stattfinden. Im zweiten Teil der Arbeit untersuchen wir das globale dynamische Phänomen selbstanhaltender Aktivität (self-sustained activity / SSA) in neuronalen Netzwerken. Wir betrachten Netzwerke mit hierarchischer und modularer Topologie , umfassend Neuronen von verschiedenen kortikalen elektrophysiologischen Zellklassen. Wir zeigen, dass SSA Zustände mit ähnlich zu den experimentell beobachteten Eigenschaften existieren. Durch Analyse der Dynamik einzelner Neuronen sowie des Phasenraums des gesamten Systems erläutern wir die Rolle der Inhibierung. Darüber hinaus zeigen wir, dass beide Netzwerkarchitektur, in Bezug auf Modularität, sowie Mischung aus verschiedenen Neuronen, in Bezug auf die unterschiedlichen Zellklassen, einen Einfluss auf die Lebensdauer der SSA haben. / In this work we study global dynamical phenomena which emerge as a result of the interplay between network topology and single-node dynamics in systems of excitable elements. We first focus on relatively small structured networks with comprehensible complexity in terms of graph-symmetries. We discuss the constraints posed by the network topology on the dynamical flow in the phase space of the system and on the admissible synchronized states. In particular, we are interested in the stability properties of flow invariant polydiagonals and in the evolutions of attractors in the parameter spaces of such systems. As a suitable theoretical framework describing excitable elements we use the Kuramoto and Shinomoto model of sinusoidally coupled “active rotators”. We investigate plane hexagonal lattices of different size with periodic boundary conditions. We study general conditions posed on the adjacency matrix of the networks, enabling the Watanabe-Strogatz reduction, and discuss different examples. Finally, we present a generic analysis of bifurcations taking place on the submanifold associated with the Watanabe-Strogatz reduced system. In the second part of the work we investigate a global dynamical phenomenon in neuronal networks known as self-sustained activity (SSA). We consider networks of hierarchical and modular topology, comprising neurons of different cortical electrophysiological cell classes. In the investigated neural networks we show that SSA states with spiking characteristics, similar to the ones observed experimentally, can exist. By analyzing the dynamics of single neurons, as well as the phase space of the whole system, we explain the importance of inhibition for sustaining the global oscillatory activity of the network. Furthermore, we show that both network architecture, in terms of modularity level, as well as mixture of excitatory-inhibitory neurons, in terms of different cell classes, have influence on the lifetime of SSA.

In dieser Arbeit werden zwei wesentliche neue Ergebnisse vorgestellt. Das erste bezieht sich auf die olfaktorische Kodierung und das zweite auf das sensorische Gedaechtnis. Beide Phaenomene werden am Beispiel des Gehirns der Honigbiene untersucht. In Bezug auf die olfaktorische Kodierung zeige ich, dass die neuronale Dynamik waehrend der Stimulation im Antennallobus duftspezifische Trajektorien beschreibt, die in duftspezifischen Attraktoren enden. Das Zeitinterval, in dem diese Attraktoren erreicht werden, betraegt unabhaengig von der Identitaet und der Konzentration des Duftes ungefaehr 800 ms. Darueber hinaus zeige ich, dass Support-Vektor Maschinen, und insbesondere Perzeptronen, ein realistisches und biologisches Model der Wechselwirkung zwischen dem Antennallobus (dem kodierenden Netwerk) und dem Pilzkoerper (dem dekodierenden Netzwerk) darstellen. Dieses Model kann sowohl Reaktionszeiten von ca. 300 ms als auch die Invarianz der Duftwahrnehmung gegenueber der Duftkonzentration erklaeren. In Bezug auf das sensorische Gedaechtnis zeige ich, dass eine einzige Stimulation ohne Belohnung dem Hebbschen Postulat folgend Veraenderungen der paarweisen Korrelationen zwischen Glomeruli induziert. Ich zeige, dass diese Veranderungen der Korrelationen bei 2/3 der Bienen ausreichen, um den letzten Stimulus zu bestimmen. In der zweiten Minute nach der Stimulation ist eine erfolgreiche Bestimmung des Stimulus nur bei 1/3 der Bienen moeglich. Eine Hauptkomponentenanalyse der spontanen Aktivitaet laesst erkennen, dass das dominante Muster des Netzwerks waehrend der spontanen Aktivitaet nach, aber nicht vor der Stimulation das duftinduzierte Aktivitaetsmuster bei 2/3 der Bienen nachbildet. Man kann deshalb die duftinduzierten (Veraenderungen der) Korrelationen als Spuren eines Kurzzeitgedaechtnisses bzw. als Hebbsche "Reverberationen" betrachtet werden. / Two major novel results are reported in this work. The first concerns olfactory coding and the second concerns sensory memory. Both phenomena are investigated in the brain of the honeybee as a model system. Considering olfactory coding I demonstrate that the neural dynamics in the antennal lobe describe odor-specific trajectories during stimulation that converge to odor-specific attractors. The time interval to reach these attractors is, regardless of odor identity and concentration, approximately 800 ms. I show that support-vector machines and, in particular perceptrons provide a realistic and biological model of the interaction between the antennal lobe (coding network) and the mushroom body (decoding network). This model can also account for reaction-times of about 300 ms and for concentration invariance of odor perception. Regarding sensory memory I show that a single stimulation without reward induces changes of pairwise correlation between glomeruli in a Hebbian-like manner. I demonstrate that those changes of correlation suffice to retrieve the last stimulus presented in 2/3 of the bees studied. Succesful retrieval decays to 1/3 of the bees within the second minute after stimulation. In addition, a principal-component analysis of the spontaneous activity reveals that the dominant pattern of the network during the spontaneous activity after, but not before stimulation, reproduces the odor-induced activity pattern in 2/3 of the bees studied. One can therefore consider the odor-induced (changes of) correlation as traces of a short-term memory or as Hebbian reverberations.