Past decades of auditory research have identified several acoustic features that influence perceptual organization of sound, in particular, the frequency of tones and the rate of presentation. One class of stimuli that have been intensively studied are sequences of tones that alternate in frequency. They are typically presented in patterns of repeating doublets ABAB… or repeating triplets ABA-ABA-... where the symbol “-” stands for a gap of silence between triplets repeats. The duration of each tone or silence is typically tens to hundreds of milliseconds, and listeners hearing the sequence perceive either one auditory object ("stream integration") or two separate auditory objects (“stream segregation”). Animal studies have characterized single- and multi- unit neural activity and event-related local field potentials while systematically varying frequency separation between tones (ΔF) or the presentation rate (PR). They found that the B tone responses in doublets were differentially suppressed with increasing PR and that the B tones responses in triplets decreased with larger ΔF. However, the neural mechanisms underlying these animal data have yet to be explained. In this study, we built an integrate-and-fire network model of the primary auditory cortex (AC) that accurately reproduced the experimental results. Then, we extended the model to account for basic spectro-temporal features of electrocorticography (ECoG) recordings from the posteriomedial part of the Heschl's gyrus (HGPM; cortical area equivalent to the AC of monkeys), obtained from humans listening to sequences of triplets ABA-. Finally, we constructed a firing rate reduced model of the proposed integrate-and-fire network and analyzed its dynamics as function of parameters. A large network of voltage-dependent leaky integrate-and-fire neurons (3600 excitatory, 900 inhibitory) was constructed to simulate neural activity from layers 3/4 of AC during streaming of tone triplets. Parameters describing synaptic and membrane properties were based on experimental data from early studies of AC. Network structure assumed spatially-dependent probability of connections and tonotopic organization. Subpopulations of neurons were tuned to different frequencies along the tonotopic map. In-silico recordings were performed during the presentation of long sequences of triplets and/or doublets. The network’s output was derived with two types of measurements in mind: spiking activity of individual neurons and/or local populations of neurons, and local field potentials. The network spiking neural activity reproduced reliably data reports, including dependence of responses to the B tone in triplets ABA- on stimulus parameter ΔF. Approximations of average evoked potentials (AEPs) from ECoG signals recorded at four depth contacts placed over human HGPM during auditory streaming of triplets were also obtained.

Neurons communicate by action potentials. This process can be described by very detailed biochemical models of neuronal membrane and its channels, or by simpler phenomenological models of membrane potential (integrate-and- fire models) or even by very abstract models when only time of spikes are considered. We took one particular description - stochastic leaky integrate-and-fire model - and compared it with recorded in-vivo intracellular activity of the neuron. We estimated parameters of this model, compared how the model simulation corresponds with a real neuron. It can be concluded that the data are generally consistent with the model. At a more abstract level of description, the spike trains are analyzed without considering exact membrane voltage and one asks how the external stimulus is encoded in the spike train emitted by neurons. There are many neuronal codes described in literature and we focused on the open problem of neural code responsible for spatial hearing in mammals. Several theories explaining the experimental findings have been proposed and we suggest a specific variant of so called slope-encoding model. Neuronal circuit mimick- ing auditory pathway up to the first binaural neuron was constructed and experimental results were reproduced. Finally, we estimated the minimal number of such...

Neurons communicate by action potentials. This process can be described by very detailed biochemical models of neuronal membrane and its channels, or by simpler phenomenological models of membrane potential (integrate-and- fire models) or even by very abstract models when only time of spikes are considered. We took one particular description - stochastic leaky integrate-and-fire model - and compared it with recorded in-vivo intracellular activity of the neuron. We estimated parameters of this model, compared how the model simulation corresponds with a real neuron. It can be concluded that the data are generally consistent with the model. At a more abstract level of description, the spike trains are analyzed without considering exact membrane voltage and one asks how the external stimulus is encoded in the spike train emitted by neurons. There are many neuronal codes described in literature and we focused on the open problem of neural code responsible for spatial hearing in mammals. Several theories explaining the experimental findings have been proposed and we suggest a specific variant of so called slope-encoding model. Neuronal circuit mimick- ing auditory pathway up to the first binaural neuron was constructed and experimental results were reproduced. Finally, we estimated the minimal number of such...

One-dimensional neural networks comprised of large numbers of Integrate-and-Fire neurons have been widely used to model electrical activity propagation in neural slices. Despite these efforts, the vast majority of these computational models have no analytical solutions. Consequently, my Ph.D. research focuses on a specific class of homogeneous Integrate-and-Fire neural network, for which analytical solutions of network dynamics can be derived. One crucial analytical finding is that the traveling wave acceleration quadratically depends on the instantaneous speed of the activity propagation, which means that two speed solutions exist in the activities of wave propagation: one is fast-stable and the other is slow-unstable. Furthermore, via this property, we analytically compute temporal-spatial spiking dynamics to help gain insights into the stability mechanisms of traveling wave propagation. Indeed, the analytical solutions are in perfect agreement with the numerical solutions. This analytical method also can be applied to determine the effects induced by a non-conductive gap of brain tissue and extended to more general synaptic connectivity functions, by converting the evolution equations for network dynamics into a low-dimensional system of ordinary differential equations. Building upon these results, we investigate how periodic inhomogeneities affect the dynamics of activity propagation. In particular, two types of periodic inhomogeneities are studied: alternating regions of additional fixed excitation and inhibition, and cosine form inhomogeneity. Of special interest are the conditions leading to propagation failure. With similar analytical procedures, explicit expressions for critical speeds of activity propagation are obtained under the influence of additional inhibition and excitation. However, an explicit formula for speed modulations is difficult to determine in the case of cosine form inhomogeneity. Instead of exact solutions from the system of equations, a series of speed approximations are constructed, rendering a higher accuracy with a higher order approximation of speed.

Redes complexas têm sido cada vez mais utilizadas para a modelagem e análise dos mais diversos sistemas da natureza. Um dos tópicos mais estudados na área de redes está relacionado com a identificação e caracterização de grupos de nós mais conectados entre si do que com o restante da rede, chamados de comunidades. Neste trabalho, mostramos que comunidades podem ser caracterizadas por quatro classes gerais de propriedades, relacionadas com a topologia interna, dinâmica interna, fronteira topológica, e fronteira dinâmica das comunidades. Verificamos como estas diferentes características influenciam em dinâmicas ocorrendo sobre a rede. Em especial, estudamos o inter-relacionamento entre a topologia e a dinâmica das comunidades para cada uma dessas quatro classes de atributos. Mostramos que certas propriedades provocam a alteração desse inter-relacionamento, dando origem ao que chamamos de comportamento específico de comunidades. De forma a apresentarmos e analisarmos este conceito nos quatro casos considerados, estudamos as seguintes combinações topológicas e dinâmicas. Na primeira, investigamos o passeio aleatório tradicional ocorrendo sobre redes direcionadas, onde mostramos que a direção das conexões entre comunidades é o principal fator de alteração no relacionamento topologia-dinâmica. Aplicamos a metodologia proposta em uma rede real, definida por módulos corticais de animais do gênero Macaca. O segundo caso estudado aborda o passeio aleatório enviesado ocorrendo sobre redes não direcionadas. Mostramos que o viés associado às transições da dinâmica se tornam cada vez mais relevantes com o aumento da modularidade da rede. Verificamos também que a descrição da dinâmica a nível de comunidades possibilita modelarmos com boa acurácia o fluxo de passageiros em aeroportos. A terceira análise realizada envolve a dinâmica neuronal integra-e-dispara ocorrendo sobre comunidades geradas segundo o modelo Watts-Strogatz. Mostramos que as comunidades podem possuir não apenas diferentes níveis de ativação dinâmica, como também apresentar diferentes regularidades de sinal dependendo do parâmetro de reconexão utilizado na criação das comunidades. Por último, estudamos a influência das posições de conexões inibitórias na dinâmica integra-e-dispara, onde mostramos que a inibição entre comunidades dá origem a interessantes variações na ativação global da rede. As análises realizadas revelam a importância de, ao modelarmos sistemas reais utilizando redes complexas, considerarmos alterações de parâmetros do modelo na escala de comunidades. / There has been a growing interest in modeling diverse types of real-world systems through the tools provided by complex network theory. One of the main topics of research in this area is related to the identification and characterization of groups, or communities, of nodes more densely connected between themselves than with the rest of the network. We show that communities can be characterized by four general classes of features, associated with the internal topology, internal dynamics, topological border, and dynamical border of the communities. We verify that these characteristics have direct influence on the dynamics taking place over the network. Particularly, for each considered class we study the interdependence between the topology and the dynamics associated with each network community. We show that some of the studied properties can influence the topology-dynamics interdependence, inducing what we call the communities specific behavior. In order to present and characterize this concept on the four considered classes, we study the following combinations of network topology and dynamics. We first investigate traditional random walks taking place on a directed network. We demonstrate that, for this dynamics, the direction of the edges between communities represents the main method for the modification of the topology-dynamics relationship. We apply the developed approach on a real-world network, defined by the connectivity between cortical regions in primates of the Macaca genus. The second studied case considers the biased random walk on undirected networks. We demonstrate that the transition bias of this dynamics becomes more relevant for higher network modularity. In addition, we show that the biased random walk can be used to model with good accuracy the passenger flow inside the communities of two airport networks. The third analysis is done on a neuronal dynamics, called integrate-and-fire, applied to networks composed of communities generated by the Watts-Strogatz model. We show that the considered communities can not only posses distinct dynamical activation levels, but also yield different signal regularity. Lastly, we study the influence of the positions of inhibitory connections on the integrate-and-fire dynamics. We show that inhibitory connections placed between communities can have a non-trivial influence on the global behavior of the dynamics. The current study reveals the importance of considering parameter variations of network models at the scale of communities.

Redes complexas têm sido cada vez mais utilizadas para a modelagem e análise dos mais diversos sistemas da natureza. Um dos tópicos mais estudados na área de redes está relacionado com a identificação e caracterização de grupos de nós mais conectados entre si do que com o restante da rede, chamados de comunidades. Neste trabalho, mostramos que comunidades podem ser caracterizadas por quatro classes gerais de propriedades, relacionadas com a topologia interna, dinâmica interna, fronteira topológica, e fronteira dinâmica das comunidades. Verificamos como estas diferentes características influenciam em dinâmicas ocorrendo sobre a rede. Em especial, estudamos o inter-relacionamento entre a topologia e a dinâmica das comunidades para cada uma dessas quatro classes de atributos. Mostramos que certas propriedades provocam a alteração desse inter-relacionamento, dando origem ao que chamamos de comportamento específico de comunidades. De forma a apresentarmos e analisarmos este conceito nos quatro casos considerados, estudamos as seguintes combinações topológicas e dinâmicas. Na primeira, investigamos o passeio aleatório tradicional ocorrendo sobre redes direcionadas, onde mostramos que a direção das conexões entre comunidades é o principal fator de alteração no relacionamento topologia-dinâmica. Aplicamos a metodologia proposta em uma rede real, definida por módulos corticais de animais do gênero Macaca. O segundo caso estudado aborda o passeio aleatório enviesado ocorrendo sobre redes não direcionadas. Mostramos que o viés associado às transições da dinâmica se tornam cada vez mais relevantes com o aumento da modularidade da rede. Verificamos também que a descrição da dinâmica a nível de comunidades possibilita modelarmos com boa acurácia o fluxo de passageiros em aeroportos. A terceira análise realizada envolve a dinâmica neuronal integra-e-dispara ocorrendo sobre comunidades geradas segundo o modelo Watts-Strogatz. Mostramos que as comunidades podem possuir não apenas diferentes níveis de ativação dinâmica, como também apresentar diferentes regularidades de sinal dependendo do parâmetro de reconexão utilizado na criação das comunidades. Por último, estudamos a influência das posições de conexões inibitórias na dinâmica integra-e-dispara, onde mostramos que a inibição entre comunidades dá origem a interessantes variações na ativação global da rede. As análises realizadas revelam a importância de, ao modelarmos sistemas reais utilizando redes complexas, considerarmos alterações de parâmetros do modelo na escala de comunidades. / There has been a growing interest in modeling diverse types of real-world systems through the tools provided by complex network theory. One of the main topics of research in this area is related to the identification and characterization of groups, or communities, of nodes more densely connected between themselves than with the rest of the network. We show that communities can be characterized by four general classes of features, associated with the internal topology, internal dynamics, topological border, and dynamical border of the communities. We verify that these characteristics have direct influence on the dynamics taking place over the network. Particularly, for each considered class we study the interdependence between the topology and the dynamics associated with each network community. We show that some of the studied properties can influence the topology-dynamics interdependence, inducing what we call the communities specific behavior. In order to present and characterize this concept on the four considered classes, we study the following combinations of network topology and dynamics. We first investigate traditional random walks taking place on a directed network. We demonstrate that, for this dynamics, the direction of the edges between communities represents the main method for the modification of the topology-dynamics relationship. We apply the developed approach on a real-world network, defined by the connectivity between cortical regions in primates of the Macaca genus. The second studied case considers the biased random walk on undirected networks. We demonstrate that the transition bias of this dynamics becomes more relevant for higher network modularity. In addition, we show that the biased random walk can be used to model with good accuracy the passenger flow inside the communities of two airport networks. The third analysis is done on a neuronal dynamics, called integrate-and-fire, applied to networks composed of communities generated by the Watts-Strogatz model. We show that the considered communities can not only posses distinct dynamical activation levels, but also yield different signal regularity. Lastly, we study the influence of the positions of inhibitory connections on the integrate-and-fire dynamics. We show that inhibitory connections placed between communities can have a non-trivial influence on the global behavior of the dynamics. The current study reveals the importance of considering parameter variations of network models at the scale of communities.

This thesis solves estimation and control problems in computational neuroscience, mathematically dealing with the first-passage times of diffusion stochastic processes. We first derive estimation algorithms for model parameters from first-passage time observations, and then we derive algorithms for the control of first-passage times. Finally, we solve an optimal design problem which combines elements of the first two: we ask how to elicit first-passage times such as to facilitate model estimation based on said first-passage observations. The main mathematical tools used are the Fokker-Planck partial differential equation for evolution of probability densities, the Hamilton-Jacobi-Bellman equation of optimal control and the adjoint optimization principle from optimal control theory. The focus is on developing computational schemes for the solution of the problems. The schemes are implemented and are tested for a wide range of parameters.

Since the mid-1990s, formal verification has become increasingly important because it can provide guarantees that a software system is free of bugs and working correctly based on a provided model. Verification of biological and medical systems is a promising application of formal verification. Human neural networks have recently been emulated and studied as a biological system. Some recent research has been done on modelling some crucial neuronal circuits and using model checking techniques to verify their temporal properties. In large case studies, model checkers often cannot prove the given property at the desired level of generality. In this thesis, we provide a model using the Coq proof assistant and prove some properties concerning the dynamic behavior of some basic neuronal structures. Understanding the behavior of these modules is crucial because they constitute the elementary building blocks of bigger neuronal circuits. By using a proof assistant, we guarantee that the properties are true in the general case, that is, true for any input values, any length of input, and any amount of time. In this thesis, we define a model of human neural networks. We verify some properties of this model starting with properties of neurons. Neurons are the smallest unit in a human neuronal network. In the next step, we prove properties about functional structures of human neural networks which are called archetypes. Archetypes consist of two or more neurons connected in a suitable way. They are known for displaying some particular classes of behaviours, and their compositions govern several important functions such as walking, breathing, etc. The next step is verifying properties about structures that couple different archetypes to perform more complicated actions. We prove a property about one of these kinds of compositions. With such a model, there is the potential to detect inactive regions of the human brain and to treat mental disorders. Furthermore, our approach can be generalized to the verification of other kinds of networks, such as regulatory, metabolic, or environmental networks.

No description available.

Um die komplexe Dynamik von Neuronen und deren Informationsverarbeitung mittels Pulssequenzen zu verstehen, ist es wichtig, die stationäre Puls-Aktivität zu charakterisieren. Die statistischen Eigenschaften von Pulssequenzen können durch vereinfachte stochastische Neuronenmodelle verstanden werden. Eine gut ausgearbeitete Theorie existiert für die Klasse der Erneuerungsmodelle, welche die statistische Unabhängigkeit der Interspike-Intervalle (ISI) annimmt. Experimente haben jedoch gezeigt, dass viele Neuronen Korrelationen zwischen ISIs aufweisen und daher nicht gut durch einen Erneuerungsprozess beschrieben werden. Solche Korrelationen können durch Nichterneuerungs-Modelle erfasst werden, welche jedoch theoretisch schlecht verstanden sind. Diese Arbeit ist eine analytische Studie von Nichterneuerungs-Modellen, die zwei bedeutende Korrelationsmechanismen untersucht: farbiges Rauschen, welches zeitlich-korrelierten Input darstellt, und negative Puls-Rückkopplung, welche Feuerraten-Adaption realisiert. Für das "Perfect-Integrate-and-Fire" (PIF) Modell, welchen durch ein allgemeines Gauss''sches farbiges Rauschen getrieben ist, werden die Statistiken höherer Ordnung der Output-Pulssequenz hergeleitet, insbesondere der Koeffizient der Variation, der serielle Korrelationskoeffizient (SCC), die ISI-Dichte und der Fano-Faktor. Weiterhin wird die Dynamik des PIF Modells mit Puls-getriggertem Adaptionsstrom und weissem Stromrauschen im Detail analysiert. Die Theorie liefert einen Ausdruck für den SCC, der für schwaches Rauschen aber beliebige Adaptions-Stärke und Zeitskale gültig ist, sowie die lineare Antwortfunktion und das Leistungsspektrum der Pulssequenz. Ausserdem wird gezeigt, dass ein stochastischer Adaptionsstrom wie ein langsames farbiges Rauschen wirkt, was ermöglicht, die dominierende Quellen des Rauschen in einer auditorischen Rezeptorzelle zu bestimmen. Schliesslich wird der SCC für das fluktuations-getriebene Feuerregime berechnet. / To understand the complex dynamics of neurons and its ability to process information using a sequence of spikes, it is vital to characterize its stationary spontaneous spiking activity. The statistical properties of spike trains can be explained by reduced stochastic neuron models that account for various sources of noise. A well-developed theory exists for the class of renewal models, in which the interspike intervals (ISIs) are statistically independent. However, experimental studies show that many neurons are not well described by a renewal process because of correlations between ISIs. Such correlations can be captured by generalized, non-renewal models, which are, however, poorly understood theoretically. This thesis represents an analytical study of non-renewal models, focusing on two prominent correlation mechanisms: colored-noise driving representing temporally correlated inputs, and negative feedback currents realizing spike-frequency adaptation. For the perfect integrate-and-fire (PIF) model driven by a general Gaussian colored noise input, the higher-order statistics of the output spike train is derived using a weak-noise analysis of the Fokker-Planck equation. This includes formulas for the coefficient of variation, the serial correlation coefficient (SCC), the ISI density and the Fano factor. Then, the dynamics of a PIF model with a spike-triggered adaptation and a white-noise current is analyzed in detail. The theory yields an expression for the SCC valid for weak noise but arbitrary adaptation strengths and time scale, and also provides the linear response to time-dependent stimuli and the spike train power spectrum. Furthermore, it is shown that a stochastic adaptation current acts like a slow colored noise, which permits to determine the source of spiking variability observed in an auditory receptor neuron. Finally, the SCC is calculated for the fluctuation-driven spiking regime by assuming discrete states of colored noise or adaptation current.