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We establish a general linear response relation for spiking neuronal networks, based on chains with unbounded memory. This relation allows us to predict the influence of a weak amplitude time-dependent external stimuli on spatio-temporal spike correlations, from the spontaneous statistics (without stimulus) in a general context where the memory in spike dynamics can extend arbitrarily far in the past. Using this approach, we show how linear response is explicitly related to neuronal dynamics with an example, the gIF model, introduced by M. Rudolph and A. Destexhe. This example illustrates the collective effect of the stimuli, intrinsic neuronal dynamics, and network connectivity on spike statistics. We illustrate our results with numerical simulations., 60 pages, 8 figures

Neuronen sind erregbare Systeme. Ihre Antwort auf Anregungen oberhalb eines bestimmten Schwellwertes sind Pulse. Häufig wird die Pulserzeugung von verschiedenen Rückkopplungsmechanismen beeinflusst, die auf langsamen Zeitskalen agieren. Das kann zu Phänomenen wie Feuerraten-Adaptation, umgekehrter Feuerraten-Adaptation oder zum Feuern von Pulsen in Salven führen. Weiterhin sind Neuronen verschiedenen Rauschquellen ausgesetzt und wechselwirken mit anderen Neuronen, in neuronalen Netzen. Doch wie beeinflusst das Zusammenspiel von Rückkopplungsmechanismen, Rauschen und der Wechselwirkung mit anderen Neuronen die Pulserzeugung? Diese Arbeit untersucht, wie die Pulserzeugung in rauschgetriebenen erregbaren Systemen von langsamen Rückkopplungsmechanismen und der Wechselwirkung mit anderen erregbaren Systemen beeinflusst wird. Dabei wird die Pulserzeugung in drei Szenarien betrachtet: (i) in einem einzelnen erregbaren System, das um einen langsamen Rückkopplungsmechanismus erweitert wurde, (ii) in gekoppelten erregbaren Systemen und (iii) in stark gekoppelten salvenfeuernden Neuronen. In jedem dieser Szenarien wird die Pulsstatistik mit Hilfe von analytischen Methoden und Computersimulationen untersucht. Das wichtigste Resultat im ersten Szenario ist, dass das Zusammenspiel von einer stark anregenden Rückkopplung und Rauschen zu rauschkontrollierter Bistabilität führt. Das erlaubt es dem System zwischen verschiedenen Modi der Pulserzeugung zu wechseln. In (ii) wird die Pulserzeugung stark von der Wahl der Kopplungsstärken und der Anzahl der Verbindungen beeinflusst. Analytische Näherungen werden abgeleitet, die einen Zusammenhang zwischen der Anzahl der Verbindungen und der Pulsrate, sowie der Pulszugvariabilität herstellen. In (iii) wird festgestellt, dass eine hemmende Rückkopplung zu sehr unregelmäßigem Verhalten der isolierten Neuronen führt, wohingegen eine starke Kopplung mit dem Netzwerk ein regelmäßigeres Feuern von Salven hervorruft. Neurons are excitable systems. Their responses to excitations above a certain threshold are spikes. Usually, spike generation is shaped by several feedback mechanisms that can act on slow time scales. These can lead to phenomena such as spike-frequency adaptation, reverse spike-frequency adaptation, or bursting. In addition to these, neurons are subject to several sources of noise and interact with other neurons, in the connected complexity of a neural network. Yet how does the interplay of feedback mechanisms, noise as well as interaction with other neurons affect spike generation? This thesis examines how spike generation in noise-driven excitable systems is influenced by slow feedback processes and coupling to other excitable systems. To this end, spike generation in three setups is considered: (i) in a single excitable system, which is complemented by a slow feedback mechanism, (ii) in a set of coupled excitable systems, and (iii) in a set of strongly-coupled bursting neurons. In each of these setups, the statistics of spiking is investigated by a combination of analytical methods and computer simulations. The main result of the first setup is that the interplay of strong positive (excitatory) feedback and noise leads to noise-controlled bistability. It enables excitable systems to switch between different modes of spike generation. In (ii), spike generation is strongly affected by the choice of the coupling strengths and the number of connections. Analytical approximations are derived that relate the number of connections to the firing rate and the spike train variability. In (iii), it is found that negative (inhibitory) feedback causes very irregular behavior of the isolated bursters, while strong coupling to the network regularizes the bursting.

International audience; As one gains more intuitions and results on the importance of concerted activity in spike trains, models are developed to extract potential canonical principles underlying spike coding. These methods shed a new light on spike train dynamics. However, they require time and expertise to be implemented efficiently, making them hard to use in a daily basis by neuroscientists or modelers. To bridge this gap, we developed the license free multiplatform software ENAS integrating tools for spike trains analysis and simulation. These tools are accessible through a friendly Graphical User Interface that avoids any scripting or writing code from user. Most of them have been implemented to run in parallel to reduce the time and memory consumption. One of the main strength of ENAS when compared to competing software is to provide statistical analysis with Maximum Entropy-Gibbs distributions taking into account both spatial and temporal correlations as constraints, allowing to introduce causality and memory in statistics. Conversely, given this analysis or other known statistics, ENAS one can also generate new spike trains. These methods result from a series of work and they have already been applied to the analysis of retina data. This comes in addition to basic visualizations and classical analysis for statistics of spike trains analysis. All these tools are generic and can be applied to any spike train. Interestingly, ENAS also includes specific tools dedicated for vision, and the retina in particular. For example, one can jointly visualize stimulus and spiking activity or estimate receptive fields. ENAS also includes a virtual retina simulator extending former Virtual Retina simulator to include lateral connections in the IPL. We hope that ENAS will become a useful tool for neuroscientists to analyse spike trains and we hope to improve it thanks to user feedback. Our goal is to progressively enrich ENAS with the latest research results, in order to facilitate transfer of new methods to the community. It is downloadable from https://enas.inria.fr where documentation, tutorials and samples of spike trains are available.

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.

International audience; The retina encodes visual scenes by trains of action potentials that are sent to the brain via the optic nerve. In this paper, we describe a new free access user-end software allowing to better understand this coding. It is called PRANAS (https://pranas.inria.fr), standing for Platform for Retinal ANalysis And Simulation. PRANAS targets neuroscientists and modelers by providing a unique set of retina-related tools. PRANAS integrates a retina simulator allowing large scale simulations while keeping a strong biological plausibility and a toolbox for the analysis of spike train population statistics. The statistical method (entropy maximization under constraints) takes into account both spatial and temporal correlations as constraints, allowing to analyze the effects of memory on statistics. PRANAS also integrates a tool computing and representing in 3D (time-space) receptive fields. All these tools are accessible through a friendly graphical user interface. The most CPU-costly of them have been implemented to run in parallel.

We consider the maximum entropy Markov chain inference approach to characterize the collective statistics of neuronal spike trains, focusing on the statistical properties of the inferred model. To find the maximum entropy Markov chain, we use the thermodynamic formalism, which provides insightful connections with statistical physics and thermodynamics from which large deviations properties arise naturally. We provide an accessible introduction to the maximum entropy Markov chain inference problem and large deviations theory to the community of computational neuroscience, avoiding some technicalities while preserving the core ideas and intuitions. We review large deviations techniques useful in spike train statistics to describe properties of accuracy and convergence in terms of sampling size. We use these results to study the statistical fluctuation of correlations, distinguishability, and irreversibility of maximum entropy Markov chains. We illustrate these applications using simple examples where the large deviation rate function is explicitly obtained for maximum entropy models of relevance in this field.