Recently-developed technologies for monitoring activity in populations of neurons make it possible for the first time, in principle, to ask many basic questions in neuroscience. However, computational tools for analyzing newly available data need to be developed. The goal of this thesis is to contribute to this effort by focusing on two specific problems. First, we used a point-process regression framework to provide a methodology for statistical assessment of the link between neural spike synchrony and network-wide oscillations. In simulations, we showed that our method can recover ground-truth relationships, and in two types of spike train data we illustrated the kinds of results the method can produce. The approach improves on methods in the literature and may be adapted to many different experimental settings. Second, we considered the problem of source extraction in calcium imaging data, i.e., the detection of neurons within a field of view and the extraction of each neuron’s activity. The data we mainly focus on are recorded with a microendoscope, which has the unique advantage of imaging deep brain regions in freely behaving animals. These data suffer from high levels of background fluorescence, as well as the potential for overlapping neuronal signals. Based on the existing constrained nonnegative matrix factorization (CNMF) framework, we developed an efficient method to process microendoscopic data. Our method utilizes a novel algorithm to initialize the spatial shapes and temporal activity of the neurons from the raw video data independently from the strong fluctuating background. This step ensures the efficiency and accuracy of solving a nonconvex CNMF problem. Our method also models the complicated background by including its low-spatial frequency structure and the locally-low-rank feature to avoid absorbing cellular signals into the background term. We developed a tractable solution to estimate the background activity using this new model. After subtracting the approximated background, we followed the CNMF framework to demix neural signals and recover denoised and deconvolved temporal activity. We optimized several algorithms in solving the CNMF problems to get accurate results. In practice, our method outperforms all existing methods and has been adopted by many experimental labs.

The activities of pairs of mammalian motor neurons (MNs) receiving varying degrees of common excitatory synaptic input were simulated to study the relationship between nearly-coincident spiking and common excitatory drive. The somatic membrane potential of each MN was modeled using a single compartment model. Each MN was modeled toreceive synaptic contacts from hundreds of pre-synaptic fibers. The percentage of pre-synaptic fibers that diverged to supply both MNs of a pair with common synaptic input could be varied from 0 (no common inputs) to 100% (all common inputs). Spikes trains on separate re-synaptic fibers were independent of one another and were modeled as realizations of renewal processes with mean firing rates (10 - 50Hz) resembling that associated with supra-spinal input. Maximum synaptic conductances and time constants were varied across synapsesto match experimentally observed somatic EPSPs. The number of active pre-synaptic fibers to each MN was adjusted in order that the firingrates of MNs were between 8 and 15 Hz. For each common input condition, 100 s of concurrent spiking activity of the MNs was usedto construct cross-correlation histograms. The sizes of the central peaks in the histograms were quantified using both the k' (Ellaway and Murthy 1985) and CIS (Nordstrom et al. 1992) indices ofsynchrony. Monotonically increasing linear relationships between the proportion of common synaptic input and the magnitude of synchronywere observed for both indices. For example, the model predicted that 10% common input would yield a CIS value of 0.27 whereas 100% commoninput would yield a CIS value of 1.5. These values are within the range of values observed experimentally. These results, therefore,provide a means to translate measures of nearly-coincident spiking into plausible renditions of synaptic connectivity.

In the stochastic modeling of neurons, the first passage time problem arises as a natural object of study when considering the inter spike interval distribution. In this report, we study some aspects of this problem as it arises in the context of neuroscience. In the first chapter we describe the basic neurophysiology required to model the neuron. In the second, we study the Poisson model, Stein’s model, and some diffusion models, calculating or indicating methods to compute the density of the first passage time random variable or its moments. In the third and fourth chapters, we study the Fokker-Planck equation, and use it to compute the first passage time in the discrete and continuous time random walk cases. In the final chapter, we study sequences of neurons and the change in the density of the waiting time distributions, and hence in the inter spike intervals, as the output spike train from one neuron is considered as the input in the subsequent neuron.

Les neurones sensoriels réagissent à des stimuli externes en émettant des séquences de potentiels d’action (“spikes”). Ces spikes transmettent collectivement de l’information sur le stimulus en formant des motifs spatio-temporels qui constituent le code neural. On observe expérimentalement que ces motifs se produisent de façon irrégulière, mais avec une structure qui peut être mise en évidence par l’utilisation de descriptions probabilistes et de méthodes statistiques. Cependant, la caractérisation statistique des données expérimentales présente plusieurs contraintes majeures: en dehors de celles qui sont inhérentes aux statistiques empiriques comme la taille de l’échantillonnage, ‘le’ modèle statistique sous-jacent est inconnu. Dans cette thèse, nous abordons le problème d’un point de vue complémentaire à l’approche expérimentale. Nous nous intéressons à des modèles neuro-mimétiques permettant d’étudier la statistique collective des potentiels d’action et la façon dont elle dépend de l’architecture et l’histoire du réseau ainsi que du stimulus. Nous considérons tout d’abord un modèle de type Intègre-et-Tire à conductance incluant synapses électriques et chimiques. Nous montrons que la statistique des potentiels d’action est caractérisée par une distribution non stationnaire et de mémoire infinie, compatible avec les probabilités conditionnelles (left interval-specification), qui est non-nulle et continue, donc une distribution de Gibbs. Nous présentons ensuite une méthode qui permet d’unifier les modèles dits d’entropie maximale spatio-temporelle (dont la mesure invariante est une distribution de Gibbs dans le sens de Bowen) et les modèles neuro-mimétiques, en fou / Sensory neurons respond to external stimulus using sequences of action potentials (“spikes”). They convey collectively to the brain information about the stimulus using spatio-temporal patterns of spikes (spike trains), that constitute a “neural code”. Since spikes patterns occur irregularly (yet highly structured) both within and over repeated trials, it is reasonable to characterize them using statistical methods and probabilistic descriptions. However, the statistical characterization of experimental data presents several major constraints: apart from those inherent to empirical statistics like finite size sampling, ‘the’ underlying statistical model is unknown. In this thesis we adopt a complementary approach to experiments. We consider neuromimetic models allowing the study of collective spike trains statistics and how it depends on network architecture and history, as well as on the stimulus. First, we consider a conductance-based Integrate-and-Fire model with chemical and electric synapses. We show that the spike train statistics is characterized by non-stationary, infinite memory, distribution consistent with conditional probabilities (Left interval specifications), which is continuous and non null, thus a Gibbs distribution. Then, we present a novel method that allows us to unify spatio-temporal Maximum Entropy models (whose invariant measure are Gibbs distributions in the Bowen sense) and neuro-mimetic models, providing a solid ground towards biophysical explanation of spatio-temporal correlations observed in experimental data. Finally, using these tools, we discuss the stimulus response of retinal ganglion cells, and the possible generalization of the co

We study two methods of data analysis which are common tools for the analysis of neuronal data. In particular, we examine how causal interactions between brain regions can be investigated using time series reflecting the neural activity in these regions. Furthermore, we analyze a method used to study the neural code that evaluates the discrimination of the responses of single neurons elicited by different stimuli. This discrimination analysis is based on the quantification of the similarity of the spike trains with time scale parametric spike train distances. In each case we describe the methods used for the analysis of the neuronal data and we characterize their specificity using simulated or exemplary experimental data. Taking into account our results, we comment the previous studies in which the methods have been applied. In particular, we focus on the interpretation of the statistical measures in terms of underlying neuronal causal connectivity and properties of the neural code, respectively. / Estudiem dos mètodes d'anàlisi de dades que són eines habituals per a l'anàlisi de dades neuronals. Concretament, examinem la manera en què les interaccions causals entre regions del cervell poden ser investigades a partir de sèries temporals que reflecteixen l'activitat neuronal d'aquestes regions. A més a més, analitzem un mètode emprat per estudiar el codi neuronal que avalua la discriminació de les respostes de neurones individuals provocades per diferents estímuls. Aquesta anàlisi de la discriminació es basa en la quantificació de la similitud de les seqüències de potencials d'acció amb distàncies amb un paràmetre d'escala temporal. Tenint en compte els nostres resultats, comentem els estudis previs en els quals aquests mètodes han estat aplicats. Concretament, ens centrem en la interpretació de les mesures estadístiques en termes de connectivitat causal neuronal subjacent i propietats del codi neuronal, respectivament.

Electrical activity is omnipresent throughout the brain, and it varies dependant on the brain region. Areal hierarchy has been suggested to be one of the main principles of the organization of the brain, but there is not a lot of evidence available related to the specialization of the brain’s regions in the temporal domain, that is, how the activity evolves over time. It has been suggested that there is a relationship between spatial location and timescale [1] and that the timescales of neuronal activity in rodents change according to the hierarchical position (derived from anatomical connectivity measurements) of the brain region [2]. Timescale is related to to the capability of a neuron to maintain the same firing rate over a time period. This firing rate can be measured as decay time constant of an auto-correlation matrix of spiking activity, referred to as the timescale of a single neuron [3]. In this thesis, timescales of spontaneous brain activity were measured in eight regions of the mouse prefrontal cortex (PFC) (data obtained in the Carlén Laboratory) and compared to the timescales of eight visual areas (Neuropixels Visual Coding dataset, Allen Institute for Brain Science) [4]. The results showed that cortical regions hold varying timescales, but that there is no clear correspondence of the timescales of spontaneous activity to the anatomical hierarchies. Instead, we show that the PFC regions have a greater variability in their respective timescales compared to visual cortical regions. The analysis was done using two different approaches, where for some regions the measured timescales significantly differs, due to the difference in the use of the magnitudes of the correlation. This work highlights how neuronal timescales measurements can be approached in cortical regions and used for the future work investigating their functional role and the mechanisms of generation of distinct neuronal timescales in the brain. / Elektrisk aktivitet är allestädes närvarande i hela hjärnan, och den varierar beroende på hjärnregionen. Arealhierarki har föreslagits vara en av huvudprinciperna för hjärnans organisation, men det finns inte mycket bevis tillgängligt relaterat till specialiseringen av hjärnans regioner i den temporala domänen, det vill säga hur aktiviteten utvecklas över tiden . Det har föreslagits att det finns ett samband mellan rumslig plats och tidsskala [1] och att tidsskalorna för neuronal aktivitet hos gnagare ändras beroende på den hierarkiska positionen (härledd från anatomiska anslutningsmätningar) i hjärnregionen [2]. Tidsskala är relaterat till förmågan hos ett neuron att bibehålla samma fyrningshastighet under en tidsperiod. Denna avfyrningshastighet kan mätas som fallstidskonstant för en autokorrelationsmatris av spikaktivitet, kallad tidsskalan för en enda neuron [3]. I denna avhandling mättes tidsskalor för spontan hjärnaktivitet i åtta regioner i musens prefrontala kortex (PFC) (data erhållen av Carlén Laboratory) och jämfört med tidsskalorna för åtta visuella områden (Neuropixels Visual Coding dataset, Allen Institute for Brain Science) [4]. Resultaten visade att kortikala regioner har olika tidsskalor, men att det inte finns någon tydlig överensstämmelse mellan tidsskalorna för spontan aktivitet med de anatomiska hierarkierna. Istället visar vi att PFC-regionerna har större variation i sina respektive tidsskalor jämfört med visuella kortikala regioner. Analysen gjordes med hjälp av två olika tillvägagångssätt, där de uppmätta tidsskalorna för vissa regioner skiljer sig avsevärt på grund av skillnaden i användning av storleken på korrelationen. Detta arbete belyser hur neuronala tidsskalemätningar kan beaktas i kortikala regioner och användas för det framtida arbetet med att undersöka deras funktionella roll och mekanismerna för generering av distinkta neuronala tidsskalor i hjärnan.

L’évolution des techniques d’acquisition de l’activité neuronale permet désormais d'enregistrer simultanément jusqu’à plusieurs centaines de neurones dans le cortex ou dans la rétine. L’analyse de ces données nécessite des méthodes mathématiques et numériques pour décrire les corrélations spatiotemporelles de la population neuronale. Une méthode couramment employée est basée sur le principe d’entropie maximale. Dans ce cas, le produit N×R, où N est le nombre de neurones et R le temps maximal considéré dans les corrélations, est un paramètre crucial. Les méthodes de physique statistique usuelles sont limitées aux corrélations spatiales avec R = 1 (Ising) alors que les méthodes basées sur des matrices de transfert, permettant l’analyse des corrélations spatio-temporelles (R > 1), sont limitées à N×R≤20. Dans une première partie, nous proposons une version modifiée de la méthode de matrice de transfert, basée sur un algorithme de Monte-Carlo parallèle, qui nous permet d’aller jusqu’à N×R=100. Dans la deuxième partie, nous présentons la bibliothèque C++ Enas, dotée d’une interface graphique développée pour les neurobiologistes. Enas offre un environnement hautement interactif permettant aux utilisateurs de gérer les données, effectuer des analyses empiriques, interpoler des modèles statistiques et visualiser les résultats. Enfin, dans une troisième partie, nous testons notre méthode sur des données synthétiques et réelles (rétine, fournies par nos partenaires biologistes). Notre analyse non exhaustive montre l’avantage de considérer des corrélations spatio-temporelles pour l’analyse des données rétiniennes; mais elle montre aussi les limites des méthodes d’entropie maximale. / Recent experimental advances have made it possible to record up to several hundreds of neurons simultaneously in the cortex or in the retina. Analyzing such data requires mathematical and numerical methods to describe the spatio-temporal correlations in population activity. This can be done thanks to Maximum Entropy method. Here, a crucial parameter is the product N×R where N is the number of neurons and R the memory depth of correlations (how far in the past does the spike activity affects the current state). Standard statistical mechanics methods are limited to spatial correlation structure with R = 1 (e.g. Ising model) whereas methods based on transfer matrices, allowing the analysis of spatio-temporal correlations, are limited to NR ≤ 20. In the first part of the thesis we propose a modified version of the transfer matrix method, based on the parallel version of the Montecarlo algorithm, allowing us to go to NR=100. In a second part we present EnaS, a C++ library with a Graphical User Interface developed for neuroscientists. EnaS offers highly interactive tools that allow users to manage data, perform empirical statistics, modeling and visualizing results. Finally, in a third part, we test our method on synthetic and real data sets. Real data set correspond to retina data provided by our partners neuroscientists. Our non-extensive analysis shows the advantages of considering spatio-temporal correlations for the analysis of retina spike trains, but it also outlines the limits of Maximum Entropy methods.