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Graphics processing units (GPUs) are widely available and have been used with great success to accelerate scientific computing in the last decade. These advances, however, are often not available to researchers interested in simulating spiking neural networks, but lacking the technical knowledge to write the necessary low-level code. Writing low-level code is not necessary when using the popular Brian simulator, which provides a framework to generate efficient CPU code from high-level model definitions in Python. Here, we present Brian2CUDA, an open-source software that extends the Brian simulator with a GPU backend. Our implementation generates efficient code for the numerical integration of neuronal states and for the propagation of synaptic events on GPUs, making use of their massively parallel arithmetic capabilities. We benchmark the performance improvements of our software for several model types and find that it can accelerate simulations by up to three orders of magnitude compared to Brian's CPU backend. Currently, Brian2CUDA is the only package that supports Brian's full feature set on GPUs, including arbitrary neuron and synapse models, plasticity rules, and heterogeneous delays. When comparing its performance with Brian2GeNN, another GPU-based backend for the Brian simulator with fewer features, we find that Brian2CUDA gives comparable speedups, while being typically slower for small and faster for large networks. By combining the flexibility of the Brian simulator with the simulation speed of GPUs, Brian2CUDA enables researchers to efficiently simulate spiking neural networks with minimal effort and thereby makes the advancements of GPU computing available to a larger audience of neuroscientists.

The brain is a complex dynamical system. It can operate in many different states of neuronal network activity and switch between them, depending on cognitive demands. To better understand functionally relevant network states and pathological activity it is beneficial to employ computational models for predictions and mechanistic analyses. In this thesis, I consider a biophysical neuronal network model which is specified by parameters that can be related to experimental data. For characterizing the stochastic network dynamics I use and extend methods mainly from statistical physics and numerical mathematics. Particularly, I derive a mean-field description in terms of the Fokker-Planck equation for a large recurrent network of adaptive exponential integrate-and-fire neurons that receive fluctuating input. The derivation is based on the diffusion approximation of the synaptic input and the separation of fast (membrane voltage) and slow (adaptation current) neuronal time scale. The resulting mean-field system allows to obtain the dynamics of the spike rate through solving a partial differential equation. It is extended in a self-consistent way to yield in addition to the spike rate also the second and higher order spiking statistics. I further derive – based on a spectral decomposition of the Fokker-Planck operator and alternatively on a linear-nonlinear cascade ansatz – efficient spike rate models that describe the network activity with only a few ordinary differential equations. These models can be rapidly integrated and allow for convenient mathematical investigations of asynchronous, oscillatory or chaotic network states because linear stability analyses and powerful related methods are readily applicable. To efficiently solve the mean-field system and the derived simpler models I provide novel numerical methods implemented in the Python programming language: (1) a simulation scheme for the Fokker-Planck model using a finite volume discretization, (2) an algorithm for solving the eigenvalue problem associated with the Fokker-Planck operator, and (3) the time integration of the low-dimensional spike rate models. The source code is available as open source software under a free license on GitHub. Applying the Fokker-Planck model I analyze how neuronal properties, particularly adaptation currents, affect the spike rate dynamics of recurrent networks and the spiking statistics of single neurons. With this approach and by exploration of the relevant parameter space I find that the type of the adaptation current has a distinct role in regulating the activity state of a recurrent network and in determining gain, threshold and variability of neuronal spiking. A spike-driven adaptation current is capable of generating spike rate oscillations, and reducing gain as well as spiking irregularity. A subthreshold adaptation mechanism on the other hand destabilizes such rhythms leading to asynchronous states, and increases the spiking threshold and its variability. This thesis demonstrates the benefits of deriving population activity descriptions using bottom-up approaches and contributes concrete models and corresponding efficient numerics, advancing the methodology of the theoretical neuroscience community., Das Gehirn ist ein komplexes dynamisches System. Es kann sich in vielen unterschiedlichen Zuständen neuronaler Netzwerkaktivität befinden und abhängig von den kognitiven Anforderungen zwischen diesen wechseln. Um funktionell wichtige Netzwerkzustände sowie pathologische Aktivität besser zu verstehen, ist es vorteilhaft mathematische Modelle einzusetzen, insbesondere für Vorhersagen und mechanistische Untersuchungen. In dieser Dissertation betrachte ich ein biophysikalisches neuronales Netzwerkmodell, dessen Parameter aus experimentellen Daten erhalten werden können. Ich charakterisiere die stochastische Netzwerkdynamik mit Methoden hauptsächlich aus der statistischen Physik sowie der numerischen Mathematik, welche ich hierfür erweitere. Insbesondere leite ich eine Mean-Field-Beschreibung in Form der Fokker-Planck-Gleichung für große rekurrente Netzwerke adaptiver exponentieller Integrate-and-Fire-Neurone, welche verrauschte Eingangssignale empfangen, her. Diese Herleitung basiert auf der Diffusionsapproximation des synaptischen Stroms sowie einer Trennung der schnellen (Membranspannung) und langsamen (Adaptationsstrom) neuronalen Zeitskala. Das Mean-Field-System ermöglicht es, durch Lösung einer partiellen Differentialgleichung, die Dynamik der Feuerrate zu berechnen. Es wird selbstkonsistent auf eine Beschreibung erweitert, welche neben der Feuerrate auch das zweite Moment sowie die Momente höherer Ordnung der Feuerstatistik enthält. Ich leite außerdem – basierend auf der Spektralzerlegung des Fokker-Planck-Operators und alternativ durch einen parametrischen Kaskadenansatz – effiziente Feuerratenmodelle her, welche die Netzwerkaktivität mit nur wenigen gewöhnlichen Differentialgleichungen beschreiben. Diese Modelle können sehr schnell integriert werden und ermöglichen detaillierte mathematische Analysen von asynchronen, oszillatorischen oder chaotischen Netzwerkzuständen, da hierfür leistungsfähige Methoden existieren. Um das Mean-Field-System und die hergeleiteten reduzierten Modelle effizient lösen zu können, stelle ich neuartige numerische Methoden, welche in der Programmiersprache Python implementiert sind, zur Verfügung: (1) Ein Simulationsschema für das Fokker-Planck-Modell auf Grundlage einer Finite-Volumen-Diskretisierung, (2) einen Algorithmus zur Lösung des Eigenwertproblems für den Fokker-Planck-Operator und (3) die Zeitintegration der niedrigdimensionalen Feuerratenmodelle. Der Quellcode ist als Open-Source-Software unter einer freien Lizenz auf GitHub verfügbar. Ich verwende das Fokker-Planck-Modell, um den Einfluss neuronaler Eigenschaften, insbesondere der Adaptationsströme auf die Feuerratendynamik rekurrenter Netzwerke und die Feuerstatistik einzelner Neurone zu untersuchen. Durch diese Methode sowie einer Exploration des relevanten Parameterraums zeige ich, dass der Typ des Adaptationsstroms maßgeblich ist, um den Aktivitätszustand eines rekurrenten Netzwerkes zu regulieren, und die Verstärkung, den Schwellwert und die Variabilität des neuronalen Feuerverhaltens zu bestimmen. Ein Adaptationsstrom, der durch Aktionspotentiale aktiviert wird, kann Feuerratenoszillationen generieren und die Verstärkung sowie die Irregularität verringern. Ein unterschwelliger Adaptationsmechanismus hingegen destabilisiert solche Rhythmen und führt zu asynchronen Zuständen, zudem erhöht er den Schwellwert und die Feuervariabilität. Diese Dissertation zeigt die Vorteile von hergeleiteten Beschreibungen der Populationsaktivität unter Verwendung von Bottom-Up-Ansätzen auf, und steuert konkrete Modelle und dazugehörige numerische Verfahren bei, was die Methodik der theoretischen Neurowissenschaften weiterentwickelt.

The spiking activity of single neurons can be well described by a nonlinear integrate-and-fire model that includes somatic adaptation. When exposed to fluctuating inputs sparsely coupled populations of these model neurons exhibit stochastic collective dynamics that can be effectively characterized using the Fokker-Planck equation. This approach, however, leads to a model with an infinite-dimensional state space and non-standard boundary conditions. Here we derive from that description four simple models for the spike rate dynamics in terms of low-dimensional ordinary differential equations using two different reduction techniques: one uses the spectral decomposition of the Fokker-Planck operator, the other is based on a cascade of two linear filters and a nonlinearity, which are determined from the Fokker-Planck equation and semi-analytically approximated. We evaluate the reduced models for a wide range of biologically plausible input statistics and find that both approximation approaches lead to spike rate models that accurately reproduce the spiking behavior of the underlying adaptive integrate-and-fire population. Particularly the cascade-based models are overall most accurate and robust, especially in the sensitive region of rapidly changing input. For the mean-driven regime, when input fluctuations are not too strong and fast, however, the best performing model is based on the spectral decomposition. The low-dimensional models also well reproduce stable oscillatory spike rate dynamics that are generated either by recurrent synaptic excitation and neuronal adaptation or through delayed inhibitory synaptic feedback. The computational demands of the reduced models are very low but the implementation complexity differs between the different model variants. Therefore we have made available implementations that allow to numerically integrate the low-dimensional spike rate models as well as the Fokker-Planck partial differential equation in efficient ways for arbitrary model parametrizations as open source software. The derived spike rate descriptions retain a direct link to the properties of single neurons, allow for convenient mathematical analyses of network states, and are well suited for application in neural mass/mean-field based brain network models. Characterizing the dynamics of biophysically modeled, large neuronal networks usually involves extensive numerical simulations. As an alternative to this expensive procedure we propose efficient models that describe the network activity in terms of a few ordinary differential equations. These systems are simple to solve and allow for convenient investigations of asynchronous, oscillatory or chaotic network states because linear stability analyses and powerful related methods are readily applicable. We build upon two research lines on which substantial efforts have been exerted in the last two decades: (i) the development of single neuron models of reduced complexity that can accurately reproduce a large repertoire of observed neuronal behavior, and (ii) different approaches to approximate the Fokker-Planck equation that represents the collective dynamics of large neuronal networks. We combine these advances and extend recent approximation methods of the latter kind to obtain spike rate models that surprisingly well reproduce the macroscopic dynamics of the underlying neuronal network. At the same time the microscopic properties are retained through the single neuron model parameters. To enable a fast adoption we have released an efficient Python implementation as open source software under a free license.

Published by BioMed Central Ladenbauer, Josef ; Augustin, Moritz ; Obermayer, Klaus : How adaptation currents and synaptic inhibition change threshold, gain and variability of neuronal spiking. - In: BMC Neuroscience. - ISSN 1471-2202 (online). - 14 (2013), suppl. 1, P299. - doi:10.1186/1471-2202-14-S1-P299.