Neuron models of the generic bifurcation type:network analysis and data modeling

Minimal nonlinear dynamic neuron models of the generic bifurcation type may provide the middle way between the detailed models favored by experimentalists and the simplified threshold and rate model of computational neuroscientists. This thesis investigates to which extent generic bifurcation type models grasp the essential dynamical features that may turn out play a role in cooperative neural behavior. The thesis considers two neuron models, of increasing complexity, and one model of synaptic interactions. The FitzHugh-Nagumo model is a simple two-dimensional model capable only of spiking behavior, and the Hindmarsh-Rose model is a three-dimensional model capable of more complex dynamics such as bursting and chaos. The model for synaptic interactions is a memory-less nonlinear function, known as fast threshold modulation (FTM). By means of a combination of nonlinear system theory and bifurcation analysis the dynamical features of the two models are extracted. The most important feature of the FitzHugh-Nagumo model is its dynamic threshold: the spike threshold does not only depend on the absolute value, but also on the amplitude of changes in the membrane potential. Part of the very complex, intriguing bifurcation structure of the Hindmarsh-Rose model is revealed. By considering basic networks of FTM-coupled FitzHugh-Nagumo (spiking) or Hindmarsh-Rose (bursting) neurons, two main cooperative phenomena, synchronization and coincidence detections, are addressed. In both cases it is illustrated that pulse coupling in combination with the intrinsic dynamics of the models provides robustness. In large scale networks of FTM-coupled bursting neurons, the stability of complete synchrony is independent from the network topology and depends only on the number of inputs to each neuron. The analytical results are obtained under very restrictive and biologically implausible hypotheses, but simulations show that the theoretical predictions hold in more realistic cases as well. Finally, the realism of the models is put to a test by identification of their parameters from in vitro measurements. The identification problem is addressed by resorting to standard techniques combined with heuristics based on the results of the reported mathematical analysis and on a priori knowledge from neuroscience. The FitzHugh-Nagumo model is only able to model pyramidal neurons and even then performs worse than simple threshold models; it should be used only when the advantages of the more realistic threshold mechanism are prevalent. The Hindmarsh-Rose model can model much of the diversity of neocortical neurons; it can be used as a model in the study of heterogeneous networks and as a realistic model of a pyramidal neuron

Classes de dynamiques neuronales et correlations structurées par l'experience dans le cortex visuel.

Neuronal activity is often considered in cognitive neuroscience by the evoked response but most the energy used by the brain is devoted to the sustaining of ongoing dynamics in cortical networks. A combination of classification algorithms (K means, Hierarchical tree, SOM) is used on intracellular recordings of the primary visual cortex of the cat to define classes of neuronal dynamics and to compare it with the activity evoked by a visual stimulus. Those dynamics can be studied with simplified models (FitzHugh Nagumo, hybrid dynamical systems, Wilson Cowan) for which an analysis is presented. Finally, with simulations of networks composed of columns of spiking neurons, we study the ongoing dynamics in a model of the primary visual cortex and their effect on the response evoked by a stimulus. After a learning period during which visual stimuli are presented, waves of depolarization propagate through the network. The study of correlations in this network shows that the ongoing dynamics reflect the functional properties acquired during the learning period.L'activité neuronale est souvent considérée en neuroscience cognitive par la réponse évoquée mais l'essentiel de l'énergie consommée par le cerveau permet d'entretenir les dynamiques spontanées des réseaux corticaux. L'utilisation combinée d'algorithmes de classification (K means, arbre hirarchique, SOM) sur des enregistrements intracellulaires du cortex visuel primaire du chat nous permet de définir des classes de dynamiques neuronales et de les comparer l'activité évoquée par un stimulus visuel. Ces dynamiques peuvent être étudiées sur des systèmes simplifiés (FitzHugh-Nagumo, systèmes dynamiques hybrides, Wilson-Cowan) dont nous présentons l'analyse. Enfin, par des simulations de réseaux composés de colonnes de neurones, un modèle du cortex visuel primaire nous permet d'étudier les dynamiques spontanées et leur effet sur la réponse à un stimulus. Après une période d'apprentissage pendant laquelle des stimuli visuels sont presentés, des vagues de dépolarisation se propagent dans le réseau. L'étude des correlations dans ce réseau montre que les dynamiques spontanées reflètent les propriétés fonctionnelles acquises au cours de l'apprentissage

Efficient Numerical Population Density Techniques with an Application in Spinal Cord Modelling

MIIND is a neural simulator which uses an innovative numerical population density technique to simulate the behaviour of multiple interacting populations of neurons under the influence of noise. Recent efforts have produced similar techniques but they are often limited to a single neuron model or type of behaviour. Extensions to these require a great deal of further work and specialist knowledge. The technique used in MIIND overcomes this limitation by being agnostic to the underlying neuron model of each population. However, earlier versions of MIIND still required a high level of technical knowledge to set up the software and involved an often time-consuming manual pre-simulation process. It was also limited to only two-dimensional neuron models. This thesis presents the development of an alternative population density technique, based on that already in MIIND, which reduces the pre-simulation step to an automated process. The new technique is much more flexible and has no limit on the number of time-dependent variables in the underlying neuron model. For the first time, the population density over the state space of the Hodgkin-Huxley neuron model can be observed in an efficient manner on a single PC. The technique allows simulation time to be significantly reduced by gracefully degrading the accuracy without losing important behavioural features. The MIIND software itself has also been simplified, reducing technical barriers to entry, so that it can now be run from a Python script and installed as a Python module. With the improved usability, a model of neural populations in the spinal cord was simulated in MIIND. It showed how afferent signals can be integrated into common reflex circuits to produce observed patterns of muscle activation during an isometric knee extension task. The influence of proprioception in motor control is not fully understood as it can be both task and subject-specific. The results of this study show that afferent signals have a significant effect on sub-maximal muscle contractions even when the limb remains static. Such signals should be considered when developing methods to improve motor control in activities of daily living via therapeutic or mechanical means

Stochastic resonance and finite resolution in a network of leaky integrate-and-fire neurons.

This thesis is a study of stochastic resonance (SR) in a discrete implementation of a leaky integrate-and-fire (LIF) neuron network. The aim was to determine if SR can be realised in limited precision discrete systems implemented on digital hardware. How neuronal modelling connects with SR is discussed. Analysis techniques for noisy spike trains are described, ranging from rate coding, statistical measures, and signal processing measures like power spectrum and signal-to-noise ratio (SNR). The main problem in computing spike train power spectra is how to get equi-spaced sample amplitudes given the short duration of spikes relative to their frequency. Three different methods of computing the SNR of a spike train given its power spectrum are described. The main problem is how to separate the power at the frequencies of interest from the noise power as the spike train encodes both noise and the signal of interest. Two models of the LIF neuron were developed, one continuous and one discrete, and the results compared. The discrete model allowed variation of the precision of the simulation values allowing investigation of the effect of precision limitation on SR. The main difference between the two models lies in the evolution of the membrane potential. When both models are allowed to decay from a high start value in the absence of input, the discrete model does not completely discharge while the continuous model discharges to almost zero. The results of simulating the discrete model on an FPGA and the continuous model on a PC showed that SR can be realised in discrete low resolution digital systems. SR was found to be sensitive to the precision of the values in the simulations. For a single neuron, we find that SR increases between 10 bits and 12 bits resolution after which it saturates. For a feed-forward network with multiple input neurons and one output neuron, SR is stronger with more than 6 input neurons and it saturates at a higher resolution. We conclude that stochastic resonance can manifest in discrete systems though to a lesser extent compared to continuous systems

Adaptive dynamical networks

It is a fundamental challenge to understand how the function of a network is related to its structural organization. Adaptive dynamical networks represent a broad class of systems that can change their connectivity over time depending on their dynamical state. The most important feature of such systems is that their function depends on their structure and vice versa. While the properties of static networks have been extensively investigated in the past, the study of adaptive networks is much more challenging. Moreover, adaptive dynamical networks are of tremendous importance for various application fields, in particular, for the models for neuronal synaptic plasticity, adaptive networks in chemical, epidemic, biological, transport, and social systems, to name a few. In this review, we provide a detailed description of adaptive dynamical networks, show their applications in various areas of research, highlight their dynamical features and describe the arising dynamical phenomena, and give an overview of the available mathematical methods developed for understanding adaptive dynamical networks

Locomotor Network Dynamics Governed By Feedback Control In Crayfish Posture And Walking

Sensorimotor circuits integrate biomechanical feedback with ongoing motor activity to produce behaviors that adapt to unpredictable environments. Reflexes are critical in modulating motor output by facilitating rapid responses. During posture, resistance reflexes generate negative feedback that opposes perturbations to stabilize a body. During walking, assistance reflexes produce positive feedback that facilitates fast transitions between swing and stance of each step cycle. Until recently, sensorimotor networks have been studied using biomechanical feedback based on external perturbations in the presence or absence of intrinsic motor activity. Experiments in which biomechanical feedback driven by intrinsic motor activity is studied in the absence of perturbation have been limited. Thus, it is unclear whether feedback plays a role in facilitating transitions between behavioral states or mediating different features of network activity independent of perturbation. These properties are important to understand because they can elucidate how a circuit coordinates with other neural networks or contributes to adaptable motor output. Computational simulations and mathematical models have been used extensively to characterize interactions of negative and positive feedback with nonlinear oscillators. For example, neuronal action potentials are generated by positive and negative feedback of ionic currents via a membrane potential. While simulations enable manipulation of system parameters that are inaccessible through biological experiments, mathematical models ascertain mechanisms that help to generate biological hypotheses and can be translated across different systems. Here, a three-tiered approach was employed to determine the role of sensory feedback in a crayfish locomotor circuit involved in posture and walking. In vitro experiments using a brain-machine interface illustrated that unperturbed motor output of the circuit was changed by closing the sensory feedback loop. Then, neuromechanical simulations of the in vitro experiments reproduced a similar range of network activity and showed that the balance of sensory feedback determined how the network behaved. Finally, a reduced mathematical model was designed to generate waveforms that emulated simulation results and demonstrated how sensory feedback can control the output of a sensorimotor circuit. Together, these results showed how the strengths of different approaches can complement each other to facilitate an understanding of the mechanisms that mediate sensorimotor integration

Understanding spiking and bursting electrical activity through piece-wise linear systems

In recent years there has been an increased interest in working with piece-wise linear caricatures of nonlinear models. Such models are often preferred over more detailed conductance based models for their small number of parameters and low computational overhead. Moreover, their piece-wise linear (PWL) form, allow the construction of action potential shapes in closed form as well as the calculation of phase response curves (PRC). With the inclusion of PWL adaptive currents they can also support bursting behaviour, though remain amenable to mathematical analysis at both the single neuron and network level. In fact, PWL models caricaturing conductance based models such as that of Morris-Lecar or McKean have also been studied for some time now and are known to be mathematically tractable at the network level. In this work we proceed to analyse PWL neuron models of conductance type. In particular we focus on PWL models of the FitzHugh-Nagumo type and describe in detail the mechanism for a canard explosion. This model is further explored at the network level in the presence of gap junction coupling. The study moves to a different area where excitable cells (pancreatic beta-cells) are used to explain insulin secretion phenomena. Here, Ca2+ signals obtained from pancreatic beta-cells of mice are extracted from image data and analysed using signal processing techniques. Both synchrony and functional connectivity analyses are performed. As regards to PWL bursting models we focus on a variant of the adaptive absolute IF model that can support bursting. We investigate the bursting electrical activity of such models with an emphasis on pancreatic beta-cells

Neural Field Models: A mathematical overview and unifying framework

Rhythmic electrical activity in the brain emerges from regular non-trivial interactions between millions of neurons. Neurons are intricate cellular structures that transmit excitatory (or inhibitory) signals to other neurons, often non-locally, depending on the graded input from other neurons. Often this requires extensive detail to model mathematically, which poses several issues in modelling large systems beyond clusters of neurons, such as the whole brain. Approaching large populations of neurons with interconnected constituent single-neuron models results in an accumulation of exponentially many complexities, rendering a realistic simulation that does not permit mathematical tractability and obfuscates the primary interactions required for emergent electrodynamical patterns in brain rhythms. A statistical mechanics approach with non-local interactions may circumvent these issues while maintaining mathematically tractability. Neural field theory is a population-level approach to modelling large sections of neural tissue based on these principles. Herein we provide a review of key stages of the history and development of neural field theory and contemporary uses of this branch of mathematical neuroscience. We elucidate a mathematical framework in which neural field models can be derived, highlighting the many significant inherited assumptions that exist in the current literature, so that their validity may be considered in light of further developments in both mathematical and experimental neuroscience.Comment: 55 pages, 10 figures, 2 table

Patient-specific modellling of cortical spreading depression applied to migraine studies.

254 p.-La migraña es un trastorno neurológico muy común. Un tercio de los pacientes que sufren migraña experimentan lo que se denomina aura, una serie de alteraciones sensoriales que preceden al típico dolor de cabeza unilateral. Diversos estudios apuntan a la existencia de una correlación entre el aura visual y la depresión cortical propagada (DCP), una onda de despolarización que tiene su origen en el córtex visual para propagarse, a continuación, por todo el córtex hacia las zonas periféricas. La complejidad y la elevada especificidad de las características del córtex cerebral sugieren que la geometría podría tener un impacto significativo en la propagación de la DCP. En esta tesis hemos combinado dos modelos existentes: un modelo neurológico pormenorizado para el componente electrofisiológico de la DCP y un modelo de reacción-difusión que tiene en consideración la difusión del potasio, el impulsor de la propagación de la DCP. Durante el proceso, hemos integrado dos aspectos de la DCP que tienen lugar en diferentes escalas de tiempo: la dinámica electrofisiológica seguiría un patrón temporal del orden de milisegundos, mientras que la dinámica del potasio extracelular que acciona las funciones de propagación de la DCP se mediría en una escala de minutos. Como resultado, obtendremos un modelo multiescalar EDP-EDO. Asimismo, hemos incorporado los datos específicos del paciente en el modelo DCP: (i) la geometría cerebral específica de un paciente obtenida a través de resonancia magnética, y (ii) los tensores de conductividad personalizados obtenidos a través de diffusion tensor images. A fin de estudiar el papel que desempeña la geometría en la propagación de la DCP, hemos definido las cantidades de interés (CdI) relacionadas con la geometría y las que dependen de la DCP y las hemos evaluado en dos casos prácticos. Si bien la geometría no parece tener un impacto significativo en la propagación de la DCP, algunas CdI han resultado ser unas candidatas muy prometedoras para facilitar la clasificación de individuos sanos y pacientes con migraña. Finalmente, para justificar la carencia de datos experimentales para la validación y selección de los parámetros del modelo, hemos aplicado diversas técnicas de cuantificación de la incertidumbre al modelo DCP y hemos analizado el impacto de las diversas elecciones de parámetros en el resultado del modelo.Migraine is a common neurological disorder and one-third of migraine patients suffer from migraine aura, a perceptual disturbance preceding the typically unilateral headache. Cortical spreading depression (CSD), a depolarisation wave that originates in the visual cortex and propagates across the cortex to the peripheral areas, has been suggested as a correlate of visual aura by several studies. The complex and highly individual-specific characteristics of the brain cortex suggest that the geometry might have a significant impact on CSD propagation. In this thesis, we combine two existing models, a detailed neurological model for the electrophysiological component of CSD and a reaction-diffusion model accounting for the potassium diffusion, the driving force of CSD propagation. In the process, we integrate two aspects of CSD that occur at different time scales: the electrophysiological dynamics features a temporal scale in the order of milliseconds, while the extracellular potassium dynamics that triggers CSD propagation features is on the scale of minutes. As a result we obtain a multi-scale PDE-ODE model. In addition, we incorporate patient-specific data in the CSD model: (i) a patient-specific brain geometry obtained from magnetic resonance imaging, and (ii) personalised conductivity tensors derived from diffusion tensor imaging data. To study the role of the geometry in CSD propagation, we define geometric and CSD-dependent quantities of interest (QoI) that we evaluate in two case studies. Even though the geometry does not seem to have a major impact on the CSD propagation, some QoI are promising candidates to aid in the classification of healthy individuals and migraine patients. Finally, to account for the lack of experimental data for validation and selection of the model parameters, we apply different techniques of uncertainty quantification to the CSD model and analyse the impact of various parameter choices on the model outcom