Ce travail de thèse concerne la modélisation mathématique et l’étude du comportement d’une population de neurones. Dans tout ce travail on s’arrêtera principalement sur une population de neurones auto-excitateurs où chaque cellule du réseau est supposée suivre la loi de l’intègre et tire. Néanmoins nous aborderons au détour d’un chapitre la modélisation d’une population de neurones inhibiteurs, et dans une dernière partie, nous discuterons la modélisation d’une population de neurones obéissant au modèle Ermentrout-Kopell aussi appelé le théta-neurone. L’angle de vue adopté dans cette thèse est donné par l’approche densité de population. Cette approche, dont nous rappellerons en détail les hypothèses et la construction, a été introduite il ya maintenant plus d’une dizaine d’années afin de faciliter la simulation d’une grande population de neurones. Dit plus précisément, une telle approche donne une équation aux dérivées partielles sur la densité de population de neurones dans l’espace d’état formé des potentiels admissibles du neurone. Nous ferons de plus l’hypothèse que la réponse d’un neurone à l’arrivée d’une impulsion est une dépolarisation instantanée, autrement dit un saut de potentiel. Comme nous le verrons,cette équation aux dérivées partielles est non linéaire (à cause du couplage de la population) et non locale (à cause du saut de potentiel). Si cette idée est compliquée et abstraite, elle anéanmoins prouvé tout au long de ces dix dernières années son importance dans la simulation numérique des grands réseaux.Il s’agit avant tout dans ce travail de thèse de donner un cadre mathématique adéquat aux équations aux dérivées partielles qui surgissent d’une telle approche. Ainsi nous discuterons,selon les différents choix de modélisation, du caractère bien posé du modèle par densité de populationet de sa possible explosion en temps fini. Nous discuterons comment la prise en compte d’hypothèses réalistes supplémentaires dans la modélisation, comme le retard entre l’émission d’un potentiel d’action et sa réception ou encore la période réfractaire peut stopper l’explosionen temps fini et garantir l’existence d’une solution globale. Un autre aspect abordé dans ce travail concerne les explications et la prédiction de la synchronisation des neurones. Deux définitions de la synchronisation seront explicitées selon encoreune fois les choix de modélisation. Nous verrons qu’en interprétant l’explosion en temps fini dela solution comme l’arrivée d’une masse de Dirac dans le taux de décharge de la populationon peut relier l’explosion à la synchronisation. Toutefois, avec des hypothèses de modélisation plus réalistes, comme les retards et la période réfractaire, ce phénomène est exclu. Nous verrons néanmoins qu’avec ces paramètres physiques supplémentaires des solutions périodiques apparaissent offrant différents rythmes de décharge de la population. Encore une fois, l’apparition de ces oscillations sera perçue comme la synchronisation de la population. / This thesis concerns the mathematical modelling and the study of the behavior of a population of neurons. In this work we will mainly consider a population of excitatory neurons whe reall the cells of the network follow the integrate-and-fire model. Nonetheless, we will tackle in a chapter the modelling of an inhibitory population of neurons, and we will discuss in the lastchapter the modelling of a population of neurons that follows the Ermentrout-Koppell model.The point of view of this thesis is given by the population density approach that has beenintroduced more than a decade ago in order to facilitate the simulation of a large assembly ofneurons. More precisely, this approach gives a partial differential equation that describes thedensity of neurons in the state space that is the set of all admissible potential of a neuron. We will assume that when receiving an action potential, the potential of the neuron makes a small jump. As we will see this partial differential equation is non linear (due to the coupling betweenneurons) and non-local (due to the potential jump). If this idea is complicated and abstract, itallows to simulate easily a large neural network.First of all, the thesis gives a mathematical framework for the equations that arise from thisthe population density approach. Then we will discuss the existence and the possible blow upin finite time of the solution. We will discuss how the consideration of more realistic modellingassumptions, as the refractory period and the delay between the emission and the reception ofan action potential can stop the blow up of the solution and give a well posed model.We will also try to caracterise the occurence of synchronization of the neural network. Twodifferent ways of seeing the synchronization will be describe. One relates the blow up in finitetime of the solution to the occurence of a Dirac mass in the firing rate of the population.Nonetheless, taking into account the delays, this kind of blow up will not be observed anymore.Nonetheless, as we will see, with this additional features the model will generate some periodicalsolutions that can also be related to the synchronization of the population.

Recentemente observou-se experimentalmente, Johnson et al. (2010), que fatias organotípicas corticais de rato são capazes de completar padrões espaço-temporais, após serem treinadas. Embora se especule que mecanismos de plasticidade sináptica e homeostática estejam por trás do fenômeno, ainda não existe nenhuma explicação detalhada sobre o assunto. Com o intuito de propor uma explicação clara e consistente para os mecanismos que permeiam a resposta da rede aos estímulos como um todo, nos propomos a estudar este fenômeno por meio de uma rede de neurônios de integração-e-disparo dotada de mecanismos de homeostase intrínseca e de plasticidade sináptica dependente de disparos. O sistema construído foi explorado, de modo a determinar em que condições a rede poderia comportar-se como o sistema real, e treinado de forma similar `a realizada experimentalmente por Johnson et al. (2010). / Recently it has been observed experimentally, Johnson et al. (2010), that organotypic cortical slices of rat are capable of completing spatio-temporal patterns after training. Although it is speculated that synaptic and homeostatic plasticity may have an important role in this phenomenon, there is still no detailed explanation about this subject. In order to propose a clear and consistent explanation for the mechanisms that underlie the network response to stimuli as a whole, we propose to study this phenomenon through a network of integrate-and-fire neurons endowed with intrinsic homeostasis and spike-timing dependent plasticity mechanisms. The constructed system was explored, aiming to determine in which conditions the network could behave as the real system, and trained in a way similar as the experimental one done by Johnson et al. (2010).

Recentemente observou-se experimentalmente, Johnson et al. (2010), que fatias organotípicas corticais de rato são capazes de completar padrões espaço-temporais, após serem treinadas. Embora se especule que mecanismos de plasticidade sináptica e homeostática estejam por trás do fenômeno, ainda não existe nenhuma explicação detalhada sobre o assunto. Com o intuito de propor uma explicação clara e consistente para os mecanismos que permeiam a resposta da rede aos estímulos como um todo, nos propomos a estudar este fenômeno por meio de uma rede de neurônios de integração-e-disparo dotada de mecanismos de homeostase intrínseca e de plasticidade sináptica dependente de disparos. O sistema construído foi explorado, de modo a determinar em que condições a rede poderia comportar-se como o sistema real, e treinado de forma similar `a realizada experimentalmente por Johnson et al. (2010). / Recently it has been observed experimentally, Johnson et al. (2010), that organotypic cortical slices of rat are capable of completing spatio-temporal patterns after training. Although it is speculated that synaptic and homeostatic plasticity may have an important role in this phenomenon, there is still no detailed explanation about this subject. In order to propose a clear and consistent explanation for the mechanisms that underlie the network response to stimuli as a whole, we propose to study this phenomenon through a network of integrate-and-fire neurons endowed with intrinsic homeostasis and spike-timing dependent plasticity mechanisms. The constructed system was explored, aiming to determine in which conditions the network could behave as the real system, and trained in a way similar as the experimental one done by Johnson et al. (2010).

O estudo de redes complexas tem despertado muita atenção nos últimos anos, principalmente pela sua capacidade de permitir a análise dos mais diversificados sistemas através de um mesmo conjunto de ferramentas matemáticas e computacionais. Até pouco tempo a ênfase nessa área era sobre o estudo das propriedades estruturas e sua influência em característica globais da dinâmica ocorrendo sobre o sistema. Recentemente foi-se percebendo a riqueza de comportamentos que podemos estudar ao olharmos para grupos de vértices e as interações que ocorrem entre esses grupos, de forma que cada vez mais se torna necessária a criação de uma forma sistemática de quantificar, a nível de nó, como uma dinâmica é influenciada (ou diferenciada) pela estrutura do sistema. Apresentamos nesse trabalho um primeiro passo nessa direção, no qual definimos uma medida que, baseada em características dinâmicas obtidas em diversas condições iniciais, busca comparar o comportamento dos nós presentes no sistema. Através dessa medida encontramos a alta capacidade de modelos de rede geográficas de produzir, por simples flutuações estatísticas, grupos de nós de dinâmica muito distinta em relação ao demais, um fato que não ocorre para uma rede definida de forma semelhante mas não geográfica. Nessas redes conseguimos verificar também se um nó de topologia local distinta dos demais consegue possuir uma dinâmica diferenciada, o que encontramos foi que mesmo em regiões da rede extremamente regulares existe a formação de grupos dinâmicos devido ao efeito da borda dessa região. Saindo de modelos de redes para apresentar aplicações práticas do método, encontramos na rede neuronal do verme Caenorhabditis elegans um conjunto de nós de alta diferenciação dinâmica, que representam os interneurônios do cordão ventral do verme. Adicionalmente encontramos na rede cortical do macaco da família Cercopithecidae uma divisão em relação a regularidade dos sinais dinâmicos, o que indica a presença de comunidades funcionais nessa rede. Além da metodologia de diferenciação desenvolvemos ainda uma ferramenta que busca encontrar de forma totalmente automatizada as características dinâmicas mais relevantes do sistema em estudo, que foi capaz de representar medidas dinâmicas tradicionalmente utilizadas na área. Todos esses resultados abrem caminho para diversas vertentes de estudo, em especial citamos a influência de uma borda irregular no interior de uma região regular, o estudo da rede Caenorhabditis elegans com dinâmicas neuronais mais precisas e a aplicação sistemática de dinâmicas para encontrar a divisão funcional de comunidades em redes direcionadas, um tema que apresenta resultados promissores. / The study of complex networks has drawn much attention over the last years, mainly by virtue of its potential to characterize the most diverse systems through the same mathematical and computational tools. Not long ago the emphasis on this field mostly focused on the effects of the structural properties on the global behavior of a dynamical process taking place in the system. Recently, some studies started to unveil the richness of interactions that occur between groups of nodes when we look at the small scale of interactions occurring in the network. Such findings call for a new systematic methodology to quantify, at node level, how a dynamics is being influenced (or differentiated) by the structure of the underlying system. Here we present a first step towards this direction, in which we define a new measurement that, based on dynamical characteristics obtained for a series of initial conditions, compares the dynamical behavior of the nodes present in the system. Through this measurement we find the high capacity of networks generated by geographical models to exhibit, by means of statistical fluctuations, groups of nodes with very distinct dynamics compared to the rest of the network, a behavior that does not occur for a similar non-geographical network. We also verify if a large topological differentiation of a node necessarily reflects on its dynamics. We find that even in very regular regions of the network the nodes tend to form dynamical groups influenced by the border effects. In addition to the network models used, we present practical applications of the methodology by using the neuronal network of the nematode Caenorhabditis elegans, where we show that the interneurons of the ventral cord presents a very large dynamical differentiation when compared to the rest of the network. We also analyze the cortical network of the Cercopithecidae monkey by means of signal communication complexity, finding that it contains two well-defined functional communities. Besides the differentiation measurement, we also presents an useful mechanism for automatically obtaining the relevant dynamical characteristics of the nodes, which showed promising results by obtaining traditional measurements of the area with little effort. All these results pave the way to a range of different studies, of which we highlight the influence of an irregular border on the dynamics taking place inside a regular network region, the study of the neurons of Caenorhabditis elegans using more robust neuronal dynamics and the systematic application of different dynamics in order to find the functional community division of directed networks.

A stochastic leaky integrate-and-fire neuron model was implemented in this study to simulate the spiking activity of the electrosensory "P-unit" receptor neurons of the weakly electric fish Apteronotus leptorhynchus. In the context of sensory coding, these cells have been previously shown to respond in experiment to natural random narrowband signals with either a linear or nonlinear coding scheme, depending on the intrinsic firing rate of the cell in the absence of external stimulation. It was hypothesised in this study that this duality is due to the relation of the stimulus to the neuron's excitation threshold. This hypothesis was validated with the model by lowering the threshold of the neuron or increasing its intrinsic noise, or randomness, either of which made the relation between firing rate and input strength more linear. Furthermore, synchronous P-unit firing to a common input also plays a role in decoding the stimulus at deeper levels of the neural pathways. Synchronisation and desynchronisation between multiple model responses for different types of natural communication signals were shown to agree with experimental observations. A novel result of resonance-induced synchrony enhancement of P-units to certain communication frequencies was also found.

Activities in the motor cortex are found to be dynamical in nature. Modeling these activities and comparing them with neural recordings helps in understanding the underlying mechanism for the generation of these activities. For this purpose, Recurrent Neural networks or RNNs, have emerged as an appropriate tool. A clear understanding of how the design choices associated with these networks affect the learned dynamics and internal representation still remains elusive. A previous work exploring the dynamical properties of discrete time RNN architectures (LSTM, UGRNN, GRU, and Vanilla) such as the fixed point topology and the linearised dynamics remains invariant when trained on 3 bit Flip- Flop task. In contrast, they show that these networks have unique representational geometry. The goal for this work is to understand if these observations also hold for networks that are more biologically realistic in terms of neural activity. Therefore, we chose to analyze rate networks that have continuous dynamics and biologically realistic connectivity constraints and on spiking neural networks, where the neurons communicate via discrete spikes as observed in the brain. We reproduce the aforementioned study for discrete architectures and then show that the fixed point topology and linearized dynamics remain invariant for the rate networks but the methods are insufficient for finding the fixed points of spiking networks. The representational geometry for the rate networks and spiking networks are found to be different from the discrete architectures but very similar to each other. Although, a small subset of discrete architectures (LSTM) are observed to be close in representation to the rate networks. We show that although these different network architectures with varying degrees of biological realism have individual internal representations, the underlying dynamics while performing the task are universal. We also observe that some discrete networks have close representational similarities with rate networks along with the dynamics. Hence, these discrete networks can be good candidates for reproducing and examining the dynamics of rate networks. / Aktiviteter i motorisk cortex visar sig vara dynamiska till sin natur. Att modellera dessa aktiviteter och jämföra dem med neurala inspelningar hjälper till att förstå den underliggande mekanismen för generering av dessa aktiviteter. För detta ändamål har återkommande neurala nätverk eller RNN uppstått som ett lämpligt verktyg. En tydlig förståelse för hur designvalen associerade med dessa nätverk påverkar den inlärda dynamiken och den interna representationen är fortfarande svårfångad. Ett tidigare arbete som utforskar de dynamiska egenskaperna hos diskreta RNN- arkitekturer (LSTM, UGRNN, GRU och Vanilla), såsom fastpunkts topologi och linjäriserad dynamik, förblir oförändrad när de tränas på 3-bitars Flip- Flop-uppgift. Däremot visar de att dessa nätverk har unik representationsgeometri. Målet för detta arbete är att förstå om dessa observationer också gäller för nätverk som är mer biologiskt realistiska när det gäller neural aktivitet. Därför valde vi att analysera hastighetsnätverk som har kontinuerlig dynamik och biologiskt realistiska anslutningsbegränsningar och på spikande neurala nätverk, där neuronerna kommunicerar via diskreta spikar som observerats i hjärnan. Vi reproducerar den ovannämnda studien för diskreta arkitekturer och visar sedan att fastpunkts topologi och linjäriserad dynamik förblir oförändrad för hastighetsnätverken men metoderna är otillräckliga för att hitta de fasta punkterna för spiknätverk. Representationsgeometrin för hastighetsnätverk och spiknätverk har visat sig skilja sig från de diskreta arkitekturerna men liknar varandra. Även om en liten delmängd av diskreta arkitekturer (LSTM) observeras vara nära i förhållande till hastighetsnäten. Vi visar att även om dessa olika nätverksarkitekturer med varierande grad av biologisk realism har individuella interna representationer, är den underliggande dynamiken under uppgiften universell. Vi observerar också att vissa diskreta nätverk har nära representationslikheter med hastighetsnätverk tillsammans med dynamiken. Följaktligen kan dessa diskreta nätverk vara bra kandidater för att reproducera och undersöka dynamiken i hastighetsnät.

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Various researchers have indicated the importance of graphs in physical sciences and the difficulties that learners may experience with graphs. More specifically, learners’ problems with motion graphs have been reported in literature. Learners’ difficulty in the application of basic concepts in graphs to solve kinematics graphs problems leads to underperformance in physical sciences. Their ability to handle problems in kinematics graphs is enhanced if they have an effective knowledge base or conceptual resources on graphs. In South Africa there seems to be a gap between the GET [General Education and Training] and FET [Further Education and Training] band’s requirements on graphs. A smooth learning progression is needed. For this reason this study selected to investigate the conceptual resources acquired by grade 10 learners from grade 9 that can be used productively for the learning of kinematics graphs in grade 10. The primary aim of the study was to determine and analyse grade 10 learners’ conceptual resources for learning kinematics graphs in physical sciences. The use of a mixed method approach was considered appropriate for this study. The mixed method depended on the quantitative method to produce precise and measurable data, while a qualitative method was to enhance the understanding of the data produced by the quantitative method. Data obtained by quantitative methods was drawn into tables and graphs, and the consistency in responses determined. Patterns and trends in learners’ reasoning were probed with the aid of qualitative method. In the study it was reported that the quantitative data in the form of a questionnaire was completed by 201 learners. Qualitative data was also obtained by interviewing three learners with varying abilities. The results showed that many learners could answer mathematics questions, but struggled with similar questions in kinematics. The results further showed that the learners did not answer the questionnaire consistently, but their responses depended on the context of the questions. In the interviews learners used everyday applications to explain scientific concepts, instead of using scientific principles. Still, some of the everyday applications may be used as resources for teaching the science concepts. From the results it can be deduced that learners’ conceptual resources can influence their understanding of kinematics graphs in physics. These resources are gained from everyday experiences and previous learning in mathematics and the natural sciences. A constraint is that many learners do not efficiently integrate their mathematics and physics knowledge. iv In the study some learners did not transfer their mathematics knowledge to physics, while others could not transfer their physics knowledge to mathematics. From the results recommendations can be made for the teaching of graphs in the GET band for easier progress into the FET band. The strategy to improve understanding of kinematics graphs is to progressively integrate mathematics and physics from grade nine. Line graphs should be treated in more detail in grade 9 to form proper conceptual resources for kinematics graphs in grade ten. / MEd (Natural Sciences Education), North-West University, Potchefstroom Campus, 2014

Various researchers have indicated the importance of graphs in physical sciences and the difficulties that learners may experience with graphs. More specifically, learners’ problems with motion graphs have been reported in literature. Learners’ difficulty in the application of basic concepts in graphs to solve kinematics graphs problems leads to underperformance in physical sciences. Their ability to handle problems in kinematics graphs is enhanced if they have an effective knowledge base or conceptual resources on graphs. In South Africa there seems to be a gap between the GET [General Education and Training] and FET [Further Education and Training] band’s requirements on graphs. A smooth learning progression is needed. For this reason this study selected to investigate the conceptual resources acquired by grade 10 learners from grade 9 that can be used productively for the learning of kinematics graphs in grade 10. The primary aim of the study was to determine and analyse grade 10 learners’ conceptual resources for learning kinematics graphs in physical sciences. The use of a mixed method approach was considered appropriate for this study. The mixed method depended on the quantitative method to produce precise and measurable data, while a qualitative method was to enhance the understanding of the data produced by the quantitative method. Data obtained by quantitative methods was drawn into tables and graphs, and the consistency in responses determined. Patterns and trends in learners’ reasoning were probed with the aid of qualitative method. In the study it was reported that the quantitative data in the form of a questionnaire was completed by 201 learners. Qualitative data was also obtained by interviewing three learners with varying abilities. The results showed that many learners could answer mathematics questions, but struggled with similar questions in kinematics. The results further showed that the learners did not answer the questionnaire consistently, but their responses depended on the context of the questions. In the interviews learners used everyday applications to explain scientific concepts, instead of using scientific principles. Still, some of the everyday applications may be used as resources for teaching the science concepts. From the results it can be deduced that learners’ conceptual resources can influence their understanding of kinematics graphs in physics. These resources are gained from everyday experiences and previous learning in mathematics and the natural sciences. A constraint is that many learners do not efficiently integrate their mathematics and physics knowledge. iv In the study some learners did not transfer their mathematics knowledge to physics, while others could not transfer their physics knowledge to mathematics. From the results recommendations can be made for the teaching of graphs in the GET band for easier progress into the FET band. The strategy to improve understanding of kinematics graphs is to progressively integrate mathematics and physics from grade nine. Line graphs should be treated in more detail in grade 9 to form proper conceptual resources for kinematics graphs in grade ten. / MEd (Natural Sciences Education), North-West University, Potchefstroom Campus, 2014

Young people are currently affected in the realm of self-esteem, identity development and relational behaviour amidst advancing technology. There is the need for research in this area. Obstacles include fragmented homes, distance job–location and back-log parental education. There is the need for spiritual direction and for the formation of Christian spiritual principles. This is evident in the attraction of young people to cults and the influences of the New Age. The study will explore the complexity of relationships, identity and behaviour in young people. The empirical research will be obtained with permission from a school in the Eastern Cape; the focus group of this study is ages 11 – 18 years. The aim of this study will be to explore interventions which can help facilitate better personal, family and peer relationships in adolescents. The aim will further be to help bring about better integration in their personalities, relationships and communication. / Practical Theology / M. Th. (Practical Theology)