This thesis focuses on an adaptive quadratic spiking model of a motoneuron that is both versatile in its ability to represent a range of experimentally observed neuronal firing patterns as well as computationally efficient for large network simulation. The objective of research is to fit membrane voltage data to the model using a parameter estimation approach involving simulated annealing. By manipulating the system dynamics of the model, a realizable model with linear parameterization (LP) can be obtained to simplify the estimation process. With a persistently excited current input applied to the model, simulated annealing is used to efficiently determine the best model parameters that minimize the square error function between the membrane voltage reference data and data generated by the LP model. Results obtained through simulation of this approach show feasibility to predict a range of different neuron firing patterns.

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Este trabalho trata sobre a transferência de frequência em neurônios de disparo, especificamente neurônios integra-e-dispara com escoamento e neurônios de Izhikevich. Através de análises matemáticas analíticas e sistemáticas simulações numéricas é obtida a função de ganho, a transferência de frequência estacionária e dinâmica dos neurônios utilizados, para diversos valores dos parâmetros do modelo. Desse modo, são realizados múltiplos ajustes às curvas obtidas, e os coeficientes estimados são apresentados. Com base em todos esses dados, são obtidas diversas características dessas relações de transferência de frequência, e como suas propriedades variam com relação aos principais parâmetros do modelo de neurônio e sinapse utilizados. Diversos resultados interessantes foram apresentados, incluindo evidências de que a função ganho do neurônio integra-e-dispara pode se comportar de modo bastante semelhante à função de ganho e transferência estacionária do neurônio de Izhikevich, dependendo dos parâmetros adotados; a divisão do plano de parâmetros do modelo integra-e-dispara de acordo com a linearidade da transferência de frequência dinâmica; o limiar da intensidade de corrente contínua e de frequência de spikes pré-sinápticos de um neurônio de Izhikevich é determinado apenas pelo parâmetro b, no intervalo de parâmetros usual; modelos de sinapses distintos tendem a não alterar a forma da transferência de frequência estacionária de um neurônio de Izhikevich. / This work is about the frequency transfer of spiking neurons, specifically integrate-and-fire neurons and Izhikevich neurons. Through analytical and systematic numerical simulations the gain function, the stationary and dynamic frequency transfer of the adopted neuron models, are obtained for several values of the model parameters. Thus, multiple fits are made to the curves obtained, and the estimated coefficients are presented. Based on all these data, several characteristics of the frequency transfer relations are obtained, and information is obtained about how their properties vary with respect the parameters of the adopted neuron and synapse model. Several interesting results have been presented, including evidences that the integrate-and-fire neuron\'s gain function can behave quite similarly to the Izhikevich neuron\'s stationary transfer and gain function, depending of the adopted parameters. We also obtained the division of the parameters plane of integrate-and-fire model according to the linearity of the dynamic frequency transfer. It was also verified that the thresholds of the presynaptic spikes\' current intensity and frequency of an Izhikevich neuron are determined only by the parameter b, in the usual parameter range. In addition, it was observed that the considered distinct synapses models tend not to depart from the stationary frequency transfer of an Izhikevich neuron.

Στην παρούσα εργασία μελετάμε τη μοντελοποίηση του υποθαλαμικού πυρήνα των βασικών γαγγλίων με χρήση του μαθηματικού νευρωνικού μοντέλου Izhikevich. Βάση της μελέτης μας αποτελούν μικροηλεκτροδιακές καταγραφές, που έχουν ληφθεί κατά τη διάρκεια νευροχειρουργικών επεμβάσεων εν τω βάθει εγκεφαλικής διέγερσης, για τη συμπτωματική θεραπεία της νόσου Πάρκινσον. Θα ξεκινήσουμε με μια εισαγωγή στην φυσιολογία του νευρικού κυττάρου και στην ανατομία των βασικών γαγγλίων. Θα αναλύσουμε τα βασικά ποιοτικά μοντέλα που ερμηνεύουν τη συμμετοχή των τελευταίων σε κινητικές διεργασίες, αλλά και την εμπλοκή τους στη νόσο Πάρκινσον. Μετά από μια σύντομη αναφορά στη μέθοδο της εν τω βάθει διέγερσης και στις μικροηλεκτροδιακές καταγραφές, θα εστιάσουμε στα δυναμικά τοπικού πεδίου και στη νευροφυσιολογική σημασία τους. Συνεχίζοντας, θα κάνουμε μια περιεκτική ανασκόπηση των βασικότερων μαθηματικών μοντέλων νευρώνων και ύστερα θα επικεντρωθούμε στον υποθαλαμικό πυρήνα, περιγράφοντας δύο πρόσφατα μοντέλα που έχουν κατασκευαστεί για την προσομοίωση των νευρώνων αυτού. Έπειτα, θα περάσουμε στην περιγραφή του μοντέλου Izhikevich και στην τροποποίησή του για την αναπαραγωγή των χαρακτηριστικών του νευρώνα του υποθαλαμικού πυρήνα. Κατόπιν, θα αναλύσουμε τη μεθοδολογία που ακολουθήσαμε στην παρούσα υλοποίηση και τις βασικές θεωρήσεις της μοντελοποίησης μας. Θα ολοκληρώσουμε με την παρουσίαση των αποτελεσμάτων, το σχολιασμό αυτών και τις ιδέες για μελλοντική επέκταση της μεθόδου μας. / The main objective of this MSc thesis is the study of subthalamic nucleus, by using the Izhikevich neuron model. Microelectrode recordings, taken during deep brain stimulation operations for Parkinson’s disease, have been used for that purpose. In chapters 1-2, there is an introduction to the physiology of the neuron and the basal ganglia anatomy. In the two following chapters, we are analyzing the basic qualitative models that describe the involvement of the basal ganglia in movements and the pathophysiology of Parkinson’s disease. We are briefly discussing the method of deep brain stimulation, microelectrode recordings processing and the extraction of local field potentials. In chapter 5, the basic mathematical neuron models are discussed. We are focusing on the subthalamic nucleus and we are describing two recently developed mathematical models of the subthalamic neuron. In chapter 6, we are outlining Izhikevich neuron model and its modification in order to describe the subthalamic neuron. In addition, we are analyzing the methodology developed for the implementation of the modeling process and our basic considerations. In chapter 7, the results of the simulation are presented and discussed, so that our conclusions provide ideas for further research.