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.

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