A contraction argument for two-dimensional spiking neuron models




Foxall, Eric

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The field of mathematical neuroscience is concerned with the modeling and interpretation of neuronal dynamics and associated phenomena. Neurons can be modeled individually, in small groups, or collectively as a large network. Mathematical models of single neurons typically involve either differential equations, discrete maps, or some combination of both. A number of two-dimensional spiking neuron models that combine continuous dynamics with an instantaneous reset have been introduced in the literature. The models are capable of reproducing a variety of experimentally observed spiking patterns, and also have the advantage of being mathematically tractable. Here an analysis of the transverse stability of orbits in the phase plane leads to sufficient conditions on the model parameters for regular spiking to occur. The application of this method is illustrated by three examples, taken from existing models in the neuroscience literature. In the first two examples the model has no equilibrium states, and regular spiking follows directly. In the third example there are equilibrium points, and some additional quantitative arguments are given to prove that regular spiking occurs.



Dynamical Systems, Mathematical Neuroscience