Homoclinic bifurcations at tonic-firing to bursting transitions: underlying mechanisms and impact on neuronal synchronization

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We propose a series of studies to elucidate the dynamical properties of a physiologically justified neuron model. Exploring the role of slow subthreshold currents in the model dynamics will play an essential role for understanding the mechanism of the neuron’s transition from a tonic firing state to a bursting state. This transition seems to be connected with the appearance of an unusual kind of homoclinic bifurcation first found in this model. Its description in terms of bifurcation theory is a central aspect of our proposed study. A more realistic theory of neuronal mechanisms has to take into account the role of noise. We will achieve this by applying Gaussian white noise to the ion flow kinetics which leads to a physiologically justifiable colored-noise perturbation of the membrane voltage. Since neurons usually act as parts of large networks, we will study the circumstances and various possible scenarios under which synchronization can be achieved or broken. Recent research hints at the aforementioned tonic-to-bursting transition to be the crucial ingredient that makes or breaks synchrony. We aim at finding out how and why this is so by building upon results obtained from our studies of the single-neuron dynamics.