Mode locking in a periodically forced "ghostbursting" neuron model

We study a minimal integrate-and-fire based model of a "ghostbursting" neuron under periodic stimulation. These neurons are involved in sensory processing in weakly electric fish. There exist regions in parameter space in which the model neuron is mode-locked to the stimulation. We analyse this locked behavior and examine the bifurcations that define the boundaries of these regions. Due to the discontinuous nature of the flow, some of these bifurcations are nonsmooth. This exact analysis is in excellent agreement with numerical simulations, and can be used to understand the response of such a model neuron to biologically realistic input.

Mode locking in a spatially extended neuron model: active soma and compartmental tree

Understanding the mode-locked response of excitable systems to periodic forcing has important applications in neuroscience. For example it is known that spatially extended place cells in the hippocampus are driven by the theta rhythm to generate a code conveying information about spatial location. Thus it is important to explore the role of neuronal dendrites in generating the response to periodic current injection. In this paper we pursue this using a compartmental model, with linear dynamics for each compartment, coupled to an active soma model that generates action potentials. By working with the piece-wise linear McKean model for the soma we show how the response of the whole neuron model (soma and dendrites) can be written in closed form. We exploit this to construct a stroboscopic map describing the response of the spatially extended model to periodic forcing. A linear stability analysis of this map, together with a careful treatment of the non-differentiability of the soma model, allows us to construct the Arnol'd tongue structure for 1:q states (one action potential for q cycles of forcing). Importantly we show how the presence of quasi-active membrane in the dendrites can influence the shape of tongues. Direct numerical simulations confirm our theory and further indicate that resonant dendritic membrane can enlarge the windows in parameter space for chaotic behavior. These simulations also show that the spatially extended neuron model responds differently to global as opposed to point forcing. In the former case spatio-temporal patterns of activity within an Arnol'd tongue are standing waves, whilst in the latter they are traveling waves.