A Geometric Approach to Phase Resetting Estimation Based on Mapping Temporal to Geometric Phase

Abstract

The membrane potential is the most commonly traced quantity in both numerical simulations and electrophysiological experiments. One quantitative measure of neuronal activity that could be extracted from membrane potential is the firing period. The phase resetting curve (PRC) is a quantitative measure of the relative change in the firing period of a neuron due to external perturbations such as synaptic inputs. The experimentally recorded periodic oscillations of membrane potential represent a one-dimensional projection of a closed trajectory, or limit cycle, in neuron’s multidimensional phase space. This chapter is entirely dedicated to the study of the relationship between the PRC and the geometry of the phase space trajectory. This chapter focuses on systematically deriving the mappings σ=σ(φ, μ) between the temporal phase φ and the geometric phase σ when some parameters μ are perturbed. For this purpose, both analytical approaches, based on the vector fields of a known theoretical models, and numerical approaches, based on experimentally recorded membrane potential, are discussed in the context of phase space reconstruction of limit cycle. The natural reference frame attached to neuron’s unperturbed limit cycle, γ breaks the perturbation of control parameter μ into tangent and normal displacements relative to the unperturbed γ. Detailed derivations of PRC in response to weak tangent and normal, perturbations of γ are provided. According to the geometric approach to PRC prediction, a hard, external perturbation forces the figurative point to cross the excitability threshold, or separatrix, in the phase space. The geometric method for PRC prediction detailed in this chapter gives accurate predictions both for hard inhibitory and excitatory perturbations of γ. The geometric method was also successfully generalized to a more realistic case of a neuron receiving multiple inputs per cycle.