Local linear approximation of the Jacobian matrix better captures phase resetting of neural limit cycle oscillators

One effect of any external perturbations, such as presynaptic inputs, received by limit cycle oscillators when they are part of larger neural networks is a transient change in their firing rate, or phase resetting. A brief external perturbation moves the figurative point outside the limit cycle, a geometric perturbation that we mapped into a transient change in the firing rate, or a temporal phase resetting. In order to gain a better qualitative understanding of the link between the geometry of the limit cycle and the phase resetting curve (PRC), we used a moving reference frame with one axis tangent and the others normal to the limit cycle. We found that the stability coefficients associated with the unperturbed limit cycle provided good quantitative predictions of both the tangent and the normal geometric displacements induced by external perturbations. A geometric-to-temporal mapping allowed us to correctly predict the PRC while preserving the intuitive nature of this geometric approach.