# Mathematical Biology Seminar

### Gemma Huguet

Polytechnical University of Catalonia

#### Phase and Amplitude response functions: mathematical tools for phase control in transient-states of spiking neurons

#### Friday, September 6, 2013, at 3:05pm

LCB 225

The phase response curve (PRC) is a powerful tool to study the effect of a perturbation
on the phase of an oscillator, assuming that all the dynamics can be explained by the phase
variable. However, factors like the rate of
convergence to the oscillator, strong forcing or high stimulation frequency may invalidate
the above assumption and raise the question of how is the phase variation away from an attractor.
In this talk, I will present a numerical method to perform the effective computation of
the phase advancement when we stimulate an oscillator which has not reached yet the
asymptotic state (a limit cycle) using the concept of isochrons. To do so, we first perform
a careful study of the theoretical grounds (the parameterization method for
invariant manifolds), which allow us to describe the isochronous sections of the limit cycle.
From it, we build up Phase Response Functions (PRF) and Amplitude Response Function (ARF)
to control changes in the transversal variables. In order to make this theoretical framework
applicable, we design an efficient numerical scheme to compute both the isochrons and the
PRSs of a given oscillator. Finally, I will show some examples of the computations we have
carried out for some well-known biological models. Finally, I will compare the predictions
given by the PRC-approach (a 1D map) to those given by the PRF-ARF-approach (a 2D map),
under pulse-train stimuli.