# PRC

The following method describes how can one obtain the phase reset curve (PRC) as a function of timings of pulses. It turns out using the PRC of a system we could basically say a lot about the interactions between neurons without actually simulating the whole differential equations taking the maps route. I thought I will write this as a note for the future stupid me wondering why I chose this route.

Let a neuron be spiking and

\begin{align} T_1&=\text{normal time period}\\ T_2&=\text{perturbed time period} \end{align}

The perturbation is happening at time $$t_s$$, therefore,

\begin{align} t_s/T_1&=\text{phase before perturbation}\\ T_2-t_s&=\text{time to next spike} \end{align}

Then, the phase has become $$(T_1-(T_2-t_s))/T_1$$

This therefore gives the following phase difference,

\begin{align} \Delta \phi &= \text{new phase} - \text{old phase}\\ &=((T_1-(T_2-t_s))/T_1)-t_s/T_1\\ &=(T_1-T_2)/T_1 \end{align}

Here, $$T_1$$, $$T_2$$ are the timings. $$t_s$$ is the time of perturbation. $$T_1$$ is the free running period and $$T_2$$ is the perturbed period.