# 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.

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