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Ardo Illaste edited res_fitfunction.md
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### Signal fitting
#### Fitting function for transients
The function used for fitting Ca2+ release events is shown on Figure \ref{fig:fit}. The shape of the function is described by 4 parameters: amplitude (\(A\)), rise and decay time constants (\(\tau_{r},\tau_{d}\)) and plateau duration (\(d\)). An additional parameter (\(\mu\)) determines the time when the maximum is reached. The function describing a transient is:
\[ \begin{equation}
g(A, d, \tau_{d}, \tau_{r}, \mu, t) = A\cdot\left\{
\begin{array}{l l}
1-\exp\left( -\frac{t-\mu}{\tau_r}\right)\cdot\exp\left(-2\right) & \quad \mu-2\tau_r\leq t<\mu\\\\
1-\exp(-2) & \quad \mu\leq t< \mu+d\\\\
\exp\left( -\frac{t-\mu-d}{\tau_d}\right)\cdot\left(1-\exp(-2)\right)& \quad t\geq\mu+d\\\\
0 &\quad \mathrm{otherwise}
\end{array}
\right.\] \right.
\end{equation}
The transient consists of four phases: zero level before the onset of the transient, an exponential increase with time constant \(\tau_r\) starging when \(t=\mu-2\tau_r\), a plateau phase of duration \(d\) starting at \(t=\mu\) and an exponential decay with time constant \(\tau_d\) starting at \(t=\mu+d\)