Appendix A — Mathematical formulas
Curve fitting functions
Monoexponential
\[ f(x) = b_0 + b_1 e^{-x / \tau} \]
Biexponential (decay)
\[ f(x) = b_0 + b_1 e^{-x / \tau_1} + b_2 e^{-x / \tau_2} \]
Biexponential (event)
\[ f(x) = b_0 + b_1 \left(1 - e^{-x / \mathrm{rise}}\right)e^{-x / \mathrm{decay}} \]
Triexponential
\[ f(x) = b_0 + b_1 e^{-x / \tau_1} + b_2 e^{-x / \tau_2} + b_3 e^{-x / \tau_3} \]
Here, \(x\) is a vector of times normalized by subtracting the first time point.
Starting estimates for exponential fits
\[ b_0 = \frac{1}{n}\sum_{i=1}^{n} D_i \]
\[ b_1 = D_1 - D_n \]
\[ \tau = \frac{t_{\mathrm{half}}}{\ln(2)} \]
Here, \(D_1\) and \(D_n\) are the first and last data values in the region being fitted, respectively, and \(t_{\mathrm{half}}\) is the time from \(D_1\) to half-amplitude.
For biexponential (decay) fits, the starting estimates are weighted as follows:
\[ \tau_1 = 0.1\tau, \qquad \tau_2 = 0.9\tau \]
For triexponential fits:
\[ \tau_1 = \tau_2 = \tau_3 = \frac{\tau}{3} \]
Action potential threshold methods
Here, \(v\) is a vector of recorded membrane-potential samples; \(v'\), \(v''\), and \(v'''\) are its first, second, and third derivatives, respectively; \(V_{\mathrm{thr}}\) is the estimated action-potential threshold; and \(LB\) is a lower bound set by the user.
Method I
Introduced by Sekerli et al. (2004):
\[ V_{\mathrm{thr}} = \underset{v}{\arg\max}\, f(v) \quad \text{where } v' \ge LB \]
\[ f(v) = \frac{v''}{v'} \]
Method II
Introduced by Sekerli et al. (2004):
\[ V_{\mathrm{thr}} = \underset{v}{\arg\max}\, f(v) \quad \text{where } v' \ge LB \]
\[ f(v) = \frac{v'''v' - (v'')^2}{(v')^3} \]
Leading inflection
\[ V_{\mathrm{thr}} = \arg\min_v v' \]
Maximum curvature
Introduced by Rossokhin and Saakian (1992):
\[ V_{\mathrm{thr}} = \arg\max_v f(v) \]
\[ f(v) = v''\left(1 + (v')^2\right)^{-3/2} \]
Polynomials for detrending
The polynomial equations used for detrending and Events detection, such as Curve thresholding, have the standard form:
\[ a_0 + a_1x + \cdots + a_nx^n \]
Spike frequency accommodation
Spike frequency accommodation (\(SFA\)) is calculated from the inter-spike intervals in a train of action potentials. \(ISI_{\mathrm{first}}\) is the first inter-spike interval in the spike train, and \(ISI_{\mathrm{last}}\) is the final inter-spike interval.
Divisor method
\[ SFA_{\mathrm{divisor}} = \frac{ISI_{\mathrm{first}}}{ISI_{\mathrm{last}}} \]
Local variance
The local variance (\(LV\)) method was introduced by Shinomoto et al. (2003):
\[ SFA_{LV} = \frac{1}{n - 1}\sum_{i=1}^{n-1}\frac{3\left(ISI_i - ISI_{i+1}\right)^2}{\left(ISI_i + ISI_{i+1}\right)^2} \]