Dynamic gain calculation

EL Elinor Lazarov MD Melanie Dannemeyer BF Barbara Feulner JE Jörg Enderlein MG Michael J. Gutnick FW Fred Wolf AN Andreas Neef

This protocol is extracted from research article:

An axon initial segment is required for temporal precision in action potential encoding by neuronal populations

**
Sci Adv**,
Nov 28, 2018;
DOI:
10.1126/sciadv.aau8621

An axon initial segment is required for temporal precision in action potential encoding by neuronal populations

Procedure

The frequency transfer function was calculated from responses to injected fluctuating current, using a method originally introduced by Bryant and Segundo (*64*) with modifications. AP time was registered when the somatic membrane voltage crossed 3 mV, which corresponds to the steepest point on the AP waveform. The STA current was calculated for each cell from ~600 spikes by averaging stimulus waveform in a temporal window of 500 ms before and after the spike. To improve signal-to-noise ratio, the STA was filtered in the frequency domain using a Gaussian window *w*(*f*′), centered at frequency *f*′ = *f*, with an SD of *f*/2π

This averages out neighboring frequency components of similar amplitude but random phase, that is, noise. Deterministic frequency components with a phase that changes only mildly within the Gaussian window are not affected by this windowing. Thus

If the train of APs is idealized as a discrete sequence of numbers with zero for empty samples and 1/*dt* for samples carrying an AP, then the product of the STA current and the firing rate ν equals the cross-correlation between input current and AP output. The frequency response function (or the dynamic gain), *G*(*f*), was then calculated as the ratio between the Fourier transform of this cross-correlation _{corr} is the correlation time of the noise.

To average the gain curves from N cells, we averaged the STA currents. To avoid overrepresentation of cells with a smaller input resistance, that is, cells that require a larger amplitude of current fluctuations, we weighted the STA curves:

For each neuron and for the population average, we calculated the confidence intervals of the gain curve and the noise floor by balanced bootstrap. The confidence interval at a given frequency *f*′ was defined by the 2.5th and 97.5th percentile of *G*_{BST}(*f*′) for 200 bootstrap gain curves calculated from 200 random samples of actual AP times. The noise floor at a certain frequency is understood as 95th percentile of

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