Monitoring Integrated Activity of Individual Neurons Using FRET-Based Voltage-Sensitive Dyes

Abstract

Pairs of membrane-associated molecules exhibiting fluorescence resonance energy transfer (FRET) provide a sensitive technique to measure changes in a cell’s membrane potential. One of the FRET pair binds to one surface of the membrane and the other is a mobile ion that dissolves in the lipid bilayer. The voltage-related signal can be measured as a change in the fluorescence of either the donor or acceptor molecules, but measuring their ratio provides the largest and most noise-free signal. This technology has been used in a variety of ways; three are documented in this chapter: (1) high throughput drug screening, (2) monitoring the activity of many neurons simultaneously during a behavior, and (3) finding synaptic targets of a stimulated neuron. In addition, we provide protocols for using the dyes on both cultured neurons and leech ganglia. We also give an updated description of the mathematical basis for measuring the coherence between electrical and optical signals. Future improvements of this technique include faster and more sensitive dyes that bleach more slowly, and the expression of one of the FRET pair genetically.

Appendix

We consider the statistical analysis for determining follower cells (Cacciatore et al. 1999; Taylor et al. 2003; Kleinfeld 2008). The significance of the spectral coherence between the response of any cell, labeled “i”, and the driven cell is used to determine if two cells are functionally related and thus are a candidate for a synaptically driven pair. The coherence is a complex function, denoted C_i(f), that it is calculated over the time period of the stimulus, denoted T. We further denote the time series of the optical signals as Vi(t) and the electrical reference drive signal as U(t). The mean value is removed to form:

$$ \delta {\mathrm{V}}_{\mathrm{i}}\left(\mathrm{t}\right)={\mathrm{V}}_{\mathrm{i}}\left(\mathrm{t}\right)-\frac{1}{\mathrm{T}}{\displaystyle {\int}_0^{\mathrm{T}}\mathrm{d}\mathrm{t}\ {\mathrm{V}}_{\mathrm{i}}\left(\mathrm{t}\right)} $$

with a similar expression for δU(t). The Fourier transform of δVi(t) with respect to the k-th Slepian window (Thomson 1982; Percival and Walden 1993), denoted w^(k)(t), is:

$$ \delta {{\tilde{\mathrm{V}}}_{\mathrm{i}}}^{\left(\mathrm{k}\right)}\left(\mathrm{f}\right)=\frac{1}{\sqrt{\mathrm{T}}}{\displaystyle \underset{0}{\overset{\mathrm{T}}{\int }}\mathrm{d}\mathrm{t}\kern0.5em {e}^{-\mathrm{i}2\uppi \mathrm{f}\mathrm{t}}{\mathrm{w}}^{\left(\mathrm{k}\right)}\left(\mathrm{t}\right)\;\delta {\mathrm{V}}_{\mathrm{i}}\left(\mathrm{t}\right)} $$

with a similar expression for δŨ^(k) (f). The use of multiple tapers allow averaging over a bandwidth that is set by the number of tapers, K, with the half-bandwidth at half-maximal response given by Δf = (1/T)(K + 1)/2. Our interest lies in the values of C_i(f) for f = f_Drive and the confidence limits for these values. We chose the bandwidth so that the estimate of |C_i(f_Drive)| is kept separate from that of the harmonic |C_i(2f_Drive)|. The choice Δf = 0.4 f_Drive works well, so that for f_Drive = 1 Hz and T = 9 s the integer part of 2 • 0.4 • 1 Hz • 9 s—1 yields K = 6 tapers (Fig. 6.3). The spectral coherence between the optical signal and the reference is give by:

$$ {\mathrm{C}}_{\mathrm{i}}\left(\mathrm{f}\right)=\frac{\frac{1}{\mathrm{K}}{\displaystyle \sum_{\mathrm{k}=1}^{\mathrm{K}}\delta {{\tilde{\mathrm{V}}}_{\mathrm{i}}}^{\left(\mathrm{k}\right)}\left(\mathrm{f}\right){\left[\updelta {\tilde{\mathrm{U}}}^{\left(\mathrm{k}\right)}\left(\mathrm{f}\right)\right]}^{*}}}{\sqrt{\left(\frac{1}{\mathrm{K}}{\displaystyle \sum_{\mathrm{k}=1}^{\mathrm{K}}{\left|\delta {{\tilde{\mathrm{V}}}_{\mathrm{i}}}^{\left(\mathrm{k}\right)}\left(\mathrm{f}\right)\right|}^2}\right)\left(\frac{1}{\mathrm{K}}{\displaystyle \sum_{\mathrm{k}=1}^{\mathrm{K}}{\left|\delta {\tilde{\mathrm{U}}}^{\left(\mathrm{k}\right)}\left(\mathrm{f}\right)\right|}^2}\right)}}. $$

To calculate the standard errors for the coherence estimates, we use the jackknife (Thomson and Chave 1991) and compute delete-one averages of coherence, denoted C_i ⁽ⁿ⁾ (f), where n is the index of the deleted taper:

$$ {\mathrm{C}}_{\mathrm{i}}^{\left(\mathrm{n}\right)}\left(\mathrm{f}\right)=\frac{\frac{1}{\mathrm{K}-1}{\displaystyle \sum_{{\scriptscriptstyle \begin{array}{l}\mathrm{k}=1\\ {}\mathrm{k}\ne \mathrm{n}\end{array}}}^{\mathrm{K}}\delta {{\tilde{\mathrm{V}}}_{\mathrm{i}}}^{\kern-0.25em \left(\mathrm{k}\right)}\left(\mathrm{f}\right){\left[\delta {\tilde{\mathrm{U}}}^{\left(\mathrm{k}\right)}\left(\mathrm{f}\right)\right]}^{*}}}{\sqrt{\left(\frac{1}{\mathrm{K}-1}{\displaystyle \sum_{{\scriptscriptstyle \begin{array}{l}\mathrm{k}=1\\ {}\mathrm{k}\ne \mathrm{n}\end{array}}}^{\mathrm{k}}{\left|\delta {{\tilde{\mathrm{V}}}_{\mathrm{i}}}^{\kern-0.25em \left(\mathrm{k}\right)}\left(\mathrm{f}\right)\right|}^2}\right)\left(\frac{1}{\mathrm{K}-1}{\displaystyle \sum_{{\scriptscriptstyle \begin{array}{l}\mathrm{k}=1\\ {}\mathrm{k}\ne \mathrm{n}\end{array}}}^{\mathrm{K}}{\left|\delta {\tilde{\mathrm{U}}}^{\left(\mathrm{k}\right)}\left(\mathrm{f}\right)\right|}^2}\right)}}\ \forall\ \mathrm{n}. $$

Estimating the standard error of the magnitude of C_i(f) requires an extra step since |C_i(f)| is defined on the interval [0, 1] while Gaussian variables exist on (−∞, ∞). Thus the delete-one estimates, |C_i ⁽ⁿ⁾(f)|, were replaced with the transformed values:

$$ \mathrm{g}\left\{\left|{\mathrm{C}}_{\mathrm{i}}\right|\right\}= log\left(\frac{{\left|{\mathrm{C}}_{\mathrm{i}}\right|}^2}{1-{\left|{\mathrm{C}}_{\mathrm{i}}\right|}^2}\right). $$

The mean of the transformed variable is:

$$ {\mu}_{\mathrm{i};\;\mathrm{M}\mathrm{a}\mathrm{g}}\left(\mathrm{f}\right)=\frac{1}{\mathrm{K}}{\displaystyle \sum_{\mathrm{n}=1}^{\mathrm{K}}\mathrm{g}\left\{{\mathrm{C}}_{\mathrm{i}}^{\left(\mathrm{n}\right)}\left(\mathrm{f}\right)\right\}} $$

and the standard error of the transformed variable is:

$$ {\sigma}_{\mathrm{i};\;\mathrm{M}\mathrm{a}\mathrm{g}}\left(\mathrm{f}\right)=\sqrt{\frac{\mathrm{K}-1}{\mathrm{K}}{\displaystyle \sum_{\mathrm{n}=1}^{\mathrm{K}}{\left|\mathrm{g}\left\{{\mathrm{C}}_{\mathrm{i}}^{\left(\mathrm{n}\right)}\left(\mathrm{f}\right)\right\}-{\mu}_{\mathrm{i};\mathrm{M}\mathrm{a}\mathrm{g}}\left(\mathrm{f}\right)\right|}^2}}. $$

The 95 % confidence interval for the coherence is thus:

$$ \left[\sqrt[-1]{1+{e}^{-\left({\mu}_{\mathrm{i};\mathrm{M}\mathrm{a}\mathrm{g}}-2{\sigma}_{\mathrm{i};\mathrm{M}\mathrm{a}\mathrm{g}}\right)}},\kern0.5em \sqrt[-1]{1+{e}^{-\left({\mu}_{\mathrm{i};\mathrm{M}\mathrm{a}\mathrm{g}}+2{\sigma}_{\mathrm{i};\mathrm{M}\mathrm{a}\mathrm{g}}\right)}}\right]. $$

We now turn to an estimate of the standard deviation of the phase of C(f). Conceptually, the idea is to compute the variation in the relative directions of the delete-one unit vectors C_i(f)/|C_i(f)|. The standard error is computed as:

$$ \begin{array}{cc}\hfill {\sigma}_{\mathrm{i};\mathrm{Phase}}\ \left(\mathrm{f}\right)=\sqrt{2\frac{\mathrm{K}-1}{\mathrm{K}}\left(\mathrm{K}-\left|{\displaystyle \sum_{\mathrm{n}=1}^{\mathrm{K}}\frac{{\mathrm{C}}_{\mathrm{i}}^{\left(\mathrm{n}\right)}\left(\mathrm{f}\right)}{\left|{\mathrm{C}}_{\mathrm{i}}^{\left(\mathrm{n}\right)}\left(\mathrm{f}\right)\right|}}\right|\right)}\hfill & \hfill \forall \kern0.5em \mathrm{n}\hfill \end{array}. $$

We graph the magnitude and phase of C_i(f_Drive) for all neurons, along with the confidence interval, on a polar plot (Fig. 6.4e). Finally, we consider whether the coherence of a given cell at f_Drive is significantly greater than zero, that is, larger than one would expect to occur by chance from a signal with no coherence. We compared the estimate for each value of |C_i(f_Drive)| to the null distribution for the magnitude of the coherence, which exceeds

$$ \left|{\mathrm{C}}_{\mathrm{i}}\left({\mathrm{f}}_{\mathrm{Drive}}\right)\right|=\sqrt{1-{\alpha}^{1/\left(K-1\right)}} $$

only in α of the trials (Hannan 1970; Jarvis and Mitra 2001). We use α = 0.001 to avoid false-positives. We also calculate the multiple comparisons of α level for each trial, given by α_multi = 1 − (1 − α)N, where N is the number of cells in the functional image, and verified that it did not exceed α_multi = 0.05 on any trial.