Detection by Deconvolution

Deconvolution is just the inverse of convolution. The convolutionmof two real-valued functions of real argument f and g, denotedm=f _∗g, results from flipping one of

the functions (i.e.f(t)→f(−t)), shifting it bys(f(−t)→f(s₋t)) and evaluating the

integral of the product with the other:

c(s) =

Z

−∞ ∞

g(t)f(s₋t) dt.

The convolution functionc(s)gives the area of overlap ofgand flippedf for all values

of the relative shifts.

The reason why the convolution operation is so important is that it models the process of acquisition of measurements through an instrument. The measurement v

is generallynot the underlying reality but the convolution of the instrument function

f with that underlying reality, g. This is denoted asc=f _∗g. Deconvolution is the

process of obtaininggfrom the measurements and some knowledge of the instrument,

or transfer, functionf. As an example, astronomical observations of stars usually show

so-called diffraction spikes that make them look like crosses. Deconvolution can be used to remove those spikes by mathematically reversing the optical transformation applied by the telescope.

What instrument f are we deconvolving from the measured signal c, when we

observe currents in voltage clamp? The requirements of helping detection of a postsy- naptic current and observing the presynaptic spikes match in this case. For detection we desire to have as compact a deconvolution as possible — to avoid precisely the kind of mixing in which fast-paced currents incur. But we also know that, as long as the current is of synaptic origin, it is caused by an action potential elsewhere, which is already a very compact function, extending for at most 1-2 ms (vs. perhaps 10 for the current it ultimately generates on the postsynaptic end).

Thus it becomes of help to consider what is the synaptic transfer function, even though we have detection and reconstruction phases that are uncoupled, i.e. we can detect with a biologically unrealistic kernel and still fit with a plausible one. This is a complex subject because if we really want to go back all the way to the presy- naptic spike, many processes are involved, from the presynaptic vesicle dynamics to the electronics of the amplifier. A good start was offered by the function in panel 5 of Figure 3.5. We adopted instead the somewhat simpler double exponential of Equa- tion 3.1, which we denote here, for short,α. The question is, given that the current

is the convolution of the synaptic kernelαand the spikes g,c=α_∗g, how to obtain

the spikes back from the current. The theory of Green functions, central to solving differential equations, offers the way to associate to a convolution kernel αits cor-

responding operator (synaptic operator) such thatαˆ[c] =g. For linear combinations

of exponentials, only derivatives are involved. In particular, for our double exponen- tial the first and second derivative of the currentcare needed. Derivatives are a double

and sharp-edged sword in signal analysis because they amplify the noise. They are often used together with filtering to compensate for that. Here we determined (by looking at the data) that the second derivative brought about by the double expo- nential kernel does not benefit the detection process and we thus selected for practical reasons the simpler synaptic operator presented above in Equation 3.2, which cor- responds to the simple mono-exponential synaptic kernel α(t) = Θ(t ₋t₀) e−(t−t0)/τ

and has only one derivative.

Analytic deconvolution operators are nice, but application of the convolution the- orem allows to solve for g even if no analytic expression forαˆ can be found. The

convolution theorem states

where_·is the usual product of functions andF is one of a number of integral trans-

forms. These statements for functions carry over naturally into the discrete domain for time series. The Fourier transform is the most commonly used of the conformant transforms and one of special interest because it is fast to numerically compute, using the fast Fourier transform algorithm (FFT;O(NlogN)instead ofO(N2_{)). The decon-}

volved signalgbefore application of the synaptic kernelαcan thus always be found as g=F−1 F[c] F[α] .

Selection of the synaptic operator parameter The choice ofτcan be, in fact, guided by the data. Richardson and Silberberg (2008) use a variational approach, whereby the deconvolution is calculated for several trial constants τ_t and tabulated. A good single-time-constant deconvolution of a multiexponential event does not retrieve a perfect impulse, but should tail-off to zero. Richardson and Silberberg exploit this requirement and select theτt minimizing the long-time tail of the deconvolved trace.

This is not an option for us, as there is often very little event tail available, and we resort here to using theτdfrom prior knowledge of the kinetics of spontaneous events

and leaving to the reconstruction step its fine-tuning, together with the other PSC parameters. Incidentally, it is also no option for them, but they can assume the kinetics to remain the same during the experiment and thus can always use the last event to find

τd. Applying the variational procedure for doublets is conceivable, but the approach

reaches its limit well below the number of tightly packed events observed in our traces.

Standardized evaluation of reconstruction algorithms We propose that future

implementations are benchmarked with regard to the three following, non-dimensional characteristics. These suggestions can be tweaked to suit the characteristics of the particular family of synaptic kernels under use, but the important point is to allow systematic comparisons by creating classes of equivalence of the input with respect to the reconstruction process.

asymmetry. the ratio between the time constant dominating the rise and that

dominating the decay, i.e. roughly, here,τr/τd. Asymmetry is critical to any detection procedure leaning on derivatives (such as deconvolution). Note that a double exponential (Eq. 3.1) with asymmetry 1, i.e. τd=τr is still rather asymmetric, which begs for a better name or a better characterization (e.g. 20-80% amplitude variations can be calculated analytically in terms of the time constants).

burst rate.the ratio between the duration of the event and the inter-event time

interval∆, i.e. roughly, here,(τr+τd)/∆. The burst rate is an expression of

the amount of shadowing of one event by the next and is key to the fitting of time constants, be it on the deconvolved trace or in the original one. In some cases it may be of interest to look rather atτr/∆, since overlapping in the rise phase poses qualitatively harder challenges.

signal to noise ratio.the quotient between event amplitude and a measure of the

noise level (such as its standard deviation σ, if it is a Gaussian process), i.e.

roughlyA/σ.

The dataset reported here presented often asymmetries 1, burst rates of 1 to 5 and variable signal-to-noise ratios reaching down to 1 (standard deviation of_∼5pA with the lowest admitted PSC size decreed to be 5 pA).