View source for Signal-to-noise ratio in neuroscience

== What is signal-to-noise ratio (in the context of a neuroscience experiment)? ==

The signal-to-noise ratio allows us to quantify the size of the applied or controlled signal relative to fluctuations that are outside experimental control. It has general applicability to the analysis of sensory discrimination (by nerve cells and by whole organisms) and to the performance of networks. 

A common use of SNR is to compare the quality of electrophysiological recordings containing events (for instance action potentials) recorded in the presence of noise. This measure is used (although often just approximately by reading off an oscilloscope by eye) to decide whether a recording location is adequate to begin spike sorting, or whether to move the electrode on. This can be quantified by the ratio of the variances of the event signal train and the noise. Another application, which we will discuss in more detail, is the use of signal-to-noise ratio to characterise the reliability of neural information transmission. For the sake of choosing language we will examine SNR in a sensory system, although a motor system could be studied the same way. __AUTOLINKER{language}

A sensory neuroscience experiment typically involves the collection of neural responses of some kind over repeated trials in which a particular stimulus signal is presented. Over the course of an experiment, a distribution of different stimuli (signals) will be presented, and a distribution of responses to each stimulus recorded. Note that the SNR values we will discuss here in general depend upon the choice of input signal used by the experimenter - they thus characterise the combination of the system plus external stimulus, rather than being an intrinsic property of the system itself. Two classes of experiments might be undertaken, and calculating SNR for each differs slightly.

==== Discrete stimuli ====

Here, the stimulus takes on discrete values because of limitations on the data collecting capacity of the experiment (for instance, where 30 degree increments of the orientation of a grating stimulus are used) or because of the fundamental nature of the experiment (in a signal detection task, for instance). Let us examine the situation where the response to discrete stimulus ''s'' (one of ''S'' stimuli in total) is <math>r_s\ .</math> In this case, the SNR is
:<math>SNR = \frac{P_S}{P_N} = \frac{E[r_s^2]}{\sigma_N^2}</math>
where <math>E[.]</math> denotes the expectation over stimuli, i.e. <math> 1/S \sum_s r_s^2</math> if stimuli are equiprobable, and <math>\sigma_N^2</math> is the noise variance, i.e. it measures the amount of variability that still occurs when the stimulus is held to a fixed value, which can be measured in practice by taking the variance of the distribution of responses across trials where that stimulus is repeated. If this noise variance does not depend upon the stimulus (but just takes a fixed value <math>\sigma_N^2</math>), then the SNR is simply as given above. If the noise variance depends upon which stimulus has been presented (which for instance will be true in a Poisson spiking scenario where the noise variance is equal to the mean response for the stimulus), then the average noise variance must be used, such that the signal-to-noise ratio becomes
:<math>SNR = \frac{1/S \sum_s r_s^2}{1/S \sum_s \sigma^2_N(s)} \ .</math>
Note that the extension to non-equal stimulus probabilities is straightforward.

In the special case of a signal detection task (two stimuli: signal present or absent), with the signal causing a change in the response <math>\Delta r\ ,</math> then the signal power becomes <math>P_S = (\Delta r)^2\ .</math> In this case the noise variance is still measured as the variance across experimental trials.

==== Stimuli continuously varying in time ====

[[Image:fig1_SRS.png|thumb|400px|right|F1|Signal transmission across a toy "synapse", which here is modeled simply as a 7th order Butterworth filter with cutoff frequency at 200 Hz . '''A''' White noise input signal (lowpass filtered at 1 kHz). '''B''' Gaussian distribution of inputs in A. '''C''' Mean signal in the response (blue line) and several traces of the noise in the response (grey lines). '''D''' Probability densities of signal (filled circles) and noise (open circles). This simulation was motivated by the experimental results of de Ruyter van Steveninck and Laughlin (1996). Source code available from the curator.]]An example of a continuous-stimulus is found in the study of signal transmission from photoreceptors through chemical synapses to the non-spiking large monopolar cells of the blowfly ''Calliphora vicina'' (de Ruyter van Steveninck and Laughlin 1996). In this scenario, a pseudo-random contrast signal is applied to the photoreceptors through a light-emitting diode, and the graded-potential output of the neuron (at the other side of the synapse) recorded. How are the signal and noise measured? Note that if the diode output were to be used directly to represent the signal, then the noise would have to be measured in the same units; this would mean referring the noise to the input (Horowitz and Hill, 1989) - see the Exemplars section below for one method. However, it is instead possible to measure both signal and noise power at the neural response recording site. Figure 1, at right, uses a computer simulation of a simplified version of the experiment to explain how this can be done. Here, a white fluctuating stimulus waveform (a short segment of which is shown in Fig. 1A) is presented on many repeated trials. In the real experiment, it controls the contrast of an LED; here, as shown in Fig. 1B, it has zero mean and unit standard deviation. In the simulation, the signal is passed through a "synapse", which in our simulation is modeled simply as a 7th order Butterworth filter with cutoff frequency at 200 Hz. The mean over trials of the output of this model is the average response shown by the solid line in Fig. 1C; the distribution of response values over the length of the stimulus sequence is shown by the filled circles in Fig. 1D. By squaring the modulus of the Fourier transform of this response, and normalizing appropriately, we arrive at the power spectral density (shown in Fig. 2A) of the signal <math>P_S(f)</math> (where ''f'' is frequency). Several examples of the fluctuations around that mean response on individual trials are shown by the grey lines in Fig. 1C; these have a probability density (over the length of the sequence and over trials) given by the open circles in 1D. By averaging together the power spectral densities of these fluctuation traces, we obtain the noise power spectral density <math>P_N(f)</math>shown in Fig. 2A. We now have everything we need to calculate the signal-to-noise ratio for this scenario. But note: to do this, we have assumed that the noise is independent from trial to trial. If this is not the case, for instance due to the presence of very slow fluctuations, our procedure will not be valid.
[[Image:fig2.png|thumb|400px|right|F2|Calculation of signal-to-noise ratio for the model synapse described in Figure 1. '''A''' Power spectral densities for the presynaptic signal (white noise low-pass filtered at 1 kHz), the average postsynaptic signal, and the postsynaptic fluctuations (noise) from the simulation. '''B''' Signal-to-noise ratio at each frequency. Source code available from the curator.]]

The signal-to-noise ratio pertaining to each frequency is
:<math>SNR(f) = \frac{P_S(f)}{P_N(f)}</math>
with <math>P_S(f)</math> and <math>P_N(f)</math> defined as described above. For the simulation of Fig. 1, this is shown in Fig. 2B. The overall signal-to-noise ratio is the ratio of the area under the signal and noise power spectral density curves,
:<math>SNR = \frac{\int df P_S(f) }{\int df P_N(f) } \ .</math>