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Figure 5.17: Interpixel variation. Here, error bars show standard deviation of mean single-element response across the 22-element array for leftward and rightward motion at different temporal fre- quencies or, equivalently, speeds
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Figure 5,18: 1/
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Spatial Frequency Spectra in Natural Scenes. Two natural images with corre- sponding spatial frequency power spectral densities. Straight lines indicate 1/P
slope.Figure 5.19: Example of random "natural" patterns. These one-dimensional stimuli have a spatial frequency power spectrum RUs) =
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(b)
Figure 5.21: Adding spatial noise. Diagrams show a one-dimensionall/
f
stimulus moving rightward at a constant velocity. In both cases, 1/f
spatial noise has been added with an effective (a) SNR of+
25 dB (b) or of -8 dB. Also shown are three "snapshots" of each image. Notice that under low SNR conditions, few obvious spatial features are apparent that could be reliably tracked to estimate image motion.Figure 5.22: Adding temporal noise. Diagrams show a I-D 1/
f
stimulus moving rightward at a constant velocity. In both cases, 1/f
temporal noise has been added. (a) SNR = +25 dB. (b) SNR= -8 dB. Also shown are three "photoreceptor signals" from each image. Notice that when the SNR is low, no obvious temporal features exist that could be used to estimate image motion.
58
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f
pattern moving at a constant velocity of 14 deg/s while 1/f
spatial noise was added. Error bars show one standard deviation of the time response, and represent residual deterministic pattern dependence such as that seen in Figure 5.2.Figure 5.24: Robustness with temporal noise. Performance of Reichardt detector array on direction discrimination with a 1/
f
pattern moving at a constant velocity of 14 deg/s while 1/f
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60
Chapter 6 Stimulus Reconstruction
In this chapter, we evaluate our sensor's ability to encode information about the velocity of a simple stimulus. We also compare our sensor's encoding ability with the encoding ability of the HS cell in the lobular plate of the blowfly Calliphora erythrocephala, which has been previously measured by other researchers. We replicated the experiments of Haag and Borst, 1997, which use stimulus reconstruction techniques to measure encoding fidelity. We shall first describe the techniques and then discuss the experimental results.
6.1 Stimulus Reconstruction Techniques
One way of measuring how well a sensor encodes a stimulus is to determine how well we can reconstruct an unknown stimulus from the sensor's response. The stimulus reconstruction technique used in the following experiments finds the linear filter that transforms the sensor response into an estimate of the stimulus that is optimal in the least-squares sense. That is, the reconstruction filter minimizes the square error between the estimate and the actual stimulus. This linear reconstruction represents a lower bound on encoding ability. Nonlinear filters could of course generate an estimate with a lower error, but we compare the sensor's response variability under repeated experiments with identical stimulus conditions to generate an upper bound on encoding ability. A more detailed description of these techniques may be found in Borst and Theunissen, 1999.
Suppose we stimulate a time-invariant system with a set of i Gaussian stimuli Si(t) and record responses ri(t). Given the frequency-domain representation of the stimuli Si(f) and the responses Ri (f), the optimal reverse reconstruction filter is given by the average cross-correlation normalized by the average autocorrelation of the responses:
(Ri(f) . Si(f)) Grev(f) = (Ri(f) . Ri(f))
Using this filter, we obtain the estimated stimulus Sesti(f) from the response:
Sesti(f)
=
Ri(f) . Grev(f)(6.1)
(6.2)
Taking the inverse Fourier transform of this signal gives us the time-domain stimulus estimate sesti ( t).
We can also calculate a filter between the stimulus Si(f) and the stimulus estimate Sesti(f).
This filter, denoted G(f), is given by
G(f) = (Si(f) . Sesti(f))
(Si(f) . Si(f)) (6.3)
and gives us an idea of how much information about the stimulus is transmitted at specific frequen- cies. It can be shown that G(f) is equivalent to the coherence function, ,2(f):
G(f)
=
2(f) = (Si(f)· Ri(f)) . (RHf)· Si(f)), (Si(f) . St(f)) (Ri(f)· Ri(f)) (6.4)
The coherence function is bounded between zero and one. A coherence value of one indicates that the stimulus is reconstructed perfectly by the reverse reconstruction filter at a certain frequency.
The coherence measure is derived from stimulus reconstruction fidelity, while expected coherence measure is derived from trial-to-trial response variability. (An identical Gaussian stimulus is used for each trial, so any variability in the responses must be due to noise in the sensor, assuming the system in time invariant.) The expected coherence is given by
(6.5) where snr is the signal-to-noise ratio. The signal is taken to be the average of all i responses, and the noise is calculated from the residual signal after subtracting the mean response from each individual response.
The coherence function is derived from a purely linear reconstruction filter technique and thus represents a lower bound on the information encoded by the sensor since nonlinear filters might better extract information about the stimulus from the response. The expected coherence function only takes into account trial-to-trial response variability and thus represents an upper bound on the information encoded by the sensor. (A broken sensor might have a constant zero-level output and achieve an expected coherence of one for all frequencies.)
6.2 Pattern Velocity Estimation by Fly Interneurons
The methods above have been applied to several biological sensory systems. Here we describe an experiment performed by Haag and Borst (Haag and Borst, 1997). A square-wave intensity grating displayed on an oscilloscope was presented to the eye of a female blowfly (Calliphora erythrocephala).
During the experiment, the grating moved with a random velocity (with a screen update rate of 200 Hz). The velocity profile had a flat ("white") spectrum up to approximately 20 Hz.
During stimulus presentation, an HS cell was recorded from using an intracellular electrode. As discussed in Chapter 2, HS cells are nonspiking cells in the lobular plate (third optic ganglion)
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