Are single-peaked tuning curves tuned for speed rather than accuracy?
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eLife assessment
This is important work that addresses a long-standing (but rarely acknowledged) question: given that multi-peaked tuning curves optimize Fisher information, why do early sensory areas typically have single-peaked tuning curves? This paper shows clearly, and convincingly, that multi-peaked tuning curves are likely to produce catastrophic errors at short times, so if speed is important, multi-peaked tuning curves should be avoided. This work should encourage neuroscientists to take into account the importance of stimulus encoding time in their formulations of models of neural coding.
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Abstract
According to the efficient coding hypothesis, sensory neurons are adapted to provide maximal information about the environment, given some biophysical constraints. In early visual areas, stimulus-induced modulations of neural activity (or tunings) are predominantly single-peaked. However, periodic tuning, as exhibited by grid cells, has been linked to a significant increase in decoding performance. Does this imply that the tuning curves in early visual areas are sub-optimal? We argue that the time scale at which neurons encode information is imperative to understand the advantages of single-peaked and periodic tuning curves, respectively. Here, we show that the possibility of catastrophic (large) errors creates a trade-off between decoding time and decoding ability. We investigate how decoding time and stimulus dimensionality affect the optimal shape of tuning curves for removing catastrophic errors. In particular, we focus on the spatial periods of the tuning curves for a class of circular tuning curves. We show an overall trend for minimal decoding time to increase with increasing Fisher information, implying a trade-off between accuracy and speed. This trade-off is reinforced whenever the stimulus dimensionality is high, or there is ongoing activity. Thus, given constraints on processing speed, we present normative arguments for the existence of the single-peaked tuning organization observed in early visual areas.
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Author Response
Reviewer #1 (Public Review):
There has been a lot of work showing that multi-peaked tuning curves contain more information than single peaked ones. If that's the case, why are single-peaked tuning curves ubiquitous in early sensory areas? The answer, as shown clearly in this paper, is that multi-peaked tuning curves are more likely to produce catastrophic errors.
This is an extremely important point, and one that should definitely be communicated to the broader community. And this paper does an OK job doing that. However, it suffers from two (relatively easily fixable) problems:
I) Unless one is an expert, it's very hard to extract why multi-peaked tuning curves lead to catastrophicerrors.
II) It's difficult to figure out under what circumstances multi-peaked tuning curves are bad. This isimportant, because …
Author Response
Reviewer #1 (Public Review):
There has been a lot of work showing that multi-peaked tuning curves contain more information than single peaked ones. If that's the case, why are single-peaked tuning curves ubiquitous in early sensory areas? The answer, as shown clearly in this paper, is that multi-peaked tuning curves are more likely to produce catastrophic errors.
This is an extremely important point, and one that should definitely be communicated to the broader community. And this paper does an OK job doing that. However, it suffers from two (relatively easily fixable) problems:
I) Unless one is an expert, it's very hard to extract why multi-peaked tuning curves lead to catastrophicerrors.
II) It's difficult to figure out under what circumstances multi-peaked tuning curves are bad. This isimportant, because there are a lot of neurons in the sensory cortex, and one would like to know whether multi-peaked tuning curves are really a bad idea there.
And here are the fixes:
I) Fig. 1c is a missed opportunity to explain what's really going on, which is that on any particular trialthe positions of the peaks of the log likelihood can shift in both phase and amplitude (with phase being more important). However Fig. 1c shows the average log likelihood, which makes it hard to understand what goes wrong. It would really help if Fig. 1c were expanded into its own large figure, with sample log likelihoods showing catastrophic errors for multi-peaked tuning curves but not for single peaked ones. You could also indicate why, when multi-peaked tuning curves do give the right answer, the error tends to be small.
We thank the reviewer for this suggestion. We have now split the first figure into two.
In the new Figure 1, we provide an intuitive explanation of local vs catastrophic errors and single-peaked / periodic tuning curves. We have also added smaller panels to illustrate how the distribution of errors changes with decoding time (using a simulated single-peaked population).
The new Figure 2 shows sampled likelihoods for 3 different populations. We hope this provides some intuitive understanding of the phase shifts. Unfortunately, it proved difficult not to normalize the “height” of each module’s likelihood as they can differ by several orders of magnitude across the modules. However, due to the multiplication, the peak likelihood values can (approximately) be disregarded in the ML-decoding. Lastly, we have also added more simulation points (scale factors) compared to what we had in the earlier version of the figure (see panels d-e).
II) What the reader really wants to know is: would sensory processing in real brains be more efficient ifmulti-peaked tuning curves were used? That's certainly hard to answer in all generality, but you could make a comparison between a code with single peaked tuning curves and a good code with multi-peaked tuning curves. My guess is that a good code would have lambda_1=1 and c around 0.5 (you could use the module ratio the grid cell people came up with -- I think 1/sqrt(2) -- although I doubt if it matters much). My guess is that it's the total number of spikes, rather than the number of neurons, that matters. Some metric of performance (see point 1 below) versus the contrast of the stimulus and the number of spikes would be invaluable.
We thank the reviewer for this comment and the suggestions. We agree, ideally such an expression would be useful. However, as you note it is a very challenging task due to the large parameter space (number of neurons, peak amplitude, spontaneous firing rate, width of tuning, stimulus dimensionality etc) and is beyond the scope of the present study. We have instead included a new figure (see Figure 7 in the manuscript) detailing the minimal decoding times for various choices of parameter values. We believe this gives an indication to how minimal decoding time scales with various parameters.
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eLife assessment
This is important work that addresses a long-standing (but rarely acknowledged) question: given that multi-peaked tuning curves optimize Fisher information, why do early sensory areas typically have single-peaked tuning curves? This paper shows clearly, and convincingly, that multi-peaked tuning curves are likely to produce catastrophic errors at short times, so if speed is important, multi-peaked tuning curves should be avoided. This work should encourage neuroscientists to take into account the importance of stimulus encoding time in their formulations of models of neural coding.
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Reviewer #1 (Public Review)
There has been a lot of work showing that multi-peaked tuning curves contain more information than single peaked ones. If that's the case, why are single-peaked tuning curves ubiquitous in early sensory areas? The answer, as shown clearly in this paper, is that multi-peaked tuning curves are more likely to produce catastrophic errors.
This is an extremely important point, and one that should definitely be communicated to the broader community. And this paper does an OK job doing that. However, it suffers from two (relatively easily fixable) problems:
I. Unless one is an expert, it's very hard to extract why multi-peaked tuning curves lead to catastrophic errors.
II. It's difficult to figure out under what circumstances multi-peaked tuning curves are bad. This is important, because there are a lot of neurons …
Reviewer #1 (Public Review)
There has been a lot of work showing that multi-peaked tuning curves contain more information than single peaked ones. If that's the case, why are single-peaked tuning curves ubiquitous in early sensory areas? The answer, as shown clearly in this paper, is that multi-peaked tuning curves are more likely to produce catastrophic errors.
This is an extremely important point, and one that should definitely be communicated to the broader community. And this paper does an OK job doing that. However, it suffers from two (relatively easily fixable) problems:
I. Unless one is an expert, it's very hard to extract why multi-peaked tuning curves lead to catastrophic errors.
II. It's difficult to figure out under what circumstances multi-peaked tuning curves are bad. This is important, because there are a lot of neurons in sensory cortex, and one would like to know whether multi-peaked tuning curves are really a bad idea there.
And here are the fixes:
I. Fig. 1c is a missed opportunity to explain what's really going on, which is that on any particular trial the positions of the peaks of the log likelihood can shift in both phase and amplitude (with phase being more important). However Fig. 1c shows the average log likelihood, which makes it hard to understands what goes wrong. It would really help if Fig. 1c were expanded into its own large figure, with sample log likelihoods showing catastrophic errors for multi-peaked tuning curves but not for single peaked ones. You could also indicate why, when multi-peaked tuning curves do give the right answer, the error tends to be small.
II. What the reader really wants to know is: would sensory processing in real brains be more efficient if multi-peaked tuning curves were used? That's certainly hard to answer in all generality, but you could make a comparison between a code with single peaked tuning curves and a _good_ code with multi-peaked tuning curves. My guess is that a good code would have lambda_1=1 and c around 0.5 (you could use the module ratio the grid cell people came up with -- I think 1/sqrt(2) -- although I doubt if it matters much). My guess is that it's the total number of spikes, rather than the number of neurons, that matters. Some metric of performance (see point 1 below) versus the contrast of the stimulus and the number of spikes would be invaluable.
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Reviewer #2 (Public Review):
The authors try to introduce the encoding time factor into theories of optimal encoding of information in the nervous system
The major strength is in the rigorous analysis and in the simple yet important take home message.
The authors achieved their aim by proving their point with rigorous analyses and the results support their conclusions
The paper makes a simple yet important addition and will likely call for neuroscientists to include more carefully the importance of stimulus encoding time in their formulations of models of neural coding and in the interpretations of results.
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