With in mind the view to go after more complex representations, in the future.

I thank the reviewers and editors for assessing the findings as important and the evidence as compelling, and more generally @elife.bsky.social for a smooth overall process.

Representations are the basis of cognition. But it seems to me that they are not always well understood, or characterized, in cognitive science and other behavioral sciences.
This paper is an attempt to do that, and more precisely to get at the principles under the representations of simple scalars.

Endogenous precision of the number sense A theory of efficient coding wherein the precision of representations is endogenous, task-dependent, and prior-dependent predicts scaling laws for imprecision and is supported by numerosity perception...

The "Version of Record" of my paper with Mike Woodford is finally there. I am very glad with this paper that points to new laws of psychophysics, which we derived from a carefully considered principle of endogenous efficient coding.

Isaiah Berlin on FDR in the 1930s:

"... to construct a regime which should provide for greater economic equality and social justice - ideals which were the best part of the tradition of American life - without altering the basis of freedom and democracy in his country."

Where is our Roosevelt!

"How much would you pay to not know?" (WTP right to ignorance)
"How much would you ask for being forced to know?" (WTA right to ignorance)
"How much would you pay to know?" (WTP right to know)
"How much would you ask for not being told?" (WTA right to know)

Also it is conceivable that some would be ready to pay for knowing, so as to be liberated from the throes of uncertainty. That would add a term in your calculation of the "utility" of knowing.

So now you have four numbers, corresponding to the answers to the questions:

The willingness to accept is usually larger than the WTP so you might get more positive amounts if you ask "how much do you ask to relinquish your right to ignorance?"...

Junior Theoretical Neuroscientists Workshop Junior Theoretical Neuroscientists Workshop on Simons Foundation

Applications are now open for our Junior Theoretical Neuroscientists Workshop which will take place July 21 - 24, 2026 at the Center for Computational Neuroscience @flatironinstitute.org

Learn more and apply by April 15 at www.simonsfoundation.org/event/jrwork...

Just posted an update of this study, where we show how the receptive fields of number-encoding neural populations shift and rescale with the prior — "distributed range adaptation" — in line with (dynamic) efficient coding, and predictive of behavior. Check it out!
www.biorxiv.org/content/10.1...

Random compressed coding with neurons Blanco Malerba et al. demonstrate that extended, irregular tuning curves, observed in the brain, efficiently compress information in a low-dimensional representation. Optimal smoothness of tuning curv...

Great work. I wonder how this relates to www.cell.com/cell-reports... which says something similar. (But with very ≠ setups in terms of modeled tuning curves. Also the metric used is different: you show how the FI scales with D, they show the prob of errors is exponential in the number of neurons.)

Fast efficient coding and sensory adaptation in gain-adaptive recurrent networks As the statistics of sensory environments often change, neural sensory systems must adapt to maintain useful representations. Efficient coding prescribes that neuronal tuning curves should be optimize...

Efficient coding theories often implicitly or explicitly assume slow changes in tuning (e.g., through synaptic plasticity). Arthur Prat-Carrabin has collected psychophysical data showing that it can be fast, and this can be explained by a gain-adaptive RNN:
www.biorxiv.org/content/10.1...