## Article Information

- Received September 15, 1997
- Revision received February 25, 1998
- Accepted March 3, 1998
- Published in print May 15, 1998.

## Author Information

## Author contributions

## Disclosures

- Received September 15, 1997.
- Revision received February 25, 1998.
- Accepted March 3, 1998.
This research was supported by National Institutes of Health Grants EY05603, RR00166, EY11378 and the McKnight Foundation. W.T.N. is an Investigator of the Howard Hughes Medical Institute.

We are grateful to Richard Olshen, Wyeth Bair, and Haim Sompolinsky for advice on mathematics, Marjorie Domenowske for help with illustrations, and Crista Barberini, Bruce Cumming, Greg DeAngelis, Eb Fetz, Greg Horwitz, Kevan Martin, Mark Mazurek, Jamie Nichols, and Fred Rieke for helpful suggestions. We are also grateful to two anonymous reviewers whose thoughtful remarks improved this paper considerably.

Correspondence should be addressed to Dr. Michael N. Shadlen, Department of Physiology, University of Washington Medical School, Box 357290, Seattle, WA 98195-7290.

^{d}Weak correlation*between*the pools representing a, b, and c would not affect the calculations substantially, because the resulting covariance terms would offset each other due to the sum and difference.↵FNa We were advised of this relationship by H. Sompolinsky and W. Bair. The area of the correlogram is: where Θ(τ) = T − ‖τ‖ is a triangular weighting function with a peak that lies at the center of the trial epoch of duration, T msec, and is the normalized cross- or autocorrelation function computed from bins of binary values, x

_{j‖k}(i), denoting the presence or absence of a spike in the i^{th}millisecond from neuron j or k. Mathematical details and a proof of Equation 1 will appear in a paper by E. Zohary, W. Bair, and W. T. Newsome (unpublished data).↵b Knowledge of λ does not imply exact knowledge of the input spike trains. Presumably there are many patterns of inputs that give rise to the same result, λ. Because our model for synaptic integration is deterministic, identical inputs would produce identical outputs. If

**x**were an exhaustive description of the input spike trains, that is, the time of every spike among all excitatory and inhibitory inputs, then var [N(T)‖**x**] = 0. To make sense of Equation 5, we need to make it clear that what we know about the inputs, as reflected in the conditional probabilities, is a scalar value that is computed from them, i.e., λ.↵FNc For many stochastic processes, including our random walk model, a change in rate is equivalent to scaling time. Hence the C

_{V}is constant. For these cases, Equation 9 also holds for time varying spike rates, λ(t), as long as the rate function can be repeated for each epoch contributing to the conditional variance. Thus, for a nonstationary Poisson process, the variance of the counts equals the mean of the counts.↵FNe One way to appreciate this is to try to reconstruct the spike intervals among a subset of input neurons from the pattern of output spikes. This is obviously impossible with just one output neuron, but it is equally hopeless with an arbitrarily large number of output neurons. The requisite information is encoded in the path of the membrane voltage between spikes, but such information is jettisoned from the spike code. If, in principle, there is no way to reconstruct such intervals, then they cannot encode information, except insofar as they reflect changes in spike rate.