1Department of Mathematics and Statistics, Boston University 2Howard Hughes Medical Institute, Janelia Research Campus 3Departments of Physiology and Psychiatry, University of California, San Francisco 4Kavli Institute for Fundamental Neuroscience, University of California, San Francisco 5Howard Hughes Medical Institute, University of California, San Francisco *Corresponding author: eric.denovellis@ucsf.edu
A state-space decoder combines what past spikes predict about a latent state with what the newest spike says. We check, spike by spike, whether these two sources of information point to overlapping regions of the state space. The result is analogous to a local, time-resolved goodness-of-fit test: it can show when a decoding model fails and suggest which of its assumptions is responsible.
State-space models relate high-dimensional neural activity to low-dimensional latent states, such as the position represented by hippocampal activity. Those states cannot be observed, so a decoder cannot be checked against ground truth. The usual validation tools, cross-validation and shuffle tests, summarize a whole recording in one number: they say whether a model beats chance or a simpler alternative, but not when during a session it fits poorly or which of its components is responsible.
Interpretable models are simplified on purpose. A place-cell decoder that ignores momentum or bursting can still produce scientifically useful trajectories. Rather than accept or reject such a model wholesale, we look inside it. At every spike, the filter already holds two distributions over the latent state: the prediction, which carries forward everything learned from past spikes, and the likelihood of the newest spike. We ask whether they are consistent.
The decoder is a recursive Bayesian filter, the cycle in Figure 1b. It keeps a posterior, its probability distribution over position given every spike so far, and advances it one time step at a time in two moves. First it predicts: the animal may have moved since the last step, so a movement model spreads the posterior into the prediction P(x). Then it updates: it multiplies the prediction by the likelihood of what the step recorded, the probability of those spikes at each position given the cells' place fields, and renormalizes the product into the new posterior. The next prediction starts from that posterior, so the prediction carries forward the evidence of every earlier spike, while the likelihood holds only the newest.
The likelihood comes from a Poisson model of firing. A cell's place field λc(x) is the number of spikes it is expected to fire in one time step when the animal is at x; the paper writes this expected count as a rate times the step length, λΔt, and here the step length is folded in. Given the position, each cell's spike count in a step is Poisson with that mean, independently of the other cells. The probability of what a step recorded, at each position, is then the product of λc(x) over the spikes that occurred, times exp(−Λ(x)), where Λ(x) = Σc λc(x) is the expected count from all the cells together and exp(−Λ(x)) is the probability that none of them fires. Factors that do not depend on x, such as 1/n!, vanish when the posterior is renormalized. A step with no spikes contributes only exp(−Λ(x)); because only a small fraction of a spike is expected per step, it is nearly flat and barely moves the posterior. A spike from cell c multiplies in λc(x) and pulls the posterior toward that cell's field.
Play through a short simulated recording as the animal runs up the track and back; positions are in arbitrary units (a.u.). At every step, the four rows show the calculation: the prediction, the cells' place fields with the one that fired highlighted, the likelihood, and the posterior. Between spikes, the prediction and posterior spread out as the animal's position grows uncertain. Near the end, two cells whose fields lie far from the animal fire together, and playback pauses there.
The recording is simulated and decoded by the paper's code, on the simulation's track and place fields. As in the paper's simulation, the decoder's movement model is a Gaussian random walk, which ignores the animal's momentum, as such decoders usually do. To make each step visible, the cells fire less often, and the random walk takes larger steps, than in the paper's simulation. The two distant spikes near the end were inserted into the simulated spike train. The prediction and posterior rows share one vertical scale, so a broader distribution is drawn lower; the likelihood is scaled to its own maximum. At a spike, the dashed curve is the posterior after the previous spike; at each no-spike step in between, the filter also multiplied by the nearly flat exp(−Λ(x)), which is not drawn. Pause and click the tracks to jump to a time step, or focus them and use ← →.
At each spike, the decoder's prediction P(x) is its probability distribution over the latent state x (here, position) given all earlier spikes: the one-step predictive distribution. The spike's likelihood Q(x) is its cell's place field λc(x) normalized to sum to one: the factor that spike contributed to the step's likelihood above, and how strongly that one spike supports each position. Each diagnostic compares P with Q, or with the spike itself. Two target consistency; KL divergence is included as the familiar reference that measures difference.
HPD: highest probability density.
Find the 95% highest-probability-density regions of the prediction and of the spike's likelihood, then divide the size of their intersection by the size of the smaller region. The value is 1 whenever one region lies inside the other and 0 when they do not meet, so a broad prediction containing a narrow spike still counts as consistent.
The predictive probability that a spike comes from cell c, fpred(c), averages each cell's place field λc(x) over the prediction and normalizes across cells. The p-value is the total predictive probability of every cell at least as unlikely as the one that fired, y. A small value means the model did not expect this spike. We call it the predictive p-value for short; the tracks below plot −log p (natural log), where p = 0.05 is about 3.
The most familiar way to compare two distributions, in nats (natural-log units). It measures difference, not consistency: it grows whenever the prediction is broad relative to the spike's likelihood, even when both agree about where the state is.
In the simulation, spikes are flagged when HPD overlap is at or below its 1st percentile, or KL divergence at or above its 99th percentile, in a well-specified baseline period; the p-value uses a fixed cutoff of 0.05. That HPD-overlap percentile is exactly 0, so in the simulation and the playground HPD overlap flags only spikes whose 95% region does not meet the prediction's at all.
The prediction aggregates many past spikes, while the likelihood comes from a single spike, so the two usually differ in shape and spread. What matters is whether they concentrate their probability in overlapping regions. Move the prediction, change its spread, and choose which cell fired: the three diagnostics update as you go.
The cells and rates are those of the paper's simulation: eleven place cells with Gaussian fields, plus, in the sparse epoch, five narrow, rarely firing cells near x = 30 a.u. while the ordinary ensemble is quiet. Curves are scaled to their own maxima. Flag thresholds are the simulation's. The diagnostics are computed in your browser by a port of the paper's code that is tested against the Python implementation.
We simulated place cells on a 100 a.u. track and decoded them with a Bayesian filter for 32 s. Three windows break the model on purpose; two controls contain unusual but internally consistent activity. Pick a condition, then hover over or tap the tracks, press Play, or click the tracks and use the arrow keys to inspect individual spikes.
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We decoded 203 units recorded from rat hippocampus during a spatial foraging task (Comrie et al., 2026) with two models. They share an observation model, built from each unit's place field, and differ in the prediction step: how they expect the represented position to move. The Continuous model uses a random walk, like the filter above, so the represented position can move only a short distance in each time bin. The Continuous–Fragmented model follows a model structure we previously used to capture replay and other nonlocal hippocampal representations (Denovellis et al., 2021). It adds a second, fragmented movement state: when the model is in, or switches into, that state, its next position can be anywhere on the track with equal probability, regardless of the current one. In each 2 ms time bin the model switches from continuous to fragmented with probability 0.02 and back with probability 0.02, inferring from the spikes which state is active along with the position. A fragmented state suits the hippocampus: while an animal is still, its hippocampal representation can jump to distant locations, as in awake replay (Karlsson & Frank, 2009). For the diagnostics, the Continuous–Fragmented prediction is summed over its two states, so both models are compared on the same position grid.
This two-second window contains a candidate replay event while the animal is still. The Continuous model's prediction cannot keep up with the rapidly moving representation, so many spikes are flagged. When the Continuous–Fragmented model switches to its fragmented state, its prediction becomes broad, the next spikes can move the estimate, and most of the HPD-overlap flags disappear. The recording has no known-good baseline period, so fixed cutoffs are used: a spike is flagged when HPD overlap is at or below 0.05 or the p-value at or below 0.05; KL divergence is not thresholded here. Inspect spikes the same way as in the simulation.
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A rescued spike is flagged under the Continuous model but not under the Continuous–Fragmented model: its likelihood's 95% region now overlaps the Continuous–Fragmented prediction's, which can be broad enough to allow a jump there. It does not mean that the prediction was centered on the spike.
Across the whole session, HPD overlap flags 18,790 spikes under the Continuous model. Once the model can jump, 92% of them are no longer flagged, while 176 spikes are flagged only under the Continuous–Fragmented model. The predictive p-value flags 33,954 spikes under the Continuous model, rescues 28% of them, and newly flags 1,706. Many spikes keep high KL divergence under the Continuous–Fragmented model, because its broad prediction differs from each spike's narrow likelihood. Together with the comparison between models and what is known about hippocampal replay, the local diagnostics support a specific refinement, a transition model that allows discontinuous jumps, rather than simply rejecting the Continuous model.
@misc{zeng2026local,
title = {Local goodness-of-fit measures for neural decoding},
author = {Zeng, Sirui and Comrie, Alison E. and Frank, Loren M. and
Eden, Uri T. and Denovellis, Eric L.},
year = {2026},
url = {https://github.com/edeno/statespacecheck-paper}
}
This entry will point to the preprint once it is posted.