Population Coding and Decoding

Each neuron responds most strongly to its preferred direction. The tuning-curves graph shows expected firing rates across all possible directions; the vertical line marks the presented stimulus. Use its arrow buttons, or click a curve, to highlight a neuron. The population-response graph samples those curves at the presented direction.

Tuning curves are non-negative von Mises functions. The displayed half-width is the angular distance from the preferred direction at which the stimulus-driven response is half its maximum. All neurons share the same baseline, amplitude and width, and their preferred directions are evenly spaced.

The population vector adds each neuron's preferred-direction vector, weighted by its response. Its angle is the decoded direction. With a dense, symmetric population of expected responses it closely matches the stimulus; sparse populations and narrow tuning can produce discretization errors.

Expected rates shows deterministic encoding. One stochastic trial independently draws each neuron's spike count from a Poisson distribution whose mean is its expected rate multiplied by the observation time. The bars show the resulting spike counts expressed as rates; the overlaid points show their expectations. Adjusting a model parameter generates a fresh random sample, as does Rerun. Display-only changes, such as graph text size, preserve the current sample.

Stochastic modes automatically display a spike raster with 250 ms of baseline activity, the stimulus interval, and 250 ms after the stimulus. Baseline spikes use the baseline firing-rate setting. During the shaded stimulus interval, the raster contains exactly the sampled spike counts used for decoding; only these stimulus-period counts enter the decoders. Conditional on a count, spike times are placed independently and uniformly within the period, as expected for a stationary Poisson process. Timing carries no additional information in this rate-code model, and firing rates change instantaneously at stimulus onset and offset.

Many stochastic trials repeats the same stimulus and compares two decoders. The population vector sums preferred-direction vectors weighted by neuronal responses. The maximum-likelihood decoder tests every whole-degree direction and chooses the one that makes the observed Poisson spike counts most probable. The polar plot shows one example trial; the accuracy graph summarizes all trials as signed circular errors.

Mean absolute error reports typical error size. Bias is the circular mean of the signed errors: values near zero indicate that over- and underestimates balance. These measures describe simulated performance under this model; they are not measurements from a biological population.

In many-trial mode, the arrow buttons select which trial is shown in the response, raster, decoder and example-statistics displays. The lower raster panel pools the highlighted neuron's spikes across every trial into a peri-stimulus time histogram (PSTH). Its expected-rate line shows the baseline and stimulus rates used by the model.

Try this

• Reduce the population to four neurons and narrow the tuning curves.
• Increase the number of neurons and watch the decoded direction approach the stimulus.
• Generate repeated trials without changing the stimulus and compare their decoding errors.
• Shorten the observation time and examine why fewer spikes make the estimate less reliable.
• Increase baseline firing and see how stimulus-independent spikes add uncertainty.
• Compare population-vector and maximum-likelihood decoding with sparse, narrowly tuned populations.
• Run more trials and observe how the estimated error distribution becomes smoother.

Model basis

• Georgopoulos, Schwartz & Kettner (1986): directional tuning and population-vector coding.
• Seung & Sompolinsky (1993): population-vector and maximum-likelihood decoding models.
• Jazayeri & Movshon (2006): likelihood-based decoding of sensory population responses.
• Ma et al. (2006): Poisson-like variability and probabilistic population codes.