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 analysis:course-w16:week10 [2016/02/14 15:24]mvdm [Preparing firing rates for decoding] analysis:course-w16:week10 [2017/01/05 15:16]mvdm [Bayesian decoding] Both sides previous revision Previous revision 2017/01/05 15:16 mvdm [Bayesian decoding] 2017/01/05 15:16 mvdm [Bayesian decoding] 2016/02/16 14:01 mvdm [Quantifying decoding accuracy] 2016/02/15 10:18 mvdm [Estimating tuning curves] 2016/02/15 10:05 mvdm [Estimating tuning curves] 2016/02/15 10:00 mvdm [Estimating tuning curves] 2016/02/14 15:51 mvdm [Spike train analysis II: tuning curves, encoding, decoding] 2016/02/14 15:50 mvdm 2016/02/14 15:50 mvdm [Challenges] 2016/02/14 15:38 mvdm [Quantifying decoding accuracy] 2016/02/14 15:24 mvdm [Preparing firing rates for decoding] 2016/02/14 15:23 mvdm [Preparing firing rates for decoding] 2016/02/14 15:21 mvdm 2016/02/14 15:17 mvdm [Spike train analysis II: tuning curves, encoding, decoding] 2016/02/14 15:17 mvdm [Estimating tuning curves] 2016/02/14 15:08 mvdm [Visual inspection] 2016/02/14 15:07 mvdm [Estimating tuning curves] 2016/02/14 15:00 mvdm [Preparing firing rates for decoding] 2016/02/14 14:58 mvdm [Preparing firing rates for decoding] 2016/02/14 14:57 mvdm [Preparing firing rates for decoding] 2016/02/14 14:55 mvdm 2016/02/14 14:47 mvdm [Preparing firing rates for decoding] 2016/02/14 14:44 mvdm [Estimating tuning curves] 2016/02/14 14:38 mvdm [Spike train analysis II: tuning curves, encoding, decoding] 2016/02/14 14:38 mvdm [Bayesian decoding] 2016/02/14 14:35 mvdm [Bayesian decoding] 2016/02/14 14:29 mvdm [Bayesian decoding] 2016/02/14 14:09 mvdm [Estimating tuning curves] 2016/02/14 14:07 mvdm [Estimating tuning curves] 2016/02/14 14:00 mvdm [Visual inspection] 2016/02/14 13:57 mvdm [Estimating place cell tuning curves (place fields)] 2016/02/13 16:53 mvdm 2016/02/13 16:18 mvdm created Next revision Previous revision 2017/01/05 15:16 mvdm [Bayesian decoding] 2017/01/05 15:16 mvdm [Bayesian decoding] 2016/02/16 14:01 mvdm [Quantifying decoding accuracy] 2016/02/15 10:18 mvdm [Estimating tuning curves] 2016/02/15 10:05 mvdm [Estimating tuning curves] 2016/02/15 10:00 mvdm [Estimating tuning curves] 2016/02/14 15:51 mvdm [Spike train analysis II: tuning curves, encoding, decoding] 2016/02/14 15:50 mvdm 2016/02/14 15:50 mvdm [Challenges] 2016/02/14 15:38 mvdm [Quantifying decoding accuracy] 2016/02/14 15:24 mvdm [Preparing firing rates for decoding] 2016/02/14 15:23 mvdm [Preparing firing rates for decoding] 2016/02/14 15:21 mvdm 2016/02/14 15:17 mvdm [Spike train analysis II: tuning curves, encoding, decoding] 2016/02/14 15:17 mvdm [Estimating tuning curves] 2016/02/14 15:08 mvdm [Visual inspection] 2016/02/14 15:07 mvdm [Estimating tuning curves] 2016/02/14 15:00 mvdm [Preparing firing rates for decoding] 2016/02/14 14:58 mvdm [Preparing firing rates for decoding] 2016/02/14 14:57 mvdm [Preparing firing rates for decoding] 2016/02/14 14:55 mvdm 2016/02/14 14:47 mvdm [Preparing firing rates for decoding] 2016/02/14 14:44 mvdm [Estimating tuning curves] 2016/02/14 14:38 mvdm [Spike train analysis II: tuning curves, encoding, decoding] 2016/02/14 14:38 mvdm [Bayesian decoding] 2016/02/14 14:35 mvdm [Bayesian decoding] 2016/02/14 14:29 mvdm [Bayesian decoding] 2016/02/14 14:09 mvdm [Estimating tuning curves] 2016/02/14 14:07 mvdm [Estimating tuning curves] 2016/02/14 14:00 mvdm [Visual inspection] 2016/02/14 13:57 mvdm [Estimating place cell tuning curves (place fields)] 2016/02/13 16:53 mvdm 2016/02/13 16:18 mvdm created Line 1: Line 1: ~~DISCUSSION~~ ~~DISCUSSION~~ - - :!: **UNDER CONSTRUCTION -- PLEASE DO NOT USE YET** :!: ===== Spike train analysis II: tuning curves, encoding, decoding ===== ===== Spike train analysis II: tuning curves, encoding, decoding ===== Line 9: Line 7: * Learn to estimate and plot tuning curves, raw and smoothed * Learn to estimate and plot tuning curves, raw and smoothed * Implement a basic Bayesian decoding algorithm * Implement a basic Bayesian decoding algorithm - * Compare decoded and actual position by exporting to a movie file + * Compare decoded and actual position by computing the decoding error Resources: Resources: Line 86: Line 84: - ENC_S = restrict(S,run_start,​run_end); + LoadMetadata;​ - ENC_pos = restrict(pos,​run_start,​run_end); + ENC_S = restrict(S,metadata.taskvars.trial_iv); + ENC_pos = restrict(pos,​metadata.taskvars.trial_iv); + % check for empties and remove % check for empties and remove keep = ~cellfun(@isempty,​ENC_S.t);​ keep = ~cellfun(@isempty,​ENC_S.t);​ ENC_S.t = ENC_S.t(keep);​ ENC_S.t = ENC_S.t(keep);​ ENC_S.label = ENC_S.label(keep);​ ENC_S.label = ENC_S.label(keep);​ + S.t = S.t(keep); S.t = S.t(keep); S.label = S.label(keep);​ S.label = S.label(keep);​ ​ - We have created ''​ENC_''​ versions of our spike trains and position data, containing only data from when the rat was running on the track (the ''​run_start''​ and ''​run_end''​ variables have been previously generated by a different script) and removed all cells from the data set that did not have any spikes on the track. + We have created ''​ENC_''​ versions of our spike trains and position data, containing only data from when the rat was running on the track (using experimenter annotation stored in the metadata; ​''​trial_iv'' ​contains the start and end times of trials) and removed all cells from the data set that did not have any spikes on the track. ☛ Plot the above scatterfield again for the restricted spike train. Verify that no spikes are occurring off the track by comparing your plot to the previous one for the full spike trains, above. ☛ Plot the above scatterfield again for the restricted spike train. Verify that no spikes are occurring off the track by comparing your plot to the previous one for the full spike trains, above. Line 116: Line 115: y_edges = SET_ymin:​SET_yBinSz:​SET_ymax;​ y_edges = SET_ymin:​SET_yBinSz:​SET_ymax;​ - occ_hist = histcn(pos_mat,​y_edges,​x_edges);​ + occ_hist = histcn(pos_mat,​y_edges,​x_edges); ​% 2-D version of histc() - no_occ_idx = find(occ_hist == 0); % NaN out bins rat never visited + no_occ_idx = find(occ_hist == 0); % NaN out bins never visited occ_hist(no_occ_idx) = NaN; occ_hist(no_occ_idx) = NaN; - occ_hist = occ_hist .* (1/30); % convert to seconds using video frame rate + occ_hist = occ_hist .* (1/30); % convert ​samples ​to seconds using video frame rate (30 Hz) subplot(221);​ subplot(221);​ Line 161: Line 160: {{ :​analysis:​course-w16:​raw_tc.png?​nolink&​900 |}} {{ :​analysis:​course-w16:​raw_tc.png?​nolink&​900 |}} - Note that from the occupancy map, you can see the rat spent relatively more time at the choice point compared to other segments of the track. However, the rough binning is not very satisfying. Let's see if we can do better with some smoothing: + Note that from the occupancy map, you can see the rat spent relatively more time at the base of the stem compared to other segments of the track. However, the rough binning is not very satisfying. Let's see if we can do better with some smoothing: Line 170: Line 169: occ_hist(no_occ_idx) = NaN; occ_hist(no_occ_idx) = NaN; - occ_hist = occ_hist .* (1/​30); ​% convert to seconds using video frame rate + occ_hist = occ_hist .* (1/30); subplot(221);​ subplot(221);​ Line 178: Line 177: % % spk_hist = histcn(spk_mat,​y_edges,​x_edges);​ spk_hist = histcn(spk_mat,​y_edges,​x_edges);​ - spk_hist = conv2(spk_hist,​kernel,'​same'​);​ + spk_hist = conv2(spk_hist,​kernel,'​same'​); ​% 2-D convolution spk_hist(no_occ_idx) = NaN; spk_hist(no_occ_idx) = NaN; Line 267: Line 266: In general, from the [[http://​en.wikipedia.org/​wiki/​Poisson_distribution | definition of the Poisson distribution]],​ it follows that In general, from the [[http://​en.wikipedia.org/​wiki/​Poisson_distribution | definition of the Poisson distribution]],​ it follows that - $P(n_i|\mathbf{x}) = \frac{(\tau f_i(\mathbf{x}))^{n_i}}{n_i!} e^{-\tau f_i (x)}$ + $P(n_i|\mathbf{x}) = \frac{(\tau f_i(\mathbf{x}))^{n_i}}{n_i!} e^{-\tau f_i (\mathbf{x})}$ $f_i(\mathbf{x})$ is the average firing rate of neuron $i$ over $x$ (i.e. the tuning curve for position), $n_i$ is the number of spikes emitted by neuron $i$ in the current time window, and $\tau$ is the size of the time window used. Thus, $\tau f_i(\mathbf{x})$ is the mean number of spikes we expect from neuron $i$ in a window of size $\tau$; the Poisson distribution describes how likely it is that we observe the actual number of spikes $n_i$ given this expectation. $f_i(\mathbf{x})$ is the average firing rate of neuron $i$ over $x$ (i.e. the tuning curve for position), $n_i$ is the number of spikes emitted by neuron $i$ in the current time window, and $\tau$ is the size of the time window used. Thus, $\tau f_i(\mathbf{x})$ is the mean number of spikes we expect from neuron $i$ in a window of size $\tau$; the Poisson distribution describes how likely it is that we observe the actual number of spikes $n_i$ given this expectation. Line 276: Line 275: $P(\mathbf{n}|\mathbf{x}) = \prod_{i = 1}^{N} \frac{(\tau f_i(\mathbf{x}))^{n_i}}{n_i!} $$P(\mathbf{n}|\mathbf{x}) = \prod_{i = 1}^{N} \frac{(\tau f_i(\mathbf{x}))^{n_i}}{n_i!} - e^{-\tau f_i (x)}$ + e^{-\tau f_i (\mathbf{x})}$$ An analogy here is simply to ask: if the probability of a coin coming up heads is $0.5$, what is the probability of two coints, flipped simultaneously,​ coming up heads? If the coins are independent then this is simply $0.5*0.5$. An analogy here is simply to ask: if the probability of a coin coming up heads is $0.5$, what is the probability of two coints, flipped simultaneously,​ coming up heads? If the coins are independent then this is simply $0.5*0.5$. Line 475: Line 474: end end - plot(Q_tvec_centers,​dec_err,'​.k'​);​ ​ Line 483: Line 481: % get trial id for each sample % get trial id for each sample trial_id = zeros(size(Q_tvec_centers));​ trial_id = zeros(size(Q_tvec_centers));​ - trial_idx = nearest_idx3(run_start,​Q_tvec_centers);​ % NOTE: on non-Windows,​ use nearest_idx.m + trial_idx = nearest_idx3(metadata.taskvars.trial_iv.tstart,​Q_tvec_centers);​ % NOTE: on non-Windows,​ use nearest_idx.m trial_id(trial_idx) = 1; trial_id(trial_idx) = 1; trial_id = cumsum(trial_id);​ trial_id = cumsum(trial_id);​ Line 498: Line 496: This yields: This yields: - {{ :analysis:cosmo2014:dec_err_1step_250ms.png?600 |}} + {{ :analysis:course-w16:dec_err.png?nolink&600 |}} - Thus, on average our estimate is 1.87 pixels away from the true position. Earlier laps seem to have some more outliers of bins where our estimate is bad (large distance) but there is no obvious trend across laps visible. + (Note, your plot might look a little different.) + + Thus, on average our estimate is 2.14 pixels away from the true position. Earlier laps seem to have some more outliers of bins where our estimate is bad (large distance) but there is no obvious trend across laps visible. ☛ How does the decoding accuracy depend on the bin size used? Try a range from very small (10ms) to very large (1s) bins, making sure to note the average decoding error for 50ms bins, for comparison with results in the next module. What factors need to be balanced if the goal is maximum accuracy? ☛ How does the decoding accuracy depend on the bin size used? Try a range from very small (10ms) to very large (1s) bins, making sure to note the average decoding error for 50ms bins, for comparison with results in the next module. What factors need to be balanced if the goal is maximum accuracy? Line 519: Line 519: This gives: This gives: - {{ :analysis:cosmo2014:2d_decerror_space_250ms.png?600 |}} + {{ :analysis:course-w16:dec_errspace.png?nolink&600 |}} It looks like the decoding error is on average larger on the central stem, compared to the arms of the maze. It looks like the decoding error is on average larger on the central stem, compared to the arms of the maze. Line 527: Line 527: ==== Challenges ==== ==== Challenges ==== - Visual inspection of the animation or movie suggests that the decoding does a decent job of tracking the rat's true location. However, especially because of the number of parameters involved in the analysis (bin size, how firing rates are computed, the Poisson and independence assumptions,​ etc.) it is important to quantify how well we are doing. + ★ Implement a decoding ​analysis on your own data. Remember that this does not necessarily requires using spiking data -- anything that you can construct ​a tuning curve for would work! In this module, we had something like 100 simultaneously recorded neurons, but even if you have only one, you can still attempt to use it for decoding. Quantify decoding performance (error) for a few relevant parameters. + + ★ How does decoding performance scale with the number of cells used? This is an important ​issue if we want to figure out if we should invest resources in attempting to record from more neurons, or if we have all we need in data sets such as this one. - ★ Modify the visualization code above to also compute ​a //decoding error// for each frame. This should be the distance between the rat's actual location ​and the location with the highest posterior probability ​(the "​maximum a posteriori"​ or MAP estimate). Plot this error over time, excluding those bins where no cells were active. How does this error change over the course ​of the session? How does it change if you reduce the bin size for decoding to 100ms? + ★ In a famous paper, [[http://science.sciencemag.org/content/318/​5852/​900 | Johnson ​and Redish ​(2007)]] showed that the hippocampus transiently represents possible future trajectories as rats appeared to deliberate between choices (left? right?) at a decision point. However, they used a controversial "​two-step"​ decoding algorithm which attracted criticism. Refer to the Methods section ​of that paper to figure out how they did the decoding, and modify the code above to implement their version. What differences do you notice?