analysis:course-w16:week10

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analysis:course-w16:week10 [2016/02/14 15:23] mvdm [Preparing firing rates for decoding] |
analysis:course-w16:week10 [2018/07/07 10:19] (current) |
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~~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 ===== | ||

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* 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: | ||

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<code matlab> | <code matlab> | ||

- | 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); | ||

</code> | </code> | ||

- | 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. | ||

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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); | ||

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{{ :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: |

<code matlab> | <code matlab> | ||

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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); | ||

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% | % | ||

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; | ||

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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. | ||

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\[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 $0.5*0.5$. | ||

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%% only include data we care about (runs on the maze) | %% only include data we care about (runs on the maze) | ||

Q_tsd = tsd(Q_tvec_centers,Q); | Q_tsd = tsd(Q_tvec_centers,Q); | ||

- | Q_tsd = restrict(Q_tsd,run_start,run_end); | + | Q_tsd = restrict(Q_tsd,metadata.taskvars.trial_iv); |

</code> | </code> | ||

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end | end | ||

- | plot(Q_tvec_centers,dec_err,'.k'); | ||

</code> | </code> | ||

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% 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); | ||

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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? | ||

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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. | ||

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==== 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? |

analysis/course-w16/week10.1455481406.txt.gz · Last modified: 2018/07/07 10:19 (external edit)

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