Multi-neuron spike trains - distances and information.

Transcription

1 Multi-neuron spike trains - distances and information. Conor Houghton Computer Science, U Bristol NETT Florence, March 204

2 Thanks for the invite In 87 Stendhal reportedly was overcome by the cultural richness of Florence he encountered when he first visited the Tuscan city. As he described in his book Naples and Florence: A Journey from Milan to Reggio: As I emerged from the porch of Santa Croce, I was seized with a fierce palpitation of the heart (that same symptom which, in Berlin, is referred to as an attack of the nerves); the well-spring of life was dried up within me, and I walked in constant fear of falling to the ground. Wikipedia article

3 Motivation Overview. Mutual information for spaces with distances. [Rederive a result due to Kraskov, Stöbauer and Grassberger (PRE 2004)] Spike trains and sets of spike trains as a space with distance.

4 Motivation Spike trains. Spiking responses in the auditory forebrain of zebra finch. A B C D

5 Motivation Classical approach I. Discretize Split into words , 0000, 0000 Bialek, de Ruyer van Steveninck, Strong and other coworkers, late 990s.

6 Motivation Classical approach II. Estimate probability of words. For example, say w 8 = 0000 then estimate p(w 8 ) # occurrences of w 8 # words Calculate H(W ) = i p(w i ) log 2 p(w i ) = log 2 p(w i ) Bialek, de Ruyer van Steveninck, Strong and other coworkers, late 990s.

7 Motivation Classical approach III. Conditional probability. s 0 s s 2 s 3 s 4 s H(W s 0 ) H(W s ) H(W s 2 ) H(W s 3 ) H(W s 4 ) H(W s 5 ) Bialek, de Ruyer van Steveninck, Strong and other coworkers, late 990s.

8 Motivation Classical approach IV. Mutual information H(W S) = H(W s i ) and I (W ; S) = H(W ) H(W S) Bialek, de Ruyer van Steveninck, Strong and other coworkers, late 990s.

9 Motivation ms scale information in blow fly spike trains. Bialek, de Ruyer van Steveninck, Strong and other coworkers, late 990s.

10 Motivation Difficulties with the classical approach. Undersampling. 00 ms words and 2 ms bins gives 2 50 = words. Lots of clever approaches to this, for example Nemenman et al. (PRE 2004, BMC Neuroscience 2007) where a cunning prior is used for p(w i ). Sampling bias. An even distribution will never give equal counts for each word, giving different p(w i ). Lots of clever approaches to this too, see Panzeri et al. (J Neurophys. 2007). Ilya Nemenman, William Bialek and Rob de Ruyter van Steveninck. Physical Review E 69.5 (2004): 056. Panzeri, Stefano, et al. Journal of Neurophysiology 98.3 (2007):

11 Motivation Many fixes but still... Neuron - neuron mutual information. Maze - neuron mutual information. Mutual information with multiple units.

12 KDE for spike trains. Approaches to probability estimation Parameteric approach. See Gillespie and Houghton (JCN 20).... or Yu et al., (Front. in Comp. Neuro. 200). Non-parametric approach Histograms. Kernel density estimation (KDE). kth nearest neighbor (knn). We will see that KDE and knn lead to more-or-less the same formula. Yunguo Yu et al. Frontiers in Computational Neuroscience 4 (200).

13 KDE for spike trains. Kernel density estimation. p(x) = k(x x i ) n i Picture from density estimation

14 KDE for spike trains. Histograms and the classical approach Histogram: binning : R Z x int (x/δx)th bin Converting spike trains to words: discretization : space of spike trains Z r w

15 KDE for spike trains. What we need for kernel density estimation. So we must have p(x) = k(x x i ) n i k(x)dx = That means we need to be able to integrate.

16 KDE for spike trains. Spike train space.

17 KDE for spike trains. Integrating in the space of spike trains I. The space of spike trains has no coordinates, but it does have a measure given by the distribution of spike trains. vol(d) = P(r D) which can be estimated by the fraction of responses in D vol(d) # spike trains in D # spike trains

18 KDE for spike trains. Integrating in the space of spike trains II. volume 3/7

19 KDE for spike trains. Basic idea Want to calculate: I (R; S) = s S R p(r, s) log 2 p(r s) p(r) dr Use p(r)to give a measure. Use KDE to estimate p(r s). Any integrals will be estimated by counting points. Kernels will be defined using volume. R. Joshua Tobin and Conor J. Houghton, Entropy 5 (203)

20 KDE for spike trains. Result After some fiddling this give I (R; S) n r i log 2 n s c(r i, s i ; n h ) n h n r number of responses. n s number of stimuli. n h size of the kernel. c(r i, s i ; n h ) number of responses to stimulus s i in the kernel around r i. R. Joshua Tobin and Conor J. Houghton, Entropy 5 (203)

21 KDE for spike trains. c(r i, s i ; n h ) c(crossed, red, 3) = 2

22 KDE for spike trains. Results I We are interested in the structure of spike train space as a metric space, so simulated test data was constructed using the sort of distribution of spike trains in metric space observed in Gillespie and Houghton (JCN 20). James Gillespie and Conor Houghton, The Journal of Computational Neuroscience, 30 (20)

23 KDE for spike trains. Results II histogram estimated (ban) true (ban) KDE estimated (ban) true (ban) n s = 0 and n t = 0. R. Joshua Tobin and Conor J. Houghton, Entropy 5 (203)

24 KDE for spike trains. Results III histogram estimated (ban) true (ban) KDE estimated (ban) true (ban) n s = 0 and n t = 200. R. Joshua Tobin and Conor J. Houghton, Entropy 5 (203)

25 KDE for spike trains. KDE and knn KDE I (R; S) n r i log 2 n s c(r i, s i ; n h ) n h knn I e (R; S) Ϝ(n k ) + Ϝ(n t n s ) Ϝ(n t ) Ϝ[C(r i, s i ; n k )] n r i They use a Kozachenko and Leonenko estimator and, by a clever choice of how they pick k for different parts of the estimate, they get all the volume-based terms to cancel. Alexander Kraskov, Harald Stögbauer and Peter Grassberger. Physical Review E 69.6 (2004):

26 Distances Euclidean metric Picture from Google maps

27 Distances Non-Euclidean metric Picture from Google maps

28 Distances Non-metric Picture from

29 Distances van Rossum metric Spike trains mapped to functions and a metric on the space of functions induces a metric on the spike train space. Mark van Rossum, Neural Computation 3.4 (200):

30 Distances Multi-unit van Rossum metric There is a multi-unit easily computed version of the van Rossum metric. It relies on a time constant and a population parameter. Houghton, Conor and Kamal Sen, Neural computation 20.6 (2008): Houghton, Conor and Thomas Kreuz, Network: (202):

31 Distances The Victor-Purpura metric I. Edit one spike train in to the other:. Insertion of a spike with a cost of one. 2. Deletion of a spike with a cost of one. 3. Moving a spike a distance δt costs q δt. The distance is the cost of the cheapest edit. Jonathan D. Victor and Keith P. Purpura, Journal of Neurophysiology 76.2 (996):

32 Distances The Victor-Purpura metric II. u u 2 u 3 spike train u qδt qδt 2 spike train v v v 2 v 3 v 4 d = 3 + q(δt + δt 2 )

33 Distances Multi-neuron Victor-Purpura I. The Victor-Purpura metric can be extended to measure a distance between a pair of population responses by adding a cost k for changing the identity of a spike. Edit one spike train in to the other:. Insertion of a spike with a cost of one. 2. Deletion of a spike with a cost of one. 3. Change the neuron label of a spike at a cost k. 4. Moving a spike a distance δt costs q δt. The distance is the cost of the cheapest edit. Dmitriy Aronov et al., Journal of Neurophysiology 89.6 (2003):

34 Distances Multi-neuron Victor-Purpura II. u u 2 u 3 spike train u qδt + k qδt 2 spike train v v v 2 v 3 v 4 d = 3 + q(δt + δt 2 ) + k

35 Distances Labelled-line and population codes If k = 0 it does not matter what neuron a spike came from, this is a population code. It s like superimposing all the spike trains in the population and then working out the distance. If k = 2 it is never worth changing the label, this is a labelled-line code. The distance between the two population responses is just the sum of the distances for the individual pairs of responses for each neuron. Dmitriy Aronov et al., Journal of Neurophysiology 89.6 (2003):

36 Distances Problems Multi-unit VP metric computationally expensive. How do we calculate q and k?

37 Distances SPIKE metric. u u 2 u 3 spike train u δt δt 2 δt 3 spike train v v v 2 d = (δt + δt 2 + δt 3 ) + (δt + δt 3 ) = 2δt + δt 2 + 2δt 3 Thomas Kreuz, Daniel Chicharro, Conor Houghton, Ralph G. Andrzejak and Florian Mormann, The Journal Neurophysiology 09 (203)

38 Distances Multi-neuron SPIKE metric. Extending SPIKE to multiple neurons is easy, just add a distance between neurons. spike train u u u 2 u 3 δt δt 2 δt 3 + k spike train v v v 2 d == 2δt + 2δt 2 + δt 3 + k

39 Distances Multi-neuron SPIKE metric - problem. Still stuck with k.

40 The end Conclusions Distance-based measures of mutual information. Multi-neuron distance functions.

41 The end Acknowledgements Collaborators. Thomas Kreuz / Josh Tobin / James Gillespie / Cathal Cooney. Funding: the James S McDonnell Foundation.