Supplementary Materials

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1 Supplementary Materials Neural population partitioning and a onurrent brain-mahine interfae for sequential motor funtion Maryam M. Shanehi, Rollin C. Hu, Marissa Powers, Gregory W. Wornell, Emery N. Brown & Ziv M. Williams Supplementary Modeling Model onstrution using the expetation-maximization (EM) algorithm We model the ativity of eah neuron under any given sequene as an inhomogeneous Poisson 49, 50 proess whose lielihood funtion (using the theory of point proesses) is given by K N i Si Si 1 pn ( 1: K S ) ( ( ) ) exp( ( ) ) i 1:1, (1) where is the time inrement taen to be small enough to ontain at most one spie, binary spie event of the th neuron in the time interval [( 1), ], ( S ) is its i N is the instantaneous firing rate in that interval, S i is the i th sequene, and K is the total number of bins in a duration K. For eah sequene and neuron, we need to estimate the firing rate ( S ) using the neuronal data observed. One way to do so is to bin the data into non-overlapping windows of fixed length during whih the firing rate is assumed to be onstant and estimated using maximum lielihood tehniques. This method is equivalent to finding the peristimulus time histogram (PSTH) that simply averages the number of spies over any given window. The main drawba of this tehnique is that unless there are a large number of training trials under eah sequene, to get a good estimate one has to pi relatively large windows. This in turn mass the fine-saled evolution of the firing rate. Also, there is no prinipled way for seleting a window size, whih the analysis is dependent on. 1 Nature Neurosiene: doi: /nn.350 i

2 One way to avoid these problems in estimating the spie rate funtion is to use a state-spae approah 4, 51 (See also alternative methods using Gaussian proesses in prior wor 54 ). This approah is used in many appliations to estimate an unobservable state proess and onsists of two models: A prior or state model that in general enfores any prior information available about the unobservable states suh as a simple ontinuity ondition and an observation model that relates the neuronal observations to these states. In the ase of estimating the spie rate funtion, and sine it is a non-negative quantity, similar to previous wor 4, 51 we tae the state at time inrement, x, to be the logarithm of the firing rate, i.e., x lo g( ( Si) ), or equivalently ( S ) exp( x ), () i and enfore a ontinuity ondition on it by assuming that it evolves aording to a linear firstorder Gaussian model 4, 51, x, x 1 where is the zero-mean white Gaussian noise with variane. The observation model is in turn given by substituting () in (1). Here, is an unnown parameter of the model and should be estimated jointly with the state. Hene we use the expetation-maximization (EM) iterative algorithm to find the maximum lielihood estimate of and in turn estimate the firing rate 4, 51, 5 () i. Denoting the estimate of in the i th iteration by, its estimate in the i 1 th iteration after the maximization step is given by K ( i 1) 1 W W 1 W, 1 K 1, (3) () i () i where W E[ x N1: K; ] and W, 1 E[ x 1x N1: ; ] are found from the forward filter, fixed-interval smoothing, and ovariane reursive algorithms in the expetation step as follows. Assuming that there are J total trials and denoting the ausal filter state estimate by x E x N and its variane by w, and the smoothed state estimate by () i [ 1: ; ] x E x N and its variane by w () i K [ 1: K ; ] forward filter reursions 4, 51, 55, K K, the reursions in the E-step are given by the Nature Neurosiene: doi: /nn.350

3 w x w x w x () i ( w Jexp( x ) ) J x 1 w N j x 1 j 1 ( ( ) exp( ) ), for 1,, K, where N ( j ) is the spie event in trial j, and by the fixed interval smoothing reursions 4, 51, 56, A w w 1 1 x x A ( x x ) K 1 K 1 w w A ( w w ), K 1 K 1 for K 1,, 0 and with initial ondition xk K and w K K from the filter reursions. We pi the initial onditions for the forward filter at eah iteration of the EM algorithm as x x ( i 1) ( i) K and w w. Finally the state-spae ovariane algorithm gives all the terms needed for the M- ( i 1) ( i) K ( i 1) step to find in (3) using these reursions 4, 51, 57, for 0,, K and W w x K, K W A w x x 1, 1 K K 1 K, for 0,, K 1. The iterations of the EM algorithm are run until onvergene. The estimated firing rate at any time bin 1,, K is in turn the smoothed estimate, ˆ ( ) exp( ) Si xk evaluated at the estimate of in the final iteration. Repeating this proedure for all neurons under eah sequene and fitting the inhomogeneous Poisson models results in a ontinuous smoothed estimate of the rate funtion for eah neuron under any given sequene and over the entire length of a trial. Our implementation of the EM algorithm is similar to prior wor 4, 51 (for a omparison with PSTH see Fig. 5 and Supplementary Figs. 3, 4). 3 Nature Neurosiene: doi: /nn.350

4 Testing the deoding performane Chane level auray For the sequene, the hane level auray is simply1/ S, where S is the number of sequenes used. For targets, however, one has to tae into aount the orrelation between the first and seond targets when alulating the hane level auray. This is beause depending on the number of sequenes used in the deoding analysis, the first and seond deoded targets may not be independent. In the ase of 1 sequenes, for example, sine both targets annot be at the same loation, information about one also implies some information about the other. These orrelation effets must therefore be taen into aount when alulating the hane level auray of the targets. We define indiator funtions for the first and seond targets, denoted by I 1 and I, that are 1 if the orresponding targets are deoded orretly and 0 otherwise. We show this analysis for the ase when 1 sequenes are used. In all other ases they an be found similarly. For 1 sequenes, using the total law of probability, the probability that the seond target is deoded orretly is given by, pi ( 1) pi ( 1 I 1) pi ( 1) pi ( 1 I 0) pi ( 0) Now if the first target is orretly deoded, the seond target ould be at one of three possible loations as the two targets annot be at the same loation. Hene the hane level auray in this ase is given by pi ( 1 I1 1) 1/3. By a similar argument, if the first target is deoded inorretly, the hane level auray of the seond target is pi ( 1 I1 0) /9. Hene the hane level auray of the seond target is given by hane 1 p p ( I1 1), (4) 9 9 and vie versa for the first target as the two targets are seleted symmetrially in the hoie of sequenes. For example, in a session where we observe a first target auray of * p in our deoding analysis, the hane level auray for the seond target is p hane * /9 p /9 as 4 Nature Neurosiene: doi: /nn.350

5 * hane opposed to simply 1/4. Note that if p 1/4, i.e., at hane level, then p 1/4 and also at hane level as expeted. Random permutation test: Testing signifiane for the divergene in the amount of information held by eah ell about the two targets To determine whether a signifiant divergene exists in the amount of information held by the premotor neurons about the two targets, we need to show that the absolute differene between the two target auraies of eah ell averaged aross the population is signifiantly larger than that of a population with the same target auray values but with no strutured relationship between eah ell s target auraies. To find the distribution of this average absolute differene in suh a population with no struture, we eep the auray values the same but randomly permute them within the population, and repeat this proess 100,000 times. This removes any possible struture between the target auraies of eah ell and hene reates a null hypothesis distribution. We an then establish the signifiane of the divergene by omparing the average absolute differene of the target auraies of the premotor neurons against this null distribution and alulate a P-value. To do so, we first orret for the orrelation effet between the first and seond target auray values, whih is the byprodut of the hoie of sequenes used in the experimental design. Sine the first and seond targets annot be at the same loation within the set of 1 sequenes, the auray of one target has a ontribution (even though fairly small) to the auray of the other even if the neuronal ativity is not enoding that target per se. Hene we also need to remove this effet to see the true representation of a target by the neuronal ativity, just as we tae it into aount in alulating the hane level auraies. This means that we subtrat from the first target auray value of eah ell, the hane level ontribution of that ell s seond target hane auray, or P1 1/4, and vie versa. We then randomly permute these first target auray values among the ells while eeping their seond target auray values the same, and repeat this proess many times. This generates a new population eah time with the same auray values but no pair-wise struture between the first and seond target auraies of eah ell. For eah new population, we ompute the average differene and after repeating this many times, Nature Neurosiene: doi: /nn.350 5

6 find the distribution of this average differene. We then use this distribution to find whether the differene of the first and seond target auraies of the ells averaged over the premotor population is signifiantly different from a population with no struture (We find that the average divergene in the premotor population is signifiantly different from that of a population with no struture, with or without applying the orrelation orretion, P < ). We orreted for the fairly small orrelation effet between the first and seond target auray values in the satter plots of Fig. 6 and Supplementary Fig. 6 as explained for the random permutation test above. 54. Cunningham, J. P., Yu, B. M., Shenoy, K. V. & Sahani, M. Inferring neural firing rates from spie trains using Gaussian proesses. In: Advanes in Neural Information Proessing Systems 0, (eds. Platt, J., Koller, D., Singer, Y. & Roweis, S.) (MIT Press, Cambridge, MA, 008). 55. Eden, U. T., Fran, L. M., Barbieri, R., Solo, V. & Brown, E. N. Dynami analysis of neural enoding by point proess adaptive filtering. Neural Comput. 16, (004). 56. Brown, E. N., Fran, L. M., Tang, D., Quir, M. C. & Wilson, M. A. A statistial paradigm for neural spie train deoding applied to position predition from ensemble firing patterns of rat hippoampal plae ells. J. Neurosi. 18, (1998). 57. Jong, P. D. & Mainnon, M. J. Covarianes for smoothed estimates in state spae models. Biometria 75, (1988). Nature Neurosiene: doi: /nn.350 6

7 Supplementary Figure 1 Mean population deoding auray aross all reorded sessions. (a) Mean deoding auray for the population aross all reorded sessions for the first target (red urve), seond target (blue urve) and the full sequene (bla urve). The figure has the same onvention used in Fig.. (b) Mean sequene deoding auray as a funtion of the time window length preeding the earliest go ue used in deoding. The bla urve shows the mean population sequene deoding auray (out of 1 possibilities) aross all sessions. Using an 800 ms window, the sequene deoding auray exeeds 95% of the maximum possible when using the neuronal ativity from the start of seond target presentation until the earliest go ue. Nature Neurosiene: doi: /nn.350 7

8 Supplementary Figure Mean population deoding auray over time for all reorded sessions in money 1 (a) and money (b). The figure has the same onvention used in Fig.. Aross all standard sessions and during the 500 ms woring memory period, the first target, seond target, and sequene auraies were 76 ± 1%, 60 ± 19%, and 48 ± 13% for the first money and 74 ± 11%, 43 ± 3%, and 36 ± 3% for the seond money, respetively (mean ± s.d.). Nature Neurosiene: doi: /nn.350 8

9 Supplementary Figure 3 Example of a first (urrently held) target seletive neuron. The subfigure at the upper left orner shows the first and seond target auraies of the ell as a funtion of time into the trial. The vertial bars/lines and their timings follow the same onvention as Fig.. In all other subfigures, eah top panel orresponds to a different sequene of movements with eah row illustrating the spiing ativity during a single trial and the bla dots indiating the spie times. Eah bottom panel indiates the orresponding mean firing rate estimates using the expetation-maximization proedure (bla urve) and the orresponding peristimulus time histogram (PSTH) (magenta urve). The arrow indiates the woring memory period. The subfigures in the same row orrespond to sequenes with the same first target loation. The subfigures in the same olumn orrespond to sequenes with the same seond target loation. Note that repeated target loations were not used in the sequenes and hene there are 3 subfigures per row/olumn. 9 Nature Neurosiene: doi: /nn.350

10 Supplementary Figure 4 Example of a neuron seletive for both targets. Figure has the same onvention used in Supplementary Fig Nature Neurosiene: doi: /nn.350

11 Supplementary Figure 5 Deoding auraies over time for three sample ells that were seletive for the first target only (a), seond target only (b), and both targets () during the woring memory period. Figure onventions are the same as in Fig.. 11 Nature Neurosiene: doi: /nn.350

12 Supplementary Figure 6 Partitioning of the population during woring memory. Satter plot of the first and seond target auraies of the ells that signifiantly enoded at least one target during the woring memory period. Statistial signifiane of the target auraies was tested here at a striter level (P < 0.001) than in Fig. 6. Red points indiate ells that signifiantly enoded only the first target and blue points indiate those that signifiantly enoded only the seond target. At this statistial level, no ell had a signifiant auray for both targets. The inset indiates the proportion of ells that signifiantly enoded only the first or only the seond target during the woring memory period with the same oloring shemes from left to right. Nature Neurosiene: doi: /nn.350 1

13 Supplementary Figure 7 Neural partitioning vs. sequene speifi seletivity. The original satter plot (Fig. 6) demonstrating the partitioning mehanism is shown on the left. On the right, ells that are seletive to a single speifi sequene (i.e., signifiantly hange their firing rate in response to a single sequene) are olored with yellow (paired t-test, P < 0.05, FDR orretion for 1 omparisons). We find that these sequene seletive ells are few in number (10% of the ells) and are among the least informative ells (i.e., have low auraies). Cells that show a signifiant hange in firing rate from baseline for at least one speifi sequene are olored in yan. As evident, this analysis by itself demonstrates that the majority of the ells in the satter plot display a hange in firing rate for at least one sequene (as expeted) but it does not reveal the population partitioning. The deoding analysis alulates deoding auray as a measure of information when onsidering all sequene ombinations olletively. It also further disambiguates the amount of information held simultaneously about the first and seond targets by eah neuron and therefore reveals the partitioning mehanism. Nature Neurosiene: doi: /nn

14 Supplementary Figure 8 Conditional deoding auraies. The onditional first target auray given the possible seond target loations (i.e., U, R, D, L) and the onditional seond target auray given the possible first target loations are shown for the population in a sample session (same session as in Fig. ). The dotted lines indiate the 99% hane upper onfidene bounds. Aross all sessions we found no signifiant differene in deoding auray of the seond target based on the loation of the first target, and vie versa (repeated ANOVA, P > 0.15). Nature Neurosiene: doi: /nn

15 Supplementary Figure 9 Comparison of first target deoding auray for the population in interleaved and non-interleaved sessions. In the interleaved dual-target/single-target session, target deoding auray on single-target trials is shown in red. In the single-target only session, target deoding auray is shown in magenta. Eah point on the urves indiates the deoding auray for the population over the preeding 500 ms window. Dotted lines indiate the 95% onfidene bounds for eah auray urve (rather than hane level). The red vertial bar indiates the time during whih the (first) target was presented, and the vertial dotted line indiates the average time of the first go ue presentation onset. The arrow indiates the time point orresponding to the deoding auray of the preeding woring memory period. Nature Neurosiene: doi: /nn

16 Supplementary Figure 10 Eletrode reording sites for eah of the two moneys. Top-view shemati of the eletrode array positions. Eah array (irle) ontains 3 eletrode ontrats. The bar in entimeters is referened in relation to interaural antero-postero oordinates and midline medio-lateral oordinates. Here, A is anterior, P posterior, M medial and L lateral. In the left panel, from money 1, the white and gray irles indiate reordings from two separate hemispheres. In the right panel, from money, the white irles indiate reordings from one hemisphere. Nature Neurosiene: doi: /nn