Supplementary information

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1 Supplementary information Quantification of predictability The main method of thi paper i quantitative etimation of the predictability of a pike train from a predictor variable, which may be any combination of patial location, theta phae, or peer activity (Figure S). We will firt eplain the method of predicting the pike train olely from poition. A -fold cro-validation procedure i ued to repeatedly divide the recorded data into a training et and a tet et. The training et i ued to contruct a predicted intenity f, i.e. a place map. a a function of pace ( ) The place map contructed from the training et i then evaluated on the tet et. The poition () t at each moment of the tet et, and the place field f ( ) contructed from the training et are ued to produce an etimated intenity at each time, f () t f ( () t ) log-likelihood denity of the actual pike train{ t } hown to be () log ( ) =. The under thi etimated ditribution can be () Lf = f t dt+ f t The training et i alo ued to contruct a mean firing rate independent of poition, f, defined a the number of pike during the training period divided by the length of the training period. The predictability on the tet et i defined to be the difference, L f L f. It therefore give the log likelihood ratio of the data under the two predicted intenity model, f ( ), and the contant f. The predictability of the entire data et i defined by a cro-validation procedure, where the data i divided into egment, each egment i in turn ued a tet et, and the log likelihood ratio for each egment are ummed and divided by the total time (Figure S). Thi predictability meaure alo ha an intuitive interpretation. Suppoe an oberver want to communicate whether a pike occurred at a given time intant. She or he can communicate thi mot conciely uing a code derived from her or hi bet etimate of the probability of pike occurrence. The number of bit needed, on average, to communicate uing thi code i the epected negative log likelihood under thi etimated ditribution. The predictability meaure etimate the number of bit aved by an oberver who i allowed to ue the intantaneou poition of the animal to contruct a code, compared to one who can only ue the mean firing rate. The ue of cro-validation guard againt artificial overetimation of pike train predictability. For eample, if firing probability i independent of poition, an oberver

2 uing poition to help encode the pike train would actually ue more bit than one who ued mean firing rate, becaue the random fluctuation in the training would reult in a wore code ued to encode the tet et pike train. However, if cro-validation were not ued, random fluctuation in the data et would lead to an illuory aving of bit needed to tranmit the pike train. Contruction of Place Field Prediction of intenity from location i equivalent to contruction of a place field. Here we contruct place field by a moothing-baed method, where we divide a moothed pike count map by a moothed occupancy map. The etimated intenity at a point i given by f ( ) dt = t t nw t ( t ) ( t ) w () Here, nt i the number of pike fired in a given time bin, i i the poition of the rat in that time bin, and dt i the time bin ize. The moothing function w i a Gauian with a variable width parameter: ( ) wd = ep( d / λ ) (3) By comparing prediction quality with varying value of the moothing width λ, we found that the optimal value for thi parameter wa ~5cm (ee figure S3 below). We note that the above method for place field computation i a form of locally weighted maimum likelihood etimation 3, where the intenity f at a point i choen by maimizing a weighted um of log-likelihood of each time bin under a Poion w : ditribution, with weight given by ( ) t L( f ) = w( ) f dt + n log ( f dt) (4) t t t Contruction of Phae Field The preferred phae of pyramidal cell pike with repect to the theta rhythm varie with the animal location in pace 4-7. We may therefore predict the pike train more accurately by allowing the predicted intenity to vary a a function of theta phae. To do thi, we mut quantify the dependence of the cell phae preference on patial poition. We do thi again uing locally weighted maimum likelihood etimation. We fit the

3 3 phae of all pike to a von Mie ditribution 8 whoe parameterθ (mean phae) and κ (modulation depth) vary with poition, to minimize the weighted likelihood ( κco( θ θ) ) πi ( κ ) ep L( θ, κ) = w( ) (5) Here, the um i over all pike, and andθ are the poition and intantaneou theta phae at the time of pike. A with place field, the local maimum likelihood etimate may be efficiently computed by a moothing method, according to the following formula: θ = arg κ = A ( t ) iθ e w ( ) w ( t ) iθ e w w( ) + (6) (7) where A y = I y I y 8. In order to regularize againt over-fitting in area where few pike were fired, a contant term of wa added to the denominator in (7). A i the ratio of Beel function ( ) ( ) ( ) The predicted intenity from poition and phae i the product of the place field term and a phae modulation term: ( ) ( κco( θt θ) ) I ( κ ) ep f = f (8) t t Prediction of unit activity from population To predict the activity of one cell from the population of peer cell, we ued a generalized linear model 9. Initially, the pike train of the peer cell are meared in the time domain with a Gauian function of variable width σ (the peer prediction timecale): tα (( t τα) σ ) = ep (9) πσ τα

4 4 Here the um run over all pike of cell α. Under the generalized linear model, the predicted intenity at a time t i given by ft = g t α w α () α The link function g(η ) had the following form: ep( η) η < g( η) = η+ η () A imple eponential wa not ued, becaue thi led to eceively high predicted intenitie in the cae when many poitively predicting peer cell were firing imultaneouly. The prediction weight training et wα were choen to maimize the penalized log-likelihood on the L = f dt + n f dt w α 4 t t log ( t ) () t The maimization i carried out by Newton method with an analytically calculated Heian matri. The penalty term helped to prevent over-fitting by reducing large weight value that did not ubtantially improve prediction quality on the training et. Prediction of unit activity from poition and population When pike train were to be predicted from patial variable, in combination with peer prediction, the peer prediction function wa multiplied with the patial prediction function: ( κco( θt θ) ) I ( κ ) ep ft = f ( t) g t αwα (3) α α Thi formula wa ued to eamine whether the prediction of the pike train, made from patial and phae variable, may be further improved by taking into account the activity of peer cell. The weight in thi cae are not necearily the ame a thoe when activity i predicted from peer cell alone, and are recomputed uing Newton method a thoe that maimize the penalized likelihood of the product of patial and peer prediction intenitie on the training et. Ue of prediction method to etimate patial cale of place field

5 5 Figure S3. Computation of place field involve the ue of a patial moothing cale. The cro-validation method may be ued to enure the ue of an optimal moothing cale. a) Place field of the ame cell, computed for three different value of moothing cale. At 3cm, the place field how a high degree of patial tructure; however, thi tructure arie from random fluctuation, rather than reliable place preference of the neuron (under-moothing). At cm, the place field i nearly circular; however, moothing at thi cale loe doe not capture the full patial tructure of the place field (over-moothing). b) Optimal moothing cale wa etimated by predicting the cell activity from pace alone, for a range of moothing cale. Peak predictability wa at 5.6cm for thi cell. If information rate wa computed by a direct method, without crovalidation, the apparent information content of the cell increaed without bound a the patial cale wa lowered, indicating that cro-validation i neceary to protect againt under-moothing. c) Acro the population, the median optimal moothing cale wa found to be 5cm. Relation of predictability to pike train characteritic Figure S4a. Dependence of upra-patial peer predictability (Peer Gain) on iolation quality of target cell. Unit iolation quality wa aeed uing the iolation ditance meaure. When all initially clutered cell are conidered, including thoe below the iolation ditance threhold (hown in black), a ignificant correlation i found between predictability and iolation quality (p<., red line), indicating that poorly iolated cell are le predictable from peer. However, if only thoe cell that paed the iolation ditance threhold of are conidered, no correlation i een (p>.5, green line). We therefore only conidered thee cell for further analyi. Figure S4b and c. Supra-patial peer predictability, meaured in bit/ec, i poitively correlated with target cell firing rate (Fig S4b; p<.). However, if the predictability meaure i normalized by the firing rate of the target cell to give a meaurement of bit per pike, a negative correlation i oberved (Fig S4c; p<.). Thi ugget that fater firing cell are more predictable imply becaue there are more pike whoe occurrence can be predicted. Furthermore, the negative correlation of target cell firing rate with bit per pike ugget a rule of diminihing return for high-firing cell. Figure S4d. Peer predictability increae with the number of peer predictor cell (red line; lope. bit/pike/cell; p<.). To enure the effect wa not unduly influenced by one animal with the larget number of cell (4 of the 89 paing the iolation criteria), the analyi wa repeated with thi animal ecluded (green line). Fit were contrained to pa through the origin (zero cell provide zero predictability). Figure S4e. Peer predictability appear to be negatively correlated with data et ize (i.e. length of recording)(red line; p<.). However, thi correlation i entirely due to a ingle animal with a large number of cell and a hort recording time. If thi recording i ecluded, the correlation diappear (green line).

6 6 Figure S4f. Peer predictability i correlated with pike train variability, a meaured by the Fano factor (p=.3). The Fano factor can be defined uing variable window ize; a ignificant poitive correlation of variability with predictability wa een for all window greater than or equal to 5m (the Fano factor diplayed here i calculated with a window of 5m). Thi obervation ugget that neuron ehibiting greater pike train variability alo how a higher gain in predictability uing peer activity over location alone. The timecale at which neuronal pike were bet predictable from population activity howed a clear mode at approimately 5m (Fig 3b, main paper). Neverthele, ome cell howed optimal prediction timecale different to thi value. To clarify which neuronal characteritic correlate with non-tandard predictability timecale, we divided cell into two categorie, thoe whoe predictability timecale wa cloe to the mode (in the range -4m), and all other cell. We performed a multiple logitic regreion analyi 9 to predict the category a cell belonged to uing the following array of predictor variable: Iolation quality, number of predictor cell, total recording length, pike train variability (Fano factor), and predictability from pace. A ignificant effect wa found for iolation quality and number of predictor cell, both of which correlated poitively with the probability that the cell would how optimal predictability in the range -4m (p=. and.8, repectively). None of the other predictor variable howed a ignificant effect. We therefore concluded that the catter in predictability timecale wa related to propertie of the etracellular recording, rather than propertie of the cell itelf, or it relation to network activity. Relation of prediction weight to location of place field Figure S5a) The phae of firing of hippocampal pyramidal cell i known to depend on the animal location in pace 4, with mean phae cloe to the negative peak of pyramidal layer theta cycle in the center of a place field, and cloe to the poitive peak in the periphery. One would therefore epect correlation between cell to depend on the patial overlap of their place field 5. b) The prediction weight indicate the degree to which correlation differ from thoe epected if pike timing wa olely determined by patial location. Prediction weight i hown a a function of the degree of place field overlap (computed a the calar product of the normalized place field map), for each pair of cell. While prediction weight are highly variable, there i a weak but ignificant correlation between prediction weight and the degree of place field overlap (r=.7; p<.), indicating decreaed or increaed ynchronization of cell beyond that predicted from imultaneou independent theta phae preceion. The large catter about the fit line ugget that non-patial factor alo play a role in determining correlation trength. Relation of prediction weight to anatomical location within the CA region Figure S6. a) Eample relation between anatomical location and prediction weight. Vertical cell location in the pyramidal layer wa etimated from the mean pike waveform for each cell recorded by the 8-ite ilicon electrode hank, wherea lateral

7 7 poition wa determined from the interhank ditance ( µm). Predictor cell (triangle) are color-coded by prediction weight (red poitive, blue negative) to a repreentative target cell (tar). No cell on the target cell hank were ued a predictor cell to avoid puriou ynchrony caued by iolation error 3. No conitent anatomical ditribution of poitively or negatively weighted cell wa een. b. Acro the population, no relation wa een between the anatomical pacing of cell (hank eparation), and prediction weight (linear regreion, red line, p=.47). However, becaue we avoided prediction of neuronal activity from etremely anatomically proimal neuron 3 (< µm), we cannot acertain whether thee etremely proimal neuron would how a reliable difference in prediction weight. Reference Lit. Ripley,B.D. Pattern Recognition and Neural Network. Cambridge Univerity Pre, Cambridge (996).. Cover,T.M. & Thoma,J.A. Element of information theory. New York, N.Y. ; Chicheter : John Wiley, (99). 3. Loader,C. Local regreion and likelihood. Springer-Verlag, New York (999). 4. O'Keefe,J. & Recce,M.L. Phae relationhip between hippocampal place unit and the EEG theta rhythm. Hippocampu 3, 37-3 (993). 5. Skagg,W.E., McNaughton,B.L., Wilon,M.A. & Barne,C.A. Theta phae preceion in hippocampal neuronal population and the compreion of temporal equence. Hippocampu 6, 49-7 (996). 6. Harri,K.D. et al. Spike train dynamic predict theta-related phae preceion in hippocampal pyramidal cell. Nature 47, (). 7. Mehta,M.R., Lee,A.K. & Wilon,M.A. Role of eperience and ocillation in tranforming a rate code into a temporal code. Nature 47, (). 8. Fiher,N.I. Statitical analyi of circular data. Cambridge Univerity Pre, New York (993). 9. Dobon,A.J. An Introduction to Generalized Linear Model. Chapman and Hall, London (99).. Skagg,W.E., McNaughton,B.L., Gothard,K.M. & Marku,E. Advance in Neural Information Proceing Sytem, Vol. 5. Hanon,S., Cowan,J. & Gile,G. (ed.), pp (Morgan Kaufmann, San Mateo,993).

8 8. Harri,K.D., Hirae,H., Leinekugel,X., Henze,D.A. & Buzaki,G. Temporal interaction between ingle pike and comple pike burt in hippocampal pyramidal cell. Neuron 3, 4-49 ().. Baddeley,R. et al. Repone of neuron in primary and inferior temporal viual cortice to natural cene. Proc. R. Soc. Lond B Biol. Sci. 64, (997). 3. Quirk,M.C. & Wilon,M.A. Interaction between pike waveform claification and temporal equence detection. J. Neuroci. Method 94, 4-5 (999).

9 Poition in Space Intantaneou Phae Population Activity (t) Place Field Phae Field Spike Time Smooth at timecale Activation Vector v(t) f() h(, ) g(v.w) Weight Vector w Predicted Firing Probability f()h(, g(v.w)? Figure S

10 Training Set Tet Set Poition y y Population Tunable Prediction Function Predicted Firing Rate Target Spike Train Oberved Spike Train Prediction Likelihood log f t f t dt Figure S

11 a 3cm 5.6cm (optimal) 5 5 b Spatial Information (bit/) 6 4 Cro Validated Direct Meaure 5 5 c Spatial Reolution (cm) 7 cm 5 5 Optimal Information (bit/) Optimal Spatial Reolution (cm) Figure S3

12 a b c 5 5 Peer Gain (Bit/) 4 3 Peer Gain (Bit/) 4 3 Peer Gain (Bit/Spk) Iolation Quality - 5 Firing Rate d e f Firing Rate Peer Gain (Bit/Spk).5.5 Peer Gain (Bit/Spk).5.5 Peer Gain (Bit/Spk) Number of predictor cell recording length (min) Fano Factor Figure S4

13 a b Non-overlapping place field => Anticorrelated firing 4 3 Overlapping place field => Correlated firing Prediction Weight Place Field Overlap Figure S5

14 a b 5 Prediction Weight Shank Separation Figure S6