Nature Neuroscience: doi: /nn Supplementary Figure 1. Spike-count autocorrelations in time.

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1 Supplemenary Figure 1 Spike-coun auocorrelaions in ime. Normalized auocorrelaion marices are shown for each area in a daase. The marix shows he mean correlaion of he spike coun in each ime bin wih he spike coun in every oher ime bin, averaged across neurons. These show ha he auocorrelaion is roughly saionary across ime during he foreperiod. Naure Neuroscience: doi: /nn.3862

2 Supplemenary Figure 2 Single neurons exhibi heerogeneous auocorrelaions. Ligh grey races show he spike-coun auocorrelaion as funcion of ime lag for single neurons, averaged across ime poins. Circles mark he populaion mean a each ime lag, and he curve shows he exponenial fi o he populaion daa. The observaion of singleneuron heerogeneiy reinforces he inerpreaion of inrinsic imescale as a characerisic a he populaion level raher han a he single-neuron level. Naure Neuroscience: doi: /nn.3862

3 Supplemenary Figure 3 Differences in mean firing raes across areas do no accoun for hierarchy of inrinsic imescales. Mean firing raes varied subsanially across daases and across areas wihin daases. There was no significan dependence of inrinsic imescale on mean firing rae (P = 0.51, (9) = 0.69, wo-ailed -es, regression slope m = 5.5 ± 7.9 ms/hz; P = 0.16, r s = 0.34, Spearman s rank correlaion, wo-ailed). Error bars mark s.e. Naure Neuroscience: doi: /nn.3862

4 . Supplemenary Figure 4 Auocorrelaion offse reflecs rial-o-rial correlaion. Trial-o-rial correlaion was calculaed as he Pearson correlaion coefficien beween he foreperiod spike coun in each rial and he spike coun in he nex rial. We hypohesized ha auocorrelaion offse would posiively correlae wih rial-o-rial correlaion, and found a significan posiive correlaion beween hem. This indicaes ha he auocorrelaion offse includes conribuions from variabiliy a imescales are comparable o or longer han he rial duraion. Colored lines show rends for individual daases. The arrow shows he slope of dependence from a regression analysis (slope m = 1.3 ± 0.3). Error bars mark s.e. Naure Neuroscience: doi: /nn.3862

5 Supplemenary Figure 5 Hierarchical ordering of areas by imescale of reward memory. In he Lee daase, we previously measured imescales of he decay of memory races for pas rewards in single-neuron firing raes, while monkeys performed a compeiive decision-making ask. (a) The cumulaive disribuion of reward imescales in LIP (n = 160), LPFC (n = 243), and ACC (n = 134). For neurons fi wih he sum of wo reward imescales, we used he harmonic mean of he wo imescales. (b) Median reward imescale for he hree areas. Error bars mark s.e. Naure Neuroscience: doi: /nn.3862

6 Supplemenary Mahemaical Noe The mahemaical framework of doubly sochasic poin processes informs our inerpreaion of spike-coun auocorrelaion. Here we presen calculaions for he variabiliy and auocorrelaion for an inhomogeneous Poisson process wih sochasic rae. We consider how rae flucuaions a differen imescales affecs he spike-coun auocorrelaion. Analysis of muliple imescales resuls in he funcional form used o fi he auocorrelaion daa (Eq. S.20). Definiions We consider a poin process ha generaes a sequence of evens a paricular imes. The coun in a given ime window (, + ) is denoed N(, ). For a homogeneous Poisson process, he even rae λ() is given by a consan µ, and he mean and variance of he coun are boh equal o µ. For an inhomogeneous Poisson process, he rae λ() changes in ime. In his case, he mean and variance are also equal, bu calculaed by he inegral of he even rae: N(, ) = N(, ) 2 N(, ) 2 = + + dsλ(s), and dsλ(s) (S.1) (S.2) If λ() = µ, his is he homogeneous Poisson process. Here we are ineresed he case of a doubly sochasic process, in which he ime-dependen rae λ() is iself a sochasic process, characerized by a mean and variance. For simpliciy, we assume he mean and variance of he even rae are consan in ime, i.e. λ() = µ and λ() 2 λ() 2 = σ 2. Toal variabiliy is due boh o variabiliy in even iming and variabiliy in even rae. Mean and variance Averaging over boh sources of sochasiciy, he oal mean of he coun is equal o: N(, ) = + ds λ(s) = µ (S.3) The mean coun is equal o ha of a Poisson process wih consan rae µ. The variance of couns will differ from ha of a Poisson process, o reflec he conribuion of rae variabiliy. The second 1 Naure Neuroscience: doi: /nn.3862

7 momen of he coun is given by: N(, ) 2 = + [ + ] 2 dsλ(s) + ds λ(s) = µ + + ds + ds λ(s)λ(s ) (S.4) We denoe δλ() λ() µ, he deviaion of he rae from is mean. The oal variance is hen given by: N(, ) 2 + N(, ) 2 = µ + ds + ds δλ(s)δλ(s ) (S.5) Here we consider ha he even rae is characerized by a given ime consan τ. We define he auocovariance of he rae o be: ( s s δλ(s)δλ(s ) = σ 2 ) exp τ For insance, an Orsein-Uhlenbeck process has his form of an auocovariance. Noe ha he linear rae model, shown in Fig. 4a and described in he Mehods, behaves as an Ornsein- Uhlenbeck process in response o whie-noise inpu and produces a rae auocovariance of he form in Eq. S.6. The variance of he coun is hen given by: N(, ) 2 [ ( N(, ) 2 = µ + 2σ 2 τ 2 exp ) ( 1 )] τ τ (S.6) (S.7) The firs erm is he conribuion of even iming variabiliy and he second erm is he conribuion of even rae variabiliy. If he ime window is small relaive o τ ( τ), he second erm is approximaely equal o σ 2 2 and independen of τ. If he window size is large relaive o τ ( τ), he second erm is approximaely equal o 2σ 2 τ. Auocorrelaion We now examine how rae variabiliy affecs correlaions in ime. We consider covariance beween couns in disjoin ime bins, each of same window size, separaed by an ineger number of windows k for a oal ime lag of k. The auocovariance C a k-lag is defined as: C(, + k ) N(, )N( + k, ) N(, ) N( + k, ) (S.8) 2 Naure Neuroscience: doi: /nn.3862

8 The even imings are independen because he inervals are disjoin (k 1), so even iming variabiliy does no conribue o auocovariance. The auocovariance is herefore: C(k ) = + ds + ds δλ(s)δλ(s ) ( ) ( = σ [2τsinh 2 exp k 2τ τ )] 2 (S.9) When k = 0, his equaion does no apply and auocovariance is simply equal o he variance. In he limi of slow rae flucuaions ( τ): ( C(k ) σ 2 2 exp k τ ) (S.10) In he limi of fas rae flucuaions ( τ): C(k ) δ 1k σ 2 τ 2 (S.11) where δ i j is he Kronecker dela funcion. For non-coniguous inervals (k > 1), C 0, and for coniguous inervals (k = 1), C σ 2 τ 2. The auocorrelaion R is given by he auocovariance divided by he variance: ( σ 2 [ 2τsinh ( )] 2 ) ( 2τ R(k ) = µ + 2σ 2 τ [ 2 exp ( ) ( )] exp k τ 1 τ τ ) (S.12) When k = 0, his equaion does no apply and R = 1 by definiion. In he limi of slow rae flucuaions ( τ): ( R(k ) 1 + µ ) ( 1 exp σ 2 k τ ) (S.13) If rae flucuaions are large (σ 2 /µ 1) he firs facor is 1. If rae flucuaions are small (σ 2 /µ 1), his firs facor is also small. In he limi of fas rae flucuaions ( τ): ( τ )( R(k ) δ 1k 1 + µ ) 1 2 2σ 2 (S.14) τ For non-coniguous inervals (k > 1), R 0, and for coniguous inervals (k = 1), R remains small. 3 Naure Neuroscience: doi: /nn.3862

9 Muliple imescales Now we consider he case where rae flucuaions are governed by muliple imescales {τ i }. We define he rae covariance as: δλ(s)δλ(s ) = i s s σ 2 i ( ) exp τ i (S.15) Each imscale τ i is weighed by a variance σ 2 i. A given componen of he variance σ2 can be i negaive, bu he oal variance σ 2 = i σ 2 is posiive. i The spike coun variance is hen given by: Var(N) = µ + n 2σ 2 i τ2 i i=1 [ ) )] [ exp ( τi (1 τi µ + 2 σ 2 i τ i + ] σ 2 i i S i L (S.16) where he ses S and L correspond, respecively, o he shor and long imescales, namely S = {i : τ i } and L = {i : τ i }. Unless he absolue variance σ i 2 of he shor imescales is much larger han ha of he long imescales, he conribuion of he shor imescales can be negleced. The spike-coun auocovariance is similarly generalized: C(k ) = i σ 2 i [ ( )] 2 2τ i sinh exp( k ) δ k1 σ 2 i 2τ i τ τ2 i + ( σ 2 i 2 exp k ) i i S i L τ i (S.17) Again, shor imescales have negligible conribuion o auocovariance unless heir variance is much larger han he long imescales. The auocorrelaion is obained by dividing Equaion S.17 by Equaion S.16: R(k ) ( ) i L σ 2 i exp k τ i µ + i L σ 2 i (S.18) Noe ha if ime consans are very large, and he value of k is no comparably large, he exponenial funcions of he corresponding erms are approximaely consan. If only one ime consan τ j shows an appreciable exponenial decay in he range of k considered, he auocorrelaion is approximaely equal o (he sums run over he long imescales only): [ R(k ) 1 + µ σ 2 j + 1 σ 2 i τ i i S σ 2 + σ 2 ] 1 [ ( i j i L σ 2 exp k ) + σ 2 ] i τ j j i L σ 2 j (S.19) 4 Naure Neuroscience: doi: /nn.3862

10 The las erm provides an effecive offse in he auocorrelaion ha reflecs he conribuions of long imescales o spike-rae flucuaions. In line wih Equaion S.19, we fi he experimenal spike-coun auocorrelaion daa wih he funcional form: [ ( R(k ) = A exp k τ ) ] + B (S.20) These hree parameers were used for he fi: A is he ampliude, τ is he inrinsic imescale, and B is he offse. Equaion S.19 suppors our inerpreaion of he auocorrelaion offse (B in Equaion S.20) as he relaive srengh of conribuions from long imescales compared o ha of he moderae inrinsic imescale. 5 Naure Neuroscience: doi: /nn.3862