Synapses with short-term plasticity are optimal estimators of presynaptic membrane potentials: supplementary note

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1 Synapses wih shor-erm plasiciy are opimal esimaors of presynapic membrane poenials: supplemenary noe Jean-Pascal Pfiser, Peer Dayan, Máé Lengyel Supplemenary Noe 1 The local possynapic poenial In he main ex we pursue he hypohesis ha synapses undergoing STP esimae he somaic membrane poenial of he presynapic cell, wih a quaniy we called he local possynapic poenial represening he resuls of his esimaion process. In his secion we define he local possynapic poenial formally. We also explain how experimenal daa, and in paricular hose recorded in STP experimens, perain o i, and how i relaes o single neuron compuaions. 1.1 Biophysical definiion Consider isolaing he i h synapic comparmen from he res of he dendriic ree. If we also ignore any non-synapic currens, he dynamics of is poenial would follow he simple differenial equaion: C m dv (i d = vres v (i R m + I (i syn( where v res is he resing membrane poenial, C m and R m are he membrane capaciance and resisance, respecively, and I syn( (i describes he curren enering hrough synapic channels. We call v (i he local possynapic poenial. (S1 1.2 Experimenal measuremen Of course, v (i is an idealizaion ha will no be expressed in eiher he dendriic or he somaic membrane poenial under normal condiions, for a leas wo reasons. Firs, he rue membrane poenial of he local dendriic comparmen, v (i, follows a differen equaion: C m dv (i d = vres v (i R m + I (i syn(+i (i oher ( (S2 where I (i oher ( includes oher curren conribuions ha are no he leak curren, nor he synapic curren. This includes currens propagaed from oher pars of he dendriic ree, and acive currens generaed wihin he same comparmen. Second, v (i is no expressed in he somaic membrane poenial of he pos-synapic neuron, v (soma, because he inervening dendrie filers he synapic signal, and he soma inegraes i wih he local possynapic poenials creaed by he many oher simulaneously acive inpus ha he pos-synapic neuron is likely o receive. Therefore, alhough v (i is a useful heoreical consruc for undersanding single neuron compuaion (as long as Eq. S4 holds, see below, i may no be a quaniy ha is immediaely apparen in membrane poenial recordings. Forunaely for our purposes, STP experimens provide very differen condiions from hose presen in he funcional circui 1, and hese make he experimenal assessmen of v (i amenable. 1

2 Pfiser e al. Synapses as opimal esimaors In such experimens, which are ofen in viro 2, normally only a single synapse, or a small subse of synapses is simulaed. This has wo effecs: he somaic membrane poenial depends only on he possynapic poenial in one dendriic comparmen, and he local dendriic membrane poenial is kep in he subhreshold regime which minimises he effecs of acive conducances (i.e. I (i oher 0. Overall, hese effecs resul in he somaic membrane poenial reflecing he local possynapic poenial well excep for he consequences of dendriic filering. Filering will mosly affec he ime consan and absolue ampliude of he somaic signal. We herefore fi only he relaive changes in somaic ampliudes (Fig. 3 of he main paper, which should be less affeced by his. Neverheless, a sronger experimenal es of he heory would require local dendriic recordings under such minimal simulaion condiions. 1.3 Compuaional use A common assumpion in models of neural circui aciviy is ha neurons need o compue some (poenially high-dimensional and complicaed funcions of heir inpus 3,4. In erms of (subhreshold somaic membrane poenials, one can formalise his as ( v (soma = F u (1,u (2,..., u (M (S3 where u (i is he somaic membrane poenial of presynapic neuron i (ou of M, v (soma is he somaic membrane poenial of he possynapic cell, and F is he non-linear funcion i compues on is inpus which depends on dendriic processing and oher inracellular processes. (Noe ha Eq. S3 can be rewrien in erms of pre- and possynapic firing raes, which are hemselves also some non-linear funcions of membrane poenials, in he same general form wih a suiable choice of F. However, since he possynapic membrane poenial can only direcly depend on he spiking oupu of presynapic cells, we propose ha he cell approximaes equaion S3 by applying is non-lineariy o reliable esimaes of he presynapic membrane poenials, û (i. These we sugges are represened by he local possynapic poenials, v (i described above: v (soma = F ( v (1,v (2,..., v (M F ( û (1, û (2,..., û (M This is a key reason for inroducing v (i as an idealizaion. In line wih sandard reduced neuron models 3,5, we assumed here ha he somaic membrane poenial can be wrien o a reasonable approximaion in he form given by equaion S4 as a funcion of v (i, raher han v (i, which means ha he effecs of I (i oher are incorporaed ino F. Crucially, he beer he individual v (i esimae u (i, he beer equaion S4 implemens equaion S3. This is illusraed in Supplemenary Figure S1. (S4 2 Esimaion under he swiching Ornsein Uhlenbeck process 2.1 Generaive process For convenience, recall here he descripion of he generaive model under he swiching Ornsein Uhlenbeck (OU process. In such a process, he resing membrane poenial u res is no fixed bu randomly swiches beween wo levels, u + ans u, corresponding o up and down saes (Supplemenary Fig. S2b P ( u res u res δ = 1 η δ if u res η δ if u res 1 η + δ if u res η + δ if u res = u + and u res δ = u+ = u and u res δ = u+ = u and u res δ = u = u + and u res δ = u where η and η + are he raes of swiching o he down and up saes, respecively. The presynapic membrane poenial evolves as an Ornsein-Uhlenbeck (OU process around he resing poenial u res which is now ime-dependen: P ( [ u u δ,u res = N u ; u δ + 1 ] ( u res u δ δ, σ 2 τ W δ (S6 (S5 2

3 Pfiser e al. Synapses as opimal esimaors Spike generaion is described by he same rule as in he non-swiching case. See Eqs. 5-7 of he main paper. 2.2 Opimal esimaor The opimal esimaor is given by he following filering equaion: P ( u,u res s 0: P(s u u res δ P ( u u δ,u res ( P u res u res ( δ P u δ,u res δ s 0: δ du δ We are primarirly ineresed in he poserior over he membrane poenial. This can be obained by marginalising Equaion S7: where P(u s 0: = u res (S7 P ( u,u res s 0: = (1 ρ p (u +ρ p + (u (S8 ρ =P ( u res = u + s 0: is he esimaed probabiliy of he resing membrane poenial being in is up sae (or he mixure raio, and p (u = P ( u u res = u,s 0: (S10 p + (u = P ( u u res = u +,s 0: (S11 express he poserior disribuion of u assuming ha u res is in is down or up sae, respecively. The mean of he poserior over he presynapic membrane poenial can be wrien as: (S9 û = (1 ρ µ + ρ µ + (S12 where µ = µ + = u p (u du u p + (u du (S13 (S14 are he condiional mean esimaes of u, assuming again ha u res is in is down or up sae, respecively. The change in he poserior mean, corresponding o possynapic poenial dynamics (in coninuous ime is hen û = [1 ρ +ɛ ] µ + ρ +ɛ µ + [ + ρ µ + ɛ µ ] ɛ (S15 Therefore, here are wo facors conribuing o changes in he mean poserior (and hus o he size of an EPSP: changes in he condiional means µ and µ +, and changes in he mixure raio ρ. These wo facors ac addiively. However, for closer correspondence wih he synapic resource and uilisaion variables (Online Mehods, Equaion S15 can be rearranged as: [ ] û = µ 1+ ρ µ (S16 where µ = [1 ρ +ɛ ] µ + ρ +ɛ µ + (S17 ρ [ = ρ µ + ɛ µ ] ɛ (S18 correspond o he wo facors. The associaions of µ and ρ wih shor-erm depression and faciliaion respecively are discussed in he wo nex secions. 3

4 Pfiser e al. Synapses as opimal esimaors 2.3 Changes in he condiional means, µ In order o undersand he condiional means, le us firs revisi he condiionalised poseriors, p (u and p + (u. In he limi when he dynamics of u res are sufficienly slow, and u and u + are no oo far apar, hese are approximaely p (u p + (u P(s u P(s u P ( u u δ = u,u res = u p δ (u du (S19 P ( u u δ = u,u res = u + p + δ (u du (S20 If p δ (u and p + δ (u are Gaussians, each of Equaions S19-S20 describes he poserior updaes for a simple OU process wih non-linear Poisson spike generaion, as derived in he main paper. This implies ha he wo condiional means, µ and µ +, evolve as he poserior mean, û in Eq. 11 of he main paper: heir updaes depend on he (here, condiionalised, or relaive poserior variances when observing a spike a ime : where µ ( σ ɛ 2, µ + ( σ + ɛ 2 (S21 ( σ 2 = u 2 p (u du ( µ 2 (S22 ( σ + 2 = u 2 p + (u du ( µ + 2 (S23 We know from Eqs. 11 and 12 of he main paper ha he proporionaliy in equaion S21 leads o depression. Thus µ, he oal change in û due o changes in he condiional means a he ime of an incoming spike, will also show depression because i is a weighed some of wo erms ha boh depress individually: µ σ2 (S24 where, from Eq. S17 σ 2 = [1 ρ +ɛ ] ( σ ɛ 2 + ρ+ɛ ( σ + ɛ 2 (S Changes in he mixure raio, ρ The dynamical equaion for he mixure raio can be derived as (see secion 2.5: ρ η + [ η + + η + γ + γ ] ρ [ γ + γ ] ρ 2 + ρ ɛ S (S26 where γ + δ =P ( s =1 u res = u +,s 0: δ and γ δ =P ( s =1 u res = u,s 0: δ (S27 are he (condiionalised poserior firing raes and are analogous o γ derived in Eq. 10 of he main ex. The mos imporan erm from Equaion S26 for us now is ρ = 1 1+ γ 1 ρ γ + ρ ρ (S28 expressing he size of he insananeous updae in ρ afer observing a spike (Fig. S3 a ime (see Supplemenary Figure S3. This quaniy is non-negaive, and highly non-linear in ρ: in paricular, i has an invered U-shape which means ha in he range of low ρ values (he lef arm of he invered-u, larger ρ values imply larger incremens in ρ hus leading o faciliaion. The greaer he difference in firing raes beween he up and down saes is, γ + /γ, he 4

5 Pfiser e al. Synapses as opimal esimaors larger he magniude bu also he smaller he window of faciliaion becomes (he peak of he ρ curve shifs up and o he lef. Also noe, ha for he range of large ρ values, depression, raher han faciliaion, is prediced. The overall effec, of course, also depends on he condiional means, because ρ = [ µ + ɛ µ ] ɛ ρ (see Eq. S18, and he balance beween µ and ρ (Eq. S Deriving ρ The mixure raio ρ = P(u res u res δ = u + s 0: a ime can be wrien as ρ = P ( u,u res = u + s 0: du (S29 By using he filering equaion of he opimal esimaor (equaion S7, we wrie ρ P ( u res = u + u res ( δ P u res δ s 0: δ u res δ u res δ P ( u res P(s u P ( u u δ,u res = u + u res ( δ P u res δ s 0: δ P ( s u δ,u res P ( u res u res δ = u + P ( u δ u res δ,s 0: δ du δ du = u + P ( u δ u res δ,s 0: δ du δ = u + u res ( ( δ P u res δ s 0: δ P s u res = u +,u res δ,s 0: δ (S30 (S31 If we assume ha he he probabiliy of having a spike a ime, given he resing membrane poenial a ime, u res and he spiking hisory s 0: δ, is independen of he resing membrane poenial a ime δ, we have: ρ P ( u res = u + u res ( ( δ P u res δ s 0: δ P s u res = u +,s 0: δ P ( s u res = u +,s 0: δ [ (1 ρ δ η + δ + ρ δ ( 1 η δ ] (S32 Similarly, he esimaed probabiliy of being in he down sae is appoximaely proporional (wih he same consan of proporionaliy o (1 ρ P ( s u res = u,s 0: δ [ (1 ρ δ ( 1 η + δ + ρ δ η δ ] (S33 Le α denoe he proporionaliy facor assumed in equaions S32 and S33. By firs noing ha ρ can be rivially rearranged as 1 ρ = 1+ α (1 ρ (S34 αρ we can inser equaions S32 and S33 o ge 1 ρ 1+ P(s u res = u,s 0: δ [(1 ρ δ (1 η + δ+ρ δ η δ] P(s u res = u +,s 0: δ [(1 ρ δ η + δ + ρ δ (1 η δ] Formally, i can be shown, ha (S35 lim ρ γ + /γ =1 ρ and lim ρ γ + /γ = ɛρ [1 ρ ] where ɛ = 1 1 γ /γ + 0. I is ineresing o noe ha in his laer limi he updae in ρ happens o be proporional o he uncerainy abou he resing membrane poenial (i.e. he variance of he associaed Bernoulli disribuion, ρ [1 ρ ] 5

6 Pfiser e al. Synapses as opimal esimaors By aking he limi of δ 0 and afer some elemenary algebra, we can express he emporal derivaive of ρ: ρ ρ δ ρ = lim δ 0 δ η + [ η + + η + γ + γ ] ρ [ γ + γ ] ρ 2 + ρ ɛs (S36 where γ + and γ denoe he condiionalised poserior firing raes of he up sae and down sae respecively: γ + δ = P ( s =1 u res = u +,s 0: δ γ δ = P ( s =1 u res = u,s 0: δ and ρ denoes he ampliude of he updae of ρ afer having observed a spike a ime : ρ = 1+ γ γ ρ ρ ρ (S37 6

7 Pfiser e al. Synapses as opimal esimaors { Supplemenary Figures and Legends 100 ms { presynapic neurons J 1 J M -('./012/*'3, #!!#! "!! #!! $!! %&'(*'+, possynapic neuron Firing rae [Hz] Time [ms] Supplemenary Figure S1. Compuaion wih M = 100 inpus. Lef: membrane poenial races u (i of he presynapic neurons. Middle-lef: presynapic spike rains generaed from he membrane poenial races. Middle: esimaed presynapic membrane poenial û (i wih a saic synapse (green a dynamical synapse (blue and he opimal esimaor (red. The black line denoes he presynapic membrane poenial iself. Middle-righ: ideally, he somaic membrane poenial of he possynapic cell compues a (poenially non-linear funcion (here, we ook v (soma = i J iu (i of is M = 100 inpus (black. Pracically, his is approximaed as i J iû (i wih esimaes û (i given by saic synapses (green, dynamical synapses (blue, or he opimal esimaor (red. Righ: firing rae of he possynapic neuron which is compued as a non-linear (here sigmoidal funcion of he somaic poenial of he possynapic neuron, f ( v (soma. Parameers for he simulaion were: βσ=2, τ =20 ms, g(u res =10 Hz, J i =1/ M, f(v=f 0 /(1 + exp( αv, wih f 0 =50 Hz and α=2 mv 1. 7

8 Pfiser e al. Synapses as opimal esimaors a.u 1... u 1 u c.s 1... s 1 s b u res 1... u res 1 u res.u 1... u 1 u :/;3./3./*26;0<4-4/=0-.3*07>?9 "!!"!#$!#"!$$!$".s 1... s 1 s *+,-./*0123*/ u $" (" '" &" %" Supplemenary Figure S2. Generaive model for presynapic membrane poenial flucuaions and spike generaion. (a Graphical model for he case wih consan resing membrane poenial. The membrane poenial u follows an Ornsein-Uhlenbeck (OU process around a consan resing poenial u res. A spike is elicied a ime (s = 1 wih probabiliy g(u δ. (b Graphical model for he case wih changing resing membrane poenial. The resing poenial u res (see Eq. S5 can randomly swich beween a down and an up sae. (c Insananeous firing rae g(u as a funcion of he membrane poenial u for differen values of he spiking deerminism parameer β (β 1 = 3 mv solid line, 10 mv do-dashed line, 1 mv doed line wih g 0 se such ha g( 60 mv=10 Hz. 8

9 Pfiser e al. Synapses as opimal esimaors 1 ρ ρ Supplemenary Figure S3. ρ, he incremen in ρ afer observing a spike, as a funcion of ρ, for differen values of γ+ γ (1 blue, 2 green, 5 orange, 10 red, - black. Doed black line shows he locaion of he peak of he curves as γ+ γ is varied 1. 9

10 Pfiser e al. Synapses as opimal esimaors References 1. Bors, J.G.G. The low synapic release probabiliy in vivo. Trends Neurosci. 33, ( Diman, J., Kreizer, A. & Regehr, W. Inerplay beween faciliaion, depression, and residual calcium a hree presynapic erminals. J. Neurosci. 20, ( Dayan, P. & Abbo, L.F. Theoreical Neuroscience (MIT Press, Cambridge, Zador, A. The basic uni of compuaion. Na. Neurosci. 3, 1167 ( Aguera y Arcas, B., Fairhall, A.L. & Bialek, W. Compuaion in a single neuron: Hodgkin and Huxley revisied. Neural Compu. 15, (

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

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