A method for decoding the neurophysiological spike-response transform

Journal of Neuroscience Methods 184 (2009) 337356

Contents lists available at ScienceDirect

Journal of Neuroscience Methods

journa l homepage: www.e lsev ier .com

A meth l sp

Estee Ste hara Fishberg Depa y Placb Institute of Ne mpusc Courant Instit Merce

a r t i c l

Article history:Received 21 AReceived in reAccepted 31 Ju

Keywords:Motor controlSpike trainsSynaptic transSynaptic plastNeuromusculaNonlinear system identicationNeurophysiological inputoutputtransformMathematical modeling

d byuilt uy temlf be

erizeerneredice syrall rme ti

other functions that characterize the spike-response transform. We avoid the need for any specic initialassumptions about these functions by using techniques of mathematical analysis and linear algebra thatallow us to solve simultaneously for all of the numerical function values treated as independent parame-ters. The functions are such that they may be interpreted mechanistically. We examine the performanceof the method as applied to synthetic data. We then use the method to decode real synaptic and muscle

1. Introdu

The spikcrete spikepropagatestotypical exelicits, in turelease of ntance, curreSabatini anstanding thfrom the facomplexityprevious onhistory of thprocesses opost-tetanicand FriesenHunterandbut implicit

CorresponE-mail add

0165-0270/$ doi:10.1016/j.contraction transforms. 2009 Elsevier B.V. All rights reserved.

ction

e is a basic organizing unit of neural activity. Each dis-stimulus elicits a discrete elementary response thatthroughvarious neurophysiological pathways. Thepro-ample is at the synapse, where each presynaptic spikern, a discrete rise in presynaptic calcium concentration,eurotransmitter, and changes in postsynaptic conduc-nt, and potential (Katz, 1969; Johnston and Wu, 1995;d Regehr, 1999). The fundamental difculty in under-e stimulusresponse relationship in such cases comesct that the spikes occur in trains of arbitrary temporal, and that each elementary response not only sums withes, but can itself be greatly modied by the previouse activityin synaptic physiology, by the well-known

f synaptic plasticity such as facilitation, depression, andpotentiation (Magleby and Zengel, 1975, 1982; Krausz, 1977; Zengel and Magleby, 1982; Sen et al., 1996;Milton, 2001;ZuckerandRegehr, 2002).Downstreamofly including theseprocesses of synaptic plasticity,when

ding author. Tel.: +1 212 241 6532; fax: +1 212 289 0637.ress: [email protected] (E. Stern).

they occur at a neuromuscular junction, is then such a responseas muscle contraction. In this paper we will analyze, in additionto synthetic data, experimental data from a slow invertebratemuscle where prolonged response summation and a highly non-linear and plastic neuromuscular transform (Brezina et al., 2000)make for a very complex, irregular response that, at rst sight,appears extremely challenging to understand and predict quan-titatively. Fig. 1 illustrates some of the factors that are responsiblefor the complexity of the spike-response transform with syntheticdata.

To achieve a predictive understanding of the spike-responsetransform is our goal here. As indicated in Fig. 1, from the observ-able data, just the spike train and overall response to it, we wish toextract a set of building-block functionsthe elementary responsekernel and other functions that describe the dependence of theresponse on the previous historythat will allow us to predictquantitatively the response not only to that particular spike train,but to any arbitrary spike train. This will constitute a completespike-level characterization of the spike-response transform.

The problem is one of nonlinear system identication. In neuro-physiology, and synaptic physiology in particular, such problemshave been approached by several methods (for a comparativeoverview see Marmarelis, 2004). A classic method is to t the datawith a specic model (e.g., Magleby and Zengel, 1975, 1982; Zengel

see front matter 2009 Elsevier B.V. All rights reserved.jneumeth.2009.07.034od for decoding the neurophysiologica

rna,, Keyla Garca-Crescionib, Mark W. Millerb, Crtment of Neuroscience, Mount Sinai School of Medicine, Box 1065, One Gustave L. Levurobiology and Department of Anatomy, University of Puerto Rico Medical Sciences Caute of Mathematical Sciences and Center for Neural Science, New York University, 251

e i n f o

pril 2009vised form 31 July 2009ly 2009

missionicityr

a b s t r a c t

Many physiological responses elicitepotentials, muscle contractionsare bthe spikes occur in trains of arbitrarsums with previous ones, but can itsein system identication is to charactfunctionsthe elementary response kdence on previous historythat will pdeveloping further and generalizing thtimes in a train and the observed ovebest estimated response and at the sa/ locate / jneumeth

ike-response transform

les S. Peskinc, Vladimir Brezinaa

e, New York, NY 10029, United States, 201 Boulevard del Valle, San Juan, PR 00901, United Statesr Street, New York, NY 10012, United States

neuronal spikesintracellular calcium transients, synapticp of discrete, elementary responses to each spike. However,poral complexity, and each elementary response not onlymodied by the previous history of the activity. A basic goalthe spike-response transform in terms of a small number ofl and additional kernels or functions that describe the depen-t the response to any arbitrary spike train. Here we do this bynaptic decoding approach of Sen et al. (1996). Given the spikeesponse, we use least-squares minimization to construct theme best estimates of the elementary response kernel and the

338 E. Stern et al. / Journal of Neuroscience Methods 184 (2009) 337356

Fig. 1. The pro ng, illsynthetic func rdingup of the trans ottomthe spike trainbottom set ofblue curve). Eathat spike timwaveforms) tosummate andbecause after ais, given only {

and Maglebbe guidedfails to genwhite-noiseWiener, orNaka, 1973Song et al., 2Although intion, the hiand interprimize the dintroduced,synaptic das far as totary responorder kerneof additionaing a modelorder nonliadopt, howMilton wermethod waple, some fuWith the fafurthermorkernel andessentiallyblem dealt with in this paper: the complex spike-response transform and its decodition forms. (A) The forms assumed here for the three functions K, H, and F that, accoform from the functions in panel A by the operation of the model, applying, from b

{ti} (blue dots along the bottom) elicits an instance of the dynamic kernel H that sums

waveforms) to give the overall waveform

iH (thick black curve).

iH is then transfo

ch spike also elicits an instance of the dynamic kernel K (the elementary single-spike ree (dark blue dots on A and arrows up to the top set of waveforms), and summed with pregive the overall response R to the spike train (blue curve). Note how, due to the slow sp

fuse, so that the shape of K cannot be discerned in R by simple inspection. Neither can Kn interspike interval that is longer than H, there is (with F(0) = 0) no response at all (e.gti} and R, to extract functions K , H, and F that are good estimates of the true functions K,

y, 1982). However, the choice of model must typicallyby the limited dataset itself, so that often the modeleralize. In a model-free approach, on the other hand,stimulation is used to determine the systems Volterra,other similar kernel expansion (e.g., Marmarelis and; Krausz and Friesen, 1977; Gamble and DiCaprio, 2003;009a,b; for reviews see Sakai, 1992;Marmarelis, 2004).principle providing a complete, general characteriza-

gher order kernels are difcult to compute, visualize,et mechanistically. To combine the strengths and min-rawbacks of these two approaches, Sen et al. (1996)and Hunter and Milton (2001) extended, the method of

ecoding. This method follows the model-free approachnd the systems rst-order, linear kernel (the elemen-se kernel), but then, rather than computing the higherls, combines the rst-order kernel with a small numberl linear kernels and simple functionsthus constitut-, but a relatively general oneto account for the highernearities. Here we adopt this basic strategy. We cannotever, the simplications that Sen et al. and Hunter ande able to make by virtue of the fact that their decodings geared toward synaptic physiology, where, for exam-nction forms were a priori more plausible than others.

st and rarely summating synaptic responses, they weree able to obtain the shape of the elementary responsethe amplitude to which it was scaled at each spike

by inspection (Hunter and Milton, 2001). In many slow

neuromuscduces no cotogether tofuse, so thalation. (Fig.and Huntersearch, requdescribe, aning methodlimitations.

Followinconvenientspike-respoand the resthen offerssense, of thtransform,

2. Method

In this smatical alg

The fundresponse wdecomposekernels. Furfunctions austrated with the model used in most of this paper and representativeto the model, build up the spike-response transform. (B) The buildingto top, rst Eq. (2) and then Eq. (1) in Section 2.1. First, each spike ofwith previous instances (as indicated by the thin black curves of thermed by the static nonlinear function F to give the waveform A (darksponse kernel), which is scaled in amplitude by Ai , the value of A at

vious instances (as indicated by the thin black curves of the top set ofeed of K relative to the typical interspike interval, the instances of K

be seen in isolation simply by making the interspike intervals longer,., at the two downward arrows). (C) The goal of the decoding methodH, and F.

ular systems, in contrast, an isolated single spike pro-ntraction at all, and when spikes are sufciently closeproduce contractions, the contractions summate andt the elementary response can never be seen in iso-1 shows this with synthetic data.) Finally, Sen et al.and Milton extracted parameters by a gradient descentiring many iterations and a good initial guess. Here wed apply to synaptic and neuromuscular data, a decod-that largely avoids such simplifying assumptions and

g Sen et al. (1996), we use the term decoding as ashorthand for theprocess of system identicationof thense transform from its input and output, the spike trainponse to it. However, the formulation of our methodthe possibility of a decoding also in the more usuale spike train from the response in which, through thethe spike train has been encoded (see Section 4.3).

s

ection we provide an overview of the methods mathe-orithm.amental assumption of the method is that the overallas, in fact, built up in such a way that it is meaningful toit again into a small number of elementary functions orthermore, we must assume some model of how thesere coupled together. However, in contrast to traditional

E. Stern et al. / Journal of Neuroscience Methods 184 (2009) 337356 339

model-basedmethodsand to someextent even thepreviousdecod-ing methods, we do not have to assume any specic forms for thesefunctions.

2.1. An illustrative model

We will illustrate how the method works with one particu-lar model, motivated by the models that are typically used forsynaptic transmission (MaglebyandZengel, 1975, 1982;Zengel andMagleby, 1982; Sen et al., 1996; Hunter and Milton, 2001). Otherpossible models are discussed in Section 3.3.9.

In formulating the model, we assume that

R(t) =i:ti


with respect to the discrete variables Ai and the continuous func-tion K(t). To simplify the problem, we rst discretize time intobins of duration t. Thus, t = nt, ti = nit, Kn = K(nt), Rexpn =Rexp(nt), where n and ni are integers. Then I0 is replaced by theRiemann su

I =

n=

[i

Wealso assory, so thatN (the lenK1, . . . , KN .N + Ns variaTo solve it,and set the

For K, thtions in N u

PmnKn = Qmwhere

Pmn =(i,j):n

Qm =Nsj=1

R

for m = 1, .is symmetrsystem hasdepends up

SimilarlyNs unknow

XkiAi = Ykwhere

Xki =

n=K

Yk =

n=R

for k = 1, .n > N, the sa nite numis symmetrsystem hasdepends on

Thus, weand a linearthe values ofollowing it

K (l+1) = (P

(l+1) = (X

for iteration

the value ofvalue of X

(see Sectionwell reversstart the ite

For l 1, the iterative scheme generates an estimate K (l) of Kand an estimate (l) of A. We use these to construct an estimate R(l)

of R according to

Ns

i=1K

e fo

=[

eachA, g

teed

I(l).

ver, s) conl) isns is

Perfoaccelwe sParzeooth, it w

(ni)

merinatan smlar ed bkenas d k aat a mwashing)hingthe

+1) 3.3

ep 2:

preap

er, wj:tj N for some integer

gth of K). Thus, the nonzero values of K are at mostThe task is therefore to minimize I with respect to thebles K1, . . . , KN , A1, . . . , ANs . This is a calculus problem.we differentiate I with respect to each of the variablesresult equal to 0.is leads (see Appendix A) to a linear system of N equa-nknowns given by

(3)

inj=mn

AiAj

expm+njAj

. . , N and n = 1, . . . , N. The matrix P (a Toeplitz matrix)ic and positive denite, guaranteeing that the lineara unique solution K1, . . . , KN . However, the solutionon the Ais, which are also unknown., for the Ais, we have a linear system of Ns equations in

ns given by

(4)

nnkKnni

expn Knnk

. . , Ns and i = 1, . . . , Ns. Since Kn = 0 for n 0 and forumsovern in these denitions ofXki andYk involve onlyber of nonzero terms. The matrix X (a banded matrix)ic and positive denite, guaranteeing that the lineara unique solution A1, . . . , ANs . However, this solutionK1, . . . , KN .have a linear system that determines K given A, Eq. (3),system that determines A given K, Eq. (4). To solve forf K and A that simultaneously minimize I, we use theerative scheme:

1Q )(l)

1Y)(l+1)

number l = 0,1, . . .. Here (P1Q )(l) is shorthand forP1Q when A = (l), and (X1Y)(l) is shorthand for the

1Y when K = K (l). To start the iterations, we arbitrarily3.3.3) set (0)

i= 1 for i = 1, . . . , Ns. [We can equally

e the order of nding K and A within each iteration andrations with K (0)n = 1 or 1/N for n = 1, . . . , N.]

R(l)n =

Then w

I(l) =

n

Sinceof K orguaran

I(l+1) Moreothat I(l

until I(

iteratio

2.2.1.To

3.3.2),lter (the smEq. (4)

i =

The nudenomGaussiparticugovernto weation wRexp anspikes3.2), ksmootsmootWhilethat I(l

Section

2.3. St

Theus fromHowev

S(t) =

In thisto the

WmnHn(l)nni

(l)i

.

llow the progress of the algorithm by monitoring

R(l)n Rexpn ]2t. (5)

step of our iterative scheme nds the optimal valueiven the current estimate of A or K, respectively, it isthat

ince I(l) is bounded frombelowby0, it is alsoguaranteedverges as l . In practice, we continue the iterationssufciently small or until a preset maximal number ofreached.

rmance-enhancing modicationerate the convergence of some decodings (see Sectionmoothed with a Gaussian kernel density estimationn, 1962; Szcs et al., 2003). In each iteration in whiching was applied, as soon as was obtained by solvingas replaced by given by

=

Nsj=1

(nj) exp((ni nj)2/22)

Nsj=1

exp((ni nj)2/22). (6)

ator of Eq. (6) is the sum of the values of and theor is the number of the values of included under theoothingkernel centeredon the spike timebinni of each

i. The standard deviation of the Gaussian kernel, , wasy a function that allowed to decreasethe smoothingwith increasing iteration number l 1. A typical func-

= Nt/klp, where Nt is the total number of time bins ofnd p are constants. For the typical datasetwithNs = 100ean spike rate r = 0.1, and so Nt 1,000 (see Section

set to 20 or 30 and p between 1 (for slowly weakeningand 2 (for more rapidly weakening smoothing). Thewas usually removed entirely after 1020 iterations.smoothing was applied, it was no longer guaranteedI(l), although this usually remained true in practice (see

.2 and Fig. 5).

nding H and F

sence in Eq. (2) of the nonlinear function F preventsplying a strategy like that in Step 1 directly to Eq. (2).e can apply it to the simpler equation

(t tj).

, a derivation like that in Step 1 (see Appendix B) leadsm

m (7)


where

Wmn =

(i,j):njni=mn1

Zm =Nsi=1

Sem

for m = 1, .the spikes wsystem of N

To use Eindeed to our model tis that F is iset S(t) = F

Like SteBefore theidentity funestimateof

F from thep

construct aBy them

(t) = A(t).all of the ertook F1 tocycle throuand H woulassumptioncurve withF is a scatteA for simila

contrary toneously F1

smoothingthe smootherror betwemaximal nu

A difcuis that, to sfunction S(tdiscretizedtime bin.)values of A(every time bthe spike timbut then lonly {Aj} ismate (t) ohowever, westimate {

2.3.1. PerfoIn practi

nicantly gr

the last termstructed in tH anew innent, H, thimplementetherefore in

Investigafound that ijust mentio

in every time bin, making the complete A(t) available to the algo-rithm, then Eq. (7) operated analogously to Eq. (3) in Step 1 andthe modication was not needed. When the spikes were sparse,however, then Eq. (7) was tting a waveform, S(t), that consisted

inathanuldnt solun, th

s a ree inem

re boe tonlateerpor in af thens. E

siveof prbitraot soee Se

Smoosmoocessost

Gausmesdisc

correto ex

H)

=

j

y dec,, toaxj([moseachimatly Nsing asparnd ththe seffecerted

1, to

o in

h twintey Eq

R1(t)xp+ni

. . , N and n = 1, . . . , N. (The matrix W simply countsith the specied interspike intervals.) This is a linearequations in the N unknowns H1, . . . ,HN .q. (7) to nd H and at the same time to deal with F,nd F, in the full Eq. (2), we add to the assumptions ofwo further assumptions about F. The rst assumptionnvertible. In that case, if we know F and so F1, we can1(A(t)).p 1, Step 2 therefore proceeds iteratively (see Fig. 2).rst iteration, we initialize the estimate of F, F , to thection, so that S(t) = A(t). We solve Eq. (7) to obtain anH, H, and construct [

jH](t).WeuseEq. (2) to construct

airs ([

jH](t), A(t)). Finally, applying F to [

jH](t),we

n estimate of A(t), (t).selves, these constructions would produce, of course,We would have t A(t) perfectly, but only by collectingror in the underlying F and H. Furthermore, if we nowbe the inverse of F in this form, the algorithm wouldgh precisely the same values in the next iteration: Fd never improve. To improve them, we make a secondabout F, namely that it is a relatively simple, smooth

no very sharp changes in slope. When plotted, the rawr of points, many of which have very different values ofr (even, to a given precision, the same) values of

jH,

this assumption. We therefore smooth F , and simulta-, using some smoothing lter (see Section 2.3.2). Theimproves F and so, in the next iteration, also H. Withed F , (t) /= A(t). The iterations continue as long as theen (t) and A(t) continues to decrease or until a presetmber of iterations is reached.lty in implementing Step 2 exactly as just describedolve Eq. (7) for the time-continuous function H(t), the) = F1(A(t)) must likewise be time-continuous. (Withtime, continuous here means having a value in eachHowever, from Step 1 we know only {Aj}, that is, thet) sampled only at the spike times {tj}, in general not inin. In each iteration we therefore use only the values ates to construct F and F1 and compute {Sj} = F1({Aj}),

l in these points by linear interpolation. Even thoughknown, the algorithm continues to construct the esti-f the entire continuous A(t). As a termination criterion,e use only the error between the known {Aj} and itsj} taken from (t).

rmance-enhancing modicationce we found that the efcacy of the algorithm was sig-eater if we modied each Sj to F1(Aj) [

iH]j , where

is the value at the spike time tj of the summed H con-heprevious iteration.Now, insteadof nding thewholeeach iteration, the algorithm found a residual compo-at corrected H from the previous iteration.We routinelyd this modication as part of the basic algorithm; it iscluded in Fig. 2.ting the reason for the efcacy of this modication, wet was connected with the incomplete knowledge of A(t)ned. If the spikes were so dense that a spike occurred

predomrathereral, coperfeciteratiotion. AimprovWith thSj s wepositivinterpothe intsmalletude oiteratiosuccestermsany arwere nof H: s

2.3.2.For

the suctype. Mwith asometiby the

of thetaken

and F (

F (H)

In earlkernel = (m100. Inthat atapproxtypicalbeginnF wereWe foumarilymajorwe invthat F

2.4. Tw

Witerwisegiven b

R(t) =ntly of the values linearly interpolated between the Sj sthe Sj s themselves, a waveform that therefore, in gen-ever perfectly reect the true summedH. Thus, even if ation forHwerehypothetically tobe found inaparticularat solution would be degraded again in the next itera-sult, without the modication, the estimate H ceased toany systematic way after a small number of iterations.odication, however, after therst iteration the residualth positive and negative, with many Sj s switching fromegative andvice versa in successive iterations; the lines

d between the Sj s often passed through zero so thatlated values were likewise positive and negative andbsolute terms than the Sj s themselves; and the ampli-whole waveform progressively decreased in successiveach value of the estimate H was thus built up over the

iterations as a converging sum of positive and negativeogressively decreasing magnitude, allowing essentiallyry shape of H to emerge fully (provided that the spikessparse that the interspike intervals exceeded the lengthction 3.3.6).

thingthing F and F1,we tested several lters and found thatof the decoding did not critically depend on the lterof the results presented in this paper were obtainedsian kernel density estimation lter, analogous to thatused in Step 1, implemented as follows. With F givenrete pairs ([

iH]j, Aj) as described above, the domain

sponding continuous smoothed function F (H) wastend from H = minj([

iH]j) to H = maxj([

iH]j),

was computed using the equation (analogous to Eq. (6))

Ns

=1Aj exp((H [

i

H]j)2/22)

Nsj=1

exp((H [

i

H]j)2/22)

. (8)

odings, we set the standard deviation of the Gaussiansomexed fraction of thewidth of the domain of F , i.e.,

iH]j) minj([

iH]j))/k, where k was typically 30 ort later decodings, however, we adaptively varied soH the value of the denominator of Eq. (8) was alwaysely equal to somexed fraction of the number of spikes,/10, Ns/30, or Ns/100. To avoid oversmoothing at thend end of the domain of F where the discrete values ofse, we sometimes set an upper limit on the value of .at, within these parameters, the value of affected pri-peedof convergenceof thedecodingalgorithm,withoutt on the nal result. After obtaining F in each iteration,

it and smoothed it by a similar procedure to ensureo, was a function.

puts

o independent inputs 1 and 2 that sum but do not oth-ract in producing the response (see Fig. 10), the models. (1) and (2) expands to

+ R2(t)


where

R1(t) =i:ti


Fig. 3. Decodiconstructed, bits decoded esKexp, Hexp, andValidation datpanel A, but itsduration t an

Note, hofunctions dadditional cthe correctlying functiabsolute scastructure ofby a constasettlesondeparison of tin Fig. 3C anand Hexp, anfurthermorsizes of K anstability an

Becausevery well ang of synthetic data. (A) The three functions Kexp, Hexp, and Fexp that were used to consty the operation of Eqs. (1) and (2) as in Fig. 1, for a train of 100 random spikes with meantimate R (red curve). The inset plots R against Rexp. The normalized percentage RMS errorFexp from panel A (blue) overlaid with their decoded estimates K , H, and F (red). The resa. For a different train of 100 random spikes (blue dots along the baseline), Rexp (blue cuestimate R (red curve) was constructed from the functions K , H, and F that were decoded amplitude is in arbitrary units.

wever, that the absolute scaling of the underlyingoes not in fact have a unique solution: without someonstraint, although the algorithm would have foundshapes and the correct relative scaling of the under-ons, it would not have been able to recover the trueling of each individual function. This is inherent in thethe model. In Eq. (1), for instance, K can be multiplied

nt, k, if {Ai} is multiplied by 1/k. Which k the algorithmpendson the initializationof {i}. Topermitdirect com-

he shapes of the true and estimated synthetic functionsd other decodings, we constrained the integrals of Kexpd then of K and H in each iteration, to be 1. We found,

e, that imposing some such constraint on the absoluted H in successive iterations signicantly increased the

d speed of convergence of the algorithm.the estimates K , H, and F were excellent, they wereble to predict the response to a different spike train

with the sastatistics.

3.3. Perform

3.3.1. NumThe mat

system of s(3)), {i} (Eqfound by a dessary. Howin the casethe other eiterations ato obtain atypical plotFig. 4. We foruct the overall response Rexp. (B) A representative Rexp (blue curve)spike rate r = 0.1 (blue dots along the baseline), then overlaid withbetween R and Rexp, ER (see Section 2.5), was 2.0%. (C) The functions

pective RMS errors were EK = 0.008%, EH = 15.0%, and EF = 2.7%. (D)rve) was again constructed from the functions Kexp, Hexp, and Fexp ind from the rst Rexp in panels B, C. ER was 4.8%. Time is in time bins of

me nominal statistics (Fig. 3D) or indeed with different

ance characteristics of the algorithm

ber of iterationsrix implementation of the algorithm solves the entireimultaneous equations for all of the values of K (Eq.. (4)), and H (Eq. (7)) directly, in one step, and F is alsoirect construction. No low-level iterative search is nec-ever, each of these estimates is conditional on one (or,

of the other models discussed below, more than one) ofstimates. To improve each estimate in turn, high-levelre still required. How many iterations were requiredgood decoding? For a decoding like that in Fig. 3, aof RMS error against iteration number can be seen inund that in Step 2 of the algorithm (see Section 2.3) the


Fig. 4. Magnitthat in Fig. 3, win Fig. 3A. (A) Tin Step 1. (B) TEA (lled squaproduced by HEF (open circlegiven as inputAexp and at t

estimates Herrors aftermates K anas well, someven more)erate the co

3.3.2. ConvIn Step

decreased iconvergencerrors in thcially EAi, cagain, as inever, the speven with teven failedthat we allcal purposeshows such80 iterationthose thatrithm achie8.5% RMS elying estim

plots, respectively) were completely incorrect (the RMS errors areplotted in Fig. 5A(left) and given in the legend).

The primary problem lay in the estimation of {i}.We calculatednorm)conditionnumbers for the relevantmatrices.Whereastrix Plative5, oe Eq

). Thiormat, maiallyationrticu, sincmbent noio itsdedimessumed tore, wted ve (Feverost ss unddy. Fthe (2-the mahad reFigs. 3{i} (seto 106

the infdataseessentinformthat paEq. (7)tion nuinherescenaris trasame tcould aaveragtherefoestimanegatimore sthus m

Thia remeudes of the RMS errors E in successive iterations of a decoding likeith 100 spikes at r = 0.1 and the same functions Kexp, Hexp, and Fexp ashe errors EK (open triangles), EAi (open squares), and ER (lled circles)he errors EH (lled triangles), EF (open circles), EAj (open squares), andres) in Step 2 alone, i.e., when it was given as input the true {Aj,exp}exp and Fexp. (C) The errors EK (open triangles), EH (lled triangles),s), and ER (lled circles) in Steps 1 and 2 combined, when Step 2 wasthe estimate {j} from Step 1. EAi , EAj , and EA are the errors betweenhe spike times {ti} or {tj} or over all time bins, respectively.

and F converged to practically stable values with lowonly a few (


Fig. 5. AccelerFig. 3A. This saiterations. Left(red) superimp91.7%, EAi 270.perfectly. EK w0.02%, EAi 0.07

much the sreasonableused in Figslowly evenallowed totoo and thebehaving as

More genecessarily1 did conveation of convergence in Step 1 by smoothing of {i}. For a train of 100 spikes at r = 0.me dataset was decoded in all three panels (A)(C). (A) With this dataset, the basic decodi, the magnitudes of the RMS errors EK (green), EAi (red), and ER (black) in successive iteraosed on Kexp and Rexp (blue), plotted as in Fig. 3B and C, and the estimate {i} plotted aga1%, and ER 8.5%. (B) With smoothing of {i} over the rst 15 iterations (gray rectangle) asas 0.004%, EAi 0.01%, and ER 0.006%. (C) A perfect decoding was also achieved when the%, and ER 0.003%. Time is in time bins of duration t and amplitude is in arbitrary units.

ame: the dataset is not decoded satisfactorily within atime.) Thus, although this is imperceptible on the scale. 5A(left), the overall error ER continued to decreaseas EK and EAi increased, and when such a decoding was

continue sufciently long EK and EAi began to decreasedecoding eventually reached the low-error solution,in Fig. 4 but over a much larger number of iterations.

nerally, investigations with a variety of datasets, whileincomplete, suggested that when the estimates in Steprge, they always converged to the same solution (with

the constrationwas reaAnd, at leaswas a globather belowbecause itdecoding stmates R, K ,neverthelesstriking wit1, Rexp was constructed from the functions Kexp, Hexp, and Fexp as inng algorithm alone failed to achieve a satisfactory decoding within 80tions, plotted as in Fig. 4. Right, the estimates K and R at 80 iterationsinst {Ai,exp} (top right; the diagonal line is the line of identity). EK wasdescribed in Section 2.2.1, the same dataset was decoded essentiallysmoothing was applied later, between iterations 15 and 25. EK was

int on K already described). For example, the same solu-ched fromdifferent random initializations of {i} (or K).t over some range of dataset parameters, that solutionl minimum since it had essentially zero error (see fur-). Finally, the smoothing was instructive in this regardperturbed, not the initial conditions from which thearted, but rather the subsequent trajectory of the esti-and {i}. Following different trajectories, the estimatess reached the same nal solution. This was particularlyh the real data in Figs. 11 and 12, where the nal solu-


Fig. 6. Effectsas in Fig. 3, eHexp, and Fexpspike rates r. (0.1), and densamplitudes inand amplitudevarious decodplotted on logEach point plodatasets at thaThe errors EK(C) The errorssquares) in Ste(open circles),the errors are

tion had sireached if ttions of {idifferent smnal values

3.3.4. SpeedIn MATL

PC, withouiterations oThus a sufof iteration

3.3.5. FormFor conv

analytic funKexp, an ex

dened as the vectors of their numerical values, the dots in Fig. 3A.Quite arbitrary functions could therefore be used to construct Rexp(see, e.g., Fig. 10). Since the algorithm treats each function valueas an independent degree of freedom, and the initialization of the

n estimates is arbitrary, the algorithm was able to decodenctions just as successfully as those in Fig. 3. We veried

ith a wide range of functions. If Kexp had multiple peaks, forle, the algorithm successfully decoded the correspondingex K , distinguishing those peaks of Rexp that were inherent

from those that arose by the summation of the Kexps ofnt spikes (see Fig. 10).

Spike densityile the algorithm successfully decoded datasets like those3, its success varied with several parameters of the dataset.f these was the spike density. For the decoding problem,ensity is signicant not in absolute terms, but relative to thefunctiosuch futhis wexampcomplto Kexpdiffere

3.3.6.Wh

in Fig.Chief ospike dof spike density on the decoding. Synthetic datasets were constructedach with 100 spikes and Rexp constructed from the functions Kexp,in Fig. 3A, but with different spike densities produced by differentA) Typical overall responses Rexp to sparse (r = 0.01), optimal (r =e (r = 1) spike trains. Note the very different absolute time scales andthe three cases. Time is in time bins of duration t, r is in spikes/ t,is in arbitrary units. (B)(D) Magnitudes of the RMS errors E in the

ing estimates, after 300 iterations, as a function of r. Both r and E arescales. The horizontal line in each panel, at log E = 0, marks 1% error.tted is the mean of three runs of the algorithm with three differentt particular r. The errors plotted and their symbols are as in Fig. 4. (B)(open triangles), EAi (open squares), and ER (lled circles) in Step 1.EH (lled triangles), EF (open circles), EAj (open squares), and EA (lledp 2 alone. (D) The errors EK (open triangles), EH (lled triangles), EFand ER (lled circles) in Steps 1 and 2 combined. In panels B and D,cut off when they decrease below 105%.

gnicant errors, allowing lower-error solutions to behey existed. Nevertheless, not only different initializa-} or K , but multiple trajectories caused by somewhatoothing functions all converged to essentially identicalof R, K , and {i}.

of the algorithmAB on a 2.3 GHz Apple MacBook Pro or an equivalentt any special optimization of the code for speed, 300f a decoding like that in Fig. 3 took several minutes.ciently good decoding, requiring only a small numbers, could often be achieved in seconds.

s of K, H, and Fenience, the functions in Fig. 3A were generated usingctionsan -function (Dayan and Abbott, 2001) forponential for Hexpbut the functions were actually

Fig. 7. Magnitudes of errors with different numbers of spikes. Synthetic datasetswere constructed as in Fig. 3, with r = 0.1 and R constructed from the functionsKexp, Hexp, anddecoded for 30errors are cutalgorithm wit90 and 190 spconverge withthe algorithmand 3.3.2) andthe smoothingalso permittedations alreadysmoothing. Hnumber of spiand ER (lled cEAj (open squatriangles), EH (2 combined.exp

Fexp in Fig. 3A, but with different numbers of spikes. Each dataset was0 iterations. The errors plotted and their symbols are as in Fig. 4. Theoff above 100%. Each point plotted is the median of three runs of theh three different datasets with that particular number of spikes. Withikes, essentially arbitrarily in this set of runs, the decoding failed toin 300 iterations two out of the three times; for those spike numbers,was therefore rerun with the Step 1 smoothing (see Sections 2.2.1those results are plotted; all other points were obtained without

. More systematic investigation showed that the Step 1 smoothingessentially perfect Step 1 decodings to be obtained within 300 iter-with 20 spikes rather than the 3040 spikes shown here without the

owever, the basic problem of decoding a dataset with a very smallkes remained. (A) RMS errors EK (open triangles), EAi (open squares),ircles) in Step 1. (B) RMS errors EH (lled triangles), EF (open circles),res), and EA (lled squares) in Step 2 alone. (C) RMS errors EK (openlled triangles), EF (open circles), and ER (lled circles) in Steps 1 and


characteristic time scales of the response, here those of Kexp andHexp. By varying the mean rate r (expressed in spikes per time bin)of the spike train relative to thexed timescalesofKexp andHexp,wegenerated sparse spike trains with overall responses Rexp in whichthe individuor, on theKexps fused(Fig. 6A(righ

Fig. 6BDducedby thtoo areplotE = 0, markthe errors in{Aj}, and thewhen it waFexp. FinallySteps 1 anddecoding likonly the est1. We can sperformanc

(1) With vereconst2, veryestimathave be(sparser

(2) With veing in StThe algoAi werewith noresponsfor a fufunctionthe spikonly aftthen {Ajthey can

(3) Betweeregion, ahere, woverall ra broadFig. 3 hasatisfacoften noon aver10% ofreconsthad todbe red

3.3.7. NumbThe dec

How manyperformedgenerated aup to 200 sdecodingwhigher thanspikes. Theof spikes w

At the sand the cor

increased the computational burden. At the core of the algorithm,Eqs. (3), (4), and (7) involve the construction and manipulationof numerical arrays whose size increases with the number ofspikes Ns and the number of time bins Nt Ns/r of Rexp (as

s theowed

Noiseof thalityWe eue ono

xp; anctioderlyonstapikeaddseentes aThe qise; tid nowasrecoisy Ryingt the. Insey ccon

batio

Otherombys, mhm,odeld by

stanci:ti


Fig. 8. Decodiin these datasfrom a Gaussiaamplitude of tpanel B, the amRexp (black do12.4%. In panethe noise-free

In contradistinct spipletely deco(blue curveto two siminputs (emping throughNeverthelesand {tj}, thecurve in panunderlying

Likewiseels of the fo

R(t) =

i

K

A1(t) =

i

A2(t) = F(ng of noisy synthetic data. As in Fig. 3, but with random noise added to the overall respoets, was rst constructed from the functions Kexp, Hexp, and Fexp in panel A, identical to tn distribution with zero mean and a given standard deviation was then added to the noihe noise, that is, the RMS difference between the noise-free and noisy Rexp, was computplitude of the noise was 22.1%. The algorithm then decoded the noisy Rexp to estimate R

ts) as well as against the underlying noise-free Rexp (red dots). The error ER of R relativel C, EK was 31.6%, EH 9.8%, and EF 1.7%. For the validation data in panel D, the noise amplRexp 11.9%. Time is in time bins of duration t and amplitude is in arbitrary units.

st, with the additional information provided by the twoke trains in Eq. (12), models of this class were com-dable. This can be seen in Fig. 10. Here the overall Rexpin panel B) was constructed as the summed response

ultaneous random spike trains from two independentty and lled blue circles along the baseline), each act-its own set of functions Kexp, Hexp, and Fexp (panel A).s, given just Rexp and the two sets of spike times {ti}algorithm reconstructed Rexp essentially perfectly (redel B) and in the process correctly extracted all six of thefunctions (panel C).completely decodable, in principle,were cascademod-rm

(t ti)A1(ti),

H1(t ti)A2(ti),

i

H2(t ti))

.

3.4. Real da

We nowphysiologicinvertebratthat of the bIn the neurdrive synapEndogenouwe stimulapatterned sone datasetthe excitatoin the heartcle contractand (2) for

3.4.1. SynaFig. 11A

273 randomand the ressegment isnse Rexp before decoding. A noise-free Rexp, for 200 spikes at r = 0.1hose in Fig. 3A. In each time bin, a different random number drawnse-free Rexp to construct a noisy Rexp (blue curve in panel B). The RMSed analogously to the RMS error ER using the equivalent of Eq. (9). In(red curve in panel B). The inset in panel B plots R against the noisyto the noisy Rexp was 25.2%, but relative to the noise-free Rexp only

itude was 24.1%, ER relative to the noisy Rexp 24.9%, and ER relative to

ta

describe the decoding of two kinds of real neuro-al data. Both sets of data were recorded in a standarde preparation, the crustacean heart (Cooke, 2002), herelue crab, Callinectes sapidus (Fort et al., 2004, 2007a,b).

ogenic crustacean heart, trains of motor neuron spikesticdepolarizationsandcontractionsof theheartmuscle.sly, the spikesoccur inbursts. Inbothdatasets, however,ted the motor neurons to produce instead white-pike trains as in the synthetic data. As the response, inwe recorded the synaptic potentialsmore correctly,ry junctional potentials (EJPs)that the spikes elicitedmuscle, and in the other dataset the subsequent mus-ions. In both cases, we used the model given by Eqs. (1)the decoding.

ptic potentials(top) shows the entire decoded dataset, consisting ofmotor neuron spikes (blue circles along the baseline)

ulting Rexp of muscle EJPs (blue curve). A representativeexpanded below, overlaidwith the estimate R after both


Fig. 9. Magnitwere construcconstructed framplitudes of300 iterationsEF (open circle2 combined. Fothe point plott

Steps 1 anddataset, thein Step 1, wing of {Aj} iseemed higwere RMS,with thesethose obtai1973; KrauSong et al.,values. By rtimes, we foand preparwas 33.0%.and prediccase to theER of the derange.

The decis overlaidtative EJP cinterpretedneuromuscand FriesenHunter andthe amountof H can beter release athe immedi2001; Zuckereects facinonsummapairs of EJPsdepressioncular junctiinterspike itic plasticity

The estimresponse to

Fig. 11C(top), ER was 39.0%, not much higher than over the originaldataset.

Muscausetrachalled dairclestracndedcurvrve).tep 1tic din Ste of, butfor tprep.3%.decd onape oe sama incularHowude,uggethe tas inatteper been bsubsprodrvalrst. Tot o

low aestiudes of errors with different amplitudes of noise. Synthetic datasetsted as in Fig. 8, with 200 spikes at r = 0.1 and the noise-free Rexpom the functions Kexp, Hexp, and Fexp in Fig. 8A, but with different RMSnoise added to construct the noisy Rexp. Each dataset was decoded for. Plotted are the RMS errors EK (open triangles), EH (lled triangles),s), and ER relative to the noise-free Rexp (lled circles) in Steps 1 andr each nominal noise amplitude, the algorithm was run three times;ed is that with the lowest ER .

2 of the complete decoding (red curve). Over the entireRMS error ER was 35.6%. Most of this error originatedhere ER was 27.2%, whereas the error in the decod-

n Step 2 was only 15.5%. At rst glance, these errorsh. In part, however, they appeared high because theyas opposed to MS, errors. Allowing for this, a decodingerrors was in fact quite satisfactory when compared toned by other methods (cf., e.g., Marmarelis and Naka,sz and Friesen, 1977; Sakai, 1992; Marmarelis, 2004;2009b). Nevertheless, we computed some benchmarkecording the response to the same spike train severalund that the intrinsic RMS variability of the recording

ation, which ER cannot be expected to improve upon,At the other end of the scale, if we had no model at allted Rexp simply by its own mean, ER, equivalent in thiscoefcient of variation of R , was 127.2%. Thus, the

3.4.2.Bec

cle conmore cdecode(blue ccle conis expa(green(red cuafter Ssynthe2 thanbecausspikesmodelof thewas 69

Theoverlaithe shhas thEJP datromusorigin.amplittions, sstep ofof H hspike pspikess betwto thewouldan inteous buburst nthat al

The

exp

coding was in fact close to the low end of the possible

oded estimates K , H, and F are shown in Fig. 11B. Kon, to show that it closely resembles, one represen-ut from Rexp. H has a complex, biphasic shape. If H isas reecting processes of presynaptic plasticity at the

ular junction (Magleby and Zengel, 1975, 1982; Krausz, 1977; Zengel and Magleby, 1982; Sen et al., 1996;Milton, 2001; Zucker and Regehr, 2002) that determineof transmitter released,A, then the early negativephaseinterpreted as a brief depression of potential transmit-fter each spike, perhaps due to temporary depletion ofately releasable pool of transmitter (Hunter andMilton,r and Regehr, 2002). The later positive phase of H then

litation of the transmitter release. With the fast, largelyting EJPs, the depression could indeed be observed insimply in the raw recording (e.g., in Fig. 11A), and bothand facilitation have been observed at this neuromus-on with regular spike patterns incorporating differentntervals such as are conventionally used to study synap-

(Fort et al., 2005, 2007a).ates K , H, and F were equally well able to predict thea new train of spikes. Over the entire new dataset in

contractiondataset in Fthe datasetdatasets) it

4. Discussi

4.1. The dec

We havecation of tneurophysiapproach omethod comand modelapproach. Lthe rst-ording higher orst-order kand nonlineacterize themethods, ostructure. Mle contractionsof the overlap and summation of themuch slowermus-tions, we expected that they would present a muchnging decoding problem. Fig. 12A(top) shows the entiretaset, consisting of 295 random motor neuron spikesalong the baseline) and the resultingRexp of heartmus-

tion amplitude (blue curve). A representative segmentbelow, overlaid with the estimate R after Step 1 only

e) and after both Steps 1 and 2 of the complete decodingOver the entire dataset, the RMS error ER was only 7.1%and 28.9% after both Steps 1 and 2. As with the sparse

atasets (Fig. 6), the error was signicantly higher in Stepep 1, and so also in the complete decoding, presumablythe incomplete sampling of A(t) in Eq. (2) by the sparsealso, perhaps, because Eq. (2) may not be the optimalhis part of the real transform. The intrinsic variabilityaration was 8.0% and the coefcient of variation of Rexp

oded estimates K , H, and F are shown in Fig. 12B. K is, and closely resembles, three segments of Rexp in whichf Kexp could be identied (see legend). H, remarkably,e complex, biphasic shape as that decoded from the

Fig. 11B. Thus, the history kernel H of the overall neu-transform has a synaptic, and probably presynaptic,ever,where the F in Fig. 11Bnever produces EJPs of zerothe F in Fig. 12B does produce zero-amplitude contrac-sting most likely a threshold nonlinearity in the nalransform from EJPs to contractions. The complex shapeteresting implications for the case of the endogenousrn, where the spikes are generated in bursts with 7urst, 40 ms between the spikes in the burst, and 2.5ursts (Fort et al., 2004). Each spike inhibits the responseequent spikes in the same burst, and the entire burstuce little contraction response were it not preceded, bywell matched to the positive phase of H, by the previ-hus at the natural frequency of the cardiac rhythm eachnly produces a contraction, but sets up the conditionscontraction to be produced by the next burst.

mates K , H, and F were equally well able to predict theresponse to a new train of spikes. Over the entire new

ig. 12C(top), ER was30.0%, andover the last 100 spikes of(reecting a noticeable nonstationarity in each of thesewas just 21.2%.

on

oding method: its advantages and limitations

described here a method for nonlinear system identi-he transform from a discrete spike train to a continuousological response. Based on the synaptic decodingf Sen et al. (1996) and Hunter and Milton (2001), ourbines elements of traditional model-based methods

-free methods such as the Volterra or Wiener kernelike the latter methods, our method begins by ndinger, linear kernel of the system (K). Instead of comput-rder kernels, however, our method then combines theernel with a small number of additional linear kernelsar functions (in our illustrative model,H and F) to char-systems nonlinearities. Like traditional model-based

ur method thus ts to the data a model with a certainathematical analysis of the model is used to nd an


Fig. 10. Decod(blue curve), cbaseline), then(blue) overlaidbins of duratio

efcient ttlinear-algebassumes stadataset (theory (nite lafter each s

The metments, andhave a numing of two inputs. As in Fig. 3, but with two spike trains, each with its own set of functonstructed from the functions in panel A for 200 random spikes with mean rate r = 0.05overlaid with the decoded estimate R (red curve). The inset plots R against Rexp. ER waswith their decoded estimates K , H, and F (red). EK was 0.13%, EH14.4%, and EF3.2% forn t and amplitude is in arbitrary units.

ing algorithm that can be optimally implemented withra techniques. Like most other methods, the methodtionarity of the systemover thedurationof thedecodedmethodnds averages over that duration), nitemem-ength of K and H), and causality (K and H only operatepike).hods combination of model-based and model-free ele-the analytical and linear-algebra techniques of solution,ber of advantages but also some limitations.

Like themethod caninput (seestrictly requheartneurospike patte

The basSince any ct only a suions Kexp, Hexp, and Fexp, shown in panel A. (B) A representative Rexpfrom each of the two inputs (empty and lled blue circles along the1.6%. (C) The two sets of functions Kexp, Hexp, and Fexp from panel A

input 1, and EK0.25%, EH7.0%, and EF1.2% for input 2. Time is in time

model-free Volterra or Wiener kernel approach, thetake advantageof thebenets ofwhite-patterned spike

Sakai, 1992; Marmarelis, 2004), yet such input is notired. For example,wehavebeen able to decode the crabmuscular transformevenwith theendogenousburstingrns (Stern et al., 2007a).ic structure of the model must be chosen in advance.hosen structure, unless it is completely general, willbset of systems, the need for a model means that the


Fig. 11. Decodin the heart mprocess with aSection 3.4.1. Tbins over whicfrom the datasfor a second d

method canof anyarbitrin particulaimplicit in tther below.choose a mmation avathen undercand, withoudifculty is,

It is in itcient technadvantageothe previouton. In contmodel is chearities suchneed to beforms emercal functionand is initiafound inoneing of real synaptic potentials. (A, top) The decoded dataset, a representative recording ouscle of Callinectes sapidus (blue curve) in response to a train of 273 random motor nenominal rate of 3 Hz. The boxed segment is expanded below. The red curve is the estimahe decoding in Step 1, the reconstruction of R, and the computation of ER was performedh K extended after each spike. The decoding in Step 2 was performed with t = 10 ms, oet in panel A. K is superimposed on a representative scaled single EJP (blue). (C) Validatataset like that in panel A.

not a priori guarantee a completely general descriptionary system.Themanner inwhichour illustrativemodel,r, departs from the completely general model that ishe Volterra or Wiener kernel approach is discussed fur-It is furthermore possible (aswe saw in Section 3.3.9) toodel that is more complex than is justied by the infor-ilable in the single overall response Rexp. The model isonstrained by the data: multiple solutions are possiblet additional information, cannot be distinguished. Thishowever, common to allmethods that involve amodel.s relative freedom from a priori assumptions and ef-iques of solution where the method compares mostusly with traditional model-based methods as well ass decoding methods of Sen et al. and Hunter and Mil-rast to traditional model-based methods, once the basicosen, apart from the assumption that its static nonlin-as F are invertible and smooth, no specic assumptions

made about the forms of its kernels and functions. Thege automatically from the computation. Each numeri-value is treated as an independent degree of freedomlly set arbitrarily. All of the values of each function arestepby solvingamatrix equationorbydirect construc-

tion, and fobe globallythe solutioners. Howevfunction vathis is likelythat some mical analysilinear matrspike resposhape in somin Eq. (1), m

To makeour methodMilton in mton used mtheir methoal. and Hunthe shape oK from Rexpinspection.than workif EJP amplitude (intracellularly recorded muscle membrane voltage)uron spikes (blue circles along the baseline) generated by a Poissonte R after both Steps 1 and 2 of the complete decoding. For errors seewith a time bin duration t = 1 ms, only over the union of the time

ver all time bins of the dataset. (B) The estimates K , H, and F decodedion data. The estimates K , H, and F in panel B were used to predict R

r that step the solution can, in many cases, be proved tooptimal. High-level iterations are still required becausefor each function is conditional on estimates of the oth-

er, extensive low-level iterations to nd the individuallues are not required. For many practical applications,to be a decisive advantage. One limitation, however, isodels may not be readily amenable to the mathemat-

s required to separate the functions into the individualix equations. For example, models in which the single-nse kernel K does not have a xed shape, but changesemanner that ismore complex than the simple scalingay be difcult to solve.this discussion more concrete, it is worth comparingto the previous methods of Sen et al. and Hunter andore detail. Since both Sen et al. and Hunter and Mil-

odels similar to our illustrative model, we will discussds in terms of our functions K, H, F, A, and R. Both Sen etter and Milton obtained the kernel K simply by takingf an isolated response to a single spike. Deconvolvingthen gave the Ais, solving our Step 1 essentially by

Hunter and Milton note that extracting the Ais ratherng with the entire continuous R reduce[d] the compu-


Fig. 12. DecodCallinectes sapof 2 Hz. The boThe time bin dindicated by thThe estimates

tation timetrain. We dgenerality:EJPs in Fig. 1Fig. 12 andthe Ais, Sena gradientble assumpthe specicforms for thorder polynin the optimet al. as anations of thof high-levmanneresto nd an ia simple futheyused gfunction, thdescribed bWith our mof the full sing of real muscle contractions. (A, top) The decoded dataset, a representative recordinidus (blue curve) in response to a train of 295 random motor neuron spikes (blue circles alxed segment is expanded below. The green curve shows the estimate R after Step 1 onluration t was 10 ms throughout. (B) The estimates K , H, and F decoded from the datasee arrow in panel A) where, after an interspike interval just slightly longer than K , Rexp haK , H, and F in panel B were used to predict R for a second dataset like that in panel A.

by two to three orders of magnitude for a 30 sec spikeid not wish to take this shortcut because of its lack ofalthough it would have been possible to do it with the1, it would have been difcult with the contractions inimpossible with any denser response waveform. Fromet al. and Hunter and Milton then found H and F by

descent optimization coupled with the earliest possi-tion, guided where possible also by prior knowledge ofsynaptic system under investigation, of likely simpleese functions, for example exponential for H and low-omial for F, whose few parameters were then foundization. Taking the less automated procedure of Sen

example, this involved a nesting of the low-level iter-e gradient descent optimization itself within a seriesel iterations. First, Sen et al. divided H in a piecewisesentially as we did hereand used gradient descentnitial approximation of each piece. Then, substitutingnction that appeared best to describe the emerging H,radient descent again to optimize theparameters of thaten of F, whichwas constructedmuch like our F and theny a simple function, and nally of bothH and F together.ethod, in contrast, we nd the best current estimateshapes of K, H, and F in one step in each high-level itera-

tion, withoany initial gbe postulatby the datadescent optsimple funcemerge. Unable to solvsolution domany indivwhat is cleaThis may bein the relat

In the fukernels are(Sakai, 199mechanistiis also postwith a basican itself bing physiolin neuromube identieg of contraction amplitude (muscle tension) of the heart muscle ofong the baseline) generated by a Poisson process with a nominal ratey, the red curve after both Steps 1 and 2. For errors see Section 3.4.2.t in panel A. K is superimposed on three scaled segments of Rexp (oned, according to the model, the exact shape of Kexp. (C) Validation data.

ut any simplication of these shapes or dependence onuess. On the other hand, essentially any model that caned, even if it is in fact more complex than is justiedand has multiple solutions, can be solved by gradientimization, in the sense that some solution, which withtion forms will furthermore appear to be plausible, willder the same circumstances, our method may not bee the matrix equations for numerical reasons, or, if aes emerge, the many degrees of freedom allowed by theidual values of the functions K, H, and F will producerly an implausible solution (as in Fig. 5A, for example).valuable, however, as an explicit warning of a problem

ionship of the model to the data.ll Volterra or Wiener kernel approach, the higher orderabstract descriptions of the structure of the dataset

2; Marmarelis, 2004) that can typically be interpretedcally only if a traditional, specic model of the systemulated (e.g., Song et al., 2009a,b). In contrast, a modelc internal structure such as is decoded by our methode immediately interpreted in terms of the underly-ogical mechanisms. Interpreting our illustrative modelscular terms, as we have done here, H can plausiblyd with the presynaptic processes of facilitation and


depression of transmitter release, A with the amount of transmit-ter actually released at each spike, and K with the postsynapticelectrical response (when the EJPs are taken as the response) andthe Ca2+ elevation and contractile processes in the muscle (whenthe musclestatic nonlinthen be puby their bephysiologiccompared aafter modicrab cardiacof these wapaper intoneuromusc

4.2. Relatio

Althoughwe have wo(1) and (2).indeed in stcascades ofbeen studiesystems andods based oin the kerne1988; KorenMarmarelis

Superc(also calleda dynamicdynamic linand Kearnements are Hpurely seriaplaces, noteach spikeeach spikeallel branchschema ontwo branchically modeand Kearneif, for betteels, we likethat case, thby a sum otime, a sequtime bin). Oequation

R(t) = t

that is, a cosimilarly.) Tproduce R,the processdecoding pfeatures macascade (itas when T(methods us

Althoughdiscrete, re

putational benets by focusing only on the times when spikesoccur, it can also solve the continuous case, such as given by Eq.(13) or the continuous generalization of Eq. (2). Therefore, ourmethod can solve the general LNL cascade. Interestingly, in achiev-

e sole intmethd ofmprEq.

o a se thac nonrelisquivN cattinghis dtiverepquiror L euctur ofegrerepretionarne

rther

ougapseour dver tn begle sacelled w000;d cans riunctie gele a rtherm96), toblei arerainsthe4). Lch an1997ng (Susinglreadus an t

S(t)s thepre

ep uscontractions are taken as the response). F reects theearities of the entire pathway. Such interpretations can

rsued experimentally. Better models can be identiedtter t to data that emphasize different aspects of theal mechanism. The forms of the model functions can becross different synapses or muscles (Sen et al., 1996) orcation by neuromodulators (Stern et al., 2007a). In thesystem,wehave already extendedour analysis in some

ys and developed the basic decodings presented in thisa more denitive characterization of the crab cardiacular transform (Stern et al., 2007a,b, 2008).

nship to LNL and other cascade structures

our method can solve various models, in this paperrked mainly with the illustrative model given by Eqs.This model is of a similar degree of generality to, andructure closely resembles, the various block-structureddynamic linear and static nonlinear elements that haved extensively as general models of nonlinear biologicalpreviously solved by various methods, notably meth-

n information about these structures that is containedls of theVolterra orWiener expansion (Korenberg et al.,berg, 1991; Sakai, 1992; Westwick and Kearney, 2003;, 2004).ially, our illustrative model appears identical to the LNLsandwich or Korenberg) cascade, a serial structure oflinear element, a static nonlinearity, and then anotherear element (Korenberg and Hunter, 1986; Westwicky, 2003; Marmarelis, 2004). In our model, these ele-, F, and K, respectively. However, our model is not al cascade. The input spike train enters the model in twoonly at the beginning of the cascade, at Eq. (2) whereelicits an instance of H, but also later at Eq. (1) whereelicits an instance of K. This creates an additional, par-in the structure (as can be seen, for example, in the

the left-hand side of Fig. 1B). Furthermore, where thees join at Eq. (1), they do not join additively as is typ-led in parallel cascades (Korenberg, 1991; Westwicky, 2003), but rather multiplicatively. This can be seenr comparison with the time-continuous cascade mod-wise generalize our model to the continuous case. Ine spike train becomes a continuous waveform, T, givenf delta functions at the spike times (or, in discretizedence of integers denoting the number of spikes in eachver all time points t, t, Eq. (1) then generalizes to the

K(t t)A(t)T(t) dt, (13)

nvolution of K with the product AT. (Eq. (2) generalizeshe spikes thus multiplicatively gate the ow of A toand conversely, when decoding R, their absence hideses that produced A from view, giving rise to some of theroblems that we encountered in Section 3. All of theseke our model signicantly different from the pure LNLbecomes a pure LNL cascade only in trivial cases, sucht) = 1) and make difcult its solution by the previoused with such cascades.our decoding method was developed for the case of

latively sparse spikes, and realizes considerable com-

ing thand thF, ourmethotively i

Ouralent tcascada statiMarmatially esolve Ltheir

As tillustracient toThat recades,last strnumbetrary dkerneldescripand Ke

4.3. Fu

Althlar synaboutwherethat caant sinof intrrecordet al., 2methotinuoutrain, fence thexamp

Fural. (19sory prstimulspike tstructsal., 200(Ringaet al.,encodiearityhave astimul

R(t) =

wheresuch awe canone stution in two separable steps, rst Step 1 to nd Kermediate function A, and then Step 2 to nd H andod would appear to be signicantly simpler than theKorenberg and Hunter (1986), for example, that itera-oves the estimates of all three functions in turn.(2) (especially in the continuous case) is fully equiv-ubset of the LNL cascade, the LN (also called Wiener)t consists just of a dynamic linear element followed bylinearity (Sakai, 1992; Westwick and Kearney, 2003;, 2004). Our method of solving Eq. (2) in Step 2 is essen-alent to that used by Hunter and Korenberg (1986) toscades, except that we have substituted smoothing forof a high-order polynomial.iscussion implies, neither a single LNL cascade nor ourmodel is completely general in the sense of being suf-resent thedynamics of anynonlinear dynamical system.es a structure of multiple parallel LNL cascades, LN cas-lements all feeding into a multi-input nonlinearity (there being called the Wiener-Bose model). A sufcientthese elements can represent any system to any arbi-e of accuracy, and has an associated Volterra or Wienersentation,which thus canprovide a completely generalof the system (Palm, 1979; Korenberg, 1991; Westwicky, 2003; Marmarelis, 2004; Song et al., 2009a).

applications

h in this paper we have used data from a neuromuscu-, there is nothing inherently synaptic or neuromuscularecoding method. We anticipate that it may be useful

rains of spikes elicit continuous responses of any kindregarded as being built up of discrete, relatively invari-pike responses. This includes, for example, responsesular signals such as cAMP and Ca2+, which can beith the requisite temporal resolution (see, e.g., KreitzerAugustine et al., 2003; Nikolaev and Lohse, 2006). Thealso be used to decode, from a spike train and a con-

ng rate function constructed from the very same spikeons likeH and F that characterize how past spikes inu-neration of future spikes in an autonomously active, foregularly bursting, neuron (Brezina et al., 2008).ore, as was pointed out for their method by Sen et

he method may also be applicable to the inverse sen-m, which seeks to understand how continuous sensoryencoded in discrete spike trains and how, from those, the central nervous system then decodes or recon-sensory stimuli again (Rieke et al., 1997; Simoncelli etinear lters such asK are the basis of reverse-correlationd Shapley, 2004) and stimulus-reconstruction (Rieke) techniques, and more elaborate models of sensoryimoncelli et al., 2004; Pillow et al., 2005) add nonlin-building blocks like those of Eqs. (1) and (2). As we

y noted, our method generalizes to cases where bothd response are continuous, such as the equation

K(t t)S(t) dt, (14)

is, say, a sensory stimulus and R(t) the response to it,spike rate of a responding neuron. Knowing S and K,

dict R (sensory encoding); knowing S and R, we can, ining a generalized Eq. (3), nd K (the problem usually


addressed by reverse correlation); and knowing K and R, we can, inone step using a generalized Eq. (4), reconstruct the stimulus S.

The inverse problem arises, indeed, even with the spike-response transforms that we have considered here in the forwarddirection. Hresponse toreconstructresponse. Aequations oreadily accocrete Eq. (4spike trainsandcontrac(Brezina et

Acknowled

SupportGM61838,implement

Appendix A

The datalist of spikethe numberexperimentan algorithm

I0 = +

[

or, replacintext,

I0 = +

[

with respectinuous funK(t) = 0 forthe spike ha

In practproblem. Thduration

t = ntti = nitKn = K(nRexpn = Rexpwhere n andI:

I =

n=

[i

The secondmemory, ththe causalitnonzero va

Our taskK1, . . . , KN ,differentiat

result equal to 0. In evaluating the derivatives, we make use of therelationships

Kn = nm

0

0

ij

ab =eren

I

m=

t

i

last shichj):ni

e un njNs

j=1Rem

nota

mnP

ng thng in

nKn

holdofN

nitimet

tive dto zelinealutionhe Aieren

I

k=

2tnere, having characterized a transform, we predicted thea given spike train. However, we may well wish to

, conversely, the spike train that has produced a givennd, by virtue of the global matrix formulation of thef the transform, this inverse computation, too, can bemplished. In preliminary work, we have used the dis-

) or its continuous version, as applicable, to reconstructfrom synthetic response waveforms as well as the EJPtion responsewaveforms recorded in the real crabheartal., 2009).

gments

ed by NIH grants NS058017, NS41497, GM087200,and RR03051. We thank Aaron Liand for help withation of the arbitrary spike timer used in Figs. 11 and 12.

. Complete derivation for Step 1, nding K and A

set used as input to Step 1 of the algorithm is an orderedtimes (t1, . . . , tNs ), where t1 < t2 < < tNs and Ns isof spikes in the experimental record, together with theal response to this spike train, Rexp(t). We describe here

to minimize

R(t) Rexp(t)]2 dt,

g R(t) with the model assumed in Eq. (1) of the main

Nsi=1

K(t ti)Ai Rexp(t)]2

dt,

t to the discrete variables Ai, i = 1, . . . , Ns, and the con-ction K(t). K(t) is assumed to be causal, that is, to satisfyt 0, since the system cannot respond to a spike befores occured, nor can it respond instantaneously.

ice, we make two simplications before tackling thee rst simplication is to discretize time into bins of

t. Thus,

t)

(nt)

ni are integers. Then I0 is replaced by the Riemann sum

Ns

=1KnniAi R

expn

]2t.

simplication is to assume that the kernel K has niteat is, that Kn = 0 for n > N for some integer N. Byy assumption, we also have Kn = 0 for n 0. Thus, thelues of K are at most K1, . . . , KN .now is tominimize Iwith respect to theN + Ns variablesA1, . . . , ANs . This is a calculus problem. To solve it, wee I with respect to each of these variables and set the

KmKnAi

=

AiKn

=

AiAj

=

whereDiff

0 = K

= 2

In then for w

Pn =(i,

with ththat ni

Qm =

In this

n=

K

Recallichangi

Nn=1

Pm

whichsystemthe deis a symis posiequalabovethe soupon t

Diff

0 = A

=1 if a = b, and ab = 0 otherwise.tiating I with respect to K, we have

2t

n=

[Nsi=1

KnniAi Rexpn

]Nsj=1

nnj,mAj

Ns

,j=1Km+njniAiAj

Nsj=1

Rexpm+njAj

.

tep, we have made use of the fact that the only value ofnnj,m is nonzero is n = m + nj . Let

nj=nAiAj

derstanding that Pn = 0 if there are no pairs (i, j) such= n, and also let

xp+njAj.

tion, the equation 0 = I/Km becomes

n = Qm.

at the only nonzero values of K are K1, . . . , KN , anddices, we obtain

= Qm, (A.1)

s for m = 1, . . . , N and therefore has the form of a linearequations in theNunknownsK1, . . . , KN . It is clear fromon of P that the matrix Pwith the elements Pmn = Pmnric Toeplitzmatrix, and it can be shown that thismatrixenite (except in the trivial case that all of the Ais arero). These properties of its matrix guarantee that ther system has a solution K1, . . . , KN , and moreover thatis unique. Note, however, that the solution depends

s, which are also unknown.tiating I with respect to the Ais, we have

2t

n=

[Nsi=1

KnniAi Rexpn

]Nsj=1

Knnj jk

=

[Nsi=1

KnniAi Rexpn

]Knnk .


Let

Xki =

n=KnnkKnni

Yk =

n=R

The sums oinvolveonlyand also forthe Ais take

Nsi=1

XkiAi =

for k = 1, . .a symmetridenite. It fexists and mdepends on

In the foK given A, EK, Eq. (A.2).minimize I,

K (l+1) = (P

(l+1) = (X

for l = 0,1,shorthand fis shorthanations, we a

For l 1mate K (l) ofan estimate

R(l)n =Nsi=1

Kn

Then we ca

I(l) =

n=[

Since eachof K or A, gguaranteed

I(l+1) I(l).Moreover, sthat I(l) conK (l) and(l)

(see main te

Appendix B

As explaStep 2 is the

S(t) =j:tj N. Expressed in terms of X and Y, the equations forthe form of a linear system:

Yk (A.2)

. , Ns. It is clear that the matrix X of this linear system isc, banded matrix, and it can be shown that X is positiveollows that a solution A1, . . . , ANs of this linear systemoreover is unique. Note, however, that this solution

K1, . . . , KN , which appears in the denitions of X and Y.regoing, we have found a linear system that determinesq. (A.1), and a linear system that determines A givenTo solve for the values of K and A that simultaneouslywe use the following iterative scheme:

1Q )(l)

1Y)(l+1)

. . .. Recall that P and Q depend on A. Thus (P1Q )(l) isor the value of P1Q when A = (l). Similarly, (X1Y)(l)d for the value of X1Y when K = K (l). To start the iter-rbitrarily set (0)

i= 1 for i = 1, . . . , Ns.

, the iterative scheme dened above generates an esti-K andanestimate(l) ofA.Wecanuse these to constructR(l) of R according to

(l)ni

(l)i

.

n follow the progress of the algorithm by monitoring

R(l)n Rexpn ]2t.

step of our iterative scheme nds the optimal valueiven the current estimate of A or K, respectively, it isthat

ince I(l) is bounded frombelowby0, it is alsoguaranteedverges as l . This does not by itself guarantee thatconverge, but that iswhatweusuallyobserve inpracticext).

. Complete derivation for nding H in Step 2

ined in the main text, a key feature of our algorithm insimplication of Eq. (2) to the equation

(t tj).

n can then be used to nd H(t) by an algorithm analo-in Step1. The input to this algorithm is anordered list of

I0 =

with rsimilar

t = ntj = njHn = HSexpn =whereI:

I =

n=

As in SmemoN. Thu

To mferentiequal t

HnHm

=

whereThu

0 = H

= 2

In then for w

Wn =(

with ththat nj

Zm =i

The eq

n=

H

Recallichangi

Nn=1

WmS(t) Sexp(t)]2 dt = +

Ns

j=1H(t tj) Sexp(t) dt

t to the continuous function H(t). We proceed with aretization of time as in Step 1:

t)

(nt)

nj are integers. Then I0 is replaced by the Riemann sum

Ns

=1Hnnj S

expn

2

t.

, we assume that the kernel H is causal and has nitethatHn = 0 forn 0andalso forn > N for some integernonzero values of H are at most H1, . . . ,HN .ize I with respect to the variables H1, . . . ,HN , we dif-

with respect to each of these variables and set the resulte make use of the relationship

= 1 if n = m, and nm = 0 otherwise.

2t

n=

Ns

j=1Hnnj S

expn

Ns

i=1nni,m

Ns

,j=1Hm+ninj

Nsi=1

Sexpm+ni

.

tep, we have made use of the fact that the only value ofnni,m is nonzero is n = m + ni. Let

ni=n1

derstanding that Wn = 0 if there are no pairs (i, j) such= n, and also let

xp+ni .

n 0 = I/Hm then becomes

n = Zm.

at the only nonzero values of H are H1, . . . ,HN , anddices, we obtain

n = Zm.


This holds for m = 1, . . . , N, and therefore has the form of a lin-ear system of N equations in the N unknowns H1, . . . ,HN . It isclear from the denition of W that the matrixWwith the elementsWmn = Wmn is a symmetric Toeplitz matrix, and it can be shownthat this matrix is positive denite. These properties of its matrixguarantee that the above linear system has a solution H1, . . . ,HN ,and moreover that the solution is unique.

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A method for decoding the neurophysiological spike-response transformIntroductionMethodsAn illustrative modelStep 1: finding K and APerformance-enhancing modification

Step 2: finding H and FPerformance-enhancing modificationSmoothing

Two inputsError evaluationExperimental methods

ResultsSynthetic dataA typical decodingPerformance characteristics of the algorithmNumber of iterationsConvergence in Step 1Uniqueness of solutionsSpeed of the algorithmForms of K, H, and FSpike densityNumber of spikesNoiseOther models

Real dataSynaptic potentialsMuscle contractions

DiscussionThe decoding method: its advantages and limitationsRelationship to LNL and other cascade structuresFurther applications

AcknowledgmentsComplete derivation for Step 1, finding K and AComplete derivation for finding H in Step 2References

Documents

A method for decoding the neurophysiological spike-response transform