{"title": "Single Neuron Model: Response to Weak Modulation in the Presence of Noise", "book": "Advances in Neural Information Processing Systems", "page_first": 67, "page_last": 74, "abstract": null, "full_text": "SINGLE NEURON MODEL: RESPONSE TO WEAK \n\nMODULATION IN THE PRESENCE OF NOISE \n\nA. R. Bu/,ara and E. W. Jaco6, \n\nNaval Ocean Syat.em.a Cenw, Materials Reaean:h Branch, San Diego, CA 92129 \n\nF.Mou \n\nPhysics Dept.., Univ. of Missouri, St. Louis, MO 63121 \n\nABSTRACT \n\nWe consider a noisy bist.able single neuron model driven by a periodic \nexternal modulation. The modulation introduces a correlated switching \nbetween st.ates driven by the noise. The information flow through the sys(cid:173)\ntem from the modulation to the output switching events, leads to a succes(cid:173)\nsion of strong peaks in the power spectrum. The signal-to-noise ratio (SNR) \nobtained from this power spectrum is a measure of the information content \nin the neuron response . With increasing noise intensity, the SNR passes \nt.hrough a maximum, an effect which has been called stochastic resonance. \nWe treat t.he problem wit.hin the framework of a recently developed approx(cid:173)\nimate theory, valid in the limits of weak noise intensity, weak periodic forc(cid:173)\ning and low forcing frequency. A comparison of the results of this theory \nwith those obtained from a linear syst.em FFT is also presented . \n\nINTRODUCTION \n\nRecently, there has been an upsurge of interest in s1ngie or few-neuron nonlinear \ndynamics (see e.g. Li and Hopfield~ 1989; Tuckwell, 1988; Paulus, Gass and Mandell, 1990; \nAihara, Takake and Toyoda, 1990). However, the precise relationship between the many(cid:173)\nneuron connected model and a single effect.ive neuron dynamics has not been examined in \ndetail. Schieve, Bulsara and Davis (1991) have considered a network of N symmetrically \ninterconnected neurons embodied} for example in the \"connectionist.\" models of Hopfield \n(1982, 1984) or Shamma (1989) \\the latter corresponding to a mammalian auditory net(cid:173)\nwork). Through an adiabatic elimination procedure, they have obtained, in closed form, the \ndynamics of a single neuron from the system of coupled differential equations describing the \nN-neuron problem . The problem has been treated both deterministically and stochastically \n(through the inclusion of additive and multiplicative noise terms). It. is important. to point. \nout that the work of Schieve, Bulsara, and Davis does not include a prion' a self-coupling \nterm, although the inclusion of such a term can be readily implemented in their theory; this \nhas been done by Bulsara and Schieve (1991) . Rather, t.heir theory results in an explicit. \nform of t.he self-coupling term, in terms of the parameters of the remaining neurons in the \nnet.work . This term, in effect, renormalizes the self-coupling t.erm in the Shamma and Hop(cid:173)\nfield models. The reduced or \"effect.ive\" neuron model is expected to reproduce some of the \ngross features of biological neurons. The fact that simple single neuron models, such as the \nmodel t.o be considered in this work, can indeed reproduce several feat.ures observed in bi~ \nlogical experiments has been strikingly demonstrated by Longtin, Bulsara and Moss (1991) \nthrough their construction of the inter-spike-interval histograms (ISIHs) using a Schmidt \ntrigger to model the neuron. The results of their simple model agree remarkably well with \ndata obtained in two different experiments (on the auditory nerve fiber of squirrel monkey \n(Rose, Brugge, Andersen and Hind, 1967) and on the cat visual cort.ex (Siegal, 1990)), \n\nIn this work, we consider such a \"reduced\" neural element subject to a weak periodic \nthe \n\nexternal modulation . The modulation int.roduces a correlat.ed switching between \n\n67 \n\n\f68 \n\nBuisara, Jacobs, and Moss \n\nbistable states, driven by the noise with the signal-to-noise ratio (SNR) obtained from the \npower spectrum, being taken as a measure of the information content in the neuron \nresponse. As the additive noise variance increases, the SNR passes through a maximum. \nThis effect has been called wstochastic resonance W and describes a phenomenon in which the \nnoise actually enhances the information content, i.e., the observability of the signal. Sto(cid:173)\nchastic resonance has been observed in a modulated ring laser experiment (McNamara, \nWiesenfeld and Roy, 1988; Vemuri and Roy, 1989) as well as in electron paramagnetic reso(cid:173)\nnanCe experiments (Gammaitoni, Martinelli, Pardi and Santucci, 1991) and in a modulated \nmagnetoselastic ribbon (Spano and Ditto, 1991). The introduction at multiplicative noise \n(in the coefficient of the sigmoid transfer function) tends to degrade this effect. \n\nTHE MODEL; STOCHASTIC RESONANCE \n\nThe reduced neuron model consists of a single Hopfield-type computational element, \nwhich may be modeled as a R-C circuit with nonlinear feedback provided by an operational \namplifier having a sigmoid transfer function. The equation (which may be rigorously \nderived from a fully connected network model as outlined in the preceding section) may be \ncast in the form, \n\ni + a x - b ta.nhx = Xo+ F(t), \n\n(1) \ndc input (which we set equal to zero for the remainder of this work) . An analysis of lIl' \nwhere F( tJ is Gaussian delta-correlated noise with zero mean and variance 2D, Xo bein9 a \n\nincluding multiplicative noise effects, has been given by Bulsara, Boss and Jacobs (1989 . \nFor the purposes of the current work , we note that the neuron may be treated as a partic e \nin a one-dimensional potential given by, \n\nU(x) = -2- - b In cosh x , \n\na x2 \n\n(2) \n\nx being the one-dimensional state variable representing the membrane potential. In gen(cid:173)\neral, the coefficients a and b depend on the details of the interaction of our reference neu(cid:173)\nron to the remaining neurons in the network (Schieve, Bulsara and Davis, 1990). The \npotential described by (2) is bimodal for '7 > 1 With the extrema occurring at (we set a=1 \nthroughout the remainder of this work), \n\nc=o, \u00b1 [1-\n\n1-ta.nhb \n1- b sech 2b \n\nj::=:bta.nhb, \n\n(3) \n\nthe approximation holding for large b. Note that the N-shaped characteristic inherent in \nthe firing dynamics derived from the Hodgkin-Huxley equations (Rinzel and Ermentrout, \n1990) is markedly similar to the plot of dV/dx vs. x for the simple bistable system (1). \nFor a stationary potential , and for D\u00ab Vo where Vo is the depth of the deterministic \npotential, the probability that a switching event will occur in unit time, i.e. the switching \nrate, is given by the Kramers frequency (Kramers, 1940), \n\n'.= \\ D l. dy .xp (U(y)/ D) ( , duxp (- U(z)/ D) r. \n\n(40) \n\nwhich, for small noise, may be cast in the form (the local equilibrium assumption of Kra(cid:173)\nmers), \n\nro::=: (271\"r l ll V(21(0) I V(21(c)]1/'2 exp (- Vo/ D), \n\n(4b) \n\nwhere V(2}(x) == d 2 V /dx 2. \n\nWe now include a periodic modulation term esinwt on the right-hand-side of (1) (note \n\nthat for \u00ab2(b-1)3/(3b) one does not observe SWitchinq in the noise-free system) . This \n\nleads to a modulation (i .e. rocking) of the potential 2) with time: an additional term \n- xesinwt is now present on the right-hand-side of (2). n this case, the Kramers rate (4) \nbecomes time-dependent: \n\n(5) \nwhich is accurate only for e\u00ab Vo and w\u00ab {VI21(\u00b1c )}1/'2 . The latter condition is referred to \nas the adiabatic approximation. It ensures that the probability density corresponding to \n\nr(t)::=:roexp(-Xisinwt/D), \n\n\fSingle Neuron Model: Response to Weak Modulation in the Presence of Noise \n\n69 \n\nthe time-modulated potential is approximately stationary (the modulation is slow enough \nthat the instantaneous probability density can \"adiabatically\" relax to a succession of \nquasi-stationary states) . \n\nWe now follow the work of McNamara and Wiesenfeld (1989), developing a two-state \nmodel by introducing a probability of finding the system in the left or right well of the \npotential. A rate equation is constructed based on the Kramers rate r(t) given by (5). \nWithin the framework of the adiabatic approximation, this rate equation may be integrated \nto yield the time-dependent conditional probability density function for finding the system \nin a given well of the potential. This leads directly to the autocorrelation function \n< :z:{t) :z:{t + 1') > and finally, via the Wiener-Khinchine theorem, to the power spectral den(cid:173)\nsity P(O). The details are given by Bulsara, Jacobs, Zhou, Moss and Kiss (1991) : \n\nP 0 = 1-\n\n[ \n\n() \n\n2rg f2C2 1 [ 8c2ro 1 \n+ \n\nD2{4rl+(2) \n\n4rl+02 \n\n47rc 4rg f2 \n\nD2(4rg+02) \n\n6 w- 0 \n( \n\n), \n\n(6) \n\nwhere the first term on the right-hand-side represents the noise background, the second \nterm being the signal strength. Taking into account the finite bandwidth of the measuring \nsyste~, ~e replace {for the I?urpose of compariso~ with e~perimental results} t~e .delta(cid:173)\nfunctlOn m (6) by the quantity (.6w)-1 where .6w IS the Width of a frequency bm m the \n(experimental) Fourier transformation. We introduce signal-to-noise ratio SNR = 10 log R in \ndecibels, where R is given by \n\nR == 1 + D24~Cr~~f:2) (.6w)-1 [1- D2 ~:~t:2w2) rJ \n\n[ 4r~gc~~2l \u00b7 \n\n(7) \n\nIn writing down the above expressions, the approximate Kramers rate (4b) has been used . \nHowever, in what follows, we discuss the effects of replacing it by the exact expression (4a). \nThe location of the maximum of the SNR is found by differentiating the above equation; It \ndepends on the amplitude f and the frequency w of the modulation, as well as the additive \nnoise variance D and the parameters a and b in the potential. \n\nThe SNR computed via the above expression increases as the modulation frequency is \nlowered relative to the Kramers frequency . Lowering the modulation frequency also shar(cid:173)\npens the resonance peak, and shifts it to lower noise values, an effect that has been demon(cid:173)\nstrated, for example, by Bulsara, Jacobs, Zhou, Moss and Kiss (1991). The above may be \nreadily explained. The effect of the weak modulating signal is to alternately raise and lower \nthe potential well with respect to the barrier height Vo. In the absence of noise and for \nl \u00ab Vo, the system cannot switch states, i.e. no information is transferred to the output. In \nthe presence of noise, however, the system can switch states through stochastic activation \nover the barrier . Although the switching process is statistical, the transition probability is \nperiodically modulated by the external signal. Hence, the output will be correlated, to some \ndegree, with the input signal (the modulation \"clocks\" the escape events and the whole pro(cid:173)\ncess will be optimized if the noise by itself produces, on average, two escapes within one \nmodulation cycle) . \n\nFigure 1 shows the SNR as a function of the noise variance 2D . The potential barrier \n\nheight Vo:;;:: 2.4 for the b = 2.5 case considered. Curves corresponding to the adiabatic expres(cid:173)\nsion (7), as well as the SNR obtained through an exact (numerical) calculation of the Kra(cid:173)\nmers rate, using (4a) are shown, along with the data points obtained via direct numerical \nsimulation of (1). The Kramers rate at the maximum (2D ~ Vo) of the SNR curve is 0.72. \nThis is much greater than the driving frequency w = 0.0393 used in this plot. The curve \ncomputed using the exact expression (4a) fits the numerically obtained data points better \nthan the adiabatic curve at high noise strengths. This is to be expected in light of the \napproximations used in deriving (4b) from (4a). Also, the expression (6) has been derived \nfrom a two-state theory (taking no account of the potential) . At low noise, we expect the \ntwo-state theory to agree with the actual system more closely . This is reflected in the reso(cid:173)\nnance curves of figure 1 with the adiabatic curve differing (at the maximum) from the data \npoints by approximately Idb. We reiterate that the SNR, as well as the agreement between \nthe data points and the theoretical curves improves as the modulation frequency is lowered \nrelative to the Kramers rate (for a fixed frequency this can be achieved by changing the \npotential barrier height via the parameters a and b in (2)). On the same plot, we show the \nSNR obtained by computing directly the Fourier transform of the signal and noise. At very \n\n\f70 \n\nBulsara, Jacobs, and Moss \n\nlow noise, the Mideal linear filter M yields results that are considerably better than stochastic \nresonance. However, at moderate-to-high noise, the stochastic resonance, which may be \nlooked upon as a Mnonlinear filter M, offers at least a 2.5db improvement for the parameters \nof the figure. As indicated above, the improvement in performance achieved by stochastic \nresonance over the \"ideal linear filterM may be enhanced by raising the Kramers frequency \nof the nonlinear filter relative to the modulation frequency w. In fact, as long as the basic \nconditions of stochastic resonance are realized, the nonlinear filter will outperform the best \nlinear filter except at very low noise. \n\nZZ.5 \n\n20.0 \n\n17.5 \n\n15.0 \n\nco \n~ \na: \nz \n\nC/) \n\no \n\no \n\n0 \n\no \n\no \n\no o o \n\n\"(cid:173) , .... \n\n' .... \n\n~ ................. . .... .......... \n........ .... .... \n\nFig 1. SNR using adiabatic theory, \neqn. (7), with (b ,w,e)= \n(2.5,0.0393,0.3) and ro given \nby (4b) (solid curve) and (4a) \n(dotted curve) . Data points \ncorrespond to SNR obtained \nvia direct simulation of (1) \n(frequency resolution =6.1x 10-6 Hz). \nDashed curve corresponds to best \npossible linear filter (see text) . \n\n12.5+----+---+--_--+----+----<----<--...... -.:.. \n\n3. 75 \n\n5.00 \n\n0.00 \n\n1.Z5 \n\nZ.50 \n\nNoise Variance 2D \n\nMultiplicative Noise Effects \n\nWe now consider the case when the neuron is exposed to both additive and multipli(cid:173)\n\ncative noise. In this case, we set b(t) = bo+ \u20ac(t) where \n\n<\u20ac(t\u00bb \n\n=0, < \u20ac(t) \n\n\u20ac(s) > =2Dm ott - s) . \n\n(8) \nIn a real system such fluctuations might arise through the interaction of the neuron with \nother neurons in the network or with external fluctuations. In fact, Schieve, Bulsara and \nDavis (1991) have shown that when one derives the Mreduced M neuron dynamics in the \nform (1) from a fully connected N-neuron network with fluctuating synaptic couplings, then \nthe resulting dynamics contain multiplicative noise terms of the kind being discussed here. \nEven Langevin noise by itself can introduce a pitchfork bifurcation into the long-time \ndynamics of such a reduced neuron model under the appropriate conditions (Bulsara and \nSchieve, 1991). In an earlier pUblication (Bulsara, Boss and Jacobs, 1989), it was shown \nthat these fluctuations can qualitatively alter the behavior of the stationary probability \ndensity function that describes the stochastic response of the neuron. In particular, the \nmultiplicative noise may induce additional peaks or erase peaks already present in the den(cid:173)\nsity (see for example Horsthemke and Lefever 1984) . In this work we maintain Dm suffi(cid:173)\nciently small that such effects are absent. \n\nIn the absence of modulation, one can write down a Fokker Planck equation for the \n\nprobability density function p (:z; ,t) describing the neuron response: \n\nE.1!. \nat=-a;[a(x)p!+z ax 2 [.8(x)pj, \n\n1 a2 \n\na \n\nwhere \n\na(x) == - x + botanhx + Dm tanhx sech 2x, \n\n.8(x);: 2(D + Dm tanh 2x) I \n\nD being the additive noise intensity . In the steady state, (9) may be solved \nMmacroscopic potential\" function analogous to the function U(:z;} defined in (2): \n\nU (x) = - 2 f .8( z) dx + In .8(:z;) . \n\n&~ \n\n(9) \n\n(10) \nto yield a \n\n(11) \n\n\fSingle Neuron Model: Response to Weak Modulation in the Presence of Noise \n\n71 \n\nFrom (11), one obtains the turning points of the potential through the solution of the tran(cid:173)\nscendental equation \n\n(12) \nThe modified Kramers rate, rpm, for this x-dependent diffusion process has been derived by \nEnglund, Snapp and Schieve lI984): \n\nx - bo tanhx + Dm tanhx sech2x = 0 . \n\nrOm = ~ [ U(21(Xl) I U(21(0) I p'\" exp [ U(XI) - U(O) I, \n\n2,.. \n\n(13) \n\nwhere the maximum of the potential occurs at x=o and the left minimum occurs at Z=ZI' \n\nIf we now assume that a weak sinusoidal modulation \u00a3sinwt is present, we may once \n\nagain introduce this term into the potential as in the preceding case, again making the adi(cid:173)\nabatic approximation . We easily obtain for the modified time-dependent Kramers rate, \n\nr \u00b1(t) = If..Ql [ U(21(xI) I U(21(0) III\", exp [ U(XI) - U(O) \u00b1 2 (0 \u00a3SID(' w) t dz ]. \n\n(14) \n\n0 \n\n(3 z \n\n4,.. \n\nFollowing the same procedure as we used in the additive noise case, we can obtain the ratio \nR = 1 + S / l::lw N, for the case of both noises being present. The result is, \n\nR = 1 + 7r\"l0'10 (.6w) \n\n2 \n\n-I [ \n\n1 -\n\n2\"1lr7J ]-1 \n2 \n2 \n\"10 + '10 \n\n' \n\n(15) \n\n(16a) \n\nwhere, \n\nand \n\n20 \n\nEO \n~ 10 \na: \nZ \nCf) \n\no \n\n'1 -~J,~o dz \n0='0 (3(z) \n\n-\n\n\u20ac \n\n2 + Dm \n(D \n\n) \n\n[xl+ml/2 tan-l (m l/2 tanhx l )], \n\n(16b) \n\nFig 2. Effect of multiplicative \nnoise, eqn . (15). (b ,w,\u20ac) = \n(2,0.31,0.4) and Dm =0 (top \ncurve), 0.1 (middle curve) and \n0.2 (bottom curve) . \n\n0.3 \n\n0.6 \n\nD \n\n0 .9 \n\n1.2 \n\n1.5 \n\nIn figure 2 we show the effects of both additive and multiplicative noise by plotting \nthe SNR for a fixed external frequency w=O.31 with (b o, \u00a3) = (2,0.4) as a function of the \nadditive noise intensity D. The curves correspond to different values of Dm, with the upper(cid:173)\nmost curve corresponding to Dm =0, i.e., for the case of additive noise only . We note that \nincreasing Dm leads to a decrease in the SNR as well as a shift in its maximum to lower \nvalues of D. These effects are easily explained using the results of Bulsara, Boss and Jacobs \n\n\f72 \n\nBuisara, Jacobs, and Moss \n\n(1989), wherein it was shown that the effect of mUltiplicative noise is to decrease, on aver(cid:173)\nage, the potential barrier height and to shift the locations of the stable steady states. This \nleads to a degradation of the stochastic resonance effect at large Dm while shifting the loca(cid:173)\ntion of the maximum toward lower D . \n\nTHE POWER SPECTRUM \n\nWe turn now to the power spectrum obtained via direct numerical simulation of the \ndynamics (1). It is evident that a time series obtained by numerical simulation of (1) would \ndisplay SWitching events between the stable states of the potential, the residence time in \neach state being a random variable. The intrawell motion consists of a random component \nsuperimposed on a harmonic component, the latter increasing as the amplitude i of the \nmodulation increases. In the low noise limit, the deterministic motion dominates . However, \nthe adiabatic theory used in deriving the expressions (6) and (7) is a two-state theory that \nsimply follows the switching events between the states but takes no account of this \nintrawell motion. Accordingly, in what follows, we draw the distinction between the full \ndynamics obtained via direct simulation of (1) and the \"equivalent two-state dynamics\" \nobtained by passing the output through a two-state filter. Such a filter is realized digitally \nby replacing the time series obtained from a simulation of (1) with a time series wherein \nthe x variable takes on the values x = \u00b1 c, depending on which state the system is in . Fig(cid:173)\nure 3 shows the power spectral density obtained from this equivalent two-state system. The \ncop curve represents the signal-free case and the bottom curve shows the effects of turning \non the signal . Two features are readily apparent : \n\n50 00 \n\n35 75 \n\n7 2S \n\nFig 3. Power spectral density via \ndirect simulation of (1). \n(b ,w ,< ,2D) = (1.6056,0.03,0.65,0.25). \nBottom curve: <=0 case. \n\nCD \n~ \n\na: z \n\nV) \n\n-700L---~~ __ -===::::::::;;;;;;~~~ \n\n011 \n\n0.15 \n\n000 \n\nO . O~ \n\nO.OB \n\nlrequency (Hz) \n\nl. The power spectrum displays odd harmonics of the modulation; this is a hallmark of sto(cid:173)\nchastic resonance (Zhou and Moss, 1990) . If one destroys the symmetry of the potential (1) \n(through the introduction of a small de driving term, for example), the even harmonics of \nthe modulation appear. \n2. The noise floor is lowered when the signal is turned on. This effect is particularly striking \nIn the two-state dynamics. It stems from the fact that the total area under the spectral \ndensity curves in figure 3 (i .e. the total power) must be conserved (a consequence of \nParseval's theorem). The power in the signal spikes therefore grows at the expense of the \nbackground noise power. This is a unique feature of weakly modulated bistable noisy sys(cid:173)\ntems of the type under consideration in this work, and ~raPhiCallY illustrates the ability of \nnoise to assist information flow to the output (the signal. The effect may be quantified on \nexamining equ~tion (6) a~ove . The noise power spectra ~density (reJ?resented by the first \nCerm on the nght-hand-slde) decreases as the term 2ro\u00a32c2{D2(4rg +(2)}-1 approaches \nunity . This reduction in the noise floor is most pronounced when the signal is of low fre(cid:173)\nquency (compared to the Kramers rate) and large amplitude . A similar, effect may be \nobserved in the spectral density correspondin~ to the full system dynamics . In this case, the \ntotal power is only approximately conserved tin a finite bandwidth) and the effect is not so \n\n\fSingle Neuron Model: Response to Weak Modulation in the Presence of Noise \n\n73 \n\npronounced. \n\nDISCUSSION \n\nIn this paper we have presented the details of a cooperative stochastic process that \noccurs in nonlinear systems subject to weak deterministic modulating signals embedded in \na white noise background. The so-called \"stochastic resonance\" phenomenon may actually \nbe interpreted as a noise-assisted flow of information to the output. The fact that such sim(cid:173)\nple nonlinear dynamic systems (e.g. an electronic Schmidt trigger) are readily realizeable in \nhardware, points to the possible utility of this technique (far beyond the application to sig(cid:173)\nnal processing in simple neural networks) as a nonlinear filter. We have demonstrated that, \nby suitably adjusting the system parameters (in effect changing the Kramers rate), we can \noptimize the response to a given modulation frequency and background noise. In a practical \nsystem, one can move the location and height of the bell-shaped response curve of figure 1 \nby changing the potential parameters and, possibly, infusing noise into the system. The \nnoise-enhancement of the SNR improves with decreasing frequency. This is a hallmark of \nstochastic resonance and provides one with a possible filtering technique at low frequency . \nIt is important to point out that all the effects reported in this work have been reproduced \nvia analog simulations (Bulsara, Jacobs, Zhou, Moss and Kiss, 1991: Zhou and Moss, 1990). \nRecently a new approach to the processing of information in noisy nonlinear dynamic sys(cid:173)\ntems, based on the probability density of residence times in one of the stable states of the \npotential, has been developed by Zhou, Moss and Jung (1990). This technique, which offers \nan alternative to the FFT, was applied by Longtin, Moss and Bulsara (1991) in their con(cid:173)\nstruction of the inter-spike-interval histograms that describe neuronal spike trains in the \ncentral nervous system. Their work points to the important role played by noise in the \nprocesing of information by the central nervous system. The beneficial role of noise has \nalready been recognized by Buhmann and Schulten (1986, 87). They found that noise, deli(cid:173)\nberately added to the deterministic equations governing individual neurons in a network \nsignificantly enhanced the network's performance and concluded that \" ... the noise ... is an \nessential feature of the information processing abilities of the neural network and not a \nmere SOurce of disturbance better suppressed ... \" \n\nAcknowledgements \n\nThis work was carried out under funding from the Office of Naval Research grant nos. \n\nNOOOI4-90-AF-OOOOI and NOOOOI4-90-J-1327 . \n\nReferences \nAihara K., Takake T., and Toyoda M., 1990; \"Chaotic Neural Networks\", Phys. Lett. \nA144, 333-340. \n\nBuhmann J., and Schulten K., 1986; \"Influence of Noise on the Behavior of an Autoassocia(cid:173)\ntive Neural Network\", in J. 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