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Effects of Fast Presynaptic Noise in Attractor Neural Networks

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Effects of Fast Presynaptic Noise in Attractor Neural Networks
    a  r   X   i  v  :  q  -   b   i  o   /   0   5   0   8   0   1   3  v   1   [  q  -   b   i  o .   N   C   ]   1   3   A  u  g   2   0   0   5 Effects of Fast Presynaptic Noise in Attractor Neural Networks J. M. Cortes †‡ , J. J. Torres † , J. Marro † , P. L. Garrido † and H. J. Kappen ‡† Institute  Carlos I   for Theoretical and Computational Physics, andDepartamento de Electromagnetismo y F´ısica de la Materia,University of Granada, E-18071 Granada, Spain. ‡ Department of Biophysics, Radboud University of Nijmegen,6525 EZ Nijmegen, The NetherlandsApril 5, 2007 To appear in Neural Computation, 2005Corresponding author: Jesus M. Abstract We study both analytically and numerically the ef-fect of presynaptic noise on the transmission of in-formation in attractor neural networks. The noiseoccurs on a very short–time scale compared to thatfor the neuron dynamics and it produces short–time synaptic depression. This is inspired in recentneurobiological findings that show that synapticstrength may either increase or decrease on a short–time scale depending on presynaptic activity. Wethus describe a mechanism by which fast presynap-tic noise enhances the neural network sensitivity toan external stimulus. The reason for this is that,in general, the presynaptic noise induces nonequi-librium behavior and, consequently, the space of fixed points is qualitatively modified in such a waythat the system can easily scape from the attrac-tor. As a result, the model shows, in addition topattern recognition, class identification and catego-rization, which may be relevant to the understand-ing of some of the brain complex tasks. 1 Introduction There is multiple converging evidence[Abbott and Regehr, 2004] that synapses de-termine the complex processing of information inthe brain. An aspect of this statement is illustratedby attractor neural networks. These show thatsynapses can efficiently store patterns that are af-terwards retrieved with only partial information onthem. In addition to this long–time effect, artificialneural networks should contain some “synapticnoise”, however. That is, actual synapses exhibitshort–time fluctuations, which seem to competewith other mechanisms during the transmissionof information, not to cause unreliability but toultimately determine a variety of computations[Allen and Stevens, 1994, Zador, 1998]. In spiteof some recent efforts, a full understanding of how the brain complex processes depend on suchfast synaptic variations is lacking —see belowand [Abbott and Regehr, 2004], for instance—.A specific matter under discussion concerns theinfluence of short–time noise on the fixed pointsand other details of the retrieval processes inattractor neural networks [Bibitchkov et al., 2002].The observation that actual synapses endureshort–time  depression   and/or  facilitation   is likelyto be relevant in this context. That is, onemay understand some observations by assumingthat periods of elevated presynaptic activity maycause either decrease or increase of the neuro-transmitter release and, consequently, that thepostsynaptic response will be either  depressed   or  facilitated   depending on presynaptic neural ac-tivity [Tsodyks et al., 1998, Thomson et al., 2002,Abbott and Regehr, 2004]. Motivated by the neu-robiological findings, we report in this paper on ef-fects of presynaptic depressing noise on the func-1  tionality of a neural circuit. We study in detail anetwork in which the neural activity evolves at ran-dom in time regulated by a “temperature” param-eter. In addition, the values assigned to the synap-tic intensities by a  learning   (e.g., Hebb’s) rule areconstantly perturbed with  microscopic   fast noise.A new parameter is involved by this perturbationthat allows for a continuum transition from depres-sion to normal operation.As a main result, this paper illustrates that,in general, the addition of fast synaptic noise in-duces a nonequilibrium condition. That is, oursystems cannot asymptotically reach equilibriumbut tend to nonequilibrium steady states whosefeatures depend, even qualitatively, on dynamics[Marro and Dickman, 1999]. This is interesting be-cause, in practice, thermodynamic equilibrium israre in nature. Instead, the simplest conditionsone observes are characterized by a steady flux of energy or information, for instance. This makesthe model mathematically involved, e.g., there isno general framework such as the powerful (equi-librium) Gibbs theory, which only applies to sys-tems with a single Kelvin temperature and a uniqueHamiltonian. However, our system still admits ana-lytical treatment for some choices of its parametersand, in other cases, we discovered the more intri-cate model behavior by a series of computer simula-tions. We thus show that fast presynaptic depress-ing noise during external stimulation may inducethe system to scape from the attractor, namely,the stability of fixed point solutions is dramaticallymodified. More specifically, we show that, for cer-tain versions of the system, the solution destabi-lizes in such a way that computational tasks such asclass identification and categorization are favored.It is likely this is the first time such a behavior isreported in an artificial neural network as a con-sequence of biologically–motivated stochastic be-havior of synapses. Similar instabilities have beenreported to occur in monkeys [Abeles et al., 1995]and other animals [Miller and Schreiner, 2000], andthey are believed to be a main feature in odor en-coding [Laurent et al., 2001], for instance. 2 Definition of model Our interest is in a neural network in whicha local stochastic dynamics is constantly influ-enced by presynaptic  noise  . Consider a setof   N   binary neurons with configurations  S  ≡{ s i  = ± 1; i  = 1 ,...,N  } .  1 Any two neurons areconnected by synapses of intensity:  2 w ij  =  w ij x j  ∀ i,j.  (1)Here,  w ij  is fixed, namely, determined in a pre-vious  learning   process, and  x j  is a stochasticvariable. This generalizes the hypothesis in previ-ous studies of attractor neural networks with noisysynapses; see, for instance, [Sompolinsky, 1986,Garrido and Marro, 1991, Marro et al., 1999]. Once  W ≡{ w ij }  is given, the state of the systemat time  t  is defined by setting  S  and  X  ≡ { x i } . These evolve with time —after the learning processwhich fixes W — via the familiar Master Equation,namely, ∂P  t ( S , X ) ∂t  = − P  t ( S , X )   X ′  S ′ c [( S , X ) → ( S ′ , X ′ )]+   X ′  S ′ c [( S ′ , X ′ ) → ( S , X )] P  t ( S ′ , X ′ ) .  (2)We further assume that the  transition rate   or prob-ability per unit time of evolving from ( S , X ) to( S ′ , X ′ ) is c [( S , X ) → ( S ′ ,  X ′ )] =  pc X [ S → S ′ ] δ  ( X − X ′ )+(1 −  p ) c S [ X → X ′ ] δ  S , S ′ .  (3)This choice [Garrido and Marro, 1994,Torres et al., 1997] amounts to consider com-peting mechanisms. That is, neurons ( S ) evolvestochastically in time under a noisy dynamicsof synapses ( X ) ,  the latter evolving (1  −  p ) /p times faster than the former. Depending on thevalue of   p,  three main classes may be defined[Marro and Dickman, 1999]:1. For  p  ∈  (0 , 1) both the synaptic fluctua-tion and the neuron activity occur on the 1 Note that such binary neurons, although a crude sim-plification of nature, are known to capture the essentials of cooperative phenomena, which is the focus here. See, forinstance [Abbott and Kepler, 1990, Pantic et al., 2002]. 2 For simplicity, we are neglecting here postsynaptic de-pendence of the stochastic perturbation. There is someclaim that plasticity might operate on rapid time–scaleson postsynaptic activity; see [Pitler and Alger, 1992]. How-ever, including  x ij  in (1) instead of   x j  would impede someof the algebra in sections 3 and 4. 2  same temporal scale. This case has alreadybeen preliminary explored [Pantic et al., 2002,Cortes et al., 2004].2. The limiting case  p  →  1 .  This corresponds toneurons evolving in the presence of a quenchedsynaptic configuration, i.e.,  x i  is constantand independent of   i.  The  Hopfield model  [Amari, 1972, Hopfield, 1982] belongs to thisclass in the simple case that  x j  = 1 , ∀  j. 3. The limiting case  p → 0 .  The rest of this paperis devoted to this class of systems.Our interest for the latter case is a consequence of the following facts. Firstly, there is adiabatic elim-ination of fast variables for  p  →  0 which decou-ples the two dynamics [Garrido and Marro, 1994,Gardiner, 2004]. Therefore, some exact analyticaltreatment —though not the complete solution— isthen feasible. To be more specific, for  p  →  0 ,  theneurons evolve as in the presence of a steady distri-bution for X .  If we write  P  ( S , X ) =  P  ( X | S ) P  ( S ) , where  P  ( X | S ) stands for the conditional probabil-ity of   X  given  S ,  one obtains from (2) and (3), after rescaling time  tp  →  t  (technical details areworked out in [Marro and Dickman, 1999], for in-stance) that ∂P  t ( S ) ∂t  = − P  t ( S )  S ′ ¯ c [ S → S ′ ]+  S ′ ¯ c [ S ′ → S ] P  t ( S ′ ) .  (4)Here,¯ c [ S → S ′ ] ≡    d X P  st ( X | S ) c X [ S → S ′ ] ,  (5)and  P  st ( X | S ) is the stationary solution that satis-fies P  st ( X | S ) =    d X ′ c S [ X ′ → X ] P  st ( X ′ | S )    d X ′ c S [ X →  X ′ ]  .  (6)This formalism will allows us for modelling fastsynaptic noise which, within the appropiate con-text, will induce sort of synaptic depression, as ex-plained in detail in section 4.The superposition (5) reflects the fact that ac-tivity is the result of competition between differ-ent elementary mechanisms. That is, different un-derlying dynamics, each associated to a differentrealization of the stochasticity  X ,  compete and,in the limit  p  →  0 ,  an  effective   rate results fromcombining  c X [ S →  S ′ ] with probability  P  st ( X | S )for varying  X .  Each of the elementary dynamicstends to drive the system to a well-defined equi-librium state. The competition will, however, im-pede equilibrium and, in general, the system willasymptotically go towards a  nonequilibrium   steadystate [Marro and Dickman, 1999]. The question isif such a competition between synaptic noise andneural activity, which induces nonequilibrium, is atthe srcin of some of the computational strategiesin neurobiological systems. Our study below seemsto indicate that this is a sensible issue. As a matterof fact, we shall argue below that  p → 0 may be re-alistic  a priori   for appropriate choices of   P  st ( X | S ) . For the sake of simplicity, we shall be concernedin this paper with sequential updating by meansof single neuron or “spin–flip” dynamics. That is,the elementary dynamic step will simply consist of local inversions  s i  →− s i  induced by a bath at tem-perature  T.  The elementary rate  c X [ S  →  S ′ ] thenreduces to a single site rate that one may write asΨ[ u  X ( S ,i )] .  Here,  u X ( S ,i ) ≡ 2 T  − 1 s i h X i  ( S ) ,  where h X i  ( S ) =   j  = i  w ij x j s j  is the net presynaptic cur-rent arriving to —or local field acting on— the(postsynaptic) neuron  i.  The function Ψ( u ) is ar-bitrary except that, for simplicity, we shall assumeΨ( u ) = exp( − u )Ψ( − u ) ,  Ψ(0) = 1 and Ψ( ∞ ) = 0[Marro and Dickman, 1999]. We shall report onthe consequences of more complex dynamics in aforthcomming paper [Cortes et al., 2005]. 3 Effective local fields Let us define a function  H  eff  ( S ) through the con-dition of detailed balance, namely,¯ c [ S → S i ]¯ c [ S i → S ] = exp  −  H  eff  ( S i ) − H  eff  ( S )  T  − 1  . (7)Here, S i stands for S  after flipping at  i, s i  →− s i . We further define the “effective local fields”  h eff  i  ( S )by means of  H  eff  ( S ) = − 12  i h eff  i  ( S ) s i .  (8)Nothing guaranties that  H  eff  ( S ) and  h eff  i  ( S ) havea simple expression and are therefore analytically3  useful. This is because the superposition (5), un-like its elements Ψ( u  X ) ,  does not satisfy detailedbalance, in general. In other words, our system hasan essential nonequilibrium character that preventsone from using Gibbs’s statistical mechanics, whichrequires a unique Hamiltonian. Instead, there ishere one energy associated with each realization of  X = { x i } .  This is in addition to the fact that thesynaptic weights  w ij  in (1) may not be symmetric.For some choices of both the rate Ψ andthe noise distribution  P  st ( X | S ) ,  the function H  eff  ( S ) may be considered as a true effec-tive Hamiltonian [Garrido and Marro, 1989,Marro and Dickman, 1999]. This means that H  eff  ( S ) then generates the same nonequilibriumsteady state than the stochastic time–evolutionequation which defines the system, i.e., equation(4), and that its coefficients have the propersymmetry of interactions. To be more explicit,assume that  P  st ( X | S ) factorizes according to P st ( X | S ) =  j P   ( x j | s j ) ,  (9)and that one also has the factorization¯ c [ S → S i ] =  j  = i    d x j  P  ( x j | s j )Ψ(2 T  − 1 s i w ij x j s j ) . (10)The former amounts to neglect some global de-pendence of the factors on  S = { s i }  (see below),and the latter restricts the possible choices forthe rate function. Some familiar choices for thisfunction that satisfy detailed balance are: theone corresponding to the Metropolis algorithm,i.e., Ψ( u ) = min[1 , exp( − u )]; the Glauber caseΨ( u ) = [1 + exp( u )] − 1 ; and Ψ( u ) = exp( − u/ 2)[Marro and Dickman, 1999]. The latter fulfillsΨ( u  +  v ) = Ψ( u )Ψ( v ) which is required by (10) 3 . It then ensues after some algebra that h eff  i  = − T   j  = i  α + ij s j  +  α − ij  ,  (11) 3 In any case, the rate needs to be properly normalized.In computer simulations, it is customary to divide Ψ( u ) byits maximum value. Therefore, the normalization happensto depend on temperature and on the number of stored pat-terns. It follows that this normalization is irrelevant for theproperties of the steady state, namely, it just rescales thetime scale. with α ± ij  ≡  14 ln ¯ c ( β  ij ;+)¯ c ( ± β  ij ; − )¯ c ( − β  ij ; ∓ )¯ c ( ∓ β  ij ; ± ) ,  (12)where  β  ij  ≡ 2 T  − 1 w ij ,  and¯ c ( β  ij ; s j ) =    dx j  P  ( x j | s j )Ψ( β  ij x j ) .  (13)This generalizes a case in the litera-ture for random  S  –independent fluc-tuations [Garrido and Munoz, 1993,Lacomba and Marro, 1994,Marro and Dickman, 1999]. In this case, onehas ¯ c ( ± κ ;+) = ¯ c ( ± κ ; − ) and, consequently, α − ij  = 0  ∀ i,j.  However, we here are concerned withthe case of  S –dependent disorder, which results ina non–zero threshold,  θ i  ≡  j  = i  α − ij   = 0 . In order to obtain a true effective Hamiltonian,the coefficients  α ± ij  in (11) need to be symmetric.Once Ψ( u ) is fixed, this depends on the choice for P  ( x j | s j ) ,  i.e., on the fast noise details. This is stud-ied in the next section. Meanwhile, we remark thatthe effective local fields  h eff  i  defined above are veryuseful in practice. That is, they may be computed—at least numerically— for any rate and noise dis-tribution. As far as Ψ( u  +  v ) = Ψ( u )Ψ( v ) and P st ( X | S ) factorizes, 4 it follows an effective transi-tion rate as¯ c [ S → S i ] = exp  − s i h eff  i  /T   .  (14)This effective rate may then be used in computersimulation, and it may also serve to be substitutedin the relevant equations. Consider, for instance,the  overlaps   defined as the product of the currentstate with one of the stored patterns: m ν  ( S ) ≡  1 N   i s i ξ  ν i  .  (15)Here,  ξ  ν  =  { ξ  ν i  =  ± 1 ,i  = 1 ,...,N  }  are  M  random patterns previously stored in the system, ν   = 1 ,...,M.  After using standard techniques[Hertz et al., 1991, Marro and Dickman, 1999]; see 4 The factorization here does not need to be inproducts  P   ( x j | s j ) as in (9). The same result (14) holds for the choice that we shall introduce in thenext section, for instance.4  also [Amit et al., 1987], it follows from (4) that ∂  t m ν  = 2 N  − 1  i ξ  ν i  sinh  h eff  i  /T   − s i  cosh  h eff  i  /T   . (16)which is to be averagedover both thermal noise andpattern realizations. Alternatively, one might per-haps obtain dynamic equations of type (16) by us-ing Fokker-Planck like formalisms as, for instance,in [Brunel and Hakim, 1999]. 4 Types of synaptic noise The above discussion and, in particular, equations(11) and (12), suggest that the system emergent properties will importantly depend on the details of the synaptic noise  X .  We now work out the equa-tions in section 3 for different hypothesis concerningthe stationary distribution (6).Consider first (9) with the following specificchoice: P  ( x j | s j ) = 1 +  s j F  j 2  δ  ( x j +Φ)+ 1 − s j F  j 2  δ  ( x j − 1) . (17)This corresponds to a simplification of the stochas-tic variable  x j .  That is, for  F  j  = 1  ∀  j,  the noisemodifies  w ij  by a factor  − Φ when the presynapticneuron is firing,  s j  = 1 ,  while the learned synap-tic intensity remains unchanged when the neuronis silent. In general,  w ij  = − w ij Φ with probability 12  (1 +  s j F  j ) .  Here,  F  j  stands for some informa-tion concerning the presynaptic site  j  such as, forinstance, a local threshold or  F  j  =  M  − 1  ν   ξ  ν j  . Our interest for case (17) is two fold, namely,it corresponds to an exceptionally simple situationand it reduces our model to two known cases. Thisbecomes evident by looking at the resulting localfields: h eff  i  = 12  j  = i [(1 − Φ) s j − (1 + Φ) F  j ] w ij .  (18)That is, exceptionally, symmetries here are suchthat the system is described by a  true   effectiveHamiltonian. Furthermore, this corresponds to theHopfield model, except for a rescaling of temper-ature and for the emergence of a threshold  θ i  ≡  j  w ij F  j  [Hertz et al., 1991]. On the other hand,it also follows that, concerning stationary prop-erties, the resulting effective Hamiltonian (8) re-produces the model as in [Bibitchkov et al., 2002].In fact, this would correspond in our notation to h eff  i  =  12  j  = i  w ij s j x ∞ j  ,  where  x ∞ j  stands for thestationary solution of certain dynamic equation for x j .  The conclusion is that (except perhaps concern-ing dynamics, which is something worth to be in-vestigated) the fast noise according to (9) with (17) does not imply any surprising behavior. In anycase, this choice of noise illustrates the utility of the effective–field concept as defined above.Our interest here is in modeling the noise consis-tent with the observation of short-time synaptic de-pression [Tsodyks et al., 1998, Pantic et al., 2002].In fact, the case (17) in some way mimics that in-creasing the mean firing rate results in decreasingthe synaptic weight. With the same motivation, amore intriguing behavior ensues by assuming, in-stead of (9), the factorization P  st ( X | S ) =  j P  ( x j | S ) (19)with P  ( x j | S ) =  ζ   (   m )  δ  ( x j  +Φ)+[1 − ζ   (   m )]  δ  ( x j − 1) . (20)Here,    m  =    m ( S )  ≡  m 1 ( S ) ,...,m M  ( S )   is the M  -dimensional overlap vector, and  ζ   (   m ) standsfor a function of     m  to be determined. The de-pression effect here depends on the overlap vec-tor which measures the net current arriving topostsynaptic neurons. The non–local choice (19)–(20) thus introduces non–trivial correlations be-tween synaptic noise and neural activity, whichis not considered in (17). Note that, therefore,we are not modelling here the synaptic depres-sion dynamics in an explicity way as, for instance,in [Tsodyks et al., 1998]. Instead, equation (20) amounts to consider fast synaptic noise which nat-urally depresses the strengh of the synapses afterrepeated activity, namely, for a high value of   ζ   (   m ) . Several further comments on the significance of (19)-(20), which is here a main hypothesis together with  p  →  0 ,  are in order. We first mention thatthe system time relaxation is typically orders of magnitude larger than the time scale for the var-ious synaptic fluctuations reported to account forthe observed high variability in the postsynapticresponse of central neurons [Zador, 1998]. On theother hand, these fluctuations seem to have differ-ent sources such as, for instance, the stochasticity5
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