Period-two cycles in a feedforward layered neural network model with symmetric sequence processing

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1 PHYSICAL REVIEW E 75, Period-two cycles in a feedforward layered neural network model with symmetric sequence rocessing F. L. Metz and W. K. Theumann Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 1551, Porto Alegre, Brazil Received 8 May 26; revised manuscrit received 1 November 26; ublished 12 Aril 27 The effects of dominant sequential interactions are investigated in an exactly solvable feedforward layered neural network model of binary units and atterns near saturation in which the interaction consists of a Hebbian art and a symmetric sequential term. Phase diagrams of stationary states are obtained and a hase of cyclic correlated states of eriod two is found for a weak Hebbian term, indeendently of the number of condensed atterns c. DOI: 1.113/PhysRevE PACS number s : e, 64.6.Cn, 7.5.Mh I. INTRODUCTION The dynamics and the stationary states of recurrent attractor neural networks that rocess sequences of atterns have been studied over some time and there has been a recent revival of interest near the storage saturation limit in large networks 1 8. Either a rocess involving a sequence of atterns 3 7 referred to in this aer as asymmetric sequence rocessing, leading to a stationary limit cycle, or a rocess involving a air of sequences, one with atterns in increasing order and the other one with atterns in decreasing order referred to as symmetric sequence rocessing in what follows, was considered in those works. Symmetric sequence rocessing cometing with attern reconstruction favored by a Hebbian term of fixed strength has also been considered 1,8. The ratio J H /J S between the strengths J H and J S of the Hebbian term and of the air of sequences, resectively, has been restricted to a mostly dominant Hebbian strength, that is, to 1 J H /J S, leading to hase diagrams that exhibit only fixed-oint solutions including nontrivial correlated attractors 1,8. These are states that indicate a selectivity in resonse to a set of reviously learned atterns in which the correlation coefficients for the attractors with increasingly distant atterns from a stimulus are decreasing functions which eventually become vanishingly small. The sequential art of the interaction induces transitions between atterns, whereas a sufficiently strong Hebbian term locks the transitions, favoring single-attern recognition. The case of network models with a weak Hebbian interaction cometing with a dominating symmetric sequential rocessing in which J H /J S 1 has, aarently, not been studied before and it is imortant to find out what kind of solutions aear in that case and their ossible biological imlications. Indeed, the connection between the information inut in a network of contiguous stimuli in a training sequence of uncorrelated atterns and correlated delay activity as an outut has been of great interest to exlain the results of exerimental recordings of a visual-memory task in the inferotemoral IT cortex of monkeys 8 13, in which correlated states lay an essential role. The results can be interreted as a connection between ersistent cortex activity and long-term associative memory described by a fixed-oint attractor dynamics, and the early model calculations that have been done are based on the cometition between attern reconstruction and symmetric sequence rocessing 1,11. There could be other attractors that may redict further behavior on visual memory in the rimate IT cortex viewed as a dynamical system, generating either limit cycles or chaotic or other kinds of behavior. Earlier model calculations on the cometition between attern reconstruction and asymmetric sequence rocessing exhibit eriodic and stationary fixed-oint attractors, besides quasistationary states 14,15. Motivated by the results that aeared in the tye of exeriments on the IT cortex in monkeys and their interretation 1,8 13, it would be interesting to see if there are eriodic attractors that aear as correlated states in a neural network model with the kind of cometition between attern reconstruction and symmetric sequence rocessing that has been studied before in a recurrent network 1,11, now with a weak Hebbian term. The resence of eriodic attractors would suggest a new kind of ersistent oscillating activity in the IT cortex. The main urose of this work is to exlore this issue and with that in mind one has to resort to a dynamical rocedure even to detect stationary cyclic behavior and to determine the emergent roerties. Since the dynamics of fully recurrent neural networks with binary units is already fairly comlicated, and even more so with graded resonse or other more realistic units, we make a drastic simlification to start with in order to get to the essentials of our issue. We are mainly interested in finding out if there are cyclic attractors and in characterizing their roerties for a sizable critical storage of atterns. Our interest is in the role of cometing interactions that favor either transitions between atterns or the recognition of secific atterns. To that end we focus on a simle attractor feedforward layered network model of binary units and atterns with a arallel dynamics and without lateral connections 16. The model is exactly solvable and has been extensively used in the ast as a model for associative memory that has all the stationary features of a recurrent network and it is articularly suited to detect stationary nonequilibrium states. We show that, desite sequential learning, the rocedure involves in ractice a finite number of recursion relations even for a macroscoically large system. The effect of the lateral connections is to change the quantitative results of the model. To make that oint clear we construct the hase diagrams that describe the network behavior for an arbitrary strength /27/75 4 / The American Physical Society

2 F. L. METZ AND W. K. THEUMANN PHYSICAL REVIEW E 75, of the Hebbian term in order to check first that we get the same qualitative behavior as that already found for a recurrent network in the case of balanced interactions or for a dominant Hebbian term. Having studied that art, one can be reasonably confident that the behavior of the layered network for dominant sequential interaction should also describe the correct qualitative behavior of a recurrent network. The fully quantitative behavior of that network is a relevant issue which is beyond the scoe of the resent aer. It will be shown that a hase of cyclic stationary solutions of eriod two is obtained, indeendently of the number of condensed atterns c with macroscoic overla with the states of the network, and that this is a hase of nontrivial correlated states. These are states that could reflect a new kind of ersistent activity in visual-memory task exeriments. On the other hand, the stability of the cyclic hase is strongly deendent on c. These features distinguish the roerties of the resent model from those for asymmetric sequence rocessing cometing with attern reconstruction where cycles of eriod c aear, for arbitrary c 15. The urose of studying a simlified model which has the qualitative features of a recurrent network with more realistic interactions is that it tells what to look for and what could be relevant in a more biological tye of network. The aer is organized as follows. In Sec. II we resent the model and outline the derivation of the dynamics with the aid of an Aendix. We resent our main results in Sec. III and conclude with a further discussion in Sec. IV. II. THE MODEL AND THE DYNAMICS The network model consists of L layers, each containing N binary units neurons in states S i l = ±1, where i denotes the unit and l the layer. The state +1 reresents a firing unit and the state 1 a unit at rest. The state of each unit on a given layer is determined in arallel by the state of all units on the revious layer, the layer label acting as a time ste, according to the stochastic rule with robability 16 P S i l +1 S l = ex S i l +1 h i l +1, 1 2 cosh h i l +1 N h i l +1 = J ij l S j l, j=1 where h i l+1 is the local field at unit i on layer l+1 due to the set of states S l of all units on layer l and J ij l is the synatic couling between unit j on layer l and unit i on layer l+1. There is no feedback in the udating of the units and the first layer has to be set externally in a given state. The arameter =T 1 controls the synatic noise such that the dynamics is fully deterministic when T and fully random when T. A macroscoic set of = N statistically indeendent and identically distributed random atterns l, =1,...,, with comonents i l = ±1 and robability 1 2 for either value, are stored on every layer indeendently of other layers, according to the learning rule 2 J ij l = 1 N, =1 i l +1 X j l. Thus, there are only interactions between airs of units on consecutive layers. Connections between units on more distant layers, as well as lateral connections between units on the same layer, are excluded. Here, X are the elements of the matrix X = A B, and A =, + 1, +1 +, 1, B = b, + 1 b, +1 +, 1 4 are the elements of the c c and c c blocks A and B resonsible for the signal and for the noise in the local field, resectively, and c is the number of condensed atterns that yield macroscoic overlas defined below. The diagonal two-block interaction matrix reflects the fact that the atterns are associated in two indeendent cycles, one for the condensed atterns c+1 l = 1 l and the other one for the noncondensed atterns +1 l = c+1 l. This guarantees the alicability of the rocedure. The first arts of A and B contribute to a Hebbian interaction J H and their second arts contribute to the symmetric sequential interaction J S. The training of the network model may be thought to roceed in two stages assuming the atterns are numbered in a given order. In one stage the set of atterns is resented to the network in random order, every attern being resented the same number of times. This builds u the Hebbian art of the learning rule, whereas the sequential art of the rule takes lace as follows in another stage. The atterns are ordered in two sequences, one sequence in increasing order in which each attern is resented with the following attern in the sequence, and the other sequence in decreasing order where every attern is resented with the revious attern. The crucial arameter is, determining the ratio J H /J S of the Hebbian to sequential interaction in the signal term of the local field. On the other hand, different sequential noise levels given by b should yield qualitatively similar results as found in revious works 1,15. This does not mean that b is an irrelevant arameter and we consider this oint below. When b=1 there is a urely Hebbian noise and for any other b there is a Hebbian lus sequential noise. In distinction to other works, the choice,b 1 enables us to exlore the full range of arameters and b. The macroscoic overla comonents m l of O 1, between the configuration S l and the condensed atterns, are given by the large-n limit of N m N l = 1 N i l S i l, =1,...,c, 5 i=1 where denotes a thermal average with Eq. 1, whereas the overlas with the remaining c noncondensed atterns are M N l =O 1/ N

3 PERIOD-TWO CYCLES IN A FEEDFORWARD LAYERED Following the standard rocedure for the layered network, one may write the local field as a sum of a signal and a noise term i l due to the condensed and the noncondensed atterns, resectively 16. The noise follows a Gaussian distribution with mean zero and a variance 2 l given by the large-n limit of N 2 l = =c+1 Q N l 2, where the overbar denotes the average over all the atterns, in which Q N l = bm N l + 1 b M 1 N l + M +1 N l. 7 The noncondensed overlas that aear here deend on all atterns as functions of the full local field. Averages over the noncondensed atterns can then be erformed by integration and we obtain first the recursion relations for the macroscoic overlas m l =(m 1 l,...,m c l ), 6 m l +1 = Dz tanh Am l + l z, 8 where Dz=e z2 /2 dz/ 2 and denotes an exlicit average over the condensed atterns which does not deend on the secific realization of the atterns. To obtain a dynamic equation for 2 l we need recursion relations not only for the average squared noncondensed overlas M N l 2 and M N ±1 l 2, which can be derived in the usual way 16, but also for the correlation of two consecutive overlas and two next-to-consecutive overlas, M N l M N ±1 l and M N 1 l M N +1 l, resectively. These generate, in turn, correlations of overlas between more distant atterns, which requires us to kee track of the general form 15 C n 2 l = =c+1 Q N l Q N +n l, in which n=,..., c 1 and C c l = l. The c recursion relations for the C n s can be obtained in a systematic way, as outlined in the Aendix, leading to a variance of the noise which deends on the sin-glass order arameter q l = S l 2, 9 q l = Dz tanh 2 Am l + l z. 1 In the case where b=1 Hebbian noise, the variance follows the simle form 2 l+1 = +K 2 l 2 l, where K l = 1 q l. In the case of the absence of stochastic noise =, the equations coincide with those for a recurrent network with a arallel dynamics. The main difference between the layered and the recurrent networks is the absence of recurrent connections in the former. The recurrent connections tend to amlify the effects roduced by the stochastic noise and the fact that the equations are the same when = is simly due to the absence of noise to amlify in the recurrent network in that case. In the absence of stochastic noise the equations are also the same for the recurrent T (a) α (b) c=12 c=16 C c=22 c=15 c= ν c=12 c=16 C PHYSICAL REVIEW E 75, c=22 c=15 c=11 network with either symmetric or asymmetric extreme dilution 2. Thus, the network dynamics is described by the recurrence relations for the vector overla 8 and for all the C n s that go into the variance of the noise. Although the equations form an infinite set in the limit, the number of significant C n s is finite, making the model solvable in ractice. The transients and the dynamic evolution of the network can be studied in full detail but here we restrict ourselves to the stationary states. III. RESULTS Consider first the T, hase diagram of stationary states for = and various numbers of condensed atterns c, shown in Fig. 1 a. The Hofield ansatz m 1 =,1, for =1,...,c, is used as initial condition. The states corresonding to fixed-oint solutions and the hase boundaries for.5 are recisely the same as those found for a recurrent network 1, since the equations for the order arameters are exactly the same in the layered and in the recurrent networks when =. The Hofield-like states H have one large condensed overla comonent and the others are either small or zero. The hase of symmetriclike states S has equal or nearly equal overla comonents, at high or low T, resec- S S D P H SG (a) ν FIG. 1. Phase diagrams for a = and b T= with a urely Hebbian noise b=1. The dotted and full lines indicate, resectively, continuous and discontinuous transitions and the hases are described in the text. Some oints on the hase boundary for c =22 are indicated with crosses. D H (b)

4 F. L. METZ AND W. K. THEUMANN PHYSICAL REVIEW E 75, tively, ending at a aramagnetic hase P, with m= and q=. There is also a hase of nontrivially correlated states D in which the correlation coefficients for the overlas with increasingly distant atterns from a stimulus become gradually smaller, as will be seen below. The hase boundaries for.5 are ractically indeendent of c. The unusual features of the hase diagrams are stationary cyclic solutions C of eriod two, for any c, in the region.5 in which m l+2 =m l. These are the only stable states below the hase boundaries for the resence of cyclic states. The main difference from the asymmetric sequence rocessing studied in an earlier work 15 is the resence in that case of cycles of eriod c, as well as quasieriodic states. Figure 1 now shows that the size of the cyclic hase decreases or increases with an increase of c, ifc is even or odd, resectively, and there are no cyclic solutions for c 7 in the latter case. These roerties have been checked by a linear stability analysis of the S hase for = in extension of earlier work 14. Clearly, the eriodic solutions are fairly robust to synatic noise T. In relation to a recent work 7, we also studied our model with a finite number of indeendent airs of sequences for = and low and found only cyclic solutions of eriod two, for any number of stored sequence airs and for any c in each sequence. The effects of stochastic noise due to a macroscoic number of atterns = N are shown in Fig. 1 b for T=, b=1 Hebbian noise and various c. For the noncondensed overlas we choose the initial C n 2 1 =, for all n, and for the condensed overlas we take again a Hofield initial condition. As usual in the layered network, there is now a singlass hase with q labeled SG. For =1 we recover the critical storage ratio c.269 for the Hebbian layered network model 16. Also here we find the same kind of stationary states as in Fig. 1 a for.5 and stationary cyclic states of eriod two in the region of low, with the absence of the latter for c 7 in the case of odd c. Again, the boundaries between hases of fixed-oint states are fairly indeendent of c, while the cyclic hase boundaries have a similar deendence on c as in the T, hase diagram. We also found that the cyclic hase boundaries almost do not vary beyond c=11 and c=22, for odd and even c, resectively. They are very close and they should be indeendent of c in the large-c limit. Although Fig. 1 gives a fair account of the hase diagram for Hebbian noise b=1, one may ask which is the effect of other noise arameters b 1 and in Fig. 2 we show the critical values c for the existence of two tyical states in each of the hases H and C as functions of b, for c=13 and T=. In the case of fixed-oint states, that is, within the hase H and also for the D hase, not shown in the figure, c increases with increasing for a given b, whereas in the case of cyclic states c decreases, as one would exect. Similar results are obtained for c=12. There is a maximum c for an otimal b.748 and the kind of noise becomes more relevant for a dominant sequential art small in the signal of the local field. We consider next the solutions for the stationary overlas that describe the long-time behavior of the network, for a weak Hebbian term. The overlas for states in the hase α c ν= diagrams bifurcate from a fixed-oint behavior in the S hase to stable stationary limit cycles on the continuous discontinuous transition from the S to the C hase, for even odd c, resectively. In order to describe one of our main results, that is, the nature of the cyclic behavior, we illustrate this for the overlas in Fig. 3 for c=13 when = for a tyical low synatic noise of T =.3. The bifurcation diagram contains the stationary values of the first seven overlas as functions of, with m 1 =,1 as an initial condition. Each overla assumes a larger and a smaller value in the cyclic hase marked with the same symbols for clarity, only the larger one is labeled, with a decreasing oscillation amlitude as we move away from the stimulated attern. For considerably higher noise levels, say T=1.25 and aroriate values of see below, all the airs of overla comonents kee oscillating between the same uer and lower values. Thus, for such high levels of T, the cyclic hase is already a hase of a air of symmetric states, one with all equal larger condensed overlas and the other with all equal smaller overlas. In addition, the overla comonents have the symmetry m +n t =m n t, where is the stimulated attern and n =1,..., c 1 /2. In the case of even c, there is first a continuous transition to a air of overlas with decreasing such that all solutions kee oscillating between an uer and a lower value. There b ν= ν=.1 FIG. 2. Critical storage ratio as a function of b for T= and c =13 in the Hofield-like hase =1 and.9 and in the cyclic hase = and m µ.2.1 m 3 m 4 m 5 m 6 m 7 m 1 m2 ν= ν FIG. 3. Overlas for the stationary cycles of eriod two, discussed in the text, that bifurcate from the symmetric hase for c =13, =, and T=

5 PERIOD-TWO CYCLES IN A FEEDFORWARD LAYERED C d is a further, discontinuous transition, for lower values of, to distinct airs of overlas that is similar to the discontinuous transition for the case of odd c, with the same symmetry between overlas now for n=1,..., c 2 /2. Thus, the behavior of the overla comonents has a rich structure which deends on whether c is even or odd. In order to demonstrate that the cyclic states may be nontrivial correlated attractors for low, we consider next the correlation coefficients between the attractors corresonding to any two condensed atterns, and a distance d= away, defined here as C d = 1 C i i i i i = tanh h tanh h tanh 2 h, 11 where i is the attractor corresonding to the initially stimulated attern and i is its mean value over the network which is zero for the unbiased atterns we are using here. We show in Fig. 4 the deendence of the correlation coefficients for the states corresonding to the larger overla comonent of each air of states with the distance d to increasingly distant condensed atterns in the sequence from a given stimulated one, for c=13 and =, that is, in the absence of stochastic noise where the results are the same for the layered and the recurrent network. We do this, as indicated, for T, =.3,.1 and also for 1.25,.1 both within the hase of cyclic states. In the first case, where each overla comonent assumes distinct larger and smaller values, there are also correlation coefficients for the smaller values, not shown in the figure, which turn out to decrease less raidly than the coefficients for the larger values. The reason for this is that there is a weaker distinction between the smaller comonents than among the larger ones. For the higher T = 1.25 the correlation coefficients are already indeendent of d due to the fact that all the airs of overla comonents are oscillating between the same uer and lower values. d T=.3 ν=.1 T=.3 ν=.625 T=1.25 ν=.1 FIG. 4. Correlation coefficients between attractors as a function of the distance d from a reference attern, defined in the text, for c=13 and = in the hases C two uer curves and D lower curve. The lines are a guide to the eye. PHYSICAL REVIEW E 75, In distinction to the fixed-oint correlated states in hase D, shown for our model by the lower set of results in the figure for T=.3 and =.625, the correlation coefficients for the cyclic states do not decrease to zero and, instead, exhibit a behavior tyical of quasisymmetric states for low T. IV. DISCUSSION To summarize our results, we obtained a closed-form attractor dynamics for a feedforward layered network model of binary units and atterns in terms of a finite number of macroscoic order arameters for the cometition between attern reconstruction and symmetric sequence rocessing. The dynamics is a arallel one, in which all units in each layer are udated simultaneously, and the work resented here is restricted to the stationary states of the dynamics which exhibits either fixed-oint or cyclic behavior of eriod two, deending on the relative strength of the interactions. Either kind of behavior is an emergent roerty of the network which is an outcome of the dynamics. Full hase diagrams of stationary network behavior were obtained, either for a finite loading of atterns or in the saturation limit for the storage of a macroscoic number of atterns. In the case of a balanced or dominant Hebbian term, that is, for.5, we obtained qualitatively the same hase diagrams as those found in work by revious authors for a recurrent attractor network exhibiting Hofield-like states, correlated states, and symmetric ordered states 1. This suggests that the layered network dynamics discussed here may also serve to make qualitative redictions about a recurrent network in the case of a weak Hebbian term. Of course, to get the right quantitative behavior exected for a recurrent network in the storage saturation limit, and for detailed comarison with exeriments, one has to start by including lateral connections between units in a layer but this involves a more comlicated dynamics which is beyond the scoe of this aer. For below.5 we have first a regime of symmetric fixed-oint behavior and for smaller values of we find the cyclic attractors discussed in this work. These are states that have decreasing correlation coefficients with increasingly distant attractors from an initially stimulated attern for a sizable range of synatic noise. The stationary overlas between the states of the network and the condensed atterns describe the long-time behavior of the system. As we saw, the overlas for states within the cyclic hase are always eriodic with eriod two, oscillating either between a distinct air of uer and lower values for each overla comonent or between the same air of values for all comonents, deending on the arity of c and on the state in the,t, hase diagram. We argue that the cyclic behavior of eriod two is a roerty that follows essentially from the nature of the interactions, that is, from the strong symmetric sequential term, rather than being an artifact of the model due to the lack of lateral connections between the units or having binary units. First, in suort of our claim, work in rogress on the arallel dynamics of a fully connected recurrent network of binary units and atterns indicates, indeed, that there are exclusively cyclic states of eriod two in a finite region of the hase diagram 17. Second,

6 F. L. METZ AND W. K. THEUMANN PHYSICAL REVIEW E 75, we checked exlicitly that numerical simulations with threshold-linear units 12 in our feedforward layered network yield only cyclic states of eriod two for tyical low values of and, att=. The secific regions of the hase diagram where distinct airs of uer and lower values or the same air of values for the overlas aear deends, of course, on the details of the model. The hase diagrams found in this work are quite different from those for asymmetric sequence rocessing 14,15. Indeed, the latter exhibit a 1 duality that aears in the form of symmetric hase diagrams with a corresondence between fixed-oint solutions for large and stationary cycles of eriod c for small. There is, aarently, no such duality in the case of symmetric sequence rocessing and whenever stable cycles aear they are of eriod two, indeendently of the number of condensed atterns c. Also, the strong c deendence of the stability of the cyclic hase is in contrast with the results for asymmetric sequence rocessing, in which the boundary of the cyclic hase in ractice does not deend on the number of condensed atterns 14,15. Although the work resented here is restricted to a layered feedforward network with no lateral interactions, it is exected to exhibit further features of a recurrent network beyond those ointed out above, in articular, the robustness to both synatic T and to stochastic noise due to the noncondensed atterns over a sizable ratio / 1 of the relative strength of the Hebbian to sequential interaction Figs. 1 a and 1 b, resectively. The robustness becomes stronger in the case of decreasing even values of the number of condensed atterns c and weaker for odd values of decreasing c. The model used here has several limitations with resect to a closer to biological network which do not allow us to make the roer quantitative redictions to comare with exeriments, mainly the binary full activity units, in lace of continuous or integrate-and-fire neurons, and unbiased highactivity atterns, without lateral interactions between units in the same layer. Desite those limitations, there is the ossibility of making extended qualitative redictions for the kind of visual-task exeriments in the IT rimate cortex and their interretation in terms of correlated states in simle models for a recurrent network Those works rovided a connection between a fixed-oint attractor dynamics in a recurrent network and ersistent activity in a biological system. Our work suggests a connection between a eriodic attractor dynamics in a recurrent network trained with atterns in a random order and with atterns in a sequence and a kind of oscillating ersistent activity in the IT cortex with the secific cyclic behavior discussed in this work. The original exeriments were based on intensive training with atterns in a random order and with atterns in a sequence. One may argue that, if the training of a rimate with visual atterns in random order, which is suosed to be a realization of a Hebbian rule, is not sufficiently strong, one may have a situation such as that described here for small with the resence of correlated cyclic states of eriod two, in which the correlation coefficients decay with increasing distance from the stimulus, u to a finite value. The results resented here should stimulate further theoretical and exerimental work for the case of weak Hebbian reenforcement of atterns. It may also lead to interesting alications in information rocessing in networks 18. ACKNOWLEDGMENTS The work of one of the authors W.K.T. was financially suorted, in art, by CNPq Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil. Grants from CNPq and FAPERGS Fundação de Amaro à Pesquisa do Estado de Rio Grande do Sul, Brazil, to the same author are gratefully acknowledged. F.L.M. acknowledges financial suort from CNPq. APPENDIX: RECURSION RELATIONS We resent next an outline of the derivation of the recursion relations for all C n s. It is based on an extension of the usual rocedure 15,16 adated to our secific synatic matrix. The main ste is the recursion relation for the correlation between any two microscoic overlas with the noncondensed atterns that enter in Eq. 9. We start with the definition of the average M N M N = 1 N 2 i j S i S j ij A1 for, =c+1,...,, in which the rimed unrimed variables refer to layer l+1 l, resectively. The averages are exlicitly calculated only with resect to rimed variables since the averages over the underlying unrimed variables in the lower layer are taken care of by means of the law of large numbers. Writing Eq. A1 in the form M N M N = 1 N 2 i i i + 1 N 2 i j S i S j A2 i j allows us to take the thermal averages, leaving, in our case of binary atterns, M N M N = N + 1 N 2 tanh i h i tanh j h, j i j A3 using the fact that the atterns are uncorrelated variables with i i =. The embedding field i h i is given by i h i = i Am + Q N + i i, A4 where i is a set of indeendent Gaussian random variables i = i Q with mean i = and width defined by Eq. 6. Incidentally, the same exression for the embedding field without the need of searating a single term on the right may be used to obtain the recursion relation for the overla comonents that yield Eq. 8. Turning now to the configurational average in Eq. A3 over rimed variables, it decoules into a roduct of averages. Making an exansion to leading order in Q N =O 1/ N we obtain

7 PERIOD-TWO CYCLES IN A FEEDFORWARD LAYERED PHYSICAL REVIEW E 75, tanh i h i = tanh i Am + i z i + Q N 1 tanh 2 i Am + i z i. A5 Averaging first with resect to the variable i and then taking the configurational average over the Gaussian variable, we obtain tanh i h i = KQ N, A6 in which K= 1 q, with q defined in Eq. 1. Substituting Eq. A6 in A3 we obtain, to O 1/N, M N M N = N + K2 Q N Q N. A7 We can now aly Eq. 9 to layer l+1 and write it in terms of the correlations M N M N with the aid of Eq. 7. Then, Eq. A7 yields the c recursion relations u to 2 = b 2 + K b 1 b K 2 C b 2 K 2 /K C 2 2, A8 C 1 2 = b 2 K 2 C b 1 b K 2 /K C b 2 K 2 3C C 2 3, A9 C 2 2 = b 2 K 2 C b 1 b K 2 C C b 2 K 2 /K C C 2 4, A1 C n 2 = b 2 K 2 C 2 n +2b 1 b K 2 2 C n b 2 K 2 2 C n C n+1 +2C 2 2 n + C n+2, A11 in which n=3,4,..., c 1, and these equations have to be solved numerically. The extension to Q-state neurons and atterns, for Q 3 is straightforward, as well as for continuous neurons. 1 L. F. Cugliandolo and M. V. Tsodyks, J. Phys. A 27, A. C. C. Coolen, in Handbook of Biological Physics IV: Neuro-Informatics and Neural Modeling, edited by F. Moss and S. Gielen Elsevier, Amsterdam, 21, A. Düring, A. C. C. Coolen, and D. Sherrington, J. Phys. A 31, K. Kitano and T. Aoyagi, J. Phys. A 31, L M. Kawamura and M. Okada, J. Phys. A 35, W. K. Theumann, Physica A 328, K. Mimura, M. Kawamura, and M. Okada, J. Phys. A 37, T. Uezu, A. Hirano, and M. Okada, J. Phys. Soc. Jn. 73, Y. Miyashita and H. S. Chang, Nature London 331, Y. Miyashita, Nature London 335, M. Griniasty, M. V. Tsodyks, and D. J. Amit, Neural Comut. 5, N. Brunel, Network Comut. Neural Syst. 5, See G. Mongillo, D. J. Amit, and N. Brunel, Eur. J. Neurosci. 18, , for a recent review. 14 A. C. C. Coolen and D. Sherrington, J. Phys. A 25, F. L. Metz and W. K. Theumann, Phys. Rev. E 72, E. Domany, W. Kinzel, and R. Meir, J. Phys. A 22, ; R. Meir and E. Domany, Phys. Rev. Lett. 59, F. L. Metz and W. K. Theumann unublished. 18 The Handbook of Brain Theory and Neural Networks, edited by M. A. Arbib MIT Press, Cambridge, MA,