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A self-organizing network in the weak-coupling limit
PHYSICA ELSEVIER
Physica D 110 (1997) 195-208
A self-organizing network in the weak-coupling limit Paul C. Bressloff Department of Mathematical Sciences, Loughborough University, Loughborough, Leics. LEl l 3TU, UK Received 5 November 1996; accepted 28 May 1997 Communicated by A.C. Newell
Abstract We prove the existence of spatially localized ground states of the diffusive Haken model. This model describes a selforganizing network whose elements are arranged on a d-dimensional lattice with short-range diffusive coupling. The network evolves according to a competitive gradient dynamics in which the effects of diffusion are counteracted by a localizing potential that incorporates an additional global coupling term. In the absence of diffusive coupling, the ground states of the system are strictly localized, i.e. only one lattice site is excited. For sufficiently small non-zero diffusive coupling c~, it is shown analytically that localized ground states persist in the network with the excitations exponentially decaying in space. Numerical results establish that localization occurs for arbitrary values of c~ in one dimension but vanishes beyond a critical coupling C~c(d), when d > 1. The one-dimensional localized states are interpreted in terms of instanton solutions of a continuum version of the model.
1. Introduction The study of spatially localized states i n extended systems is an area of great current interest. Much of this interest has focused on spatially localized periodic oscillations in networks of coupled non-linear oscillators - so-called discrete breathers. The existence of these discrete breathers has been demonstrated both numerically [1,2] and analytically [3-5]. A number of applications of such localized structures have been proposed including D N A dynamics in molecular biology [6]. One method for proving the existence o f discrete breathers in weakly coupled systems is based on the uniform continuation o f periodic solutions constructed in the uncoupled limit [3,7] (als o known as the anti-integrable or anti-continuum limit). More generally, investigating how certain features of network dynamics persist from the uncoupled limit has become a
powerful method for studying weakly coupled systems [8,9]. Examples include the study of ground state systems in condensed matter physics [10-12] and multistability and wave propagation failure in networks of bistable elements [ 13,14]. A distinct but very important example of localized structures in networks is found in neurobiology. In many regions of the cortex, groups of adjacent neurons appear to form higher functional units that serve to analyse some particular stimulus feature such as the orientation of an edge of an image or velocity of movement (see [15] and references therein). The association of groups of neurons with local properties of visual images varies in a regular way with the location of the neurons in the visual cortex. Thus, a socalled topographic map of various features of an image is built up. Another example o f such a phenomenon is found in the auditory cortex of the bat where a
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topographic map of signal amplitude versus pitch is formed. This generates a frequency spectrogram of an auditory signal. In addition, bats possess maps representing the time difference between two acoustic events, which plays an important role in the sonar orientation of the animal [16]. Topographic maps are not limited to sensory regions but also occur in the motor cortex. While sensory maps generate spatially localized regions of excitation whose location represents the signal features being analysed, motor maps create from spatially localized excitations an activity pattern in space and time that triggers a particular movement [171. Neural network models of the formation of coherent structures in brain function generally involve two aspects: (i) a selection mechanism for determining the centre of a localized excitation in response to an input, and (ii) an interaction mechanism that spreads the response over a neighbouring region of the network leading to the formation of a topographic map. A network typically learns the appropriate form of interaction by an adaptive process [19,20]. The selection mechanism, on the other hand, usually involves a sequential search for the optimal response. Exploiting certain similarities between pattern formation in synergetic systems and pattern recognition, Haken [21] has constructed a simple neural network model that implements this selection process using a form of competitive gradient dynamics. (Such a model does not itself have a direct biological interpretation, but is rather a caricature of more realistic models.) The ground states of the system consist of strictly localized states in which only one neuron is excited and the remainder quiescent. In other words, the network dynamically realises a winnertake-all strategy. An interesting extension of this type of network is the diffusive Haken model in which the introduction of a diffusive interaction between the neurons leads to a delocalization of the original model's ground states [22]. When there exists a balance between the effects of diffusion and localization, it is possible to obtain new ground states that are localized excitations (or bubbles) distributed over many neurons. Such ground states are more robust than those of the simple Haken model and also better reflect the
kind of coherent structures found in neurobiological systems. One of the major claims of Ref. [22] was that a network arranged on a d-dimensional square lattice with standard nearest-neighbour diffusive coupling possesses localized ground states when d = 1 but not for d > 1, even for arbitrarily small diffusive coupling. This claim was based on numerical simulations of the lattice model together with some variational calculations of a continuum model obtained in the limit of large diffusive coupling. In this paper, we prove analytically that such a claim is false: the diffusive Haken model supports localized ground states in any finite dimension provided that the diffush:e coupling is sufficiently small. (Such a result also holds for more general choices of coupling provided that the strength of coupling decays exponentially with distance on the lattice.) Our approach is based on a uniform continuation from the case of zero diffusive coupling. The structure of the paper is as follows. In Section 2, we introduce the diffusive Haken model and describe the ground states of the system in the absence of diffusive coupling. The weak coupling limit is then analysed in Section 3, expanding upon our previous work in Ref. [23]. Numerical results are presented in Section 4, which establish that localization occurs for arbitrary values of a in one dimension but vanishes beyond a critical coupling otc(d), when d > 1. Finally, in Section 5, we show how the one-dimensional localized states can be interpreted in terms of instanton solutions of a continuum version of the diffusive Haken model.
2. Diffusive Haken model Consider a network of N d elements arranged on a d-dimensional lattice F . Denote the state of the ith element by qi c • with i ~ / ' . In the diffusive Haken model, the state of the network at time t, Q(t) = {qi(t), i ~ F}, evolves according to the gradient dynamical rule [22]: q~ -
OV Oqi
(2.1)
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for which the lattice structure does not play a role. Eq. (2.4) then becomes
with a potential V of the form
V [ Q , a ] = - ~oe Z ( q j
1 D [Q] - qi) 2 - -~
qi = q3 + (1 -- 2D)qi.
(i,j) + -1D[Q]
2
1
4
(2.2)
i
where (i, j ) denotes neighbour pairs and
D[Q] = Z
summation
over
q.2.
nearest-
(2.3)
The first term on the right-hand side of Eq. (2.2) represents a nearest-neighbour diffusive interaction with coupling constant ot whilst the remaining terms correspond to the localizing potential introduced by Haken [21]. Substituting Eq. (2.2) into (2.1) gives the equation of motion:
(]i ~ Ol ~-~[qj -- qi] -}- (1 -- 2 D [ q ] + q2)qi,
(2.4)
(jli}
where ( j [i) denotes summation over all nearest neighbours j of i. Eq. (2.4) is invariant under the transformation Q ~ - Q . Moreover, qi(t) >_ 0 for all t > 0 and i ~ F if qi (0) > 0 for all i E F . For, suppose that qi (t) -~ 0 and qj (t) > 0 for all j ¢ i. Setting qi = 0 on the right-hand side o f Eq. (2.4) shows that gli (t) > O. That is, qi cannot cross over to the negative real axis. Since we are interested in finite energy states of the network, we shall also assume that D[Q] < ec. In other words, each state Q is taken to be squaresummable. Hence, we can restrict our discussion to solutions in the domain 7-{-----[QENMI
IQl2
The equilibria of Eq. (2.5), which we denote by 0 , satisfy qi = 0 o r qi = 2 % / ~ -- 1. Hence, the stationary states can be divided into M + 1 classes determined by the number m of excited sites [22]. For a given m, D = m / ( 2 m - 1) and the corresponding potential is V (m) = _ (m/4) (2m - 1 ) - 1. We shall establish below that the stationary states m ---- 1 are stable, whereas all other stationary states are either unstable (m = 0) or saddle points (m > 1). For m = 1 there exists a single excited site, i0 say, such that qi = (~io,i. Moreover, D = 1 and V (1) = - 1 / 4 . This is one of the N strictly localized ground states of the network. There are two homogeneous stationary states given by the vacuum state m = 0 and the dissipative state m -----M. The former satisfies qi = 0 for all i and V (°) = 0 and the latter has qi = 1 / 4 ~ - ~ - - 1 for all i and V (M) = --M/(8M4). In the large M limit, the dissipative state becomes pointwise identical to the vacuum state but has lower energy, V (~) = - 1 / 8 . Also note that for an infinite lattice the dissipative state is marginally stable. In order to determine the stability o f a stationary state ~), we need to consider the eigenvalues )~ of the Jacobian
Jij --
0 2 V 0 = 6 e~=0 • Oqi Oqj
(2.6)
A stationary state will be stable if, and only if, all the eigenvalues are strictly negative. Substituting Eq. (2.2) into (2.6) gives Jij -~ --4qiqj,
where M = N d and IQI2 = x/~//q?, is the 12 norm on the vector space o f network states 7-/. The network converges to one of the stationary states of the potential V, i.e., OV/Oqi = 0 for all i ~ F . (Note that V is bounded from below.) Such a state m a y be obtained by setting qi = 0 in Eq. (2.4), solving the resulting time-independent equation for fixed D and then determining D self-consistently using Eq. (2.3). This procedure can be carried out explicitly in the case of zero diffusive coupling (~ = 0)
(2.5)
i ¢ j,
Jii = _ ~ 2 + 1 -- 2 0 .
(2.7) (2.8)
In the case m = 0, the matrix J reduces to the identity matrix so that )v = 1 is M - f o l d degenerate and hence the vacuum state is unstable. On the other hand, when m = 1 the matrix J is diagonal with an ( M -- 1)-fold degenerate eigenvalue ;v = --1 and a non-degenerate eigenvalue ), = - 2 ; singly localized states are thus stable. Finally, consider the case m > 1. By an appropriate re-ordering of the matrix J , it is simple to show
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that the stability of the system will be determined by the solutions )v of the m x m determinant [21] a - £ 2a 2a
2a a -- ~ 2a
2a
2a
2a 2a a --
2a
... ... ...
...
2a 2a 2a
= 0,
(2.9)
a--)~
where a = - 2 / ( 2 m - 1). One finds that there is an (m - 1)-fold degenerate positive eigenvalue )v = - a and a non-degenerate negative eigenvalue )~ ---- (2m 1)a. (The remaining M - m eigenvalues of J satisfy )~ = a/2.) Hence, multiply localized states are saddles. We conclude that, in the case of zero diffusive coupling, the ground states of the system consist of strictly localized states in which only one site is excited and the remainder quiescent; the particular ground state selected depends on the initial data a n d / o r additional applied inputs. In other words, the network dynamically realises a winner-take-all strategy. Now suppose that the diffusive interaction between neighbouring sites is switched on. This leads to a delocalization of the original ground states [22]. For sufficiently small coupling o~, there exists a balance between the effects of diffusion and localization so that it is possible to obtain new ground states that are localized excitations (or lattice instantons) distributed over many lattice sites. A simple heuristic argument for the existence of localized ground states proceeds as follows [23]. We shall assume for concreteness that the network is infinite in extent. Suppose that the initial state of the network is one of the strictly localized states qi (0) = (~i,io. Substitution into Eq. (2.2) then shows that the potential at t = 0 satisfies V < - 1 / 8 foroe < 1/8d. Since d V / d t <_ 0 f o r all t, the final state cannot be the dissipative state. Moreover, we expect this final state to be localized since D < oc for finite values of V so that the equilibrium lattice configuration must decay sufficiently fast with distance on the lattice from i0. Although the above energy argument supports the existence o f localized ground states, it is still necessary to establish their relationship to the ground states at a = 0, their stability properties, how they decay with spatial location on the lattice and also the struc-
tural stability of the system. A l l of these issues can be tackled in terms of a uniform continuation from the uncoupled limit as shown in Section 3. The analytical results for small oe will be supplemented by numerical simulations in Section 4, where the behaviour of the system for larger values of the diffusive coupling will be studied. We shall find that localized states persist for all values of oe in one dimension but vanish beyond a critical value e~c(d) for d > 1.
3. Weak-coupling limit 3.1. Existence of localized states Stationary s t a t e s of the diffusive Haken model satisfy (1 - 2D)qi + q3 + ot ~_,(qj - qi) = O.
(3.1)
(jli)
For the moment, we shall consider both ot and D as free parameters; later on D will be determined selfconsistently using Eq. (2.3). In terms of the network state Q ~ 7-/, Eq. (3.1) can be rewritten in the form
F ( Q , D) + otK(Q) =-- G ( Q , ~, D) = 0,
(3.2)
where
[F(Q, D)]i = q3 ÷ (1 - 2D)qi,
(3.3)
[K(Q)]i = ~_a(qj - qi).
(3.4)
(j[i)
The condition for an equilibrium is then
G ( Q , a, D) ----0,
G
"
7-[ × R 2 ~
~'~.
(3.5)
Eq. (3.5) is formally very similar to the coupled system of bistable elements studied by M a c k a y and Sepulchre [13]; such a system would be obtained under the transformation F ( Q , D) ---> - F ( Q , D). The stability properties of the system for fixed D are very different, however, from the full Haken model with D determined self-consistently by Eq. (2.3). For a given D = Do, Do > 1/2, and o~ = 0 the equilibria of Eq. (3.5) satisfy qi = 0 or c]i = 4 - ~ - 1 (if negative solutions are included). Denote the Jacobian OG/OQ by 6G. Since [6G(~), 0, Do)]i j -= (]i,j)v i with
199
PC. Bressloff/Physica D 110 (1997) 195-208 )~i ~ - - ( 2 D 0 - 1 ) if~i = 0 andZi = 2(2D0--1) ifc~i 7~ 0, 3G is invertible at the stationary point (~), 0, Do). Hence, one can use the IFT to prove the existence of a local continuation of each Q for sufficiently small ot.
Theorem 1. There exists oto, 8o > 0 such that for Iotl < oto and [D -- Dol < ao there is a locally unique continuation Q(ot, D) with G(Q(ot, D), ot, D) = 0
0Q OD
-- [ r G ( Q , ot, D ) I - I K ( Q ) ,
(3.10)
with D(0) = 1 and qi(O, 1) = 8i,io. Differentiate Eq. (3.10) with respect to ot:
dD dot -
2(Q]OQ/Oot) 1 - 2(QlOQ/OD)"
(3.11)
(QIQ') = ~'~qiq~
(3.12)
(3.7) -- 218G(Q, ot, D ) ] - I Q .
Hence, the solution Q(ot, D) is differentiable with respect to ot and D as long as 8G has a bounded inverse. Recall that, given a norm [ • [ on a vector space ~ , a linear operator L acting on 7-/ is bounded if there exits some real constant c such that
[LQI~clQI
sup I t O l . Q~7~ IQI=I
(3.8)
Later on it will be useful to consider both the 12 norm introduced previously and the sup norm on 7-( defined by I O l ~ = sup Iqi[. i~F
such that 7-{ becomes a Hilbert space. Substituting Eq. (3.7) into (3.11) gives
dD 2(QISG-1]K[O]) dot = 4(QISG-1IQ) - 1"
(3.9)
We shall denote the corresponding norms on the space of bounded linear operators by ]1 • 112 and II • I1, respectively. We now substitute the solution Q (ot, D) of (3.5) into the self-consistency condition (2.3) and apply the IFT a second time with ~) taken to be a strictly localized state.
(3.13)
At the stationary point (Q, ot, D) = ( 0 , 0, 1) we have dD = -2d, dot
forall Q 6 7 - {
We can then define a corresponding norm II - II on the space of bounded linear operators L: 7-[ --+ 7-{ according to Iltll =
q/2(ot, O(ot)) = D(ot) i
We have introduced an inner product on the vector space 7-[ according to
Differentiating Eq. (3.6) shows that
aot
Z
(3.6)
and Q(0, Do) = Q where Q is a given equilibrium.
aQ
Theorem 2. There exists otl > 0 such that, for Iotl < there is a local continuation D(ot) and Q(ot) = Q(ot, D(ot)) E 7-( such that G(Q(ot), ot, D(ot)) = 0 and
otl,
(3.14)
where d is the dimension of the lattice. It follows from Eqs. (3.7) and (3.13) and the relation
dQ dot
----
OQ Oot
OQ dD OD dot
+ - - - -
(3.15)
that the solution Q (ot) remains differentiable provided that (i) 8G has a bounded inverse, (ii) 4(Q[3G-1IQ) > 1. Condition (ii) can be written in a more useful form using some basic properties of bounded symmetric linear operators on the Hilbert space ~ [24]: II t 112~11 t II, inf I(QILQ)r Q # 0 (QIQ) -
(3.16)
1 I[ L - 1 I1~
(3.17)
Setting L = 8G -1 we have
I(QISG-I[Q)I > {OlO______~). II 8 6 II
(3.18)
PC. Bressloff/Physica D 110 (1997) 195-208
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Hence, condition (ii) becomes
where q+ are the critical points of f , i.e., f ( q + ) and
(ii') II 3G II< 4D. We shall now use conditions (i) and (if) to estimate a lower bound Oil for the existence of local continuations Q(~) of the strictly localized states () present in the case of zero coupling. We shall then determine whether or not these localized states are candidate ground states, i.e. V < - 1 / 8 in the case of an infinite network, and also whether or not they are stable.
( 2 D - 1 ~ 3/2 f ( q + ) = qz2 k, 3 ]
q~_ = -t-~/(2D -- 1)/3y,
(3.20) (see Fig. 1). It is easy to establish that qi ( ~ ) E [0, q + ) for all i 5~ i0 and qi(oz) E (q+, qmax), where qmax is the positive root of the equation f ( q ) = f ( q _ ) , i.e., qmax = 2q+. This follows from Fig. 1 and the result f ( v ) < 0 and f ( w ) > O, where v = infi qi(o~) and w = supi qi(oe). Hence, Eq. (3.19) holds if
3.2. Estimation of oq
2D- 1 o~ < - 6d
We first consider condition (i), namely whether or not ~ G has a bounded inverse. For the moment assume that D is fixed with 1/2 < D < 1. (Eq. (3.14) shows that D < 1, at least for sufficiently small a. The lower bound for D occurs for the dissipative state.) Suppose that each lattice site i is to be assigned a value qi (o0 --- 0, which is a continuation of the strictly localized state qi (~i,io~ 1. This can be achieved uniquely, by taking qi ( ~ ) tO be the middle root (if i 7~ i0) or the largest root (if i = i0) of the equation f ( q i ) = q3 + (1 - 2D)qi = -or Y~(jli)(qJ - qi), provided that =
= 0
(3.21)
Condition (3.21) ensures that 8G is invertible. To show this, separate out the diagonal and off-diagonal parts of 3G by writing ~G(c~) -----A + oiL1, where
A i j = ( f ' ( q i ) - 2doe)~ij,
(Zl)ij
=~
~j,k.
(kli)
(3.22)
-
Then II~a(c~) -1 II <
1 IIA-111-1 - ~llZill'
(3.23)
provided ~IILi II < 11-4-1 I1-1. Since e~sup ~ ( q j ( o t )
-- qi(ol))
(3.19)
< If(q+)[,
i~F (jli)
IlL1 II = sup ~
ILl lij = 2d,
iEF'j~ F
f(q)
IIA - l II = sup [f'(qi) - 2du1-1, icF it follows that ~G(Q(o0, e) is invertible when
[ - 2do~ + ft(qi(ot))l > 2dot
I
I
q m
,
Fig. 1. Cubic non-linearity f ( q ) = q3 + (1 -- 2D)q for fixed D, 1/2 < D < 1. Critical points are q+ = - t - 2 4 ( 2 D - 1)/3 and qmax = 2q+.
(3.24)
for all i ~ F . Condition (3.24) is immediately seen to hold for all i 7~ i0 since qi (or) ~ [0, q+), so that f'(qi(ot)) < 0 (see Fig. 1). In the case i = i0, we have qio(Ot) E (q+, qmax), SO that f'(qio(Ot)) > f t ( q + ) = 2 ( 2 D - 1). Hence, Eq. (3.24) holds for i = i0, provided fr(q+(ot)) > 40td, which reduces to the condition ~ < (2D - 1)/2d. We now recall that D is itself dependent on oe. Using Eq. (3.14), we have that for oe << 1, D(ot) ~ 1 - 2 d o t . Substituting this approximation into
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P.C Bressloff/Physica D 110 (1997) 195-208
Eq. (3.21) then yields the result that 8G is invertible when
1 oe < tel -- 10d"
(3.25)
It is simple to show that condition (ii:) is also satisfied when (3.25) holds. For [ISGII ~ 4doe + sup(3qi(oe) 2 ÷ 1 - 2D) i 2
< 4doe + 3qmax + 1 - 2 D = 4doe + 3(2D -- 1). Condition (ii') requires that cz < (3 - 2 D ) / 4 d which is a weaker condition than (3.21) (and hence (3.25)) since 1/2 < D < 1. Note that our estimate (3.25) for the value of diffusive coupling below which localized states are proven to exist is close to the value 1 / 8 d obtained using the energy argument at the end of Section 2.
3.3. Energy and stability
The unique state Q(oe) obtained as a local continuation of the strictly localized state Q (Theorem 2) is in fact a candidate ground state. For Taylor expand the energy function V of Eq. (2.2) to first order in oe and use the fact that Q is a stationary point of V, O V / O Q = 0 at Q = (~: V[Q(oe), oe] ~ - ~ 1 + ~oe E ( ~ j
__ ~/i)2 = - ~ 1 + doe.
(i,j)
(3.26) Eq. (3.26) shows that (at least to this level of approximation) V < - 1 / 8 if oe < l / 4 d . Thus the localized state state Q(oe) with oe < Otl has lower energy than the dissipative state. In order to investigate the linear stability of Q(oe) we linearize equation (2.4) about Q(oe). Setting ui = qi -- qi (oe) and expanding to first-order in ui gives the linear equation tii = E [ S G ( ~ ) i j (J]i)
- 4qi (c~)q] (oe)luj,
(3.27)
where we have set G[Q(oe), oe, D(o0] = G(~). Hence, the condition for linear stability is that all eigenvalues of the matrix J(oe) = 3G(oe) - 4Q(oe) ® Q(oe)
(3.28)
are bounded below zero. The matrix J(0) is given by Eqs. (2.7) and (2.8) with qi 3i,io. Its spectrum consists of a single non-degenerate eigenvalue )~ = - 2 and a single degenerate eigenvalue )~ = - 1 . Since the spectrum of J(oe) is real, it follows that Q(oe) will remain linearly stable as ot increases from zero provided that J (oe) remains invertible. We shall now prove that conditions (i) and (ii) of Section 3.1 ensure that J(oe) is invertible. First write =
j(oe)-I ___[3G(ot) - 4Q(oe) ® Q(oe)]-i = 3G(oe) -1 T(oe) -1,
(3.29)
where T(oe) = I -- 4[Q(o0 ® Q(oe)]3G(oe) -1
(3.30)
and I is the identity matrix. Suppose that T(ot) -~ exists and for an arbitrary P c 7-[ consider the linear equation R = T(oe)-lP.
(3.31)
Eqs. (3.30) and (3.31) imply P = T(oe)R = R -- 4Q(oO(Q(oe)13G(ot) -1 [R).
(3.32) Eq. (3.32) shows that the unknown vector R must be of the form R = P + ~cQ for some real constant x. Substituting this into (3.32) allows us to determine ~c so that 4(Q(oe)16G(oe) -1 [P) K = 1 -- 4(Q(oe)lSG(oe) l[Q(oe))'
(3.33)
Since P is arbitrary, we conclude from (3.33) that T (cO and hence J(oe) have bounded inverse provided that conditions (i) and (ii) hold. Finally, note that substituting Eqs. (3.7) and (3.13) into (3.15) and comparing the result with Eqs. (3.31)-(3.33), when P = K [ Q ] , yields d Q = _ j (oe)_i K[Q(oe)].
doe
(3.34)
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This equation also follows from applying the IFT directly to Eq. (3.1) after substituting for D using Eq. (2.3). One could then work backwards to derive Conditions (i) and (ii).
for any z ~, [1, l/L] such that fl(z) < [laG -1[[-1, where
3.4. Localization length
From the decomposition of 6G given by Eq. (3.22), we see that
We shall now show that the continuation Q(ot) of a strictly localized state is exponentially localized and determine an upper bound for its localization length. We shall follow the analysis of Baesens and Mackay [25], who treat the issue of exponentially localized network states within a general framework. For convenience, we consider an infinite lattice, F = N d, and take the distance d(i, j ) between sites i = (il . . . . . id) d and j = (jx . . . . . jd) to be d(i, j ) = ~ p = l lip - Jpl. Without loss of generality, the strictly localized state is assumed to be located at the origin of the lattice i=0. Suppose that Q (ot) is exponentially localized about the origin. In other words,
fi(z) = 2dot(z - 1)
Iqi(ot)l --< C(ot) Ld(i'O)
(3.35)
for some C(ot) and 0 < L < 1. Eq. (3.35) holds for ot = 0 by construction; one needs to verify that the form of Eq. (3.35) is preserved as ot increases from zero according to Eq. (3.34). From Eqs. (3.29)-(3.34), we have
dqi < ~_~ [aG~jlIIK[Q] j + Kqjl dot - j~U
(3.36)
with x given by Eq. (3.33) for P = K[Q]. Eq. (3.35) implies that
IK[Q]i + Kqil <_ ~(ot)£d(i,o),
(3.37)
where C(ot) ----C(ot)ld(£ + £-1 _ 2) + x[.
(3.38)
fi(z) = sup ~
[3Gijl(z d(i'j) - 1).
(3.40)
i~F j~ F
(3.41)
and from Eqs. (3.23) and (3.24), 1
IlaG -x II -< - -
K(ot)'
K(ot) = inf ]f'(qi) - 2dot[ - 2dot.
(3.42)
Substituting Eqs. (3.41) and (3.42) into (3.39) then gives
dqi
dC Ld(i,O)
(3,43)
~d -
for any £ < 1 such that fi(1/L) < K(ot) and dC C(ot) d ~ -----K(ot) - fi(1/L)"
(3.44)
We can determine a lower bound L* for the exponential decay rate by solving fi(1/L) = K(ot). Using Eq. (3.41)
K(ot)l_1 L* =
1 + 2dot _1
"
(3.45)
It follows that the state Q(ot) decays at least as fast as L*, and hence an upper bound for the localization length is l / I n ( I / L * ) . The amplitude C(ot) can also be calculated, if needed, using Eq. (3.44). However, this is complicated by the fact that h: itself depends on C(ot). Alternatively, we could use the fact that for small ot, D(ot) ~ 1 - 2dot. Now from Eqs. (2.3) and (3.35) we obtain
D[QI = Z
q] < C(ot)2~_~) 2d(i,0).
(3.46)
i~F
From Eqs. (3.36) and (3.37) and [25, Theorem 1], it can be shown that if the matrix 6G is invertible then
Furthermore,
~ i~F
< C(ot) 116G_1 i1_~_/~(z)
(3.39)
oo L2d(i'°) = 1 + N(L),
N(L) = ~ . / k f ( n ) L 2n,
n=l (3.47)
PC. Bressloff/Physica D 110 (1997) 195-208
where N ' ( n ) is the number of lattice sites at a distance of n from the central site. For example, N ' ( n ) = 2 in one dimension and N ' ( n ) = 4n in two dimensions. Thus )~2 N()~) = 1 - 2 ~ 4~2 N ( ) 0 -- (1
(d = 1), (3.48) (d ----2).
-- )~2)2
It follows that, for small ~, we can take C(o0 ~
~1
- 2dee
1 + N()~)
.
(3.49)
3.5. Structural stability
So far we have shown that for sufficiently small coupling ~ there exist stable localized states Q(oe) of the diffusive Haken model that are local continuations of the global ground states of the system with zero coupling. We would like to identify the states Q(ee) as the ground states of the coupled system. However, there still remains an important question: are additional equilibria created on adding a small coupling? In other words, is the uncoupled system structurally stable? The creation of new equilibria could, in principle, introduce a new ground state o f the system that is not a local continuation of one of the zero-coupling ground states. Although we do not wish to explore this issue in any detail here, we expect the system to be structurally stable. This is certainly the case for a finite-dimensional network where one can appeal to the fact that the system is uniformly hyperbolic [9].
203
the choice of boundary condition for sufficiently large N. Typical results for a one-dimensional network are shown in Fig. 2. The stationary state of a network of N = 100 lattice sites is displayed for a range of values of the diffusive coupling c~; oe = 0.1, 0.5, 1.0. The initial data is scattered in the range [0, 0.5] except for the lattice site at i = 50 which is chosen to have the value 0.6. It is clear from the figure that, for each value of oe, the stationary solution is localized around i = 50. Moreover, one finds that the energy is less than the dissipative state, V < - 1 / 8 . The value of o~ for which localized states persist is only limited by the size of the network since increasing increases the localization length so that boundary effects become increasingly important. The behaviour of the network as a function of oe differs considerably in higher dimensions. This is illustrated in Fig. 3 for a two-dimensional network consisting of 50 x 50 lattice sites. The initial data is taken to be a strictly localized state at the centre of the lattice. One finds that a localized stationary state now only occurs for small values of ~, oe < tic where eec 0.098. There is a sharp phase transition at ~c beyond which the network converges to the dissipative state. Another difference between one and two dimensions is that in the latter case localized ground states have relatively small basins of attraction whose size is very sensitive to the value of ee. For example, scattered initial data along the lines of Fig. 2 only converges to a localized state for values of ~e well below Oec. We shall consider such issues in more detail elsewhere. It would appear, therefore, that in two or more dimensions the delocalizing effects of diffusion dominate.
4. Numerical results 5. Continuum limit In order to explore the behaviour of the diffusive Haken model away from the weak-coupling limit, it is necessary to consider numerical solutions of Eqs. (2.3) and (2.4). We use a simple finite difference scheme, namely Euler's direct method. Note that applying nearest-neighbour coupling to boundary sites in a finite network corresponds to a zero-flux condition (the analogue of a Neumann boundary condition in continuum models); our results are not sensitive to
In Section 3, we proved that for sufficiently small coupling ~ there exist localized ground states of the diffusive Haken model in any finite dimension d. We also showed numerically in Section 4 that, for d > 1, there exists a critical coupling ~ec(d) beyond which localized ground states cease to exist. We shall now investigate to what extent these various results apply to a continuous version of the Haken model. Introduce
204
RC.
Bressloff/Physica D 110 (1997) 195-208 (a)
0.6
(b)
0.8 0.5 • • • ". • •. " •
" ..•
• • ° •
••°
•
.•" °° ° ° °• °°
"
0.6
0,4 0.3
0.4
,
0.2
.
•
,.o
•"
•
°
•'.
0.2 0.i • .
"°
20
.
40
60
•
,
80
".
2'0. . . .
i00
go
s'o
6'0'
'
0.25[ 0.2
(c)
'
16o
(d)
0.25 0.2 0.15
0.1 0.1
0.05
0.05
20
40
60
80
0
i00
20
60
40
80
i00
Fig. 2. Localized steady-state of the one-dimensional diffusive Haken model for a range of values of the diffusive coupling ~: (a) initial data with lattice site at i = 50 the most excited, (b) final state when c~ = 0.1, (c) final state when ~ = 0.5, and (d) final state when o~ = 1.0.
(a)
~--~--~2o
(b)
~ o
5 0.8
0.6
0.6
0,4
0.4 .2 .2
i0 "
50
30
20
20
.
~
40
10
30
50
Fig. 3. Localized steady-state of the two-dimensional diffusive Haken model for (a) ~ = 0.05, and (b) c~ = 0.097. Initial data consists of a strictly localized state at the centre of the lattice.
the lattice spacing a and assume that the diffusive c o u p l i n g constant scales as 1/a 2. W e thus consider ot -+ ot/a 2 w i t h ot taken to be a - i n d e p e n d e n t . Set x = ai and write qi = q (x). Taking the c o n t i n u u m limit a --+ 0 in Eq. (2.2) then yields the potential or "action":
+ -~
12
dxq(x) 2
,
(5.1)
w h e r e the integration d o m a i n is S ___ ~d. The timei n d e p e n d e n t states of the c o n t i n u u m m o d e l are n o w given by stationary solutions o f the potential V, and these satisfy the E u l e r - L a g r a n g e equations
vt~,~=f
dx[2(Vq(x))2 s
q(x)44
q(2)2]
3V 8V V ~ V q (x~ -- ~q ( x ) '
(5.2)
PC. Bressloff/Physica D 110 (1997) 195-208 where 3/3q(x) denotes Eqs. (5.1) and (5.2) give
a functional
o~VSq(x) = (2D[q] -- 1)q(x) - q ( x ) 3
derivative.
(5.3)
with
D[q] = f
dxq(x) 2.
(5.4)
s First, consider constant solutions. These are simply the continuous versions of the homogeneous states of the original discrete model (Section 2). The vacuum state is given trivially b y q(x) = O, D[q] = 0 and V[q, or] = 0. For a finite system with periodic boundary conditions, the dissipative state satisfies 1
q(x) = V[q, a] --
2a"2 - 1' £2
D[q]
presence of a diffusion term introduces a smoothness constraint so that the previous choice of discontinuous test functions is no longer appropriate. Instead, we shall consider normalized Gaussians of the form q(x) = qoe -c~21xl2, where Ix[ is the Euclidean norm on ~d and 1/o-2 determines the width of the Gaussian. Since q is radially symmetric, the potential (5.1) reduces to OO
V[q, or] = gd
2a'2~'
f
drr d-1
ot \ - & r )
o
1
1
- ~D[q] ÷ ~ D [ q ] 2,
(5.6)
where
S2 --
205
OO
(5.5)
8S2 -- 1
As in the discrete model, we restrict ourselves to the case of positive solutions. Note that £2 = fs dx determines the size of the system and we assume S-2 > 1/2 so that the dissipative state exists. In the thermodynamic limit S-2 - + co, the dissipative state becomes pointwise identical to the vacuum state but has lower energy V = - 1/8. It follows that the ground states of an infinite system can only be spatially structured (e.g. localized) if there exist states with energy V < - 1 / 8 . A major difference between the continuum and discrete models is that, in the absence of diffusion (o~ = 0), the potential V of the former is unbounded from below. This is a consequence of the fact that the continuum model can have states localized to a region whose support has arbitrarily small measure, whereas in the discrete model the smallest localized region is a single lattice site. Consider, as an example, the state q(x) = qo for all x 6 So C S and q ( x ) = 0 otherwise. Let q0 = 1 / V / - ~ where S-20 = fso dx so that D[q] = 1. Substitution into Eq. (5.1) with a = 0 then gives V = - 1 / 4 £ 2 0 with V - + - o c as £2o ~ 0. Thus, for ot = 0 there do not exist lowest energy states (ground states) that are global minima of the potential V since V is unbounded from below. One can use similar energy arguments to show that V remains unbounded for non-zero coupling o~. The
D[q] = Yd /
drrd-lq(r) 2
(5.7)
~t
o and Yd = drca/2/F(1 + d/2) is the surface area of a unit sphere in d dimensions. We choose q~ = a(20-2/rc) a/2 so that D = a. Substitution of the Gaussian function into (5.6) gives a
V(0-) = - - ~ + -~- - ~ -
+ ~0-2a.
(5.8)
It is clear from Eq. (5.8) that V - + - ~ as a - + oo for arbitrary coupling ~, provided that 0-d > 2~--2~d. (Note that such low energy states were not taken into account in the analysis of Ref. [22].) Although global minima of the continuum model do not exist, one can still ask whether or not there exist local minima of the potential with energy less than the dissipative state. From our investigation of the discrete model in Sections 3 and 4, we know that weakly localized stationary solutions distributed over many lattice sites exist in one dimension but not in higher dimensions. This is a consequence of the fact that localized states persist for large o~ only when d = 1. Since weakly localized states should be reasonably well approximated by a continuum version of the model, we expect the latter to possess localized states corresponding to local minima of the potential (5.1) for d = 1 but not for d > 1. Following Ref. [22], we shall establish the existence of such states
P.C. Bressloff/Physica D 110 (1997) 195-208
206
U(q)
for arbitrarily large values of a in the one-dimensional continuum model (d = 1) by finding explicit instantonlike solutions to the Euler-Lagrange equation (5.3). We shall proceed by solving the equation d2q t ~ x 2 ----(2D - 1)q(x) - q ( x ) 3
E>0
(5.9)
E=0
for fixed D using standard phase-plane analysis with D then determined self-consistently using Eq. (5.4). Integrating Eq. (5.9) with D fixed gives the "energy" equation
oe \ - ~ x ]
+ U ( q ( x ) ) = E,
(5.10)
Fig. 4. Undamped motion of a particle in the auxiliary potential U(q) for 1/2 < D _< 1: (a) periodic solution when E < 0 or E > 0, and (b) instanton solution when E = 0.
where U is the auxiliary potential
U(q) --
2 D - 1 q2 2
q4 + --. 4
(5.11)
We require candidate finite energy solutions to be square-integrable, f d x q ( x ) 2 < ~ , so that D[q] < ec. This implies that q ( x ) -+ 0 as x --+ -t-oc; such finite energy configurations will be called instantons. Eq. (5.10) formally describes an undamped particle of energy E moving in a potential U with q re-interpreted as a particle "position" and x as a continuous "time". A n instanton configuration then corresponds to a homoclinic orbit of the particle through the fixed point at q = 0. For D < 1/2 the potential U(q) is monostable with a single minimum at q = 0. All nonhomogeneous solutions are periodic functions of x and are thus not square-integrable. On the other hand, when D > 1/2 the potential U(q) is bistable and possesses an instanton solution when E = 0 (see Fig. 4). Setting E = 0 in Eq. (5.10) and integrating with respect to x yields the following analytical expression for the instanton (centred at x = 0):
q ( x ) = q o F c o s h ( q°~x~]] -1 . L \~/2oe/J
(5.12)
The amplitude of the instanton is q0 = ~/(2D -- 1)2. Substituting Eq. (5.12) into (5.4) leads to the selfconsistency condition for D D = 4~/e~(2D - 1).
(5.13)
This has real solutions provided that e~ > 6~ = 1/16. Keeping only the lower energy solution, we find that the amplitude of the instanton as a function of the coupling e~ is q0(cQ = 2 ~ 2 ~ ( 1
- ~/1 - & / a ) .
(5.14)
The associated potential is V ( a ) = -~/2a~q0 + 4oeqg
~ ~ - q 0 3-
(5.15)
V is a monotonically increasing function of o~ with V(60 = - 1 / 6 and lim~_+~ V(eQ = - 1 / 8 [22]. We deduce that the one-dimensional continuum model has a localized stationary state of the form (5.14) for all values of the diffusive coupling a such that o~ > 6e. We interpret these states as continuous versions of the weakly localized solutions found in the discrete onedimensional model. The analysis of the Euler-Lagrange Eq. (5.3) is more difficult in higher dimensions (d > 1). In particular, it does not appear possible to generate explicit solutions. However, there is good evidence to suggest that there are no stationary solutions of the continuous model in more than one dimension with energy lower than the dissipative state. We shall show this for a restricted class o f solutions (see Ref. [22] for more details). Given the various symmetries of the potential V in Eq. (5.1), a reasonable assumption concerning a candidate low energy state is that it has radial symmetry, q(x) = q(r), where r = Ix[ is the radial
P.C. Bressloff/Physica D 110 (1997) 195-208
distance from the origin. Under such an assumption, one reduces the search to that of finding local minima of the potential (5.6). Such minima satisfy the EulerLagrange equation
Vd2q d-ldq]
o~ L - ~ r2 +
r
drr
dU
--
dq "
(5.16)
Eq. (5.18) formally describes the dynamics of a particle moving in a potential U in the presence of damping, with q once again interpreted as the particle position and r now acting as the time variable. Substituting the energy equation (5.11) into (5.18) shows that the energy monotonically decreases along a particle trajectory according to
dE dr
d--1 ( dq~ 2 -- --~
r
\-~-rJ
.
(5.17)
The right-hand side vanishes in the one-dimensional case (zero damping). Since the energy of an instanton solution is expected to increase with the number of turning points or nodes (see Fig. 5), the search for low energy local minima reduces to the question whether or not the unique lowest order self-consistent instanton solution has energy lower than the dissipative state. Numerical calculations carried out in Ref. [22] show that such solutions in fact have energy higher than the dissipative state. This is consistent with our findings for the discrete model, namely that there are no weakly localized ground states (see Fig. 3). U(q)
q21"-" . . . . . \
........
~1
........ I- - ' ~ . . . . .
" t ql -..~ q0 q
Fig. 5, Damped motion of a particle in the auxiliary potential U(q) for 1/2 < D < 1: instanton solutions for different starting points q0, ql, q2 are shown. The lowest energy solution is the one with the m i n i m u m number of turning points.
207
6. Discussion
The analysis of the weak-coupling limit (Section 3) can be extended to more general choices of coupling than discussed in this paper. This includes the nonstandard diffusive coupling of Ref. [22], for which Eq. (3.4) is replaced by g[Q]i
= ~--~(qj - qi) 2n+l
(6.1)
(Jli)
for n = 1, 2 . . . . One can also consider long-range interactions provided that the strength of the coupling decays exponentially with distance on the lattice, for example
K[Qli = Z y d ( i , j ) ( q j jeF
_
qi)
(6.2)
such that 0 < g < 1. This raises an interesting question concerning the persistence of localized ground states away from the weak-coupling limit. It has been established elsewhere that non-standard diffusion increases the tendency for localization with weakly localized ground states occurring in more than one dimension [22]. It would be very interesting to carry out a more detailed comparison of different forms of interaction to understand, in more general terms, the conditions necessary for localization. Another issue we hope to explore elsewhere is the dependence of the basin of attraction of a localized stationary state on the coupling. Finally, the method of uniform continuation from the uncoupled limit (Section 3) is becoming an increasingly powerful tool for studying a wide range of network dynamical systems. As mentioned in Section 1, an important application is the analysis of discrete breathers in systems of coupled non-linear oscillators [3]. We briefly indicate here how the analysis of the diffusive Haken model for fixed D (Theorem 1), can be used to prove the existence of discrete breathers for the discrete non-linear Schr6dinger equation (NLS) on a d-dimensional lattice: i
d~Pi dt
= 1/till~til2 -~o/~--~(~tj (i,j)
- ~/ti) .
(6.3)
P.C. Bressloff/Physica D 110 (1997) 195-208
208
F o l l o w i n g M a c k a y and A u b r y [3], we change variables b y going to a rotating frame, lPi (t) = q~i(t)e -'°~t
(6.4)
so that dq~i i dt = -co,hi q-q~ilq~il 2 q-oe Z ( ~ b j - ~bi). (j[i}
(6.5)
Eq. (6.5) m a y be formulated in terms of a H a m i l t o n i a n system b y introducing the conjugate variables Pi = -~(qSi) and qi = ~(q~i); the associated H a m i l t o n i a n is
e _[_q2)2 H(q, p) = c°~--~ (p2 + q { ) + 1 Z (Pi i~F iEF O[ Z [ ( p j -- pi) 2 + (qj -- qi)2].
+g
(i,j} (6.6) T i m e - i n d e p e n d e n t solutions of Eq. (6.5) in the case of zero coupling (c~ = 0) satisfy ~bi = 0 or IqSi12 = w. Suppose we break the degeneracy on the circle b y restricting attention to real-valued 4~i's. Stationary solutions of (6.5) then satisfy q~3 __ O)~bi _[_ 0/ ~
(~bj -- ~i) : 0
(6.7)
(jli) and are given b y the configurations 4~i = 0 or 4~i = :t=v~. E q u a t i o n is identical in form to Eq. (3.1) with co = 2 D - 1 for constant D. Therefore, we can immediately apply the implicit function theorem along similar lines to T h e o r e m 1 of Section 3: all stationary solutions of the t i m e - i n d e p e n d e n t NLS at ~ = 0 have locally u n i q u e continuations for sufficiently small oe. M o v i n g back to the original non-rotating frame shows that these solutions are time periodic, spatially localized oscillations - discrete breathers. For a m u c h more general discussion of discrete breathers in the weakcoupling limit see [3,7].
References [1] S. Takeno, K. Kisoda and M. King, Prog. Theor. Phys• Suppl. 94 (1988) 242-269. [2] D.K. Campbell and M. Peyrard, in: Chaos, ed. D.K. Campbell (American Institute of Physics, 1990) 305-334. [3] R.S. Mackay and S. Aubry, Nonlinearity 7 (1994) 16231643. [4] S. Flach, Phys. Rev. E 51 (1995) 1503-1507. [5] D. Bambusi, Nonlinearity 9 (1996) 433-457. [6] T. Dauxois, M. Peyrard and A.R. Bishop, Physica D 57 (1992) 267-282• [7] J.-A. Sepulchre and R.S. Mackay, Nonlinearity 10 (1997) 679-713. [8] S. Auhry, Physica D 86 (1995) 284296. [9l R.S. Mackay, Dynamics of networks: features which persist from the uncoupled limit, in: Stochastic and Spatial Structures of Dynamical Systems, eds. S.J. van Strein and S.M. Verduyn Lunel (North-Holland, Amsterdam, 1996). [10] S. Aubry and G. Abramovici, Physica D 43 (1990) 199219. [11] S. Aubry, G. Abramovici and J.-L. Raimbault, J. Stat. Phys. 67 (1992) 675-780. [12] C. Baesens and R.S. Mackay, Nonlinearity 7 (1994) 59-84. [13] R.S. Mackay and J.-A. Sepulchre, Physica D 82 (1996) 243-254. [14] RC. Bressloff and G. Rowlands, Exact travelling wave solutions of an "integrable" discrete reaction--diffusion equation, Physica D 106 (1997) 255-269. [15] N.V. Swindale, Network 7 (1996) 161-247. [16] N. Suga and P.H. Jen, Science 194 (1976) 542-544. [17] A.P. Georgopoulos, M. Taira and A. Lukashin, Science 260 (1993) 47-52. [18] C.H. vonder Marlsburg and D.J. Willshaw, Exp. Brain Res. Suppl. 1 (1976) 463-469. [19] T. Kohonen, Self-Organization and Associative Memory (Springer, Heidelberg, 1984). [20] K. Obermayer, H. Ritter and K. Schulten, Proc. Nat. Acad. Sci. USA 87 (1990) 8345-8349. [21] H. Haken, Synergetic Computers and Cognition (Springer, Berlin, 1991) [22] M. Schmutz and W. Banzhaf, Phys. Rev. A 45 (1992) 4132-4145. [23] P.C. Bressloff, Phys. Rev. Lett. 75 (1995) 962-965. [24] E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1989) [25] C. Baesens and R.S. Mackay, Finite coherence length for invertible linear maps on networks with exponentially decaying coupling (1996), preprint.
Physica D 110 (1997) 195-208
A self-organizing network in the weak-coupling limit Paul C. Bressloff Department of Mathematical Sciences, Loughborough University, Loughborough, Leics. LEl l 3TU, UK Received 5 November 1996; accepted 28 May 1997 Communicated by A.C. Newell
Abstract We prove the existence of spatially localized ground states of the diffusive Haken model. This model describes a selforganizing network whose elements are arranged on a d-dimensional lattice with short-range diffusive coupling. The network evolves according to a competitive gradient dynamics in which the effects of diffusion are counteracted by a localizing potential that incorporates an additional global coupling term. In the absence of diffusive coupling, the ground states of the system are strictly localized, i.e. only one lattice site is excited. For sufficiently small non-zero diffusive coupling c~, it is shown analytically that localized ground states persist in the network with the excitations exponentially decaying in space. Numerical results establish that localization occurs for arbitrary values of c~ in one dimension but vanishes beyond a critical coupling C~c(d), when d > 1. The one-dimensional localized states are interpreted in terms of instanton solutions of a continuum version of the model.
1. Introduction The study of spatially localized states i n extended systems is an area of great current interest. Much of this interest has focused on spatially localized periodic oscillations in networks of coupled non-linear oscillators - so-called discrete breathers. The existence of these discrete breathers has been demonstrated both numerically [1,2] and analytically [3-5]. A number of applications of such localized structures have been proposed including D N A dynamics in molecular biology [6]. One method for proving the existence o f discrete breathers in weakly coupled systems is based on the uniform continuation o f periodic solutions constructed in the uncoupled limit [3,7] (als o known as the anti-integrable or anti-continuum limit). More generally, investigating how certain features of network dynamics persist from the uncoupled limit has become a
powerful method for studying weakly coupled systems [8,9]. Examples include the study of ground state systems in condensed matter physics [10-12] and multistability and wave propagation failure in networks of bistable elements [ 13,14]. A distinct but very important example of localized structures in networks is found in neurobiology. In many regions of the cortex, groups of adjacent neurons appear to form higher functional units that serve to analyse some particular stimulus feature such as the orientation of an edge of an image or velocity of movement (see [15] and references therein). The association of groups of neurons with local properties of visual images varies in a regular way with the location of the neurons in the visual cortex. Thus, a socalled topographic map of various features of an image is built up. Another example o f such a phenomenon is found in the auditory cortex of the bat where a
0167-2789/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S0167-2789(97)00129-2
196
PC. Bressloff/ Physica D 110 (1997) 195-208
topographic map of signal amplitude versus pitch is formed. This generates a frequency spectrogram of an auditory signal. In addition, bats possess maps representing the time difference between two acoustic events, which plays an important role in the sonar orientation of the animal [16]. Topographic maps are not limited to sensory regions but also occur in the motor cortex. While sensory maps generate spatially localized regions of excitation whose location represents the signal features being analysed, motor maps create from spatially localized excitations an activity pattern in space and time that triggers a particular movement [171. Neural network models of the formation of coherent structures in brain function generally involve two aspects: (i) a selection mechanism for determining the centre of a localized excitation in response to an input, and (ii) an interaction mechanism that spreads the response over a neighbouring region of the network leading to the formation of a topographic map. A network typically learns the appropriate form of interaction by an adaptive process [19,20]. The selection mechanism, on the other hand, usually involves a sequential search for the optimal response. Exploiting certain similarities between pattern formation in synergetic systems and pattern recognition, Haken [21] has constructed a simple neural network model that implements this selection process using a form of competitive gradient dynamics. (Such a model does not itself have a direct biological interpretation, but is rather a caricature of more realistic models.) The ground states of the system consist of strictly localized states in which only one neuron is excited and the remainder quiescent. In other words, the network dynamically realises a winnertake-all strategy. An interesting extension of this type of network is the diffusive Haken model in which the introduction of a diffusive interaction between the neurons leads to a delocalization of the original model's ground states [22]. When there exists a balance between the effects of diffusion and localization, it is possible to obtain new ground states that are localized excitations (or bubbles) distributed over many neurons. Such ground states are more robust than those of the simple Haken model and also better reflect the
kind of coherent structures found in neurobiological systems. One of the major claims of Ref. [22] was that a network arranged on a d-dimensional square lattice with standard nearest-neighbour diffusive coupling possesses localized ground states when d = 1 but not for d > 1, even for arbitrarily small diffusive coupling. This claim was based on numerical simulations of the lattice model together with some variational calculations of a continuum model obtained in the limit of large diffusive coupling. In this paper, we prove analytically that such a claim is false: the diffusive Haken model supports localized ground states in any finite dimension provided that the diffush:e coupling is sufficiently small. (Such a result also holds for more general choices of coupling provided that the strength of coupling decays exponentially with distance on the lattice.) Our approach is based on a uniform continuation from the case of zero diffusive coupling. The structure of the paper is as follows. In Section 2, we introduce the diffusive Haken model and describe the ground states of the system in the absence of diffusive coupling. The weak coupling limit is then analysed in Section 3, expanding upon our previous work in Ref. [23]. Numerical results are presented in Section 4, which establish that localization occurs for arbitrary values of a in one dimension but vanishes beyond a critical coupling otc(d), when d > 1. Finally, in Section 5, we show how the one-dimensional localized states can be interpreted in terms of instanton solutions of a continuum version of the diffusive Haken model.
2. Diffusive Haken model Consider a network of N d elements arranged on a d-dimensional lattice F . Denote the state of the ith element by qi c • with i ~ / ' . In the diffusive Haken model, the state of the network at time t, Q(t) = {qi(t), i ~ F}, evolves according to the gradient dynamical rule [22]: q~ -
OV Oqi
(2.1)
197
P.C. Bressloff/Physica D 110 (1997) 195-208
for which the lattice structure does not play a role. Eq. (2.4) then becomes
with a potential V of the form
V [ Q , a ] = - ~oe Z ( q j
1 D [Q] - qi) 2 - -~
qi = q3 + (1 -- 2D)qi.
(i,j) + -1D[Q]
2
1
4
(2.2)
i
where (i, j ) denotes neighbour pairs and
D[Q] = Z
summation
over
q.2.
nearest-
(2.3)
The first term on the right-hand side of Eq. (2.2) represents a nearest-neighbour diffusive interaction with coupling constant ot whilst the remaining terms correspond to the localizing potential introduced by Haken [21]. Substituting Eq. (2.2) into (2.1) gives the equation of motion:
(]i ~ Ol ~-~[qj -- qi] -}- (1 -- 2 D [ q ] + q2)qi,
(2.4)
(jli}
where ( j [i) denotes summation over all nearest neighbours j of i. Eq. (2.4) is invariant under the transformation Q ~ - Q . Moreover, qi(t) >_ 0 for all t > 0 and i ~ F if qi (0) > 0 for all i E F . For, suppose that qi (t) -~ 0 and qj (t) > 0 for all j ¢ i. Setting qi = 0 on the right-hand side o f Eq. (2.4) shows that gli (t) > O. That is, qi cannot cross over to the negative real axis. Since we are interested in finite energy states of the network, we shall also assume that D[Q] < ec. In other words, each state Q is taken to be squaresummable. Hence, we can restrict our discussion to solutions in the domain 7-{-----[QENMI
IQl2
The equilibria of Eq. (2.5), which we denote by 0 , satisfy qi = 0 o r qi = 2 % / ~ -- 1. Hence, the stationary states can be divided into M + 1 classes determined by the number m of excited sites [22]. For a given m, D = m / ( 2 m - 1) and the corresponding potential is V (m) = _ (m/4) (2m - 1 ) - 1. We shall establish below that the stationary states m ---- 1 are stable, whereas all other stationary states are either unstable (m = 0) or saddle points (m > 1). For m = 1 there exists a single excited site, i0 say, such that qi = (~io,i. Moreover, D = 1 and V (1) = - 1 / 4 . This is one of the N strictly localized ground states of the network. There are two homogeneous stationary states given by the vacuum state m = 0 and the dissipative state m -----M. The former satisfies qi = 0 for all i and V (°) = 0 and the latter has qi = 1 / 4 ~ - ~ - - 1 for all i and V (M) = --M/(8M4). In the large M limit, the dissipative state becomes pointwise identical to the vacuum state but has lower energy, V (~) = - 1 / 8 . Also note that for an infinite lattice the dissipative state is marginally stable. In order to determine the stability o f a stationary state ~), we need to consider the eigenvalues )~ of the Jacobian
Jij --
0 2 V 0 = 6 e~=0 • Oqi Oqj
(2.6)
A stationary state will be stable if, and only if, all the eigenvalues are strictly negative. Substituting Eq. (2.2) into (2.6) gives Jij -~ --4qiqj,
where M = N d and IQI2 = x/~//q?, is the 12 norm on the vector space o f network states 7-/. The network converges to one of the stationary states of the potential V, i.e., OV/Oqi = 0 for all i ~ F . (Note that V is bounded from below.) Such a state m a y be obtained by setting qi = 0 in Eq. (2.4), solving the resulting time-independent equation for fixed D and then determining D self-consistently using Eq. (2.3). This procedure can be carried out explicitly in the case of zero diffusive coupling (~ = 0)
(2.5)
i ¢ j,
Jii = _ ~ 2 + 1 -- 2 0 .
(2.7) (2.8)
In the case m = 0, the matrix J reduces to the identity matrix so that )v = 1 is M - f o l d degenerate and hence the vacuum state is unstable. On the other hand, when m = 1 the matrix J is diagonal with an ( M -- 1)-fold degenerate eigenvalue ;v = --1 and a non-degenerate eigenvalue ), = - 2 ; singly localized states are thus stable. Finally, consider the case m > 1. By an appropriate re-ordering of the matrix J , it is simple to show
P.C. Bressloff/Physica D 110 (1997) 195-208
198
that the stability of the system will be determined by the solutions )v of the m x m determinant [21] a - £ 2a 2a
2a a -- ~ 2a
2a
2a
2a 2a a --
2a
... ... ...
...
2a 2a 2a
= 0,
(2.9)
a--)~
where a = - 2 / ( 2 m - 1). One finds that there is an (m - 1)-fold degenerate positive eigenvalue )v = - a and a non-degenerate negative eigenvalue )~ ---- (2m 1)a. (The remaining M - m eigenvalues of J satisfy )~ = a/2.) Hence, multiply localized states are saddles. We conclude that, in the case of zero diffusive coupling, the ground states of the system consist of strictly localized states in which only one site is excited and the remainder quiescent; the particular ground state selected depends on the initial data a n d / o r additional applied inputs. In other words, the network dynamically realises a winner-take-all strategy. Now suppose that the diffusive interaction between neighbouring sites is switched on. This leads to a delocalization of the original ground states [22]. For sufficiently small coupling o~, there exists a balance between the effects of diffusion and localization so that it is possible to obtain new ground states that are localized excitations (or lattice instantons) distributed over many lattice sites. A simple heuristic argument for the existence of localized ground states proceeds as follows [23]. We shall assume for concreteness that the network is infinite in extent. Suppose that the initial state of the network is one of the strictly localized states qi (0) = (~i,io. Substitution into Eq. (2.2) then shows that the potential at t = 0 satisfies V < - 1 / 8 foroe < 1/8d. Since d V / d t <_ 0 f o r all t, the final state cannot be the dissipative state. Moreover, we expect this final state to be localized since D < oc for finite values of V so that the equilibrium lattice configuration must decay sufficiently fast with distance on the lattice from i0. Although the above energy argument supports the existence o f localized ground states, it is still necessary to establish their relationship to the ground states at a = 0, their stability properties, how they decay with spatial location on the lattice and also the struc-
tural stability of the system. A l l of these issues can be tackled in terms of a uniform continuation from the uncoupled limit as shown in Section 3. The analytical results for small oe will be supplemented by numerical simulations in Section 4, where the behaviour of the system for larger values of the diffusive coupling will be studied. We shall find that localized states persist for all values of oe in one dimension but vanish beyond a critical value e~c(d) for d > 1.
3. Weak-coupling limit 3.1. Existence of localized states Stationary s t a t e s of the diffusive Haken model satisfy (1 - 2D)qi + q3 + ot ~_,(qj - qi) = O.
(3.1)
(jli)
For the moment, we shall consider both ot and D as free parameters; later on D will be determined selfconsistently using Eq. (2.3). In terms of the network state Q ~ 7-/, Eq. (3.1) can be rewritten in the form
F ( Q , D) + otK(Q) =-- G ( Q , ~, D) = 0,
(3.2)
where
[F(Q, D)]i = q3 ÷ (1 - 2D)qi,
(3.3)
[K(Q)]i = ~_a(qj - qi).
(3.4)
(j[i)
The condition for an equilibrium is then
G ( Q , a, D) ----0,
G
"
7-[ × R 2 ~
~'~.
(3.5)
Eq. (3.5) is formally very similar to the coupled system of bistable elements studied by M a c k a y and Sepulchre [13]; such a system would be obtained under the transformation F ( Q , D) ---> - F ( Q , D). The stability properties of the system for fixed D are very different, however, from the full Haken model with D determined self-consistently by Eq. (2.3). For a given D = Do, Do > 1/2, and o~ = 0 the equilibria of Eq. (3.5) satisfy qi = 0 or c]i = 4 - ~ - 1 (if negative solutions are included). Denote the Jacobian OG/OQ by 6G. Since [6G(~), 0, Do)]i j -= (]i,j)v i with
199
PC. Bressloff/Physica D 110 (1997) 195-208 )~i ~ - - ( 2 D 0 - 1 ) if~i = 0 andZi = 2(2D0--1) ifc~i 7~ 0, 3G is invertible at the stationary point (~), 0, Do). Hence, one can use the IFT to prove the existence of a local continuation of each Q for sufficiently small ot.
Theorem 1. There exists oto, 8o > 0 such that for Iotl < oto and [D -- Dol < ao there is a locally unique continuation Q(ot, D) with G(Q(ot, D), ot, D) = 0
0Q OD
-- [ r G ( Q , ot, D ) I - I K ( Q ) ,
(3.10)
with D(0) = 1 and qi(O, 1) = 8i,io. Differentiate Eq. (3.10) with respect to ot:
dD dot -
2(Q]OQ/Oot) 1 - 2(QlOQ/OD)"
(3.11)
(QIQ') = ~'~qiq~
(3.12)
(3.7) -- 218G(Q, ot, D ) ] - I Q .
Hence, the solution Q(ot, D) is differentiable with respect to ot and D as long as 8G has a bounded inverse. Recall that, given a norm [ • [ on a vector space ~ , a linear operator L acting on 7-/ is bounded if there exits some real constant c such that
[LQI~clQI
sup I t O l . Q~7~ IQI=I
(3.8)
Later on it will be useful to consider both the 12 norm introduced previously and the sup norm on 7-( defined by I O l ~ = sup Iqi[. i~F
such that 7-{ becomes a Hilbert space. Substituting Eq. (3.7) into (3.11) gives
dD 2(QISG-1]K[O]) dot = 4(QISG-1IQ) - 1"
(3.9)
We shall denote the corresponding norms on the space of bounded linear operators by ]1 • 112 and II • I1, respectively. We now substitute the solution Q (ot, D) of (3.5) into the self-consistency condition (2.3) and apply the IFT a second time with ~) taken to be a strictly localized state.
(3.13)
At the stationary point (Q, ot, D) = ( 0 , 0, 1) we have dD = -2d, dot
forall Q 6 7 - {
We can then define a corresponding norm II - II on the space of bounded linear operators L: 7-[ --+ 7-{ according to Iltll =
q/2(ot, O(ot)) = D(ot) i
We have introduced an inner product on the vector space 7-[ according to
Differentiating Eq. (3.6) shows that
aot
Z
(3.6)
and Q(0, Do) = Q where Q is a given equilibrium.
aQ
Theorem 2. There exists otl > 0 such that, for Iotl < there is a local continuation D(ot) and Q(ot) = Q(ot, D(ot)) E 7-( such that G(Q(ot), ot, D(ot)) = 0 and
otl,
(3.14)
where d is the dimension of the lattice. It follows from Eqs. (3.7) and (3.13) and the relation
dQ dot
----
OQ Oot
OQ dD OD dot
+ - - - -
(3.15)
that the solution Q (ot) remains differentiable provided that (i) 8G has a bounded inverse, (ii) 4(Q[3G-1IQ) > 1. Condition (ii) can be written in a more useful form using some basic properties of bounded symmetric linear operators on the Hilbert space ~ [24]: II t 112~11 t II, inf I(QILQ)r Q # 0 (QIQ) -
(3.16)
1 I[ L - 1 I1~
(3.17)
Setting L = 8G -1 we have
I(QISG-I[Q)I > {OlO______~). II 8 6 II
(3.18)
PC. Bressloff/Physica D 110 (1997) 195-208
200
Hence, condition (ii) becomes
where q+ are the critical points of f , i.e., f ( q + ) and
(ii') II 3G II< 4D. We shall now use conditions (i) and (if) to estimate a lower bound Oil for the existence of local continuations Q(~) of the strictly localized states () present in the case of zero coupling. We shall then determine whether or not these localized states are candidate ground states, i.e. V < - 1 / 8 in the case of an infinite network, and also whether or not they are stable.
( 2 D - 1 ~ 3/2 f ( q + ) = qz2 k, 3 ]
q~_ = -t-~/(2D -- 1)/3y,
(3.20) (see Fig. 1). It is easy to establish that qi ( ~ ) E [0, q + ) for all i 5~ i0 and qi(oz) E (q+, qmax), where qmax is the positive root of the equation f ( q ) = f ( q _ ) , i.e., qmax = 2q+. This follows from Fig. 1 and the result f ( v ) < 0 and f ( w ) > O, where v = infi qi(o~) and w = supi qi(oe). Hence, Eq. (3.19) holds if
3.2. Estimation of oq
2D- 1 o~ < - 6d
We first consider condition (i), namely whether or not ~ G has a bounded inverse. For the moment assume that D is fixed with 1/2 < D < 1. (Eq. (3.14) shows that D < 1, at least for sufficiently small a. The lower bound for D occurs for the dissipative state.) Suppose that each lattice site i is to be assigned a value qi (o0 --- 0, which is a continuation of the strictly localized state qi (~i,io~ 1. This can be achieved uniquely, by taking qi ( ~ ) tO be the middle root (if i 7~ i0) or the largest root (if i = i0) of the equation f ( q i ) = q3 + (1 - 2D)qi = -or Y~(jli)(qJ - qi), provided that =
= 0
(3.21)
Condition (3.21) ensures that 8G is invertible. To show this, separate out the diagonal and off-diagonal parts of 3G by writing ~G(c~) -----A + oiL1, where
A i j = ( f ' ( q i ) - 2doe)~ij,
(Zl)ij
=~
~j,k.
(kli)
(3.22)
-
Then II~a(c~) -1 II <
1 IIA-111-1 - ~llZill'
(3.23)
provided ~IILi II < 11-4-1 I1-1. Since e~sup ~ ( q j ( o t )
-- qi(ol))
(3.19)
< If(q+)[,
i~F (jli)
IlL1 II = sup ~
ILl lij = 2d,
iEF'j~ F
f(q)
IIA - l II = sup [f'(qi) - 2du1-1, icF it follows that ~G(Q(o0, e) is invertible when
[ - 2do~ + ft(qi(ot))l > 2dot
I
I
q m
,
Fig. 1. Cubic non-linearity f ( q ) = q3 + (1 -- 2D)q for fixed D, 1/2 < D < 1. Critical points are q+ = - t - 2 4 ( 2 D - 1)/3 and qmax = 2q+.
(3.24)
for all i ~ F . Condition (3.24) is immediately seen to hold for all i 7~ i0 since qi (or) ~ [0, q+), so that f'(qi(ot)) < 0 (see Fig. 1). In the case i = i0, we have qio(Ot) E (q+, qmax), SO that f'(qio(Ot)) > f t ( q + ) = 2 ( 2 D - 1). Hence, Eq. (3.24) holds for i = i0, provided fr(q+(ot)) > 40td, which reduces to the condition ~ < (2D - 1)/2d. We now recall that D is itself dependent on oe. Using Eq. (3.14), we have that for oe << 1, D(ot) ~ 1 - 2 d o t . Substituting this approximation into
20l
P.C Bressloff/Physica D 110 (1997) 195-208
Eq. (3.21) then yields the result that 8G is invertible when
1 oe < tel -- 10d"
(3.25)
It is simple to show that condition (ii:) is also satisfied when (3.25) holds. For [ISGII ~ 4doe + sup(3qi(oe) 2 ÷ 1 - 2D) i 2
< 4doe + 3qmax + 1 - 2 D = 4doe + 3(2D -- 1). Condition (ii') requires that cz < (3 - 2 D ) / 4 d which is a weaker condition than (3.21) (and hence (3.25)) since 1/2 < D < 1. Note that our estimate (3.25) for the value of diffusive coupling below which localized states are proven to exist is close to the value 1 / 8 d obtained using the energy argument at the end of Section 2.
3.3. Energy and stability
The unique state Q(oe) obtained as a local continuation of the strictly localized state Q (Theorem 2) is in fact a candidate ground state. For Taylor expand the energy function V of Eq. (2.2) to first order in oe and use the fact that Q is a stationary point of V, O V / O Q = 0 at Q = (~: V[Q(oe), oe] ~ - ~ 1 + ~oe E ( ~ j
__ ~/i)2 = - ~ 1 + doe.
(i,j)
(3.26) Eq. (3.26) shows that (at least to this level of approximation) V < - 1 / 8 if oe < l / 4 d . Thus the localized state state Q(oe) with oe < Otl has lower energy than the dissipative state. In order to investigate the linear stability of Q(oe) we linearize equation (2.4) about Q(oe). Setting ui = qi -- qi (oe) and expanding to first-order in ui gives the linear equation tii = E [ S G ( ~ ) i j (J]i)
- 4qi (c~)q] (oe)luj,
(3.27)
where we have set G[Q(oe), oe, D(o0] = G(~). Hence, the condition for linear stability is that all eigenvalues of the matrix J(oe) = 3G(oe) - 4Q(oe) ® Q(oe)
(3.28)
are bounded below zero. The matrix J(0) is given by Eqs. (2.7) and (2.8) with qi 3i,io. Its spectrum consists of a single non-degenerate eigenvalue )~ = - 2 and a single degenerate eigenvalue )~ = - 1 . Since the spectrum of J(oe) is real, it follows that Q(oe) will remain linearly stable as ot increases from zero provided that J (oe) remains invertible. We shall now prove that conditions (i) and (ii) of Section 3.1 ensure that J(oe) is invertible. First write =
j(oe)-I ___[3G(ot) - 4Q(oe) ® Q(oe)]-i = 3G(oe) -1 T(oe) -1,
(3.29)
where T(oe) = I -- 4[Q(o0 ® Q(oe)]3G(oe) -1
(3.30)
and I is the identity matrix. Suppose that T(ot) -~ exists and for an arbitrary P c 7-[ consider the linear equation R = T(oe)-lP.
(3.31)
Eqs. (3.30) and (3.31) imply P = T(oe)R = R -- 4Q(oO(Q(oe)13G(ot) -1 [R).
(3.32) Eq. (3.32) shows that the unknown vector R must be of the form R = P + ~cQ for some real constant x. Substituting this into (3.32) allows us to determine ~c so that 4(Q(oe)16G(oe) -1 [P) K = 1 -- 4(Q(oe)lSG(oe) l[Q(oe))'
(3.33)
Since P is arbitrary, we conclude from (3.33) that T (cO and hence J(oe) have bounded inverse provided that conditions (i) and (ii) hold. Finally, note that substituting Eqs. (3.7) and (3.13) into (3.15) and comparing the result with Eqs. (3.31)-(3.33), when P = K [ Q ] , yields d Q = _ j (oe)_i K[Q(oe)].
doe
(3.34)
P.C. Bressloff/Physica D 110 (1997) 195-208
202
This equation also follows from applying the IFT directly to Eq. (3.1) after substituting for D using Eq. (2.3). One could then work backwards to derive Conditions (i) and (ii).
for any z ~, [1, l/L] such that fl(z) < [laG -1[[-1, where
3.4. Localization length
From the decomposition of 6G given by Eq. (3.22), we see that
We shall now show that the continuation Q(ot) of a strictly localized state is exponentially localized and determine an upper bound for its localization length. We shall follow the analysis of Baesens and Mackay [25], who treat the issue of exponentially localized network states within a general framework. For convenience, we consider an infinite lattice, F = N d, and take the distance d(i, j ) between sites i = (il . . . . . id) d and j = (jx . . . . . jd) to be d(i, j ) = ~ p = l lip - Jpl. Without loss of generality, the strictly localized state is assumed to be located at the origin of the lattice i=0. Suppose that Q (ot) is exponentially localized about the origin. In other words,
fi(z) = 2dot(z - 1)
Iqi(ot)l --< C(ot) Ld(i'O)
(3.35)
for some C(ot) and 0 < L < 1. Eq. (3.35) holds for ot = 0 by construction; one needs to verify that the form of Eq. (3.35) is preserved as ot increases from zero according to Eq. (3.34). From Eqs. (3.29)-(3.34), we have
dqi < ~_~ [aG~jlIIK[Q] j + Kqjl dot - j~U
(3.36)
with x given by Eq. (3.33) for P = K[Q]. Eq. (3.35) implies that
IK[Q]i + Kqil <_ ~(ot)£d(i,o),
(3.37)
where C(ot) ----C(ot)ld(£ + £-1 _ 2) + x[.
(3.38)
fi(z) = sup ~
[3Gijl(z d(i'j) - 1).
(3.40)
i~F j~ F
(3.41)
and from Eqs. (3.23) and (3.24), 1
IlaG -x II -< - -
K(ot)'
K(ot) = inf ]f'(qi) - 2dot[ - 2dot.
(3.42)
Substituting Eqs. (3.41) and (3.42) into (3.39) then gives
dqi
dC Ld(i,O)
(3,43)
~d -
for any £ < 1 such that fi(1/L) < K(ot) and dC C(ot) d ~ -----K(ot) - fi(1/L)"
(3.44)
We can determine a lower bound L* for the exponential decay rate by solving fi(1/L) = K(ot). Using Eq. (3.41)
K(ot)l_1 L* =
1 + 2dot _1
"
(3.45)
It follows that the state Q(ot) decays at least as fast as L*, and hence an upper bound for the localization length is l / I n ( I / L * ) . The amplitude C(ot) can also be calculated, if needed, using Eq. (3.44). However, this is complicated by the fact that h: itself depends on C(ot). Alternatively, we could use the fact that for small ot, D(ot) ~ 1 - 2dot. Now from Eqs. (2.3) and (3.35) we obtain
D[QI = Z
q] < C(ot)2~_~) 2d(i,0).
(3.46)
i~F
From Eqs. (3.36) and (3.37) and [25, Theorem 1], it can be shown that if the matrix 6G is invertible then
Furthermore,
~ i~F
< C(ot) 116G_1 i1_~_/~(z)
(3.39)
oo L2d(i'°) = 1 + N(L),
N(L) = ~ . / k f ( n ) L 2n,
n=l (3.47)
PC. Bressloff/Physica D 110 (1997) 195-208
where N ' ( n ) is the number of lattice sites at a distance of n from the central site. For example, N ' ( n ) = 2 in one dimension and N ' ( n ) = 4n in two dimensions. Thus )~2 N()~) = 1 - 2 ~ 4~2 N ( ) 0 -- (1
(d = 1), (3.48) (d ----2).
-- )~2)2
It follows that, for small ~, we can take C(o0 ~
~1
- 2dee
1 + N()~)
.
(3.49)
3.5. Structural stability
So far we have shown that for sufficiently small coupling ~ there exist stable localized states Q(oe) of the diffusive Haken model that are local continuations of the global ground states of the system with zero coupling. We would like to identify the states Q(ee) as the ground states of the coupled system. However, there still remains an important question: are additional equilibria created on adding a small coupling? In other words, is the uncoupled system structurally stable? The creation of new equilibria could, in principle, introduce a new ground state o f the system that is not a local continuation of one of the zero-coupling ground states. Although we do not wish to explore this issue in any detail here, we expect the system to be structurally stable. This is certainly the case for a finite-dimensional network where one can appeal to the fact that the system is uniformly hyperbolic [9].
203
the choice of boundary condition for sufficiently large N. Typical results for a one-dimensional network are shown in Fig. 2. The stationary state of a network of N = 100 lattice sites is displayed for a range of values of the diffusive coupling c~; oe = 0.1, 0.5, 1.0. The initial data is scattered in the range [0, 0.5] except for the lattice site at i = 50 which is chosen to have the value 0.6. It is clear from the figure that, for each value of oe, the stationary solution is localized around i = 50. Moreover, one finds that the energy is less than the dissipative state, V < - 1 / 8 . The value of o~ for which localized states persist is only limited by the size of the network since increasing increases the localization length so that boundary effects become increasingly important. The behaviour of the network as a function of oe differs considerably in higher dimensions. This is illustrated in Fig. 3 for a two-dimensional network consisting of 50 x 50 lattice sites. The initial data is taken to be a strictly localized state at the centre of the lattice. One finds that a localized stationary state now only occurs for small values of ~, oe < tic where eec 0.098. There is a sharp phase transition at ~c beyond which the network converges to the dissipative state. Another difference between one and two dimensions is that in the latter case localized ground states have relatively small basins of attraction whose size is very sensitive to the value of ee. For example, scattered initial data along the lines of Fig. 2 only converges to a localized state for values of ~e well below Oec. We shall consider such issues in more detail elsewhere. It would appear, therefore, that in two or more dimensions the delocalizing effects of diffusion dominate.
4. Numerical results 5. Continuum limit In order to explore the behaviour of the diffusive Haken model away from the weak-coupling limit, it is necessary to consider numerical solutions of Eqs. (2.3) and (2.4). We use a simple finite difference scheme, namely Euler's direct method. Note that applying nearest-neighbour coupling to boundary sites in a finite network corresponds to a zero-flux condition (the analogue of a Neumann boundary condition in continuum models); our results are not sensitive to
In Section 3, we proved that for sufficiently small coupling ~ there exist localized ground states of the diffusive Haken model in any finite dimension d. We also showed numerically in Section 4 that, for d > 1, there exists a critical coupling ~ec(d) beyond which localized ground states cease to exist. We shall now investigate to what extent these various results apply to a continuous version of the Haken model. Introduce
204
RC.
Bressloff/Physica D 110 (1997) 195-208 (a)
0.6
(b)
0.8 0.5 • • • ". • •. " •
" ..•
• • ° •
••°
•
.•" °° ° ° °• °°
"
0.6
0,4 0.3
0.4
,
0.2
.
•
,.o
•"
•
°
•'.
0.2 0.i • .
"°
20
.
40
60
•
,
80
".
2'0. . . .
i00
go
s'o
6'0'
'
0.25[ 0.2
(c)
'
16o
(d)
0.25 0.2 0.15
0.1 0.1
0.05
0.05
20
40
60
80
0
i00
20
60
40
80
i00
Fig. 2. Localized steady-state of the one-dimensional diffusive Haken model for a range of values of the diffusive coupling ~: (a) initial data with lattice site at i = 50 the most excited, (b) final state when c~ = 0.1, (c) final state when ~ = 0.5, and (d) final state when o~ = 1.0.
(a)
~--~--~2o
(b)
~ o
5 0.8
0.6
0.6
0,4
0.4 .2 .2
i0 "
50
30
20
20
.
~
40
10
30
50
Fig. 3. Localized steady-state of the two-dimensional diffusive Haken model for (a) ~ = 0.05, and (b) c~ = 0.097. Initial data consists of a strictly localized state at the centre of the lattice.
the lattice spacing a and assume that the diffusive c o u p l i n g constant scales as 1/a 2. W e thus consider ot -+ ot/a 2 w i t h ot taken to be a - i n d e p e n d e n t . Set x = ai and write qi = q (x). Taking the c o n t i n u u m limit a --+ 0 in Eq. (2.2) then yields the potential or "action":
+ -~
12
dxq(x) 2
,
(5.1)
w h e r e the integration d o m a i n is S ___ ~d. The timei n d e p e n d e n t states of the c o n t i n u u m m o d e l are n o w given by stationary solutions o f the potential V, and these satisfy the E u l e r - L a g r a n g e equations
vt~,~=f
dx[2(Vq(x))2 s
q(x)44
q(2)2]
3V 8V V ~ V q (x~ -- ~q ( x ) '
(5.2)
PC. Bressloff/Physica D 110 (1997) 195-208 where 3/3q(x) denotes Eqs. (5.1) and (5.2) give
a functional
o~VSq(x) = (2D[q] -- 1)q(x) - q ( x ) 3
derivative.
(5.3)
with
D[q] = f
dxq(x) 2.
(5.4)
s First, consider constant solutions. These are simply the continuous versions of the homogeneous states of the original discrete model (Section 2). The vacuum state is given trivially b y q(x) = O, D[q] = 0 and V[q, or] = 0. For a finite system with periodic boundary conditions, the dissipative state satisfies 1
q(x) = V[q, a] --
2a"2 - 1' £2
D[q]
presence of a diffusion term introduces a smoothness constraint so that the previous choice of discontinuous test functions is no longer appropriate. Instead, we shall consider normalized Gaussians of the form q(x) = qoe -c~21xl2, where Ix[ is the Euclidean norm on ~d and 1/o-2 determines the width of the Gaussian. Since q is radially symmetric, the potential (5.1) reduces to OO
V[q, or] = gd
2a'2~'
f
drr d-1
ot \ - & r )
o
1
1
- ~D[q] ÷ ~ D [ q ] 2,
(5.6)
where
S2 --
205
OO
(5.5)
8S2 -- 1
As in the discrete model, we restrict ourselves to the case of positive solutions. Note that £2 = fs dx determines the size of the system and we assume S-2 > 1/2 so that the dissipative state exists. In the thermodynamic limit S-2 - + co, the dissipative state becomes pointwise identical to the vacuum state but has lower energy V = - 1/8. It follows that the ground states of an infinite system can only be spatially structured (e.g. localized) if there exist states with energy V < - 1 / 8 . A major difference between the continuum and discrete models is that, in the absence of diffusion (o~ = 0), the potential V of the former is unbounded from below. This is a consequence of the fact that the continuum model can have states localized to a region whose support has arbitrarily small measure, whereas in the discrete model the smallest localized region is a single lattice site. Consider, as an example, the state q(x) = qo for all x 6 So C S and q ( x ) = 0 otherwise. Let q0 = 1 / V / - ~ where S-20 = fso dx so that D[q] = 1. Substitution into Eq. (5.1) with a = 0 then gives V = - 1 / 4 £ 2 0 with V - + - o c as £2o ~ 0. Thus, for ot = 0 there do not exist lowest energy states (ground states) that are global minima of the potential V since V is unbounded from below. One can use similar energy arguments to show that V remains unbounded for non-zero coupling o~. The
D[q] = Yd /
drrd-lq(r) 2
(5.7)
~t
o and Yd = drca/2/F(1 + d/2) is the surface area of a unit sphere in d dimensions. We choose q~ = a(20-2/rc) a/2 so that D = a. Substitution of the Gaussian function into (5.6) gives a
V(0-) = - - ~ + -~- - ~ -
+ ~0-2a.
(5.8)
It is clear from Eq. (5.8) that V - + - ~ as a - + oo for arbitrary coupling ~, provided that 0-d > 2~--2~d. (Note that such low energy states were not taken into account in the analysis of Ref. [22].) Although global minima of the continuum model do not exist, one can still ask whether or not there exist local minima of the potential with energy less than the dissipative state. From our investigation of the discrete model in Sections 3 and 4, we know that weakly localized stationary solutions distributed over many lattice sites exist in one dimension but not in higher dimensions. This is a consequence of the fact that localized states persist for large o~ only when d = 1. Since weakly localized states should be reasonably well approximated by a continuum version of the model, we expect the latter to possess localized states corresponding to local minima of the potential (5.1) for d = 1 but not for d > 1. Following Ref. [22], we shall establish the existence of such states
P.C. Bressloff/Physica D 110 (1997) 195-208
206
U(q)
for arbitrarily large values of a in the one-dimensional continuum model (d = 1) by finding explicit instantonlike solutions to the Euler-Lagrange equation (5.3). We shall proceed by solving the equation d2q t ~ x 2 ----(2D - 1)q(x) - q ( x ) 3
E>0
(5.9)
E=0
for fixed D using standard phase-plane analysis with D then determined self-consistently using Eq. (5.4). Integrating Eq. (5.9) with D fixed gives the "energy" equation
oe \ - ~ x ]
+ U ( q ( x ) ) = E,
(5.10)
Fig. 4. Undamped motion of a particle in the auxiliary potential U(q) for 1/2 < D _< 1: (a) periodic solution when E < 0 or E > 0, and (b) instanton solution when E = 0.
where U is the auxiliary potential
U(q) --
2 D - 1 q2 2
q4 + --. 4
(5.11)
We require candidate finite energy solutions to be square-integrable, f d x q ( x ) 2 < ~ , so that D[q] < ec. This implies that q ( x ) -+ 0 as x --+ -t-oc; such finite energy configurations will be called instantons. Eq. (5.10) formally describes an undamped particle of energy E moving in a potential U with q re-interpreted as a particle "position" and x as a continuous "time". A n instanton configuration then corresponds to a homoclinic orbit of the particle through the fixed point at q = 0. For D < 1/2 the potential U(q) is monostable with a single minimum at q = 0. All nonhomogeneous solutions are periodic functions of x and are thus not square-integrable. On the other hand, when D > 1/2 the potential U(q) is bistable and possesses an instanton solution when E = 0 (see Fig. 4). Setting E = 0 in Eq. (5.10) and integrating with respect to x yields the following analytical expression for the instanton (centred at x = 0):
q ( x ) = q o F c o s h ( q°~x~]] -1 . L \~/2oe/J
(5.12)
The amplitude of the instanton is q0 = ~/(2D -- 1)2. Substituting Eq. (5.12) into (5.4) leads to the selfconsistency condition for D D = 4~/e~(2D - 1).
(5.13)
This has real solutions provided that e~ > 6~ = 1/16. Keeping only the lower energy solution, we find that the amplitude of the instanton as a function of the coupling e~ is q0(cQ = 2 ~ 2 ~ ( 1
- ~/1 - & / a ) .
(5.14)
The associated potential is V ( a ) = -~/2a~q0 + 4oeqg
~ ~ - q 0 3-
(5.15)
V is a monotonically increasing function of o~ with V(60 = - 1 / 6 and lim~_+~ V(eQ = - 1 / 8 [22]. We deduce that the one-dimensional continuum model has a localized stationary state of the form (5.14) for all values of the diffusive coupling a such that o~ > 6e. We interpret these states as continuous versions of the weakly localized solutions found in the discrete onedimensional model. The analysis of the Euler-Lagrange Eq. (5.3) is more difficult in higher dimensions (d > 1). In particular, it does not appear possible to generate explicit solutions. However, there is good evidence to suggest that there are no stationary solutions of the continuous model in more than one dimension with energy lower than the dissipative state. We shall show this for a restricted class o f solutions (see Ref. [22] for more details). Given the various symmetries of the potential V in Eq. (5.1), a reasonable assumption concerning a candidate low energy state is that it has radial symmetry, q(x) = q(r), where r = Ix[ is the radial
P.C. Bressloff/Physica D 110 (1997) 195-208
distance from the origin. Under such an assumption, one reduces the search to that of finding local minima of the potential (5.6). Such minima satisfy the EulerLagrange equation
Vd2q d-ldq]
o~ L - ~ r2 +
r
drr
dU
--
dq "
(5.16)
Eq. (5.18) formally describes the dynamics of a particle moving in a potential U in the presence of damping, with q once again interpreted as the particle position and r now acting as the time variable. Substituting the energy equation (5.11) into (5.18) shows that the energy monotonically decreases along a particle trajectory according to
dE dr
d--1 ( dq~ 2 -- --~
r
\-~-rJ
.
(5.17)
The right-hand side vanishes in the one-dimensional case (zero damping). Since the energy of an instanton solution is expected to increase with the number of turning points or nodes (see Fig. 5), the search for low energy local minima reduces to the question whether or not the unique lowest order self-consistent instanton solution has energy lower than the dissipative state. Numerical calculations carried out in Ref. [22] show that such solutions in fact have energy higher than the dissipative state. This is consistent with our findings for the discrete model, namely that there are no weakly localized ground states (see Fig. 3). U(q)
q21"-" . . . . . \
........
~1
........ I- - ' ~ . . . . .
" t ql -..~ q0 q
Fig. 5, Damped motion of a particle in the auxiliary potential U(q) for 1/2 < D < 1: instanton solutions for different starting points q0, ql, q2 are shown. The lowest energy solution is the one with the m i n i m u m number of turning points.
207
6. Discussion
The analysis of the weak-coupling limit (Section 3) can be extended to more general choices of coupling than discussed in this paper. This includes the nonstandard diffusive coupling of Ref. [22], for which Eq. (3.4) is replaced by g[Q]i
= ~--~(qj - qi) 2n+l
(6.1)
(Jli)
for n = 1, 2 . . . . One can also consider long-range interactions provided that the strength of the coupling decays exponentially with distance on the lattice, for example
K[Qli = Z y d ( i , j ) ( q j jeF
_
qi)
(6.2)
such that 0 < g < 1. This raises an interesting question concerning the persistence of localized ground states away from the weak-coupling limit. It has been established elsewhere that non-standard diffusion increases the tendency for localization with weakly localized ground states occurring in more than one dimension [22]. It would be very interesting to carry out a more detailed comparison of different forms of interaction to understand, in more general terms, the conditions necessary for localization. Another issue we hope to explore elsewhere is the dependence of the basin of attraction of a localized stationary state on the coupling. Finally, the method of uniform continuation from the uncoupled limit (Section 3) is becoming an increasingly powerful tool for studying a wide range of network dynamical systems. As mentioned in Section 1, an important application is the analysis of discrete breathers in systems of coupled non-linear oscillators [3]. We briefly indicate here how the analysis of the diffusive Haken model for fixed D (Theorem 1), can be used to prove the existence of discrete breathers for the discrete non-linear Schr6dinger equation (NLS) on a d-dimensional lattice: i
d~Pi dt
= 1/till~til2 -~o/~--~(~tj (i,j)
- ~/ti) .
(6.3)
P.C. Bressloff/Physica D 110 (1997) 195-208
208
F o l l o w i n g M a c k a y and A u b r y [3], we change variables b y going to a rotating frame, lPi (t) = q~i(t)e -'°~t
(6.4)
so that dq~i i dt = -co,hi q-q~ilq~il 2 q-oe Z ( ~ b j - ~bi). (j[i}
(6.5)
Eq. (6.5) m a y be formulated in terms of a H a m i l t o n i a n system b y introducing the conjugate variables Pi = -~(qSi) and qi = ~(q~i); the associated H a m i l t o n i a n is
e _[_q2)2 H(q, p) = c°~--~ (p2 + q { ) + 1 Z (Pi i~F iEF O[ Z [ ( p j -- pi) 2 + (qj -- qi)2].
+g
(i,j} (6.6) T i m e - i n d e p e n d e n t solutions of Eq. (6.5) in the case of zero coupling (c~ = 0) satisfy ~bi = 0 or IqSi12 = w. Suppose we break the degeneracy on the circle b y restricting attention to real-valued 4~i's. Stationary solutions of (6.5) then satisfy q~3 __ O)~bi _[_ 0/ ~
(~bj -- ~i) : 0
(6.7)
(jli) and are given b y the configurations 4~i = 0 or 4~i = :t=v~. E q u a t i o n is identical in form to Eq. (3.1) with co = 2 D - 1 for constant D. Therefore, we can immediately apply the implicit function theorem along similar lines to T h e o r e m 1 of Section 3: all stationary solutions of the t i m e - i n d e p e n d e n t NLS at ~ = 0 have locally u n i q u e continuations for sufficiently small oe. M o v i n g back to the original non-rotating frame shows that these solutions are time periodic, spatially localized oscillations - discrete breathers. For a m u c h more general discussion of discrete breathers in the weakcoupling limit see [3,7].
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