Coupled Dynamics of Fast Neurons and 
Slow Interactions 
A.C.C. Coolen R.W. Penney D. Sherrington 
Dept. of Physics - Theoretical Physics 
University of Oxford 
i Keble Road, Oxford OX1 3NP, U.K. 
Abstract 
A simple model of coupled dynamics of fast neurons and slow inter- 
actions, modelling self-organization in recurrent neural networks, 
leads naturally to an effective statistical mechanics characterized 
by a partition function which is an average over a replicated system. 
This is reminiscent of the replica trick used to study spin-glasses, 
but with the difference that the number of replicas has a physi- 
cal meaning as the ratio of two temperatures and can be varied 
throughout the whole range of real values. The model has inter- 
esting phase consequences as a function of varying this ratio and 
external stilnuli, and can be extended to a range of other models. 
1 
A SIMPLE MODEL WITH FAST DYNAMIC 
NEURONS AND SLOW DYNAMIC INTERACTIONS 
As the basic archetypal model we consider a system of Ising spin neurons rri C 
{-1, 1}, i C {1,..., N}, interacting via continuous-valued symmetric interactions, 
Jij, which themselves evolve in response to the states of the neurons. The neurons 
are taken to have a stochastic field-alignment dynamics which is fast compared with 
the evolution rate of the interactions Jij, such that on the time-scale of Jij-dynamics 
the neurons are effectively in equilibrium according to a Boltzmann distribution, 
P{.,} ({o'i}) cr exp [-H{.,} ({o'i})] 
(1) 
447 
448 Coolen, Penney, and Sherrington 
where 
- (2) 
<j 
and the subscript {Ji} indicates that the {Jj} are to be considered as quenched 
variables. In practice, several specific types of dynamics which obey detailed balance 
lead to the equilibrium distribution (1), such as a Markov process with single-spin 
flip Glauber dynamics [1]. The quantity  is an inverse temperature characterizing 
the stochastic gain. 
For the Jij dynamics we choose the form 
d 1 1 
rJij = {aiaj}{ao} - yJij + ?ij(t) (i < j) (3) 
where {..-){ao} refers to a thermodynamic average over the distribution (1) with 
the effectively instantaneous {Jj}, and ?j(t) is a stochastic Gaussian white noise 
of zero mean and correlation 
(rlij(t)rlkt(t')} - 2r/-l S(ij),(kl)5(t - t') 
The first term on the right-hand side of (3) is inspired by the Hebbian process in 
neural tissue in which synaptic efficacies are believed to grow locally in response to 
the simultaneous activity of pre- and post-synaptic neurons [2]. The second term 
acts to limit the magnitude of Zij; / is the characteristic inverse temperature of 
the interaction system. (A related interaction dynamics without the noise term, 
equivalent to/ = c, was introduced by Shinomoto [3]; the anti-Hebbian version of 
the above coupled dynamics was studied in layered systems by Jonker et al. [4, 5].) 
Substituting for Io'io'j) in terms of the distribution (1) enables us to re-write (3) as 
d c 
NrJii = -c2Ji-'7/ ( {Jii)) + x/rlii(t) (4) 
where the effective Hamiltonian 7/({Ji}) is given by 
1 N 
_1 lnZ; ({Jij}) + Y  Ji} (5) 
where Z ({Jij}) is the partition function associated with (2): 
({Jij})= Tr exp . 
{o'i} 
2 COUPLED SYSTEM IN THERMAL EQUILIBRIUM 
We now recognise (4) as having the form of a Langevin equation, so that the equilib- 
rium distribution of the interaction system is given by a Boltzmann form. Hence- 
forth, we concentrate on this equilibrium state which we can characterize by a 
partition function  and an associated 'free energy' /: 
/i< j ' [ 1~ I --/-lln2 
2 --- dJij [Z({Jij})] exp -ttN Ji} 10 = (6) 
<j 
Coupled Dynamics of Fast Neurons and Slow Interactions 449 
where n --//. We may use  as a generating functional to produce thermody- 
namic averages of state variables (I> ({rri}; {Jq}) in the combined system by adding 
suitable infinitesimal source terms to the neuron Itamiltonian (2): 
lira 
-0 0h 
fl-[cidJi 
-- (7) 
where the bar refers to an average over the asymptotic {Jij) dynamics. 
The form (6) with n - 0 is immediately reminiscent of the effective partition 
function which results from the application of the replica trick to replace In Z by 
lim,_0 (Z ' - 1) in dealing with a quenched average for the infinite-ranged spin- 
glass [6], while n = i relates to the corresponding annealed average, although we 
note that in the present model the time-scales for neuron and interaction dynamics 
remain completely disparate. These observations correlate with the identification 
of n with//3, which implies that n - 0 corresponds to a situation in which the 
interaction dynamics is dominated by the stochastic term rli j (t), rather than by the 
behaviour of the neurons, while for n - i the two characteristic temperatures are 
the same. For n - oe the influence of the neurons on the interaction dynamics 
dominates. In fact, any real n is possible by tuning the ratio between the two fi's. 
In the formulation presented in this paper n is always non-negative, but negative 
values are possible if the Itebbian rule of (3) is replaced by an anti-Itebbian form 
with (rrrrj) replaced by -(rrirrj) (the case of negative n is being studied by Mzard 
and co-workers [7]). 
The model discussed above is range-free/infinite-ranged and can therefore be an- 
alyzed in the thermodynamic limit N  oc by the replica mean-field theory as 
devised for the Sherrington-Kirkpatrick spin-glass [6, 8, 9]. This can be developed 
precisely for integer n [6, 8, 9, 10] and analytically continued. In the usual manner 
there enters a spin-glass order parameter 
where the superscripts are replica labels. qV* is given by the extremum of 
F({qV*}): 2fi_rt2 y [qV*] 2 +in Tr exp 
while 2 is proportional to exp [NextrF ({qV*})]. In the replica-symmetric region 
(or ansatz) one assumes qV* = q. 
We will first choose as the independent variables n and/ and briefly discuss the 
phase picture of our model (full details can be found in [11]). The system exhibits 
a transition from a paramagnetic state (q = 0) to an ordered state (q > 0) at a 
critical/(n). For n _< 2 this transition is second order at/ = 1, down to the SK 
450 Coolen, Penney, and Sherrington 
spin-glass limit, n - 0, but for n > 2 the coupled dynamics leads to a qualitative, 
as well as quantitative, change to first order. Replica symmetry is stable above a 
critical value no(/3), at which there is a de Almeida-Thouless (AT) transition (c.f. 
Kondor [12]). As expected from spin-glass studies, no(/3) goes to zero as /3  1 
but rises for larger /7, having a maximum of order 0.3 at /3 of order 2. Thus, for 
n > n(max) m 0.3 there is no instability against small replica-symmetry breaking 
fluctuations, while for smaller n there is re-entrance in this stability. The transition 
from a paramagnetic to an ordered state and the onset of local RS instability for 
various temperatures is shown in Figure 1. 
3 EXTERNAL FIELDS 
Several simple modifications of the above model are possible. One consists of adding 
external fields to the spin dynamics and/or to the interaction dynamics, by making 
the substitutions 
in (2) and (5) respectively. These external fields may be viewed as generating fields 
in the sense of (7); for example 
Ob:ij - Jij OKijOKkt -/3 [JijJk - Jij 
For neural network models a natural first choice for the external fields would be 
Oi  hi and Kij -- Kij, i  {-1, 1}, where the i are quenched random vari- 
ables corresponding to an imposed pattern. Without loss of generality all the i 
can be taken as +1, via the gauge transformation o'i  o'ii, Jij  Jijij. Hence- 
forth we shall make this choice. The neuron perturbation field h induces a finite 
'magnetization' characterized by a new order parameter 
which is independent of a in the replica-symmetric assumption (which turns out 
to be stable with respect to variation in this parameter). As in the case of the 
spin-glass, there is now a critical surface in (h, n,/3) space characterizing the onset 
of replica symmetry breaking. In introducing the interaction perturbation field K 
we find that K/lu is the analogue of the mean exchange J0 in the SK spin-glass 
model, f2 -- (/3nit) -t being the analogue of the variance. If large enough, this field 
leads to a spontaneous 'ferromagnetic' order. 
Again we find further examples of both second and first order transitions (details 
can be found in [11]). For the paramagnetic (P; m = 0, q = 0) to ferromagnetic 
(F; m  0, q  0) case, the transition is second order at the SK value fiJ0 = 1 so 
long as (/3f)-2 _> 3n - 2. Only when (/3f)-2 < 3n - 2 do the interaction dynamics 
Coupled Dynamics of Fast Neurons and Slow Interactions 451 
PARAMAGNET .. .. ......... ' ...... 
1 
0.6 MATTIB GLABB 
T 
0.6 
0.4 
0.2  SPllq 
O0 1 2 3 
Figure 1: Phasediagram for j = 1. Dotted line: first order transition, solid line: 
second order transition. The separation between Mattis-glass and spin-glass phase 
is defined by the de Almeida-Thouless instability 
452 Coolen, Penney, and Sherrington 
influence the transition, changing it to first order at a lower temperature. Regarding 
the ferromagnetic to spin-glass (SG; m = 0, q  0) transition, this exhibits both 
second order (lower Jo) and first order (higher J0) sections separated by a tricritical 
point for n less than a critical value of the order of 3.3. This tricritical point exhibits 
re-entrance as a function of n. 
4 
COMPARISON BETWEEN COUPLED DYNAMICS 
AND SK MODEL 
In order to clarify the differences, we will briefly summarize the two routes that 
lead to an SK-type replica theory: 
Coupled Dynamics: 
Fast Ising spin neurons + slow dynamic interactions, 
d i K 
j = (j)j,j) +  - j +GWN 
Free energy: 
Define: 
Thermodynamics: 
1 
/3nextr G ({qV*}; {my}) + const. 
SK spin-glass: 
Ising spins + fixed random interactions, 
P(Jij) -= [2rJ2]-e -['J'-]/ 
Free energy: 
Sell-averaging: 
Physical scaling: 
Thermodynamics: 
N --- o<>: 
1 1 lim 1 [Z_ 1] 
f --= 3N log Z = -/3- ,-0 n 
J0 = o/V, J = /v 
- 1 
f = - lim -extr G ({qV*}; {my}) + const. 
r--+ 0 
Coupled Dynamics of Fast Neurons and Slow Interactions 453 
5 DISCUSSION 
We have obtained a solvable model with which a coupled dynamics of fast stochas- 
tic neurons and slow dynamic interactions can be studied analytically. Furthermore 
it presents the replica method from a novel perspective, provides a direct inter- 
pretation of the replica dimension n in terms of parameters controlling dynamical 
processes and leads to new phase transition characters. As a model for neural learn- 
ing the specific example analyzed here is however only a first step, with h and K 
as introduced corresponding to only a single pattern. Its adaptation to treat many 
patterns is the next challenge. 
One type of generalization is to consider the whole system as of lower connectivity 
with only pairs of connected sites being available for interaction upgrade. For 
example, the system could be on a lattice, in which case the corresponding coupled 
partition function will have the usual greater complication of a finite-dimensional 
system, or randomly connected with each bond present with a probability C/N, 
in which case there results an analogue of the Viand-Bray [13] spin-glass. In each 
of these cases the explicit factors involving N in the {Jij} dynamics (3) should 
be removed (their presence or absence being determined by the need for statistical 
relevance and physical scaling). 
Yet another generalization is to higher order interactions, for example to p-neuron 
ones: 
S{j} ({(Ti})"-" -- Z Jil,...:ip(Til(Ti9...(Tip 
ii,...,ip 
with corresponding interaction dynamics 
or to more complex neuron types. 
If the symmetry-breaking fields Kij in the interaction dynamics are choosen at 
random, we obtain a curious theory in which we find replicas on top of replicas (the 
replica trick would be used to deal with the quenched disorder of the Kij, for a 
model in which replicas are already present due to the coupled dynamics). 
Finally, our approach can in fact be generalized to any statistical mechanical system 
which in equilibrium is described by a Boltzmann distribution in which the Hamilto- 
nian has (adiabatically slowly) evolving parameters. By choosing these parameters 
to evolve according to an appropriate Langevin process (involving the free energy 
of the underlying fast system) one always arrives at a replica theory describing the 
coupled system in equilibrium. 
Acknowledgements 
Financial support from the U.K. Science and Engineering Research Council under 
grants 9130068X and GR/H26703, from the European Community under grant 
S/SC1'915121, and from Jesus College, Oxford, is gratefully acknowledged. 
454 Coolen, Penney, and Sherrington 
References 
[1] Glauber R.J. (1963) J. Math. Phys. 4 294 
[2] Hebb D.O. (1949) 'The Organization of Behaviour' (Wiley, New York) 
[3] Shinomoto S. (1987) J. Phys. A: Math. Gen. 20 L1305 
[4] Jonker It.J.J. and Coolen A.C.C. (1991) J. Phys. A: Math. Gen. 24 4219 
[5] Jonker H.J.J., Coolen A.C.C. and Denier van der Gon J.J. (1993) J. Phys. A: 
Math. Gen. 26 2549 
[6] Sherrington D. and Kirkpatrick S. (1975) Phys. Rev. Left. 35 1792 
[7] M6zard M. private communication 
[8] Kirkpatrick S. and Sherrington D. (1978) Phys. Rev. B 17 4384 
[9] M6zard M., Parisi G. and Virasoro M.A. (1987) 'Spin Glass Theory and Be- 
yond' (World Scientific, Singapore) 
[10] Sherrington D. (1980) J. Phys. A: Math. Gen. 13 637 
[11] Penney R.W., Coolen A.C.C. and Sherrington D. (1993) J. Phys. A: Math. 
Gen. 26 3681-3695 
[12] Kondor I. (1983) J. Phys. A: Math. Gen. 16 L127 
[13] Viana L. and Bray A.J. (1983) J. Phys. C 16 6817 
