Phase-coupling in Two-Dimensional 
Networks of Interacting Oscillators 
Ernst Niebur, Daniel M. Kammen, Christof Koch, 
Daniel Ruderman  & Heinz G. Schuster 2 
Computation and Neural Systems 
Caltech 216-76 
Pasadena, CA 91125 
ABSTRACT 
Coherent oscillatory activity in large networks of biological or artifi- 
cial neural units may be a useful mechanism for coding information 
pertaining to a single perceptual object or for detailing regularities 
within a data set. We consider the dynamics of a large array of 
simple coupled oscillators under a variety of connection schemes. 
Of particular interest is the rapid and robust phase-locking that 
results from a "sparse" scheme where each oscillator is strongly 
coupled to a tiny, randomly selected, subset of its neighbors. 
1 INTRODUCTION 
Networks of interacting oscillators provide an excellent model for numerous physical 
processes ranging from the behavior of magnetic materials to models of atmospheric 
dynamics to the activity of populations of neurons in a variety of cortical locations. 
Particularly prominent in the neurophysiological data are the 40-60 Hz oscillations 
that have long been reported in the rat and rabbit olfactory bulb and cortex on the 
basis of single-and multi-unit recordings as well as EEG activity (Freeman, 1978). 
In addition, periodicities in eye movement reaction times (PSppel and Logothetis, 
1986), as well as oscillations in the auditory evoked potential in response to single 
click or a series of clicks (Madler and PSppel, 1987) all support a 30 - 50 Hz 
framework for aspects of cortical activity. Two groups (Eckhorn et al., 1988, Gray 
Permanent address: Department of Physics, University of California, Berkeley, CA 94720 
Permanent address: Institut fiir Theoretische Physik, Universitt Kiel, 2300 Kiel 1, Germany. 
123 
124 Niebur, Kammen, Koch, Ruderman, and Schuster 
and Singer, 1989; Gray et al., 1989) have recently reported highly synchronized, 
stimulus specific oscillations in the 35 - 85 Hz range in areas 17, 18 and PMLS of 
anesthetized as well as awake cats. Neurons with similar orientation tuning up to 
7 ram apart show phase-locked oscillations with a phase shift of less than 1 rasec 
that have been proposed to play a role in the coding of visual information (Crick 
and Koch, 1990, Niebur et al. 1991). 
The complexity of networks of even relatively simple neuronal units - let alone 
"real" cortical cells - warrants a systematic investigation of the behavior of two 
dimensional systems. To address this question we begin with a network of mathe- 
matically simple limit-cycle oscillators. While the dynamics of pairs of oscillators are 
well understood (Sakaguchi, et al. 1988, Schuster and Wagner, 1990a,b), this is not 
the case for large networks with nontrivial connection schemes. Of general interest 
is the phase-coupling that results in networks of oscillators with different coupling 
schemes. We will summarize some generic features of simple nearest-neighbor cou- 
pled models, models where each oscillator receives input from a large neighborhood, 
and of "sparse" connection geometries where each cell is connected to only a tiny 
fraction of the units in its neighborhood, but with large coupling strength. The 
numerical work was performed on a CM-2 Connection Machine and involved 16,384 
oscillators in a 128 by 128 square grid. 
2 The Model 
The basic unit in our networks is an oscillator whose phase 0i1 is 2r periodic and 
which has the intrinsic frequency wid. The dynamics of an isolated oscillator are 
described by: 
dOij _ 
_ . (1) 
The influence of the network can be expressed as an additional interaction term, 
dOij 
at = + fq(Oo,O, ...o,). (2) 
The coupling function, fir we used is expressed as the sum of terms, each one 
consisting of the product of a coupling strength and the sine of a phase difference 
(see below, eq. 3). The sinusoidal form of the interaction is, of course, linear for 
small differences. 
This system, and numerous variants, has received a considerable amount of attention 
from solid state physicists (see, e.g. Kosterlitz and Thouless 1973, and Sakaguchi e! 
al. 1988), although primarily in the limit of t --+ cx. With an interest in the possible 
role of networks of oscillators in the parsing or segregating of incident signals in 
nervous systems, we will concentrate on short time, non-equilibrium, properties. 
We shall confine ourselves to two generic network configurations described by 
a - . ), 
dt - wid + Jid,sin(Oid - (3) 
ki 
Phase-coupling in Two-Dimensional Networks of Interacting Oscillators 125 
where a designates the global strength of the interaction, and the geometry of the 
interactions is incorporated in Jij,kt. 
The networks are all defined on a square grid and they are characterized as follows: 
1: Gaussian Connections. The cells are connected to every oscillator within a 
specified neighborhood with Gaussian weighted connections. Hence, 
1((i-k) +(j-l)) (4) 
Jij,kl ---- 2raeXp 2a 2  
We truncate this function at 2a, i.e. Jii,k = 0 if (i- k) 2 + (j - 0 2 _> (2rr) 2. While 
the connectivity in the nearest neighbor case is 4, the connectivity is significantly 
higher for the Gaussian connection schemes: Already a = 2 yields 28 connections 
per cell, and the largest network we studied, with rr = 6, results in 372 connections 
per cell. 
2: Sparse Gaussian Connections. In this scheme we no longer require sym- 
metric connections, or that the connection pattern is identical from unit to unit. A 
given cell is connected to a fixed number, n, of neighboring cells, with the probability 
of a given connection determined by 
1((i-k)2+(j-l)2) (5) 
ij,kl = 274'0' exp 2a 2 . 
Jij,kt is unity with probability ij,kl and zero otherwise. This connection scheme 
is constructed by drawing for each lattice site n coordinate pairs from a Gaussian 
distribution, and use these as the indices of the cells that are connected with the 
oscillator at location (i, j). Therefore, the probability of making a connection de- 
creases with distance. If a connection is made, however, the weight is the same as 
for all other connections. We typically used n = 5, and in all cases 2 _< n _< 10. 
For all networks, the sum of the weights of all connections with a given oscillator i, j 
was conserved and chosen as c Y-}l Jii,la = 10  , where  is the average frequency 
of all N oscillators in the system,  =  --ii "' this procedure, the total 
impact of the interaction term is identical in all WcSes. By 
3 RESULTS 
Perhaps the most basic, and most revealing, comparison of the behavior of the 
models introduced above is the two-point correlation function of phase-coupling, 
which is defined as 
c(R,t) =< cos [%(0 - ok,(t)] >, (6) 
where R is defined as the separation between a pair of cells, R = [rij - rkl]. We 
compute and then average C(R, t) over 10,000 pairs of oscillators separated by R in 
the array. In all cases, the frequencies wij are chosen randomly, with a Gaussian dis- 
tribution with mean 0.5 and variance 1. In Figure 1 we plot C(R, t) for separations 
126 Niebur, Kammen, Koch, Ruderman, and Schuster 
of R = 20, 30, 40, 50, 6, and 70 oscillators. Time is measured in oscillation peri- 
ods of the mean oscillator frequency, . At t = 0, phases are distributed randomly 
between 0 and 2r with a uniform distribution. The case of Gaussian connectivity 
with a = 6 and hence 372 connection per cell is seen in Figure l(a), and the sparse 
connectivity scheme with a = 6 and n = 5 is presented in Figure l(b). The most 
striking difference is that correlation levels of over 0.9 are rapidly achieved in the 
sparse scheme for all cases, even for separations of 70 oscillators (plotted as aster- 
isks, *), while there are clear separation-dependent differences in the phase-locking 
behavior of the Gaussian model. In fact, even after t = 10 there is no significant 
locking over the longer distances of R = 50, 60, or 70 units. For local connectivity 
schemes, like Gaussian connectivity with a = 2 or nearest neighbors connections, 
no long-range order evolves even at larger times (data not shown). 
Data in Fig. I were computed with a uniform phase distribution for t = 0. An in- 
teresting and robust feature of the dynamics emerges when the influence of different 
types of initial phase distributions are examined. In Figure 2 we plot the probabil- 
ity distribution of phases at different early times. In Figure 2(a) the distribution 
of phases is plotted at t = 0 (diamonds), t = 0.2 ("plus signs, +) and at t = 0.4 
(squares) for the sparse scheme with a uniform initial distribution. In Figure 2(b), 
the evolution of a Gaussian initial distribution centered at 0 = r of the phases 
is plotted. Note the slight curve in the distribution at t = 0, indicating that the 
Gaussian initial seeding is rather slight (variance rr = 2r). Remarkably, however, 
this has a dramatic impact on the phase-locking as after two-tenth of an average 
cycle time ("plus" signs) there is already a pronounced peak in the distribution. At 
t = 0.4 (squares) the system that started with the uniform distribution begins to 
only exhibit a slight increase in the phase-correlation while the system with Gaus- 
sian distributed initial phases is strongly peaked with virtually no probability of 
encountering phase values that differ significantly from the mean. 
4 DISCUSSION 
The power of the sparse connection scheme to rapidly generate phase-locking through- 
out the network that is equivalent, or superior, to that of the massively intercon- 
nected Gaussian scheme highlights a trade-off in network dynamics: massive av- 
eraging versus strong, long-range, connections. With n = 5, the sparse scheme 
effectively "tiles" a two-dimensional lattice and tightly phase-locks oscillators even 
at opposite corners of the array. Similar results are obtained even with n = 2 (data 
not shown). 
In many ways the Gaussian and sparse geometries reperent opposing avenues to 
achieve global coherence: exhaustive local coupling or distributed, but powerful 
long-range coupling. The amount of wiring necessary to implement these schemes 
is, however, radically different. 
Phase-coupling in Two-Dimensional Networks of Interacting Oscillators 127 
Acknowledgement 
EN is supported by the Swiss National Science Foundation through Grant No. 8220- 
25941. DMK is a recipient of a Weizman Postdoctoral Fellowship. CK acknowledges 
support from the Air Force Office of Scientific Research, a NSF Presidential Young 
Investigator Award and from the James S. McDonnell Foundation. HGS is sup- 
ported by the Volkswagen Foundation. 
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128 Niebur, Kammen, Koch, Ruderman, and Schuster 
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Figure 1: Two-point correlation functions, C(R, t), for various separations, R, in 
(a) the a = 6 Gaussian scheme with 372 connections per cell and (b) the sparse 
connection scheme with a = 6 and n = 5 connections per cell. Separations of 
R = 20 (diamonds), R = 30 ("plus"signs, +), R = 40 (squares), R = 50 (crosses, 
x), R = 60 (triangles), and R = 70 (asterisks, *) are shown. Note the rapid locking 
for all lengths in the sparse scheme (b) while the Gaussian scheme (a) appears far 
more "diffusive,"with progressively poorer and slower locking as R increases. 
Phase-coupling in Two-Dimensional Networks of Interacting Oscillators 129 
0 
14 
22 
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2 
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Figure 2: Snapshots of the distribution of phases in the sparse scheme 
(n = 5, a = 6) when the system begins from (a) uniform and (b) a Gaussian 
"biased" initial distribution. The figures show the probability P(O) to find a phase 
between 0 and 0 + dO (bin size r/10). At t = 0, the distribution is flat (a) or very 
slightly curved (b); see text. The difference in the time evolution can clearly be 
seen in the state of the system after t = 0.2 ("plus" signs, +) and t = 0.4 (squares). 
