76 Kammen, Koch and Holmes 
Collective Oscillations 
Visual Cortex 
in the 
Daniel Karomen & Christof Koch 
Computation and Neural Systems 
Caltech 216-76 
Pasadena, CA 91125 
Philip J. Holmes 
Dept. of Theor. & Applied Mechanics 
Cornell University 
Ithaca, NY 14853 
ABSTRACT 
The firing patterns of populations of cells in the cat visual cor- 
tex can exhibit oscillatory responses in the range of 35 - 85 Hz. 
Furthermore, groups of neurons many tom's apart can be highly 
synchronized as long as the cells have similar orientation tuning. 
We investigate two basic network architectures that incorporate ei- 
ther nearest-neighbor or global feedback interactions and conclude 
that non-local feedback plays a fundamental role in the initial syn- 
chronization and dynamic stability of the oscillations. 
I INTRODUCTION 
40 -- 60 Hz oscillations 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, 1972; Wilson & Bower 1990). Recently, two groups (Eckhorn e! 
el., 1988 and Gray e! el., 1989) have 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 mm apart show 
phase-locked oscillations, with a phase shift of less than 3 msec. We address here 
the computational architecture necessary to subserve this process by investigating 
to what extent two neuronal architectures, nearest-neighbor coupling and feedback 
from a central "comparator", can synchronize neuronal oscillations in a robust and 
rapid manner. 
Collective Oscillations in the Visual Cortex 77 
It was argued in earlier work on central pattern generators (Cohen et al., 1982), that 
in studying coupling effects among large populations of oscillating neurons, one can 
ignore the details of individual oscillators and represent each one by a single periodic 
variable: its phase. Our approach assumes a population of neuronal oscillators, 
firing repetitively in response to synaptic input. Each cell (or group of tightly 
electrically coupled cells) has an associated variable representing the membrane 
potential. In particular, when 0i - r, an action potential is generated and the 
phase is reset to its initial value (in our case to -r). The number of times per unit 
time 0i passes through r, i.e. dOi/dt, is then proportional to the firing frequency of 
the neuron. For a network of n + I such oscillators, our basic model is 
dOi 
-- wi + fi(0o,Ol,...,O,), (1) 
dt 
where wi represents the synaptic input to neuron i and f, a function of the phases, 
represents the coupling within the network. Each oscillator i in isolation (i.e. with 
fi - 0), exhibits asymptotically stable periodic oscillations; that is, if the input 
is changed the oscillator will rapidly adjust to a new firing rate. In our model wi 
is assumed to derive from neurons in the lateral geniculate nucleus (LGN) and is 
purely excitatory. 
2 FREQUENCY AND PHASE LOCKING 
Any realistic model of the observed, highly synchronized, oscillations must account 
for the fact that the individual neurons oscillate at different frequencies in isolation. 
This is due to variations in the synaptic input, wi, as well as in the intrinsic prop- 
erties of the cells. We will contrast the abilities of two markedly different network 
architectures to synchronize these oscillations. The "chain" model (Fig. I top ) con- 
sists of a one-dimensional array of oscillators connected to their nearest neighbors, 
while in the alternative "comparator" model (Fig. I middle), an array of neurons 
project to a single unit, where the phases are averaged (i.e. (l/n) 'i"--0 0i(t)). This 
average is then feed back to every neuron in the network. In the continuum limit 
(on the unit interval) with all fi - f being identical, the two models are 
(Chain Model) O0(x, t) = w(x) + - (c) 
Ot n 
(Comparator Model) O0(x,t) = w(x) + f(O(x,t) - fo  
ot 
(2) 
O(s,t)ds), (3) 
where 0 < x < 1 and b is the phase gradient, b =  os 
_ _ ;. In the chain model, we 
require that f be an odd function (for simplicity of analysis only) while our analysis 
of the comparator model holds for any continuous function f. We use two spatially 
separated "spots" of width 5 and amplitude a as visual input (Fig. I bottom). This 
pattern was chosen as a simple version of the double-bar stimulus that (Gray et al. 
1989) found to evoke coherent oscillatory activity in widely separated populations 
of visual cortical cells. 
78 Kammen, Koch and Holmes 
Oi=o(t) 
Oi=n(t) 
to(O) to(n) 
to(O) to(n) 
o(x) 
x 
Figure 1: The linear chain (top) and comparator (middle) architectures. The 
spatial pattern of inputs is indicated by ozi(x). See equs. 2 & 3 for a mathematical 
description of the models. The "two spot" input is shown at bottom and represents 
two parts of a perceptually extended figure. 
We determine under what circumstances the chain model will develop frequency- 
locked solutions, such that every oscillator fires at the same frequency (but not 
necessarily at the same time), i.e. OO/OxO! = O. We prove (Kammen, et al. 1990) 
that frequency-locked solutions exist as long as In(x - f o:(s)ds)l does not exceed 
the maximal value of f, fmax (with  = fo  o(s)ds the mean excitation level). 
Thus, if the excitation is too irregular or the chain too long (n >> 1), we will not 
find frequency-locked solutions. Phase coherence between the excited regions is not 
generally maintained and is, in fact, strongly a function of the initial conditions. 
Another feature of the chain model is that the onset of frequency locking is slow 
and takes time of order V . 
The location of the stimulus has no effect on phase relationships in the comparator 
model due to the global nature of the feedback. The comparator model exhibits 
two distinct regimes of behavior depending on the amplitude of the input, a. In the 
case of the two spot input (Fig. I bottom ), if c is small, all neurons will frequency- 
lock regardless of location, that is units responding to both the "figure" and the 
background ("ground") will oscillate at the same frequency. They will, however, 
fire at different times, with Ofi  Onct. If  is above a critical threshold, the units 
responding to the "figure" will decouple in frequency as well as phase from the 
background while still maintaining internal phase coherency. Phase gradients never 
exist within the excited groups, no matter what the input amplitude. 
Collective Oscillations in the Visual Cortex 79 
We numerically simulated the chain and comparator models with the two spot input 
for the coupling function f(r) -- sin(r). Additive Gaussian noise was included in the 
input, cvi. Our analytical results were confirmed; frequency and phase gradients were 
always present in the chain model (Fig. 2A) even though the coupling strength was 
ten times greater than that of the comparator model. In the comparator network 
small excitation levels led to frequency-locking along the entire array and to phase- 
coupled activity within the illuminated areas (Fig. 2B), while large excitation levels 
led to phase and frequency decoupling between the "figure" and the "background" 
(Fig. 2C). The excited regions in the comparator settle very rapidly - within 2 to 
3 cycles - into phase-locked activity with small phase-delays. The chain model, on 
the other hand, exhibits strong sensitivity to initial conditions as well as a very slow 
approach to coherence that is still not complete even after 50 cycles (See Fig. 2). 
A 
B 
C 
Figure 2: The phase portrait of the chain (A), weak (B) and strongly (C) excited 
comparator networks after 50 cycles. The input, indicated by the horizontal lines, 
is the two spot pattern. Note that the central, unstimulated, region in the chain 
model has been "dragged along" by the flanking excited regions. 
3 STABILITY ANALYSIS 
Perhaps the most intriguing aspect of the oscillations concerns the role that they 
may play in cortical information processing and the labeling of cells responding to a 
single perceptual object. To be useful in object coding, the oscillations must exhibit 
some degree of noise tolerance both in the input signal and in the stability of the 
population to variation in the firing times of individual cells. 
The degree to which input noise to individual neurons disrupts the synchronization 
of the population is determined by the ratio input noise -- w/- For small per- 
couplintl strentlth  ' 
turbations, w(t) - wo d- e(t), the action of the feedback, from the nearest neighbors 
in the chain and from the entire network in the comparator, will compensate for 
the noise and the neuron will maintain coherence with the excited population. As 
e is increased first phase and then frequency coherence will be lost. 
In Fig. 3 we compare the dynamical stability of the chain and comparator models. 
In each case the phase, 0, of a unit receiving perturbated input is plotted as the 
deviation from the average phase, 00, of all the excited units receiving input w0. The 
chain in highly sensitive to noise: even 10% stochastic noise significantly perturbs 
the phase of the neuron. In the comparator model (Fig. 3B) noise must reach the 
80 Kammen, Koch and Holmes 
40% level to have a similar effect on the phase. As the noise increases above 0.30w0 
even frequency coherence is lost in the chain model (broken error bars). Frequency 
coherence is maintained in the comparator for  = 0.60w0. 
A) 
+0.10 
+O. OS 
0.00 
..O.OS 
-0.10 
0.0 20 40 60 0.0 20 40 60 
 (% of m 0) 
Figure 3: The result of a perturbation on the phase, 0, for the chain (A) and 
comparator (B) models. The terminus of the error bars gives the resulting devia- 
tion from the unperturbed value. Broken bars indicate both phase and frequency 
alecoupling. 
The stability of the solutions of the comparator model to variability in the activity 
of individual neurons can easily be demonstrated. For simplicity consider the case 
of a single input of amplitude w superposed on a background of amplitude w0. The 
solutions in each region are: 
dOo 
dt --- o+f 2 (4) 
d0 (01-00) 
dt = a, + f 2 ' (5) 
We define the difference in the solutions to be b(t) -- 01(t)--OO(t ) and Aw = w -w0. 
We then have an equation for the rate the solutions converge or diverge: 
dt = Aw + -- . 
If the solutions are stable (of constant velocity) then dO1/dt = dOo/dt and 01 = Oo+c 
with c a constant. We then have the stable solution 4* = c dq*/dt = Aw + f()- 
f(-- ) = O. Stability of the solutions can be seen by perturbing O to O = O0 + e + e 
with Id < 1. The perturbed solution, 4 = 4* + e, has the derivative d/dt = de/dr. 
Developing f() into a Taylor series around * and neglecting terms on the order 
of c 2 and higher, we arrive at 
= f'( )+ f'( )' (7) 
Collective Oscillations in the Visual Cortex 81 
If f(b) is odd then f'() is even, and eq. (7) reduces to 
de , c 
 -- ef (). ($) 
Thus, if if(c/2) < 0 the perturbations will decay to zero and the system will main- 
tain phase locking within the excited regions. 
4 THE FREQUENCY MODEL 
The model discussed so far assumes that the feedback is only a function of the 
phases. In particular, this implies that the comparator computes the average phase 
across the population. Consider, however, a model where the feedback is propor- 
tional to the average firing frequency of a group of neurons. Let us therefore replace 
phase in the feedback function with firing frequency, 
oo(, t) _ + f 
Ot 
ot 
with oo 
$7 -  fo  O(s, t)ds - o This is a very special differential equation as can be 
seen by setting v(x,t) - O0(x, t)/Ot. This yields an algebraic equation for v with 
no explicit time dependency: 
v(x) - w(x) -9 f(v(x) - '(x)) (10) 
and, after an integration, we have, 
O(x,t) - v(x)dt - v(x)t -- Oo(x). (11) 
Thus, the phase relationships depend on the initial conditions, Oo(x), and no phase 
locking occurs. While frequency locking only occurs for w(x) = 0 the feedback can 
lead to tight frequency coupling among the excited neurons. 
Reformulating the chain model in terms of firing-frequencies, we have 
O0(x,t) I (Ow(x) 10  
under the assumption that f(-x) - -f(x). With 7(x, t) - 
at a stationary algebraic equation 
l(0wxw I 0 ))) 
(x) = n n O  ' 
and 
fo t 
= = + 4o(x) 
(12) 
. ., we again arrive 
(13) 
In other words, the system will develop a time-dependent phase gradient. Frequency 
os = 0 everywhere only occur if w(x) = 0 everywhere. 
locked solutions of the sort  
Thus, the chain architecture leads to very static behavior, with little ability to either 
phase- or frequency-lock. 
(14) 
82 Kammen, Koch and Holmes 
5 DISCUSSION 
We have investigated the ability of two networks of relaxation oscillators with dif- 
ferent connectivity patterns to synchronize their oscillations. Our investigation 
has been prompted by recent experimental results pertaining to the existence of 
frequency- and phase-locked oscillations in the mammalian visual cortex (Gray et 
el., 1989; Eckhorn e! el., 1988). While these 35-85 Hz oscillations are induced by 
the visual stimulus, usually a flashing or moving bar, they are not locked to the fre- 
quency of the stimulus. Most surprising is the finding that cells tuned to the same 
orientation, but separated by up to 7 turn, not only exhibit coherent oscillatory 
activity, but do so with a phase-shift of less than 3 rnsec (Gray et al., 1989).  
We have assumed the existence of a population of cortical oscillators, such as those 
reported in cortical slice preparations (Llins, 1988; Chagnac-Amitai and Connors, 
1989). The issue is then how such a population of oscillators can rapidly begin to 
fire in near total synchrony. Two neuronal architectures suggest themselves. 
As a mechanism for establishing coherent oscillatory activity the comparator model 
is far superior to a nearest-neighbor model. The comparator rapidly (within 1 - 3 
cycles) achieves phase coherence, while the chain model exhibits a far slower onset 
of synchronization and is highly sensitive to the initial conditions. Once initiated, 
the oscillations in the two models exhibit markedly different stability characteris- 
tics. The diffusive nature of communication in the chain results in little ability to 
regulate the firing of individual units and consequently only highly homogeneous 
inputs will result in collective oscillations. The long-range connections present in 
the comparator, however, result in stable collective oscillations even in the presence 
of significant noise levels. Noise uniformly distributed about the mean firing level 
will have little effect due to the averaging performed by the comparator unit. 
A more realistic model of the interconnection architecture of the cortex will cer- 
tainly have to take both local as well as global neuronal pathways into account and 
the ever-present delays in cellular and network signal propagation (Kammen, et 
el., 1990). Long range (up to 6 ram) lateral excitatory connections have been 
reported (Gilbert and Wiesel, 1983). However, their low conduction velocities 
( I mm/msec) would lead to significant phase-shifts in contrast to the data. 
While the cortical circuitry contains both local as well as global connection, our 
results imply that a cortical architecture with one or more "comparator" neurons 
driven by the averaged activity of the hypercolumnar cell populations is an attrac- 
tive mechanism for synchronizing the observed oscillations. 
We have also developed a model where the firing frequency, and not the phase is 
involved in the dynamics. Coding based on phase information requires that the 
cells track the time interval between incident spikes whereas the firing frequency 
is available as the raw spike rate. This computation can be readily implemented 
1 Note that this result is obtained by averaging over many trials. The phase-shift for individual 
trial may possibly be larger, but could be randomly distributed from trial to trial around the 
origin. 
Collective Oscillations in the Visual Cortex 83 
neurobiologically and is entirely consistent with the known biophysics of cortical 
cells. 
Von der Malsburg (1985) has argued that the temporal synchronization of groups of 
neurons labels perceptually distinct objects, subserving figure-ground segregation. 
Both firing frequency and inter-cell phase (timing) relationships of ensembles of 
neurons are potential channels to encode the signatures of various objects in the 
visual field. Perceptually distinct objects could be coded by groups of synchronized 
neurons, all locked to the same frequency with the groups only distinguished by 
their phase relationships. We do not believe, however, that phase is a robust enough 
variable to code this information across the cortex, A more robust scheme is one in 
which groups of synchronized neurons are locked at different firing frequencies. 
Acknowledgement 
D.K. is a recipient of a Weizman Postdoctoral Fellowship. P.H. acknowledges sup- 
port from the Sherman Fairchild Foundation and C.K. from the Air Force Office 
of Scientific Research, a NSF Presidential Young Investigator Award and from the 
James S. McDonnell Foundation. We would like to thank Francis Crick for useful 
comments and discussions. 
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