Hansel 1992 Synchronization
Hansel 1992 Synchronization
Hansel 1992 Synchronization
68, NUMBER 5
LETTERS
PH YSICAL REVI E%
Synchronization
and Computation
in
3 FEBRUARY 1992
D. Hansel
Centre de Ph]tsique Theorique, Ecole Pol],'tec'hnique,
Ml/28 Palaiseau,
Franc. e
H. Sompolinsky
Racah institute
Jerusalem,
Israel 91904
Chaos generated by the internal dynamics of a large neural network can be correlated over large spatial scales. Modulating the spatial coherence of the chaotic fluctuations by the spatial pattern of the
external input provides a robust mechanism for feature segmentation and binding, which cannot be accomplished by networks of oscillators with local noise. This is demonstrated by an investigation of synchronized chaos in a network model of bursting neurons responding to an inhornogeneous stimulus.
PACS numbers:
Coherent
of weakly coupled
range connections. Desynchronization
neurons is achieved by local noise.
Both the experimental results and the suggested relevance of synchronization to global operations, such as object segmentation, imply that there is an eScient mechanism for rapid desynchronization
of the relative te~;.poral
phase of large internally synchronized groups of neurons.
However, systems of noisy oscillators are incapable of
generating such a large-scale rapid and reversible desynchronization.
This is because the amplitude
of the
effective noise that acts on the phase of an internally synchronized group of say lV oscillators is only 6/J!V, where
6 is its local amplitude, and thus is neg1igible for large A'.
Consequently, even weak coupling between two large assemblies of oscillators will eventually synchronize them.
More importantly, even when such assemblies are not interacting at all, the time taken to desynchronize the initial relative phase between them is extremely long, i.e. , of
O(lV/8-). To overcome this problem within the context
of oscillatory networks one has to introduce an ad hoc
spatially correlated noise [13,14].
In this Letter we study a chaotic neural network model
that exhibits synchronization
at large spatial scales,
modulated by the distributed features of the extern ~1
stimulus. The advantage of chaos over external noise is
the fact. that the spatial correlations of the deterministic
noise are not fixed but depend on the dynamic state of the
system. As a consequence, the external stimulus can
modulate the spatial scale of the dynamic noise. In particular, depending on the pattern of the input the system
can break into large weakly coupled clusters, each exhibiting a globally chaotic activity. The chaotic fluctuations
rapidly desynchronize the relative phases of the different
clusters.
neu-
rons
X; = I';
. dA; V;,
Y; =c.
Z;
(2)
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0.08
0.04
FIG. l. The time-averaged firing rates plotted against the local external inputs. The continuous line is for =0.5. The plateau at 0 corresponds to neurons in a quiescent state. The next
plateaus correspond to periodically bursting neurons with 1, 2,
3, and 4 spikes per burst, respectively. The nonsmooth regime
corresponds to chaotic neurons. The last, linear part corresponds to repetitively firing neurons. The dashed line is for
J =5.0. Results from simulations of Eqs. (1)-(3) with iV =800.
For parameters see text. Inset: The power spectrum of I,,, for
=5.0 (in arbitrary units).
an aperiodic component which, ho~ever, is not synchronized across the system, and therefore contributes only a
small finite-size noisy component to 1(t).
(iii) Synchronized chaos
When . the coupling is
strong, 3.5& J, all the active neurons are chaotic and
furthermore their chaotic fluctuations are spatially correlated. An example is shown in Fig. 2(b) where the syn-
(a)
J~3.
-2
(b)
JJ
-3
4000
4250
4500
4750
5000
J=0.
719
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chronized
in
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720
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25
0~
20os/
o.6I
15~ 0.4!
30
5
-400
-200
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