Period-two cycles in a feedforward layered neural network model
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)Abstract
There has been renewed interest in the dynamics and the stationary
states of attractor neural networks that process sequences of
patterns with or without pattern reconstruction. The competition
between either symmetric or asymmetric sequence processing and a
Hebbian process of strength $J_H$ that favors pattern
reconstruction has been studied in recent works. In the case of
symmetric sequence processing of strength $J_S$, that favors a
transition to both the next and to the previous pattern in the
sequence with equal strength, the works done so far are within the
dominant Hebbian regime in which $1J_H/J_Słeqınfty$,
leading to phase diagrams of stationary states which only exhibit
fixed-point solutions. In the present work 1 we study the
long-time dynamics and the stationary states, in a signal-to-noise
procedure, for all $J_H$ and $J_S$ including the regime of
dominant sequential interactions where $J_H/J_S<1$. This is
done in an exactly solvable feedforward layered neural network
model of binary units and patterns with interactions between each
unit in one layer and all units in the next one, without lateral
connections between units in the same layer, near the limit of
pattern saturation, that extends an earlier model 2. Recursion
relations for the relevant order parameters are obtained in which
the local field at a unit is a sum of a signal and a Gaussian
noise. Although an infinite number of recursion relations for the
noise is generated in the saturation limit, only a finite set
turns out to be numerically significant. The effects of the noise
and variable interaction strength on the performance of the model
are analyzed and phase diagrams of stationary states are obtained
with fixed-point solutions describing a retrieval, a symmetric and
a spin-glass phase, including a phase of correlated states 3. The
new feature is the presence of a phase of cyclic correlated states
of period two, where the correlation coefficients for
low temperature decay, but do not vanish, with the distance from a
stimulated pattern. This and the other features of the model could
have applications for a visual-memory task in the inferotemporal
cortex of monkeys 4.
1) F. L. Metz and W. K. Theumann, Phys. Rev. E 75 (2007),
in press. \\
2) E. Domany, W. Kinzel and R. Meir, J. Phys. A:Math.
Gen. 22, 2081 (1989). \\
3) L. F. Cugliandolo and M. V.
Tsodyks, J. Phys. A 27, 741 (1994). \\
4) G. Mongillo, D. J.
Amit and N. Brunel, Eur. J. Neurosci. 18, 2011 (2003), for a
review.\\
This work was partially supported by the Brazilian agencies CNPq
and FAPERGS.