17,581 research outputs found
Symmetric sequence processing in a recurrent neural network model with a synchronous dynamics
The synchronous dynamics and the stationary states of a recurrent attractor
neural network model with competing synapses between symmetric sequence
processing and Hebbian pattern reconstruction is studied in this work allowing
for the presence of a self-interaction for each unit. Phase diagrams of
stationary states are obtained exhibiting phases of retrieval, symmetric and
period-two cyclic states as well as correlated and frozen-in states, in the
absence of noise. The frozen-in states are destabilised by synaptic noise and
well separated regions of correlated and cyclic states are obtained. Excitatory
or inhibitory self-interactions yield enlarged phases of fixed-point or cyclic
behaviour.Comment: Accepted for publication in Journal of Physics A: Mathematical and
Theoretica
Instability of frozen-in states in synchronous Hebbian neural networks
The full dynamics of a synchronous recurrent neural network model with Ising
binary units and a Hebbian learning rule with a finite self-interaction is
studied in order to determine the stability to synaptic and stochastic noise of
frozen-in states that appear in the absence of both kinds of noise. Both, the
numerical simulation procedure of Eissfeller and Opper and a new alternative
procedure that allows to follow the dynamics over larger time scales have been
used in this work. It is shown that synaptic noise destabilizes the frozen-in
states and yields either retrieval or paramagnetic states for not too large
stochastic noise. The indications are that the same results may follow in the
absence of synaptic noise, for low stochastic noise.Comment: 14 pages and 4 figures; accepted for publication in J. Phys. A: Math.
Ge
Period-two cycles in a feed-forward layered neural network model with symmetric sequence processing
The effects of dominant sequential interactions are investigated in an
exactly solvable feed-forward layered neural network model of binary units and
patterns near saturation in which the interaction consists of a Hebbian part
and a symmetric sequential term. Phase diagrams of stationary states are
obtained and a new phase of cyclic correlated states of period two is found for
a weak Hebbian term, independently of the number of condensed patterns .Comment: 8 pages and 5 figure
Spatially structured oscillations in a two-dimensional excitatory neuronal network with synaptic depression
We study the spatiotemporal dynamics of a two-dimensional excitatory neuronal network with synaptic depression. Coupling between populations of neurons is taken to be nonlocal, while depression is taken to be local and presynaptic. We show that the network supports a wide range of spatially structured oscillations, which are suggestive of phenomena seen in cortical slice experiments and in vivo. The particular form of the oscillations depends on initial conditions and the level of background noise. Given an initial, spatially localized stimulus, activity evolves to a spatially localized oscillating core that periodically emits target waves. Low levels of noise can spontaneously generate several pockets of oscillatory activity that interact via their target patterns. Periodic activity in space can also organize into spiral waves, provided that there is some source of rotational symmetry breaking due to external stimuli or noise. In the high gain limit, no oscillatory behavior exists, but a transient stimulus can lead to a single, outward propagating target wave
Short term synaptic depression improves information transfer in perceptual multistability
Competitive neural networks are often used to model the dynamics of
perceptual bistability. Switching between percepts can occur through
fluctuations and/or a slow adaptive process. Here, we analyze switching
statistics in competitive networks with short term synaptic depression and
noise. We start by analyzing a ring model that yields spatially structured
solutions and complement this with a study of a space-free network whose
populations are coupled with mutual inhibition. Dominance times arising from
depression driven switching can be approximated using a separation of
timescales in the ring and space-free model. For purely noise-driven switching,
we use energy arguments to justify how dominance times are exponentially
related to input strength. We also show that a combination of depression and
noise generates realistic distributions of dominance times. Unimodal functions
of dominance times are more easily differentiated from one another using
Bayesian sampling, suggesting synaptic depression induced switching transfers
more information about stimuli than noise-driven switching. Finally, we analyze
a competitive network model of perceptual tristability, showing depression
generates a memory of previous percepts based on the ordering of percepts.Comment: 26 pages, 15 figure
Global analysis of parallel analog networks with retarded feedback
We analyze the retrieval dynamics of analog ‘‘neural’’ networks with clocked sigmoid elements and multiple signal delays. Proving a conjecture by Marcus and Westervelt, we show that for delay-independent symmetric coupling strengths, the only attractors are fixed points and periodic limit cycles. The same result applies to a larger class of asymmetric networks that may be utilized to store temporal associations with a cyclic structure. We discuss implications for various learning schemes in the space-time domain
The complexity of dynamics in small neural circuits
Mean-field theory is a powerful tool for studying large neural networks.
However, when the system is composed of a few neurons, macroscopic differences
between the mean-field approximation and the real behavior of the network can
arise. Here we introduce a study of the dynamics of a small firing-rate network
with excitatory and inhibitory populations, in terms of local and global
bifurcations of the neural activity. Our approach is analytically tractable in
many respects, and sheds new light on the finite-size effects of the system. In
particular, we focus on the formation of multiple branching solutions of the
neural equations through spontaneous symmetry-breaking, since this phenomenon
increases considerably the complexity of the dynamical behavior of the network.
For these reasons, branching points may reveal important mechanisms through
which neurons interact and process information, which are not accounted for by
the mean-field approximation.Comment: 34 pages, 11 figures. Supplementary materials added, colors of
figures 8 and 9 fixed, results unchange
Relaxation, closing probabilities and transition from oscillatory to chaotic attractors in asymmetric neural networks
Attractors in asymmetric neural networks with deterministic parallel dynamics
were shown to present a "chaotic" regime at symmetry eta < 0.5, where the
average length of the cycles increases exponentially with system size, and an
oscillatory regime at high symmetry, where the typical length of the cycles is
2. We show, both with analytic arguments and numerically, that there is a sharp
transition, at a critical symmetry \e_c=0.33, between a phase where the
typical cycles have length 2 and basins of attraction of vanishing weight and a
phase where the typical cycles are exponentially long with system size, and the
weights of their attraction basins are distributed as in a Random Map with
reversal symmetry. The time-scale after which cycles are reached grows
exponentially with system size , and the exponent vanishes in the symmetric
limit, where . The transition can be related to the dynamics
of the infinite system (where cycles are never reached), using the closing
probabilities as a tool.
We also study the relaxation of the function ,
where is the local field experienced by the neuron . In the symmetric
system, it plays the role of a Ljapunov function which drives the system
towards its minima through steepest descent. This interpretation survives, even
if only on the average, also for small asymmetry. This acts like an effective
temperature: the larger is the asymmetry, the faster is the relaxation of ,
and the higher is the asymptotic value reached. reachs very deep minima in
the fixed points of the dynamics, which are reached with vanishing probability,
and attains a larger value on the typical attractors, which are cycles of
length 2.Comment: 24 pages, 9 figures, accepted on Journal of Physics A: Math. Ge
Diversity of emergent dynamics in competitive threshold-linear networks: a preliminary report
Threshold-linear networks consist of simple units interacting in the presence
of a threshold nonlinearity. Competitive threshold-linear networks have long
been known to exhibit multistability, where the activity of the network settles
into one of potentially many steady states. In this work, we find conditions
that guarantee the absence of steady states, while maintaining bounded
activity. These conditions lead us to define a combinatorial family of
competitive threshold-linear networks, parametrized by a simple directed graph.
By exploring this family, we discover that threshold-linear networks are
capable of displaying a surprisingly rich variety of nonlinear dynamics,
including limit cycles, quasiperiodic attractors, and chaos. In particular,
several types of nonlinear behaviors can co-exist in the same network. Our
mathematical results also enable us to engineer networks with multiple dynamic
patterns. Taken together, these theoretical and computational findings suggest
that threshold-linear networks may be a valuable tool for understanding the
relationship between network connectivity and emergent dynamics.Comment: 12 pages, 9 figures. Preliminary repor
Synchronization of electrically coupled resonate-and-fire neurons
Electrical coupling between neurons is broadly present across brain areas and
is typically assumed to synchronize network activity. However, intrinsic
properties of the coupled cells can complicate this simple picture. Many cell
types with strong electrical coupling have been shown to exhibit resonant
properties, and the subthreshold fluctuations arising from resonance are
transmitted through electrical synapses in addition to action potentials. Using
the theory of weakly coupled oscillators, we explore the effect of both
subthreshold and spike-mediated coupling on synchrony in small networks of
electrically coupled resonate-and-fire neurons, a hybrid neuron model with
linear subthreshold dynamics and discrete post-spike reset. We calculate the
phase response curve using an extension of the adjoint method that accounts for
the discontinuity in the dynamics. We find that both spikes and resonant
subthreshold fluctuations can jointly promote synchronization. The subthreshold
contribution is strongest when the voltage exhibits a significant post-spike
elevation in voltage, or plateau. Additionally, we show that the geometry of
trajectories approaching the spiking threshold causes a "reset-induced shear"
effect that can oppose synchrony in the presence of network asymmetry, despite
having no effect on the phase-locking of symmetrically coupled pairs
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