Retrieval Properties of a Hopfield Model with Random Asymmetric Interactions

2000 ◽  
Vol 12 (4) ◽  
pp. 865-880 ◽  
Author(s):  
Zhang Chengxiang ◽  
Chandan Dasgupta ◽  
Manoranjan P. Singh
Keyword(s):  
Fixed Points ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Hopfield Model ◽  
Convergence Time ◽  
Memory State ◽  
Antisymmetric Component ◽  
Memory States ◽  
Pattern Retrieval ◽  
Asymmetric Interactions

The process of pattern retrieval in a Hopfield model in which a random antisymmetric component is added to the otherwise symmetric synaptic matrix is studied by computer simulations. The introduction of the antisymmetric component is found to increase the fraction of random inputs that converge to the memory states. However, the size of the basin of attraction of a memory state does not show any significant change when asymmetry is introduced in the synaptic matrix. We show that this is due to the fact that the spurious fixed points, which are destabilized by the introduction of asymmetry, have very small basins of attraction. The convergence time to spurious fixed-point attractors increases faster than that for the memory states as the asymmetry parameter is increased. The possibility of convergence to spurious fixed points is greatly reduced if a suitable upper limit is set for the convergence time. This prescription works better if the synaptic matrix has an antisymmetric component.

Fixed points in a Hopfield model with random asymmetric interactions

1995 ◽  
Vol 52 (5) ◽  
pp. 5261-5272 ◽  
Author(s):  
Manoranjan P. Singh ◽  
Zhang Chengxiang ◽  
Chandan Dasgupta
Keyword(s):  
Fixed Points ◽  
Hopfield Model ◽  
Asymmetric Interactions

scholarly journals External Stimuli on Neural Networks: Analytical and Numerical Approaches

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 1034
Author(s):  
Evaldo Mendonça Curado ◽  
Nilo Barrantes Melgar ◽  
Fernando Dantas Nobre
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
External Environment ◽  
Basins Of Attraction ◽  
Hopfield Model ◽  
New Approach ◽  
The Neural Network ◽  
Asymmetric Interactions ◽  
External Stimuli ◽  
Numerical Approaches

Based on the behavior of living beings, which react mostly to external stimuli, we introduce a neural-network model that uses external patterns as a fundamental tool for the process of recognition. In this proposal, external stimuli appear as an additional field, and basins of attraction, representing memories, arise in accordance with this new field. This is in contrast to the more-common attractor neural networks, where memories are attractors inside well-defined basins of attraction. We show that this procedure considerably increases the storage capabilities of the neural network; this property is illustrated by the standard Hopfield model, which reveals that the recognition capacity of our model may be enlarged, typically, by a factor 102. The primary challenge here consists in calibrating the influence of the external stimulus, in order to attenuate the noise generated by memories that are not correlated with the external pattern. The system is analyzed primarily through numerical simulations. However, since there is the possibility of performing analytical calculations for the Hopfield model, the agreement between these two approaches can be tested—matching results are indicated in some cases. We also show that the present proposal exhibits a crucial attribute of living beings, which concerns their ability to react promptly to changes in the external environment. Additionally, we illustrate that this new approach may significantly enlarge the recognition capacity of neural networks in various situations; with correlated and non-correlated memories, as well as diluted, symmetric, or asymmetric interactions (synapses). This demonstrates that it can be implemented easily on a wide diversity of models.

scholarly journals Behavioral Time Scale Plasticity of Place Fields: Mathematical Analysis

2021 ◽  
Vol 15 ◽  
Author(s):  
Ian Cone ◽  
Harel Z. Shouval
Keyword(s):  
Synaptic Plasticity ◽  
Fixed Points ◽  
Time Scales ◽  
Convergence Time ◽  
Simple Mathematical Model ◽  
System Parameters ◽  
Temporal Correlations ◽  
Postsynaptic Activity ◽  
Naturally Occurring ◽  
Place Fields

Traditional synaptic plasticity experiments and models depend on tight temporal correlations between pre- and postsynaptic activity. These tight temporal correlations, on the order of tens of milliseconds, are incompatible with significantly longer behavioral time scales, and as such might not be able to account for plasticity induced by behavior. Indeed, recent findings in hippocampus suggest that rapid, bidirectional synaptic plasticity which modifies place fields in CA1 operates at behavioral time scales. These experimental results suggest that presynaptic activity generates synaptic eligibility traces both for potentiation and depression, which last on the order of seconds. These traces can be converted to changes in synaptic efficacies by the activation of an instructive signal that depends on naturally occurring or experimentally induced plateau potentials. We have developed a simple mathematical model that is consistent with these observations. This model can be fully analyzed to find the fixed points of induced place fields and how these fixed points depend on system parameters such as the size and shape of presynaptic place fields, the animal's velocity during induction, and the parameters of the plasticity rule. We also make predictions about the convergence time to these fixed points, both for induced and pre-existing place fields.

scholarly journals Fixed Points, Symmetries, and Bounds for Basins of Attraction of Complex Trigonometric Functions

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Kasey Bray ◽  
Jerry Dwyer ◽  
Roger W. Barnard ◽  
G. Brock Williams
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Basins Of Attraction ◽  
Trigonometric Functions ◽  
Fractal Image ◽  
Fractal Images

The dynamical systems of trigonometric functions are explored, with a focus on tz=tanz and the fractal image created by iterating the Newton map, Ftz, of  tz. The basins of attraction created from iterating  Ftz are analyzed, and some bounds are determined for the primary basins of attraction. We further prove x- and y-axis symmetry of the Newton map and explore the nature of the fractal images.

scholarly journals Robustness of Supercavitating Vehicles Based on Multistability Analysis

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Yipin Lv ◽  
Tianhong Xiong ◽  
Wenjun Yi ◽  
Jun Guan
Keyword(s):  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Small Range ◽  
Practical Application ◽  
Stable States ◽  
Supercavitating Vehicles ◽  
Control Gain ◽  
Stable Periodic ◽  
Setting Parameters

Supercavity can increase speed of underwater vehicles greatly. However, external interferences always lead to instability of vehicles. This paper focuses on robustness of supercavitating vehicles. Based on a 4-dimensional dynamic model, the existence of multistability is verified in supercavitating system through simulation, and the robustness of vehicles varying with parameters is analyzed by basins of attraction. Results of the research disclose that the supercavitating system has three stable states in some regions of parameters space, namely, stable, periodic, and chaotic states, while in other regions it has various multistability, such as coexistence of two types of stable equilibrium points, coexistence of a limit cycle with a chaotic attractor, and coexistence of 1-periodic cycle with 2-periodic cycle. Provided that cavitation number varies within a small range, with increase of the feedback control gain of fin deflection angle, size of basin of attraction becomes smaller and robustness of the system becomes weaker. In practical application, robustness of supercavitating vehicles can be improved by setting parameters of system or adjusting initial launching conditions.

On-Off Intermittency and Riddled Basins of Attraction in a Coupled Map System

1998 ◽  
Vol 01 (02n03) ◽  
pp. 161-180 ◽  
Author(s):  
J. Laugesen ◽  
E. Mosekilde ◽  
Yu. L. Maistrenko ◽  
V. L. Maistrenko
Keyword(s):  
Periodic Orbits ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Chaotic State ◽  
One Dimensional ◽  
Type Iii ◽  
Different Types ◽  
One Dimensional Maps ◽  
The Basin Of Attraction

The paper examines the appearance of on-off intermittency and riddled basins of attraction in a system of two coupled one-dimensional maps, each displaying type-III intermittency. The bifurcation curves for the transverse destablilization of low periodic orbits embeded in the synchronized chaotic state are obtained. Different types of riddling bifurcation are discussed, and we show how the existence of an absorbing area inside the basin of attraction can account for the distinction between local and global riddling as well as for the distinction between hysteric and non-hysteric blowout. We also discuss the role of the so-called mixed absorbing area that exists immediately after a soft riddling bifurcation. Finally, we study the on-off intermittency that is observed after a non-hysteric blowout bifurcaton.

Sharp upper bounds on perfect retrieval in the Hopfield model

1999 ◽  
Vol 36 (03) ◽  
pp. 941-950 ◽  
Author(s):  
Anton Bovier
Keyword(s):  
Fixed Points ◽  
Lower Bounds ◽  
Upper Bound ◽  
Random Variables ◽  
Upper Bounds ◽  
Negative Association ◽  
Hopfield Model ◽  
Gradient Dynamics ◽  
One Step ◽  
The One

We prove a sharp upper bound on the number of patterns that can be stored in the Hopfield model if the stored patterns are required to be fixed points of the gradient dynamics. We also show corresponding bounds on the one-step convergence of the sequential gradient dynamics. The bounds coincide with the known lower bounds and confirm the heuristic expectations. The proof is based on a crucial idea of Loukianova (1997) using the negative association properties of some random variables arising in the analysis.

scholarly journals Attractor-state itinerancy in neural circuits with synaptic depression

2019 ◽  
Author(s):  
Bolun Chen ◽  
Paul Miller
Keyword(s):  
Fixed Points ◽  
Synaptic Depression ◽  
Basins Of Attraction ◽  
Dependent Manner ◽  
Neural Populations ◽  
History Of ◽  
Point Attractor ◽  
Dynamical Features ◽  
The Stability ◽  
Multiple Fixed Points

AbstractNeural populations with strong excitatory recurrent connections can support bistable states in their mean firing rates. Multiple fixed points in a network of such bistable units can be used to model memory retrieval and pattern separation. The stability of fixed points may change on a slower timescale than that of the dynamics due to short-term synaptic depression, leading to transitions between quasi-stable point attractor states in a sequence that depends on the history of stimuli. To better understand these behaviors, we study a minimal model, which characterizes multiple fixed points and transitions between them in response to stimuli with diverse time- and amplitude-dependences. The interplay between the fast dynamics of firing rate and synaptic responses and the slower timescale of synaptic depression makes the neural activity sensitive to the amplitude and duration of square-pulse stimuli in a non-trivial, history-dependent manner. Weak cross-couplings further deform the basins of attraction for different fixed points into intricate shapes. Our analysis provides a natural explanation for the system’s rich responses to stimuli of different durations and amplitudes while demonstrating the encoding capability of bistable neural populations for dynamical features of incoming stimuli.

scholarly journals Basins of Attraction for Various Steffensen-Type Methods

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Alicia Cordero ◽  
Fazlollah Soleymani ◽  
Juan R. Torregrosa ◽  
Stanford Shateyi
Keyword(s):  
Computer Algebra System ◽  
Dynamical Behavior ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Numerical Tests ◽  
Derivative Free ◽  
Iteration Functions ◽  
The Stability ◽  
Programming Package ◽  
The Basin Of Attraction

The dynamical behavior of different Steffensen-type methods is analyzed. We check the chaotic behaviors alongside the convergence radii (understood as the wideness of the basin of attraction) needed by Steffensen-type methods, that is, derivative-free iteration functions, to converge to a root and compare the results using different numerical tests. We will conclude that the convergence radii (and the stability) of Steffensen-type methods are improved by increasing the convergence order. The computer programming package MATHEMATICAprovides a powerful but easy environment for all aspects of numerics. This paper puts on show one of the application of this computer algebra system in finding fixed points of iteration functions.

scholarly journals Computing and Controlling Basins of Attraction in Multistability Scenarios

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
John Alexander Taborda ◽  
Fabiola Angulo
Keyword(s):  
Fixed Point ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Control Technique ◽  
Unstable Periodic Orbits ◽  
Control Effort ◽  
Coexistence Of Attractors ◽  
And Control

The aim of this paper is to describe and prove a new method to compute and control the basins of attraction in multistability scenarios and guarantee monostability condition. In particular, the basins of attraction are computed only using a submap, and the coexistence of periodic solutions is controlled through fixed-point inducting control technique, which has been successfully used until now to stabilize unstable periodic orbits. In this paper, however, fixed-point inducting control is used to modify the domains of attraction when there is coexistence of attractors. In order to apply the technique, the periodic orbit whose basin of attraction will be controlled must be computed. Therefore, the fixed-point inducting control is used to stabilize one of the periodic orbits and enhance its basin of attraction. Then, using information provided by the unstable periodic orbits and basins of attractions, the minimum control effort to stabilize the target periodic orbit in all desired ranges is computed. The applicability of the proposed tools is illustrated through two different coupled logistic maps.

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