Strong attractors of Hopfield neural networks to model attachment types and behavioural patterns

Abstract

Abstract-We study the notion of a strong attractor of a Hopfield neural model as a pattern that has been stored multiple times in the network, and examine its properties using basic mathematical techniques as well as a variety of simulations. It is proposed that strong attractors can be used to model attachment types in developmental psychology as well as behavioural patterns in psychology and psychotherapy. We study the stability and basins of attraction of strong attractors in the presence of other simple attractors and show that they are indeed more stable with a larger basin of attraction compared with simple attractors. We also show that the perturbation of a strong attractor by random noise results in a cluster of attractors near the original strong attractor measured by the Hamming distance. We investigate the stability and basins of attraction of such clusters as the noise increases and establish that the unfolding of the strong attractor, leading to its breakup, goes through three different stages. Finally the relation between strong attractors of different multiplicity and their influence on each other are studied and we show how the impact of a strong attractor can be replaced with that of a new strong attractor. This retraining of the network is proposed as a model of how attachment types and behavioural patterns can undergo change.

Introduction to self-attachment and its neural basis

“…In order to represent a deeply or repeatedly learned pattern or prototype, the notion of a strong pattern of the discrete Hopfield network was introduced in [32], [33] as a pattern that has been multiply stored in the network. These patterns result in strong and stable attractors with a large basin of attraction.…”

“…It was shown in [32] that the energy of the strongest pattern stored in a network decreases linearly with the degree (multiplicity) of the strong pattern. For simplicity let's focus on the case that ξ µ i for µ = 1, .…”

“…Then, the distribution of error in the stability of ξ 1 when N → ∞ and d 1 p is given by (cf. [32]):…”

“…In [32], [33], in order to use the Hopfield network as a model for attachment and behavioural prototypes, the creation of a strong pattern is identified as the formation of a prototype and the increase in the degree of the strong pattern as the strengthening of the prototype, which takes place as a result of repetition of training, corresponding to long term potentiation. For example consider the following experiment.…”

“…As soon as the multiplicity of the happy face exceeds that of the sad face, it would become more likely that the happy face is retrieved by any initial random pattern. Change of attachment or behavioural types has been modelled in this way in [32], [33]. It can also be used to model the process of psychotherapy in which a more positive cognitive and emotional prototype is gradually entrenched by repetition and will eventually outweigh an existing negative prototype.…”