Capacity of strong attractor patterns to model behavioural and cognitive prototypes
We solve the mean field equations for a stochastic Hopfield network with temperature (noise) in the presence of strong, i.e., multiply stored, patterns, and use this solution to obtain the storage capacity of such a network. Our result provides for the first time a rigorous solution of the mean filed equations for the standard Hopfield model and is in contrast to the mathematically unjustifiable replica technique that has been used hitherto for this derivation. We show that the critical temperature for stability of a strong pattern is equal to its degree or multiplicity, when the sum of the squares of degrees of the patterns is negligible compared to the network size. In the case of a single strong pattern, when the ratio of the number of all stored pattens and the network size is a positive constant, we obtain the distribution of the overlaps of the patterns with the mean field and deduce that the storage capacity for retrieving a strong pattern exceeds that for retrieving a simple pattern by a multiplicative factor equal to the square of the degree of the strong pattern. This square law property provides justification for using strong patterns to model attachment types and behavioural prototypes in psychology and psychotherapy.
In this paper the authors consider applying a mean 
      field approach to a Hopfield model which has presented with strong 
      patterns (patterns that have been presented to the model more than once). 
      By using a mean field approach and applying the Lyapunov condition they 
      show that they can arrive at mean field equations for two cases: 
      1) When the model is presented with a finite collection of strong 
      patterns and a collection of simple patterns such that the total number of 
      stored patterns over the total number of neurons tends to zero as the 
      number of neurons becomes large. 2) When the model is presented 
      with a single strong pattern and a collection of simple patterns such that 
      the total number of stored patterns over the total number of neurons tends 
      to some constant alpha as the number of neurons becomes large. In 
      both cases they arrive at self consistent equations, in the first case 
      they then evaluate how noise level changes the models ability to recall 
      the strong patterns and in the second they look at how the constant alpha 
      changes the recovery of the strong patterns. Quality 
      ------------------------------------------------------------ There 
      a few typos and bad sentences in this paper (see below) but on the whole 
      the quality of the writing is OK. My main problems, however, are with the 
      proof/statement of lemma 4.1 and the use of lemma 4.1 in the proof of 
      theorem 4.3. Proof/Statement of lemma 4.1 
      --------------------------------- I think that lemma 4.1 should 
      have m_mu instead of m_nu for it to be applied later to theorem 4.3 and 
      I'm going to assume this in what follows. I think the proof of lemma 4.1 
      does not work. In this proof the authors assume that \xi^1, \xi_i^\mu and 
      m_\mu are independent, this is not true since m_{\mu} is constructed from 
      a sum including \xi_i^\mu, granted this is a dependency that becomes 
      weaker with increasing N, but it is still present and should be addressed. 
      Secondly the authors state that m_\nu are independent, I think this is an 
      assumption rather than a statement since eqn 11 shows that one may be 
      written as a function of the other m_\mu. Proof of Theorem 4.3 
      -------------------------------------- I think that eqn 15 should 
      have as the sum index mu\neq 1 and not mu\neq 1,\nu. In which case either 
      lemma 4.1 cannot be applied at eqn 16 or the statement of lemma 4.1 must 
      be changed. If this can be addressed then I agree that the rest of the 
      proof of 4.3 follows. Typos etc -------------------------- 
      - Index should be j around eqns 2 and 3 - I would like the authors 
      to mention for some constant m or similar after the first eqn in section 3 
      - Statement of lemma 4.1 (see above) - square at end of eqn 13 
      should be at the end of the proof - brackets around the expectation in 
      eqn 13 - lower limits of the integrals in eqn 12 and the text after 
      eqn 13 - The paragraph above theorem 4.3 is missing a word or two I 
      think - Eqn 15 (see theorem 4.3 above) - The + symbol in the 
      variance just above and in eqn 18. Clarity 
      --------------------------------------- Mistakes, typos, lemma 4.1 
      and the beginning of the proof of theorem 4.3 aside the paper is clear and 
      well written. I think that if the problems outlined above were addressed 
      the paper would present a concise and comprehensible discussion on the 
      storage of strong patterns in the Hopfield model. Originality 
      --------------------------------------- As far as I know this 
      material is original. Significance 
      ---------------------------------------------- The results seem 
      significant enough to be published.