Storage capacity of a neural network with state-dependent synapses

A replica approach to the state-dependent synapses neural network

“…These two assumptions are incompatible for η = 0. Indeed, following closely the derivation in [5,6] by starting from the mean-field equations for the thermal average of the overlap with a non-condensed pattern, m ν N , and expanding it in a Taylor series to first order we arrive at…”

“…The energy function (5) has been used in [5] to derive fixed-point equations for the relevant order parameters using the statistical mean-field scheme [6,7]. The key idea of the latter calculation is to assume that the noise to which the small overlaps with the non-condensed patterns add up is Gaussian.…”

“…Comparing the fixed-point equations (13)- (14) with those of the statistical meanfield scheme derived in [5] (see egs. (23)-(26)) we find that they are different as soon as η = 0.…”

“…In order to lift this limitation of the Hebb rule a model with state-dependent synapses has been discussed recently [4,5]. The idea thereby is to introduce a threshold η cutting out of the Hebb rule all patterns whose correlations with the state of the system are smaller than this threshold [5]. These authors propose an energy function for this state-dependent synapse (SDS) model and derive the corresponding fixed-point equations using the so-called heuristically motivated statistical mean-field scheme developed in [6,7] (see also [8]).…”

“…In section 2 the SDS-model is shortly reviewed. In section 3 the RS approach to this model at zero temperature is outlined and a detailed comparison with the statistical method used in [5] is made. Section 4 contains a discussion of the RSB1 solution.…”