We consider the storage properties of temporal patterns, i.e. cycles of finite lengths, in neural networks represented by (generally asymmetric) spin glasses defined on random graphs. Inspired by the observation that dynamics on sparse systems have more basins of attractions than the dynamics of densely connected ones, we consider the attractors of a greedy dynamics in sparse topologies, considered as proxy for the stored memories. We enumerate them using numerical simulation and extend the analysis to large systems sizes using belief propagation. We find that the logarithm of the number of such cycles is a non monotonic function of the mean connectivity and we discuss the similarities with biological neural networks describing the memory capacity of the hippocampus.
We introduce a bipartite, diluted and frustrated, network as a sparse restricted Boltzman machine and we show its thermodynamical equivalence to an associative working memory able to retrieve multiple patterns in parallel without falling into spurious states typical of classical neural networks. We focus on systems processing in parallel a finite (up to logarithmic growth in the volume) amount of patterns, mirroring the low-level storage of standard Amit-Gutfreund-Sompolinsky theory. Results obtained trough statistical mechanics, signal-to-noise technique and Monte Carlo simulations are overal…
The comprehension of the mechanisms at the basis of the functioning of complexly interconnected networks represents one of the main goals of neuroscience. In this work, we investigate how the structure of recurrent connectivity influences the ability of a network to have storable patterns and in particular limit cycles, by modeling a recurrent neural network with McCulloch-Pitts neurons as a content-addressable memory system. A key role in such models is played by the connectivity matrix, which, for neural networks, corresponds to a schematic representation of the "connectome": the set of chem…
We consider a generalization of the Hopfield model, where the entries of patterns are Gaussian and diluted. We focus on the high-storage regime and we investigate analytically the topological properties of the emergent network, as well as the thermodynamic properties of the model. We find that, by properly tuning the dilution in the pattern entries, the network can recover different topological regimes characterized by peculiar scalings of the average coordination number with respect to the system size. The structure is also shown to exhibit a large degree of cliquishness, even when very spars…
In this paper we study, via equilibrium statistical mechanics, the properties of the internal energy of an Hopfield neural network whose patterns are stored continuously (Gaussian distributed). The model is shown to be equivalent to a bipartite spin glass in which one party is given by dichotomic neurons and the other party by Gaussian spin variables. Dealing with replicated systems, beyond the Mattis magnetization, we introduce two overlaps, one for each party, as order parameters of the theory: The first is a standard overlap among neural configurations on different replicas, the second is a…
Neural networks are dynamical systems that compute with their dynamics. One example is the Hopfield model, forming an associative memory which stores patterns as global attractors of the network dynamics. From studies of dynamical networks it is well known that localized attractors also exist. Yet, they have not been used in computing paradigms. Here we show that interacting localized attractors in threshold networks can result in universal computation. We develop a rewiring algorithm that builds universal Boolean gates in a biologically inspired two-dimensional threshold network with randomly…
An analogy between crowd synchrony and multi-layer neural network architectures is proposed. It indicates that many non-identical dynamical elements (oscillators) communicating indirectly via a few mediators (hubs) can synchronize when the number of delayed couplings to the hubs or the strength of the couplings is large enough. This phenomenon is modeled using a system of semiconductor lasers optically delay-coupled in either a fully connected or a diluted manner to a fixed number of non-identical central hub lasers. A universal phase transition to crowd synchrony with hysteresis is observed, …
Storing Infinite Dynamical Attractors in Nonreciprocal Associative Neural Networks
BCAM – Basque Center for Applied Mathematics（巴斯克应用数学中心）；IKERBASQUE, Basque Foundation for Science（伊克尔巴斯基克科学基金会）；Biofisika Institute (CSIC, EHU)（生物物理研究所（西班牙国家研究委员会、巴斯克大学））
