2017/02/10 by Diego Fasoli, Fasoli, Diego, Anna Cattani +3
Computer Science · Engineering · Neuroscience · #Advanced Memory and Neural Computing #FOS: Biological sciences #Neural Networks and Applications #Neural dynamics and brain function #Neurons and Cognition (q-bio.NC)
paper · pdf · doi:10.48550/arxiv.1702.03183
openalex publication_date 2017/02/10 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
Exactly solvable neural network models with asymmetric weights are rare, and\nexact solutions are available only in some mean-field approaches. In this\narticle we find exact analytical solutions of an asymmetric spin-glass-like\nmodel of arbitrary size and we perform a complete study of its dynamical and\nstatistical properties. The network has discrete-time evolution equations,\nbinary firing rates and can be driven by noise with any distribution. We find\nanalytical expressions of the conditional and stationary joint probability\ndistributions of the membrane potentials and the firing rates. The conditional\nprobability distribution of the firing rates allows us to introduce a new\nlearning rule to store safely, under the presence of noise, point and cyclic\nattractors, with important applications in the field of content-addressable\nmemories. Furthermore, we study the neuronal dynamics in terms of the\nbifurcation structure of the network. We derive analytically examples of the\ncodimension one and codimension two bifurcation diagrams of the network, which\ndescribe how the neuronal dynamics changes with the external stimuli. In\nparticular, we find that the network may undergo transitions among multistable\nregimes, oscillatory behavior elicited by asymmetric synaptic connections, and\nvarious forms of spontaneous symmetry-breaking. On the other hand, the joint\nprobability distributions allow us to calculate analytically the higher-order\ncorrelation structure of the network, which reveals neuronal regimes where,\nstatistically, the membrane potentials and the firing rates are either\nsynchronous or asynchronous. Our results are valid for networks composed of an\narbitrary number of neurons, but for completeness we also derive the network\nequations in the mean-field limit and we study analytically their local\nbifurcations. All the analytical results are extensively validated by numerical\nsimulations.\n