Kinetic lattice models of disorder

Neural networks with fast time-variation of synapses

“…As an extension of previous work on disordered systems [16,20,26], we have presented and studied a model for associative memory. This is a generalized, kinetic version of the Hopfield neural network in the sense that the synapse intensities do not remain constant after the learning process but fluctuate with time during neuron activity.…”

“…This corresponds to a situation in which, once the learning process is complete, the synapses intensities change (fluctuate) very fast as compared with neuron changes, in such a way that one may distinguish two well defined timescales. That is, there is a microscopic timescale, τ , for the fluctuations of the synapses, in which neurons do not appreciably evolve, and a different scale, t = pτ for p → 0 and τ → ∞, in which neurons evolve under a steady distribution for the synapses; we refer elsewhere [16] for a detailed study of such separation of timescales. Under such condition, which, as discussed in section 1, can be interpreted as an oversimplification of the actual situation in neurobiology, (6) transforms into…”

“…In other words, the system reduces in the limit p → 0 to the Markovian process (15) with effective rate (16). Note that such a result relies on the assumption that (J xy → J xy ) in (6) is independent of the current neuron configuration; therefore, the stationary solution of (14) for the fast variables does not depend on the more slowly varying neurons [16]. The only motivation for our restriction to such a non-generic case here is simplicity.…”

Effects of Fast Presynaptic Noise in Attractor Neural Networks

“…This choice (Garrido & Marro, 1994;Torres, Garrido, & Marro, 1997) amounts to considering competing mechanisms. That is, neurons (S) evolve stochastically in time under a noisy dynamics of synapses (X), the latter 1 Note that such binary neurons, although a crude simplification of nature, are known to capture the essentials of cooperative phenomena, which is the focus here.…”

“…First, there is adiabatic elimination of fast variables for p → 0, which decouples the two dynamics (Garrido & Marro, 1994;Gardiner, 2004). Therefore, an exact analytical treatment-though not the complete solution-is then feasible.…”