Life on the Edge: Latching Dynamics in a Potts Neural Network

Abstract

Abstract:We study latching dynamics in the adaptive Potts model network, through numerical simulations with randomly and also weakly correlated patterns, and we focus on comparing its slowly and fast adapting regimes. A measure, Q, is used to quantify the quality of latching in the phase space spanned by the number of Potts states S, the number of connections per Potts unit C and the number of stored memory patterns p. We find narrow regions, or bands in phase space, where distinct pattern retrieval and duration of latching combine to yield the highest values of Q. The bands are confined by the storage capacity curve, for large p, and by the onset of finite latching, for low p. Inside the band, in the slowly adapting regime, we observe complex structured dynamics, with transitions at high crossover between correlated memory patterns; while away from the band latching transitions lose complexity in different ways: below, they are clear-cut but last so few steps as to span a transition matrix between states with few asymmetrical entries and limited entropy; while above, they tend to become random, with large entropy and bi-directional transition frequencies, but indistinguishable from noise. Extrapolating from the simulations, the band appears to scale almost quadratically in the p − S plane, and sublinearly in p − C. In the fast adapting regime the band scales similarly, and it can be made even wider and more robust, but transitions between anti-correlated patterns dominate latching dynamics. This suggest that slow and fast adaptation have to be integrated in a scenario for viable latching in a cortical system. The results for the slowly adapting regime, obtained with randomly correlated patterns, remain valid also for the case with correlated patterns, with just a simple shift in phase space.

Latching dynamics as a basis for short-term recall

“…Moreover, we see that whether we consider only very slow or only very fast inhibition, as in previous analytical studies [ 19 , 22 ], or a more plausible balance of the two, the network behaves similarly in terms of short-term memory function. Based on this observation, hereafter we only concentrate on the balanced, or intermediate regime ( γ A = 0.5).…”

“…We first ask ourselves: once the network is cued with a given pattern, what elicits the retrieval of the next one? In previous studies [ 19 , 22 ], it was shown that transitions occur most frequently between highly correlated patterns, when the Potts model serves a long-term memory function. We confirmed that this is also the case when the Potts model serves a short-term memory function, as in the current study ( S9 Fig ).…”

“…The quality of latching is evaluated by means of d 12 − Q . d 12 is the difference between the largest overlap and the next largest one, averaged over time and over so called quenched variables [ 22 ], while is the average overlap with the next L patterns, since . is the overlap of the network activity with a pattern μ i and μ 1 , …, μ L −1 are the L − 1 patterns having largest overlaps excluding the maximum overlap.…”

“…The unit-specific thresholds and describe local inhibition, which in the cortex is relayed by at least 3 main classes of inhibitory interneurons [ 21 ] acting on GABA A and GABA B receptors, with widely different time courses, from very short to very long. In the Potts network it has proved convenient, in order to separate time scales, to consider either very slow or very fast inhibition [ 19 , 22 ]. Here, we consider a more realistic case in which both slow and fast inhibition are taken into account.…”

Continuous attractors for dynamic memories

“…The production of sequences of discrete memories can be implemented with a heteroassociative component ( Sompolinsky and Kanter, 1986 ), usually dependent on the time integral of the instantaneous activity, that brings the network out of equilibrium and to the next step in the sequence. A similar effect can be obtained with an adaptation mechanism in a coarse grained model of cortical networks ( Kropff and Treves, 2005 ), with the difference that in this case the transitions are not imposed, but driven by the correlations between the memories in so-called latching dynamics ( Russo et al, 2008 ; Kang et al, 2017 ). Moreover, adaptation-based mechanisms have been used to model the production of random sequences on continuous manifolds ( Azizi et al, 2013 ), and shown to be crucial in determining the balance between retrieval and prediction in a network describing CA3-CA1 interactions ( Treves, 2004 ).…”