A Nonequilibrium-Potential Approach to Competition in Neural Populations

Abstract

Energy landscapes are a useful aid for the understanding of dynamical systems, and a valuable tool for their analysis. For a broad class of rate models of neural networks, we derive a global Lyapunov function which provides an energy landscape without any symmetry constraint. This newly obtained "nonequilibrium potential" (NEP) predicts with high accuracy the outcomes of the dynamics in the globally stable cases studied here. Common features of the models in this class are bistability-with implications for working memory and slow neural oscillations-and "population burst", also relevant in neuroscience. Instead, limit cycles are not found. Their nonexistence can be proven by resort to the Bendixson-Dulac theorem, at least when the NEP remains positive and in the (also generic) singular limit of these models. Hopefully, this NEP will help understand average neural network dynamics from a more formal standpoint, and will also be of help in the description of large heterogeneous neural networks.

Enhancing energy harvesting by coupling monostable oscillators

“…We have studied the effect of coupling between units and sought the optimal configuration (number N of coupled units) in order to enhance the system's energy harvesting. In agreement with [23], whilst diffusive coupling between units reduces the system's performance, anti-diffusive couplings cause an enhancement. Next we found that it does so via the mechanism of a diffusive instability.…”

“…After confirming our preliminary evidence [23] that the performance of antidiffusively coupled units can be notably enhanced as compared with the non-coupled case, we ask ourselves what the optimal number N of (antidiffusively) coupled units is. For our surprise however (see Fig.…”

“…coupling strength k < 0), outperforms in-phase one (i.e. k > 0, diffusive or ferromagnetic like), [23].…”

“…Several recent works [7,8,9,10,11] have stressed non-linearity as a resource to overcome this problem, specially if the vibration statistics is broader than Gaussian [12,13,14]. Considering moreover that for the phenomenon of stochastic resonance [15,16]-a paradigm of the constructive interplay between stochasticity and nonlinearity-coupling between units enhances the collective response [17,18,19,20,21,22], here too one finds that coupling can boost the overall efficiency [23,24,25].…”

The nonequilibrium potential today: A short review

“…Among its virtues, it yields a visual criterion for the EW-KPZ crossover. (2) The Wilson-Cowan model of the neocortex-describing the competition between excitatory and inhibitory neural populations-is generically a non-relaxational potential system admitting a bona-fide NEP [63]. In Section 4, we illustrate the usefulness of the NEP concept to draw results on stochastic thermodynamics by deriving a Jarzynski equality [64] in the Wilson-Cowan model and a thermodynamic uncertainty relation (TUR) [65] in the KPZ equation.…”