Transition to chaos separates learning regimes and relates to measure of consciousness in recurrent neural networks.

Recurrent neural networks exhibit chaotic dynamics when the variance in their connection strengths exceed a critical value. Recent work indicates connection variance also modulates learning strategies; networks learn "rich" representations when initialized with low coupling and "lazier" solutions with larger variance. Using Watts-Strogatz networks of varying sparsity, structure, and hidden weight variance, we find that the critical coupling strength dividing chaotic from ordered dynamics also differentiates rich and lazy learning strategies. Training moves both stable and chaotic networks closer to the edge of chaos, with networks learning richer representations before the transition to chaos. In contrast, biologically realistic connectivity structures foster stability over a wide range of variances. The transition to chaos is also reflected in a measure that clinically discriminates levels of consciousness, the perturbational complexity index (PCIst). Networks with high values of PCIst exhibit stable dynamics and rich learning, suggesting a consciousness prior may promote rich learning. The results suggest a clear relationship between critical dynamics, learning regimes and complexity-based measures of consciousness.

Edge of Chaos and Avalanches in Neural Networks with Heavy-Tailed Synaptic Weight Distribution.

We propose an analytically tractable neural connectivity model with power-law distributed synaptic strengths. When threshold neurons with biologically plausible number of incoming connections are considered, our model features a continuous transition to chaos and can reproduce biologically relevant low activity levels and scale-free avalanches, i.e., bursts of activity with power-law distributions of sizes and lifetimes. In contrast, the Gaussian counterpart exhibits a discontinuous transition to chaos and thus cannot be poised near the edge of chaos. We validate our predictions in simulations of networks of binary as well as leaky integrate-and-fire neurons. Our results suggest that heavy-tailed synaptic distribution may form a weakly informative sparse-connectivity prior that can be useful in biological and artificial adaptive systems.

Erratum: Subdiffusive continuous-time random walks with stochastic resetting [Phys. Rev. E 99, 052116 (2019)].

This corrects the article DOI: 10.1103/PhysRevE.99.052116.

Time-dependent density of diffusion with stochastic resetting is invariant to return speed.

The canonical Evans-Majumdar model for diffusion with stochastic resetting to the origin assumes that resetting takes zero time: upon resetting the diffusing particle is teleported back to the origin to start its motion anew. However, in reality getting from one place to another takes a finite amount of time which must be accounted for as diffusion with resetting already serves as a model for a myriad of processes in physics and beyond. Here we consider a situation where upon resetting the diffusing particle returns to the origin at a finite (rather than infinite) speed. This creates a coupling between the particle's random position at the moment of resetting and its return time, and further gives rise to a nontrivial cross-talk between two separate phases of motion: the diffusive phase and the return phase. We show that each of these phases relaxes to the steady state in a unique manner; and while this could have also rendered the total relaxation dynamics extremely nontrivial, our analysis surprisingly reveals otherwise. Indeed, the time-dependent distribution describing the particle's position in our model is completely invariant to the speed of return. Thus, whether returns are slow or fast, we always recover the result originally obtained for diffusion with instantaneous returns to the origin.

Robust random search with scale-free stochastic resetting.

A new model of search based on stochastic resetting is introduced, wherein rate of resets depends explicitly on time elapsed since the beginning of the process. It is shown that rate inversely proportional to time leads to paradoxical diffusion which mixes self-similarity and linear growth of the mean-square displacement with nonlocality and non-Gaussian propagator. It is argued that such resetting protocol offers a general and efficient search-boosting method that does not need to be optimized with respect to the scale of the underlying search problem (e.g., distance to the goal) and is not very sensitive to other search parameters. Both subdiffusive and superdiffusive regimes of the mean-squared displacement scaling are demonstrated with more general rate functions.

Subdiffusive continuous-time random walks with stochastic resetting.

We analyze two models of subdiffusion with stochastic resetting. Each of them consists of two parts: subdiffusion based on the continuous-time random walk scheme and independent resetting events generated uniformly in time according to the Poisson point process. In the first model the whole process is reset to the initial state, whereas in the second model only the position is subject to resets. The distinction between these two models arises from the non-Markovian character of the subdiffusive process. We derive exact expressions for the two lowest moments of the full propagator, stationary distributions, and first hitting time statistics. We also show, with an example of a constant drift, how these models can be generalized to include external forces. Possible applications to data analysis and modeling of biological systems are also discussed.

Thermodynamics of Superdiffusion Generated by Lévy-Wiener Fluctuating Forces.

Scale free Lévy motion is a generalized analogue of the Wiener process. Its time derivative extends the notion of "white noise" to non-Gaussian noise sources, and as such, it has been widely used to model natural signal variations described by an overdamped Langevin stochastic differential equation. Here, we consider the dynamics of an archetypal model: a Brownian-like particle is driven by external forces, and noise is represented by uncorrelated Lévy fluctuations. An unperturbed system of that form eventually attains a steady state which is uniquely determined by the set of parameter values. We show that the analyzed Markov process with the stability index α < 2 violates the detailed balance, i.e., its stationary state is quantified by a stationary probability density and nonvanishing current. We discuss consequences of the non-Gibbsian character of the stationary state of the system and its impact on the general form of the fluctuation-dissipation theorem derived for weak external forcing.

Learning with three factors: modulating Hebbian plasticity with errors.

Synaptic plasticity is a central theme in neuroscience. A framework of three-factor learning rules provides a powerful abstraction, helping to navigate through the abundance of models of synaptic plasticity. It is well-known that the dopamine modulation of learning is related to reward, but theoretical models predict other functional roles of the modulatory third factor; it may encode errors for supervised learning, summary statistics of the population activity for unsupervised learning or attentional feedback. Specialized structures may be needed in order to generate and propagate third factors in the neural network.

Emergence of Lévy Walks from Second-Order Stochastic Optimization.

In natural foraging, many organisms seem to perform two different types of motile search: directed search (taxis) and random search. The former is observed when the environment provides cues to guide motion towards a target. The latter involves no apparent memory or information processing and can be mathematically modeled by random walks. We show that both types of search can be generated by a common mechanism in which Lévy flights or Lévy walks emerge from a second-order gradient-based search with noisy observations. No explicit switching mechanism is required-instead, continuous transitions between the directed and random motions emerge depending on the Hessian matrix of the cost function. For a wide range of scenarios, the Lévy tail index is α=1, consistent with previous observations in foraging organisms. These results suggest that adopting a second-order optimization method can be a useful strategy to combine efficient features of directed and random search.

Optimal first-arrival times in Lévy flights with resetting.

We consider the diffusive motion of a particle performing a random walk with Lévy distributed jump lengths and subject to a resetting mechanism, bringing the walker to an initial position at uniformly distributed times. In the limit of an infinite number of steps and for long times, the process converges to superdiffusive motion with replenishment. We derive a formula for the mean first arrival time (MFAT) to a predefined target position reached by a meandering particle and we analyze the efficiency of the proposed searching strategy by investigating criteria for an optimal (a shortest possible) MFAT.

"Cargo-mooring" as an operating principle for molecular motors.

Navigating through an ever-changing and unsteady environment, and utilizing chemical energy, molecular motors transport the cell׳s crucial components, such as organelles and vesicles filled with neurotransmitter. They generate force and pull cargo, as they literally walk along the polymeric tracks, e.g. microtubules. What we suggest in this paper is that the motor protein is not really pulling its load. The load is subject to diffusion and the motor may be doing little else than rectifying the fluctuations, i.e. ratcheting the load׳s diffusion. Below we present a detailed model to show how such ratcheting can quantitatively account for observed data. The consequence of such a mechanism is the dependence of the transport׳s speed and efficacy not only on the motor, but also on the cargo (especially its size) and on the environment (i.e. its viscosity and structure). Current experimental works rarely provide this type of information for in vivo studies. We suggest that even small differences between assays can impact the outcome. Our results agree with those obtained in wet laboratories and provide novel insight in a molecular motor׳s functioning.

First Order Transition for the Optimal Search Time of Lévy Flights with Resetting.

We study analytically an intermittent search process in one dimension. There is an immobile target at the origin and a searcher undergoes a discrete time jump process starting at x_{0}≥0, where successive jumps are drawn independently from an arbitrary jump distribution f(η). In addition, with a probability 0≤r<1, the position of the searcher is reset to its initial position x_{0}. The efficiency of the search strategy is characterized by the mean time to find the target, i.e., the mean first passage time (MFPT) to the origin. For arbitrary jump distribution f(η), initial position x_{0} and resetting probability r, we compute analytically the MFPT. For the heavy-tailed Lévy stable jump distribution characterized by the Lévy index 0<µ<2, we show that, for any given x_{0}, the MFPT has a global minimum in the (µ,r) plane at (µ^{*}(x_{0}),r^{*}(x_{0})). We find a remarkable first-order phase transition as x_{0} crosses a critical value x_{0}^{*} at which the optimal parameters change discontinuously. Our analytical results are in good agreement with numerical simulations.