Adaptive balancing of exploration and exploitation around the edge of chaos in internal-chaos-based learning

Learning performance of cerebellar circuit depends on diversity and chaoticity of spiking patterns in granule cells: A simulation study

The cerebellum, composed of numerous neurons, plays various roles in motor control. Although it is functionally subdivided, the cerebellar cortex has a canonical structural pattern in neuronal circuits including a recurrent circuit pattern formed by granule cells (GrCs) and Golgi cells (GoCs). The canonical circuital pattern suggests the existence of a fundamental computational algorithm, although it remains unclear. Modeling and simulation studies are useful for verifying hypotheses about complex systems. Previous models have shown that they could reproduced the neurophysiological data of the cerebellum; however, the dynamic characteristics of the system have not been fully elucidated. Understanding the dynamic characteristics of the circuital pattern is necessary to reveal the computational algorithm embedded in the circuit. This study conducted numerical simulations using the cerebellar circuit model to investigate dynamic characteristics in a simplified model of cerebellar microcircuits. First, the diversity and chaoticity of the patterns of spike trains generated from GrCs depending on the synaptic strength between the GrCs and GoCs were investigated based on cluster analysis and the Lyapunov exponent, respectively. Then the effect of synaptic strength on learning tasks was investigated based on the convergence properties of the output signals from Purkinje cells. The synaptic strength for high learning performance was almost consistent with that for the high diversity of the generated patterns and the edge of chaos. These results suggest that the learning performance of the cerebellar circuit depends on the diversity and the chaoticity of the spiking patterns from the GrC-GoC recurrent circuit.

Chaotic recurrent neural networks for brain modelling: A review

Even in the absence of external stimuli, the brain is spontaneously active. Indeed, most cortical activity is internally generated by recurrence. Both theoretical and experimental studies suggest that chaotic dynamics characterize this spontaneous activity. While the precise function of brain chaotic activity is still puzzling, we know that chaos confers many advantages. From a computational perspective, chaos enhances the complexity of network dynamics. From a behavioural point of view, chaotic activity could generate the variability required for exploration. Furthermore, information storage and transfer are maximized at the critical border between order and chaos. Despite these benefits, many computational brain models avoid incorporating spontaneous chaotic activity due to the challenges it poses for learning algorithms. In recent years, however, multiple approaches have been proposed to overcome this limitation. As a result, many different algorithms have been developed, initially within the reservoir computing paradigm. Over time, the field has evolved to increase the biological plausibility and performance of the algorithms, sometimes going beyond the reservoir computing framework. In this review article, we examine the computational benefits of chaos and the unique properties of chaotic recurrent neural networks, with a particular focus on those typically utilized in reservoir computing. We also provide a detailed analysis of the algorithms designed to train chaotic RNNs, tracing their historical evolution and highlighting key milestones in their development. Finally, we explore the applications and limitations of chaotic RNNs for brain modelling, consider their potential broader impacts beyond neuroscience, and outline promising directions for future research.

Chaotic neural dynamics facilitate probabilistic computations through sampling

Cortical neurons exhibit highly variable responses over trials and time. Theoretical works posit that this variability arises potentially from chaotic network dynamics of recurrently connected neurons. Here, we demonstrate that chaotic neural dynamics, formed through synaptic learning, allow networks to perform sensory cue integration in a sampling-based implementation. We show that the emergent chaotic dynamics provide neural substrates for generating samples not only of a static variable but also of a dynamical trajectory, where generic recurrent networks acquire these abilities with a biologically plausible learning rule through trial and error. Furthermore, the networks generalize their experience in the stimulus-evoked samples to the inference without partial or all sensory information, which suggests a computational role of spontaneous activity as a representation of the priors as well as a tractable biological computation for marginal distributions. These findings suggest that chaotic neural dynamics may serve for the brain function as a Bayesian generative model.

Fractal Analysis of Local Activity and Chaotic Motion in Nonlinear Nonplanar Vibrations for Cantilever Beams

Many problems in practical engineering can be simplified as the cantilever beam model, which is generally studied by theoretical analysis, experiment, and numerical simulation. This paper discusses the local activity of the nonlinear nonplanar motion of a cantilever beam at the equilibrium point. Firstly, the equilibrium point of the model and the Jacobian matrix have been calculated. The stability of the characteristic root corresponding to the characteristic polynomial has been analyzed. Secondly, the corresponding complexity function of the model at the equilibrium point has been given. Then, the local activity region of the model at the equilibrium point can be obtained by using the theory of the local activity. Based on the actual engineering research background, the damping coefficient is generally taken as 0 < c < 1. The cantilever beam model is the local activity at the equilibrium point only if the parameters of the model satisfy a certain condition. In the numerical simulation, it is found that when the proper parameters are selected in the local activity region, the cantilever beam can exhibit different types of chaotic motion. The local activity theory provides a theoretical basis for the parameter selection of the chaotic motion in the cantilever beam.

Sensitivity - Local index to control chaoticity or gradient globally

Here, we introduce a fully local index named "sensitivity" for each neuron to control chaoticity or gradient globally in a neural network (NN). We also propose a learning method to adjust it named "sensitivity adjustment learning (SAL)". The index is the gradient magnitude of its output with respect to its inputs. By adjusting its time average to 1.0 in each neuron, information transmission in the neuron changes to be moderate without shrinking or expanding for both forward and backward computations. That results in moderate information transmission through a layer of neurons when the weights and inputs are random. Therefore, SAL can control the chaoticity of the network dynamics in a recurrent NN (RNN). It can also solve the vanishing gradient problem in error backpropagation (BP) learning in a deep feedforward NN or an RNN. We demonstrate that when applying SAL to an RNN with small and random initial weights, log-sensitivity, which is the logarithm of RMS (root mean square) sensitivity over all the neurons, is equivalent to the maximum Lyapunov exponent until it reaches 0.0. We also show that SAL works with BP or BPTT (BP through time) to avoid the vanishing gradient problem in a 300-layer NN or an RNN that learns a problem with a lag of 300 steps between the first input and the output. Compared with manually fine-tuning the spectral radius of the weight matrix before learning, SAL's continuous nonlinear learning nature prevents loss of sensitivities during learning, resulting in a significant improvement in learning performance.