Fig 1.
The noise repetition–detection task proposed by Agus et al. (2010).
(a) A conceptual diagram of the task and stimuli types. Participants listen to a 1 s white noise stimulus and answer whether the first and second 0.5 s segments are the same. There are four stimuli types, divided by the presence or absence of repetition within the stimulus (RN/N) and the presence or absence of repetition throughout the session (Referenced). As RN and RefRN have a repetitive structure, the correct answer is “Yes”, whereas for N and RefN, the correct answer is “No”. (b) The concept art of repetition detection score time course in the NRD task. The detection score for RefRN improves to a nearly perfect score immediately, whereas the score for RN remains around chance–level. As the participants do not know about the presence of the referenced stimuli and the feedback for their answers is not given, this task can be regarded as implicit unsupervised perceptual learning.
Fig 2.
(a) The process of input signal formation. There are four stimuli types: N and RefN stimuli consist of 1 s of white noise, whereas RN and RefRN stimuli consist of 0.5 s repetitions of white noise. The sampling frequency is 44 kHz. Subsequently, each time series is passed through an A–weighting filter, which reflects human auditory characteristics, peaking around 3,000 Hz, with high frequencies attenuated. The middle figure shows the resulting power spectra of before (gray) and after (black) A–weighting filter used in the simulation. After filter adaptation, each stimulus was resampled at 2,000 Hz to reduce computational costs. (b) An overview of the model. The resampled time series are presented to the neuron in the input layer as stimuli. W, the reservoir weights matrix is dynamic and maintained by Oja’s Hebbian plasticity rule. Wout, the weights between the reservoir and neurons in the output layer are optimized using the gradient descent method. The model’s output target is one step ahead of prediction of the input time series.
Fig 3.
Representative examples of output time–series data.
These are outputs for RefRN of Hebbian networks, although the similar tendency was observed for different network types and stimulus types. The results for three distinct spectral radii (ρ = 0.1, 1.2, 2.0) are plotted separately. Each graph plots the overlaid output values on the vertical axis against the first 200 time points on the horizontal axis. The different colors correspond to three different output trials. Time is not plotted in its entirety, but rather, the first 200 points are magnified and plotted. When ρ is low, the lines converge following the transient period, indicating the identical response trajectory regardless of variations in the initial states. As ρ increases, the network no longer has ESP and behaves completely differently across distinct trials.
Fig 4.
The evaluation of selective consistency.
The consistency was evaluated by correlation between the first and second segment time series for each test run with repeated noise (RN; left) and referenced repeated noise (RefRN; right). (a) The evaluation at the level of the output neuron. The violin plots show probability density distributions and interquartile ranges of Hebbian (right side; magenta and brown) and non–Hebbian (left side; green and cyan), respectively (****; PR < 0.01%, p < 0.001). The colored line plots connect the mean values for each condition. The black lines in the bottom windows show the difference between Hebbian and non–Hebbian models. The horizontal axis represents the spectral radius of the evaluated networks. (b) The evaluation of the five randomly selected reservoir neurons. Each dotted line represents five distinct neurons. The solid lines represent the mean value for these five neurons.
Fig 5.
The edge of chaos and selective consistency.
Each bin of the histograms shows averaged inter–segment correlation for four conditions (magenta; RN of Hebbian network, green; RN of non–Hebbian network, brown; RefRN of Hebbian network, cyan; RefRN of non–Hebbian network). The histograms represent results from networks with spectral radii of 0.9, 1.4, and 1.9, from left to right—which can be described as stable, edge of chaos, and chaotic, respectively (****; PR < 0.01%, p < 0.001). The error bars are representing 95% confidence intervals. Notably, the edge of chaos region was chosen for its strong observation of selective consistency (see Fig 4). The statistical significances were tested with the nonparametric rank–order test based on the surrogate data model technique. It was found that in networks that are either stable or conversely chaotic, there was no difference between conditions, and differences were observed only in the edge of chaos region.
Fig 6.
The evaluation of selective consistency without optimizing the readout weights.
The figure styles are the same as Fig 4.The consistency was evaluated by the correlation between the first and second segment time series for each test run for repeated noise (RN; left) and referenced repeated noise (RefRN; right). The violin plots show probability density distributions and interquartile ranges of Hebbian (right side; magenta and brown) and non–Hebbian (left side; green and cyan) models, respectively (****; PR < 0.01%, p < 0.001). The colored line plots connect the mean values for each condition. The black lines in the bottom windows show the difference between Hebbian and non–Hebbian models. The horizontal axis represents the spectral radius of the evaluated networks.
Fig 7.
Prediction error with varying spectral radii.
(a) Root mean squared error (RMSE) series and (b) normalized root mean square error (NRMSE) series. The plots represent the averaged RMSE or NRMSE between network prediction and the correct future time series. Non–plastic and plastic models are shown on the left and right, respectively. The results of RN and RefRN are represented by cyan and brown lines, respectively. The 95% confidence intervals are depicted as light cyan (RN) and pink (RefRN) filled areas.