Figures and data
Unsupervised prior learning in a recurrent neural network.
(a) A schematic of a network model is shown. The interconnected circles denote the model neurons, of which the activities are controlled by two types of inputs: feedforward (FF) and recurrent (REC) inputs. Colored circles indicate active neurons. Here, W denotes FF, and M and G denote REC connections. We considered two modes of activity (i.e., evoked and spontaneous activity). In the evoked mode, the membrane potential u of a network neuron was calculated as a linear combination of inputs across all different connections (vW, vM, and vG). This evoked mode is considered during the learning phase, when all synapses attempt to predict the network activity, as we will explain in the main text. Once all synapses are sufficiently learned, all FF inputs are removed, and the network is driven spontaneously (spontaneous mode). Our interest lies in the statistical similarity of the network activity in these two modes. (b) The gain and threshold of output response function was controlled by a dynamic variable, h, which tracks the history of the membrane potential. (c) A schematic of the learning rule for a network neuron is shown (top). During learning, for each type of connection on a postsynaptic neuron, synaptic plasticity minimizes the error between output (gray diamond) and synaptic prediction (colored diamonds). Note that all types of synapses share the common plasticity rule, where weight updates are calculated as the multiplication of the error term and the presynaptic activities (bottom). Our hypothesis is that such plasticity rule allows a recurrent neural network to spontaneously replay the learned stochastic activity patterns without external input.
Formation of stimulus-selective assemblies in a recurrent network.
(a) Example dynamics of neuronal output and synaptic predictions are shown before (left) and after (right) learning. Colored bars at the top of the figures represent periods of stimulus presentations. (b) Example dynamics of feedforward connection W and inhibitory connection G are shown. W-connections onto neurons organizing to encode the same or different input patterns are shown in red and blue, respectively. Similarly, the same colors are used to represent G connections within and between assemblies. (c) Dynamics of the mean connection strengths are shown on neuron in cell assembly 1. Shaded areas represent SDs. In the schematic, triangles indicate input neurons and circles indicate network neurons. The color of each neuron indicates the stimulus preference of each neuron. (d) Example dynamics of the averaged dynamical variable 
Priors coded in spontaneous activity.
An nDL network was trained with five probabilistic inputs. (a) Stimulus 1 appeared twice as often as the other four stimuli during learning. The example empirical probabilities of the stimuli used for learning are shown. (b) The spontaneous activity of the trained network shows distinct assembly structures. (c) The mean ratio of the population-averaged firing rate of assembly 1 to those of the other assemblies is shown for different values of the occurrence probability of stimulus 1. Vertical bars show SDs over five trials. A diagonal dashed line is a ground truth. (d) Similarly, the mean ratios of the size of assembly 1 to those of the other assemblies are shown. (e) The mean ratios of the total activities of neurons in assembly 1 to those of the other assemblies are shown. (f) Five stimuli occurring with different probabilities were used for training the nDL model. (g) The population firing rates are shown for five self-organized cell assemblies encoding the stimulus probabilities shown in (f).
Probability encoding by learned within-assembly synapses.
(a) Two input stimuli were presented in two protocols: uniform (50% vs. 50%) or biased (30% vs. 70%). (b) The total incoming synaptic strength on each neuron was calculated within each cell assembly. (c) left, The distributions of incoming synaptic strength are shown for the learned assemblies in the 50-vs-50 case. right, Same as in the left figure, but in the 30-vs-70 case. (d) left, The empirical probabilities of stimuli 1 and 2 and the normalized excitatory incoming weights within assemblies are compared in the 50-vs-50 case. right, Same as in the left figure, but in the 30-vs-70 case.
Simulations of biased perception of visual motion coherence.
(a) The network model simulated perceptual decision-making of coherence in random dot motion patterns. In the network shown here, network neurons have already learned two assemblies encoding leftward or rightward movements from input neuron groups L and R. The firing rates of input neuron groups were modulated according to the coherence level Coh of random dot motion patterns (Materials and Methods). (b) The choice probabilities of monkeys (circles) and the network model (solid lines) are plotted against the motion coherence in two learning protocols with different prior probabilities. The experimental data were taken from Hanks et al. (2011). In the 50:50 protocol, moving dots in the “R” (Coh = 0.5) and “L” (Coh = -0.5) directions were presented randomly with equal probabilities, while in the 80:20 protocol, the “R” and “L” directions were trained with 80% and 20% probabilities, respectively. Shaded areas represent SDs over 20 independent simulations. The computational and experimental results show surprising coincidence without curve fitting. (c) Spontaneous and evoked activities of the trained networks are shown for the 50:50 (left) and 80:20 (right) protocols. Evoked responses were calculated for three levels of coherence: Coh = -50%, 0%, and 50%. In both protocols, the activity ratio in spontaneous activity matches the prior probability and gives the baseline for evoked responses. In the 80:20 protocol, the biased priors of “R” and “L” motion stimuli shift the activity ratio in spontaneous activity to an “R”-dominant regime.
A network model with distinct excitatory and inhibitory connections.
(a) Strong excitatory connections were formed within assemblies. (b) The first type of recurrent inhibitory connections, Minh, became stronger between assemblies, enhancing assembly competition. (c) The second type of inhibitory connections G were strengthened within assemblies to balance the strong excitatory inputs.
The DL model of excitatory and inhibitory cell assemblies.
(a) This model consists of distinct excitatory and inhibitory neuron pools, obeying Dale’s law. (b) Each inhibitory neuron projects to another neuron X through two inhibitory paths, path 1 and path 2, where the index X refers to an excitatory or an inhibitory postsynaptic neuron. Hexagons represent minimal units for prediction and learning in the neuron model and may correspond to dendrites, which were not modeled explicitly. (c) The probability ratios estimated by numerical simulations are plotted for the assembly activity ratios (purple), firing rate ratios (cyan), and assembly size ratios (green) as functions of the true probability ratio of external stimuli. Error bars indicate SEs calculated over five simulation trials with different initial states of neurons and synaptic weights in each parameter setting. (d) Inhibitory connection matrices are shown for path 1 and path 2. (e) The mean weights of self-organized synapses on excitatory and inhibitory postsynaptic neurons are shown. (f) Within-assembly and between-assembly connectivity patterns of excitatory and inhibitory neurons are shown. Colors indicate three cell assemblies self-organized. (g) The strengths of lateral inhibitions within-(W/N) and between-assemblies (B/N) are shown for paths 1 and 2. Horizontal bars show the medians and quartiles. (h) The resultant connectivity pattern suggests an effective competitive network between excitatory assemblies with self-(within-assembly) and lateral (between-assembly) inhibition.
