Fig 1.
Neural implementations of the single-layer covPCNs using error neurons.
xa and εa denotes the a-th value/error neuron in the networks. For simplicity we show the case where the input patterns have only 2 dimensions, corresponding to 2 value and error neurons. A: explicit covPCN originally proposed in [16]. B: Implicit covPCN. C: Dendritic covPCN with direct structural mapping to a pyramidal cell. Unlabeled connections have strengths 1.
Fig 2.
We use the single-layer implicit/dendritic covPCN to model the hippocampus, and a hierarchical PCN from [12] to model the sensory cortex and neocortex. Neurons and synapses in the hierarchical layers follow the dynamic rules in [12]. For clarity of demonstration, only one layer of our neocortex model is shown. Expanded boxes show the detailed computations within individual neurons and related synapses specified in Eqs 20 and 22, where denotes the a-th neuron in the lth layer, and Θab and Wab denote the individual weights from the ath to the bth neurons. Dog image in this figure is obtained from Wikimedia Commons under a CC BY 4.0 license.
Fig 3.
Performance of covPCNs in AM of random patterns, and the equivalence between them.
A: A subset of 5 × 5 random patterns memorized by all 3 models. After training, we corrupted the bottom 2 rows (10 pixels) and let the networks run inference on the corrupted parts for retrieval. B: Retrieval MSEs of the models when corrupted with different mask sizes. Experiments in A and B are performed with networks with d = 25 neurons. C: Sample covariance of a random 2-dimensional dataset and the learned weight matrices of an explicit model and an implicit/dendritic model on this dataset. D: The random 2-dimensional dataset to memorize, and the linear retrieval obtained by masking the second dimension x2 by all 3 models, as well as the theoretical retrieval line. All the lines overlap as they are equivalent in theory. Experiments in C and D are performed with networks with d = 2 neurons.
Fig 4.
Performance of the single-layer covPCNs in AM of structured images.
A: Examples of retrieved MNIST (top) [19] and grayscale CIFAR10 (bottom)[20] images by explicit, implicit and dendritic models. All models here are trained to memorize 64 images. For MNIST, the networks have d = 784 neurons; for grayscale CIFAR10, d = 1024. B: Retrieval mean squared errors (MSEs) of the single-layer models across multiple numbers of training memories (N). C: Evolution of the retrieval MSEs of the implicit and dendritic models when N = 256. D: Example eigenspectra of the weight matrices defining the inferential dynamics for the dendritic (left) and implicit (right) covPCNs. Error bars obtained by 5 different seeds for image sampling. Please see main text for an explanation of the matrix M.
Fig 5.
Performance of the multi-layer models.
A: Demonstration of how we keep the number of parameters across different models to be roughly the same. B: Retrieval mean squared errors (MSEs) of the multi-layer models across multiple numbers of training memories (N). The curve for the implicit model is the same as the one in Fig 4. C: Evolution of the retrieval MSEs of the implicit and dendritic hybrid models when N = 256. Error bars obtained by 5 different seeds for image sampling.
Fig 6.
Comparison of linear and nonlinear implicit covPCN.
A: Performance of linear and nonlinear implicit models in the completion task with varying Ns. B: Same as A, but with the denoising task, where cues are memories with Gaussian noise of variance 0.1. C: A simple 3-dimensional example, where stars are data points the networks were trained to memorize. After training we ran inference on both linear and nonlinear models, initialized with grid test data drawn from the range [−1, 1]3. The position of the test data at convergence of inference indicates the shape of attractors. Images taken from the CIFAR10 dataset [20].
Table 1.
Summary of covPCNs discussed in this work.