Recurrent predictive coding models for associative memory employing covariance learning

Abstract

The computational principles adopted by the hippocampus in associative memory (AM) tasks have been one of the most studied topics in computational and theoretical neuroscience. Recent theories suggested that AM and the predictive activities of the hippocampus could be described within a unitary account, and that predictive coding underlies the computations supporting AM in the hippocampus. Following this theory, a computational model based on classical hierarchical predictive networks was proposed and was shown to perform well in various AM tasks. However, this fully hierarchical model did not incorporate recurrent connections, an architectural component of the CA3 region of the hippocampus that is crucial for AM. This makes the structure of the model inconsistent with the known connectivity of CA3 and classical recurrent models such as Hopfield Networks, which learn the covariance of inputs through their recurrent connections to perform AM. Earlier PC models that learn the covariance information of inputs explicitly via recurrent connections seem to be a solution to these issues. Here, we show that although these models can perform AM, they do it in an implausible and numerically unstable way. Instead, we propose alternatives to these earlier covariance-learning predictive coding networks, which learn the covariance information implicitly and plausibly, and can use dendritic structures to encode prediction errors. We show analytically that our proposed models are perfectly equivalent to the earlier predictive coding model learning covariance explicitly, and encounter no numerical issues when performing AM tasks in practice. We further show that our models can be combined with hierarchical predictive coding networks to model the hippocampo-neocortical interactions. Our models provide a biologically plausible approach to modelling the hippocampal network, pointing to a potential computational mechanism during hippocampal memory formation and recall, which employs both predictive coding and covariance learning based on the recurrent network structure of the hippocampus.

Predictive Coding Networks for Temporal Prediction

“…x and Σ −1 y could be encoded in the learnt A and C matrices, similarly as it has been shown in static predictive coding models [48]. Thus, the tPC model can represent the covariance matrices implicitly in its synaptic weights, without needing to implement them in explicit synaptic weights.…”

“…Although earlier works have proposed to encode the noise precision matrices Σ −1 x and Σ −1 y into additional connections explicitly [9], this approach would introduce extra complexity into the neural implementation of tPC. On the other hand, it has been shown that in the static case, the noise precision matrix can be implicitly encoded in recurrent connections similar to the A matrix in our tPC model [48], without needing to represent the precision matrix explicitly as in [9]. Therefore, here we investigate whether A and C can encode the precision matrices of the process and observation noise respectively after learning.…”

Predictive coding networks for temporal prediction

“…The update rules for the A , B , and C weight matrices ( Eq 11 ) are also precisely Hebbian, since they are outer products between the prediction errors and the value neurons of the layer above which, crucially, are also precisely the pre and post-synaptic activities of the neurons where the synapses implementing these weight matrices are located. Moreover, we empirically demonstrate in the Results sections that scaling by the inverse covariance matrices and could be encoded in the learnt A and C matrices, similarly as it has been shown in static predictive coding models [ 48 ]. Thus, the tPC model can represent the covariance matrices implicitly in its synaptic weights, without needing to implement them in explicit synaptic weights.…”

“…Although earlier works have proposed to encode the noise precision matrices and into additional connections explicitly [ 9 ], this approach would introduce extra complexity into the neural implementation of tPC. On the other hand, it has been shown that in the static case, the noise precision matrix can be implicitly encoded in recurrent connections similar to the A matrix in our tPC model [ 48 ], without needing to represent the precision matrix explicitly as in [ 9 ]. Therefore, here we investigate whether A and C can encode the precision matrices of the process and observation noise respectively after learning.…”

“…Overall, these results suggest that the synaptic plasticity of tPC can encode the noise covariance in the generative process, without representing them explicitly in the dynamics and circuit implementations. It is also interesting to investigate the exact theoretical relationship between the weights and noise covariance, similar to the analytical relationship in [ 48 ], and we intend to investigate it in future explorations.…”