Predictive Coding Theories of Cortical Function

Predictive coding is a unifying framework for understanding perception, action and neocortical organization. In predictive coding, different areas of the neocortex implement a hierarchical generative model of the world that is learned from sensory inputs. Cortical circuits are hypothesized to perform Bayesian inference based on this generative model. Specifically, the Rao-Ballard hierarchical predictive coding model assumes that the top-down feedback connections from higher to lower order cortical areas convey predictions of lower-level activities. The bottom-up, feedforward connections in turn convey the errors between top-down predictions and actual activities. These errors are used to correct current estimates of the state of the world and generate new predictions. Through the objective of minimizing prediction errors, predictive coding provides a functional explanation for a wide range of neural responses and many aspects of brain organization

Expressive probabilistic sampling in recurrent neural networks

In sampling-based Bayesian models of brain function, neural activities are assumed to be samples from probability distributions that the brain uses for probabilistic computation. However, a comprehensive understanding of how mechanistic models of neural dynamics can sample from arbitrary distributions is still lacking. We use tools from functional analysis and stochastic differential equations to explore the minimum architectural requirements for $\textit{recurrent}$ neural circuits to sample from complex distributions. We first consider the traditional sampling model consisting of a network of neurons whose outputs directly represent the samples (sampler-only network). We argue that synaptic current and firing-rate dynamics in the traditional model have limited capacity to sample from a complex probability distribution. We show that the firing rate dynamics of a recurrent neural circuit with a separate set of output units can sample from an arbitrary probability distribution. We call such circuits reservoir-sampler networks (RSNs). We propose an efficient training procedure based on denoising score matching that finds recurrent and output weights such that the RSN implements Langevin sampling. We empirically demonstrate our model's ability to sample from several complex data distributions using the proposed neural dynamics and discuss its applicability to developing the next generation of sampling-based brain models

Dynamic Predictive Coding Explains Both Prediction and Postdiction in Visual Motion Perception

Due to transmission delays, the perceptual information our brain can access quickly becomes outdated as events unfold in real-time. We suggest our perceptual system learns internal representations that encode sequences (or timelines) rather than single points to compensate for transmission de- lays. Specifically, we investigate the dynamic predictive coding (DPC) model in which high-level states predict the transition dynamics of lower-level states and represent lower-level state sequences. We show that a two-level DPC network trained to predict videos captures several aspects of the well-known flash-lag illusion and exhibits both predictive and postdictive effects resembling those observed in human visual motion processing. Our results support the view that visual perception relies on temporally abstracted representations that encode sequences (or timelines) rather than single time points

Comparing Alternative Computational Models of the Stroop TaskUsing Effective Connectivity Analysis of fMRI Data

Methodological advances have made it possible to generatefMRI predictions for cognitive architectures, such as ACT-R, thus expanding the range of model predictions and mak-ing it possible to distinguish between alternative models thatproduce otherwise identical behavioral patterns. However, fortasks associated with relatively brief response times, fMRI pre-dictions are often not sufficient to compare alternative models.In this paper, we outline a method based on effective connec-tivity, which significantly augments the amount of informationthat can be extracted from fMRI data to distinguish betweenmodels. We show the application of this method in the caseof two competing ACT-R models of the Stroop task. Althoughthe models make, predictably, identical behavioral and BOLDtime-course predictions, patterns of functional connectivity fa-vor one model over the other. Finally, we show that the samedata suggests directions in which both models should be re-vised

Cue-triggered activity recall in the DPC model.

(a) The experimental setup of Xu et al. (adapted from [1]). A bright dot stimulus moved from START to END repeatedly during conditioning. Activities of neurons whose receptive fields (colored ellipses) were along the dot’s trajectory were recorded. (b) Generative model combining an associative memory and DPC. The red part denotes the augmented memory component that binds the initial content vector r0 and the dynamics vector rh to encode an episodic memory. (c) Depiction of the memory encoding process. The presynaptic memory activity and postsynaptic prediction error jointly shape the memory weights G. (d) Depiction of the recall process. Prediction error on the partial observation drives the convergence of the memory estimates and recalls the higher-level dynamics vector as a top-down prediction. The red dotted box depicts the prediction error between the missing observations for rh and the prediction ; this error is ignored during recall, implementing a form of robust predictive coding [49]. (e) The image sequence used to simulate conditioning and testing for our memory-augmented DPC network. (f) Responses of the lower-level neurons of the network. Colored lines represent the five most active lower-level neurons at each step. Left to right: neural responses during conditioning, testing the network with a single start frame, middle frame, and end frame. (g, h) Normalized pairwise cross correlation of (g) primary visual cortex neurons (adapted from [1]) and (h) the lower-level model neurons. Top: during conditioning; middle two: testing with the starting stimulus, before and after conditioning; bottom: the differences between cross correlations, “After” minus “Before” conditioning.</p

Three-level DPC model learns progressively more abstract temporal representations.

(a) Generative model for three-level DPC. (b) Schematic depiction of an inference process. Observation nodes are omitted for clarity. (c) Inference for an example Moving MNIST sequence with “straight bouncing” dynamics. Red time steps mark the moments when the first-level prediction error exceeded the threshold, causing the network to transition to a new second-level state (see Methods). For these time steps, the predictions (second row) are by the second-level neurons, while the rest are by the first-level neurons as in Fig 3. (d) The network’s responses to the Moving MNIST sequence in (c). Left to right: first-level responses, second-level responses, third-level responses, first-level modulation weights, second-level modulation weights. (e) Same as (d) but with “clockwise bouncing” dynamics. (f) Same as (d) but for the sequence in (e). (g) Third-level responses to the Moving MNIST sequences visualized in the 2D space of the first two principal components. Left: responses colored according to bouncing type; right: responses colored according to motion direction. (h) Comparison of decoding performance for bouncing type versus motion direction using the modulation weights generated by the second and third level. Error bars: ±1 standard deviation from 10-fold cross validation. Orange: chance accuracies.</p

Flash-lag illusion and object representations in apparent motion.

(a) The flash-lag test conditions used by [26]. The moving ring could have an initial trajectory (top) or no trajectory (bottom). At the time of the flash (bright disk), the ring could move along the initial trajectory, stop, or reverse its trajectory. Adapted from [26]. (b) Two test conditions (left) regarding initial trajectories of the moving object (a digit) in the flash-lag experiment with the model, and four test conditions (right) for the moving object. The flashed object was shown at time t and turned off at time t + 1 (same as the “Terminate” condition). (c & d) Psychophysical estimates for human subjects reported by [26] when the moving object had initial trajectories (c) or no initial trajectory (d). (e) Perceived location of the flashed object in the DPC model at time t + 1. The error bar indicates ±1 standard deviation (measured across presentations of different digits). (f) Perceived displacement between the moving object (with initial trajectories) and the flashed object in the DPC model for the four test conditions. (g) Same as (f) but with no initial trajectory for the moving object. (h) Illustration of the prediction-error-driven dynamics of the perception of the moving object in the model when the trajectory reversed at time t + 1. Red ellipsis between panels denotes the prediction error minimization process. (i) Interference pattern during human apparent motion perception with continuous motion (left) and reversed motion (right) at short latency (fast detection task). Brighter color denotes more interference. Dashed arrows represent object motion direction. Adapted from [29]. (j) Same as (i) but at long latency (slow discrimination task) [29]. (k) Perceived location of the moving object in the DPC model at time t + 1 probed at short versus long latency during prediction error minimization. Positive values denote distance along the original trajectory. Negative values denote distance along the reversed trajectory. Short and long latency correspond to “Early percept” and “Late percept” respectively in part (h). (l) Perceived location of the digit at all latencies during the prediction error minimization process in part (h).</p

Hierarchical temporal representation with different timescales.

(a) Autocorrelation of the lower- and higher-level responses in the trained network with natural videos. Shaded area denotes ±1 standard deviation. Dotted lines show fitted exponential decay functions. Left: response recorded during natural video stimuli; right: white noise stimuli. (b) Autocorrelation of the neural responses recorded from MT and LPFC of monkeys. Adapted from Murray et al. [6] (c) Inference for an example Moving MNIST sequence in a trained network. The red dashed boxes mark the time steps when the dynamics of the input changed. (d) The network’s responses to the input Moving MNIST sequence in (c). Note the changes in the higher-level responses after the input dynamics changed (red dashed boxes); this gradient-based change helps to minimize prediction errors. (e) Higher-level responses to the Moving MNIST sequences visualized in the 2D space of the first two principal components. Left: responses colored according to motion direction; right: responses colored according to digit identities. (f) Comparison of decoding performance for motion direction versus digit identity using lower- and higher-level neural responses. Error bars: ±1 standard deviation from 10-fold cross validation. Orange: chance accuracies.</p

Supporting information.

Fig A. Improvement on test set loss saturates as the number of transition matrices increases. (a) Test set loss as training proceeded. Shaded area denotes ±1 standard deviation computed over eight runs with random initialization for each K. K = 1 shows the performance of the single-layer model. (b) Best test loss as K increases. Error bars denote ±1 standard deviation. Fig B. Cue-triggered recall is cue-specific. Four examples of cue-specific sequence recall by the associative memory model after training on different sequences, when given the first frame as the cue. In each quadrant: top: the original image sequence; bottom: cue-triggered recall of the stored sequence. Fig C. Prediction error threshold robustly finds changes of dynamics. (a) The distribution of first-level prediction errors in the two-level DPC model on the Moving MNIST training set. The red dashed line denotes the threshold ρ = 0.73, where the cumulative density reaches 0.75. (b) Examples of input sequences in the test set. The red arrows mark time steps when the first-level prediction errors exceeded ρ, corresponding to changes in input dynamics. Table A. DPC generative model parameters and values. Table B. Optimizers and learning rates used for inference and learning in the DPC experiments. Here Δ denotes the difference in rt or rh from before and after the current iteration of gradient descent. Table C. Memory model parameters and values. Table D. Optimizers and learning rates used for inference and learning in the memory model. Here Δ denotes the difference in m from before and after the current iteration of gradient descent. Table E. Additional parameters and values for the three-level DPC model. Table F. Additional optimizers and learning rates used for inference and learning in the three-level DPC experiments. Algorithm A. Inference & learning process. Algorithm B. Inference & learning process for the three-level DPC model. (PDF)</p