using Go and Python language at the prior link. These same equations and
standard parameters have been used to simulate over 40 different models in
O’Reilly . (2012); O’Reilly Munakata (2000), and a number of other research
models. Thus, the model can be viewed as an instantiation of a systematic
modeling framework using standardized mechanisms, instead of constructing new
mechanisms for each model (O’Reilly ., 2016).

The neurons use a rate code version of the adaptive exponential (AdEx)
conductance-based point neuron model (Brette Gerstner, 2005), with the
standard RC circuit equations:

$\Delta V_{m}(t)=\tau\sum_{c}g_{c}(t)\overline{g_{c}}(E_{c}-V_{m}(t)),$ (2)

where $c$ represents excitatory, inhibitory, and leak channels. Inhibition is
driven by simulated interneurons in proportion to feedforward and feedback
dynamics, producing sparse distributed representations, and controlling the
effects of bidirectional excitatory connections between layers.

Each neuron learns using a more biologically-based version of the Contrastive
Hebbian Learning (CHL) algorithm, as shown in Figure 2:

$\Delta_{chl}=x^{+}y^{+}-x^{-}y^{-}$ (3)

where $x$ is the sending activation, $y$ is the receiving activation, and the
$+$ superscript indicates activations in the plus phase, and $-$ those in the
minus phase. The actual learning equations, detailed at
https://github.com/emer/leabra and in the online textbook:
https://CompCogNeuro.org produce a combination of error-driven and self-
organizing factors, which emerge out of a single learning rule that was
derived from a biologically detailed model of synaptic plasticity by (Urakubo
., 2008), and is closely related to the Bienenstock, Cooper & Munro (BCM)
algorithm (Bienenstock ., 1982).

####  9.4.1 Deep Context

This section describes in detail the equations that are specific to the Deep
version of Leabra that implements the specific predictive learning additions
to the general algorithm. Like the simple recurrent network (SRN) (JL. Elman,
1990; Jordan, 1989) which the deep predictive learning model functionally
resembles, the primary computational specialization required is the
maintenance of prior temporal context in the CT layer. In addition, the
pulvinar layers have to be driven by the bottom-up inputs in the plus phase,
after being driven by the CT inputs in the minus phase.

Computationally, the CT layer is specialized for maintaining context from the
previous alpha cycle, to generate the prediction over the pulvinar layer. At
the end of every plus phase, a new CT context excitatory input is computed
from the normalized dot product of the context weights times the sending
activations, just as in the standard net input used in Leabra:

$\eta_{j}=\langle x_{i}w_{ij}\rangle=\frac{1}{n}\sum_{i}x_{i}w_{ij}$ (4)

where $x_{i}$ are the sending activations and $w_{ij}$ are the weights. This
net input is then added in with the standard net input at each cycle of
processing during the subsequent alpha cycle.

The relative strength of these context layer inputs was set progressively
larger for higher layers in the network, with a maximum of 4 in V4, TEO, and
TE. In addition, TEO and TE received _self_ context projections which provide
an extended window of temporal context into the prior 200 ms interval,
consistent with multiple sources of neural data (Chaudhuri ., 2015). These
self projections were connected only within the narrower Pool level of units,
enabling these neurons to develop mutually-excitatory loops to sustain
activations over the multiple trials when the same object was present. We
hypothesize that these modifications correspond to biological adaptations in
IT cortex that likewise support greater sustained activation of object-level
representations.

Learning of the context weights occurs as normal, but using the sending
activation states from the prior time step’s activation.

####  9.4.2 Computational and Biological Details of SRN-like Functionality

Predictive auto-encoder learning has been explored in various frameworks, but
the most relevant to our model comes from the application of the SRN to a
range of predictive learning domains (JL. Elman, 1990; J. Elman ., 1996). One
of the most powerful features of the SRN is that it enables error-driven
learning, instead of arbitrary parameter settings, to determine how prior
information is integrated with new information. Thus, SRNs can learn to hold
onto some important information for a relatively long interval, while rapidly
updating other information that is only relevant for a shorter duration. This
same flexibility is present in our DeepLeabra model. Furthermore, because this
temporal context information is hypothesized to be present in the deep layers
throughout the entire neocortex (in every microcolumn of tissue), the
DeepLeabra model provides a more pervasive and interconnected form of temporal
integration compared to the SRN, which typically just has a single temporal
context layer associated with the internal “hidden” layer of processing units.

An extensive computational analysis of what makes the SRN work as well as it
does, and explorations of a range of possible alternative frameworks, has led
us to an important general principle: subsequent outcomes determine what is
relevant from the past. At some level, this may seem obvious, but it has
significant implications for predictive learning mechanisms based on temporal
context. It means that the information encoded in a temporal context
representation cannot be learned at the time when that information is
presently active. Instead, the relevant contextual information is learned on
the basis of what happens next.

This explains the peculiar power of the otherwise strange property of the SRN:
the temporal context information is preserved as a direct copy of the state of
the hidden layer units on the previous time step (Figure 19), and then learned
synaptic weights integrate that copied context information into the next
hidden state (which is then copied to the context again, and so on). This
enables the error-driven learning taking place in the current time step to
determine how context information from the previous time step is integrated.
And the simple direct copy operation eschews any attempt to shape this
temporal context itself, instead relying on the learning pressure that shapes
the hidden layer representations to also shape the context representations. In
other words, this copy operation is essential, because there is no other
viable source of learning signals to shape the nature of the context
representation itself (because these learning signals require future outcomes,
which are by definition only available later).

Figure 19: How the DeepLeabra temporal context computation compares to the SRN
mathematically. a) In a standard SRN, the context (deep layer biologically) is
a copy of the hidden activations from the prior time step, and these are held
constant while the hidden layer (superficial) units integrate the context
through learned synaptic weights. b) In DeepLeabra, the deep layer performs
the weighted integration of the soon-to-be context information from the
superficial layer, and then holds this integrated value, and feeds it back as
an additive net-input like signal to the superficial layer. The context net
input is pre-computed, instead of having to compute this same value over and
over again. This is more efficient, and more compatible with the diffuse
interconnections among the deep layer neurons. Layer 6 projections to the
thalamus and back recirculate this pre-computed net input value into the
superficial layers (via layer 4), and back into itself to support maintenance
of the held value.

The direct copy operation of the SRN is however seemingly problematic from a
biological perspective: how could neurons copy activations from another set of
neurons at some discrete point in time, and then hold onto those copied values
for a duration of 100 ms, which is a reasonably long period of time in neural
terms (e.g., a rapidly firing cortical neuron fires at around 100 Hz, meaning
that it will fire 10 times within that context frame). However, there is an
important transformation of the SRN context computation, which is more
biologically plausible, and compatible with the structure of the deep network
(Figure 19). Specifically, instead of copying an entire set of activation
states, the context activations (generated by the phasic 5IB burst) are
immediately sent through the adaptive synaptic weights that integrate this
information, which we think occurs in the 6CC (corticortical) and other
lateral integrative connections from 5IB neurons into the rest of the deep
network.

The result is a pre-computed net input from the context onto a given hidden
unit (in the original SRN terminology), not the raw context information
itself. Computationally, and metabolically, this is a much more efficient
mechanism, because the context is, by definition, unchanging over the 100 ms
alpha cycle, and thus it makes more sense to pre-compute the synaptic
integration, rather than repeatedly re-computing this same synaptic
integration over and over again (in the original feedforward backpropagation-
based SRN model, this issue did not arise because a single step of activation
updating took place for each context update — whereas in our bidirectional
model many activation update steps must take place per context update).

There are a couple of remaining challenges for this transformation of the SRN.
First, the pre-computed net input from the context must somehow persist over
the subsequent 100 ms period of the alpha cycle. We hypothesize that this can
occur via NMDA and mGluR channels that can easily produce sustained excitatory
currents over this time frame. Furthermore, the reciprocal excitatory
connectivity from 6CT to TRC and back to 6CT could help to sustain the initial
temporal context signal. Second, these contextual integration synapses require
a different form of learning algorithm that uses the sending activation from
the prior 100 ms, which is well within the time constants in the relevant
calcium and second messenger pathways involved in synaptic plasticity.

##  10 Backpropagation Model Methods

Figure 20: Learning curves for the backpropagation version of the WWI model.
Although it achieves better predictive accuracy than the DeepLeabra version,
it fails to acquire abstract object category structure, indicating a potential
tradeoff between simplifying and categorizing inputs, versus predicting
precisely where the low-level visual features will move.

The backpropagation version of the WWI model has exactly the same layer sizes
and feedforward patterns of connectivity as the DeepLeabra version.
Topographically, the V1p and V1hp pulvinar layers serve as output layers at
the highest level of the network, receiving all the various connections from
deep layers as shown in Table 1. Likewise, the LIPp served as a target output
layer for the Where pathway. To achieve predictive learning, the V1 pulvinar
targets were from the scene at time $t$, while the V1s inputs were from the
scene at time $t-1$. We also ran a comparison auto-encoder model that had
inputs and target outputs from the same time step, and it showed even less
systematic organization of its higher-level representations, further
supporting the notion that predictive learning is important, across all
frameworks. The learning curve for the predictive version is shown in Figure
20, which shows better overall prediction accuracy compared to the DeepLeabra
model. However, as the RSA showed, this backpropagation model failed to learn
object categories that go beyond the input similarity structure, indicating
that perhaps it was paying too much “attention” in learning to this low-level
structure, and lacked the necessary mechanisms to enable it to impose a
simplifying higher-level structure on top of these inputs.

##  11 PredNet Model Methods

The PredNet architecture was designed to incorporate principles from
predictive coding theory into a neural network model for predicting the next
frame in a video sequence. Details of the model can be found in the original
paper (Lotter ., 2016), but here we provide a brief overview of the
architecture.

###  11.1 Architecture

PredNet is a deep convolutional neural network that is composed of layers
containing discrete modules. The lowest layer generates a prediction of
incoming inputs (i.e. the pixels in the next frame), while each of the higher
layers attempts to predict the errors made by the previous layer. Each layer
contains an input convolutional module ($A_{l}$), a recurrent representational
module ($R_{l}$), a prediction module ($\hat{A}_{l}$), and a representation of
its own errors ($E_{l}$). The input convolutional module ($A_{l}$) transforms
its input with a set of standard convolutional filters, a rectified linear
activation function, and a max-pooling operation. The recurrent representation
module ($R_{l}$) is a convolutional LSTM, which is a recurrent convolutional
network that replaces the matrix multiplications in the standard LSTM
equations with convolutions, allowing it to maintain a spatially organized
representation of its inputs over time. The prediction module ($\hat{A_{l}}$)
consists of another standard convolutional layer and rectified linear
activation that is used to generate predictions from the output of $R_{l}$.
These predictions are then compared against the output of the input
convolutional module ($A_{l}$). The errors generated in this comparison are
represented explicitly in $E_{l}$, which applies a rectified linear activation
to a concatenation of the positive ($A_{l}-\hat{A}_{l}$) and negative
($\hat{A}_{l}-A_{l}$) prediction errors. These errors then become the inputs
to the next layer.

$A_{l}^{t}=\begin{cases}x_{t},&\text{if }l=0\\\
MaxPool(ReLU(Conv(E_{l-1}^{t}))),&\text{if }l>0\end{cases}$ (5)
$\hat{A}_{l}^{t}=ReLU(Conv(R_{l}^{t}))$ (6)
$E_{l}^{t}=[ReLU(A_{l}^{t}-\hat{A}_{l}^{t});ReLU(\hat{A}_{l}^{t}-A_{l}^{t})]$
(7) $R_{l}^{t}=ConvLSTM(E_{l}^{t-1},R_{l}^{t-1},UpSample(R_{l+1}^{t}))$ (8)

At each time step in the video sequence, PredNet generates a prediction of the
next frame. This is done as follows: first, the $R_{l}$ is computed for each
layer starting from the top of the hierarchy (because each $R_{l}^{t}$ depends
on input from $R_{l+1}^{t}$), and then the $A_{l}^{t}$, $\hat{A}_{l}^{t}$ and
$E_{l}^{t}$ are computed in a feed-forward fashion (becauase each $A_{l}^{t}$
depends on input from the layer below, $E_{l-1}^{t}$).

All analyses in the RSA were conducted using the representations from the
$R_{l}$ layers.

Figure 21: Learning curves for the PredNet model. This model achieves the best
overall prediction performance but also has the least well differentiated,
categorical representations.

###  11.2 Implementation details

All experiments with the PredNet architecture were performed using PyTorch. An
informal hyperparameter search was conducted to find the settings that
maximized representational similarity to the human judgments. This was done by
conducting RSA on each layer for each hyperparameter setting, and computing,
according to the Centroid categories derived from the human data, the
difference between the average within-category similarity and the average
between-category similarity. Our final architecture had 6 layers with 3, 16,
32, 64, 128, and 256 filters in the $A_{l}$ and $R_{l}$ modules, and 3x3
kernels throughout the whole network. We also found that using sigmoid and
tanh activation functions in fully-connected convolutional LSTMs slightly
improved performance, so these were used for all experiments.

The weights in the PredNet model are trained using error backpropagation.
Predictions are generated and errors are computed at all levels of the
hierarchy, but the model performs better when only the lowest layer’s errors
are backpropagated (Lotter ., 2016). We confirmed these results with
experiments that backpropagated the errors in higher layers, in which
performance (in terms of mean squared error) was marginally reduced but the
RSA results were similar. For this reason, all reported experiments used a
PredNet that was trained by only backpropagating the lowest level error.

The model was trained using a batch size of 8 and an Adam optimizer with a
learning rate of 0.0001, with no scheduler, for 150,000 batches. A training
curve is shown in Figure 21, showing that it achieves the best overall
prediction accuracy of any model we tested, and yet does not have
representations that are as differentiated or categorical as our biologically
based model, as shown in the main paper.

###  11.3 Regularization experiments

As discussed in the main paper, our biologically based model includes a number
of important biologically motivated properties that may be contributing to the
development of its categorical representations. These properties, including
excitatory bidirectional connections, inhibitory competition, and an
additional form of Hebbian learning, may be acting as regularizers that
encourage categorical learning. We therefore tested whether standard
regularization methods used in deep learning would have similar effects on the
representations developed in the PredNet architecture. We tested 1) batch
normalization, 2) dropout (0.1, 0.3, and 0.5), and 3) weight decay
(0.01,0.001,0.0001,0.00001). All experiments with batch normalization and
weight decay showed reduced performance (in terms of both prediction error on
the test set and within-category correlation). As shown in figure 22, dropout
marginally improved the within-category correlation while also slightly
improving prediction accuracy, so a dropout rate of 0.1 was used for the
comparison to our biologically based model in the main paper.

Figure 22: Effect of dropout in PredNet on RSA, as measured by the difference
between the average within-category correlation and the average between
category correlation (using the Centroid categories derived from human data).
Dropout marginally improves the category structure learned in PredNet.

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