Constrained Predictive Coding as a Biologically Plausible Model of the Cortical Hierarchy
Anonymous Author(s)
AfﬁliationAddress email
Abstract
Predictive coding (PC) has emerged as an inﬂuential normative model of neuralcomputation, with numerous extensions and applications.
As such, much effort hasbeen put into mapping PC faithfully onto the cortex, but there are issues that remainunresolved or controversial.
In particular, current implementations often involveseparate value and error neurons and require symmetric forward and backwardweights across different brain regions.
These features have not been experimentallyconﬁrmed.
In this work, we show that the PC framework in the linear regime canbe modiﬁed to map faithfully onto the cortical hierarchy in a manner compatiblewith empirical observations.
By employing a disentangling-inspired constrainton hidden-layer neural activities, we derive an upper bound for the PC objective.
Optimization of this upper bound leads to an algorithm that shows the same per- formance as the original objective and maps onto a biologically plausible network.
The units of this network can be interpreted as multi-compartmental neurons withnon-Hebbian learning rules, with a remarkable resemblance to recent experimentalﬁndings.
There exist prior models which also capture these features, but theyare phenomenological, while our work is a normative derivation.
Notably, thenetwork we derive does not involve one-to-one connectivity or signal multiplexing,which the phenomenological models required, indicating that these features are notnecessary for learning in the cortex.
The normative nature of our algorithm in thesimpliﬁed linear case allows us to prove interesting properties of the framework andanalytically understand the computational role of our network’s components.
Theparameters of our network have natural interpretations as physiological quantitiesin a multi-compartmental model of pyramidal neurons, providing a concrete linkbetween PC and experimental measurements carried out in the cortex.
Introduction
Over the past decades, predictive coding (PC), a normative framework for learning representationsthat maximize predictive power, has played an important role in computational neuroscience [1,2].
Initially proposed as an unsupervised learning paradigm in the retina [3], it has since been expandedto the supervised regime [4] with arbitrary graph topologies [5, 6].
The PC framework has beenanalyzed in many contexts [7,8] and has found many applications, from clinical neuroscience [9,10]to memory storage and retrieval [11].
We refer the reader to [12,13] for recent reviews.
Predictive coding is viewed as a possible theory of cortical computation, and many parallels havebeen drawn with the known neurophysiology of cortex [12].
While the initial works proposed a
[Figure] [Figure]
interneurons biologically plausible network [1,2], the connection with cortex was more closely examined in [14],where the PC module was mapped onto a cortical-column microcircuit.
However, there are aspects ofthis mapping that have proved controversial.
Among these are the requirement of multiple redundantcortical operations, the symmetric connectivity pattern, the one-to-one connectivity of value and errorneurons, and also the requirement that feedback connections be inhibitory [12,15,16], as sketched in
Fig.
1a .
The presence of separate error and value neurons has itself been called into question [15].
The PC-based neural circuits also do not account for more recent experimental ﬁndings highlightingthe details of computation in the cortex [17–26].
For example, the learning dynamics of multicompartmental pyramidal neurons has been closely investigated.
In particular, it was observed thatthe plasticity of the synapses of the basal compartment is driven by the activity in the apical tuftof the neuron by so-called calcium plateau potentials [19,20,24], leading to non-Hebbian learningrules [25].
These experiments have motivated the development of several models of microcircuits withmulti-compartmental neurons [27–33].
In a number of cases, it has been shown that these models canreplicate learning similar to the backpropagation algorithm under speciﬁc assumptions [30,34,35].
However, because of their rather phenomenological nature, detailed analysis of these models isin many cases challenging and one must resort to numerical simulations, rendering the task ofunderstanding the role of various neurophysiological quantities difﬁcult.
To this date, a normativeframework (PC or otherwise) that can explain these experimental ﬁndings is still lacking.
In this work, we show how the PC framework can be made compatible with the aforementionedexperimental observations.
Inspired by prior work which explored the effects of ﬁnding decorrelatedrepresentations [36–41], we add a decorrelating inequality constraint to the covariance of the latentrepresentations of PC.
Using this constraint, we derive an upper bound for the PC objective.
Byworking in the linear regime, we can prove interesting properties of our algorithm.
We showthat the learning algorithm derived from this upper bound does not suffer from the issues of priorimplementations and naturally maps onto the hierarchical structure of the cortical pyramidal neurons.
Contributions
• By imposing a decorrelation-inspired inequality constraint on the latent space, we ﬁnd an
analytic upper bound to the PC objective.
• We introduce BioCCPC, a novel biologically plausible algorithm derived from this upper
bound and show that it closely matches the known physiology of the cortical hierarchy.
• We interpret the different parameters of the algorithm in terms of the conductances and
leaks of the separate compartments of the pyramidal neuron.
We ﬁnd that the neural
compartmental conductances encode the varainces of the PC framework, and the somatic
leak maps onto a thresholding mechanism of the associated eigenvalue problem.
Related work and review of predictive coding
The backpropagation algorithm [34, 42] is the predominant tool in the training of deep neuralnetworks but is generally not considered biologically plausible [43].
Over the years, many authorshave explored biologically plausible approximations or alternatives to backpropagation [30,31,44–58]
(for a more complete review see [59]).
These approaches generally fall into two categories.
First arenormative approaches, such as Predictive Coding [1–3], Target Propagation and variations [44,45,47],
Equilibrium Propagation [49, 53] and others [51, 54–56], where one starts from a mathematicallymotivated optimization objective.
These methods, by virtue of their normative derivation, have aﬁrm grounding in theory; however, they do not fully conform to the experimental observations inthe brain (see below and Sec. 4).
The second approach is driven by biology, with network structuresand learning rules inspired by experimental ﬁndings [28,30,31,48,60].
While these works mostlyconform to the experimentally observed ﬁndings, they are more challenging to analyze because oftheir conjectured phenomenological nature.
The goal of the present paper is not to propose yet another biologically plausible alternative tobackpropagation.
It is rather to demonstrate that the normative framework of predictive coding,when combined with a constraint, can indeed closely match experimental observations.
For thisreason, in this work, we focus on comparing our method with previous implementations of predictivecoding and do not concern ourselves with other biologically plausible alternatives to backpropagation.
The relationship between the PC framework and backpropagation was explored in [4,61–64].
Theadvantages of PC over backpropagation were highlighted in [13].
Notation.
Bold upper case M and lower case variables v denote matrices and vectors, respectively.
By upper case letter X , Y , Z , we denote data matrices in R d ⇥ T , where d and T are the dimensionsof the relevant variable and the number of samples.
Lower case x , y , z , denote the relevant quantitiesof a single sample, and x t , y t , z t , denote the t th sample.
k M k 2 F denotes the Frobenius norm of M .
2.1 Review of predictive coding
Probabilistic model.
In this section, we review the supervised predictive coding algorithm [4].
Thederivation starts from a probabilistic model for supervised learning, which parallels the architectureof an artiﬁcial neural network (ANN) with n + 1 layers.
In this model, the neurons of each layerare random variables z ( l ) (denoting the vector of activations in the l th layer) with layers 0 and n ,respectively, denoting the input and output layers of the network.
We assume that the joint probabilityof the latent variables factorizes in a Markovian structure
[Equation]
with the relationship between the random variables of adjacent layers given by:
[Equation]
The mean of the probability density on layer l mirrors the activity of the analogous ANN given byµ ( l ) =
W ( l  1) f
( z ( l  1) ) , where W ( l  1) are the weights connecting layers l  1 and l .
The objectivefunction is then given by the negative log-likelihood of the joint distribution function:
[Equation]
where we have switched to the data matrix notation for brevity and assumed the variances  ( l )2 areﬁxed hyperparameters.
In the following, we refer to L , Eq.
(2), without the constant term.
Learning.
Learning takes place in two steps.
First, the values of the random variables are determinedby ﬁnding the most probable conﬁguration of the joint distribution function when both the input andoutput layers are conditioned on the given input and output ( z (0) =
x and z ( n ) =
y ):
[Equation]
The solution to this minimization problem can be found via gradient descent, which we evaluatecomponent-wise for clarity as:
[Equation]
where ⌘ is the gradient descent step size.
The second step is then to minimize the objective withrespect to the weights while keeping the previously obtained neuron values ﬁxed.
This corresponds tooptimizing the value of the loss at the MAP estimate and can also be carried out by gradient descent,leading to:
[Equation]
where · ⇤ denotes equilibrium values of the latent variable dynamics obtained in the ﬁrst step (Eq.
This algorithm can be implemented by a biologically plausible network as described in [4]; see
Figure 1a.
However, as discussed in Section 1, its mapping onto the cortex has proved controversial.
For further details regarding these steps see [4].
A constrained predictive coding framework
In this section, we introduce and discuss our novel, covariance-constrained predictive coding (CCPC)model within the supervised PC paradigm of [4].
Our model also straightforwardly extends to the unsupervised learning paradigm discussed in [1,2].
For simplicity we work in the linear regime ( f ( x ) = x )which allows us to prove different properties of our framework.
3.1 Derivation of upper bound objective
Reduction to a sum of objectives.
We start by reducing the optimization problem, Eq.
(2), into aset of overlapping sub-problems, which will allow us to break the symmetry between feedforwardand feedback weights.
To do so, we ﬁrst introduce a copy of the terms containing the weights W (1)to W ( n  2) , denoted by W ( l ) a and W ( l ) b respectively, as
[Equation]
0)
[Equation]
[Equation]
(
[Equation]
For consistency, we rename W (0) , W ( n  1) to W (0) b , W ( n  1) a , respectively.
Introducing these copiesdoes not change the optimization 1 but will help us avoid weight sharing in the steps below.
We nowpair the terms two by two as
[Equation]
[Equation] [Equation]
with g ( n  1) a = 1 /  ( n )2 , g (1) b = 1 /  (1)2 , and g ( l  1) a = g ( l ) b = 1 / (2  ( l )2 ) for l = 2 , . . .
, n  1 .
Weight sharing occurs here from terms like z ( l +1) >
Wz ( l ) , obtained from expanding the squarednorms in Eq.
Indeed, the gradient descent dynamics with respect to z ( l +1) (resp.
z ( l ) ) leads toterms of the form
Wz ( l ) in ˙ z
( l +1)
(resp.
W > z ( l +1) in ˙
z ( l ) ), which use the same weights W .
Thanksto the doubling of the weights, we can avoid this problem by optimizing each term in the sum in
Eq.
(7) separately.
In other words,
[Equation]
where ˆ L ( l ) ⌘ 1n
[Equation]
=
[Equation]
This equality holds simply because we are no longer ﬁnding the minimum of the full objective ˆ L .
Weare instead ﬁnding the minimum of each component separately, and then evaluating
[Equation]
This splits the ( n + 1) -layer optimization problem into a set of 3 -layer optimizations, in each ofwhich only the middle layer is being optimized.
Note, however, that these are overlapping, so thedifferent optimization problems need to be solved self-consistently.
We make this precise in thesupplementary materials section and show that it provides an upper bound for our objective L (SM
Sec.
A).
Separating the objective function in this manner eliminates the weight sharing problem for
W b , but the problem remains for W a .
We address this problem in the following.
Whitening constraint.
The idea of decorrelating internal representation has been widely usedfor unsupervised tasks, often motivated by neuroscience [36–38].
In the case of deep learning, themain motivations were improved convergence speed and generalization [39–41].
Decorrelation hasalso been used to circumvent the weight transport problem [55].
Inspired by these observations,we introduce the constraint 1 T Z ( l ) Z ( l ) >
I , imposing an upper bound on the eigenvalues of thecovariance matrix:
[Equation]
The ﬁrst inequality results from the fact that the parameters of the new optimization problem area subset of the parameters of the original optimization problem.
In the second inequality we haveexpanded the second term of ˆ L ( l ) and used the constraint 1 T Z ( l ) Z ( l ) >  I to replace Z ( l ) Z ( l ) >with T , making the term larger, resulting again in an upper bound.
The inequality can be implemented by using a positive-deﬁnite Lagrange multiplier Q ( l ) > Q ( l ) :
[Equation]
where we have added an additional quadratic term in Z as a regularizer.
This can also be interpretedas an extra term in the prior of the latent space.
Equation (9) is the ﬁnal form of the objective whichwe then optimize using gradient descent/ascent.
3.2 Neural dynamics and learning rules.
Similar to the PCN of [4], the dynamics of our network during learning proceeds in two steps.
Firstthe neural dynamics is derived by taking gradient steps of ˆ L ( l ) from Eq.
(9) with respect to z ( l ) :
[Equation]
where we have deﬁned the variables n ( l )
= (1 /g ( l ) a ) Q ( l )
z ( l ) for each layer of the network.
The weight updates are derived via stochastic gradient descent of the loss given in Eq.
(9) after theneural dynamics have reached equilibrium.
These are given by
[Equation]
We used the neural dynamics equilibrium equation for z ( l ) to simplify the weight update for W ( l ) b .
This yields our online algorithm (Alg. 1 in SM Sec. E), with the architecture shown in Fig.
1b.
Thealgorithm can be implemented in a biologically plausible neural network as in Fig. 2; see Sec. 4.
Biological implementation and comparison with experimental observations
In this section, we introduce a biologically plausible neural circuit that implements the CCPCalgorithm, denoted by BioCCPC.
We also demonstrate that the details of this circuit resemble theneurophysiological properties of pyramidal cells in the neocortex and the hippocampus.
Neural architecture.
The algorithm for BioCCPC (Alg. 1) summarized by the neural dynamicsfrom Eq.
(10) and weight update rules from Eqs.
(11) can be implemented by a neural circuit withschematic shown in Fig.
The activity of the n  1 hidden layers, { z ( l ) } n  1 l =1 , is encoded as the outputs of n  1 sets of neurons,representing pyramidal neurons of different cortical regions.
The matrices W ( l ) a and W ( l ) b areencoded as the weights of synapses between the pyramidal neurons of adjacent layers.
Explicitly,the matrix W ( l ) a (resp.
W ( l  1) b ) is the efﬁcacy of the synapse connecting z ( l +1) (resp.
z ( l  1) ) tothe pyramidal neurons z ( l ) .
Because of the disjoint nature of these two inputs, we model theseas synapsing respectively onto the distal (apical tuft) and proximal (mostly basal) dendrites of thepyramidal neurons, respectively; see Fig.
This is reminiscent of cortical pyramidal neurons, whichalso have two integration sites, the proximal compartment comprised of the basal and proximal apicaldendrites providing inputs to the soma, and the distal compartment comprised of the apical dendritictuft [22,65].
These two compartments receive excitatory inputs from two separate sources [18,66].
Similarly, the auxiliary variables n ( l ) are represented by the activity of interneurons in each corticalregion.
2 The Q ( l ) synaptic weights are encoded in the weights of synapses connecting n ( l ) to z ( l ) ,while Q ( l ) > models the weights of synapses from z ( l ) to n ( l ) .
In a biological setting, the impliedequality of weights of synapses from z ( l ) to n ( l ) and the transpose of those from n ( l ) to z ( l ) can beguaranteed by applying the same Hebbian learning rule [55].
However, unlike previous work, we donot require interneurons of one layer to be connected to the pyramidal neurons of another layer.
Neural dynamics and canonical components.
The neural dynamics of the pyramidal neurons ofour circuit given by Eq.
(10) can be recast as the dynamics of a three compartment neuron:
[Equation]
where we have deﬁned the apical and basal compartmental membrane potentials
[Equation]
Here we deﬁned the leak conductance g ( l ) lk = c ( l )  g ( l ) a .
Interestingly, this is the same neuraldynamics of a three-compartment neuron posited previously in [28, 30], which we derived in anormative way.
Here, g a (resp.
g b ) are conductances between the apical (resp. basal) compartments [Figure] and the soma, and g lk is the somatic leak.
Our derivation clariﬁes the relationship between thephysiological quantities and the objective function.
Indeed, the basal and apical conductancesplay the role of the inverse variances in the predictive coding objective.
These inverse variances,generally referred to as ‘precisions’ in the PC literature, have previously been argued to be encodedin lateral inhibitions [2] or top-down attention [69,70].
In our model, they are simply encoded in thecompartmental conductances.
In Sec.
5 we show that the leak term also has a functional role anddetermines a threshold for dynamically learning more diverse variables [71,72].
Synaptic weight updates.
Matching observations in the cortex, in our circuit [18,22,65,73,74], thebasal and apical weights ( W b and W a ) of our algorithm are updated differently.
The basal synapticweights of the pyramidal neurons, given by the elements of W ( l ) b , are updated by the product of twofactors represented in the corresponding post- and pre-synaptic neurons (Eq. 11a):
[Equation]
The ﬁrst term in parentheses, g ( l ) a v ( l ) a , is proportional to the total apical current, given by the differencebetween the excitatory synaptic current in the apical tuft, W ( l ) >
a z ( l +1) , and the inhibitory currentinduced by interneurons synapsing onto the distal compartment, Q ( l ) > n ( l ) .
Biologically, this factorcan be approximated by the calcium plateau potential traveling down the apical shaft.
The calciumplateau potential has been experimentally seen to drive plasticity of the basal synapses matchingthe derived update rule of our circuit [18,20,23–25].
The second factor is simply the neural output.
Because the total update is not purely dependent on the action potentials of the pre- and postsynaptic neurons, such plasticity is called non-Hebbian [25].
The synaptic learning rule for the apicalweights W ( l ) a (Eq. 11b) and the synapses connecting the interneurons and the pyramidal neurons
Q ( l ) (Eq. 11c) are simply Hebbian, also matching experimental observations in the cortex [19,75].
Features and limitations.
Here we summarize the main differences between the biological implementations of CCPC and PC.
In PC, the forward and backward weights (in our notation W b and W > a ) are symmetric, whereas in CCPC, they are not constrained to be symmetric and are genericallynot so (see Sec. 6).
Another difference is that in PC, the connection between value neurons and errorneurons is one-to-one and ﬁxed; this constraint does not exist between the pyramidal and interneuronsof our model, which are no longer one-to-one, and have plastic connectivity.
While we have shown the improvement in biological realism of our model over the traditional PCnetwork, this was done in the linear case.
However, recent work on deep linear networks [76],has provided many insights into the learning dynamics of deep networks.
Some of the propertiesdiscovered in the deep linear network, like “balancedness" of weights [77,78], generalize for certainnonlinear networks [78].
We believe several of our observations will generalize in a similar fashion.
There are other aspects that can also be improved.
For example, the connectivity between pyramidalneurons and interneurons of CCPC is required to be symmetric.
This symmetry can be achievedvia Hebbian learning [55]; however, it would be interesting to explore if such symmetry is indeedrequired.
In CCPC, symmetric connectivity is only required between neurons of the same corticalregion, which is a much less stringent biological requirement than the that of symmetric weightsbetween different cortical regions, as in PC.
Another shortcoming of our model is that while theteaching signal for the basal synapses in CCPC is signed and graded, in the cortex these signals aregenerally believed to be stereotypical [18].
Theoretical arguments
We summarize some of the theoretical features of our framework.
All proofs given in SM Sec.
B.
The errors are implicitly computed in v ( l ) a .
In backpropagation as well as in PC, the learningalgorithm computes a loss (or a local error) which is then used to compute updates to the weights.
In
CCPC, no error or loss is explicitly computed.
So how does CCPC learn?
In the following proposition,we show that in the g a !
limit, the algorithm implicitly computes an error in its weight updates.
This quasi feed-forward or weak nudging limit, has been used to explore the learning dynamics ofbiologically plausible networks [30] and is related to the ‘ﬁxed prediction’ assumption [4,61,62].
Proposition 1.
Assume that the learning rules (Eqs. 11) are at equilibrium
and we receive a newdatapoint given by x T +1 , y T
+1 .
For c ( l ) = 0 and in the limit of ✏ ⌘ g a /g b ! 0 , the leading term in✏ for the forward weight updates  W ( l  1) b for this new sample is given by
[Equation]
where ˜ y T +1 =
YZ ( l  1) >
Z ( l  1) Z ( l
[Equation]
) >   1 z ( l  1) T +1 is the optimal linear inferred value for y T +1 .
This implies that v ( l ) a , which we argued can be implemented as the calcium plateau potential,implicitly encodes an error signal.
This is in spirit similar to difference target propagation [47], wherethe difference between the forward pass and the target is explicitly computed and backpropagated.
Adaptive latent dimension discovery for a single module.
For optimization within a singlehidden layer, we show that the system performs adaptive latent dimension discovery by thresholdingeigenvalues of a Gramian including contributions from both adjacent layers.
If we take the lossfunction in (9), setting g a = 1 , g b = ✏ , and optimizing over W a , W b , we get the objective
Tr[( c +✏ ) Z > Z  Z > Z ⇥ ✏ X > ( XX > )  1 X + Y > Y ⇤ for Z .
The next theorem indicates the solution:
Theorem 1.
Let the concatenated matrix ⌅ =
[ ✏ 1 / 2 C  1 / 2 x
X , Y ] 2 R ( d X + d Y ) ⇥
T have SVD ⌅ =
P ( d X + d Y ) ↵ = u ↵  ↵ v > ↵ with { u ↵ 2 R ( d X + d Y ) } and { v ↵ 2 R T } both being sets of orthonormal vectors,with the convention that (  ↵ ) are sorted in decreasing order.
We consider the minimization
[Equation]
Then one of the optimal solutions is given by ˆ Z =
P
D ↵ =1 w ↵ (  ↵ > c + ✏ ) v > ↵ where { w ↵ 2 R d } isan arbitrary set of orthonormal vectors, with D = min( d, d X + d Y ) .
Thus, the latent modes are found by thresholding the eignevalues [72] of the sum of Gramians [71] in
Eq.
This implies that the quantity c ( l ) that we showed is encoded in the somatic compartment leakconductance, biologically sets an adaptive threshold for the dimensionality of the latent representation.
From this theorem, we derive the computational limits of ✏ ! 1 and ✏ ! 0 in the supplementarymaterials.
We also show that the neurons of the narrowest layer get whitened and the other layersbecome low-rank.
For further details of these statements and all proofs see SM Sec.
B.
Numerical experiments
[Figure]
We tested our BioCCPC algorithm on the MNIST dataset [79], with one-hot labels.
We centered theinput images and normalized by the standard deviation.
We compared our results with a standardpredictive coding network [4] with linear activation (PC) and with a version that includes a covarianceconstraint (PC+cons; Sec. C).
We used unit variances,  ( l )2 = 1 , in the predictive-coding networks,and matching conductances in our algorithm, g (1) b = g ( n ) a = 1 , g ( l ) a = g ( l ) = 1 / 2 for other l .
Our method ﬁnds solutions that are almost optimal in terms of predictive-coding loss (see Fig.
3a)and are generally better than the solutions obtained from simply adding the covariance constraint to apredictive-coding network.
Learning is slower in our network than in unconstrained PCN, as it takestime for the covariance constraint to be enforced.
It is possible that different initialization schemescould improve the learning speed, but we leave this for future work.
The covariance constraint is saturated in the narrowest layer—in our example, the second hiddenlayer, of size 5—see Fig. 3c.
The activation in the other hidden layer is rank-restricted by thenarrowest layer, as described above, and so it is only whitened within a 5D subspace, Fig.
3b.
In thesupplementary materials, we provide further numerical experiments, where we look at the effectsof the thresholding, the size of the network, and performance on other datasets.
We also show thatthe forward and backward weights, W b and W > a , are not symmetric, in accordance with biologicalobservations, and we explore their relation.
See SM Sec.
C for these and other simulation details.
Conclusion
We have derived a biologically plausible algorithm for covariance-constrained predictive coding andhave shown that it avoids the criticisms of PC and has many features in common with recent experimental observations.
Our algorithm does not have symmetric forward and backward connectivitymatrices, and does not need one-to-one connectivity between any neurons.
Furthermore, we showedthat the learning rules of our algorithm closely match experimental observations, and we connectedour parameters to neurophysiological quantities.
Using the simplicity afforded to us because of thelinearity, we showed a number of interesting properties of our algorithm, including the effect of thesomatic leak term, which acts as an adaptive thresholding mechanism.
We also showed how ouralgorithm implicitly computes a difference term that it uses for learning.
We argued that this termcould be encoded in the calcium plateau potential, which is known to affect the learning of basalsynapses.
We hope that this concrete connection between our normative model and observation canfurther our understanding of cortical computation by guiding future experiments.
Broader impact
We do not foresee any potential societal harm caused by our work.
We hope our contribution movesforward the discussion on the relationship between multi-layer networks and cortical structure.
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