Constrained Predictive Coding as a Biologically Plausible Model of the Cortical Hierarchy
Authors: Siavash Golkar
Tiberiu Tesileanu
Yanis Bahroun
Anirvan M. Sengupta
Dmitri Chklovskii
Venue: NeurIPS 2022 Accept (2022)
Keywords: predictive coding
biologically plausible
calcium plateau
cortical hierarchy
biologically realistic

Abstract: Predictive coding (PC) has emerged as an influential normative model of neural computation with numerous extensions and applications. As such, much effort has been put into mapping PC faithfully onto the cortex, but there are issues that remain unresolved or controversial. In particular, current implementations often involve separate value and error neurons and require symmetric forward and backward weights across different brain regions. These features have not been experimentally confirmed. In this work, we show that the PC framework in the linear regime can be modified to map faithfully onto the cortical hierarchy in a manner compatible with empirical observations. By employing a disentangling-inspired constraint on 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 performance as the original objective and maps onto a biologically plausible network. The units of this network can be interpreted as multi-compartmental neurons with non-Hebbian learning rules, with a remarkable resemblance to recent experimental findings. There exist prior models which also capture these features, but they are phenomenological, while our work is a normative derivation. This allows us to determine which features are necessary for the functioning of the model. For instance, the network we derive does not involve one-to-one connectivity or signal multiplexing, which the phenomenological models require, indicating that these features are not necessary for learning in the cortex. The normative nature of our algorithm in the simplified linear case also allows us to prove interesting properties of the framework and analytically understand the computational role of our network's components. The parameters of our network have natural interpretations as physiological quantities in a multi-compartmental model of pyramidal neurons, providing a concrete link between PC and experimental measurements carried out in the cortex.

Constrained Predictive Coding as a Biologically Plausible Model of the Cortical Hierarchy
Siavash Golkar"!
Tiberiu Tesileanu"
Yanis Bahroun
Anirvan M. Sengupta 233,4
Dmitri Chklovskii 1.5
Center for Computational Neuroscience, Flatiron Institute
Center for Computational Mathematics, Flatiron Institute
Center for Computational Quantum Physics, Flatiron Institute
Department of Physics and Astronomy, Rutgers University
Neuroscience Institute, NYU Medical Center
{sgolkar, tttssteennnnn aitesslleauuybbahrooo anirvans. anirvans.physicsseeeee
org
Abstract
Predictive coding (PC) has emerged influential normative model of neural computation with numerous extensions and applications. As such, much effort has been put into mapping PC faithfully onto the cortex, but there issues that remain unresolved or controversial. In particular, current implementations often involve separate value and error neurons and require symmetric forward and backward weights across different brain regions. These features have not been experimentally confirmed. In this work, show that the PC framework in the linear regime be modified to map faithfully onto the cortical hierarchy in a manner compatible constraint
with empirical observations. By employing disentangllinginspiin on hidden-layer neural activities, we derive an upper bound for the PC objective. Optimization of this upper bound leads 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 calliiicompatttettall neurons with non-Hebbian learning rules, with remarkable resemblance to recent experimental findings. There exist prior models which also capture these features, but they are phenomenological, while our work is normative derivation. This allows us determine which features are necessary for the functioning of the model. For instance, the network we derive does not involve one-to-one connectivity or signal multiplexing, which the phenomenological models require, indicating that these features are not necessary for learning in the cortex. The normative nature of our algorithm in the simplified linear case also allows us to prove interesting properties of the framework and analytically understand the computational role of our net- work's 's components. The parameters of our network have natural interpretations physiological quantities in nulti-compartmental model of pyramidal neurons, providing a concrete link between PC and experimental measurements carried out in the cortex.
'Equal contribution
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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Introduction
Over the past decades, predictive coding (PC), normative framework for learning representations that 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 expanded to the supervised regime [4] with arbitrary graph topologies [5, 6]. The PC framework has been analyzed 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.
(a)
(b)
error neurons
Tt
interneurons
W
Wa W
y 5
5
y 5
input
value neurons
supervision
input
supervision
Figure 1: Schematic architecture of the predictive coding network (PCN) and our covariance--onnsrriind network (BioCCPC). (a) PCN from [4], figure adapted from [13]. The intra-layer connectivity is to be one-to- one, while the inter-layer connectivity is symmetric. (b) Our BioCCPC network. There no requirement for symmetric weights across layers, and the connectivity within layers can diffuse.
Predictive coding is viewed as a possible theory of cortical computation, and many parallels have been drawn with the known neurophysiology of cortex [12]. While the initial works proposed 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 of this mapping that have proved controversial. Among these are the requirement of multiple redundant cortical operations, the symmetric connectivity pattern, the one-to-one connectivity of value and error neurons, and also the requirement that feedback connections be inhibitory [12, 15, 16], as sketched in Fig. la. 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 highlighting the details of computation in the cortex [17-26]. For example, the learning dynamics of multi- compartmental pyramidal neurons has been closely investigated. In particular, it was observed that the plasticity of the synapses of the basal compartment is driven by the activity in the apical tuft of the neuron by so-called calcium plateau potentials [19,20, 24]. leading to non-Hebbian learning rules [25]. These experiments have motivated the development of several models of microcircuits with multi-compartmenttl neurons [27-33]. In number of cases, it has been shown that these models can replicate learning similar to the backpropagation algorithm under specific assumptions [30, 34, 35]. However, because of their rather phenomenological nature, detailed analysis of these models is in many cases challenging and one must resort to numerical simulations, rendering the task of understanding the role of various neurophysiological quantities difficult. To this date, a normative framework (PC or otherwise) that can explain these experimental findings is still lacking.
In this work, we show how to modify the PC framework to make it compatible with the aforementioned experimental observations while maintaining much of the analytical control afforded to use by normative approach. Inspired by prior work which explored the effects of finding decorrelated representations [36-41], we add decorrelating inequality constraint the covariance of the latent representations of PC. Using this constraint, we derive an upper bound for the PC objective. By working the linear regime, we can prove several useful properties of our algorithm. We show that the learning algorithm derived from this upper bound does not suffer from the issues of prior implementations and naturally maps onto the hierarchical structure of the cortical pyramidal neurons.
Contributions
We introduce BioCCPC, novel biologically plausible algorithm derived as modification of predictive coding (PC), and show that it closely matches the known physiology of the cortical hierarchy.
We find analytic upper bound to the PC objective by imposing decomrelationiinspiee inequality constraint on the latent space.
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
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compartmental conductances encode the variances of the PC framework, and the somatic leak maps onto a thresholding mechanism of the associated eigenvalue problem.
2 Related work and review of predictive coding
The backpropagation algorithm [34, 42] is the predominant tool in the training of deep neural networks but is generally not considered biologically plausible [43]. Over the years, many authors have explored biologically plausible approximations or alternatives to backpropagation 130,31,44-55 (for more complete review see [59]). These approaches generally fall into two categories. First are normative approaches, such as Predictive Coding [1-3], Target Propagation and variations 45,44,,47, Equilibrium Propagation [39,53] and others [51 54-56], where one starts from a mathematically motivated optimization objective. These methods, by virtue of their normative derivation, have firm grounding in theory; however, they do not fully conform to the experimental observations in the brain (see below and Sec. 4). The second approach is driven by biology, with network structures and learning rules inspired by experimental findings [28, 30, 31,48, 60]. While these works mostly conform to the experimentally observed findings, they are more challenging to analyze because of their conjectured phenomenological nature.
The goal of the present paper is not to propose yet another biologically plausible alternative to backpropagation. It is rather to demonstrate that the normative framework of predictive coding, when combined with a constraint, can indeed closely match experimental observations. For this reason, this work, we focus on comparing our method with previous implementations of predictive coding 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.. The advantages of PC over backpropagation were highlighted in [13]. For other work addressing the realism problems of PC see [65,66].
We now turn to a brief review of the predictive coding framework from [4] before delving into the details of our method.
Notation. Bold upper case M and lower case variables denote matrices and vectors, respectively. By upper case letter X, Y, Z, we denote data matrices in R where d and T are the dimensions of the relevant variable and the number of samples. Lower case x, y. z, denote the relevant quantities of single sample, and xt, Yt. 2e, denote the 1th sample. [ denotes the Frobenius norm of M.
2.1 Review of predictive coding
Probabilistic model. In this section, we review the supervised predictive coding algorithm from [4]. The derivation starts from a probabilistic model for supervised learning, which parallels the archi- tecture of an artificial neural network (ANN) with + 1 layers. In this model, the neurons of each layer are random variables (t) (denoting the vector of activations in the 1hh layer) with layers 0 and respectively, denoting the input and output layers of the network. We assume that the joint probability of the latent variables factorizes in Markovian structure
pzz (11) .... ()) p(z( (2 (=-1)) p((("-1)\/(nn
x p(z!0))
with the relationship between the random variables of adjacent layers given by:
p(z (((1)) =V) ((), I()), with f f(1
E(1 =0 112.. (1)
The mean of the probability density on layer I mirrors the activity of the analogous ANN given by (x()) ((l-)) where ((I1) are the weights connecting layers / - and l. The objective function is then given by the negative log-likelihood of the joint distribution function:
Z
W(Z-)((Z(-))) const, (()2
L. Σ log
(n)
Σ
where we have switched to the data matrix notation for brevity and assumed the variances (3)2 are fixed hyperparameters. In the following, we refer to L, Eq. (2), without the constant term.
3

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Learning. Learning in the Whittingoon and Bogacz framework takes place in two steps [4]. First, the values of the random variables are determined by finding the most probable configuration of the joint distribution function when both the input and output layers are conditioned on the given input and output (z x and 2!n) y):
(1)
(ln
arg min L(z = x, x, (n) = =y)
(3)
The solution to this minimization problem can be found via gradient descent, which we evaluate component-win -oiiee for clarity as:
= = - n155 + Σ c D (( with
L
(4)
U12
where y/ is the gradient descent step size. The second step is then minimize the objective with respect to the weights while keeping the previously obtained neuron values fixed. This corresponds to optimizing the value of the loss the MAP estimate and can also be carried out by gradient descent, leading to:
sw & aL ((+1) 1(350).
ow [0) E, ij
(5)
where the star superscript denotes equilibrium values of the latent variable dynamics obtained in the first step (Eq. (3)). This algorithm can be implemented by a biologically plausible network as described in [4]; see Figure 1a. However, discussed in Section 1, its mapping onto the cortex has proved controversial. For further details regarding these steps see [4].
3 A constrained predictive coding framework
In this section, we introduce and discuss novel covariance-constrr model that builds upon the supervised PC paradigm of [4]. Our model also straightforwardly extends the unsupervised PC framework of [1,2]. For simplicity, we work in the linear regime (f(xx) = x), which allows us to prove different properties of our framework.
predictive coding (CCPC)
3.1 Derivation of an upper bound objective
Reduction to sum of objectives. We start by reducing the optimization problem, Eq. (2), into set of overlapping sub-problems, which will allow us to break the symmetry between feedforward and feedback weights. To do so, we first introduce a copy of the terms containing the weights W(1)
to (n-)) denoted by w! and (( respectively, as
Z
(((-1)
min L Z.W
min Z.W2
min W.W. W,
0022
1=1
1 |Z
w! (0))
|| (((n (n-1) 1) (n()))
L
min
(6)
2. W.,.W.
2
r1)2
x(n)2
w (1 1)
- ||
|Z()) w!! (1-)) Z (11) 2
o (1)
g()2
For consistency, rename ( W(n to w (0) (-1
respectively. Introducing these copies
does not change the optimization but will help avoid weight sharing in the steps below. We now pair the terms two by two as
min Z. W.W...W W.
i 1- -
Σ
(1) 9b
IZ
(0-11 9 -
(7)
This can be directly verified by finding the optima for W's before and after the change. We have W ((()) w. = (0) (0(0((2())))))) (0) Plugging these back into Eq. (6) we see that the equality holds. However, in the next step, since treat Wa's and Ws's differently, W. We will no longer hold.

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1/(20(12) for = .2 1. (n-1) = 1//(m), 0/ = 1/0 and g/ 9b (1) with Da
(1-1)
Weight sharing occurs here from terms like T obtained from expanding the squared norms in Eq. (7). Indeed, the gradient descent dynamics with respect to 2(++1) (resp. z()) leads to terms of the form Wz!!) in ((++)) (resp. WTz(( in ) which use the same weights W. Thanks to the doubling of the weights, we can avoid this problem by optimizing each term in the sum in Eq. (7) separately. In other words,
min Z.W W. 1=1 2 w
L
S
min
1
(8)
w
where 0
1
w! 1) 11 + 000h 2241) W
9b
2
This inequality holds simply because we are no longer finding the minimum of the full objective L. We are instead finding the minimum of each component separately and then evaluating Σ, L. This splits the (n + -layer optimization problem into a set of 3-layer optimizations, in each of which only the middle layer is being optimized. Note, however, that these overlapping, so the different optimization problems need to be solved self-consistently. We make this precise in the supplementary 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 Wa. but the problem remains for W a. This can be seen by noting that the gradient descent equations only depend on W but depend on both W. a and W We address this problem next.
Whitening constraint. The idea of decorrelating internal representation has been widely used for unsupervised tasks, often motivated by neuroscience [36-38]. In the case of deep learning, the main motivations were improved convergence speed and generalization [39-41]. Decorrelation has also been used to circumvent the weight transport problem [55]. Inspired by these observations, we introduce the inequality constraint 4 (z) 3 I, imposing an upper bound on the eigenvalues of the covariance matrix. In Sec. 5 and SM Sec. B we show that under some assumptions this inequality constraint leads to whitening the latent variables. Adding this inequality constraint to the objective we have:
1 S
min 0
min Z W W!
0 W w
z 0 XIXI
2 (1) - (((-))
+ ( [z() |
min 20 w (
2
z z (
0 Tr -2 (++1) w!'z"") + g ( !!! w
The first inequality results from the fact that the parameters of the new optimization problem are subset of the parameters of the original optimization problem. In the second inequality we have I to replace (()
expanded the second term of (() and used the constraint (ZOZ(OT with T, making the term larger, resulting again in an upper bound. This replacement eliminates the weight sharing problem in W... as as the gradient descent equations now only depend on WT and not on W. (cf. Eqs. (10) and (11)).
The incquality can be implemented by using a positive-definite Lagrange multiplier QQTQQ" [67]:
1 (1)
w
1)
+ 1Z
min max- 200, W. ( Q99
9b
+ ( Tr -2Z (++)T
+ 9 (T M w (
+ Tr
O TI) const
(9)
where we have added an additional quadratic term in Z as a regularizer. This can also be interpreted as an extra term in the prior of the latent space. Equation (9) is the final form of the objective, which we then optimize using gradient descent/ascent.
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3.2 Neural dynamics and learning rules.
Similar to the PCN of [4], the dynamics of our network during learning proceeds two steps. First the neural dynamics is derived by taking gradient steps of f,0) from Eq. (9) with respect to (())
1 9b (1-111 ((-1) + 9a
(9) w!!) z ((+1)
(gb (1)
+ c
Q "
(1)
(10)
where we have defined the variables n (1) = (1/9g) ()) Q zz)z())))) for each layer of the network.
The weight updates are derived via stochastic gradient descent of the loss given in Eq. (9) after the neural dynamics have reached equilibrium. These are given by
9a
((+1)
(W(++1] ((+22) Q ((+ Q(t)) ((++)))(+)) 2(0T, (0T
(11a)
& 29 ( W
(11b)
(
x (1) (0) - Q"
(11c)
We used the neural dynamics equilibrium equation for ( to simplify the weight update for w This yields our online algorithm (Alg. in SM See. E), with the architecture shown in Fig. 1b. The algorithm can be implemented in biologically plausible neural network as in Fig. 2; see Sec. 4.
b
4 Biological implementation and comparison with experimental observations
In this section, we introduce a biologically plausible neural circuit that implements the CCPC algorithm, denoted by BioCCPC. We also demonstrate that the details of this circuit resemble the neurophysiological properties of pyramidal cells in the neocortex and the hippocampus.
The algorithm for BioCCPC (Alg. I) summarized by the neural dynamics from Eq. (10) and weight update rules from Eqs. (11) can be implemented by a neural circuit with schematic shown in Fig. 2.
Neural architecture.
The activity of the n - hidden layers, {z M= encoded as the outputs of n - sets of neurons,
representing pyramidal neurons of different cortical regions. The matrices ( and w encoded as the weights of synapses between the pyramidal neurons of adjacent layers. Explicitly, the matrix w (resp. w((-")) is the efficacy of the synapse connecting (!++)) (resp. z to the pyramidal neurons (0) Because of the disjoint nature of these two inputs, model these as synapsing respectively onto the distal (apical tuft) and proximal (mostly basal) dendrites of the pyramidal neurons, respectively;
Fig. 2. This is reminiscent of cortical pyramidal neurons, which also have two integration sites, the proximal compartment comprised of the basal and proximal apical dendrites providing inputs to the soma, and the distal compartment comprised of the apical dendritic tuft [22, 68]. These two compartments receive excitatory inputs from two separate sources [18, 69].
Similarly, the auxiliary variables (() are represented by the activity of interneurons in each cortical region. The Q() synaptic weights are encoded in the weights of synapses connecting (!)) to z())
while Q models the weights of synapses from (1)) to (!) In biological setting, the implied equality of weights of synapses from ( to n and the transpose of those from n z(!) can be
guaranteed by applying the same Hebbian learning rule [55]. However, unlike previous work, we do not require interneurons of one layer be connected the pyramidal neurons of another layer.
Neural dynamics and canonical components. The neural dynamics of the pyramidal neurons of circuit given by Eq. (10) can be recast the dynamics of three compartment neuron:
T2 (1)
91k z - (v z ()) + 9b (v
(1) (1)
z "))
(12)
where we have defined the apical and basal compartmental membrane potentials
(1) (++1) Q TT (1)
w (1-1),, (1-1) (1-1)
(13)
2 'There are multiple types of interneurons targeting pyramidal cells 170, 71J. The interneurons of BioCCPC most closely resemble the somatostatin expresssinii interneurons, which preferentially inhibit the apical dendrites.

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Supervisory irput
Pyram dal netron to interneuron weightss
Layer n
T1
interneuron output
interneuron to Pyram cal nouron weights
Wa
Layer
Q
pyramidel neuron output
W apical synaplic weights
W
Wi basei synaptic weights
Layer
V/ apical compartment membrane potential
Intereuron
Pyramidal neuron
irpul
Figure 2: Schematic of the biological implementation the BioCCPC algorithm. Left: overall connectivity pattern, showing all-to-all connections between pyramidal neurons in consecutive layers, and between pyramidal and interneurons in the same layer. Right: zoom in on neuron and its immediate neighbors, showing the membrane potentials and synaptic weights that relevant for algorithm.
Interestingly. this is the same neural dynamics a three-compartment neuron posited previously in [28, 30], which we derived a normative way. Here, Ia (resp. 9b) are conductances between the apical (resp. basal) compartments and the soma, and Oik is the somatic leak. Our derivation clarifies the relationship between the physiological quantities and the objective function. Indeed, the basal and apical conductances play the role the inverse variances the predictive coding objective. These inverse variances, generally referred as 'precisions' in the PC literature, have previously been argued to be encoded in lateral inhibitions [2] or top-down attention [72, 73]. In our model, they are simply encoded the compartmental conductances. In Sec. we show that the leak term also has functional role and determines a threshold for dynamically learning more diverse variables [74, [74,75].
Here we defined the leak conductance 9ik (1) c0) (1)
Synaptic weight updates. Matching observations in the cortex, in our circuit [18, 22, 68, 77, the basal and apical weights (W, and Wa) our algorithm are updated differently. The basal synaptic weights of the pyramidal neurons, given by the elements of W are updated by the product of two factors represented in the corresponding post- and pre-synaptic neurons (Eq. 11a):
sw(())
ga v
cl) (111))T
(14)
( The first term in parentheses, Da va is proportional to the total apical current, given by the difference between the excitatory synaptic current in the apical tuft, w z((+1), and the inhibitory current induced by interneurons synapsing onto the distal compartment, Q(TT Biologically, this factor can be approximated by the calcium plateau potential traveling down the apical shaft. The calcium plateau potential has been experimentally seen drive plasticity of the basal synapses matching the 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 post- synaptic neurons, such plasticity is called non-Hebbian [25]. The synaptic learning rule for the apical weights w(( (Eq. lbb) and the synapses connecting the interneurons and the pyramidal neurons Q()) (Eq. 1 11c) are simply Hebbian, also matching experimental observations in the cortex [19, 78).
Features and limitations. Here we summarize the main differences between the biological imple- mentations of CCPC and PC. In PC, the forward and backward weights (in our notation W, and WT)) are symmetric, whereas, in CCPC, they are not constrained to be symmetric and are generically not so (see Sec. 6). Another difference is that in PC, the connection between value neurons and error neurons one-to-one and fixed; this constraint does not exist between the pyramidal and interneurons of our model, which are no longer one-to-one, and have plastic connectivity.
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While we have shown the improvement in biological realism of our model over the traditional PC network, this was done in the linear case. However, recent work on deep linear networks [79] has provided many insights into the learning dynamics of deep networks. Some of the properties discovered in the deep linear network, like "balancedness" of weights [80, 8,],, generalize for certain nonlinear networks [81]. We believe several of our observations will generalize in similar fashion.
There are other aspects that can also be improved. For example, the connectivity between pyramidal neurons and interneurons of CCPC is required to be symmetric. This symmetry can be achieved via Hebbian learning [55]; however, would be interesting to explore if such symmetry is indeed required. In CCPC, symmetric connectivity is only required between neurons of the same cortical region, which a much less stringent biological requirement than that of symmetric weights between different cortical regions, as in PC. Another shortcoming of our model is that while the teaching signal for the basal synapses in CCPC is signed and graded, in the cortex these signals are generally believed to be stereotypical [18).
5 Theoretical arguments and interpretation
In this section, we summarize some of the salient theoretical features of our framework and describe their biological significance. Further details and proof's are given in SM Sec. B.
5.1 Implicit error computation
In backpropagation, as well as in PC, the learning algorithm computes 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 Ja -> 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 of biologically plausible networks [30] and is related to the 'fixed prediction' assumption [4, 61, Proposition 1. Assume that the learning rules (Eqs. 11) are at equilibrium and we receive a new datapoint given by XT+1,YT+1. For c!!) and in the limit of = ga/9b + O, the leading term in e for the forward weight updates (t for this new sample is given by
(1-1)T = W ... w( (yT+1-1 - -T- 1T-1 4-1)T 8 Sw! 11 X Va.T+12T+1
+ +((
Y-IT (Z -1 TT+1 is the optimal linear inferred value for yr-1-
where ýT+1
Biological interpretation Recall that the learning rule for the basal synapses is proportional to !! (Eq. (14)). We argued in the previous section that this quantity can be implemented as the calcium plateau potential. In this light, Prop. shows that the calcium plateau potential is implicitly encoding the prediction error. This is remarkable as in our algorithm, neither the prediction nor the prediction error is explicitly computed. However, via the combined learning dynamics of the synapses and the neural dynamics of the interneurons, the final apical signal implicitly encodes the prediction error. This is in spirit similar to difference target propagation (DTP) [47], however, in DTP, the difference between the forward pass and the target is explicitly computed and backpropagated.
5.2 Adaptive latent dimension
For optimization within a single hidden layer, we show that the system performs adaptive latent dimension discovery by thresholding eigenvalues of a Gramian including contributions from both and optimizing over adjacent layers. If we take the loss function in (9), setting Ga = 1. go Y for Z. The next
W.. a: WA, b, we get the objective Tr((c c)Z Z Z Z e
Y]
theorem indicates the solution:
Theorem 1. Let the concatenated matrix = = 1/2 C/ /2, YJ E Rldx+ have SVD E =
1/2
[(dx-dy) V with {ua E +dy) and {va. E R both being sets of orthonormal vectors, with the convention that (Xa) are sorted in decreasing order. We consider the minimization
min Z
Tr [[c T Z Z x
-X
(15)
41a
8

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Then one of the optimal solutions is given by 2 - - Wal(Ao > c e)v. where {wo E R is an arbitrary set of orthonormal vectors, with D= = min(d, (d,ndx dx + dy).
Thus, the latent modes are found by thresholding the eignevalues [75] of the sum of Gramians [74] in Eq. (15). From this theorem, we derive the computational limits of e - oc and e + 0 in the supplementary materials. We also show that the neurons of the narrowest layer get whitened and the other layers become low-rank. For further details of these statements and all proof's see SM Sec. B.
Biological interpretation In our model, the number of neurons in the intermediate layers is fixed. However, the dimensionality of the representation encoded in these intermediate layers is determined dynamically according to the spectrum of the eigenvalue problem specified in Thm. 1. In particular, it is the quantity ct which encodes the somatic compartment leak conductance, that biologically sets an adaptive threshold for the dimensionality of the latent representation. In other words, lower (resp. higher) somatic leak corresponds to higher (resp. lower) sensitivity to the small eigenvalues of the above eigenvalue problem. For details of the interpretation of the eigenvalue problem different limits (e > oo and - 0), see SM Sec. B.
Numerical experiments
(a)
PC
(b)
covariance evals
3
first
0.3
(c)
covariance evals
0.31
second
0.1
PC cons BloCCPC
hidden layer
hidden layer
PC loss
0.0
0.2
0.0
50
-0.
0.1
0.1
0.0
10a
10a
10a
10
10
10a
10
iteration
iteration
iteration
Figure 3: Our algorithm (BioCCPC) compares well to less biologically plausible predictive-coding (PC) imple- mentations while enforcing an inequality constraint on the covariance of hidden-layer activations. Simulations run the MNIST dataset using network with two hidden layers. sizes 50 and 5. The shaded area represents the 95% range out of 40 runs. (a) Evolution of the the covariance matrix of hidden-layer activations the two
setiictvve-odding loss, Eq. (6),
validation set during
learning. (b), (c) Evolution of the cigenvalues hidden layers. The dashed lines show the scale of the constraint; see SM Sec. A. Inset: covariance matrix at the end of training. Note how the activity is whitened in the second hidden layer (c), and low-rank in the first hidden layer (b).
We tested our BioCCPC algorithm the MNIST [82] as well as other datasets." 4 We compared our results with a standard predictive coding network [4] with linear activation (PC) and with a version that includes a covariance constraint (PC+cons; Sec. C). We used unit variances, (0) 1, in the PC networks, and matching conductances in our algorithm, 9b
(1) ga (m) = 1, ga ( = 7 for other I.
Our method finds solutions that are almost optimal in terms of predictive-coding loss a held-out validation set (see Fig. 3a) and are generally better than the solutions obtained from simply adding predictive-coding network. Learning is slower in our network than
the covariance constraint in unconstrained PCN, as it takes time for the covariance constraint to be enforced. It is possible that different initialization schemes could improve the learning speed, but we leave this for future work. Note that since we focus on linear networks, metrics based on prediction accuracy are not very informative for the datasets we consider, which are not linearly separable.
The covariance constraint saturated the narrowest layer-in our example, the second hidden layer, of size 5esee Fig. 3c. The activation in the other hidden layer is rank-restricted by the narrowest layer, as described above, and so it is only whitened within a 5D subspace, Fig. 3b. In SM Sec. C. we provide further numerical experiments, exploring the effects of thresholding, the size of the network, and performance on other datasets. In all tested scenarios, BioCCPC achieves the same validation loss as PC (Figs. 5 and 6 of SM Sec. C). We also show that the forward and backward weights, W, and WT, a are not symmetric, in accordance with biological observations, and explore their relationship.
'The code is available at https: http::///ithhbb wwtp://ttetteeee/nttiaaaaaaaeeae

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In SM Sec. D we compare PC and BioCCPC with regards to generalization. We find that there is very little generalization gap for either case (Fig. 9). This is due to the linearity of our model. which acts as strong regularizer. Because of this, the potential generalization benefits of whitening the latent variables are not observed. Therefore, the primary benefit of the covariance constraint this work is the elimination of weight sharing in W. (ef. Sec. 3.1).
7 Conclusion
We have derived a biologically plausible algorithm for covariance-constrainee predictive coding and have shown that it avoids the criticisms of PC and has many features in common with recent experi- mental observations. Our algorithm does not have symmetric forward and backward connectivity matrices, and does not need one-to-one connectivity between any neurons. Furthermore, we showed that the learning rules of our algorithm closely match experimental observations, and we connected our parameters to neurophysiological quantities. Using the simplicity afforded to us because of the linearity, we showed number of interesting properties of our algorithm, including the effect of the somatic leak term, which acts as an adaptive thresholding mechanism. We also showed how our algorithm implicitly computes a difference term that it uses for learning. We argued that this term could be encoded in the calcium plateau potential, which is known to affect the learning of basal synapses. We hope that this concrete connection between our normative model and observation can further our understanding of cortical computation by guiding future experiments.
Acknowledgments and Disclosure of Funding
We would like to thank Colin Bredenherg for guidance and helpful comments throughout the project. SG, TT, YB, and DC were internally funded by the Simons Foundation. AS was partly funded by Simons Foundation Neuroscience grant SF 626323 during this work.
Broader impact
We do not foresee any potential societal harm caused by our work. We hope our contribution moves forward the discussion on the relationship between multi-layer networks and cortical structure.
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15


Reviews:
[{"review_id": "zj-jPd7gjuS", "reviewer": "Reviewer_UwcZ", "summary": "This paper proposes a variant of predictive coding (a biologically inspired framework for neural computation and learning) that better aligns with the known physiology and anatomy of cortex. This is achieved by applying a decorrelation constraint, yielding an upper bound on the PC objective. The resulting architecture is proposed to map onto the cortical microcircuitry, with each of the 3 main terms affecting a separate compartment of the pyramidal neurons. Numerical experiments show that this BioCCPC model can approach the performance of vanilla PC on MNIST. ", "questions": "* Some of the mathematical derivations seem to skip a number of steps, making them difficult to follow. A clear example is Equation (9), whose form is very difficult to relate to any of the previous equations. Similarly, Equation (4) doesn’t naturally follow from the previous ones.\n* Should the $v_a$ and $n$ updates be switched in Algorithm 1? $v_a$ is updated using $n$, which isn’t assigned until the next step…\n* Line 244, should it be the optimal linear inferred value for $y_{T+1}$ instead of $y_T$ ?\n", "limitations": "The main limitation due to the linearity assumption is properly acknowledged.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n* Potentially important variant of predictive coding that could be of interest to neuroscientists and ML scientists.\n* Clear biological motivation\n* Thorough theoretical framework\n\nWeaknesses:\n* All results are obtained under an assumption of linearity that could limit the usefulness of the findings in ML applications\n* The theoretical derivations are not always easy to follow\n* Experiments are restricted to a toy model and dataset (MNIST). \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "wer3paeS9fU", "reviewer": "Reviewer_9q1E", "summary": "The main objective of this article is to propose a theoretical extension of the Predictive Coding (PC) algorithm, called the Covariance Constrained Predictive Coding (CCPC), to better ground PC in neurophysiology. The 2 main theoretical novelties compared to PC are: 1) the weights are separated to break symmetry between the feedforward and the feedback weights 2) the eigenvalues of the covariance matrix are upper-bonded using a Lagrange multiplier decomposition. These theoretical changes allow the authors to re-interpret CCPC in light of physiology.", "questions": "* The authors decided to derive the PC formulation from the probabilistic interpretation (which is good, on my opinion). But the probabilistic description of the model is incomplete as the authors do not define the factorization of the joint distribution. This lack of rigor leads to a misunderstanding : the authors are trying to minimize the negative log likelihood of the joint distribution (as this is done in Rao and Ballard [1]) as in a classical generative model, but here the authors seem to tackle another problem which is inferring the label knowing the image (i.e.  $p(z^n | z^0)$). Given Eq. 2, I imagine the considered factorization is the following one : $p(z^0, z^1, … z^n) = p(z^n | z^{n-1}) * p(z^{n-1} | z^{n-2}) … * p(z^0)$. And $p(z^0)$ is the probability of the input distribution (which is hard to assess). Actually the log probability the authors are minimizing in the right term of Eq 2 (which is also 100% fine) is log $p(z^n | z^{n-1}, z^{n-2}, ..,  z^{0})$, which is equivalent (provided markovian hypothesis, which seems to be the case here) to $p(z^n | z^0)$. I would suggest to update the model section of the article to make it more accurate (just keep in mind the reader does not have to guess the authors hypothesis !).\n\n* In general, I think the section 3.1 is the more important one (this is where the novelty of this paper is). But this part is not rigorous enough for 2 reasons : 1- The authors suggest that splitting $W$ into $W_a$ and $W_b$, doesn’t change the optimization problem. This specific point should be demonstrated. The gradient (w.r.t. $z^l$) depends now on $W_a$ and $W_b$. Meaning that if $W_a \\neq W_b$ (I guess this is the case, otherwise I don’t see the point of splitting), the gradient has therefore a different value, which suppose that the optimization problem is not the same anymore. I think this part is crucial, it would deserve a more rigorous demonstration. 2 - I have trouble to understand the inequality presented in Eq. 8. In the right term of Eq 8 there is a minimization w.r.t $W_{b}^{(l)}$, but as expected just after this equation, $L^{(l)}$ does not depend on $W_{b}^{(l)}$ but rather on $W_{b}^{(l-1)}$. Is it a typo in your inequality in Eq.8 ? Even if it is a typo, this inequality would deserve more explanation. \n \n* In Eq.9, the authors call the term $||z^{(l)}||^2$ a ‘prior’.‘Prior’ has a probabilist connotation, and here it doesn’t fit into your probabilist description of the model (nowhere, you have exhibited a normal prior for every $z^{(i)}$ in section 2.1). You can easily circumvent that by calling the term $||z^{(l)}||^2$ a ‘regularization term’ which has no probabilist connotation (I know I am being picky, but as this article is almost fully theoretical, it should be as rigorous as possible)\n\n* Section 6: I do not understand the authors experimental comparison. It seems that the authors are comparing the training loss of PC, PC + cons, BioCCPC (cause on my understanding you have to give both the input and the one-hot to the model). Is it True ? Purely in terms of (training) loss, it seems that the BioCCPC performs not as good as PC (see Fig 3). Then what's the point of BioCCPC ? It is only a more bio-plausible formulation (if yes, it should be fitted to physiological data, see the last bullet point).\n\n* Section 6 again: I do not think the training loss is a good measure to compare algorithms. As the authors already mention (line 141), the whitening regularization added in the CCPC should improve the generalization abilities. But with the training loss only, how is it possible to assess the generalization gap (different between training and testing ? ). Why don’t you show the testing accuracy / testing loss ?\n\n* The authors suggest that their CCPC allows to interpret the variable in terms of neurophysiology. Then I would suggest to conduct quantitative comparison with neurophysiological data to back the authors interpretation.\n\n\nMinor :\n\nLine 100 : f is not defined at this point. The first time it is defined is line 118. Specifying f is an activation function when you use it, would improve the understanding of the article. \n\nLine 129 : with respect $z^{(l+1)}$ —> with respect to $z^{(l+1)}$\n\n[1] : Rao, Rajesh PN, and Dana H. Ballard. \"Predictive coding in the visual cortex: a functional interpretation of some extra-classical receptive-field effects.\" Nature neuroscience 2.1 (1999): 79-87.", "limitations": "In general, I think have properly discussed the limitations of their work in their article.", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "This article is clear and well written. It is easy to follow and solve most of the ’neuro-plausibility’ issue raised by the PC algorithm. This article is mainly grounded in theory (which is 100% fine), but in the same time it do not propose a huge theoretical breakthrough. On my opinion, the 2 main drawbacks of this article are 1) The theoretical part is not rigorous enough, 2) the comparison with neurophysiology is interesting but it more a matter of interpretation than a rigorous quantitative comparison with neurophysiological data (see section \"Questions\" for more information).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "nExAaC7iL0y", "reviewer": "Reviewer_9PPr", "summary": "This paper derives an upper bound for the PC objective by imposing a decorrelation-inspired inequality constraint on the latent space, which later comes to a biologically plausible algorithm BioCCPC. The paper shows that the proposed algorithm matches the known physiology of the cortical hierarchy.\n", "questions": "Major:\nPlease refer to the comments above in the Weakness section for the major concerns.\n\nMinor:\nThe related work section could be added more theoretical or application works that include predictive coding. Simply listing a review paper may not be a good option.\n", "limitations": "See comments above.\n", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n(1) The paper is well written, the structure and presentation are good and clear.\n\n(2) The idea and theoretical analysis seem sound.\n\nWeakness:\n\n(1) After reading [4,1,2], the level of novelty contribution is not very clear, especially in the current linear setting.\n\n(2)  I would like to see if the authors could provide additional experimental results on more (complicated) data, and the comparison between different PC baselines.\n\n(3) I would also suggest that the authors can provide a way to interpret the results, to show a better way to understand the model.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept