Phase transitions in when feedback is useful
Authors: Lokesh Boominathan
Xaq Pitkow
Venue: NeurIPS 2022 Accept (2022)
Keywords: Bayesian Inference
Predictive Coding
Efficient Coding
Linear Quadratic Gaussian

Abstract: Sensory observations about the world are invariably ambiguous. Inference about the world's latent variables is thus an important computation for the brain. However, computational constraints limit the performance of these computations. These constraints include energetic costs for neural activity and noise on every channel. Efficient coding is one prominent theory that describes how such limited resources can best be used. In one incarnation, this leads to a theory of predictive coding, where predictions are subtracted from signals, reducing the cost of sending something that is already known. This theory does not, however, account for the costs or noise associated with those predictions. Here we offer a theory that accounts for both feedforward and feedback costs, and noise in all computations. We formulate this inference problem as message-passing on a graph whereby feedback serves as an internal control signal aiming to maximize how well an inference tracks a target state while minimizing the costs of computation. We apply this novel formulation of inference as control to the canonical problem of inferring the hidden scalar state of a linear dynamical system with Gaussian variability. The best solution depends on architectural constraints, such as Dale's law, the ubiquitous law that each neuron makes solely excitatory or inhibitory postsynaptic connections. This biological structure can create asymmetric costs for feedforward and feedback channels. Under such conditions, our theory predicts the gain of optimal predictive feedback and how it is incorporated into the inference computation. We show that there is a non-monotonic dependence of optimal feedback gain as a function of both the computational parameters and the world dynamics, leading to phase transitions in whether feedback provides any utility in optimal inference under computational constraints.

Phase transitions in when feedback is useful
Lokesh Boominathan Department of ECE Rice University Houston, TX 77005
Xaq Pitkow
Dept. of Neuroscience, Dept. of ECE Baylor College of Medicine, Rice University Houston, TX 77005 xaq@rice. edu
lokesh. www..ootatnnaanninc....... edu
Abstract
Sensory observations about the world are invariably ambiguous. Inference about the world's 's latent variables is thus an important computation for the brain. How- ever, computational constraints limit the performance of these computations. These constraints include energetic costs for neural activity and noise on every channel. Efficient coding is one prominent theory that describes how such limited resources can best be used. In one incarnation, this leads to a theory of predictive coding, where predictions are subtracted from signals, reducing the cost of sending some- thing that is already known. This theory does not, however, account for the costs or noise associated with those predictions. Here offer theory that accounts for both feedforward and feedback costs, and noise all computations. We formulate ceessage-passingg
this inference problem an internal control signal aiming to maximize how well an inference tracks a target state while minimizing the costs of computation. We apply this novel formulation inference as control to the canonical problem of inferring the hidden scalar state of linear dynamical system with Gaussian variability. The best solution depends on architectural constraints, such as Dale's 's law, the ubiquitous law that each neuron makes solely excitatory or inhibitory postsynaptic connections. This biological structure can create asymmetric costs for feedforward and feedback channels. Under such conditions, our theory predicts the gain of optimal predictive feedback and how it is incorporated into the inference computation. We show that there is a monooonc dependence optimal feedback gain function of both the computational parameters and the world dynamics, leading to phase transitions in whether feedback provides any utility in optimal inference under computational constraints.
graph whereby feedback serves
Introduction
A critical computation for the brain is to infer the world's latent variables from ambiguous observa- tions. Computational constraints, including metabolic costs and noisy signals, limit the performance of these inferences. Efficient coding [1] is prominent theory that describes how limited resources can be used best. In one incarnation, this leads to the theory of predictive coding [2), which posits that predictions are sent along feedback channels be subtracted from signals lower cortical areas; only the difference returns to the higher areas along feedforward channels, reducing the metabolic or informational cost of sending redundant signals already known to the higher areas. This theory does not, however, account for the additional costs or noise associated with the feedback. Depending on the costs for sending predictions and the reliability of signals encoding those predictions, we expect different optimal strategies to perform computationally constrained inferences. For example, if the feedback channel is too unreliable and expensive, hypothesize that is not worth sending any predictions at all. Here we offer more general theory of inference that accounts for the costs and reliabilities of the feedback and feedforward channels, and the relative importance of good inferences
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

---

about the latent world state. We formulate the inference problem as control via message- passiing on a graph, maximizing how well an inference tracks a target state while minimizing the message costs. Messages become control actions with their own costs to reduce while improving how well an inference tracks target state. We call this method inference as control, as it flips the interesting perspective of viewing optimal control as an inference problem [3]. We solve this problem under (LQG) assumptions: Linear dynamics, Quadratic state and control costs, and Gaussian noise for the process, observations, and messages. Our theory enables us determine the optimal predictions and how they are integrated into computationally constrained inference. This analysis reveals phase transitions in when feedback is helpful, as we change the computation parame- ters or the world dynamics. Finally, we connect our theory's constraints to biology by providing a simplified example of how feedback/feedforward pathways with different anatomical structures can yield different optimal strategies.
2 Related work
Our work brings together several related theories: Bayesian inference, efficient coding, and predictive coding. The idea that the brain performs Bayesian inference about latent variables amid uncertain sensory observation has been long studied in neuroscience [4-7]. Bayesian inference involves optimally combining prior information from the past or from surrounding context with the likelihood of current sensory observations [8-11]. The theory of efficient coding [1. 12-15] focuses encoding the sensory observations themselves, capturing maximal information subject to limited biological resources. The theory of predictive coding [2] tackles such resource constraints by using top-down feedback predictions to suppress the part of sensory observations that is already known to the brain, and sending only the novel part of the observation back to the brain [16-21]. Understanding the utility of predictive feedback has also shown promise in gaining insights into deep learning algorithms and improving their performance, recently drawing wide interest in this topic [22-29].
Park et al. [30] brought together the ideas of a Bayesian brain and efficient coding, by identifying efficient neural codes as optimizing a posterior dis- tribution while accounting for limited firing rates.
A World state
Inference
Chalk al. [18] unified theories of predictive cod- ing, efficient coding, and sparse coding by showing how these regimes emerge in three dimensional constraint space described by channel capacity, past information content about the world state, and the time point to which the estimate is targeted. Mly- narski et al. [31] investigated how sensory coding can adapt over time to account for the trade-off of an inference cost against a metabolic cost for high- fidelity encoding. Aitchison et al. [32] argued that Bayesian inference can be achieved using predictive coding, but is not necessary. While these works made important contributions towards unifying different encoding schemes that are optimal under different circumstances, they all assume that information costly but computation is free. In particular, none of them have explicitly accounted for biological con- straints along the feedback channel well. In our work, we discuss how the optimal strategy changes when balancing inference performance against ener- getic costs, when there is noise in both feedforward and feedback pathways.
Observation
Estimate
X
B
Inference as control
World state
C
Observation
0
0
E
Residual
A
D
Prediction (Control)
P
p
L
H
Estimate
Figure 1: Resource constrained inference mod- eled as a control problem. A: Graphical model of the inference problem, tracking a latent state in Hidden Markov Model. B: Expanded inference problem, indicating prediction as control.
3 Defining the problem
We consider an inference task in which the brain tracks latent world state xf based on its noisy sensory observations o' (Fig 1A). The trajectory of the world state follows a known stochastic linear

---

dynamical system (Eq 1) with Gaussian process noise and Gaussian observations (Eq 2). At each time step, the brain sends a top-down prediction p' based on its best estimate . based on evidence up until the previous time step (Eq 3). The prediction is then sent through an additive Gaussian noise feedback channel to the sensory level (Eq 4). The noisy prediction je is then combined with the new observations of to form a residual A (Eq 5). The residual is then sent through an additive Gaussian noise feedforward channel to the brain (Eq 6). Based on this noisy residual A and the prediction it had just sent, the brain updates its estimate 4 (Eq 7). Fig IB shows trellis diagram of the dynamics.
Structurally, these dynamics are equivalent to a Kalman filter. The most common representation of this filter might even be viewed as predictive coding, as the update step uses the residual between the predicted and actual observation. However, the Kalman filter has no costs aside from the final inference, and no computational noise except the observation. Biological computation therefore may weigh fundamentally different tradeoffs in its inferences. Our goal is to find the parameters that minimize a weighted combination of inference loss, feedback energy cost, and feedforward energy cost (Eq 8) given limitations caused by computational noise.
We optimize the following parameters: the gain L on the previous inference that is integrated into the new prediction; the prediction multiplier D and observation multiplier E that describe how the noisy prediction and the observation are weighted form the residual; and the parameters F, G. and H that determine how the inference is updated in light of the new noisy residual. The optimization done at steady state, assuming the observer must continually update its estimate in a stationary dynamic environment. For mathematical tractability, we assume that all messages are linear functions of their inputs, the three losses are quadratic, the weight on the inference loss a scalar, and all noise is independent Gaussian white noise. The equations governing this problem are:
x² x''
vpp
state dynamics
0 = C x¹ + 1oo p¹ I, T 11111
observation
(2)
prediction
(3)
F¹ p¹ + 1/b
noisy prediction, feedback
(4)
A' D p¹ +Eo
residual
(5)
Á' A' + m/
noisy residual, feedforward
(6)
a = F 11-1
+ G
estimation
(7)
Cost = lim Σ
| s4ll 2
Wh F'
We A
(8)
T-000 T t=1
Costant
Cost,
Costy
We consider how the optimal computational strategy varies with cost weithweights (W,, Wi) that determine the relative importance of feedback, feedforward, and inference costs (Costi, Costr, and Costint). Up 7oo !/8, and тr represent the process noise, observation noise, feedback noise, and feedforward noise with variances of oh of and of respectively. We will see below that certain combinations of parameters determine the system's 's behavior at transition points.
4 Method: Inference as Control
We adopt two-step approach to solve the optimization problem in Eq 8. First we fix the parameters D and E. that determine how the feedback is used to subtract predictions, and solve in closed form for the optimal feedback gain (L) and optimal integration of residuals (F,, G. H!) as function of the fixed parameters. We then numerically optimize for D and E, given the optimal feedback. Mathematically. we write the two-step minimization as below, with an argmin determined analogously.
min D.E.L.E...H
Costent
min min Costoot D.E L,F.G...
In order to find the closed form solution for fixed D and E, we recast the minimization as an LQG control problem [33] where the prediction is treated as an internal control. The LQG dynamical Eqs 10-11 are obtained by concisely writing Eqs 1-2, and 4-6 in terms of an augmented state =²= [xx A+ (see Appendix A. Augmented dynamics, control, and measurement matrices Aang. Bang. Caug, and the noise vector yanng are expressed in terms of D and F. The feedforward and feedback energy costs are expressed the LQG state and control costs respectively (Eq 12,
3

---

0
0
derived in Appendix A.2), where Q
and W6-
0
Wi
Aaug 2t-1 1-1 + Baug p' + Tang
(10)
Â
=Caug 2t
(11)
min
lim
Q Rp
(12)
t=1
control cost
Note that the inference cost is not explicitly added in the LQG objective function (Eq 12). However, using the separation principle [34], we show in Appendix A.3 that for a fixed and E the LQG solution automatically minimizes the inference cost well. The separation principle states that at each instant the observer needs to make an optimal estimate of the world state to form the optimal control. Furthermore, we show in Appendix A.4-A.6 that the LQG solution also provides the optimal F, G, H, L, and Cost in terms of D and E. These are denoted as F", G¹, H', L', and Cost respectively. Finally, numerically optimize Cost with respect to D and E (as in Eq 9).
Results
A
B
Observation
Multiplier (E)
In this paper we analyze our system for one-dimensional world state to gain precise mathematical insight into the core compu- tational problem. One useful perspective that the feedback signal functions as kind of self-control for the inference system. As we formulated the problem, this control has its own "action cost," and this allows to use the known solutions for controllable systems to identify the optimal feedback. The control gain is therefore a fundamen- tal parameter,
Control Gain (L)
log
Foedforward Noise
log( Fost ffowwadd Weight
Prediction Multiplier (D)
C
D
Fehlbaak Weight
Feedback Noise
it indicates whether it is best to send a prediction (L * 0) or not (L= 0). The control gain always takes the opposite sign as the product of prediction- multiplier and iisservationmuuliige ing that the prediction is subtracted from the observation, which thereby reduces the feed- forward cost. However, when the optimal control gain is the inadiction-multiilinn D also becomes 0 so feedback noise does not corrupt the observation if no predic- tion is sent. Fig 2 shows how the optimal control gain, prediction-muliplac observation-multipe
E
F
log Gain
Measurement
log Process Noisej
ensur-
G
H
log
Obegrvation)
World Slate Timeseale
Noise
D and E change with dif- ferent parameters that define the constraints involved. We observe that there are cases where feedback and feedforward messages useful, and others where messages are harmful.
The best use of predictions and observations de- pends on the world state dynamics and channels' noise costs. For tolerable feedbuck noise cost, A: feedforward noise, or the feedforward weight, the opti- mal strategy transitions from sending only feedforward, to predictive coding, back to only feedforward, and finally to messages. For moderate feedforward noise cost, as increase either the C: feedback noise, or the feedback weight, the optimal strategy transitions from predictive cod- ing to feedforward messages only. When we increase either the E: measurement gain, or the F: process noise, the H: world state timescale, the optimal strategy transitions from silence, to only feedforward messages, to predictive coding. G: Increasing the observation noise leads to the opposite sequence.
Figure 2:
increase either the
5.1 Conditions
for the Messages to be Useful
If the costs and noise are high, then it is not useful to send messages forward. To understand this quantitatively, we consider what happens without feedback (D 0). In

---

A
Reedback Noise Crest
Indifferent to Foedback
Feedbbakk Noise Cost.
uan Indifferent to Foedforward
Predictability
Efficient Coding
No
Messages
Predictive
Coding
Feedforward Noise Cost
Foedforward Noise Cost
B
Foodbaek Noiso Con
Noise Cost
Feedback
Foedforgard
Cost
Foedbeck Noise Cost
World State
Feedbeck Noise Cost
Measurement Gain
Timescale
Foodforward Noise Crest.
Foodforward Noise Cost.
Foodforward Noise Crest
Figure 3: Boundary curves divide the space of feedback and feedforward noise costs into regions of categorically different optimal strategies. A: Predictive coding is favored below the solid yellow line, feedforward efficient coding is favored above it, and messages are favored to the right of the dashed line. B: The boundary curve determining whether feedback is useful lies below the line (cyan) along which the feedback and feedforward noise costs are equal. implying that for predictions to useful the feedback channel must be cheaper and/or less noisy than the feedforward channel. C: With increasing predictability (thinner lines) of the world to be inferred, boundaries between coding strategies shift upwards and rightwards. As the predictubility increuses for fixed value of channel parameters (dot), the optimal strategy transitions from sending messages, to example
sending only feedforward messages, to sending and subtracting predictive feedback messages. of the shift in boundary curve with increasing predictability is shown for different world state timescales. As the timescale increases, the memory of the world state increases along with the SNR at the observation level, thereby increasing the predictability. E: A similar example is shown for different measurement gains. As the measurement gain increoses, the observation SNR increases, thereby increasing the predictability.
the absence of feedback, feedforward messages are worth its cost when the optimal observation multiplier E at D = is non-zero. We know that the derivative of the total cost at the optimal non-zero E, when it exists, should be O. We set this derivative to 0 and then use Descartes' rule of signs to find when such a non-zero root exists. This yields the condition Eq 13. Sending even an infinitesimal feedforward message about the observation to the brain is worth its feedforward energy cost and only if
in U
(1 - A2)
(13)
SNR.
where Un. Woo? is the feedforward noise cost, i.e. the cost incurred by the noise variance in of C the
of feedforward messages. Ug- (1 (2) the steady state signal power, and SNR, (1-A2) oz observation Signal to Noise Ratio. When the left and right hand sides Eq 13 are equal, the optimal strategy is indifferent between sending feedforward messages or not.
Next we consider the case where feedback is allowed. the optimal prediction rediction-multiplier is zero then it is the same as no feedback, so by assumption the optimal observation- observation-multillier is non-zero non-zero,
and only feedforward is useful (Eq 13 holds). the optimal eredictioo-multipil then naturally the feedforward evidence must also be non-zero to have any signal worth predicting. Furthermore, when sending feedback is optimal, must worth sending some feedforward messages small
even if feedback is cut off. These purely feedforward messages might need to be smaller as the residuals when we had feedback - but sending at least some information is evidently worth the cost, since it was worth paying that feedforward cost even when noisy feedback corrupted the inference. Thus, mathematically, feedforward messages are useful if and only if Eq 13 is true, both with and without feedback. Eq 13 implies that feedforward messages are worth their energy cost when either the observations are very reliable (high SNR.). or when the relative cost of sending an arbitrarily small feedforward message is low (small Un, (U.).
Similarly, as shown in Appendix A.7, we find that feedback is valuable when
Unn < {UUm, A, A C o²)
(14)
5

---

log( Feedforward Noise Cosr
Cuutrol Gain
Cost(D)Coott1
gres
feedback
feedforward
Ouly
Only fendforwanc forward
Indifferent
Prediction Multiplier (I)
Favors feedback
Faxslback
Noise Cost
Predictions versus channel parameter space
Costs near indifference poinl
Figure 4: Phase transition the value of feedback. Left: Optimal control gain varies with the feedback and feedforward noise costs. As the feedforward noise cost increases (black line), the optimal strategy transitions non non-monotonically from sending only feedforward messages, to predictive coding with suppressive feedback, back to sending only feedforward messages, and eveniually to sending panel the phuse transition for feedbuck: the locus of points where the optimal strategy is indifferent to sending feedback (green area) or not (red area). Near the critical point (colored circles), the loss function goes through a phase transition [35], seen at Right: The cost finction has a minimum at Prediction muloliplleler D = O for low feedforward noise cost, favoring no feedback (red). But as the feedforward noise cost increases from low moderate values, this extremum switches a maximum, with nearby minima that favor feedback with nonzero D (green). Right at the transition point, the system becomes indifferent to feedback (yellow). Another phase fransition occurs in the opposite direction as the feedforward noise cost increases further from moderate to high.
messages. The yellow curve the
where & is a complicated function that involves solving for a quartic equation. Below in section 5.2 will interpret the behavior of this function. Uba Wo of is the feedback noise cost, i.e. the cost incurred by the noise variance in feedback messages. In Fig 3A, ob (solid yellow line) is plotted as a function of the feedforward noise cost. Where Eq 14 holds as an equality (feedback and feedforward noise costs that correspond to points on the yellow line), an optimal observer is indifferent about sending feedback messages. Sending feedback is useful below that line, and harmful above it.
As stated above, when sending feedback is optimal (Eq 14 satisfied), it must be worth sending feedforward messages even if feedback is cut off (Eq 13 holds truc). This is indeed observed in Fig 3A, where the dashed line is the locus of points where the optimal strategy is indifferent to sending feedforward messages: to the left, sending feedforward messages is useful, and to the right, it is not. As expected, the region where feedback is useful is also where feedforward messages are useful. Also, for very small feedback noise cost, the only case where sending feedback becomes harmful is when sending even an infinitesimal feedforward message is not useful either. As a result, the feedback indifference curve ends exactly at the precise feedforward noise cost where the optimal strategy is indifferent to sending feedforward messages. Fig 3B shows a magnified version where feedforward and feedback products are near zero. At the origin, where the products equal zero, feedforward and feedback messages are free and/or noiseless. A noiseless message can be arbitrarily small and still convey information, so the cost can be arbitrarily small. In this case, there nothing to gain or lose by subtracting predictions, so even free feedback makes no difference to costs or inferences. Thus the feedback indifference curve starts at the origin. Fig 3B also shows that for feedback to be useful, the feedback channel should be cheaper and/or less noisy than the feedforward channel. Note that this is true regardless of sensory parameter values (world state timescale, measurement gain, etc.).
The exact optimal multipliers for the optimization in Eq 9 depends on all system parameters (Eqs 1-7): feedforward and feedback noise variances and weights, the measurement gain, world state timescale, the process and observation noise variances. However, from Eqs 13-14, we see that the optimal categorical strategy - which multipliers are nonzero - depends only on certain combinations of these parameters. For example, the phase transition depends on the measurement gain and observation noise only through their ratio, (SNR.), which determines the quality of observations. Specifically, in Appendix A.8-A.9, we prove that the feedforward (feedback) noise variance and feedforward (feedback) weights affect the phase transition only through their product, the feedforward (feedback) noise cost. The core reason is the following: If any (possibly suboptimal) multiplier is nonzero for one value of noise and cost weight, then there is a whole whmlefamily of nonzero multipliers for noises and cost weights that are scaled inversely, keeping their product (the noise cost) fixed - where every member of this family has identical costs and signals. Thus if we optimize the multipliers for one noise and weight, and find it is nonzero, then the optimal multipliers will still be nonzero for this

---

entire family of inversely scaled noises and weights. In other words, whether feedforward or feedback useful depends on the noise costs.
5.2 Transitions in the Optimal Strategy
Having seen above how there are categorically different optimal strategies for computationally constrained inference, we now examine how individual parameters move the system between these strategies. We broadly group parameters into three categories: feedforward, feedback, and sensory.
We first consider the case of low to moderate feedback noise cost. Fig 4 shows the transition between optimal strategies as a function of the feedforward and feedback noise costs. The black line on Fig 4 left traces the transition as we increase the feedforward noise cost for a fixed feedback noise cost. For low values of the feedforward noise cost, feedforward messages are almost free, so the system does not save appreciable resources by sending predictions; even worse, noisy feedback would corrupt the signal with noise. Thus for low feedforward noise costs it is optimal to send no predictions. As this cost increases, at some point it becomes equally valuable to send or withhold feedback message. For higher feedforward noise costs, cross the point of indifference, to where feedforward messages are important yet their channel is not economical by itself. Predictive feedback then becomes preferable, even when accounting for additional feedback noise.
Feedforward parameters:
As feedforward noise cost increases, reliable transmission through the channel becomes less afford- able. As consequence, the inference degrades. Upon increasing the feedforward noise cost beyond a certain point, the inference becomes so poor that it is no longer possible to make a good prediction worthy of the cost it incurs. Thus the system resorts to sending only feedforward messages. For similar reasons, as the world becomes more predictable (Fig 3C), there is a wider range of feedforward and feedback noise costs for which sending predictions is optimal. The predictability can be increased either by lengthening the world state timescale, or by enhancing the observation SNR, enabling better inferences and thereby better predictions. Figs 3D-E demonstrates this for different world state timescales and measurement gains respectively. However, for large enough feedforward noise cost, it is best to remain silent sending even feedforward messages alone becomes too expensive/noisy.
Fig 2A and B show how the optimal control gain L, prediction-multiphe D and observation observation-mulliiiirr E change with feedforward noise and feedforward weight respectively. As only the product of noise variance and weight determines the transition in optimal strategy, we observe the same trend as either one of them increases. However, these parameters actually exhibit different effects away from the points of transitions. For example, when feedforward noise is close to 0, the system can save costs by attentuating the signal (E - 0) and still beat the negligible feedforward noise. In contrast, in the extreme case that the feedforward messages are nearly free (W; - 0), the observation mullippeer increases arbitrarily (E - x) to dominate any feedforward noise, improving the inference at no cost. These limiting cases provide helpful intuitions. Naturally, the challenging regime is in the middle, where there are real tradeoffs to make, and where feedback becomes relevant.
For the limiting case where feedback noise cost is extremely high, the feedback channel is too expensive/noisy to be used. Hence, as we slowly increase the feedforward noise cost from zero, we start by sending just the feedforward messages until the feedforward channel is too expensive/noisy, at which point no longer send any messages.
Feedback parameters: Analogous to the analysis described for feedforward parameters, here we discuss how the optimal strategy changes as we increase the feedback noise cost, for different levels of feedforward noise cost. We first consider the interesting case of low to moderate feedfoward noise cost. For low values of feedback noise cost we are in the regime where feedback is cheap and/or noiseless, and hence it is beneficial to send both feedback and feedforward messages. But when the feedback noise cost becomes too high, sending predictions become too expensive/noisy, it is best send only feedforward messages. This is shown in Fig 2C, D. Analogous to the feedforward case, although the same transition in optimal strategy is observed as we increase either the feedback noise or weight, these parameters actually exhibit different effects away from that transition. For example, L > 0 when feedback noise is close to 0, and L + -o when feedback weight is close to 0. Finally, in the case of extremely high feedfoward noise cost, it is not worth sending even an infinitesimal feedforward message, so the optimal strategy is to not send any messages.
Sensory parameters: Fig 3C shows the boundary between strategies for three systems with different levels of predictability. For fixed value of feedback and feedforward noise costs (dot), the optimal
7

---

strategy changes with the predictabiliity. With unpredictable dynamics (thick curve), the dot lies in the region where sending no messages optimal, since the feedforward channel is relatively poor (right of the thick dashed line). For a slightly higher predictability, the dot lies within the region where feedforward-only messages are optimal (above medium curve, but left of the corresponding dashed line). And for even higher predictabilitty, the dot lies in the region where predictive coding is optimal (below the thin curve). Hence, as the predictability increases, there is a transition from sending no messages, to sending just feedforward, to sending both feedback and feedforward messages.
Similarly, if either the measurement gain or the process noise increases. it would yield a higher observation SNR, which would then improve the inference and thereby predictability. Figs 2E-F show the resultant transitions from no messages, to feedforward-only messages. then to predictive coding, with increasing measurement gain or process noise. Increasing observation noise has the opposite effect, making observations less reliable and thereby reducing the predictability. This is shown in Fig 2G. Finally, increasing the world state timescale increases the predictability since it increases both the memory of the system, and the observation SNR. Fig 2H reveals the familiar sequence of transitions in strategy, supporting our core intuitions about when feedback is valuable.
5.3 Anatomical and functional differences can lead to different channel noise costs
We characterized the properties of feedforward/feedback channel in terms of the channel noise and weight of energetic cost, and showed that the phase transition depends on the feedfor- ward/feedback channel noise costs. But why would channels have different noise costs? Bio- logically, asymmetries in noise costs can arise from anatomical or functional constraints. For example, Dale's law [36] states that each neuron makes only excitatory or inhibitory synapses onto its targets. However, long-range feedback connections in the brain are excitatory [37, 38). This implies that to subtract predictions using long-range inhibitory feedback, the signals must pass through two neurons: long-range excitation that is inverted by short-range inhibition. In contrast, feedforward signals can proceed directly to their targets. The sign inversion neu- rons induce additional noise in the feedback pathway. Below we review biological evidence that feedback and feedforward channels have distinct anatomical and functional properties.
To see how channels with different anatomical structures can yield different channel noise costs, contrast two versions of a channel with equal numbers of neurons, all having identical energetic weights and noise variances. One version has sin-
Observation Multiplier EE)
Total Cost
Control Gain (L)
lop
Population Size Rotio
gle population of neurons that linearly encodes the message to be sent; the channel output simply decodes this population's 's representation. We call this channel the non-sequential case (Appendix Fig 9A). The second version is the sequential case (Appendix Fig 9B), and uses two populations sequentially con- nected in series. one after the other, like long-range inhibition comes from long-range excitation followed by short-range in- hibition. The first population linearly encodes the message to be sent, which is then decoded and then encoded by the second population; the channel's output is decoding of the second population.
Prediction Multiplier (D)
Figure 5: Allocation neurons be-
tween two sequentially connected pop- ulations of the feedback charnel offects when sending feedback is useful.
Appendix 1. shows that for the non-sequential case with one population, we scale up the number of neurons, the channel's weight scales up and the noise scales down inversely. Their product, the channel noise cost, does not change with the number of neurons, yielding the same optimal categorical strategy for a given noise and weight per neuron. For the sequential case with two populations, we show in Appendix 10.2 that the channel noise cost again does not depend on the total number of neurons, but does depend on the ratio of population sizes. Furthermore, the total cost changes non-monotyn with population size ratio for the sequential case, and increases monotonically with total number of neurons for both sequential and non pon-eeeulttall cases. Fig shows the optimal strategy changing with the population size ratio of the feedback channel, for fixed total number of neurons. We see that the feedback is not useful when the ratio is strongly skewed (away from 1).
From the above example, we saw that channels with different anatomical structures can yield different noise costs, and thereby different optimal strategies. Biologically, there exist differences between the
8

---

anatomy or functional properties of the feedback and feedforward channels. Anatomically, feedback projections generally outnumber feedforward ones by 2:1, but the feedforward pathways compensate with greater weight, dominating at short cortical distances and leveling off at longer distances [39]. Functionally, the sparser, stronger feedforward channels therefore have less ability to average away the noise, leading to a relatively higher amount of metabolically expensive activity caused by noise, and thus to a higher noise cost. This could establish conditions under which feedback is useful.
6 Discussion
In this paper, we define a new class of dynamic optimization tasks that reflect essential biological constraints on inference in the brain, by including cost and noise for each recurrently connected computational element. We solve this optimization problem by modeling inference as control problem with prediction as self-control. The resultant optimization provides nontrivial predictions for when we expect suppressive feedback as a function of biological constraints, computational costs, and world dynamics.
Predictive coding is a promising theory for brain computations [40-44), and a variety of experimental studies have indeed shown predictive suppression effects in neural responses [45-52]. However, neural responses vary in how much they are suppressed depending on the context and SNR. Variants aiming explain these effects all neglect the computational costs associated with predictive coding, even though those costs were part of the motivation for saving metabolic costs in the first place.
Past work deseribed temporal decorrelation as optimal encoding strategy, subtracting slow, pre- dictable component from the recent measured signal, implementing a form of predictive coding [15, 53]. Like our results, they too observed that this strategy changes with the SNR. Efficient coding shows that mean responses are fully subtracted at all SNRs, but with different temporal kernels [15, 53). Later studies of predictive coding observed that for low SNR, temporal decorrelation is no longer the best strategy [18]. Our work generalizes these results to account not only for feedforward noise and costs in sensory coding, but also for noise and costs in recurrent computation. In particular, the nonmonotonic dependence of optimal feedback on feedforward noise costs is a novel consequence of our framework with analog in previous studies. We could also imagine generalizing our results to adaptive strategies, like [31]. where optimal adaptation could implement our predicted changes in the optimal suppression as sensory statistics change. Conversely. by considering the computational costs of adaptation itself (and not just the activity resulting from adapted computation), we may find conditions where adapting is not beneficial.
Although we contend that it is costly to send predictions, if these predictions are based on in- ferences that already being computed for task, then is it really extra cost send them as predictions? Yes, because distinct neurons are used to send feedforward and feedback sig- nals [54], so the brain might pay costs twice for the same inference. When beneficial, the brain could avoid the duplicate feedback costs by silencing the feedback neurons, and send its infer- ences to higher brain areas only through its feedforward neurons. related question arises for feedback suppression, which is presumably implemented through inhibitory neurons. What is the benefit of turning on an inhibitory neuron just in order turn off an excitatory neuron? Ex- citatory neurons outnumber inhibitory neurons by 4:1 [55], so few inhibitory neurons can sup- press many excitatory ones, amounting to a potentially substantial savings. These biological constraints could be used in future elaborations of computationally constrained inference models.
Our mathematical system's crucial theoretical parameters relate to biological quantities in three ways: as context-dependent param- eters (time constant of the stimulus dynamics, sensory SNR), as neural parameters that could potentially be manipulated experimen- tally (feedforward and feedback SNR), and as developmentally -ixed parameters of the system (feedforward and feedback architecture). Testing our predicted dependence on these three types of parameters requires different considerations. First, the context- deeendent pa- rameters are by far the easiest to manipulate experimentally. One can control the stimulus to adjust the observations' SNR and the world state timescale, and measure whether any response suppression is modulated by these controlled parameters as shown in Fig 6. Second, the internal computational parameters like neural noise may be controllable through stimulus changes or experimental techniques
Suppression Ratio in I8s At.
SNR,
World State Timescale
Figure 6: World timescale and observation SNR affect predictive response suppression. SNR in- creases with line thickness.

---

of causal manipulation. For example, electrical or optogenetic stimulation could directly inject neural variability. Another approach could be based on the fact that neural noise variance tends to increase with firing rate, so one could test our predictions about computational noise by elevating firing rates. Such methods could include providing a background sensory stimulus, or direct neural stimulation to increase baseline activity. In any case, it would be important to apply either natural manipulations or chronic unnatural ones to give the brain enough time and experience to optimize its computations. And finally, for the developmentally fixed parameters like architecture, we cannot easily alter the system experimentally, and may be unreasonable to assume that properties in a real biological system can be unambiguously mapped to any particular value in our abstract theoretical system. However, we can compare between brain systems or species with architectural difference, and test whether functional properties covary with those differences as predicted by our theory. Thus it may be fruitful to compare between brain areas with different architectures, such as 3-layer paleocortex (e.g. olfactory cortex) versus 6-layer neocortex (e.g. auditory visual cortex), or within neocortical areas with different properties. It may also be fruitful to compare between organisms with different architectures (e.g. mammals versus reptiles). These architectures have different microcircuits with distinct feedforward and feedback projection neurons, For example, feedback in different visual and motor cortical systems tends to target different inhibitory cells [56] which have different noise levels [57, 58). We predict that systems with more feedback noise (e.g. SOM versus VIP) would have less predictive suppression, as observed experimentally [56].
Our theory of inference as control could also be used in designing energy-efficient systems in other domains where the estimation takes place under constraints. For example, low-power applications of inference and control, such as drones or remote sensing, often place constraints on dynamic range and noise (e.g. quantization noise) for all signals, and could thus benefit from optimizations based on our model.
theory. The proposed inference as control method can be extended to the more general case of observer taking external actions in addition to the internal predictions, in order to maximize external rewards while minimizing both external action costs (c.g. movements) and internal computational costs. This could be modeled as typical LQG control problem for external variables by including appropriate entries in matrices Q, R. and Baug in addition to the internal controls we use for predictive coding. Though our results limited to one-dimensional linear dynamics with uncorrelated noise, our future work will explore these computational constraints in more complex multivariate and nonlinear tasks. Accounting for graph-structured inferences [59] and controllability [60] may provide additional constraints on the brain's inference processes, and additional predictions for experiments.
Limitations. There are several interesting avenues for generalizing
Residuals between predictions and observations are useful not just for improving inferences, but also for learning, which this paper does not address. In principle, a multivariate generalization could explain such computations as hierarchical inferences or adaptation where slower changes also subject to prediction. However, unlike adaptation, learning occurs out of equilibrium with the environment, and accounting for transient responses under nonstationary statistics will require an extension to our theory. Overall, this work points a way towards expanding theories of predictive coding and efficient coding unify inference, learning, and control in biological systems.
Acknowledgments. The authors thank Itzel Olivos Castillo, Ann Hermundstad, Krešimir Josić, and Paul Schrater for useful discussions. This work was supported in part by NSF CAREER grant 1552868, the McNair Foundation, and AFOSR grant F-9550021-110422 the Cognitive and Computational Neuroscience program.
Competing interests. XP is a founder of Upload AI, LLC, company in which he has related financial interests.
References
[1] Horace Barlow et al. Possible principles underlying the transformation of sensory messages. Sensory communication, 1:217-234, 1961.
[2] Rajesh PN Rao and Dana H Ballard. Predictive coding in the visual cortex: functional interpretation of some extra-classical receptive-fiele effects. Nature neuroscience, 2(1):79-87, 1999.
10

---

[3] Hilbert J Kappen, Vicenç Gómez, and Manfred Opper. Optimal control as graphical model inference problem. Machine learning, 87(2):159-182, 2012.
[4] Hermann Von Helmholtz. Handbuch der physiologischen Optik: mit 213 in den Text einge- druckten Holzschmitten und 11 Tafeln, volume 9. Voss, 1867.
[5] David C Knill and Alexandre Pouget. The bayesian brain: the role of uncertainty in neural coding and computation. TRENDS in Neurosciences, 27(12):712-71,, 2004.
[6] Wei Ji Ma and Mehrdad Jazayeri. Neural coding of uncertainty and probability. Annual review of neuroxcience, 37:205-220, 2014.
[7] Wei Ma, Jeffrey M Beck, and Alexandre Pouget. Spiking networks for bayesian inference and choice. Current opinion in neurobiology, 18(2):11-222, 2008.
[8] Stephen W Kuffler. Discharge patterns and functional organization of mammalian retina. Journal of neurophysiology, 16(1):37-68, 1953.
[9] Haldan Keffer Hartline. The response of single optic nerve fibers of the vertebrate eye to illumination of the retina. American Journal of Physiology-Legacy Content, 121(2):400-15,, 1938.
[10] Peter Kok, Janneke FM Jehee, and Floris De Lange. Less is more: expectation sharpens representations in the primary visual cortex. Neuron, 75(2):265 270, 2012.
[11] Christopher Summerfield and Floris P De Lange. Expectation in perceptual decision making: neural and computational mechanisms. Nature Reviews Neuroscience, (5(11):745-756, 2014.
[12] J Hans van Hateren. Real and optimal neural images in early vision. Nature, 360(6399):68-70. 1992.
[13] Joseph J Atick and A Norman Redlich. Towards a theory of early visual processing. Neural computation, (3):308-320) 1990.
[14] Simon Laughlin. A simple coding procedure enhances neuron's information capacity. Zeitschrift für Naturforschung c. 3609-10):910-1121 1981.
[15] Mandyam Veerambudi Srinivasan, Simon Barry Laughlin, and Andreas Dubs. Predictive coding: a fresh view of inhibition in the retina. Proceedings of the Royal Society of London. Series B. Biological Sciences, 216(1205):427-455 459, 1982.
[16] Michael W Spratling. Predictive coding as model of response properties in cortical area v1. Journal of neuroscience, 3019)33531-3543, 2010..
[17] Andre M Bastos, W Martin Usrey, Rick A Adams, George R Mangun, Pascal Fries, and Karl J Friston. Canonical microcireuits for predictive coding. Neuron, 76(4):695-711, 2012.
[18] Matthew Chalk, Olivier Marre, and Gasper Tkačik. Toward unified theory of efficient, predictive. and sparse coding. Proceedings of the National Academy of Sciences, 115(1):186- 191. 2018.
[19] Karl Friston and Stefan Kiebel. Predictive coding under the free-energy principle. Philosophical Transactions of the Royal Society B: Biological Sciences, 364(15211:1211-1221, 2009.
[20] Karl Friston. The free-energy principle: a unified brain theory? Nature reviews neuroscience,
1 11(2):127-138, 2010.
[21] Yanping Huang and Rajesh PN Rao. Predictive coding. Wiley Interdisciplinary Reviews: Cognitive Science, 2(5):580-592: 2011.
[22] Robert Rosenbaum. On the relationship between predictive coding and backpropagation. arXiv preprint 2rXiv:2106.13082 2021.
[23] Haiguang Wen, Kuan Han, Junxing Shi, Yizhen Zhang, Eugenio Culurciello, and Zhongming Liu. Deep predictive coding network for object recognition. In International Conference on Machine Learning, pages 5266-5275. PMLR, 2018.
11

---

[24] William Lotter, Gabriel Kreiman, and David Cox. Deep predictive coding networks for video prediction and unsupervised learning. arXiv preprint 1.5.::6600000004 2016.
[25] Abdullahi Ali, Nasir Ahmad, Elgar de Groot, Marcel AJ van Gerven, and Tim C Kietzmann. Predictive coding is consequence of energy efficiency in recurrent neural networks. bioRxiv, 2021.
[26] Kuan Han, Haiguang Wen, Yizhen Zhang, Di Fu, Eugenio Culurciello, and Zhongming Liu. Deep predictive coding network with local recurrent processing for object recognition. arXiv preprint arXiv: 1805.0752, 2018.
[27] Michael W Spratling. A hierarchical predictive coding model of object recognition in natural images. Cognitive computation, 92):1511167, 2017.
[28] Michael W Spratling. A review of predictive coding algorithms. Brain and cognition, 112:92-97 97, 2017.
[29] Rakesh Chalasani and Jose C Principe. Deep predictive coding networks. arXiv preprint 130 2013.
[30] Memming Park and Jonathan W Pillow. Bayesian efficient coding. BioRxiv, page 178418. 2017.
[31] Wiktor F Mtynarski and Ann M Hermundstad. Adaptive coding for dynamic sensory inference. Elife, 7:e32055, 2018.
[32] Laurence Aitchison and Máté Lengyel. With or without you: predictive coding and bayesian inference in the brain. Current opinion neurobiology, 46:219-227, 2017.
[33] Rudolf Emil Kalman et al. Contributions to the theory optimal control. Bol. soc. mat. mexicana, 102)1102-119 1960.
[34] Mark Davis, Stochastic modelling and control. Springer Science & Business Media, 2013.
[35] Lev Davidovich Landau and Evgenii Mikhailovich Lifshitz. Course of theoretical physics. Elsevier, 2013.
[36] Piergiorgio Strata, Robin Harvey, et al. Dale's principle. Brain research bulletin, 505):344-350. 1999.
[37] Peter Somogyi, Gabor Tamas, Rafael Lujan, and Eberhard H Buhl. Salient features of synaptic organisation in the cerebral cortex. Brain reseurch reviews, 26(2-3):113-135, 1998.
[38] György Buzsáki. feed-forward inhibition in the hippocampal formation. Progress in neurobiol- ogy, 22(2):131-153, 1984.
[39] Nikola T Markov, Julien Vezoli, Pascal Chameau, Arnaud Falchier, René Quilodran, Cyril Huissoud, Camille Lamy, Pierre Misery, Pascale Giroud, Shimon Ullman, et al. Anatomy of hierarchy: feedforward and feedback pathways in macaque visual cortex. Journal of Comparative Neurology, 522(1):225-259) 2014.
[40] Karl Friston. Does predictive coding have future? Nature neuroscience, 2119:1011-1101, 2018.
[41] Xiao Liu, Xiaolong Zou, Zilong Ji, Gengshuo Tian, Yuanyuan Mi, Tiejun Huang, Michael Kwok Yee Wong, and Si Wu. Push-pull feedback implements hierarchical information retrieval efficiently. 2019.
[42] Janneke FM Jehee, Constantin Rothkopf, Jeffrey M Beck, and Dana H Ballard. Learning receptive fields using predictive feedback. Journal of Physiology-Paris 10051-3):225-132 2006.
[43] Samuel Gershman. What does the free energy principle tell us about the brain? arXiv preprint 1011::190. 2019.
12

---

[44] Andreja Bubic, D Yves Von Cramon, and Ricarda Schubotz. Prediction, cognition and the brain. Frontiers in human neuroscience, 4:25, 2010.
[45] Toshihiko Hosoya, Stephen A Baccus, and Markus Meister. Dynamic predictive coding by the retina. Nature, 436(7047):71-77, 2005.
[46] Jan Homann, Sue Ann Koay, Alistair M Glidden, David W Tank, and Michaell J Berry. Predictive coding of novel versus familiar stimuli in the primary visual cortex. BioRxiv, page 197608, 2017.
[47] Scott O Murray, Daniel Kersten, Bruno A Olshausen, Paul Schrater, and David L Woods. Shape perception reduces activity in human primary visual cortex. Proceedings of the National 2002.
Academy of Sciences, 99(23):15144-5516
[48] Ryszard Auksztulewiez and Karl Friston. Repetition suppression and its contextual determinants in predictive coding. cortex, 80: 80:125-140, 2016.
[49] Tai Sing Lee and David Mumford. Hierarchical bayesian inference in the visual cortex. JOSA
A, 20(7):1434-1488, 2003.
[50] Rajesh PN Rao and Dana H Ballard. Dynamic model of visual recognition predicts neural response properties in the visual cortex. Neural computation, 9(4):721-763, 1997.
[51] David Mumford. On the computational architecture of the neocortex. Biological cybernetics, 66(3):241225 1992.
[52] Marta I Garrido, James M Kilner, Klaas E Stephan, and Karl J Friston. The mismatch negativity: a review underlying mechanisms. Clinical neurophysiology, 120(3)455-463, 2009.
[53] J Hans Van Hateren. Spatiotemporal contrast sensitivity of early vision. Vision research,
33(2):257-267, 1993.
[54] João D Semedo, Anna Jasper, Amin Zandvakili, Aravind Krishna, Amir Aschner, Christian K Machens, Adam Kohn, and Byron M Yu. Feedforward and feedback interactions between visual cortical areas use different population activity patterns. Nature communications, 13(1):1-14, 2022.
[55] Ryan Shewcraft, Heather Dean, Margaret M Fabiszak. Maureen A Hagan, Yan T Wong, and Bijan Pesaran. Excitatory/inhiiitory responses shape coherent neuronal dynamics driven by optogenetic stimulation in the primate brain. Journal of Neuroscience, 40(10):256-2068,
2020.
[56] Shan Shen, Xiaolong Jiang, Federico Scala, Jiakun Fu, Paul Fahey, Dimitry Kobak, Zhenghuan Tan, Jacob Reimer, Fabian Sinz, and Andreas S Tolias. Distinct organization of two cortico- cortical feedback pathways. bioRxiv, 2020.
[57] Wen-pei Ma, Bao-hua Liu, Ya-tang Li, Z. Josh Huang. Li I Zhang, and Huizhong W Tao. Visual representations by cortical somatostatin inhibitory neurons selectivv but with weak and delayed responses. Journal of Neuroscience, 30443):1437 2010.
[58] Daniel J Millman, Gabriel Koch Ocker, Shiella Caldejon, Josh D Larkin, Eric Kenji Lee, Jennifer Luviano, Chelsea Nayan, Thuyanh V Nguyen, Kat North, Sam Seid, al. Vip interneurons in mouse primary visual cortex selectively enhance responses to weak but specific stimuli. Elife, 9:e55130, 2020.
[59] Xaq Pitkow and Dora E Angelaki. Inference in the brain: statistics flowing in redundant population codes. Neuron, 94(5):943 953, 2017.
[60] Shi Gu, Fabio Pasqualetti, Matthew Cieslak, Qawi K Telesford, B Yu Alfred, Ari E Kahn, John D Medaglia, Jean M Vettel, Michael B Miller, Scott T Grafton, et al. Controllability of structural brain networks. Nature communications, 6(1):1-10, 2015.
13


Reviews:
[{"review_id": "iPkTI7eLgE", "reviewer": "Reviewer_15Ei", "summary": "This paper formulates an interesting tradeoff in optimal coding theory, in the context of (potential) predictive coding where there are feedforward messages, and top-down predictions can be sent as feedback to reduce feedforward signal cost. Previous works considered the cost of feedforward information but ignored the cost of feedback computation; this paper considers the noise cost of the feedback pathway, as well as the feedforward, and investigates the tradeoff as the noise costs in the feedforward/feedback channels vary. Interestingly, the paper shows that there is a nontrivial transition between three regimes: where predictive coding is preferred (top-down prediction feedback is useful), where efficient coding is preferred (feedback is not worth the cost, so just send feedforward signals), and where sending any messages is not worth the cost at all. The paper also provides biological evidences to argue that it is indeed plausible to have different noise costs in the feedforward/feedback pathways.\n", "questions": "i) The theory is developed in the simple LQG model setting. Can you say whether the results in this limit corresponds to any bounds to the non-linear, non-gaussian cases?\n\nii) One of the most interesting points in this paper is that the non-monotonic shape of the boundary curve $\\Phi$ as a function of the feedforward noise cost (Eq 14, Fig 3). And as soon as I saw it, I was eager to understand why it was so. When $ \\Phi $ first appears in the paper, the authors only say that it is a \"complicated function\", but it would be nice to provide some more insights on why it increases at first and decreases later. Looking at Appendix A7 (as referred to from Eq 14) did not help much. I eventually found relevant text later in the paper, in section 5.2 (paragraphs starting L232 and L242). I suggest that this material be moved up, so that the reader gets a clearer picture early on. It was nice that the authors explain it with a fixed, low-to-moderate feedback noise cost, and talk about small / intermediate / large feedforward noise regimes. It would help to add a horizontal line to Fig 3A and describe it right there.\n\niii) Minor comments:\n\n- I felt that the text was rather dense in my initial read, and the main messages were not easily located by quick skimming. Hard to pinpoint a specific suggestion, but in general, having the main point at the beginning of each paragraph is really helpful.\n- In the figures, red-green-yellow will make it difficult for color-blind readers. Suggest replacing with other color palettes.\n- The 3-d plot in Fig 4 left was difficult to understand at first, because the contour lines wasn't visible in the normal pdf view. (I realized after zooming in that there were contour lines.)\n", "limitations": "The authors have addressed the limitations of their modeling assumptions and constraints.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "This is an important and interesting paper that studies a fundamental tradeoff in the theory of neural computations. I believe that this paper provides a conceptual advance and will be a significant contribution to the community's understanding of optimal coding. The neurobiological discussion at the end is valuable as well. The paper is original in their consideration of the feedback noise cost, and the discovery of the nontrivial structure of the tradeoff.\n\nI find the mathematical formulation very reasonable and the logical developments convincing, although I have not checked all steps carefully.\n\nAs for the weaknesses of the study, the theory is formulated in a simplified model setting, and it is still to be tested how relevant the lessons are in the actual neural systems. Also, this paper provides no direct connection to real neural data. But this is how theories should start, and the authors also discuss the limitations in the paper, with plans for future work.\n\nPresentation-wise, the paper is nicely contextualized with respect to the existing ideas and models. Regarding writing/visualization, I eventually felt that the paper is clear enough, in that it describes all important points, and touches on most of my questions that came up over the course of reading. That said, in the initial read I had to go back and forth between the figures and the text paragraphs more than I would usually do. I will provide more comments below.\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": "gdb9PsU7dTq", "reviewer": "Reviewer_8Atc", "summary": "This paper presents a theory of optimal inference in presence of noisy\nsensation, prediction and feedback, under linear-quadratic-gaussian\nassumptions, with separate costs for feedforward and feedback\nsignals. The theory is used to show how the usefulness of feedback\nundergoes phase transitions depending on the structure of the\nenvironment, architectural constraints, and computational costs.\n", "questions": "Here are some passages that I found harder to read than necessary:\n- Line 182: I was confused by the sentence \"Furthermore, we also see\n  that when sending feedback is optimal, it must be worth sending some\n  feedforward messages even if feedback is cut off\". In that context,\n  it is not clear whether the reader is supposed to \"see\" that fact\n  from some of the equations above, or this is supposed to be\n  self-evident based on the consideration made in the rest of the\n  sentence. Moreover, I was surprised by the use of \"cut off\" -\n  somehow it made me doubt that I had missed a component of the model\n  that accounted for specific causal interventions on the system.\n- Lines 206--209, series of sentences starting with \"At the\n  origin...\": this same idea was much clearer when repeated below,\n  around lines 235--238 (even if for a slightly different case). In my\n  opinion, adopting the phrasing of 235 the first time that the\n  concept is presented would make for smoother reading.\n- Unless I'm missing something, lines 262--265 seem redundant with\n  266--270. The two passages follow immediately one another, but they\n  are in different paragraphs. Please consider merging these passages\n  if they are indeed redundant, or signposting better their difference\n  if they are not.\n", "limitations": "In my opinion, the paper gives a fair account of its limitations.\n", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\nThis paper tackles an important problem in theoretical\nneuroscience, hitting a sweet spot between level of detail and\ntractability of the system under exam. The method and the results\nprovide meaningful progress towards understanding efficient coding in\ndynamical contexts and its relationship with predictive coding, a\nresearch area that has seen intense effort and several high-quality\ncontributions in the last few years (such as those by Chalk & Tkacik\nand Mlynarski & Hermundstad, which the paper prominently acknowledges).\n\n### Weaknesses\nIn the central part of the paper (sections 5.1, 5.2), some\npassages could be made clearer/easier to follow. Please see\n\"questions\" for details.\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": "U1SDuuGGYWT", "reviewer": "Reviewer_v9qs", "summary": "This paper inspects the consequences of considering the costs associated with sending messages to represent univariate variables in the brain. The authors propose a dynamical system taking into account the inherent noise while updating the estimation of the latent world state. They subsequently derive a theory from the minimization (supposed to take place in the brain) of a cost function taking into account the accuracy of the latent state estimation as well as the feedforward and feedback costs. The main part of the paper presents a mathematical analysis of the system, that is, shows phase transitions between predictive coding (feedforward + feedback messages), efficient coding (feedforward messages only) and silence (no messages), depending on the values of parameters. Finally, the theoretical framework is put into an experimental perspective, through parallels with the properties of the biological brain and discussion about how to test the theoretical predictions.", "questions": "1. Is it \"just\" a model of the brain, or could it be useful in another domain? I feel like this theory could be used in designing energy-efficient systems where the estimation takes place under constraints (for instance energetic). This could be acknowledged in the discussion\n2. How could the abstract message-passing in the graph from Fig 1B relate to the messages sent from excitatory/inhibitory populations? \n3. Mathematical conventions: to relate more to predictive coding, shouldn't L be positive and D negative, instead of the reverse? Potentially also replace D with -D, in order to have a minus sign in Eq 5\n\n----------\nComments: \n1. I think Fig 9 should be moved from the appendix to the main text\n2. I didn't understand much Fig 4 with the legend. For instance, which color represents the absence of messages? None, right?", "limitations": "Limitations: The authors carefully address the limitations of the work, opening the way for generalizations of the theory (taking actions / multivariate variables / nonlinear tasks / learning).\n\nSocietal impact: The authors did not address any potential negative societal impact of their work, which is fine given the subject of the paper (understanding the brain)", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. Interesting topic, well in the scope of the NeurIPS conference\n2. Very pedagogical figures and legends\n3. The paper has a very clear and intuitive message\n4. The paper is well-grounded mathematically\n5. The link from the very theoretical framework to the biological brain is very interesting. More particularly, I enjoyed the parts about the excitatory/inhibitory processes in the brain, especially the explanation about the potential higher cost of long-range inhibitory feedback as a consequence of the anatomical constraints.\n\nWeaknesses:\n1. The authors put some effort into thinking of ways to test their theory, but the discussion remains a bit too conceptual and vague, especially the part about the comparison between brain systems or species. Add references and explicit further your predictions\n2. The relation with previous theoretical work is only superficial (see section \"Related work'). I would expect more developments in the discussion part, in light of the paper's results", "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": "4Q6SvJ2zipx", "reviewer": "Reviewer_2STF", "summary": "The present paper studies the inference of a hidden Markov model, specifically a Kalman filter-like problem. Compared with a common Kalman filter and canonical predictive coding framework, the paper considers that the feedback input (prediction) is associated with a computational cost, and how the feedback cost affects the gain of observation and predictions.", "questions": "Please see my other comments.", "limitations": "A strong requirement of a network implementation of the Kalman filter is that a network model with fixed parameters is able to infer the latent variables with different uncertainties, e.g., different observation noise or feedback noise. Not sure whether the current model could achieve this or not. It might be good to discuss this point.", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The formulation of the problem by considering a cost associated with feedback is novel, and the whole paper is structure-wise, well-written. Nevertheless, considering the neural implementation of the Kalman filter has been widely studied before, the formulation of the paper seems a bit technically simplified, and it is not clear whether the novelty of significance of the present paper is enough even though the consideration of feedback cost is new. Also, the paper directly deals with the Gaussian variables (Eq. 1-7) while we know the neural variability is Poisson-like (multiplicative). It is not clear when considering the multiplicative noise whether the main conclusions of the paper can be held. Moreover, the previous studies of neural implementation of Kalman filters typically embed the Gaussian dynamics (Eqs. 1-7) into nonlinear, biologically plausible network dynamics (e.g., Deneve, J. Neurosci., 2007; Wilson and Finkel, NeurIPS 2009; Kutschireiter 2015), hence it might be more interesting to study how the feedback cost influences the network dynamics.", "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": ["I don’t know"]}, "extra_text": {}}]

Author rebuttal:

Decision: Accept