Decomposing stimulus-specific sensory neural information via diffusion models
Authors: Steeve Laquitaine
Simone Azeglio
Carlo Paris
Ulisse Ferrari
Matthew Chalk
Venue: NeurIPS 2025 spotlight ()
Keywords: Neuroscience
Neural coding
Information theory
Efficient coding
Diffusion models
Primary area: neuroscience_and_cognitive_science
TL;DR: We propose a stimulus-wise decomposition of the mutual information that is (1) principled (axiomatic justification), (2) tractable (estimated via diffusion models), and then we demonstrate it's application on a model of visual neurons.

Abstract: A central question in sensory neuroscience is how much, but also what information neurons transmit about the world. While Shannon’s information theory provides a principled framework to quantify the amount of information neurons encode about all stimuli, it does not reveal which stimuli contribute most, or what stimulus features are encoded.  As a concrete example, it is known that  neurons in the early visual cortex are 'sensitive' to stimuli in a small region of space (their receptive field). However, it is not clear how such simple intuitions carry to more complex scenarios, e.g. with large, noisy & non-linear population of neurons and high-dimensional stimuli.

Several previous measures of neural sensitivity have been proposed. For example, the Fisher information quantifies the sensitivity of neural responses to infinitesimal stimulus perturbations. However, as the Fisher is not a valid decomposition of the mutual information it cannot say how different stimuli contribute to the total encoded information. On the other hand, previous works have proposed stimulus dependent decompositions of mutual information, which define a function $ I(x) $ such that $ I(R; X) = \mathbb{E}[I(x)] $. 
However, this decomposition is inherently ill-posed: infinitely many functions $I(x)$  satisfy the constraint, with no principled way to select among them. Further, different decompositions behave in qualitatively different ways, making it hard to interpret what are they are telling us. Finally, most proposed decompositions are computationally intractable for the high-dimensional stimuli and non-linear encoding models relevant for neuroscience.

To resolve these limitations, we propose a set of axioms that any stimulus specific and feature-specific information decomposition should satisfy in order to serve as a meaningful and interpretable measure of neural sensitivity. These axioms formalize intuitive desiderata: that the information assigned to each stimulus, and stimulus feature, should be non-negative, and additive with respect to repeated measurements.  We  also require  the decomposition  to respect  a form of locality: changes in how a neuron responds to a stimulus $ x $ should not affect the information attributed to a distant stimulus $ x' $. Finally,  the attribution must  be insensitive to irrelevant features, which do not contribute to the total information. Together, these constraints ensure that the decomposition is both interpretable and theoretically grounded.

 We show that existing decompositions violate one or more of these axioms, limiting their interpretability and use as information theoretic measures of neural sensitivity. We then introduce a novel decomposition that satisfies all of our axioms. It generalizes Fisher information by capturing neural sensitivity to both infinitesimal and finite stimulus perturbations. Moreover, it supports further decomposition across individual stimulus features (e.g., image pixels), enabling fine-grained analysis of neural representations.


Beyond satisfying our theoretical axioms, our decomposition is computationally tractable for large neural populations and high-dimensional naturalistic stimuli, through the use of diffusion models.  We demonstrate the power of our method by quantifying the information encoded by a model of visual neurons about individual images and pixels. Our approach uncovers aspects of the neural code that are not picked up by standard methods, such as the Fisher information, and opens the door to similar analyses in higher-order sensory areas, and artificial neural networks.

Decomposing stimulus-specific sensory neural information via diffusion models
Steeve Laquitaine"
Institut de la Vision, INSERM, CNRS Sorbonne Université 17 Rue Moreau, Paris 75012
Simone Azeglio*
Institut la Vision, INSERM, CNRS, & Laboratoire des Systèmes Perceptifs Sorbonne University Écolle Normale Supérieure PSI. 17 Rue Moreau, Paris 75012 & 29 Rue d'Ulm Paris 75005
Carlo Paris
Ulisse Ferrarit Institut de la Vision, INSERM, CNRS Sorbonne Université 17 Rue Moreau, Paris 75012
Institut de la Vision, INSERM, CNRS Sorbonne Université 17 Rue Moreau, Paris 75012
Matthew Chalk
Institut de la Vision, INSERM, CNRS Sorbonne Université 17 Rue Moreau, Paris 75012 matthew, chalk@inserm. fr
Abstract
understand sensory coding, we must ask not only how much information neurons encode, but also what that information is about. This requires decomposing mutual information into contributions from individual stimuli and stimulus features a fun- damentally ill-posed problem with infinitely many possible solutions. We address this by introducing three core axioms axioms-addiriviry positivity, and locality that any meaningful stimulus- wise decomposition should satisfy. We then derive decom- position that meets all three criteria and remains tractable for nigh-dimensionn be efficiently estimated using diffusion models,
stimuli. Our decomposition allowing for scaling up to complex, structured and naturalistic stimuli. Applied to model of visual neurons, method quantifies how specific stimuli and features contribute to encoded information. Our approach provides scalable, interpretable tool for probing representations in both biological and artificial neural systems.
Introduction
A central question in sensory neuroscience is how much, but also what information neurons transmit about the world. While Shannon's information theory provides principled framework to quantify the amount of information neurons encode about all stimuli, it does not reveal which stimuli contribute most, or what stimulus features are encoded 25 As a concrete example, known that
"quual contribution.
i Equal senior author contribution.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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neurons in the early visual cortex are 'sensitive' to stimuli in small region of space (their receptive field). However, it not clear how such simple intuitions carry to more complex scenarios, e.g. with large, noisy & non-linear population of neurons and high-dimensional stimuli.
Several previous measures of neural sensitivity have been proposed. For example, the Fisher information quantifies the sensitivity of neural responses to infinitesimal stimulus perturbations 88 D] However, as the Fisher is not a valid decomposition of the mutual information it cannot say how different stimuli contribute to the total encoded information. On the other hand, previous works have proposed stimulus dependent decompositions of mutual information, which define a function [(x) such that (R:X) [x]()) [8 4 17 However, this decomposition is inherently ill- pol-posed: infinitely many functions I [(x) satisfy the constraint, with no principled way to select among them. Further, different decompositions behave in qualitatively different ways, making it hard to interpret what are they are telling us. Finally, most proposed decompositions are computationally intractable for the high-dimensional stimuli and non-linear encoding models relevant for neuroscience.
23 3 31
To resolve these limitations, we propose a set of axioms that any stimulus specific and feature-specific information decomposition should satisfy in order serve as a meaningful and interpretable measure of neural sensitivity. These axioms formalize intuitive desiderata: that the information assigned to each stimulus, and stimulus feature, should be non-negatee non-negative, and additive with respect to repeated measurements. We also require the decomposition to respect form of locality: changes in how a neuron responds to stimulus r should not affect the information attributed to a distant stimulus r'. Finally, the attribution must be insensitive to irrelevant features, which do not contribute to the total information. Together, these constraints ensure that the decomposition both interpretable and theoretically grounded.
We show that existing decompositions violate one or more of these axioms, limiting their inter- pretability and use as information theoretic measures of neural sensitivity. We then introduce a novel decomposition that satisfies all of our axioms. It generalizes Fisher information by capturing neural sensitivity to both infinitesimal and finite stimulus perturbations. Moreover, it supports further decomposition across individual stimulus features (e.g., image pixels), enabling fine-grained analysis of neural representations.
Beyond satisfying our theoretical axioms, our decomposition is computationally tractable for large neural populations and high-dimensional naturalistic stimuli, through the use of diffusion models. We demonstrate the power of method by quantifying the information encoded by model of visual neurons about individual images and pixels. Our approach uncovers aspects of the neural code that not picked up by standard methods, such as the Fisher information, and opens the door similar analyses in higher-order sensory areas, and artificial neural networks.
Desired properties of stimulus-specific decomposition of information
We aim to decompose the mutual information I(R: X) between a stimulus X and a neural response R into local attributions [xx), assigning to each stimulus z a measure of its contribution to the total information. By construction, such a decomposition must satisfy:
* Axiom 1: Completeness. The average of local attributions must recover the total mutual information:
XR; = [x)]
This constraint alone does not uniquely determine the function [x), many decompositions satisfy completeness. To further constrain the attribution, we impose additional desiderata that reflect desirable properties of local information measures.
First, for T(x:) to serve as an interpretable, stimulus- specific measure of neural sensitivity, it should satisfy a locality principle: perturbations to the likelihood &/or prior in neighbourhood of xp should have a vanishing influence on [2x) for distant stimuli x # Zg. Without this property, changes in (x)) may reflect changes in neural sensitivity to any stimuli, undermining interpretability.
Axiom 2: Locality. Let p( R, X) and p( R, X) be two joint distributions over the response, R, and stimulus, X. Suppose there exists finite e > 0 and Tn such that:
p(R, x) p(R,z) for all x & (xo)),
(2)
2

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where B, (z)) is the open ball of radius centered at Ty. Assuming X has infinite support on an unbounded or remi-infinite domain, we require that, for every d there should exist some finite value d, such that:
[(z) - I(x) = for all z & Ba(zz):
where I (a) and I (xx) denote the corresponding information decompositions derived from p x)) and RR, X), respectively. That is, local perturbations to the likelihood or prior near 20 should not affect the information assigned to distant stimuli, .r.
Mutual information is globally non-negative: [R; X) 2 O. To preserve this property in our decomposition, we require the pointwise contributions / (x:) to be non-negative as well. Intuitively, observing neural response can only refine our beliefs about the stimulus- it cannot undo information. In addition, negative attributions can harm interpretability, since they can cancel out, obscuring how different stimuli contribute to the total information.
Axiom 3: Positivity. Local information attributions must be non-negative:
I (xx 2 for all z ee X.
(4)
Finally, Shannon 1948 posited additivity fundamental property of mutual information: the total information from multiple sources should equal the sum of their individual contributions. We extend this principle pointwise, requiring that information combine additively across measurements.
* Axiom 4: Additivity, For two responses R and R', the local attribution should decompose as:
IR. (x) IR(x) + [ for all I I X
(5)
where we have renamed ((x) as (2) here, to make explicit its dependence on the response, R. I [r]n(()) the conditional pointwise information from R' given R, and LR (x) denotes the pointwise information from observing both responses. By construction, we require: 1(R'; X X()) Es [z)] and (R, R¹; X) E. [R
Remark 1: Local data processing inequality. Any decomposition that fulfils both additivity and positivity, as stated above, also obeys a local form the data processing inequality. That is, post- processing should not increase information, even at the level of the individual attributions. Formally,
If X R', then IR(x) (x) for all Be X
(6)
To prove this we use additivity to write IR R (x) in two ways:
In(ø) + [(] = T Tne + (z)
Positivity gives In 2 for all x, so:
In(x) + I ( 2 In
(8)
Now, if R' is independent of X given R. then I(R'; R) = Ex [R()]] = 0. Since if J (z) 2 O pointwise, this implies [z)) 0 for all æ. Substituting into the inequality above yields the desired result: Tn(x) 2 (x).
Remark 2: Invariance to invertible transformations. The data processing inequality implies that local information attributions are invariant under invertible transformations of the response variable. That is, for any invertible function @(R), we have:
(x) = [x()).
(9)
This follows from the fact that I(R: X) () and hence Ex (x)) (x)] Supposing, for contradiction, that (x) < In(xx) for some x, equality of expectations would require [o(R) > In(x) for some other æ, violating the pointwise data processing inequality. This highlights why our axioms are important for interpretability: they imply that / (x) quantifies how information is transmitted through the system X - R, independently of how the responses are parameterized.
3

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Table 1: Satisfaction of axioms by different information theoretic measures of neural sensitivity. Only our proposed decomposition, Tocal, fulfils all the axioms.
Axiom (x)
Isp(x)
(ssi(
lsurp(x)
lcissr
Loocal (x)
Completeness
X
Locality
v
x
X
X
x
Positivity
X
Additivity
V
V
V
V
Table 2: Satisfaction of axioms by different information theoretic measures of neural sensitivity. Only our proposed decomposition, Treass, fulfils all the axioms.
Axiom
Jitsher (x)
Isp(z)
IsSI(
lowr(())
lciss(())
Completeness
x
Locality
x
X
Positivity
V
x
X
V
Additivity
v
V
Previous stimulus-dependent decompositions
Several previous works have proposed decompositions of mutual information into stimulus-specific contributions, such as the stimulus-specific information Lşsı [4) the specific information Lsp. the
stimulus-specific surprise surp. and the coordinate-invarian stimulus-speciffc information
Icsss 17
Iep(x)
H(R) (R(x))
(10)
Issi(x)
H(X) ERI [HXXR= = r]]
(11)
lxurp(z) DKL (p(R))))llp(P))
(12)
Iciss((a) (x)
ERI [7KL (p(X |=+))l(()))
(13)
While these decompositions satisfy some of proposed axioms (Table Z), none satisfy locality. This is because they all depend on global terms such as the marginal response distribution p(r) (((rl'))(zz") dx', the posterior p(=(r) p(r(=))(())/ influenced by changes to the likelihood &/or prior for any stimulus. Further, the requirement that (oissr(*) is invariant to invertible transformations of x incompatible with locality; both axioms can't be fulfilled simultancously. As discussed above, this limits the interpretability of previous decompositions as measures of neural sensitivity, since a non-zero attribution could be due to changes in neural sensitivity anywhere in the stimulus space.
((r[='))((x) de', both of which can be
Unlike the above decompositions, the Fisher information is inherently local. However, while pre- vious authors found a relation between the Fisher and mutual information 3 29 this only holds approximately, and in certain limits (e.g. low-noise). Therefore, the Fisher information cannot be used to quantify how different stimuli contribute to the total encoded information (i.e. it fails the completeness axiom).
Recently Kong et al. 2024 used diffusion models to decompose the mutual information into contribu- tions from both the stimulus, and the response, r. Two different decompositions, ((r, were proposed. However, averaging these over pl R|z) does not give stimulus wise decompositions that fulfil axioms (Appendix B). In one case, obtain (surp(2) (z) (Eqn 12 which non-local; in the other case, the decomposition is not additive, which a fundamental information theoretic constraint.
In the following we propose new stimulus- wise decomposition of the mutual information which fulfils all our axioms, combining the advantages of the Fisher information (locality, positivity additivity) while being valid decomposition of the mutual information.

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Diffusion-based information decomposition
We first derive an expression for the mutual information, (Rt X), that can be used to construct stimulus-dependent decomposition that fulfils all of the above axioms.
4.1 Exact relation between Shannon information and Fisher information
We consider a population of neurons which show a response, R, to stimulus, X. with probability pl R) ).). We then consider noise-corrupted version of the stimulus, X, vJZ, where
Z ~ (0, From the fundamental theorem of calculus we can write:
dI(X;; dy (RR; X.)) lim (R); = (nn);
(14)
d7
7-000
since in the limit ạ - 00, X, is just noise, and thus I(R; X,) - It follows that,
X00) = dI / (RP(()
0
dd((X)lr)
dl((x)) dy
dy
d7
(15)
d-
dy
70
where h (Xx)) --E(x,) p(x)) and (X,1r) = - Bp(X [log (X,, Assuming that p(X) and PR) have finite second order moments, we can apply de Bruijn's identity for all y > to obtain,
I(R; XTo
1
Trace
Ep(X,,
V2 x.
log (X.)R)
V2, (ogg(X,)) dr
70
Trace
Ep(X,
V2 By
log (R|x,) l
d7
2
= Trace
Ex(X,) [T (X+))
(16)
2
where J (xx.) is the Fisher information with respect to a noise-corrupted stimulus, X.,. Finally, since 1om / (R; Xx.) (R:; X), and (Txce(Exxx,13XXx),) always non-negative, monotone convergence yields:
IRR; X) - Trace / Ep(x,) [J (X+))
1
(17)
This is general result that holds for both discrete and continuous X and R, so long as p(X) and (X)R) have finite first and second moments.
The above identity provides a direct relation between the mutual information T R: X) and the Fisher information, (x,)). Further, also admits natural interpretation in terms of de-noising diffusion models trained to predict X from a noisy observation X.,. To see this, first we use an alternative formulation of the Fisher information, terms of the mean-squared score:
1
|| log (R(X,) dy
I(R;X) =
E Ex( R)
2 Jo
1
Ex(,,,
Ve, logp (X, Va, logp (x,)) dy
(18)
2
2
Next, from Tweedie's formula
20 14 we have:
1
I(R; X)
2n2 Bx(x, [I(XX, +(x,))1 dy
(19)
where ((x,)) E X [ and ((r,,,)) x| r). These conditional means can be approximated
using denoising diffusion models trained to sample from the prior, p(X), and posterior p(X R). respectively (26).
5

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tuning curve
B
tuning curve
Hxca(())
f(x)
x
E
F
G
m
A
Isso(())
Cissy (x)
Figure 1: Demonstration of locality. (A) A neuron responds to ID stimulus with tuning curve f(z). We compare two cases: (i) tuning changes only near x -3 (black), and (ii) additional changes near 6 (red dashed). (B-C) The Fisher information, J (x) and local information Iocal()) converge as we go far from the region where the two tuning curves differ. (C-F) For other decompositions (Eqs. 10-13), local changes in the tuning curve result in non-local changes to the information attribution, for x.
4.2 Local stimulus-specific information
Building on the integral representation of mutual information Eqn 17 we define stimulus-specific decomposition:
1 Σ / Ex(X, (J,(M)) dy, 2
Llocal (I)
(20)
where X. = V72, with NN 1). and
Bp(() ( log p(R xz)) the
7th diagonal element of the the Fisher information matrix, evaluated Zyy, This expression parallels Eq. [7] with the key distinction that the expectation is now conditioned on a fixed stimulus rather than averaging over the full stimulus distribution g Consequently, the decomposition satisfies the completeness axiom, since by construction, [RR; X) [1))
Next we outline why our decomposition fulfils the locality axiom (for the formal proof, see Appendix A). Recall that the Fisher information matrix J (z) characterizes the local curvature of the log- likelihood. log p( z). and thus quantifies the local sensitivity of the response changes in the stimulus 23 The term Ex,
Ex,,L Jar [] (X-)) generalizes this notion, measuring neural sensitivity when
we only observe noise-perturbed versions of the stimulus, X,, N(x, 7/). For finite 7, this term is dominated by values of X., close to x, and thus, it depends only on the local shape of the likelihood and prior around z. It receives vanishingly small contribution from changes to the likelihood and/or prior for distant stimuli z', || x|| >> V7. Meanwhile, as 7 - oo, X., becomes pure noise and thus Fxcx. (x) [T(X,) - 0 for all æ. Taken together, this implies that our decomposition, obtained by integrating Ex(X (m) [J(x))] over all > O, satisfies the locality axiom: local perturbations to ((, affect Toocat (x') only for nearby z', while their influence vanishes as ||ær - 2|| + oc.
The remaining axioms follow directly from standard properties of the Fisher information matrix. Positivity follows from the fact that the Fisher information is positive semi-definite. Additivity follows from the identity Tne re(x,,) Jn(xx,) + JR. R(x.)) (32. Both properties are preserved when we average the Fisher information over X(Xx, (r) and integrate over 7 0, to obtain (z))
Note that the exchange of integrals over and to from Eqn 17 to Eqn 20 is justified by the fact that the integrand, T. (xx)). is Lebesgue integrable, since it is always positive and integrates finite value.

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A
D
gaussian
gaussian
p(X)
PRX1 d
Locaa
bimodal
bimodal
p p(X x)
Y- 1-P881)) puR
1- piR (x)
gaussian
gaussian
F
bimodal
bimodal
a-
casss
p(X
(ss
11
X
X
noise. Middle and bottom: posterior p(X) r) given r ~ p(R æ1) (blue) or r ~ p(R 222) (red), for stimuli 21, 22 separated by 8. The posterior is equally sensitive to near 0 (middle) than near 1 (bottom). (B) Same as A, but with bimodal prior (top). The posterior is more sensitive to near (middle) than near (bottom). (C) The local information peaks near dashed), where the posterior is most sensitive, and is flat for Gaussian prior (black), where posterior shape constant. (D-F) Other decompositions behave differently:
Figure 2: Effect of prior. (A) A gaussian prior (top) with neuron responding
for the bimodal prior (red
(D) 4a flat for both priors; (E) Isst is minimal near = for the bimodal prior; (F) Icissi is quadratic under a Gaussian prior. Lourp (not shown) behaves similarly to Lassi-
4.3 Feature-Wi Wise Decomposition
We assume the stimulus is vector of image features x; (e.g. image pixels). Given that the local stimulus information (Eqn. 20) is expressed as a sum over diagonal elements of the Fisher information matrix, it is natural to decompose [toca( into feature-wise contributions (xx).
As with the stimulus-wizg wise decomposition, this problem ill-posed: infinitely many decompo- sitions exist in theory. However. the same axioms constrain the feature-wise decomposition. To satisfy additivity, Tx(x) must be a linear combination of Fisher diagonal terms: 7,(x) E3 0ij Jo Ex,1= (X,)] dry. Completeness requires Σ; aij = 1, while positivity enforces as 2 O. Finally, to fully specify the weights, aıj, we need to introduce one further axiom, which ensures that the attribution [((x) is zero for irrelevant stimulus features, X,, which are independent of the response, R.
Axiom 5: Insensitivity to irrelevant features. If X, is independent of R, then I,xx) 0 Vr E X.
For this axiom hold, we need set ai ōig. so that =x) So [x,,=[][(X,)] dy. If, on the contrary, ai * d.y. then neuron's sensitivity to other features (i.e. Jzs(2) > could 'leak over' to make (xx) > 0 even when X; is independent of it violating the axiom.
5
Results
5.1
Locality
To illustrate the implication of the locality axiom, analyzed the responses of model neuron to die-emnnsional stimulus drawn from a Gaussian prior. The neuron's 's response was modeled as a Gaussian random variable with mean f(x) and fixed standard deviation. We compared two tuning curves: one with a single peak at x 3 (Fig. 1A, black), and another with an additional peak at (Fig. 1A, red).
With gaussian noise, Fisher information scales with (a(x),, and thus peaked where the tuning curves steepest (Fig, IB). Similar qualitative behaviour observed for (z))(()) (Fig 1C). Crucially,
7

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mean respon
R
R'
|R]
local
p(2)
R
|R|
X
X
E
R R
R
R
R'
|R|
SSI())
R R'
0
Figure Local data processing inequality. (A) Bimodal prior. (B) Neural responses:
(C) Our decomposition satisfies the pointwise data processing inequality (DPI): J (2) (red) is always less than equal to IR [œ(z) (black). (D-F) sup also satisfies pointwise DPI, while Issi and sp do not. Ioiss (not shown) behaves similarly to Luurp.
+ noise: transformed response r'= |r|. The non-invertible transform reduces information.
the difference in (x) between the two tuning curves vanished outside the region where they differ, consistent with the locality axiom. In contrast, existing attribution methods (Fig. ID-G) showed global sensitivity: adding second peak at 6 altered the attributed information across the entire stimulus space, including far from the added feature.
5.2 Effect of Prior
We next examined how Iteent (1) responds to changes in the shape of the stimulus prior. For this, we considered two different stimulus priors: a zero-mean gaussian, and a bimodal mixture of two gaussians with peaks at i 1 and 2 1 (Fig. 2A-B. upper panels). To isolate the effect of the prior, we used a simple Garssiann likelihood model: r V(x,o2) Under this model, the neuron's sensitivity is uniform across all stimuli, any variation in attributed information must arise solely from the prior.
Intuitively, one can assess neural sensitivity by measuring how much the posterior distribution p(X r)) changes, on average, in response to small perturbations the stimulus x. With the bimodal prior (Fig. 2B), the posterior is highly sensitive near æ = O, where the two modes compete (Fig. 2B, middle panel), and relatively stable near the modes themselves, e.g., around x = 1 (Fig. 2B, lower panel). Our attribution measure (x) reflects this structure, peaking at x O. and decaying elsewhere (Fig. 2C). For the Gaussian prior, where posterior sensitivity is constant, [zcaa(z)) is flat. In contrast, previously proposed attribution methods behave inconsistently: some remain constant across both priors (Fig. 2D), others respond in the opposite direction (Fig. 2E), and some varied strongly even under Gaussian prior, where the posterior sensitivity is uniform (Fig. 2F).
5.3 Data Processing Inequality
Next we illustrate how Treea (x) respects the data processing inequality while certain other attribution methods do not. For this, we used bimodal prior (Fig. 3A) and compared linear-Gaussial neuron (r (xx a²)) to a downstream neuron with response = |r]. Since this transformation is non- invertible, information must be lost. Consistent with the inequality. I1cca((r)) decreased at every (Fig 3C). In contrast, both /ssi and Lap increased at some x and decreased at others, violating the pointwise data processing inequality (Fi 3D-E). Tsups by comparison, respected the inequality (Fig 3F).
5.4 Scaling to high-dimensions
We can use Eqn to write the Ceature-wise decomposition terms the outputs an unconditional and conditional diffusion model. trained to output ((xx)) E [[[x]] and 2(z,, -) E [X r],
8

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respectively:
/ 1 [(x)= Jo 27,2 12, (i,(X,, {(x))))² dr
(21)
To obtain a Monte-carlo approximation of this expression, we need sample from T. ~ p (x) followed by, ~ p (,) Sampling from (R(),,) is prohibitively expensive (since it requires first sampling from æ p(zul (y)) which requires full backward pass of the diffusion model). To get around this, we adopt an approximation used by Chung et al. 2023] instead approximating [x(x) using samples r ~ p @ (x)), which can be computed efficiently using one pass through the de-noising network. The integral over 7 was approximated numerically with evenly spaced (see Appendix C). Since the Fisher decays to zero for large 7, truncating the integral has little effect on our approximation of the integral.
5.5 Pixel-wise decomposition of encoded information
We applied our method to identify which regions of an image contribute most the total information encoded by population of visual neurons. For illustrative purposes, we used stimuli from the MNIST dataset and modelled a simple population of neurons with mean responses given by Af + b), where A and are constants, f(-) is sigmoid nonlinearity, and w is a linear filter representing the neuron's 's receptive field (RF). Neural responses were corrupted by Poisson noise. We simulated 49 neurons with RFs arranged in a uniform grid (Fig. 4A).
As baseline, first evaluated neural sensitivity using the diagonal of the Fisher information matrix (Fig. 4B-C, E-F). In this model, the Fisher information reduces to a weighted sum of squared RFs, where each neuron's is contribution is scaled by its activation level. This yields characteristic "blob-like" patterns, with each blob centered on the neuron's 's RF.
We then used a diffusion model trained on MNIST to estimate the pixel-wise information decomposi- tion, Iucaa (x) (Fig 4D, G; additional images are shown in Supp Fig 2). This decomposition revealed concentrated along object gages-regions where the decoded images (i.e. sam- ples form the posterior) are most sensitive to small changes in the presented stimulus (cf. Fig. 2A). Unlike the Fisher information, their RFs) and the statistical structure of the input (captured by the diffusion model), contribute towards the total encoded information.
that information
measure integrates how both the local sensitivity of neurons (via
Later, we investigated the behavior of our information decomposition on a diffusion model trained on natural images, with model of recorded ganglion cell responses from the retina [15] (Appendix section C.6, and supplementary Figure 3). We observed qualitatively similar behavior to before, with Iocal peaking in regions of high local spatial contrast, around the edges of objects.
6 Discussion
We introduced principled, cooomattion-hhorretie measure of neural sensitivity to stimuli. This measure satishies a core set of axioms that ensure interpretability and theoretical soundness. Crucially, the measure can be estimated using diffusion models, making it scalable to high-dimensions inputs and complex, non-linear neural populations. We empirically demonstrated how each axiom shapes interpretability through simple, illustrative examples. Finally, we show how the method can be applied in a high-dimensional setting to quantify the information encoded by neural population about visual stimuli.
Kong al. 2024 recently proposed two decompositions of the mutual information between visual stimuli æ and text prompts y. 1 [(, y). which can be efficiently estimated using diffusion models. These were used identify image regions most informative about accompanying text. Dewan et al. 2024 extended this approach to assess pixel-wise redundancy and synergy with respect to the prompt. However, these frameworks do not directly yield a stimulus-wis neural sensitivity measure [(x), which was our goal (Appendix B). Moreover, simply averaging the decompositions of Kong et al. over neural responses does not produce stimulus- wiie and feature-wise decompositions that satisfy our axioms: in one case the resulting decomposition is non-local and can be negative; in the other case, the decomposition not additive, and thus violates key information theoretic property pointwise.

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B
X
C
(x)
D
(x)a((x)
6
A
receptive fields
E
F
G
Figure 4: Pixel-wise information decomposition with MNIST stimuli. (A) Simulated population of finear-nonlinearPooison (LNP) neurons with circular receptive fields arranged in a grid. (B) Presented visual stimulus. (C) Diagonal of the Fisher information, given by weighted sum of neural receptive fields. (D) Pixel-wise decomposition of mutual information, (zoca(()) E-G Same as panels A-B, but for different stimulus.
Our work extends a classical result from Brunel & Nadal 1998 who showed that mutual information can be approximated in the low-noise limit using Fisher information [29]. This approximation was later used by Wei & Stocker 2013 to explain a wide range of perceptual phenomena under
the efficient coding hypothesis [21]. However, their approximation, which depends on the log determinant of the Fisher, becomes very inaccurate certain cases, such as at high noise, or where there more stimulus dimensions than neurons (in which case it returns minus infinity) [1] [3] Here, we instead identify an exact relation between mutual information and Fisher information with respect noise-corrupted stimulus. This opens new potential test theories of efficient coding of high-dimensional stimuli, and in the presence of noise.
In the future, will use method to investigate more realistic neural models, fitted on biological data, as well as diffusion models trained on natural image datasets. Here, to further improve efficiency could use zero-shot methods that sample directly from the posterior, without requiring a trained conditional diffusion model [22 6). Such approaches have recently achieved state-of-the-an performance in decoding visual scenes from retinal ganglion cell responses (30) and could enable rapid assessment of how changes to the neural model affect encoded stimulus information.
Finally, our axiomatic framework has close parallels with integrated gradients (26). method developed to attribute the output of deep neural networks to individual input features. Both our method and integrated gradients address ill-posed attribution problems by enforcing natural axioms. However, one limitation of our method, in contrast to integrated gradients, is that we do not prove the uniqueness of our attribution method, as following from our axioms. This will be interesting to investigate in the future. There are also key differences between both approaches: for example, our attribution explicitly accounts for noisy responses and does not require specification of an arbitrary baseline. These distinctions make our method particularly well-suited to neural data, and suggest potential utility principled attribution tool for analyzing deep networks-identeee stimuli, or stimulus features, different units or layers are sensitive to.
which
Acknowledgements
Ulisse Ferrari and Simone Azeglio acknowledge this work was done within the framework of the PostGenAL @ Paris project with the reference ANR-23-IACL-00077 Ulisse Ferrari and Simone Azeglio benefited from financial support by the Agence Nationale de Recherche (ANR) by grants NatNetNoise (ANR-21 CE37-0024). IHU FORESIGHT (ANR-18- 18--H11-) The PhD position of Simone Azeglio was funded by the Sorbonne Center for Artificial Intelligence (SCAI), through IDEX Sorbonne Université, project reference ANR- 1--IDEX-0004 Matthew Chalk and Steeve Laquitain
10

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are funded by grant RetNet4EC KANR-22-CE92-0011 cofunded by the ANR and DFG, Our lab is part of the DIM C-BRAINS, funded by the Conseil Régional d' Fr-ace-France.
We would also like to thank Tobias Kuhn and Olivier Rioul for interesting discussion and input, which helped us towards this paper.
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A Proof of locality axiom
Let p(R, .) and p(R, X) be two joint distributions, defined over the response, R. and stimulus, X with infinite support in an unbounded (e.g. X E (Rd) semi-bounded domain (e.g. o0)))
We restrict PR, X) to only differ from p(R, X) locally, in the vicinity of æn- Formally, suppose there exists finite > and Zo such that:
p(H,x) (Ht x) for all x & B.(x)),
(22)
where B. (2o) is the open ball of radius centered at ro-
Recall that our stimulus-wise decomposition of information is defined as:
Trace [ Ep(x,12) (I(x)) dy
T(x) = 2
(23)
where X, = V=2, Z ~ (NO, I). and [(x,) = Ep(R( [- V2 V2, logp(R R xn)) is the Fisher
information matrix Try-
We will consider the absolute difference between the stimulus-wise information, computed with p(r, æ) and p(r, z) respectively, (1(x) - [(x)). To prove the locality axiom, we separate the integral
over y into two parts as follows,
- = 2 Trace /." Baxx110) -(x))-J(xx)) d7
[xxx) I(x)]
(24)
S f(x, ma) + (g( mm):
(25)
where
Trace ( Bp(x) - d7
(x)
(26)
g(x, 7h) =
Trace Ep(Xy1x) [[(x)) T(x.)) d7 2
(27)
Th
In sections A.I and A.2 we show that f(x, (1) and g( m.) have the following limiting behavior:
flx
o
(e कमn (10=20l?))
as ||æ 2o11 + 0o
(28)
g(a,
o
uniformly over 7hh 00.
(29)
Thh
Thus, for any d > O, we can first pick 7/, large enough so that g(x, m) S 5/2,, for all æ. Then, for fixed Th. choose d such that [12 7m) < 6/2 whenever ||æ 12 d.
This guarantees that for any 5 > 0, there exists a finite constant such that,
[(x) - [(x) < 8 for all ||x z0ll 2 2 d. completing our proof of the locality axiom.
(30)
A.1 Integral from = 0 to t 7hh
Here, prove the limiting behavior of (x), as || -2ol2 co.
We start by writing out the Fisher information as function of the noise-corrupted stimulus, xny:
1
Trace( (x,,)) = 722 FRIe, (11a(1)))) a (xy,r) )
(31)
Writing out & (xx)) F|X x.,) explicitly,
s''p(x') @ Fr-z"
da'
V
t(x,)
(32)
an-aa
dxa
VT
SWEB, I' d'p(x')
du' + SPEB x' x'p(x') " Φi
(''22)
de'
V
(33)
(VGBAloo)P(a p(x1) 1b
de' + SHEB. p(x') db (2'22
dx
20

---

where 6(x) - 11=r"" and we have separated the integrals in the numerator and denominator
into contributions from inside and outside the region B. (xa) Taking just the second term in the numerator of Eqn
J
d'p(x')
x -xy VY
-x1) de'
sup (xo)
(
xt'
J d'D d'(('') da!
VT
I00
xzll E-
x'p (xr') de
V7
('EEB((Z))
by I|z0 (i
1 x. || 00 (34)
as
since, by construction, p(z) has finite moments, The same logic also applies to the second term in the denominator of Eqn 33
J
p(x') Φ (o Izy dr' O(e 25 Illoo-ap 11m0-a=13))
as
|| - x,y 12 ac.
(35)
Substituting this asymptotic behaviour back into Eqn
33
we have
# (xx,) (2 ('p((') (e Φ (WEB.(Z.) p(x))
a'
dx'
V7
+ +O(e
2n 1:- =1,0-11,11)))
(36)
dx
where the first term only depends on x in the region outside the region (x())). Using the same arguments for (( r), we can write:
f (xx) f(x r) alx,,, 20)) Ole - -1000
(37)
where
r,0) ( x'p(x')
('-B)) dx'
SWERL(OO) z'p(r,z') @ (0' (='=z)) (
dx
(38)
Sargna(ma) p(z') ø
x'-za dx
Segn z(, Φ dx'
x'-xx x.
only depends on p(r, x)) in the region & Be (2o). Taking the square, and averaging over (x,), have
Trace (I(z)) = ER [lla(2,, 2n) m0 + Ole =1000-111
1 72
(39)
1 Jay p(rlx)p(x))))
|1a(E,, 20)
+ Ole # 1200 11200
(40)
72
I [(x)
dx
Note that if Rt is discrete, the above integral over R is simply replaced by sum. We then follow the exact same procedure before, separating the integrals over into parts that inside and outside of B, (2o), to obtain
Trace (J(xz)) -/2 1b(2,, zo)ll + O(E 2 1110-2211]
1
(41)
where b( x.,)) is function that only depends on p(r, x) in the region x & (xo). Since, by construction p(r, x) = =p(r x) ) in the region & B (Tu), we can then write:
Trace (J(z))- 3 (Tm)) -( = O(e + 11,0,,, as ||:00 - x,, + oo
(42)
where J(I,) is the Fisher information obtained with p(r,x))
Averaging over p(X,lz)= = [x--x)
gives:
Ex(,)12) Trace(J - {I,))) =0(e 1o-+lee =0(e-till00-00 =1100-1111) as || - xol12 - oc.
(43)
21

---

Finally, integrating from = 0 to Th, we have:
f(x, (b)
1
Trace (Rx,11 [(Xx)) -(xx,)) dr
(44)
2
1
Th
Trace (Ex,1= [((x)) T(xx,)) an
2 J
(45)
1
Trace (Ex,1z [((x,) - J x((x,))
2 7h
sup
(46)
(0(0.
0 (c 154 (1==z==l))
|| zol12 + 00:
(47)
as stated in Eqn 28
A.2 Integral from y 7h to 00
Next, we show the limiting behaviour of g(v, (h))
We know from Eqn 42 that for any given 7 < oc, [Trace(][xr)) - -
+ a |x,|1 co. Thus
the maximum of (Trace([[ax)) - I (x,,) ))l must be obtained for some finite x.y. As a result (Ir,) - [xx))) s sup Traco - [xx))]
(48)
[Tracee] I (x - (x7,)),. where |2 < 00
(49)
32 Fetrles) (1(((1))) + (x) 11²)
-Bpole;) (1ā(x;) - (I),, r) 1?)
where | < 0o
(50)
with ã(x,) and r(z,,r) defined as the expectations over X obtained with the perturbed distribution, P(R, X). Now since, by construction, p(z), p(z r). p(z), and p(z | r) all have finite first and second moments, then for finite n", all the expectations in the above expression are finite, and we have:
[Tacee( (2,,)) S 73
for all 2
(51)
where < oo is some finite constant, independent of Tyy
Integrating over and averaging over p(X,1z) (x) gives
gx,, m)
1
Trace
(xx,) d7
(52)
2
74
1
Bx,
Trace (J(X,) F(X,)) dr
(53)
2
m
C
for all æ
(54)
2mh
0
uniformly in x,
(55)
7h
as stated Eqn 29
B
Comparison with previous decompositions using diffusion models
Recently Kong et al. 2024 proposed two different decompositions of the mutual information into terms that depend both the response and stimulus, which can be computed using diffusion models. Their decompositions take the following form:
poc
T 1Kong
2- Fx,, [||= - 10 r)12 || =({,))13] d7
(56)
[Kong.((X,T) = Jo
roc 1
Ex [1(((1))) - 2(29, dy
(57)
272
22

---

IRAR (20
1x (x)
R-1 (X) + 1x((0)
A
B
mean response
Rn (X)
k (
Local
neuron 1
neuron 2
Figure 5: Additivity, with respect to measurements from multiple neurons. (A) We considered two neurons, with sigmoidal tuning curves, f(x), and 1-d stimulus with gaussian prior. (B) We confirmed that [ocal (ar) satisfies additivity, by checking that the point-wise information from both neurons together (I,, Ra (z), red) equals IR, (a)) + |2] Ra (I). (C) For the measure derived from the work of Kong et al., which does not satisfy additivity, the curves are non-overlapping.
To obtain stimulus-wise decompositions from these expressions, that we can compare with our work, averaged both of these decompositions with respect p(R)xz), to obtain: [ ERIz [Konng (z, R and (xong.((x) ERIz [ R)].
Despite the apparent similarity with our proposed decomposition, neither Konng (2) or oggg (a) fulfill our axioms, limiting their potential use as principled & interpretable measures of neural sensitivity.
In their paper, Kong al. show that Kongg r) log p(rlz)
p(r) Taking the average over p(]|z).
gives: /Kong, (x) = ER [Xxong, (2., r]] = Dxn 1 (p(R)(()))))))))))))))) [x) Ipp( (t)). This is identical to the expression for the specific surprise, L [z)rp(2), ), in the main text (Equ 3). As shown in the main text (Fig IE), this measure does not fulfill our locality axiom. This is because sarp(()) (1) depends on p(r) ('rr'='' which can be influenced by changes to the likelihood or prior with respect to any stimulus, x'. Further, following the same prescription in the main text obtain feature-wise decomposition of Kong. (z), results in feature-wise attributions that can be negative.
To understand the second decomposition, we use Tweedie's 's law to write:
2 (x) En [ (x, r]] 2 2
Ex, [IV log p(][X)))1 dr
(58)
This differs from expression, due to the fact that it involves taking the average over x() rather than p R, ). However, while this difference may seem subtle it has important consequences for the resulting ttimulus-wiss decomposition.
To see that there are large qualitative differences between TKang.2(2) (x:) and Itoeat we first consider their behavior in the case where p( R, X) is jointly gaussian. Here, Tiocat (1) is constant across (Fig 2B, black), reflecting that the fact that the posterior is also independent of x (up to linear translation). In contrast, (x)) scales quadratically with the distance of from the mean (similar to Leurp (x)). Fig 2E).
More importantly. (z)) violates additivity point-wise. Supp Fig 1 illustrates this in a simple 1-d example. To see why it is the case, we can expand the expression for the local information from two neurons, R and R':
Kong.2 R.R
(x)
E R [IV log logp(R'|R,X,, Va, log (o( dy
(59)
Now when we expand the square we see that the cross-terms don't It cancel out. As a result, Kongg is not linear in log p( RR X.)) and log p())X,), violating additivity.
This contrasts with the behaviour of Loeea, which can be expressed as follows:
/
plocal R (x)
E Ex, x [ER,
2 log p( (R, + v2 logp(R(X,)]} dy (60)
This equation linear log p( R' R, X,) and log p( R R(X,), and thus we can easily confirm that additive with respect to repeated measurements.
23

---

Additivity is fundamental property that underpins the definition of mutual information [25]. It requires that information from multiple measurements (e.g., different neurons) combine linearly, such that their individual contributions sum to the total local information (Fig. 5 A point-wise measure that doesn't 't respect additivity could thus lead to misleading attributions as they don't respect how different measurements (or neurons) combine additively to generate the total mutual information.
C Diffusion model details
C.1 Dataset Preparation
We utilized the MNIST dataset [9]. which consists of 60,000 grayscale images of handwritten digits for training and 10,000 for testing. Each image, originally 28 x 28 pixels, preprocessed by normalizing to the range [-1, 1] and then rescaled to 32 x 32 pixels to align with the architectural requirements of our diffusion models.
C.2 Neural Encoding Model
We simulated a population of 49 neurons with spatially localized receptive fields (RFs) arranged in 7 x grid, mimicking the retinotopic organization of early visual cortex. Image pixels were mapped to 2D coordinates in visual space, (z, v) E [-1, forming a uniform grid. Each neuron's 's RF was modeled as 2D Gaussian filter centered at (2i4
(z x2)2
(y v1
wr y) exp
(61)
20
where a = 0.1 defines the spatial extent of the RF, and (xi, We) specifies the center of the i-th neuron's RF. These centers were evenly spaced across the image grid.
Neural firing rates were computed by projecting the input image / onto the receptive field weights Wi, followed by sigmoid nonlinearity:
fi = A 1 + exp(g w. (I - 0))
(62)
where A = 40 is the amplitude, 0.4 is the gain and 0 = 0.9 is the threshold. This yields the mean firing rate of neuron in response to image
To capture neural variability, we modeled spike counts as samples from a Poisson distribution with mean ft.
C.3 Diffusion Models
We trained two denoising diffusion probabilistic models (DDPMs) based on the UNer-2D architecture 24], using the implementation provided by the HuggingFace diffusers library (27].
The first model was trained to generate MNIST digits from standard Gaussian noise Z ~ ~N I). Following the DDPM framework [12], we simulated a forward diffusion process that progressively adds Gaussian noise to an image producing sequence of noisy images {xt}. The process is defined as:
Fe / 1 B It-1 B 2t, 2 N I).
(63)
where Br denotes the noise schedule. By recursively applying this equation, one obtains:
It ~ N (Voz No, (1 - -x)))) with ar II (1-B)).
(64)
Although this differs from the isotropic Gaussian noise model used in the main text, 3n = 2p + / the two formulations are equivalent up to reparameterization, with 1-a, Thus, both can be used interchangeably to estimate the information decomposition (z)c

---

We used linear noise schedule with 3t increasing from 1 x 10-4 to 0.02 over 1,000 time steps. Given noisy image 2t, the model was trained predict the noise component & (2+). which can be used to reconstruct an estimate of the clean image:
St
Ct 2x(3)))
M(x+)
(65)
Vos
second DDPM was trained using the same architecture, but conditioned on simulated neural responses r (see previous section). This model learned to predict the noise given both the noisy image and response:
f(x;, r) Zt vI ag 2x(2),
(66)
We used these models to estimate the pixel-wise information decomposition described in the main text (Eq. 9). To approximate the integral over 7 we applied additive Gaussian noise according to the diffusion schedule and evaluated f(xz) and ((z;,r) over 24 diffusion steps (indexed as [40: :000]] using 50 noisy samples of z, and r at each On average, approximating Atooa (1) for one image took approximately I minute on an NVIDIA GeForce RTX 4080 GPU.
C.4 Training Details
Both models were implemented in PyTorch (version 2.7.0) with Cuda 12.6 and trained using the HuggingFace Diffusers library (27 and Accelerate library [ for distributed training on 7 NVIDIA A100 GPUs (40 GB VRAM).
The training procedure for both models was as follows:
Optimizer: Adam W with learning rate 1 x 10-1, B1 0.9, 32 = 0.999
Learning rate scheduler: Cosine schedule with warmup (500 steps)
Batch size: 256
Training duration: 150 epochs
Loss function: Mean squared enror (MSE) between predicted and true noise
Precision: Mixed precision training with bíloat16
Training Time: 25/30 minutes per model
C.5 Decomposition for additional digits
We repeated the analysis described in Figure 4 originally performed for two digits on six additional digits using exactly the same linear-nonlinear Poisson (LNP) model, receptive field configuration, and information decomposition procedure. For each new digit, we computed the Fisher information and the pixel-wise information [oca(()) following identical preprocessing, stimulus presentation, and estimation steps in the main analysis.
C.6 Decomposition for natural images
To demonstrate that our approach generalizes more complex and biologically realistic neural encoder models as well as natural images, we trained the diffusion models on 60,000 64x64 natural images from Hugging Face's Tiny ImageNet dataset and trained deep neural network model of the
retina [15] to replicate the responses of 41 retinal ganglion cells (RGCs) from [10]. We extended the model by tiling the stimulus space with a 7x7 square lattice mosaic of receptive fields, where the resulting 49-field mosaic reproduced the response pattern of one of the fitted cells. As with the MNIST digits, we then applied pixel-wise information decomposition to natural images (Fig. 7 Consistent with our findings for the MNIST stimuli, the resulting local information maps, /ocat (x), exhibited the highest values in regions of high spatial contrast, particularly along object boundaries, indicating that these areas contribute most strongly to the encoded information about the stimulus.
25

---

f(x)
(x
f(x)
A
3
3
3
B
E
4
C
2
2
5
F
5
Figure 6: Information decomposition across additional digits. In the main text, present pixel-wise information decomposition for two example digits. Here, we provide several additional examples illustrate the decomposition patterns across other digits.
F()
A
B
C
D
E
G
H
Figure 7: Pixel-wise information decomposition with natural image stimuli. (A) Presented visual stimulus. (B) Fisher information was computed as the sum of each neuron i's pointwise contribution, (f(())) ff)x). quantifying the local neural sensitivity to infinitesimal changes in the intensity of pixel x. (C) Pixel-wise dccomposition of mutual information, (z). D-I Same as panels A-B, but for different stimuli.
26

---

A
B
C
D
Stimulus
Original RF
Perturbed RF
Naaca
Aaruro
1.0
1.0
0.5
0.5
0.5
0.5
0.0
0.0
0055
0.0
0.0
Figure 8: Illustration of the locality of (T) compared to Isurp (2).. (A) The visual stimulus MNIST digit positioned on the center-left side of the visual field. (B) Receptive field of LNP model neuron in the unperturbed condition, located on the center-left side the visual field (the remaining 48 receptive fields are unperturbed and not shown). (C) The receptive field of neuron 21 is modified by adding a sharp 2D Gaussian perturbation (highlighted by red circle) on the opposite side of the stimulus. (D) Difference in (1) and (E) difference in surp (2:) between the original and perturbed receptive field conditions. Tocal (x) remained confined near the stimulus and the perturbed receptive field, reflecting its local nature, while (u)p exhibited broader, spatially distributed changes.
C.7 Demonstration of locality for MNIST stimuli
To further illustrate the locality properties of the information decomposition, we examined how small, spatially localized perturbation to single receptive field affects [o))(()) and Isurp (2). We used the same linear-nonlin nonlinear-Poisson Poisson (LNP) model described in the main text, with receptive fields arranged across the visual field (Supplementary Figure (B). In the perturbed condition, introduced a sharp two-dimensional Gaussian bump to the receptive field of a single neuron, located on the opposite side of the visual stimulus (Figure 8C).
We then computed the pixel-wise differences in (1) and (arp)(r) between the perturbed and unperturbed conditions. As shown in Figure 8D-E, /tocal remained confined to the vicinity of the stimulus and the affected receptive field, reflecting its inherently local nature. In contrast, surp (2) exhibited broader, spatially distributed changes, highlighting its global dependence on the overall structure of the neural population code.
The code repository to reproduce figure 4, 6. and 8 can be found at nnnunnllllinee neeral-inio-decomp
27


Reviews:
[{"review_id": "BEcdSCxx8L", "reviewer": "Reviewer_1oJo", "summary": "The paper proposes a new measure of neuronal sensitivity generalizing the Fisher information, focusing on the separate contributions of specific stimulus features or dimensions to evoked neuronal activity. After giving the criteria that such a measure should meet and defining a specific measure that satisfies those desiderata, the paper shows how to use both unconditional and conditional denoising diffusion models to actually compute the defined measure.  An initial experiment shows that the measure appears to serve its purpose fairly well when computed on MNIST.", "questions": "In lines 59-60, the statement of Axiom 1, what stimulus distribution does the expectation integrate over?\n\nSince the problem of estimating the local information is ultimately one of approximating an integral over noise levels, do the authors know something other than a diffusion model that could potentially compute the approximation?  If a better neural architecture than diffusion models were found for denoising, could that be integrated into the estimation here?\n\nWhat would happen if I_{\\mathrm{local}}(\\mathbf{x}) were evaluated over spontaneous or intrinsically generated neuronal spiking, rather than over stimulus-evoked activity?", "limitations": "yes", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strength: axioms are well-motivated for extending the intuitive properties of Fisher information and mutual information\n\nStrength: a method is given to approximately compute the information-theoretic quantity defined using extent computational tools\n\nWeakness: experiments are conducted entirely on rate-coded neurons rather than spiking ones.\nWeakness: related to the above, experiments are on model neurons rather than on a real dataset", "was_revised": true, "final_justification": "The authors have specifically justified their motivation in using conditional diffusion models and added an experiment on natural data rather than MNIST.  I would like to champion this paper.", "rebuttal": "We thank reviewer 1ojo for the time taken to write a detailed review. Below we go through the specific points raised, and questions.\n\n=== Highlighted weaknesses===\n\n- We chose rate coding neurons in our simulations for simplicity, to better isolate and illustrate the core properties of the decomposition. However, the method is general: the mathematical framework applies equally to spiking neurons. Just as Shannon information theory is agnostic to the neural code, our decomposition can be applied to either rate- or spike-based representations.\n\n- We presented results from a simple LNP model responding to MNIST digits. This setup was sufficient to demonstrate the qualitative properties of the decomposition and its scalability to high-dimensional inputs and multi-neuron populations. \n\nTo address the reviewer’s concern about the inclusion of real-world data, we have tested our method on a model trained with real retinal data (Klindt et al., NeurIPS 2017), in addition to the simpler MNIST dataset. We chose to highlight the MNIST dataset in the main text to demonstrate our method’s applicability to a broad audience using a well-known, accessible dataset. However, we can now include results from the retinal data model in a new Appendix for the revised manuscript, to demonstrate the robustness of our approach across diverse datasets, directly addressing the reviewer’s concern\n\n=== Questions ====\n\n1. As with Shannon mutual information, our decomposition depends on the assumed stimulus distribution, $p(X)$. In practice, when modeling neural coding under naturalistic conditions, it is standard to use the natural image distribution (e.g., Simoncelli & Olshausen, 2001). That said, our framework does not require this assumption; any stimulus distribution can be used depending on the experimental context.\n\n2. Diffusion models are a natural fit for our decomposition because they directly provide the required conditional expectations, $E[X \\mid X_\\gamma]$ and $E[X \\mid X_\\gamma, R]$, across a range of noise levels $\\gamma$. However, other architectures capable of similar denoising (e.g., Kadkhodaie & Simoncelli's \"bias-free\" network, NeurIPS 2019) could also be used. Our decomposition is not tied to a specific model class.\n\n3. Our work focuses on decomposing the mutual information between a stimulus $X$ and a response $R$, i.e., $I(R; X)$, which is not well-defined in the absence of a stimulus. For spontaneous or intrinsic activity, where $X$ is undefined or not present, the current decomposition does not apply."}, {"review_id": "AeZpGAZMmH", "reviewer": "Reviewer_U4Li", "summary": "This paper presents a method for decomposing mutual information between stimuli and neural responses into stimulus- and feature-specific components. By introducing three core axioms - additivity, positivity, and locality - the paper derives a meaningful decomposition. Empirical results demonstrate the approach's effectiveness in some simple scenarios.", "questions": "1. The completeness axiom relies on a diagonal approximation of the Fisher information matrix. How would alternative approximations—such as block-diagonal or Kronecker-factored variants—affect the satisfaction of the axioms (completeness, locality, positivity, additivity)?\n\n2. What are the key challenges in applying this decomposition framework to high-dimensional or natural image datasets? Are the axioms still meaningful or applicable in such contexts?\n\n3. How does the computational cost of the decomposition scale with the number of neurons or layers in modern neural networks? Is the method tractable for large-scale models?\n\n4. The paper draws connections between Fisher information and diffusion processes in theory (Section 4). In the MNIST experiments, can the authors clarify the underlying relationship between theory and empirical observations ？", "limitations": "see weakness and questions", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "***Strengths*** :\n\nThe paper introduces an interesting axiomatic framework for decomposing neural information, offering a useful perspective that could advance probing and interpretability techniques in neural networks.\n\n***Weaknesses*** :\n\n1. The assumption of locality appears too restrictive for practical applications. It implicitly requires neural representations to be disentangled, which rarely holds in practice - neither in high - performing artificial neural networks nor in biological brains. For instance, conjunctive neurons, which encode multiple features jointly, are prevalent and critical for generalization in biological brains. This undermines the generality and applicability of the proposed axioms.\n\n2. While the paper claims to quantify stimulus or stimulus feature-wise contributions to encoded information, the experimental validation is limited to toy settings. There is no comparison with established interpretability methods on more realistic datasets. For instance, the MNIST experiment lacks ablation studies and quantitative evaluation, making it difficult to assess the practical value and necessity of the three proposed axioms.", "was_revised": true, "final_justification": "All my concerns have been addressed, including: 1) overly restrictive locality assumption; 2). insufficient empirical validation", "rebuttal": "We thank reviewer 1ojo for the time taken to write a detailed review. Below we go through the specific points raised, and questions.\n\n============ Weaknesses ==========\n\n1. We believe there is some confusion regarding the locality principle, which we will try to clarify here.\n\nOur locality assumption imposes a constraint on  the information decomposition, not the neural network under study. Specifically, it does *not* impose any constraints on how neurons respond to stimuli—e.g., it does not assume disentangled or independent responses. \n\nThe locality axiom states that the stimulus-dependent information, $I(x)$, should depend only on how neurons respond to stimuli near $x$ in stimulus space—not on their responses to distant stimuli $x'$. This is crucial for interpretability. For example, consider two neural populations: one responding to trees, the other to faces. Without locality, the information attributed to a face stimulus, $I(x_{\\text{face}})$, could incorrectly depend on how tree-selective neurons respond, even if the two populations are functionally independent.\n\nConversely, if the populations are entangled—e.g., responses to faces are modulated by responses to trees—this coupling will naturally be reflected in $I(x)$, as it should.\n\nIn summary, the locality axiom is not restrictive, but rather ensures that $I(x)$ provides a meaningful, interpretable decomposition of information encoded by the network. We can add an additional sentence to the manuscript to clarify this if the reviewer believes it will be helpful.\n\n2i: comparison with litterature\n\nWe respectfully clarify that we are not aware of established interpretability methods that directly address our problem. Classic saliency methods like gradient attribution, integrated gradients, or CAM have been designed for discriminative models (classifiers or regressors, with examples in computational neuroscience as well see Tanaka et al, NeurIPS 2019, and Maheswaranathan et al, Neuron 2023) and single-output settings. They cannot be straightforwardly applied to mutual information or generative sensory coding tasks (see, for example, Zheng et al. ICLR 2024, for a discussion on the challenges of developing interpretability methods for generative diffusion models).\n\nFor our problem, of decomposing the mutual information into contributions from each stimulus/feature, the only direct baselines are previously proposed stimulus-specific information measures such as $I_{\\text{SSI}}$, the specific information $I_{\\text{sp}}$, surprise $I_{\\text{surp}}$, and coordinate-invariant SSI $I_{\\text{CiSSI}}$, which we extensively discuss and theoretically compare in Section 3 and Table 1. However, these previous measures are either intractable in high-dimensions or violate key interpretability properties (e.g. $I_{\\text{sp}}$ and $I_{\\text{SSI}}$ can become negative and all these measures are non-local, meaning a change in model behaviour at one stimulus can affect attributions everywhere; see table 1, Fig. 1C–F and Fig. 3D–E). We illustrate this in simple situations, showing that our measure reduces to known quantities in special cases (e.g. it recovers the Fisher information in the small-noise limit and matches total mutual information when averaged over stimuli) and by contrasting its qualitative behaviour against previous measures in simplified scenarios (Figures 1–3). \n\nTo address the reviewers comment, asking for a comparison with previous methods on high-d dataset, we will add a new Appendix, showing how a feature-wise decomposition of $I_{surp}(x)$ (which we can estimate using a diffusion model, as shown in Appendix B) behaves on the MNIST dataset. Since the feature-wise decomposition of $I_{surp}(x)$ can be both negative and non-local, this should allow us to further highlight the importance of our axioms for interpretability on high-d datasets.\n\n2ii: application to realistic data-sets\n\nTo address the reviewers' comments, we have performed additional simulations using a model fitted to real retinal data (Klindt et al., NeurIPS 2017), and a diffusion model trained on natural images. The qualitative behaviour in this setting closely mirrors that observed with MNIST: the decomposition highlights regions of the image with high local spatial contrast. This can be included in an Appendix for the camera ready version.\n\n============ Questions ============\n\n1. To clarify, we do not make any assumptions about the Fisher Information matrix (e.g. we don't assume that it is diagonal). Instead, the multivariate de Brujin identity relates the derivative of the entropy (with respect to the variance of an additive gaussian noise perturbation) to the trace of the Fisher information, as shown in Eqns 5. We will add a citation (to Rioul et al. 2010) to clarify this.\n\n2. Figure~4 illustrates the behavior of the decomposition in a moderately high-dimensional setting (images with $32 \\times 32 = 1024$ pixels and 50 neurons). In the revised appendix, we will include additional results from simulations using a model trained on real retinal data, along with a diffusion model trained on natural images. For natural images, we used 64x64 resolution, compared to 32x32 for MNIST, while the training dataset size remained consistent at 60,000 samples, equivalent to MNIST.  In our experiments, we needed neither more image samples nor additional diffusion steps compared to MNIST to identify informative areas in natural images.\n\nIn high-d settings, our locality axiom becomes even more critical than in the 1D examples shown in Figs. 1-3. Without locality, the attribution $I(x)$ could be influenced by the network's response to *any* stimulus—even those far from $x$ in stimulus space. In low dimensions, one can visualize how $I(x)$ varies across stimuli, making such non-local effects apparent. But in high dimensions, this is no longer feasible: one cannot easily determine which stimulus-response pairs contribute to the attribution at a given $x$. Enforcing locality ensures that $I(x)$ reflects only the behaviour of the system in a neighborhood around $x$, preserving interpretability. To show this we will include an additional figure panel in the Appendix, showing the behaviour of $I_{surp}(x)$ (which doesn't respect locality) on a high-d dataset (see previous comment).\n\n3. The computational cost of our decomposition method depends on the cost of training a conditional diffusion model, that is able to sample from the posterior distribution $p(X|r)$. In our case, we used a multilayer perceptron that transforms the model neural responses into a 256-embedding, which we feed as an input to the conditional diffusion model. The complexity thus scales linearly as $O(n_{neurons}\\times 256)$. In practice, since conditional diffusion models can be trained on high-d inputs, the bottleneck for analysing real biological neural populations will typically come from the experimental dataset (i.e. how many neurons were recorded), rather than the diffusion model.\n\n4. In section 4, the relation between the Fisher information about a noise corrupted stimulus, $\\mathcal{J}(x_\\gamma)$, and the expectations, $\\hat{x}(X|x_\\gamma,r)$ & $\\hat{x}(X|x_\\gamma)$, follows directly from Tweedie's law, which relates the score in a diffusion model to these conditional expectations. Tweedie’s law is a standard result which has been used extensively  in previous work on high-d datasets (e.g. Kadkhodaie & Simoncellie Neurips 2021). \n\nIn Figure 4 we plot the Fisher information (with respect to the true stimulus, not the noise corrupted stimulus), $\\mathcal{J}(x)$, alongside $I_{local}(x)$, for the  MNIST digits. In this setting, where the stimulus dimension is higher than the number of neurons, the Fisher information has limited relevance for measuring how well we can extract information about the stimulus from neural responses (e.g. the Cramer-Rao bound, which relates the Fisher to the optimum decoding error, diverges). In contrast, due to the completeness axiom, $I_{local}(x)$ is directly related to the total information encoded by the network, $I(R;X)$."}, {"review_id": "vK2GO56M3l", "reviewer": "Reviewer_f3nf", "summary": "The paper derives a stimulus-dependent decomposition of mutual information, $I_{local}$. For this the mutual information between a noise corrupted stimulus and neural population response is expressed via an expectation over the Fisher information given the stimulus distribution. To obtain a stimulus-specific estimate of the information the authors propose to condition the Fisher information on a fixed stimulus, rather than averaging over the full stimulus distribution. The authors show that this decomposition respects the axioms of completeness, additivity, positivity and locality.\n\nSince the Fisher information can be written in terms of a mean squared-score, it can be approximated by training denoising diffusion models to sample from the prior and posterior distributions. This also means that $I_{local}$ can be estimated for high dimensional stimuli.\n\nThe authors compare to other existing stimulus-specific decompositions of mutual information and demonstrate their method allows them to estimate pixel-wise information on MNIST for a population of visual neurons.", "questions": "- In L144, where does the mean-squared score come from? I am not familiar with this formulation. If it is obvious would you mind giving a brief explanation? Otherwise it would be nice to have a citation there.\n- Does this method also work for discrete responses?\n- Are there other features / axioms of information decompositions like uniqueness that the proposed method violates?\n- L542. What is the purpose of A and why did the method require an A=2? If you could give a reason for this choice, that would be helpful.", "limitations": "Some limitations where discussed, although potentially a bit brief. The authors could expand on this, but I lack the knowledge to propose additional limiting factors to discuss.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strengths\n- The paper is well structured and clearly written. Despite not being an expert on decomposition of mutual information, I was able to follow the derivation and arguments quite well. \n- The derived decomposition seems very useful in practice, especially because it can scale to high dimensional stimulus paradigms such as those commonly found in visual decoding experiments.\n\n## Weaknesses\n- The choice of colors in Figure 2 are a bit confusing. I feel like in panel A you could have added a Gaussian (black) and colored the bimodal in red (maybe even all of them as solid lines). Also the red in panel A could suggest it has something to do with the bimodal distribution. You might want to consider changing it to something else.\n- The proposed decomposition $I_{local}$ seems most similar to that of Kong et al. 2024, yet it not compared to in the main paper (Sec. 3 for example). Why is this the case? Which are the axioms it violates (L408)? How do they compare (L405, visually, similar to Fig 1.)? Related: in L425 instead of explaining, it would be nice to be able to see this in a plot.\n- While the behavior of the Fisher information for a Gaussian was mentioned in L197 it would be nice to also show this as part of Fig. 1B\n- L218 a reference to the data processing inequality would be nice (Remark 1). Also which are the concrete methods that do not respect this inequality?\n- L239 If I understand this correctly, this is just a LNP model. Could you state that in the text?", "was_revised": true, "final_justification": "The paper is well written, technically solid and makes a notable contribution that I would like to see presented at the conference. The authors also have answered all my questions during the rebuttal and clarified any remaining concerns I had. I recommend this paper for acceptance.", "rebuttal": "We thank reviewer 1ojo for the time taken to write a detailed review. Below we go through the specific points raised, and questions.\n\n======= Weaknesses =======\n\n1. We decided to only show the bimodal distribution in the top part of Fig 2A, since the middle and lower plots only show the posteriors obtained with this prior distribution. We feel that adding the gaussian distribution on top of these plots as well would be confusing. However, if the reviewer thinks it will improve clarity,  we could additional panel beforehand, similar to the current panel 2A but showing the behaviour with a gaussian prior.\n\nWe agree that it would be a good idea to change the colour of the red plots in panel 2A, so as to not confuse with panels 2B-D where red colour indicates the bimodal distribution. We can do this for the camera ready submission.\n\n2. Kong et al. proposed two different decompositions of the mutual information $I(R;X)$, into functions of both the response and stimulus, $I(r,x)$ (Eqns 18 & 19, Appendix B). In contrast, we sought to decompose the information into a function of just the stimulus, $I(x)$. We thus decided to compare our decomposition in the main paper to previously proposed stimulus-dependent decompositions of the mutual information that take the same form as ours, and only discuss Kong et al.'s method textually, while keeping more detailed comparison with their work to appendix B.\n\nIn appendix B, we looked at the behaviour of stimulus dependent decompositions, obtained by averaging Kong et al's decompositions over $p(R|x)$. We show that when we do this with first of their decompositions (Eqn 20) it becomes equivalent to $I_{surp}(x)$ (Eqn 3), investigated in the main text (e.g.~Fig 1E), which violates the locality axiom (Table 1). The second  decomposition (Eqn 21), on the other hand, violates the additivity axiom, as we show in the Appendix B, Figure 5. This is discussed in lines 435-440 of Appendix B and lines 123-127 and 262-271 of the main text.\n\n3. This is a good suggestion. We can add the Fisher information to Figure 1 for the camera-ready version.\n\n4. The data processing inequality is a standard result and can be found for example in most textbooks on information theory (e.g. Cover; Thomas (2012). Elements of information theory). We will add this reference to the camera ready paper. In figure 3 of our paper we show that, while the data processing inequality is always respected on average for the mutual information, both the stimulus specific information (Eqn 2) and specific surprise (Eqn 3) violate the data processing inequality point-wise,  for individual stimuli (3D-E).\n\n5. Yes, we use is an LNP model. This was written in the figure caption (Fig 4), but we can also add it to the main text for clarity, for the camera ready version.\n\n========= Questions =========\n\n1. The mean-square expression (Eqn 6) arises from the standard identity for Fisher Information:\n$$\\mathcal{J}(x) = -E_{R|x} \\left[ \\nabla_x^2 \\log p(R|x) \\right] = \\mathbb{E}_{R|x} \\left[ \\left( \\nabla_x \\log p(R|x) \\right)^2 \\right].$$\nThis is a standard result, covered in texts such as \"Lehmann and Casella (1998), Theory of Point Estimation\". We will add this citation in the camera-ready version.\n\n2. Yes, our method works for both discrete and continuous responses. For example, our 1-d simulations (Figs 1-3) assume that $R$ is continuous, with gaussian noise, while our high-d simulation (Fig 4) assumes discrete responses, with Poisson noise.\n\n3. Unlike the coordinate-invariant stimulus-specific information (CiSSI; Eqn 4), our decomposition is not invariant under arbitrary invertible non-linear transformations of the stimulus variable $X$. Achieving such invariance would require sacrificing our locality axiom—an axiom we consider more important for interpretability, for the reasons discussed in the paper. We are happy to clarify this in the paper if the reviewer feels it would improve transparency.\n\n4. The reviewer drew our attention to a mistake in the appendix: the constant $A$ should appear as a multiplicative factor in Eqn 24 (not Eqn 23), and its value should be 40, not 2. This constant reflects the maximum spike count in our model, chosen to match the order of magnitude observed in V1 recordings over a 0.75 s window, where neurons typically reach firing rates of ~50 Hz. Importantly, our qualitative results are robust to this choice: they remain unchanged if $A$ is rescaled by a factor of 2 for example."}, {"review_id": "1Nluwzuq1W", "reviewer": "Reviewer_2L2J", "summary": "This paper proposes a new measure of neural sensitivity $I_{local}(x)$ that satisfies certain desired properties of completeness, locality, positivity, additivity, and insensitivity to irrelevant features. The authors use de-Brujin's identity to derive a decomposition of mutual information between response and stimuli, $I(R;X)$, into a stimulus-specific quantity $I_{local}(x)$. The proposed $I_{local}(x)$ is essentially a conditionally-averaged non-parametric Fisher information of a gaussian-noise corrupted version of the stimuli. The authors claim that their proposed decomposition satisfies all the desired properties whereas all other existing decompositions don't, particularly, the locality axiom. Furthermore, authors show that their proposed decomposition can be easily computed using a conditional and an unconditional DDPM models. The authors compare their decomposition with existing decompositions in a toy example and simulated visual neurons with MNIST dataset. The authors conclude by arguing that their decomposition is more interpretable.", "questions": "1. Why other decomposition methods were not used to study the decomposition in the MNIST dataset?", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "*I have provisionally rejected based on the technical flaws listed in the weakness of the paper section. if authors are able to satisfactorily address all my concerns below, I am happy to raise my recommendation to accept.*\n\n**Strength of the paper**: \n\n1. The paper is well-written and properly formatted.\n2. The paper attempts to follow a principled axiomatic approach to derive a measure of information, ensuring that the derived measure is theoretically grounded and interpretable.\n3. The paper does a fairly good job of exposing previous literature.\n4. I especially enjoyed the inclusion of intuitive text/english descriptions accompanying the math.\n5. The simulation results are illustrating although the results would have been more stronger if there was real data. Any particular reasons why Allen brain visual dataset was not used (see Venkatesh et al, NeurIPS'23)?\n6. The paper provides sufficient details to understand the computational experiments and does a good job of discussing its limitations.\n\n\n\n**Weaknesses of the paper**: The main weakness of the paper is the lack of rigor in the mathematical/theoretical analysis. A lot of theoretical statements are made heuristically and w/o assumptions. A simple example is that authors use Fisher information to define their decomposition, but do not state under what assumptions (e.g., smoothness, continuity on the pdfs) the Fisher information is well-defined. Fisher information requires fairly strong regularity conditions (such as existence of a second derivative) to be well-defined. Such assumptions should be clearly mentioned somewhere. I understand in practice such assumptions are typically satisfied as we work with nice distributions. Regardless, it is important to mention these assumptions as the main content of this paper is theoretically-focused. \n\nI briefly state the most concerning issues with the theoretical analysis below:  \n\n1. **Use of de-Brujin's identity to conditional differential entropy**: \n\nUsing author's notation, we can summarize the vanilla version of the de-Brujin's identity as follows:\n \n$$ \\frac{dh(X+\\sqrt{\\gamma}Z)}{d\\gamma} = \\frac{dh(X_{\\gamma})}{d\\gamma}  =\\frac{1}{2}J(X_{\\gamma})= -\\frac{1}{2}\\text{Trace}\\left[\\mathbb{E}\\left[\\nabla^2\\log\\left(p(X_{\\gamma})\\right)\\right]\\right]; Z\\sim\\mathcal{N}(\\mathbf{0},\\mathbf{I}), J(\\cdot)\\text{ is the Fisher information}.$$\n\nThe above equation (i.e., the vanilla de-Brujin's identity) only relates the differential entropy to the non-parametric Fisher information using a Gaussian channel. It **does not** relate the conditional differential entropy $h(X_{\\gamma}|R)$ to the Fisher information. Note that the conditional differential entropy and differential entropy are different mathematical entities. To illustrate that, we can explicitly write out the formulas for $h(X_{\\gamma})$ and $h(X_{\\gamma}|R)$ (assuming the underlying pdfs satisfy the regularity conditions):\n\n$$h(X_{\\gamma}) = -\\int p(X_{\\gamma})\\log(p(X_{\\gamma}))dX_{\\gamma}\\text{  and  }h(X_{\\gamma}|R) = -\\int p(R)p(X_{\\gamma}|R)\\int\\log(p(X_{\\gamma}|R))dX_{\\gamma}dR.$$\n\nNote that the term inside and outside of the $\\log(\\cdot)$ is the same in the integrand of the $h(X_{\\gamma})$. This is not the case in $h(X_{\\gamma}|R)$ (note the extra $p(R)$). Hence, I am skeptical the vanilla version of de-Brujin's identity can be applied to $h(X_{\\gamma}|R)$, as is done in equation (5) of the paper. To elaborate, let us analyze the term $\\frac{dh(X_{\\gamma}|R)}{d\\gamma}$. Assuming the underlying pdfs satisfy all the required regularity condition, we can expand the term as follows:\n\n$$\n \\frac{dh(X_{\\gamma}|R)}{d\\gamma} =- \\frac{d}{d\\gamma}\\int p(R)\\int p(X_{\\gamma}|R)\\log(p(X_{\\gamma}|R))dX_{\\gamma}dR\n$$\nUsing chain rule and exchanging the differential and integration operators: \n$$\n \\frac{dh(X_{\\gamma}|R)}{d\\gamma} =- \\int \\frac{dp(R)}{d\\gamma}\\left[\\int p(X_{\\gamma}|R)\\log(p(X_{\\gamma}|R))dX_{\\gamma}\\right]dR-\\int p(R) \\underbrace{\\frac{d\\left[\\int p(X_{\\gamma}|R)\\log(p(X_{\\gamma}|R))dX_{\\gamma}\\right]}{d\\gamma}}_{\\text{term 1}}dR\n$$\n\nNow, it may be possible to apply de-Brujin's identity to \"term 1\" as the  terms inside and outside of $\\log(\\cdot)$, i.e. $p(X_{\\gamma}|R)$, are the same. Technically, one would also need to show that Z and X are conditionally independent given R, (*which they may not be*) to apply de-Brujin's identity to term1. Assuming the (best case scenario) that  Z and X are conditionally independent given R, we can apply de-Brujin's identity in the previous equation to obtain: \n\n$$\n \\frac{dh(X_{\\gamma}|R)}{d\\gamma} =- \\int \\frac{dp(R)}{d\\gamma}\\left[\\int p(X_{\\gamma}|R)\\log(p(X_{\\gamma}|R))dX_{\\gamma}\\right]dR+\\frac{1}{2}\\text{Trace}\\left[ \\mathbb{E}\\left[\\nabla^2_{x_{\\gamma}}p(X_{\\gamma}|R)\\right]\\right]\n$$\n\nFor equation (5) in the manuscript to be correct, the authors need to show that first term on the RHS of the above equation is **zero**. Based on my reading of the manuscript, it seems the underlying (natural) assumption is $p(R|X=x)=p(R|X_{\\gamma}=x)$, as $X_{\\gamma}$ is thought of as a noisy stimulus being provided to the system (Please correct me if my assumption is wrong). Hence, $\\frac{dp(R)}{d\\gamma}$ would not be zero as $p(R)=\\int p(R|X_{\\gamma}=x)p(X_{\\gamma}=x)dx$ should depend on $\\gamma$. \n\nI would recommend authors: \n\ni. To either point me to a citation that shows a version of de-Brujin's identity that explicitly relates the conditional differential entropy and Fisher information (I skimmed through Section D. of [Rioul, 2010] and could not find it), or \n\nii. To rigorously show that vanilla de-Brujin's identity can be applied to conditional differential entropy, with all the assumptions and proof steps clearly stated. \n\nNote that if the application of de-Brujin's identity on conditional differential entropy cannot be rigorously proven, then the proposed $I_{local}(x)$ does not satisfy the completeness axiom, and leaves the manuscript on shaky ground.\n\n2. **Lack of mathematical rigor in defining Axiom 2: Locality**\n\ni. My first issue is with the equation $p(R,x)=\\tilde{p}(R,x)$ for all $x\\notin B_{\\epsilon}(x_0)$. This is too strong a statement. The perturbations are not constrained and can be anything inside the $B_{\\epsilon}(x_0)$. This is problematic, as I can define $\\tilde{p}(R,x)$ to be a distribution that is, e.g. not differentiable, in the region contained in $B_{\\epsilon}(x_0)$. Consequently, the Fisher information for $\\tilde{p}(R,x)$ would not be defined in $B_{\\epsilon}(x_0)$, making the decomposition $I_{local}(x)$ undefined for $\\tilde{p}(R,x)$. For example, choose $X$ and $R$ to be scalar, and $\\epsilon$ to be 1. Define $\\tilde{p}(X|R) = A(R)|W(x)|$ for $|X|\\leq 1$ and $\\tilde{p}(X|R) = p(X|R)$ for $|X|\\geq 1$, where $A(R)$ is a normalizing factor to ensure the probability distribution  integrates to one and W(x) is the Weierstrass function. Furthermore, keep $p(R)=\\tilde{p}(R)$. Now, $\\tilde{p}(X|R)$ is not differentiable for $|x|\\leq 1$ and hence the term $\\nabla^2\\log(\\tilde{p}(X|R))$ would be undefined for $|x|\\leq 1$. Therefore, we are unable to compute Fisher information and $I_{local}(x)$ for $\\tilde{p}(X,R)$. This example demonstrates the need for the perturbations to be constrained as one can create **pathological examples** to show that $I_{local}(x)$ may not be meaningfully *defined* much less satisfy the locality axiom.\n\nii.  The limiting procedure, i.e., $|I(x)-\\tilde{I}(x)|\\rightarrow 0$ when $\\|x-x_0\\|\\gg d$ is awkwardly phrased. Typically, the way to propose a limiting procedure is to show an \"input\" ($x$ moving along a certain direction while the output ($|I(x)-\\tilde{I}(x)|$) is converging to a quantity. The condition $\\|x-x_0\\|\\gg d$ does not show a movement of the input $x$, which makes the limiting operation a bit hard to interpet. I would encourage authors to use the more conventional way of defining limits. For example, for every $\\delta>0$ there exists a $d_{\\delta}$ such that $|I(x)-\\tilde{I}(x)|\\leq \\delta$ when $\\|x-x_0\\|\\geq d_{\\delta}$. \n\niii. The proposed Axiom 2 is problematic for finite domains. For example, let us assume the domain of $X$ is finite, then I can always choose an $\\epsilon$ big enough such that $B_{\\epsilon}(x_0)$ contains the whole domain of $X$. Hence, I can choose $\\tilde{p}(R,X)$ that is completely different than $p(R,X)$. Consequently, no possible decomposition could ever satisfy the Locality Axiom, as I could always find $\\tilde{p}(R,X)$ (for example by adding noise everywhere) such that $|I(x)-\\tilde{I}(x)|>\\epsilon\\ \\forall x$.\n\nAxioms are the heart of any mathematical theory. It is important to ensure that the axioms are defined and written properly such that the pathological examples (as above) do not occur. I would encourage authors to re-think and re-work their Axiom 2.\n\n3. **Lack of rigor in the Proof presented in Appendix A**. \n\ni. My first concern is from equation (11)-(13). I am interpreting their statement (above equation (11)) as $\\gamma\\geq d$, where $d$ is large (it is a bit confusingly written: \"large $\\gamma>d$\"). Now it seems that authors are arguing in equation (12) that $E[X|x_{\\gamma, R}]\\approx E[X|R]$ amd $E[X|x_{\\gamma}]\\approx E[X]$, but provide no reasoning as to why it should be true. I am assuming that the rationale is that $X_{\\gamma}$ for large $\\gamma$ is just noise and consequently independent from $X$. There are concerns with that line of reasoning:\n\na. $X$ being approximately independent of $X_{\\gamma}$ does not necessarily imply $X$ is approximately **conditionally independent**  of $X_{\\gamma}$. The approximate conditional independence needs to be rigorously shown by authors. Please look at the famous XOR example to gain intuition regarding the difference between conditional independence and independence. \n\nb. The change of $\\gamma^2$ to $d^2$ from equation (11) to equation (12) is abrupt and without explanation.\n\nc. Please do not use $\\approx$ in a proof. Please be precise in your statements. For example, use statements such as $\\mathcal{J}(x_{\\gamma}) = \\frac{1}{d^2}\\Sigma_{X|R}+\\text{error term}$, where the error term goes to zero as $\\gamma$ goes to infinity.\n\nd. Assuming equation (12) is true, i.e., $\\mathcal{J}(x_{\\gamma}) \\approx \\frac{1}{d^2}\\Sigma_{X|R}$, the integral $\\text{Trace}[\\int_{d}^{\\infty} E_{X_{\\gamma}|x}[J(x_{\\gamma})] d\\gamma]$ can be simplified as follows:\n \n$$\\text{Trace}[\\int_{d}^{\\infty} E_{X_{\\gamma}|x}[J(x_{\\gamma})] d\\gamma] \\approx \\frac{1}{d}\\text{Trace}[ \\int_{d}^{\\infty} E_{X_{\\gamma}|x}[\\Sigma_{X|R}]d\\gamma] = \\frac{1}{d}\\int_{d}^{\\infty} \\text{Trace}[\\Sigma_{X|R}]d\\gamma,$$\n\nwhere I substituted the approximation proposed in equation (12) followed by the fact that $\\text{Trace}[\\Sigma_{X|R}]$ does not depend on $X_{\\gamma}$ and hence we can remove the expectation operator. There has to be some error in this approximation as $ \\frac{1}{d}\\int_{d}^{\\infty} \\text{Trace}[\\Sigma_{X|R}]d\\gamma$ for any positive posterior covariance ( $\\text{Trace}[\\Sigma_{X|R}]$) would be infinite and not $\\frac{1}{d}\\text{Trace}[\\Sigma_{X|R}]$ as stated in equation (13). \n\ne.  In equation (12), authors are equating a *scalar* with a *vector*:  $\\|\\| E[X|R]-E[X]\\|\\|^2= \\Sigma_{X|R}$. I guess the authors meant $\\|\\| E[X|R]-E[X]\\|\\|^2= Trace[\\Sigma_{X|R}]$? There is also a typo regarding the factor of $\\frac{1}{2}$, the factor of $\\frac{1}{2}$ should appear in equation (13).\n\nii. Statement, \"This is a convolution with Gaussian, with width $\\gamma$ ... vanishing influence on $\\log(p(R|x_{\\gamma}))$\". Don't state but prove this statement either through a citation or a proof. The goal is indeed to show that this statement is true.\n\niii. Equation (15) provides an approximation for $\\|\\| x_{\\gamma}-x_{0}\\|\\|^2\\gg \\gamma$ whereas eqns (11)-(13) provide an approximation for $\\gamma>d$. Why is the analysis of the region between $\\|\\| x_{\\gamma}-x_{0}\\|\\|^2\\gg \\gamma$ and $\\gamma>d$, i.e., when $\\sqrt(\\gamma)$ and $d$ are roughly of the same order, **omitted**. Note that the integration goes from 0 to $\\infty$. Hence, an approximation for that region is also needed. \n\niv There is a typo, it should be $\\gamma$ in equation (14).\n\nI would encourage authors to clearly define the approximations for each segment of the integral and then show that $I_{local}(x)$ satisfies some desired locality properties.\n\n**Minor comments**\n\ni. Please do not change the order of intergrals, differential, and limiting operations without stating the appropriate regularity conditions. For example, to show that $E_{X}[I_{local}(x)] = I(R;X)$, you need dominated convergence theorem to move the expectation operator $(E_{X})$ inside the integral in the definition of $I_{local}(x)$ (provided in equation (8)) to obtain the term $E_{X}[\\mathcal{J}_{ii}(X\\gamma)]$ . Note that it not always possible to change the order of intergals, differential operators when infinite limits are involved.\n\nii. Please do not confuse Entropy with differential entropy. Differential entropy is denoted by $h(\\cdot)$ whereas the entropy function is denoted by $H(\\cdot)$. Differential entropy and entropy are different mathematical entities which is underscored by the fact that differential entropy can be negative whereas entropy is always positive. \n\niii. Please clearly state assumptions on the pdf that allow application of de-Brujin's identity, existence of differential entropy $h(X)$. For example, consult [Rioul, 2010] for the required assumptions.", "was_revised": true, "final_justification": "I have updated my score based on the author's rebuttal. Author's were able to successfully answer my most pressing concerns.", "rebuttal": "We thank reviewer 2L2J for their detailed & insightful review. We agree that the rigor of some proofs and mathematical statements can be improved, and we appreciate the reviewer’s comments, which will help us strengthen the final version. In particular, we acknowledge that Appendix A was not sufficiently clear, and we thank the reviewer for their comments, which will help us improve it. That said, the key results presented in the paper are correct, as clarified below. We hope our responses address the reviewer’s concerns.\n\n=============== Comment 1 ==============\n\nWe believe the confusion arises from a misunderstanding of the model: specifically, the reviewer assumes that $p(R|X=x) = p(R| X_\\gamma=x)$, which is not correct in our setting. We thank the reviewer for highlighting this point and allowing us to clarify.\n\nThe response $R$ is generated from the original stimulus $X$, \\& not from $X_\\gamma$, according to $p(R|X)$. The noise corrupted stimulus, $X_\\gamma$, equals the stimulus $X$, plus gaussian noise, $Z$. Thus, $R$ and $X_\\gamma$ are conditionally independent given $X$, and the joint distribution factorizes as $$p(R,X,X_\\gamma)=p(R | X)p(X_\\gamma | X)p(X).$$\n\nMarginalizing over $X$ and $X_\\gamma$, we find that $p(R)$ is independent of $\\gamma$, i.e.,\n$$\\frac{d}{d\\gamma}p(R)=0.$$\nAs a result, the first term in the reviewer’s derivation vanishes.\n\nWe are left with what the reviewer calls `term 1' which includes: $$\\frac{d h(X+\\sqrt{\\gamma}Z|R=r)}{d\\gamma}=-\\frac{d}{d\\gamma}\\int p(X+\\sqrt{\\gamma}Z |R=r)\\log p(X+\\sqrt{\\gamma} Z|R=r)dXdZ.$$ Now, as (from above) $R-X-X_\\gamma$ forms a Markov chain, we can write $p(X_\\gamma,X|r)=p(X_\\gamma|X)p(X|r)$, which also implies, $p(Z, X|r)=p(Z|X)p(X|r)$. Since, by construction, $p(Z|X)=p(Z)$, we establish that $Z$ is also *conditionally* independent of $X$ given $r$, as required for de-Brujin's identity.\n\n=============== Comment 2 ==============\n\ni. We can exclude the pathological example raised by the reviewer by stating that both $p(R,X)$ and $\\tilde p(R,X)$ satisfy standard regularity conditions for Fisher Information. Specifically, we will state that $p(R,X)$ and $\\tilde p(R,X)$ are doubly differentiable for all $X$, with sufficient decay at infinity for $\\mathcal{J}(x)$ to exist and be finite everywhere (Rioul 2010). We will add this statement to the camera-ready version.\n\nii. We agree, and we will present the limiting procedure more rigorously in the revised version, as suggested by the reviewer: namely, that for every $\\delta>0$ there exists a $d_\\delta$ such that $|I(x)-\\tilde I(x)|\\leq \\delta $ when $||x-x_0||^2\\geq d_\\delta$.\n\niii. We assume in the paper that $x\\in\\mathbb{R}^d$, and thus has infinite support (line 68). We will clarify in revision.  This is a reasonable assumption for most distributions of interest in sensory neuroscience. However, we will add a sentence to the discussion to state this limitation, and that future work could deal with bounded variables.\n\n=============== Comment 3  ==============\n\nia. The reviewer is correct in stating that we reason that, in the high-$\\gamma$ limit, $X$ becomes independent of $X_\\gamma$, since $X_\\gamma$ is just noise. Moreover, for our model, where $R$ is conditionally independent of $X_\\gamma$ given $X$, the joint factorises as $p(X, R, X_\\gamma) = p(R|X)p(X_\\gamma|X)p(X)$ (See comment 1). Thus, when $X_\\gamma$ becomes independent of $X$ (so that $p(X_\\gamma|X)=p(X_\\gamma)$), $X$ also becomes *conditionally independent* of $X_\\gamma$, given $R$. \n\nib-e. Agreed, we mistakenly replaced $\\gamma$ with $d$ in Eqn 12. Instead we should state that, in the high-$\\gamma$ limit,$$\\mathcal{J}(x_\\gamma) = \\frac{1}{\\gamma^2} \\Sigma_{X|R} + o\\left(\\frac{1}{\\gamma^2}\\right)$$. Thus, the integration becomes,$$\\int_{d}^{\\infty}\\frac{1}{\\gamma^2}\\mathrm{Tr}(\\Sigma_{X|R})d\\gamma =\\frac{1}{d}\\mathrm{Tr}(\\Sigma_{X|R}),$$which is the result stated in the appendix.\n\nii. We agree this statement needs proved. We will add the following to the appendix.\n\nConsider Equation 14 from the paper:\n$$\\log p(r|x_\\gamma)=\\log \\frac{\\int_{x'}p(r, x')\\phi\\left( \\frac{x'-x_\\gamma}{\\sqrt{\\gamma}}\\right)dx'}{\\int_{x'}p(x')\\phi\\left( \\frac{x_\\gamma-x'}{\\sqrt{\\gamma}} \\right)dx'},$$\nwhere $\\phi(\\cdot)$ denotes the standard multivariate normal density. We aim to show that $\\log p(r \\mid x_\\gamma)$ is unaffected by perturbations to $p(r,x)$ that are localized within the domain $x \\in B_\\epsilon(x_0)$, in the limit where $||x_\\gamma-x_0||^2 \\gg \\gamma$.\n\nFor this, we split the integrals (numerator and denominator) into 2 regions: $x'\\notin B_\\epsilon(x_0)$ (where $p(r,x')=\\tilde{p}(r, x')$), \\& $x' \\in B_\\epsilon(x_0)$ (where $p(r,x') \\ne \\tilde{p}(r,x')$). \n\nThen:\n$$\\log p(r|x_\\gamma) = \\log \\frac{\\int_{x'\\notin B_\\epsilon(x_0)} p(r, x')\\phi\\left( \\frac{x_\\gamma - x'}{\\sqrt{\\gamma}} \\right) dx'+\\int_{x' \\in B_\\epsilon(x_0)}p(r, x')\\phi\\left(\\frac{x_\\gamma - x'}{\\sqrt{\\gamma}} \\right) dx'\n}{\\int_{x' \\notin B_\\epsilon(x_0)}p(x')\\phi\\left(\\frac{x_\\gamma -x'}{\\sqrt{\\gamma}}\\right)dx'+\\int_{x'\\in B_\\epsilon(x_0)}p(x')\\phi\\left( \\frac{x_\\gamma -x'}{\\sqrt{\\gamma}} \\right)dx'}$$\n\nThe first terms in both numerator and denominator are independent of changes to $p(r,x)$ within $B_\\epsilon(x_0)$. To complete the argument, we thus need to show that the second terms tend to zero as $||x_\\gamma - x_0||^2 \\gg \\gamma$.\n\nIf $p(r,x)$ is locally bounded in $B_\\epsilon(x_0)$ (which follows from the regularity assumptions stated in comment 2i, that the distribution is both integrable and differentiable everywhere), i.e., there exists $M< \\infty$ such that\n$$\\sup_{x'\\in B_\\epsilon(x_0)}p(r, x')\\leq M.$$\nThen:\n$$\\int_{x'\\in B_\\epsilon(x_0)} p(r, x')\\phi\\left( \\frac{x_\\gamma-x'}{\\sqrt{\\gamma}} \\right) dx'\n\\leq M\\int_{x' \\in B_\\epsilon(x_0)}\\phi\\left( \\frac{x_\\gamma-x'}{\\sqrt{\\gamma}} \\right) dx'.$$\n\nThe right-hand side is the mass of a Gaussian centered at $x_\\gamma$ over a ball that is far from its mean—this decays rapidly as $||x_\\gamma-x_0||^2\\gg \\gamma$. Therefore, both the numerator and denominator correction terms vanish in this limit.\n\nThis shows that $\\log p(r|x_\\gamma)$ becomes insensitive to local perturbations in $p(r,x)$ far from $x_\\gamma$, as claimed.\n\niii. We thank the reviewer for highlighting this point—we agree that our original explanation lacked sufficient clarity. However, while we acknowledge the proof could be made more rigorous, we maintain that our claimed result is true. For the camera ready version, the proof can be made significantly more rigorous, with minor modifications to the mathematical statements and notation, as set out below.\n\nRecall that goal is to show that\n$$I_{local}(x)=\\frac{1}{2}\\int_{0}^{\\infty}E_{X_\\gamma |x}[ \\mathcal{J}(X_\\gamma)]d\\gamma$$\nis unaffected by perturbations to $p(r, x')$ restricted to the region $x' \\in B_\\epsilon(x_0)$, provided that $||x-x_0||^2$ is sufficiently large (the locality axiom; see comment 2ii).\n\nTo show this, we propose the following reformulatoin of the proof in Appendix A: \n\n1. First, we will show that we can always choose sufficiently large $\\gamma=\\gamma_h$ such that the tail end of the integral, $\\frac{1}{2}\\int_{\\gamma_h}^{\\infty}E_{X_\\gamma |x}[ \\mathcal{J}(X_\\gamma)]d\\gamma$,  becomes arbitrarily small for all $x$ (since it decays as $\\frac{1}{\\gamma_h}$; see lines 386-389, and comment i above).\n\n2. Next we will show that, for finite $\\gamma\\leq \\gamma_h$, we can always choose $d \\equiv ||x_0- x_\\gamma||^2$ so that the integrand $E_{X_\\gamma |x}\\left[ \\mathcal{J}(X_\\gamma) \\right]$ is unaffected by perturbations to $p(r,x')$ in the ball $B_\\epsilon(x_0)$ (lines 390-397, and comment ii above).\n\nTogether, this establishes locality, since we can say that for any $\\delta>0$  there exists a $d_\\delta$ such that $|I(x)-\\tilde I(x)|\\leq \\delta $ when $||x-x_0||^2\\geq d_\\delta$.\n\n\niv. We will correct this typo.\n\n=============== Minor comments ==============\n\ni.  In in the main text (line 151, and Eqn 8), we claim that\n$$I(R; X)=\\frac{1}{2}E_{X} \\left[ I_{\\text{local}}(X)\\right] =\\frac{1}{2}E_{X, X_\\gamma}\\left[ \\int_{0}^{\\infty} \\mathcal{J}(X_\\gamma)d\\gamma \\right].$$\nFor this to follow from Eqn 5 in the main text, requires that:\n$$I(R; X)=\\frac{1}{2}E_{X_\\gamma}\\left[ \\int_{0}^{\\infty}\\mathcal{J}(X_\\gamma) d\\gamma\\right]=\\frac{1}{2}\\int_{0}^{\\infty} E_{X_\\gamma} \\left[ \\mathcal{J}(X_\\gamma) \\right] d\\gamma,$$\nThe reviewer raises a valid question about whether this interchange of integration over $\\gamma$ and $X_\\gamma$ is justified.\n\nFrom Fubini's theorem, we can switch the order of integration so long as the integrand is Lebesgue integrable, such that $\\frac{1}{2}\\int_{0}^{\\infty}E_{X_\\gamma} \\left[| \\mathcal{J}(X_\\gamma) |\\right] d\\gamma<\\infty$. This is indeed the case in our setup, since $\\mathcal{J}(x_\\gamma)$ is always non-negative (so taking its absolute value doesn't change anything), while the integral is equal to $I(R;X)< \\infty$.\n\nii. We will replace $H$ with $h$ to denote differential entropy, as suggested.\n\niii. From Rioul 2010, for de Brujin's identity to apply, we require that $p(X)$ and $p(X|R)$ are sufficiently smooth and decay to zero as $X$ goes to infinity, so that the Fisher information exists. We will state this in the camera ready text (see also comment 2i).\n\n=============== Additional questions & comments ==============\n\nThe reviewer asked why we didn't analyse real neural data. We used a simple model of visual neurons responding to MNIST digits to show the behavior of our decomposition clearly. However, we have also applied our method to a model of retinal neurons (Klindt et al. 2017), with a diffusion model trained on natural images. This gave similar qualitative results to the MNIST dataset, and can be added to the appendix.\n\nIn the new appendix, we will add a figure comparing our decomposition to  alternative decompositions (e.g. $I_{surp}$, whose feature decomposition is non-negative and non-local) to further highlight why our decomposition is both clear and interpretable."}]

Author rebuttal:

Meta-review: a) The authors propose an axiomatic construction of stimulus specific information and propose a way to estimate it using diffusion models.

b) The axioms are sound and justified. The authors details other previously proposed definitions of stimulus specific information and show that they do not satisfy all axioms. They propose a method to construct a version that satisfies all axioms and use the method to estimate the information capture by a model of visual neurons (using MNIST).

c) No neural data. Initial concerns about the mathematical rigor (resolved!)

d) Estimating mutual information and other type of information theoretic quantity is a difficult task. The authors provide a axiomatic definition and a way to estimate the quantity that verifies their axiom. This paper deserves to be discussed as the proposed axioms can be questioned. A spotlight is great opportunity to trigger some discussion at the conference.

e) Two reviewers were initially skeptical about acceptance. The rebuttal period allowed to answer to their main concerns and both reviewers change their mind and are now in favor of accepting the paper.
Decision: Accept (spotlight)