Title: Decomposing stimulus-specific sensory neural information via diffusion models
Venue: NeurIPS 2025
Accepted: yes
Keywords: Neuroscience

Neural coding

Information theory

Efficient coding

Diffusion models
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

Abstract

To 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 fundamentally ill-posed problem with infinitely many possible solutions. We address this by introducing three core axiomsadditivity , positivity , and locality -that any meaningful stimulus-wise decomposition should satisfy. We then derive a decomposition that meets all three criteria and remains tractable for high-dimensional stimuli. Our decomposition can be efficiently estimated using diffusion models, allowing for scaling up to complex, structured and naturalistic stimuli. Applied to a model of visual neurons, our method quantifies how specific stimuli and features contribute to encoded information. Our approach provides a scalable, interpretable tool for probing representations in both biological and artificial neural systems.

1 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 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 [25, 2, 8]. 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 [23, 3, 31, 18, 9]. 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 ) = E [ I ( x )] [8, 4, 5, 17]. 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.

2 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 I ( x ) , assigning to each stimulus x 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:

This constraint alone does not uniquely determine the function I ( x ) , as many decompositions satisfy completeness. To further constrain the attribution, we impose additional desiderata that reflect desirable properties of local information measures.

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First, for I ( 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 a neighbourhood of x 0 should have a vanishing influence on I ( x ) for distant stimuli x = x 0 . Without this property, changes in I ( 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 ϵ > 0 and x 0 such that:

where B ϵ ( x 0 ) is the open ball of radius ϵ centered at x 0 . Assuming X has infinite support on an unbounded or semi-infinite domain, we require that, for every δ > 0 there should exist some finite value d , such that:

where I ( x ) and ˜ I ( x ) denote the corresponding information decompositions derived from p ( R,x ) and ˜ p ( R,X ) , respectively. That is, local perturbations to the likelihood or prior near x 0 should not affect the information assigned to distant stimuli, x .

Mutual information is globally non-negative: I ( R ; X ) ≥ 0 . To preserve this property in our decomposition, we require the pointwise contributions I ( x ) to be non-negative as well. Intuitively, observing a 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:

Finally, Shannon 1948 posited additivity as a 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:

where we have renamed I ( x ) as I R ( x ) here, to make explicit its dependence on the response, R . I R ′ | R ( x ) is the conditional pointwise information from R ′ given R , and I R,R ′ ( x ) denotes the pointwise information from observing both responses. By construction, we require: I ( R ′ ; X | R ) = E x [ I R ′ | R ( x ) ] and I ( R,R ′ ; X ) = E x [ I R,R ′ ( x )] .

Remark 1: Local data processing inequality. Any decomposition that fulfils both additivity and positivity, as stated above, also obeys a local form of the data processing inequality. That is, postprocessing should not increase information, even at the level of the individual attributions. Formally,

To prove this we use additivity to write I R ′ ,R ( x ) in two ways:

Positivity gives I R | R ′ ( x ) ≥ 0 for all x , so:

Now, if R ′ is independent of X given R , then I ( R ′ ; X | R ) = E X [ I R ′ | R ( x )] = 0 . Since if I R ′ | R ( x ) ≥ 0 pointwise, this implies I R ′ | R ( x ) = 0 for all x . Substituting into the inequality above yields the desired result: I R ( x ) ≥ I R ′ ( 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 ϕ ( R ) , we have:

This follows from the fact that I ( R ; X ) = I ( ϕ ( R ); X ) , and hence E X [ I ϕ ( R ) ( x )] = E X [ I R ( x )] . Supposing, for contradiction, that I ϕ ( R ) ( x ) < I R ( x ) for some x , equality of expectations would require I ϕ ( R ) ( x ) > I R ( x ) for some other x , violating the pointwise data processing inequality. This highlights why our axioms are important for interpretability: they imply that I ( x ) quantifies how information is transmitted through the system X → R , independently of how the responses are parameterized.

3 Previous stimulus-dependent decompositions

Several previous works have proposed decompositions of mutual information into stimulus-specific contributions, such as the stimulus-specific information I SSI [4], the specific information I sp , the stimulus-specific surprise I surp , [8], and the coordinate-invariant stimulus-specific information I CiSSI [17]:

While these decompositions satisfy some of our proposed axioms (Table 2), none satisfy locality. This is because they all depend on global terms such as the marginal response distribution p ( r ) = ∫ p ( r | x ′ ) p ( x ′ ) dx ′ , or the posterior p ( x | r ) = p ( r | x ) p ( x ) / ∫ p ( r | x ′ ) p ( x ′ ) dx ′ , both of which can be influenced by changes to the likelihood &/or prior for any stimulus. Further, the requirement that I CiSSI ( x ) is invariant to invertible transformations of x is incompatible with locality; both axioms can't be fulfilled simultaneously. As discussed above, this limits the interpretability of previous decompositions as measures of neural sensitivity, since a non-zero attribution at x could be due to changes in neural sensitivity anywhere in the stimulus space.

Unlike the above decompositions, the Fisher information is inherently local. However, while previous 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 contributions from both the stimulus, x , and the response, r . Two different decompositions, I ( r, x ) , were proposed. However, averaging these over p ( R | x ) does not give stimulus-wise decompositions that fulfil our axioms (Appendix B). In one case, we obtain I surp ( x ) (Eqn 12), which is non-local; in the other case, the decomposition is not additive, which is a fundamental information theoretic constraint.

In the following we propose a 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 a valid decomposition of the mutual information.

4 Diffusion-based information decomposition

We first derive an expression for the mutual information, I ( R ; X ) , that can be used to construct a 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 a stimulus, X , with probability p ( R | X ) . We then consider a noise-corrupted version of the stimulus, X γ = X + √ γZ , where Z ∼ N (0 , I ) . From the fundamental theorem of calculus we can write:

since in the limit γ →∞ , X γ is just noise, and thus I ( R ; X γ ) → 0 . It follows that,

where h ( X γ ) ≡ -E p ( X γ ) [log p ( X γ )] and h ( X γ | r ) ≡ -E p ( X γ | r ) [log p ( X γ | r )] . Assuming that p ( X ) and p ( X | R ) have finite second order moments, we can apply de Bruijn's identity for all γ > 0 , to obtain,

where J ( x γ ) is the Fisher information with respect to a noise-corrupted stimulus, X γ . Finally, since lim γ 0 → 0 I ( R ; X γ 0 ) = I ( R ; X ) , and Trace( E p ( X γ ) [ J ( X γ )]) is always non-negative, monotone convergence yields:

This is a general result that holds for both discrete and continuous X and R , so long as p ( X ) and p ( X | R ) have finite first and second moments.

The above identity provides a direct relation between the mutual information I ( R ; X ) and the Fisher information, J ( x γ ) . Further, it also admits a natural interpretation in terms of de-noising diffusion models trained to predict X from a noisy observation X γ . To see this, first we use an alternative formulation of the Fisher information, in terms of the mean-squared score:

Next, from Tweedie's formula [20, 14] we have:

where ˆ x ( x γ ) = E [ X | x γ ] and ˆ x ( x γ , r ) = E [ X | x γ , 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].

Figure1: local sensitivity

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4.2 Local stimulus-specific information

Building on the integral representation of mutual information in Eqn 17, we define a stimulus-specific decomposition:

where X γ = x + √ γZ , with Z ∼ N (0 , I ) , and J ii ( x γ ) = E p ( R | x γ ) [ -∇ 2 ( x γ ) i log p ( R | x γ ) ] is the i th diagonal element of the the Fisher information matrix, evaluated at x γ . This expression parallels Eq. 17, with the key distinction that the expectation is now conditioned on a fixed stimulus x , rather than averaging over the full stimulus distribution 3 . Consequently, the decomposition satisfies the completeness axiom , since by construction, I ( R ; X ) = E [ I local ( x )] .

Next we outline why our decomposition fulfils the locality axiom (for the formal proof, see Appendix A). Recall that the Fisher information matrix J ( x ) characterizes the local curvature of the loglikelihood, log p ( R | x ) , and thus quantifies the local sensitivity of the response to changes in the stimulus [23]. The term E X γ | x [ J ( X γ )] generalizes this notion, measuring neural sensitivity when we only observe noise-perturbed versions of the stimulus, X γ ∼ N ( x, γI ) . For finite γ , this term is dominated by values of X γ close to x , and thus, it depends only on the local shape of the likelihood and prior around x . It receives a vanishingly small contribution from changes to the likelihood and/or prior for distant stimuli x ′ , if ∥ x ′ -x ∥ ≫ √ γ . Meanwhile, as γ →∞ , X γ becomes pure noise and thus E p ( X γ | x ) [ J ( X γ )] → 0 for all x . Taken together, this implies that our decomposition, obtained by integrating E p ( X γ | x ) [ J ( X γ )] over all γ > 0 , satisfies the locality axiom : local perturbations to p ( R,x ) affect I local ( x ′ ) only for nearby x ′ , while their influence vanishes as ∥ x ′ -x ∥ → ∞ .

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 J R ′ ,R ( x γ ) = J R ( x γ ) + J R ′ | R ( x γ ) [32]. Both properties are preserved when we average the Fisher information over p ( X γ | x ) and integrate over γ > 0 , to obtain I local ( x ) .

Figure 2: e ff

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4.3 Feature-Wise Decomposition

We assume the stimulus x is a vector of image features x i (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 I local ( x ) into feature-wise contributions I i ( x ) .

As with the stimulus-wise decomposition, this problem is ill-posed: infinitely many decompositions exist in theory. However, the same axioms constrain the feature-wise decomposition. To satisfy additivity , I i ( x ) must be a linear combination of Fisher diagonal terms: I i ( x ) = 1 2 ∑ j a ij ∫ ∞ 0 E X γ | x [ J jj ( X γ )] dγ . Completeness requires ∑ i a ij = 1 , while positivity enforces a ij ≥ 0 . Finally, to fully specify the weights, a ij , we need to introduce one further axiom, which ensures that the attribution I i ( x ) is zero for irrelevant stimulus features, X i , which are independent of the response, R .

Axiom 5: Insensitivity to irrelevant features. If X i is independent of R , then I i ( x ) = 0 ∀ x ∈ X .

̸

For this axiom to hold, we need to set a ij = δ ij , so that I i ( x ) = 1 2 ∫ ∞ 0 E X γ | x [ J ii ( X γ )] dγ . If, on the contrary, a ij = δ ij , then a neuron's sensitivity to other features (i.e. J jj ( x ) > 0 ) could 'leak over' to make I i ( x ) > 0 even when X i is independent of R , violating the axiom.

5 Results

5.1 Locality

To illustrate the implication of the locality axiom, we analyzed the responses of a model neuron to a one-dimensional stimulus drawn from a Gaussian prior. The neuron'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 x = 6 (Fig. 1A, red).

With gaussian noise, Fisher information scales with f ′ ( x ) 2 , and thus peaked where the tuning curves were steepest (Fig. 1B). Similar qualitative behaviour was observed for I local ( x ) (Fig 1C). Crucially,

Figure 3: data-processing inequality

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the difference in I local ( x ) between the two tuning curves vanished outside the region where they differ, consistent with the locality axiom. In contrast, existing attribution methods (Fig. 1D-G) showed global sensitivity: adding a second peak at x = 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 I local ( x ) 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 x = -1 and x = 1 (Fig. 2A-B, upper panels). To isolate the effect of the prior, we used a simple linear-Gaussian likelihood model: r ∼ N ( x, σ 2 ) . Under this model, the neuron's sensitivity is uniform across all stimuli, so 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 in the stimulus x . With the bimodal prior (Fig. 2B), the posterior is highly sensitive near x = 0 , 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 I local ( x ) reflects this structure, peaking at x = 0 , and decaying elsewhere (Fig. 2C). For the Gaussian prior, where posterior sensitivity is constant, I local ( x ) 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 a Gaussian prior, where the posterior sensitivity is uniform (Fig. 2F).

5.3 Data Processing Inequality

Next we illustrate how I local ( x ) respects the data processing inequality while certain other attribution methods do not. For this, we used a bimodal prior (Fig. 3A) and compared a linear-Gaussian neuron ( r ∼ N ( x, σ 2 r ) ) to a downstream neuron with response r ′ = | r | . Since this transformation is noninvertible, information must be lost. Consistent with the inequality, I local ( x ) decreased at every x (Fig 3C). In contrast, both I SSI and I sp increased at some x and decreased at others, violating the pointwise data processing inequality (Fi 3D-E). I surp, by comparison, respected the inequality (Fig 3F).

5.4 Scaling to high-dimensions

We can use Eqn 20 to write the feature-wise decomposition in terms of the outputs of an unconditional and conditional diffusion model, trained to output ˆ x ( x γ ) = E [ X | x γ ] and ˆ x ( x γ , r ) = E [ X | x γ , r ] ,

respectively:

To obtain a Monte-carlo approximation of this expression, we need to sample from x γ ∼ p ( X γ | x ) , followed by, r ∼ p ( R | x γ ) . Sampling from p ( R | x γ ) is prohibitively expensive (since it requires first sampling from x ∼ p ( x 0 | x γ ) , which requires a full backward pass of the diffusion model). To get around this, we adopt an approximation used by Chung et al. 2023, instead approximating I i ( x ) using samples r ∼ p ( R | ˆ x ( x )) , which can be computed efficiently using one pass through the de-noising network. The integral over γ was approximated numerically with evenly spaced γ (see Appendix C). Since the Fisher decays to zero for large γ , 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 to the total information encoded by a 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 ( w · x + b ) , where A and b are constants, f ( · ) is a sigmoid nonlinearity, and w is a linear filter representing the neuron'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 a baseline, we 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 contribution is scaled by its activation level. This yields characteristic 'blob-like' patterns, with each blob centered on the neuron's RF.

We then used a diffusion model trained on MNIST to estimate the pixel-wise information decomposition, I local ( x ) (Fig 4D, G; additional images are shown in Supp Fig 2). This decomposition revealed that information was concentrated along object edges-regions where the decoded images (i.e. samples form the posterior) are most sensitive to small changes in the presented stimulus (cf. Fig. 2A). Unlike the Fisher information, our measure integrates how both the local sensitivity of neurons (via their RFs) and the statistical structure of the input (captured by the diffusion model), contribute towards the total encoded information.

Later, we investigated the behavior of our information decomposition on a diffusion model trained on natural images, with a 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 I local peaking in regions of high local spatial contrast, around the edges of objects.

6 Discussion

We introduced a principled, information-theoretic measure of neural sensitivity to stimuli. This measure satisfies 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-dimensional 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 a neural population about visual stimuli.

Kong et al. 2024 recently proposed two decompositions of the mutual information between visual stimuli x and text prompts y , I ( x, y ) , which can be efficiently estimated using diffusion models. These were used to 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-wise neural sensitivity measure I ( x ) , which was our goal (Appendix B). Moreover, simply averaging the decompositions of Kong et al. over neural responses does not produce stimulus-wise 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 is not additive, and thus violates a key information theoretic property pointwise.

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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 2015 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 in certain cases, such as at high noise, or where there are more stimulus dimensions than neurons (in which case it returns minus infinity) [1, 31, 13]. Here, we instead identify an exact relation between mutual information and Fisher information with respect to a noise-corrupted stimulus. This opens new potential to test theories of efficient coding of high-dimensional stimuli, and in the presence of noise.

In the future, we will use our 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 we 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-art 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], a 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 as a principled attribution tool for analyzing deep networks-identifying which stimuli, or stimulus features, different units or layers are sensitive to.

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A Proof of locality axiom

Let p ( R,X ) and ˜ p ( R,X ) be two joint distributions, defined over the response, R , and a stimulus, X with infinite support in an unbounded (e.g. X ∈ R d ) or semi-bounded domain (e.g. [0 , ∞ ) d ).

We restrict ˜ p ( R,X ) to only differ from p ( R,X ) locally, in the vicinity of x 0 . Formally, we suppose there exists finite ϵ > 0 and x 0 such that:

where B ϵ ( x 0 ) is the open ball of radius ϵ centered at x 0 .

Recall that our stimulus-wise decomposition of information is defined as:

where X γ = x + √ γZ , Z ∼ N (0 , I ) , and J ( x γ ) = E p ( R | x γ ) [ -∇ 2 x γ log p ( R | x γ ) ] is the Fisher information matrix at x γ .

We will consider the absolute difference between the stimulus-wise information, computed with p ( r, x ) and ˜ p ( r, x ) respectively, | I ( x ) -˜ I ( x ) | . To prove the locality axiom, we separate the integral over γ into two parts as follows,

1

2

∣

∣

∣

∣

0

∣

∣

∣

∣

In sections A.1, and A.2 we show that f ( x, γ h ) and g ( x, γ h ) have the following limiting behavior:

Thus, for any δ > 0 , we can first pick γ h large enough so that g ( x, γ h ) ≤ δ/ 2 , for all x . Then, for fixed γ h , we choose d such that f ( x, γ h ) ≤ δ/ 2 whenever ∥ x -x 0 ∥ 2 ≥ d .

This guarantees that for any δ > 0 , there exists a finite constant d such that,

completing our proof of the locality axiom.

A.1 Integral from γ = 0 to γ = γ h

Here, we prove the limiting behavior of f ( x, γ h ) , as ∥ x -x 0 ∥ 2 →∞ .

We start by writing out the Fisher information as a function of the noise-corrupted stimulus, x γ :

Writing out ˆ x ( x γ ) ≡ E [ X | x γ ] explicitly,

∫

∞

[

]

where

|

I

(

x

)

-

˜

I

(

x

)

|

=

Trace

E

p

(

X

γ

|

x

)

J

(

X

γ

)

-

˜

J

(

X

γ

)

dγ

(24)

where ϕ ( x ) ≡ 1 √ 2 π e -1 2 | x | 2 , and we have separated the integrals in the numerator and denominator into contributions from inside and outside the region B ϵ ( x 0 ) . Taking just the second term in the numerator of Eqn 33,

since, by construction, p ( x ) has finite moments. The same logic also applies to the second term in the denominator of Eqn 33,

Substituting this asymptotic behaviour back into Eqn 33, we have

where the first term only depends on x in the region outside the region B ϵ ( x 0 ) .

Using the same arguments for ˆ x ( x γ , r ) , we can write:

where

only depends on p ( r, x ) in the region x ̸∈ B ϵ ( x 0 ) .

Taking the square, and averaging over p ( r | x γ ) , we have

Note that if R is discrete, the above integral over R is simply replaced by a sum. We then follow the exact same procedure as before, separating the integrals over x into parts that are inside and outside of B ϵ ( x 0 ) , to obtain

where b ( x , x ) is a function that only depends on p ( r, x ) in the region x ̸∈ B ( x ) .

0 γ ϵ 0 Since, by construction ˜ p ( r, x ) = p ( r, x ) in the region x ̸∈ B ϵ ( x 0 ) , we can then write:

where ˜ J ( x γ ) is the Fisher information obtained with ˜ p ( r, x ) .

Finally, integrating from γ = 0 to γ h , we have:

as stated in Eqn 28.

A.2 Integral from γ = γ h to ∞

Next, we show the limiting behaviour of g ( x, γ h ) .

We know from Eqn 42 that for any given γ < ∞ , | Trace( J ( x γ ) -˜ J ( x γ )) |→ 0 as | x γ |→∞ . Thus the maximum of | Trace( J ( x γ ) -˜ J ( x γ )) | must be obtained for some finite x ∗ γ . As a result

with ˜ x ( x γ ) and ˜ x ( x γ , r ) defined as the expectations over X obtained with the perturbed distribution, ˜ p ( R,X ) . Now since, by construction, p ( x ) , p ( x | r ) , ˜ p ( x ) , and ˜ p ( x | r ) all have finite first and second moments, then for finite x ∗ γ all the expectations in the above expression are finite, and we have:

where C < ∞ is some finite constant, independent of x γ .

Integrating over γ and averaging over p ( X γ | x ) gives

as stated in 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 on both the response and stimulus, which can be computed using diffusion models. Their decompositions take the following form:

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To obtain stimulus-wise decompositions from these expressions, that we can compare with our work, we averaged both of these decompositions with respect to p ( R | x ) , to obtain: I Kong, 1 ( x ) ≡ E R | x [ I Kong, 1 ( x, R )] and I Kong, 2 ( x ) ≡ E R | x [ I Kong, 2 ( x, R )] .

Despite the apparent similarity with our proposed decomposition, neither I Kong, 1 ( x ) or I Kong, 2 ( x ) fulfill our axioms, limiting their potential use as principled & interpretable measures of neural sensitivity.

In their paper, Kong et al. show that I Kong, 1 ( x, r ) = log p ( r | x ) p ( r ) . Taking the average over p ( R | x ) , gives: I Kong, 1 ( x ) ≡ E R | x [ I Kong, 1 ( x, r )] = D KL ( p ( R | x ) ∥ p ( R )) . This is identical to the expression for the specific surprise, I surp ( x ) , in the main text (Eqn 3). As shown in the main text (Fig 1E), this measure does not fulfill our locality axiom. This is because I surp ( x ) depends on p ( r ) = ∫ p ( r | x ′ ) p ( r ′ ) dx ′ , which can be influenced by changes to the likelihood or prior with respect to any stimulus, x ′ . Further, following the same prescription as in the main text to obtain a feature-wise decomposition of I Kong, 1 ( x ) , results in feature-wise attributions that can be negative.

To understand the second decomposition, we use Tweedie's law to write:

This differs from our expression, due to the fact that it involves taking the average over p ( R | x ) , rather than p ( R | x γ ) . However, while this difference may seem subtle it has important consequences for the resulting stimulus-wise decomposition.

To see that there are large qualitative differences between I Kong, 2 ( x ) and I local ( x ) we first consider their behavior in the case where p ( R,X ) is jointly gaussian. Here, I local ( x ) is constant across x (Fig 2B, black), reflecting that the fact that the posterior is also independent of x (up to a linear translation). In contrast, I Kong, 2 ( x ) scales quadratically with the distance of x from the mean (similar to I surp ( x ) , Fig 2E).

More importantly, I Kong , 2 ( x ) 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 ′ :

Now when we expand the square we see that the cross-terms don't cancel out . As a result, I Kong, 2 is not linear in log p ( R ′ | R,X γ ) and log p ( R | X γ ) , violating additivity.

This contrasts with the behaviour of I local , which can be expressed as follows:

This equation is linear in log p ( R ′ | R,X γ ) and log p ( R | X γ ) , and thus we can easily confirm that it is additive with respect to repeated measurements.

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Additivity is a 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 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 [19], which consists of 60,000 grayscale images of handwritten digits for training and 10,000 for testing. Each image, originally 28 × 28 pixels, was preprocessed by normalizing to the range [ -1 , 1] and then rescaled to 32 × 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 a 7 × 7 grid, mimicking the retinotopic organization of early visual cortex. Image pixels were mapped to 2D coordinates in visual space, ( x, y ) ∈ [ -1 , 1] 2 , forming a uniform grid. Each neuron's RF was modeled as a 2D Gaussian filter centered at ( x i , y i ) :

where σ = 0 . 1 defines the spatial extent of the RF, and ( x i , y i ) 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 I onto the receptive field weights w i , followed by a sigmoid nonlinearity:

where A = 40 is the amplitude, g = 0 . 4 is the gain and θ = 0 . 9 is the threshold. This yields the mean firing rate of neuron i in response to image I .

To capture neural variability, we modeled spike counts as samples from a Poisson distribution with mean f i .

C.3 Diffusion Models

We trained two denoising diffusion probabilistic models (DDPMs) based on the UNet-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 (0 , I ) . Following the DDPM framework [12], we simulated a forward diffusion process that progressively adds Gaussian noise to an image x 0 , producing a sequence of noisy images { x t } . The process is defined as:

where β t denotes the noise schedule. By recursively applying this equation, one obtains:

Although this differs from the isotropic Gaussian noise model used in the main text, x t = x 0 + √ γZ , the two formulations are equivalent up to reparameterization, with γ = 1 -¯ α t ¯ α t . Thus, both can be used interchangeably to estimate the information decomposition I local ( x ) .

We used a linear noise schedule with β t increasing from 1 × 10 -4 to 0 . 02 over 1,000 time steps. Given a noisy image x t , the model was trained to predict the noise component ˆ z t ( x t ) , which can be used to reconstruct an estimate of the clean image:

A 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:

We used these models to estimate the pixel-wise information decomposition described in the main text (Eq. 9). To approximate the integral over γ , we applied additive Gaussian noise according to the diffusion schedule and evaluated ˆ x ( x t ) and ˆ x ( x t , r ) over 24 diffusion steps (indexed as [40:40:1000] ) using 50 noisy samples of x t and r at each t . On average, approximating I local ( x ) for one image took approximately 1 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 [11] for distributed training on 7 NVIDIA A100 GPUs (40 GB VRAM).

The training procedure for both models was as follows:

Optimizer: AdamW with learning rate 1 × 10 -4 , β 1 = 0 . 9 , β 2 = 0 . 999

Learning rate scheduler: Cosine schedule with warmup (500 steps)

Batch size: 256

Training duration: 150 epochs

Loss function: Mean squared error (MSE) between predicted and true noise

Precision: Mixed precision training with bfloat16

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 I local ( x ) following identical preprocessing, stimulus presentation, and estimation steps as in the main analysis.

C.6 Decomposition for natural images

To demonstrate that our approach generalizes to more complex and biologically realistic neural encoder models as well as natural images, we trained the diffusion models on 60,000 64×64 natural images from Hugging Face's Tiny ImageNet dataset and trained a 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 7×7 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, I local ( 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.

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C.7 Demonstration of locality for MNIST stimuli

To further illustrate the locality properties of the information decomposition, we examined how a small, spatially localized perturbation to a single receptive field affects I local ( x ) and I surp ( x ) . We used the same linear-nonlinear-Poisson (LNP) model described in the main text, with receptive fields arranged across the visual field (Supplementary Figure 8B). In the perturbed condition, we 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 I local ( x ) and I surp ( x ) between the perturbed and unperturbed conditions. As shown in Figure 8D-E, I local ( x ) remained confined to the vicinity of the stimulus and the affected receptive field, reflecting its inherently local nature. In contrast, I surp ( x ) 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, 7 and 8 can be found at neural-info-decomp.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\n*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.\n\nQUESTIONS:\n1. Why other decomposition methods were not used to study the decomposition in the MNIST dataset?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-16"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\n## 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?\n\nQUESTIONS:\n- 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.\n\nLIMITATIONS:\nSome 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.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\n***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.\n\nQUESTIONS:\n1. 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 ？\n\nLIMITATIONS:\nsee weakness and questions", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}

comments: 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.

STRENGTHS AND WEAKNESSES:
Strength: axioms are well-motivated for extending the intuitive properties of Fisher information and mutual information

Strength: a method is given to approximately compute the information-theoretic quantity defined using extent computational tools

Weakness: experiments are conducted entirely on rate-coded neurons rather than spiking ones.
Weakness: related to the above, experiments are on model neurons rather than on a real dataset

QUESTIONS:
In lines 59-60, the statement of Axiom 1, what stimulus distribution does the expectation integrate over?

Since 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?

What 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
TITLE: Official Review
CLARITY: 3
IMPACT: 4
ORIGINALITY: 4
DATE: 2025-07-02