Title: Jacobian-Based Interpretation of Nonlinear Neural Encoding Model
Venue: NeurIPS 2025
Accepted: yes
Keywords: fMRI

neural encoding model

nonlinearity

Jacobian matrix
Abstract: In recent years, the alignment between artificial neural network (ANN) embeddings and blood oxygenation level dependent (BOLD) responses in functional magnetic resonance imaging (fMRI) via neural encoding models has significantly advanced research on neural representation mechanisms and interpretability in the brain. However, these approaches remain limited in characterizing the brain’s inherently nonlinear response properties. To address this, we propose the Jacobian-based Nonlinearity Evaluation (JNE), an interpretability metric for nonlinear neural encoding models. JNE quantifies nonlinearity by statistically measuring the dispersion of local linear mappings (Jacobians) from model representations to predicted BOLD responses, thereby approximating the nonlinearity of BOLD signals. Centered on proposing JNE as a novel interpretability metric, we validated its effectiveness through controlled simulation experiments on various activation functions and network architectures, and further verified it on real fMRI data, demonstrating a hierarchical progression of nonlinear characteristics from primary to higher-order visual cortices, consistent with established cortical organization. We further extended JNE with Sample-Specificity (JNE-SS), revealing stimulus-selective nonlinear response patterns in functionally specialized brain regions. As the first interpretability metric for quantifying nonlinear responses, JNE provides new insights into brain information processing. Code available at https://github.com/Gaitxh/JNE.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces the Jacobian-based nonlinearity evaluation (JNE), a metric for quantifying the nonlinearity of neural encoding models.  The authors first argue that the common practice of comparing the predictive accuracy ($R^2$) of linear versus nonlinear models is an unreliable method for assessing nonlinearity, especially when using pre-trained features.  The proposed JNE metric quantifies the statistical variability of a model's local linear mappings (its Jacobian matrices) across a diverse set of input stimuli.  Applying this metric to fMRI data reveals two primary contributions: first, it provides quantitative evidence for a hierarchical gradient of increasing nonlinearity from the primary visual cortex to higher-order visual areas.  Second, it demonstrates that this nonlinearity is stimulus-selective, with regions such as the fusiform face area (FFA) and the parahippocampal place area (PPA) showing the most pronounced nonlinear responses to their preferred stimulus categories (faces and scenes, respectively), thereby linking a region's computational properties to its known functional specialization.\n\nSTRENGTHS AND WEAKNESSES:\nThe work is well-executed and addresses a timely and important problem\n\nA major strength is the empirical demonstration that comparing the predictive accuracy ($R^2$) of linear and nonlinear models is an unreliable method for assessing nonlinearity. By showing that both model types achieve nearly identical performance when using powerful features from foundation models like CLIP-ViT, the authors effectively invalidate a common heuristic and establish a clear need for a new approach.\n\nThe JNE metric is conceptually novel. While the Jacobian matrix is a standard mathematical tool, the proposal to use the statistical dispersion of Jacobians across a sample population as a direct, non-parametric measure of nonlinearity is an insightful formulation.  Its simplicity is noteworthy when compared to more complex learnable frameworks like LinBridge, which require training an auxiliary model to decompose the Jacobian.\n\nThe JNE metric is built on the principle that a truly linear system must have an input-invariant Jacobian. The authors then rigorously validate this premise through a series of well-designed simulation experiments. The network-level validation further confirms that JNE can reliably detect and quantify nonlinearity in more complex, multi-layer systems.\n\nThe introduction clearly articulates the problem and motivates the need for the JNE metric. The four-step computational workflow of JNE is presented with mathematical precision and is well-supported by a clear diagram (Fig. 2a,c).\n\nThe most critical limitation is that the JNE metric is applied to a model of the fMRI BOLD signal, not directly to neural activity. The BOLD signal is an indirect and sluggish hemodynamic measure that has its complex nonlinear relationship with the underlying neural events. These nonlinearities can be vascular in origin (e.g., blood flow refractory effects) or related to neural phenomena like adaptation, and are highly sensitive to experimental design, such as stimulus timing.\n\nThe paper notes that JNE lacks a defined threshold to distinguish a \"linear\" from a \"nonlinear\" region in absolute terms. This restricts the metric's utility to relative comparisons. It could, however, be quantified with simulation experiments and an ROC analysis, at least in an idealized setting.\n\nWhile the method is novel, the neuroscientific findings could be viewed as incremental. The discussion of limitations, including the critical BOLD confound, should not be in the appendix, but this can be fixed in the camera-ready version.\n\nQUESTIONS:\nI suggest that you generate a large number of simulated linear and nonlinear networks, calculate the JNE score for each network and\ntreat \"linear\" and \"nonlinear\" as the two ground-truth classes and the JNE score as the classifier's output. Could you then plot an ROC curve to see how well the JNE score distinguishes between the two classes across all possible thresholds? The AUC would quantify the metric's discriminative power in this idealized environment.\n\nLIMITATIONS:\nThe discussion of limitations, including the critical BOLD confound, should not be in the appendix, but this can be fixed in the camera-ready version.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose a non-linearity metric (Jacobian Non-linearity Evaluation – JNE) to characterize the non-linear response properties of the brain. The JNE for mapping Clip-VIT output activations to brain responses (from the Natural Scenes Dataset) showed increasing JNE scores from primary visual cortex, to intermediate (V2-V4), to higher level visual cortex (e.g., EBA, FFA, PPA, etc.). Moreover, JNE scores were higher for preferred categories in higher visual cortex (e.g., images of outdoor scenes evoked stronger JNE in PPA and RSC; images of people and faces evoked stronger JNE in FFA and EBA). The authors conclude that JNE characterizes the degree of non-linearity in nonlinear neural encoding models, and describes the degree of nonlinearity across different brain regions.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:  \n- Well-written, clear presentation. \n- Well-motivated, strong methodological approach. \n- Intriguing results. \n\nWeaknesses:  \n-  The authors aim to understand non-linearities in biological neural processing, but the methods measure non-linearities in the mapping between spaces (model-output-space and voxel-space), and it’s not clear the measure can speak directly to non-linearities in brain responses.  \n- The work is not fully situated in the visual neuroscience literature, and what we already know about non-linear hierarchical processing in visual cortex, and hierarchical correspondence between models and brains. Authors need to clarify what have we learned about biological vision that we didn’t already know from prior work…It can’t be that non-linearities increase as you progress in the visual processing stream (already established in prior neurophysiology work; already inferred in hierarchical correspondence between models and brains). The authors can argue that JNE provides the first direct measurement of this putative model/brain correspondence, but then it needs to be more clearly demonstrated that JNE scores are directly attributable to properties of the target-space (brain responses). If the JNE scores are model-dependent, or model-layer-dependent, then the JNE scores can only be understood as a measure of the non-linearity in the mapping from model-layer-A to brain-region-B (a property of the mapping between two spaces).\n\nQUESTIONS:\n1)\tThe JNE score measures the degree of non-linearity in the “mapping” from model-space to voxel-space. Is it the case that high JNE scores can be observed if-an-only-if the target space exhibits high non-linearity? If this is not a provable constraint on JNE scores, we should be cautious about interpreting the mapping-JNE scores as a measure of the encoding-target-space-nonlinearity. \n2)\tOn this same topic, I’m not sure what we can/should conclude about the brain regions from these JNE scores. To illustrate my concern, let’s pretend that some Clip-ViT output units respond more to “profile-faces” and others to “frontal views of faces”, and we are predicting FFA responses (face-selective cortex). Given that feature-response preference, I assume the JNE score would be higher if you presented a diverse image set (mixture of profile vs. frontal views), and lower if you presented a homogenous set (frontal views only). The brain region itself has “the same non-linearity” no matter what image set you present, but the JNE scores would vary. \n3)\tOne more question on interpretation. The fact that the linear and non-linear encoders show equivalent performance mapping between CLIP-output and brain responses, it really seems like JNE scores are capturing the non-linearity of the mapping (which itself seems like it would depend on the variability of the dataset along dimensions captured by the encoder; see previous question), rather than non-linearity of the brain-response per se. \n4)\tI found myself wondering whether/how the JNE scores would vary as a function of model-layer. Typically earlier model layers show a stronger alignment with primary and intermediate visual cortex, whereas later model layers show stronger alignment with higher visual cortex. To what extent do JNE scores depend on the model-layer that feeds into the encoding model? Is the low JNE score for primary visual cortex due to the mismatch between model-space (CLIP-output activations)?\n5) In theory, with a powerful enough encoder, it should be possible to map even an early model layer to higher-visual cortex, presumably requiring greater non-linearity in the mapping. Does this mean higher JNE scores? If so, given that the target brain regions aren’t changing, that variability would imply that JNE is a measure of the non-linearity in the mapping, not the target brain region.\n6)\tIt would be helpful if the authors could situate this work more fully in the context of what is known about the biological visual processing hierarchy, it’s known increase in complexity (particular selectivity profiles + invariances), and non-linearity in visual processing (providing some clarity on how this is measured and quantified in biological systems). It would also be helpful to relate the current approach/findings to what has been demonstrated in prior work, particularly work showing a hierarchical correspondence between model activations and brain responses (early model layers predicting early brain regions better, later model layers predicting later model layers better), which has also been proposed to reflect comparable non-linear stages in artificial and biological visual systems. What new insights do we gain from the current approach?\n7)\tIf the goal is to characterize properties of the brain, independent of the encoding model, it seems important to present results for encoding models mapping multiple model layers (from early to late to output layers) and multiple models (various architectures, visual diets, and objectives) to ensure that the conclusions about the brain are general/consistent across layers and models. My guess is that there will be significant variability based on layer (and possibly model depending on the suite drawn from), and that more nuanced conclusions regarding non-linearities in the mapping from model-space to brain-space are required.\n8)\tIn the Discussion the authors propose that the present results support a bottom-up processing mechanism in the brain’s visual system. I’m not sure I understand exactly what the authors mean here, and what the alternative is. The biological visual system is known to have ubiquitous feed-forward, feed-back, and skip connections, and fMRI responses are too sluggish to isolate purely feed-forward processing, so the signal being modeled here is necessarily a result of both feed-forward and feed-back processing. If the authors mean that the input signal (image) is the primary driver of responses along the ventral processing stream, I would agree with that, but I’m not sure the present results speak to that much. \n\nI want to emphasize that I think this is very solid work and that the results are intriguing enough for me to think the authors are onto something here. The hierarchical progression of JNE scores with brain-depth, and the category-dependent JNE scores (faces in FFA; scenes in PPA) are very interesting results which suggest the measure has potential to provide some new insight. I’m just not sure we can safely interpret the JNE values as a measure of non-linear processing in the target brain region. If not, if they are a measure of the non-linearity in mapping model-layer-A to brain-region-B, then I’m not sure exactly what they tell us about biological visual processing. In that case, I think we need more results to learn something new, some interesting pattern across model layers or across model architectures or models trained on different objectives. And then we’d need a framework for thinking about what “non-linearity in the mapping between spaces” means. Does higher JNE mean two spaces are “less natively aligned” and thus less similar? Does the perfect “model-layer for brain region B” show low JNE or high JNE when mapped to that region? (maybe that could be answered by aligning human-to-human?).\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a measure (called JNE) to functionally compare output dimensions that are nonlinear functions of the input, focusing on nonlinear encoding models of biological sensory neurons. Using simple simulations, they show that JNE increases as model nonlinearities increase, and they also show JNE computed on models of visual cortical areas reflects hierarchical (and increasingly nonlinear) computations along visual areas.\n\nSTRENGTHS AND WEAKNESSES:\n### Strength\n\n- Technically sound derivation; tractable and well-motivated; reasonable evaluation on simulations and hierarchy\n- Generally well-written, logical flow, readable\n- Addresses an important topic—quantifying single-unit nonlinearity with a scalar, model-agnostic metric\n- The proposed approach is novel in that it aims to provide a direct measure of nonlinearity that can be applied across different models, filling a gap not directly addressed by model comparison or geometry-based approaches.\n\n### Weaknesses\n\n- There are some conceptual issues (scale sensitivity, isoresponse inconsistency); some design choices are not clear (e.g. why L1 norm?)\n- Some unclear phrasing (\"stimulus sensitivity\", “parameter fluctuations”); some definitions and the relation of different terms could be improved (e.g., JNE vs JNE-SS).\n- Practical utility depends on resolving open issues (input distribution characteristics, scale sensitivity). The impact/significance of the method depends on how these are addressed.\n- The interpretability of the sample-specific MoM analysis is unclear; some stimulus clusters with similar content (e.g., faces) elicit inconsistent responses in expected regions (e.g., FFA), raising questions about the robustness or specificity of the metric.\n- Overlap with existing conceptual directions (intrinsic dimensionality, representational geometry); novelty could be better articulated relative to prior work.\n\nQUESTIONS:\n### Below are the questions for which I am happy to increase the rating given that they are convincingly addressed:\n\n1\\. How is the scale-dependence of the method justified? The current formulation conflates response gain with degree of nonlinearity, which is problematic: due to its scale dependence, the metric does not accurately reflect the intrinsic dimensionality or structure of curvature. This can yield identical scores for functionally distinct mappings through arbitrary rescaling—conflating response gain with actual nonlinearity. To illustrate this, consider two functions with inputs $x_1, x_2 \\sim \\mathcal{N}(0, 1)$:\n\n* $f_1(x_1, x_2) = a x_1^2 \\Rightarrow \\nabla f_1 = [2a x_1,\\ 0]$\n* $f_2(x_1, x_2) = x_1^2 + x_2^2 \\Rightarrow \\nabla f_2 = [2x_1,\\ 2x_2]$\n\nThen the JNE (defined as the standard deviation of the L1 norm of the Jacobian across inputs) becomes:\n\n* $\\text{JNE}(f_1) = \\mathrm{std}(|2a x_1|) = 2a \\cdot \\mathrm{std}(|x_1|)$\n* $\\text{JNE}(f_2) = \\mathrm{std}(|2x_1| + |2x_2|) = 2 \\cdot \\mathrm{std}(|x_1| + |x_2|)$\n\nMatching the JNEs yields: $ 2a \\cdot \\mathrm{std}(|x_1|) = 2 \\cdot \\mathrm{std}(|x_1| + |x_2|) \\Rightarrow a = \\frac{\\mathrm{std}(|x_1| + |x_2|)}{\\mathrm{std}(|x_1|)}$\n\nThis shows that **a 1D nonlinear function** can be made to appear **equally nonlinear** as a 2D one by simply amplifying its output. I am not convinced that response gain should be entangled with nonlinearity in a single metric, as they capture distinct computational aspects. This could be mitigated by standardizing Jacobian entries across inputs (e.g., subtracting mean and dividing by std), rather than just centering them. Why did the authors choose not to do this?\n\n2\\. Why use L1 norm and not L2? The proposed Jacobian-based metric aims to quantify a neuron's nonlinearity by aggregating input-specific gradients. However, for neurons with symmetric and rotationally invariant tuning profiles (i.e., invariance under rotations of the input space), the use of the L1 norm introduces a directional bias that breaks isoresponse symmetry. For example, consider a neuron with a 2D Gaussian tuning function: $f(x_1, x_2) = \\exp\\left(-\\frac{x_1^2 + x_2^2}{2\\sigma^2}\\right)$. The gradient is $\\nabla f = -\\frac{f(x)}{\\sigma^2} [x_1, x_2]$, and the **L2 norm** becomes: $\\lVert \\nabla f \\lVert_2 = \\frac{f(x)}{\\sigma^2} \\sqrt{x_1^2 + x_2^2}$. But since $x_1^2 + x_2^2 = -2\\sigma^2 \\log f(x)$, this yields: $\\lVert \\nabla f \\lVert_2 = \\sqrt{\\frac{-2 f(x)^2 \\log f(x)}{\\sigma^2}}$, which depends only on the function value $f(x)$ — and thus is constant along isoresponse contours. In contrast, the **L1 norm** is: $\\lVert\\nabla f\\lVert_1 = \\frac{f(x)}{\\sigma^2} (|x_1| + |x_2|)$, which varies across directions even when $f(x)$ is constant (i.e., on the same contour). In other words, equally nonlinear points are scored differently simply due to orientation. Using L2 could address this specific issue, and I wonder why the authors chose L1 norm.\n\n3\\. Why is the method \"scale-normalized\" as mentioned in the paper? which operation/step is related to this scale normalization? and why does that mean “the magnitude of standard deviation directly encodes biological sensitivity differences in neural responses”?\n\n4\\. I like that this approach is architecture-agnostic and can be applied on any model as long as the Jacobian matrix can be computed, but its dependence on the input introduces other challenges/questions: How and to what degree does JNE depend on the input distribution? How many inputs are needed? how to select the inputs? how does the number of inputs (and their selection) change with different number of output dimensions, and their degree of nonlinearity?\n\n5\\. While the method recovers the expected hierarchical trend in visual areas—consistent with prior findings—it is not immediately clear to me what new insights it enables beyond this. What does the metric reveal beyond \"this area/neuron is more/less nonlinear than another\"? The sample-specific analysis seems to hint at deeper insights, but ultimately still confirms known properties (e.g., early areas encode simple features). Could the authors clarify the kinds of new questions or discoveries this metric enables?\n\n6\\. Regarding the sample-specific analysis, I am not sure how much one can rely on or interpret these results. For instance, you state that “stimuli involving people or faces induce pronounced nonlinear fluctuations in FFA and EBA.” Looking at cluster 14 in Figure 12, this seems to hold—there are faces in the images, and FFA shows high MoM in the heatmap (Figure 8c). However, cluster 9 also contains many face stimuli, yet FFA does not exhibit similarly high MoM. Could the authors clarify this discrepancy or comment on the robustness of these observations?\n\n---\n\n### Additional questions\n1\\. The interpretation of JNE in terms of “stimulus sensitivity” was unclear to me. The authors write that \"stimulus-sensitive regions (e.g., early visual cortex) exhibit greater parameter fluctuations, which mathematically aligns with the sample-dependent characteristics of nonlinear mechanisms\" seemingly equating higher Jacobian variability (across input dimensions) and increased JNE. Could the authors clarify what does “parameter fluctuation” mean, in this context? and how does that “mathematically align with the sample-dependent characteristics of nonlinear mechanisms”?\n\n2\\. In figure 3d,e combination 2 and 3 have different JNE values, and you do mention it as an interesting observation but I could not find an explanation of why this happens. Intuitively in both cases you have linear → nonlinear → linear, and it is not clear to me why they should have different degrees of nonlinearity.\n\n3\\. In figure 7c, the caption says: where entry $(i, j)=1$ that means cortex i exhibits significantly lower JNE than cortex j. This means for instance HVC ($i=3$) exhibits significantly lower JNE than PVC ($j=1$). Shouldn’t it be the other way around? i.e. $(i, j)=1$ indicates statistically significantly “higher” as opposed to “lower”?\n\n4\\. What do the colors in figure 7d correspond to? Unfortunately, I did not get that from the caption. Is this a voxel-specific coloring or the voxels are based on the area label and the color is simply coming form the statistical test of difference between the areas? If it is the latter I am not sure what is the contribution of this figure exactly.\n\n5\\. I found the appendix defining JNE-SS a bit of an overkill. Isn’t JNE simply the square-root of JNE-SS averaged across stimuli? Changing the order to define JNE-SS first and then getting to JNE seems more intuitive to me, simply because JNE is an aggregate measure of JNE-SS, and maybe it could help simplify the definition/explanation.\n\nLIMITATIONS:\nSome limitations are acknowledged by the authors and could serve as directions for future work. However, key limitations/questions—such as the method’s sensitivity to scale, and dependence on the input distribution were not discussed. These aspects are important for interpreting the utility and generalizability of the proposed metric and it is important to communicate them.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe key contribution of this paper is to provide a new metric for quantifying non-linearity in representations. In particular, they refer to this metric as Jacobian nonlinearity evaluation index (JNE). It measures how the Jacobian of the mapping from inputs to outputs changes over the input over—for a linear model this should be 0, while for a non-linear model we should see higher values for higher sensitivity to the space of inputs being considered. They apply this metric to study the amount of non-linearity in visual representations in different parts of the human visual cortex, by mapping embeddings of images to their voxel responses. \n\nAligning with previous results, they find that high-order visual orders have the most non-linearity, and that nonlinearity is hierarchically organized in the visual cortex. Separately, they also compute sample selectivity of representations in different regions and again consistent with literature they find that regions like FFA show pronounced non-linearity in response to faces (which they are known to be encoding).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. I absolutely love the figures and how the paper is written, everything is very clean and well-organized.\n2. The metric is interesting, it is a novel way of quantifying non-linear between ANNs and brain encoding that goes beyond prediction performance.\n3. Such mapping is widely studied in the human neuroscience community across modalities: audio, vision, etc and so this metric could be influential in extracting similar information about encodings in other parts of the brain.\n\nWeaknesses:\n1. This is minor but the limitation section should really go into the main text, and I'd urge the authors to do so if the paper gets accepted (in which case they will have an extra page).\n2. Computing the Jacobian for a large model for various values of the input is a computationally expensive operation (as the authors detail in the limitations). The computation really depends on the input-output dimensionality, and size of sample space, which may restrict applicability. \n3. A crucial part of computing this metric is sampling the right set and number of inputs over which the Jacobian is computed. This hasn't really been explored in the paper as of yet.\n\nQUESTIONS:\nFollowing up on the weaknesses listed above:\n\n1. I would like to know computational costs for these experiments to get a sense of how feasible they are to be used across domains.\n2. While for the real experiments, it makes sense to stick to the data collected, it would be nice to know how sensitive the metric is to input samples in the simulations in order to strongly trust the real data results. \n3. I am also curious about the choice of L1 norm here to measure deviation from the mean, and would like more clarification on that.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}