We can appreciate that a cat is more similar to a dog than to a truck. The combined activity of millions of neurons in the brain somehow captures these everyday similarities, and this activity can be measured using imaging techniques such as functional magnetic resonance imaging (fMRI). However, fMRI scanners are not particularly precise – they average together the responses of many thousands of neurons over several seconds, which provides a blurry snapshot of brain activity. Nevertheless, the pattern of activity measured when viewing a photograph of a cat is more similar to that seen when viewing a picture of a dog than a picture of a truck. This tells us a lot about how the brain codes information, as only certain coding methods would allow fMRI to capture these similarities given the technique’s limitations.

There are many different models that attempt to describe how the brain codes similarity relations. Some models use the principle of neural networks, in which neurons can be considered as arranged into interconnected layers. In such models, neurons transmit information from one layer to the next.

By investigating which models are consistent with fMRI’s ability to capture similarity relations, Guest and Love have found that certain neural network models are plausible accounts of how the brain represents and processes information. These models include the deep learning networks that contain many layers of neurons and are popularly used in artificial intelligence. Other modeling approaches do not account for the ability of fMRI to capture similarity relations.

As neural networks become deeper with more layers, they should be less readily understood using fMRI: as the number of layers increases, the representations of objects with similarities (for example, cats and dogs) become more unrelated. One question that requires further investigation is whether this finding explains why certain parts of the brain are more difficult to image.

Neuroimaging and especially functional magnetic resonance imaging (fMRI) has come a long way since the first experiments in the early 1990s. These impressive findings are curious in light of fMRI’s limitations. The blood-oxygen-level-dependent (BOLD) response measured by fMRI is a noisy and indirect measure of neural activity (Logothetis, 2002, 2008; O'Herron et al., 2016) from which researchers try to infer neural function.

The BOLD response trails neural activity by 2 s, peaks at 5 to 6 s, and returns to baseline around 10 s, whereas neural activity occurs on the order of milliseconds and can be brief (Huettel et al., 2009). In terms of spatial resolution, the BOLD response may spill over millimeters away from neural activity due to contributions from venous signals (Turner, 2002). Likewise, differences in BOLD response can arise from incidental differences in the vascular properties of brain regions (Ances et al., 2009). Such sources of noise can potentially imply differences in neural activity in regions where there should not be.

The data acquisition process itself places limits on fMRI measurement. Motion artefacts (e.g., head movements by human subjects) and non-uniformity in the magnetic field reduce data quality. In analysis, three-dimensional images are constructed from slices acquired at slightly different times. Once collected, fMRI data are typically smoothed during analyses (Carp, 2012). All these factors place limits on what fMRI can measure.

Despite these weaknesses, fMRI has proved to be an incredibly useful tool. For example, we now know that basic cognitive processes involved in language (Binder et al., 1997) and working memory (Pessoa et al., 2002) are distributed throughout the cortex. Such findings challenged notions that cognitive faculties are in a one-to-one correspondence with brain regions.

Advances in data analysis have increased what can be inferred by fMRI (De Martino et al., 2008). One of these advances is multivariate pattern analysis (MVPA), which decodes a pattern of neural activity in order to assess the information contained within (Cox and Savoy, 2003). Rather than computing univariate statistical contrasts, such as comparing overall BOLD activity for a region when a face or house stimulus is shown, MVPA takes voxel patterns into account.

Using MVPA, so-called ‘mind reading’ can be carried out — specific brain states can be decoded given fMRI activity (Norman et al., 2006), revealing cortical representation and organization in impressive detail. For example, using these analysis techniques paired with fMRI we can know whether a participant is being deceitful in a game (Davatzikos et al., 2005), and we can determine whether a participant is reading an ambiguous sentence as well as infer the semantic category of a word they are reading (Mitchell et al., 2004).

Representational similarity analysis (RSA), another multivariate technique, is particularly suited to examining representational structure (Kriegeskorte et al., 2008; Kriegeskorte, 2009). We will focus on RSA later in this contribution, so we will consider this technique in some detail. RSA directly compares the similarity (e.g., by using correlation) of brain activity arising from the presentation of different stimuli. For example, the neural activity arising from viewing a robin and sparrow may be more similar to each other than between a robin and a penguin.

These pairwise neural similarities can be compared to those predicted by a particular theoretical model to determine correspondences. For example, Mack et al. (2013) identified brain regions where the neural similarity structure corresponded to that of a cognitive model of human categorization, which was useful in inferring the function of various brain regions. The neural similarities themselves can be visualized by applying multidimensional scaling to further understand the properties of the space (Davis et al., 2014). RSA has been useful in a number of other endeavors, such as understanding the role of various brain areas in reinstating past experiences (Tompary et al., 2016; Mack and Preston, 2016).

Given fMRI’s limitations in measuring neural activity, one might ask how it is possible for methods like RSA to be successful. The BOLD response is temporally and spatially imprecise, yet it appears that researchers can infer general properties of neural representations that link sensibly to stimulus and behavior. The neural code must have certain properties for this state of affairs to hold. What kinds of models or computations are consistent with the success of fMRI? If the brain is a computing device, it would have to be of a particular type for fMRI to be useful given its limitations in measuring neural activity.

For fMRI to recover neural similarity spaces, the neural code must display certain properties. Firstly, the neural code cannot be so fine-grained that fMRI’s temporal and spatial resolution limitations make it impossible to resolve representational differences. Second, a notion of functional smoothness, which we will introduce and define, must also be partially satisfied.

Voxel inhomogeneity across space and time

The BOLD response summates neural activity over space and time, which places hard limits on what fMRI can measure. To make an analogy, $3+5$ and $6+2$ both equal $8$ through different routes. If different ‘routes’ of neural activity are consequential to the neural code and are summated in the BOLD response, then fMRI will be blind to representational differences.

To capture representational differences, voxel response must be inhomogeneous both between voxels and within a voxel across time. Consider the fMRI analogues shown in Figure 1; paralleling neurons with pixels and voxels with the squares on the superimposed grid. The top-left image depicts neural activity that smoothly varies such that the transitions from red to yellow occur in progressive increments. Summating within a square, i.e., a voxel, will not dramatically alter the high-level view of a smooth transition from red to yellow (bottom-left image). Voxel response is inhomogeneous, which would allow decoding by fMRI (cf. Kamitani and Tong, 2005; Alink et al., 2013). Altering the grid (i.e., voxel) size will not have a dramatic impact on the results as long as the square does not become so large as to subsume most of the pixels (i.e., neurons). This result is in line with basic concepts from information theory, such as the Nyquist-Shannon sampling theorem. The key is that the red and yellow pixels/neurons are topologically organized: their relationship to each other is for all intents and purposes invariant to the granularity of the squares/voxels (for more details see: Chaimow et al., 2011; Freeman et al., 2011; Swisher et al., 2010).

Figure 1

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In contrast, the center-top image in Figure 1 involves dramatic representational changes within voxel. Each voxel (square in the grid), in this case, will produce a homogenous orange color when its contents are summated. Thus, summating the contents of a voxel in this case obliterates the representational content: red and yellow; returning instead squares/voxels that are all the same uniform color: orange. This failure is due to sampling limits that could be addressed by smaller voxels (see rightmost column). Unfortunately, arbitrarily small voxels with high sampling rates is not a luxury afforded to fMRI.

The success of fMRI given its sampling limits is consistent with proposed neural coding schemes, such as population coding (Averbeck et al., 2006; Panzeri et al., 2015; Pouget et al., 2000) in cases where neurons with similar tunings spatially cluster (e.g., Maunsell and Van Essen, 1983). In population coding, neurons jointly contribute to represent a stimulus in much the same way as pixels were contributing to represent different colors in the leftmost column of Figure 1. When this inhomogeneity breaks down, similarity structures should be difficult to recover using fMRI. Indeed, a recent study with macaque monkeys which considered both single-cell and fMRI measures supports this viewpoint — stimulus aspects which were poorly spatially clustered in terms of single cell selectivity were harder to decode from the BOLD response (Dubois et al., 2015).

The same principles extend from the spatial to the temporal domain. The BOLD response will be blind to the differences between representations to the extent that the brain relies on the precise timing of neural activity to code information. For example, in burstiness coding, neural representations are distinguished from one another not by their average firing rate but by the variance of their activity (Fano, 1947; Katz, 1996). Under this coding scheme, more intense stimulus values are represented by burstier units, not units that fire more overall. Neural similarity is not recoverable by fMRI under a burstiness coding scheme. Because the BOLD signal roughly summates through time (Boynton et al., 1996), firing events will sum together to the same number irrespective of their burstiness.

Likewise, BOLD activity may be a composite of synchronized activity at multiple frequencies. Although gamma-band local field potential is most associated with BOLD response, oscillations at other frequency bands may also contribute to the BOLD response (Magri et al., 2012; Scheeringa et al., 2011). If so, fMRI would fail to distinguish between representational states that are differentiated by the balance of contributions across bands, much like the arithmetic example at the start of this subsection in which different addends yield the same sum. As before, basic concepts in information theory, such as the Nyquist-Shannon sampling theorem, imply that temporally demanding coding schemes will be invisible to fMRI (cf. Nevado et al., 2004).

The success of fMRI does not imply that the brain does not utilize precise timing information, but it does mean that such temporally demanding coding schemes cannot be the full story given fMRI’s successes in revealing neural representations. Instead, the neural code must include in its mixture at least some coding schemes that are consistent with fMRI's successes. For example, rate coding (Adrian and Zotterman, 1926) in which the frequency at which neurons fire is a function of the intensity of a stimulus is consistent with the success of fMRI because changes in firing rate for a population of cells should be recoverable by fMRI as more blood flows to more active cells (O'Herron et al., 2016).

These examples make clear that the neural code must be somewhat spatially and temporally smooth with respect to neural activity (which is several orders of magnitude smaller than voxels) for fMRI to be successful. Whatever is happening in the roughly one million neurons within a voxel (Huettel et al., 2009) through time is being partially reflected by the BOLD summation, which would not be the case if each neuron was computing something dramatically different for in-depth discussion, see: (Kriegeskorte et al., 2010).

In this section, we consider whether neural networks with random weights are consistent with the success of fMRI given its limitations in measuring neural activity. Simulations in the next section revisit these issues through the lens of a deep learning model trained to classify photographs of real-world categories, such as moped, tiger, guitar, robin, etc.

Each simulation is analogous to performing fMRI on the candidate neural code. These simple simulations answer whether in principle neural similarity can be recovered from fMRI data taken from certain neural coding schemes. Stimuli are presented to a model while its internal representations are measured by a simulated fMRI scanner.

The methods were as follows. The stimuli consist of 100-dimensional vectors that were distortions of an underlying prototype. As noise is added to the prototype and the distortion level increases, the neural similarity (measured using Pearson’s correlation coefficient $\rho$) between the prototype and its member should decrease. The question is whether we can recover this change in neural similarity in our simulated fMRI scanner.

First, each fully-connected network was initialized to random weights drawn from a Gaussian distribution ($\mu =0,\sigma =1$). Then, a prototype was created from 100 draws from a Gaussian distribution ($\mu =0,\sigma =1$). Nineteen distortions of the prototype were created by adding levels of Gaussian noise with increasing standard deviation ($\sigma ={\sigma }_{prev}+0.05$) to the prototype. Finally, each item was re-normalized and mean centered, so that $\mu =0$ and $\sigma =1$ regardless of the level of distortion. This procedure was repeated for 100 random networks. In the simulations that follow, the models considered involved some portion of the randomly-initialized 8-layer network ($8\times {100}^2$ weights).

The coding schemes that follow are important components in artificial neural network models. The order of presentation is from the most basic components to more complex configurations of networks. To foreshadow the results shown in Figure 2, fMRI can recover the similarity structure for all of these models to varying degrees with the simpler models faring better than the more complex models.

Figure 2

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Matrix multiplication coding

This model performs more sophisticated computations on the input stimuli. In line with early connectionism and Rescorla-Wagner modeling of conditioning, this model receives an input vector and performs matrix multiplication on it, i.e., computes the weighted sums of the inputs to pass on to the output layer (Knapp and Anderson, 1984; Rescorla and Wagner, 1972). These simple one-layer neural networks can be surprisingly powerful and account for a range of complex behavioral findings (Ramscar et al., 2013). As we will see in later subsections, when a non-linearity is added (e.g., ${\displaystyle \mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}}$), one-layer networks can be stacked on one another to build deep networks.

This neural coding scheme takes an input stimulus (e.g., an image of a dog) and multiplies it by a weight matrix to create an internal representation, as shown in Figure 3. Interestingly, as shown in Figure 3, the internal representation of this coding scheme is completely nonsensical to the human eye and is not super-voxel smooth, yet it successfully preserves similarity structure (see Figure 2). Matrix multiplication maps similar inputs to similar internal representations. In other words, the result is not super-voxel smooth, but it is functionally smooth which we conjecture is critical for fMRI to succeed.

Figure 3

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Deep learning networks (DLNs) have led to a revolution in machine learning and artificial intelligence (Krizhevsky et al., 2012; LeCun et al., 1998; Serre et al., 2007; Szegedy et al., 2015a). DLNs outperform existing approaches on object recognition tasks by training complex multi-layer networks with millions of parameters (i.e., weights) on large databases of natural images. Recently, neuroscientists have become interested in how the computations and representations in these models relate to the ventral stream in monkeys and humans (Cadieu et al., 2014; Dubois et al., 2015; Guclu and van Gerven, 2015; Hong et al., 2016; Khaligh-Razavi and Kriegeskorte, 2014; Yamins et al., 2014; Yamins and DiCarlo, 2016). For these reasons, we choose to examine these models, which also allow for RSA at multiple representational levels.

In this contribution, one key question is whether functional smoothness breaks down at more advanced layers in DLNs as it did in the untrained random neural networks considered in the previous section. We address this question by presenting natural image stimuli (i.e., novel photographs) to a trained DLN, specifically Inception-v3 GoogLeNet (Szegedy et al., 2015b), and applying RSA to evaluate whether the similarity structure of items would be recoverable using fMRI.

Deep learning network simulation

We consider whether functional smoothness declines as inputs are processed by the more advanced layers of Inception-v3 GoogLeNet. If so, fMRI should be less successful in brain regions that instantiate computations analogous to the more advanced layers of such networks. Unlike the previous simulations, we present novel photographs of natural categories to these networks. The key question is whether items from related categories (e.g., banjos and guitars) will be similar at various network layers. The $40$ photographs (i.e., stimuli) are divided equally amongst $8$ subordinate categories: banjos, guitars, mopeds, sportscars, lions, tigers, robins, and partridges, which in turn aggregate into $4$ basic-level categories: musical instruments, vehicles, mammals, and birds; which in turn aggregate into $2$ superordinates: animate and inanimate.

We consider how similar the internal network representations are for pairs of stimuli by comparing the resulting network activity, which is analogous to comparing neural activity over voxels in RSA. Correlations for all possible pairings of the $40$ stimuli were calculated for both a mid and a later network layer (see Figure 4).

Figure 4

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The middle layer (Figure 4A) reveals cross-category similarity at both the basic and superordinate level. For example, lions are more like robins than guitars. However, at the later layer (Figure 4B) the similarity structure has broken down such that subordinate category similarity dominates (i.e., a lion is like another lion, but not so much like a tiger). Interestingly, the decline in functional smoothness is not a consequence of sparseness at the later layer as the Gini coefficient, a measure of sparseness (Gini, 1909), is $0.947$ for the earlier middle layer (Figure 4A) and $0.579$ for the later advanced layer (Figure 4B), indicating that network representations are distributed in general and even more so at the later layer. Thus, the decline in functional smoothness at later layers does not appear to be a straightforward consequence of training these networks to classify stimuli, although it would be interesting to compare to unsupervised approaches that can perform at equivalent accuracy levels (no such network currently exists).

These DLN results are directly analogous to those with random untrained networks (see Figure 2). In those simulations, similar input patterns mapped to orthogonal (i.e., dissimilar) internal representations in later layers. Likewise, the trained DLN at later layers can only capture similarity structure within subordinate categories (e.g., a tiger is like another tiger) which the network was trained to classify. The effect of training the network was to create equivalence classes based on the training label (e.g., tiger) such that members of that category are mapped to similar network states. Violating functional smoothness, all other similarity structure is discarded such that a tiger is no more similar to a lion than to a banjo from the network’s perspective. Should brain regions operate in a similar fashion, fMRI would not be successful in recovering similarity structure therein. In the Discussion, we consider the implications of these findings on our understanding of the ventral stream and the prospects for fMRI.

Neuroscientists would rightly prefer a method that had both excellent spatial and temporal resolution for measuring brain activity. However, as we demonstrate in this article, the fact that fMRI has proven useful in examining neural representations, despite limitations in both its temporal and spatial resolution, says something about the nature of the neural code. One general conclusion is that the neural code must be smooth, both at voxel (such that voxel responses are inhomogeneous across time and space) and functional levels.

The latter notion of smoothness is often overlooked or confused with super-voxel smoothness, but is necessary for fMRI to recover similarity spaces in the brain. Coding schemes, such as factorial and hash coding, are useful in numerous real-world applications and have an inverse function (i.e., one can go backwards from the internal representation to recover the unique stimulus input). However, these schemes are incompatible with the success of fMRI because they are not functionally smooth. For example, if the brain solely used such coding schemes, the neural representation of a robin would be no more similar to that of a sparrow than to that of a car. The fact that such neural similarities are recoverable by fMRI suggests that the neural code differs from these schemes in many cases.

In contrast, we found that the types of representations used and generated by artificial neural networks, including deep learning networks, are broadly compatible with the success of fMRI in assessing neural representations. These coding schemes are functionally smooth in that similar inputs tend toward similar outputs, which allows item similarity to be reflected in neural similarity (as measured by fMRI). However, we found that functional smoothness breaks down as additional network layers are added. Specifically, we have shown that multi-layer networks eventually converge to something akin to a hash function, as arbitrary locations in memory correspond to categories of inputs.

These results take on additional significance given the recent interest in deep artificial neural networks as computational accounts of the ventral stream. One emerging view is that the more advanced the layers of these models correspond to more advanced regions along the ventral stream (Cadieu et al., 2014-12; Dubois et al., 2015; Guclu and van Gerven, 2015; Hong et al., 2016; Khaligh-Razavi and Kriegeskorte, 2014; Yamins et al., 2014; Yamins and DiCarlo, 2016).

If this viewpoint is correct, our results indicate that neural representations should progressively become less functionally smooth and more abstract as one moves along the ventral stream (recall Figure 2). Indeed, neural representations appear to become more abstract, encoding whole concepts or categories, as a function of how far along the ventral stream they are located (Bracci and Op de Beeck, 2016; DiCarlo and Cox, 2007; Riesenhuber and Poggio, 1999, 2000; Yamins and DiCarlo, 2016). For example, early on in visual processing, the brain may extract so-called basic features, such as in broadly-tuned orientation columns (Hubel and Wiesel, 1959, 1968). In contrast, later on in processing, cells may selectively respond to particular individual stimulus classes i.e., Jennifer Aniston, grandmother, concept, or gnostic cells (Gross, 2002; Konorski, 1967; Quiroga et al., 2005), irrespective of orientation, etc.

Likewise, we found that Inception-v3 GoogLeNet’s representations became symbol-like at advanced network layers such that items sharing a category label (e.g., tigers) engendered related network states, while items in other categories engendered orthogonal states (recall Figure 4). Our simulations of random networks also found reduced functional smoothness at advanced network layers, suggesting a basic geometric property of multi-layer networks. The effect of training seems limited to creating network states in which stimuli that share the same label (e.g., multiple viewpoints of Jennifer Aniston) become similar and items from all other categories (even if conceptually related) become orthogonal. If so, areas further along the ventral stream should prove less amenable to imaging (recall Figure 2). Indeed, a recent object recognition study found that the ceiling on observable correlation values becomes lower as one moves along the ventral stream (Bracci and Op de Beeck, 2016).

Here, we focused on using fMRI to recover non-degenerate similarity spaces (i.e., where there are similarities beyond self-similarities). However, functional smoothness is also important for other analysis approaches. For example, MVPA decoders trained to classify items (e.g., is a house or a face being shown?) based on fMRI activity will only generalize to novel stimuli when functional smoothness holds. Likewise, univariate clusters (e.g., a house or face area) will most likely be found and generalize to novel stimuli when functional smoothness holds because functional smoothness implies similar activation profiles for similar stimuli. Functional smoothness should be an important factor in determining how well classifiers perform and how statistically robust univariate clusters of voxels are.

In cognitive science, research is often divided into levels of analysis. In Marr’s levels, the top level is the problem description, the middle level captures how the problem is solved, and bottom level concerns how the solution is implemented in the brain (Marr, 1982). Given that the ‘how’ and ‘where’ of cognition appear to be merging, some have questioned the utility of this tripartite division (Love, 2015).

Our results suggest another inadequacy of these three levels of description, namely that the implementation level itself should be further subdivided. What is measured by fMRI is at a vastly more abstract scale than what can be measured in the brain. For example, major efforts, like the European Human Brain Project and the Machine Intelligence from Cortical Networks project (Underwood, 2016), are chiefly concerned with fine-grained aspects of the brain that are outside the reach of fMRI (Chi, 2016; Frégnac and Laurent, 2014). Likewise, models of spiking neurons e.g., (Wong and Wang, 2006) are at a level of analysis lower than where fMRI applies.

Nevertheless, fMRI has proven useful in understanding neural representations that are consequential to behavior. Perhaps this success suggests that the appropriate level for relating brain to behavior is close to what fMRI measures. This does not mean lower-level efforts do not have utility when the details are of interest. However, fMRI’s success might mean that when one is interested in the nature of computations carried out by the brain, the level of analysis where fMRI applies may be preferred. To draw an analogy, one could construct a theory of macroeconomics based on quantum physics, but it would be incredibly cumbersome and no more predictive nor explanatory than a theory that contained abstract concepts such as money and supply. Reductionism, while seductive, is not always the best path forward.

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Author details

1. Olivia Guest

   Experimental Psychology, University College London, London, United Kingdom

   Contribution

   OG, Conceptualization, Data curation, Software, Formal analysis, Visualization, Writing—original draft, Writing—review and editing

   For correspondence

   o.guest@ucl.ac.uk

   Competing interests

   The authors declare that no competing interests exist.

   "This ORCID iD identifies the author of this article:" 0000-0002-1891-0972

2. Bradley C Love

   1. Experimental Psychology, University College London, London, United Kingdom
   2. The Alan Turing Institute, London, United Kingdom

   Contribution

   BCL, Conceptualization, Resources, Supervision, Funding acquisition, Writing—original draft, Project administration, Writing—review and editing

   For correspondence

   b.love@ucl.ac.uk

   Competing interests

   The authors declare that no competing interests exist.

   "This ORCID iD identifies the author of this article:" 0000-0002-7883-7076

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