Artificial neural network models aim at explaining human cognition and behavior in terms of the emergent consequences of a large number of simple, subcognitive processes (McClelland et al., 2010). Within this framework, the pattern seen in overt behavior (macroscopic dynamics of the system) reflects the coordinated operations of simple biophysical mechanisms (microscopic dynamics of the system), such as the propagation of activation and inhibition among elementary processing units. Though this general tenet is shared by all connectionist models, there is large variability in processing architectures and learning algorithms, which turns into varying degrees of psychological and biological realism (e.g., Thorpe and Imbert, 1989; O'Reilly, 1998). When the aim is to investigate high-level cognitive functions, simplification is essential (McClelland, 2009) and the underlying processing mechanisms do not need to faithfully implement the neuronal circuits supposed to carry out such functions in the brain. However, modelers should strive to consider biological plausibility if this can bridge different levels of description (Testolin and Zorzi, 2016).

Recent theoretical and technical progress in artificial neural networks has significantly expanded the range of tasks that can be solved by machine intelligence. In particular, the advent of powerful parallel computing architectures based on Graphic Processing Units (GPUs), coupled with the availability of “big data,” has allowed to create and train large-scale, hierarchical neural networks known as deep neural networks (LeCun et al., 2015, for review). These powerful learning systems achieve impressive performance in many challenging cognitive tasks, such as visual object recognition (Krizhevsky et al., 2012), speech processing (Mohamed et al., 2012) and natural language understanding (Collobert et al., 2011). However, while the impact of deep learning for engineering applications is undisputed, its relevance for modeling neural information processing in biological systems still needs to be fully evaluated (for seminal attempts, see Stoianov and Zorzi, 2012; Khaligh-Razavi and Kriegeskorte, 2014; Güçlü and van Gerven, 2015).

One critical aspect of most deep learning systems is the reliance on a feed-forward architecture trained with error backpropagation (Rumelhart et al., 1986), which has been repeatedly shown to yield state-of-the-art performance in a variety of problems (LeCun et al., 2015). However, the assumptions that learning is largely discriminative (e.g., classification or function learning) and that an external teaching signal is always available at each learning event (i.e., all training data is “labeled”) are clearly implausible from both a cognitive and a biological perspective (Zorzi et al., 2013; Cox and Dean, 2014). Reinforcement learning is a valuable alternative and it has already shown promising results when combined with deep learning (Mnih et al., 2015; Silver et al., 2016), but there is a broad range of situations where learning seems to be fully unsupervised and its only objective is that of discovering the latent structure of the input data in order to build rich, internal representations of the environment (Hinton and Sejnowski, 1999). We argue that more realistic neurocognitive models should therefore also exploit unsupervised forms of deep learning, where the objective is not to explicitly classify the input patterns but rather to discover internal representations by fitting a hierarchical generative model to the sensory data (Hinton, 2007, 2013; Zorzi et al., 2013). Compared to its supervised counterpart, this modeling approach emphasizes the role of feedback, recurrent connections (Sillito et al., 2006), which carry top-down expectations that are gradually adjusted to better reflect the observed data (Hinton and Ghahramani, 1997; Friston, 2010) and which can be used to implement concurrent probabilistic inference along the whole cortical hierarchy (Lee and Mumford, 2003; Gilbert and Sigman, 2007). Notably, top-down processing is also relevant for understanding attentional mechanisms in terms of modulation of neural information processing (Kastner and Ungerleider, 2000).

A powerful class of stochastic neural networks that learn a generative model of the data is that of Restricted Boltzmann Machines (RBMs), which can efficiently discover internal representations (i.e., latent features) using Hebbian-like learning mechanisms (Hinton, 2002). RBMs constitute the building block of hierarchical generative models such as Deep Belief Networks (Hinton and Salakhutdinov, 2006) and Deep Boltzmann Machines (Salakhutdinov, 2015). These unsupervised deep learning models have been successfully used to simulate a variety of cognitive functions, such as numerosity perception (Stoianov and Zorzi, 2012), letter perception (Testolin et al., under review), location-invariant visual word recognition (Di Bono and Zorzi, 2013), and visual hallucinations in psychiatric syndromes (Reichert et al., 2013). A similar approach has been used to simulate how early visual cortical representations are adapted to statistical regularities in natural images, in order to predict single voxel responses to natural images and identify images from stimulus-evoked multiple voxel responses (Güçlü and van Gerven, 2014). A temporal extension of RBMs has also been recently used to model sequential orthographic processing and spontaneous pseudoword generation (Testolin et al., 2016).

Unsupervised deep learning can be implemented using an alternative architecture based on autoencoders (Bengio et al., 2007), which are deterministic, feed-forward networks whose learning goal is to accurately reconstruct the input data into a separate layer of output units. Single-layer autoencoders are trained using error backpropagation, and can be stacked in order to build more complex, multi-layer architectures. However, despite the common view that RBMs and autoencoders could be considered equivalent (Ranzato et al., 2007), we note that their underlying architectural and learning assumptions are significantly different. In this study we empirically compare RBMs and autoencoders in terms of the type of internal encoding emerging in the hidden neurons. Moreover, we investigate how additional learning constraints, such as sparsity and limitation of computational resources (i.e., hidden layer size), could influence the representations developed by the networks. As a case study, we focus on the problem of learning visuospatial coding for sensorimotor transformations, which is a prominent example of how the emergentist approach based on learning in artificial neural networks has offered important insights into the computations performed by biological neurons (Zipser and Andersen, 1988).

Sensorimotor transformations refer to the process by which sensory stimuli are converted into motor commands. For example, reaching requires to map visual information, represented in retinal coordinates, into a system of coordinates that is centered on the effector. Coordinate transformations can be accomplished by combining sensory information with extra-retinal information, such as postural signals representing the position of eyes, head, or hand, thereby obtaining abstract representations of the space interposed between the sensory input and the motor output (Pouget and Snyder, 2000). Single-neuron recordings from monkey posterior parietal cortex have shown that the response amplitude of many neurons indeed depends on the position of the eyes, thereby unveiling a fundamental coding principle used to perform this type of signal integration (Andersen et al., 1985). The term gain field was coined to describe this gaze-dependent response of parietal neurons, and since then the notion of gain modulation has been generalized to indicate the multiplicative control of one neuron's responses by the responses of another set of neurons (Salinas and Thier, 2000). Another fundamental property unveiled by neuronal recordings is that the encoding of space used for coordinate transformations involves a variety of different, complementary frames of reference. For example, although many parietal neurons are centered on retinal coordinates (Andersen et al., 1985; Duhamel et al., 1992), others represent space using body-centered (Snyder et al., 1998) or effector-centered (Sakata et al., 1995) coordinate systems. Moreover, some neurons exhibit multiple gain modulation (Chang et al., 2009), suggesting more complex forms of spatial coding. For example, postural information related to both eye and head positions can be combined in order to encode “gaze” direction (Brotchie et al., 1995; Stricanne et al., 1996; Duhamel et al., 1997).

From a computational perspective, the seminal work of Zipser and Andersen (1988) showed that gain modulation could spontaneously emerge in supervised, feed-forward neural networks trained to explicitly map visual targets into head-centered coordinates, giving as input any arbitrary pair of eye and retinal positions. Similar results have been observed using more biologically-plausible learning settings, such as reinforcement learning (Mazzoni et al., 1991) and predictive coding (De Meyer and Spratling, 2011). Note that these learning settings assume that gain modulation emerges because the task implies to establish a mapping between different reference frames. However, it is unclear whether the form of modulation and the distribution of neuronal tuning functions is influenced by the type of learning algorithm and/or by the nature of the learning task (i.e., learning input-output mappings vs. unsupervised learning of internal representations). We also note that a popular alternative framework for modeling sensorimotor transformations is not based on learning, but rather stipulates that parietal neurons represent a set of basis functions that combine visual and postural information (for review, see Pouget and Snyder, 2000).

In summary, space coding represents an interesting case study for testing the adequacy of different neural network architectures and learning algorithms, because it provides a wealth of neurophysiological data (both at the population and single-neuron levels), and it departs from the classic problem of visual object recognition investigated in the large majority of deep learning research.

In this section we describe the space coding tasks used in our simulations, including training and test stimuli, the different learning architectures, and the procedures for analyzing the emergent neural representations.

Learning always converged for all models. For unsupervised models, convergence was monitored by measuring the mean reconstruction error on the whole training set. Autoencoders required more learning epochs to converge, but also achieved a lower reconstruction error compared to RBMs. This is probably due to the fact that autoencoders are natively real-valued. Existing real-valued extensions of RBMs (Cho et al., 2011) assume that the input values are normally distributed, which was not our case, so we preferred to use standard RBMs. Learning in the feed-forward, supervised models required almost 20 times more epochs to converge (the number of epochs required by each learning architecture is reported in Table 1).

A first, qualitative analysis shows that RBMs and autoencoders developed different types of receptive fields. As shown in Figure 3, autoencoders learned homogeneous, location-specific receptive fields that uniformly covered the central regions of the visual input. On the other hand, while some neurons in the RBMs learned location-specific receptive fields resembling those of autoencoders, other neurons developed more complex receptive fields covering larger regions of the visual fields, sometimes also simultaneously covering symmetrical portions of the input image.

The quantitative analyses (see Section Testing Procedure) allowed to group hidden neurons into different categories according to their response profiles. In line with empirical findings (Duhamel et al., 1997), there were always some neurons that did not exhibit any form of gain modulation (i.e., “purely visual” neurons), that is, they responded to visual stimuli at a given spatial location regardless of eye- or effector- positions. However, the majority of neurons developed gain fields, which in some cases were modulated exclusively by either eye or effector position (see, for example, top panels of Figure 4), while in other cases were modulated by both eye and effector position, resulting in multiple gain fields (bottom panels of Figure 4).

In this study we investigated the role of architectural and learning constraints in neural network models that learned to encode spatial information resulting from the combination of visual and postural signals. Results showed that, compared to the supervised architecture originally proposed by Zipser and Andersen (1988), unsupervised architectures like Restricted Boltzmann Machines (RBMs) and autoencoders discover space codes that more closely reproduce the distribution of neuronal tuning functions observed in neurophysiological experiments. In particular, the majority of hidden neurons of RBMs and autoencoders exhibited gain modulation, which in some cases only depended either on eye or effector position, while in other cases depended on both eye and effector positions, thereby resulting in multiple gain fields. In fact, all unsupervised models developed a much higher percentage of gain modulated neurons compared to the supervised models. Although the precise distribution of gain field types in the cerebral cortex depends on the exact recording site (Colby and Goldberg, 1999), our simulations suggest that this efficient form of encoding emerges more naturally if the task requires to reconstruct the whole sensory input, rather than to simply discover a feed-forward mapping to a target motor program. In other words, gain field coding might be useful when the goal is to discover “good” internal representations of the input data, that is, when the aim is to unveil and more explicitly encode the latent factors underlying the input data distribution.

As a general principle, the quality of an internal representation should reflect how well the learned features disentangle as many factors of variation as possible, at the same time discarding as little information about the data as is practical (Bengio et al., 2013). In the specific case of sensorimotor transformations, it has been proposed that good internal representations should have a variety of properties, such as the ability to combine the input signal in a nonlinear way, the ability to fully cover the range of possible input values, and the ability to represent multiple reference frames simultaneously within the same neurons (Pouget and Snyder, 2000). Populations of gain modulated neurons satisfy these requirements, allowing to encode visual space using a flexible set of basis functions. Notably, our simulations showed that this allows to learn coordinate transformations in two separate stages, by first learning the set of basis functions in a completely unsupervised way, and then learning appropriate mappings to target motor commands by relying on explicit supervision or reinforcement signals (Pouget and Snyder, 2000).

Our analyses also highlighted several differences in the spatial codes learned by RBMs and autoencoders, despite the fact that these two unsupervised architectures are often considered similar, if not equivalent (Ranzato et al., 2007; Coates et al., 2011). Even from a simple, qualitative analysis of the visual receptive fields, it turned out that these models developed different internal representations. Subsequent analyses conducted to investigate the emergence of gain fields further revealed that the distribution of hidden neurons' tuning functions in RBMs and autoencoders was similar only for a very narrow choice of the hyper-parameters. An important finding was that RBMs spontaneously exhibited a remarkable level of sparseness, which made them insensitive to external sparsity constraints, and which encouraged the emergence of compressed forms of spatial coding based on gain modulation. The spontaneous level of sparseness in RBMs could be manipulated only within a narrow range, by imposing an extreme sparsity constraint and jointly reducing the size of the hidden layer. This forced the internal representations to rely on even fewer neurons, and produced an increase in the percentage of multiple gain fields. These findings are consistent with the intuition that reducing the computational resources forces the networks to discover more complex (and compressed) forms of encoding, such as those resulting from the combination of many sensory/postural variables into multiple gain fields. Notably, for RBMs this was the case even when the task did not involve any coordinate transformations, which implied that postural variables were orthogonal. In other words, despite the fact that eye and effector positions were varied independently across training patterns, the RBMs with fewer active neurons often combined these signals together, resulting in an increase of multiple gain fields. Nevertheless, unlike autoencoders, RBMs always dedicated some representational resources also to encode eye and effector positions independently.

Autoencoders turned out to rely on much more distributed representations compared to RBMs, and were therefore extremely sensitive to external sparsity constraints. This implies that, compared to RBMs, autoencoders have an additional hyper-parameter that must be carefully tuned. Notably, when the sparsity pressure was reduced hidden neurons in the autoencoders did not develop any form of gain modulation. Only for specific values of sparsity constraints autoencoders could reproduce the variety of gain field types observed in neurophysiological data (Brotchie et al., 1995; Graziano et al., 1997; Snyder et al., 1998; Chang et al., 2009), with a distribution compatible with that of RBMs. However, in autoencoders the underlying sparseness indexes did not seem to be systematically related to the complexity of the emergent spatial codes. Though these findings alone do not allow to adjudicate between models, they call for a more systematic investigation of these different learning architectures, possibly spanning other domains and using a more direct comparison to neurophysiological data.

A plausible explanation for the striking differences in the spontaneous level of sparseness between RBMs and autoencoders can be found when considering the different processing dynamics embedded in these two neural network models. Indeed, in autoencoders the activation of each hidden neuron is deterministic, and simply corresponds to the (possibly graded) value returned by the non-linear, logistic activation function. In RBMs, instead, the value returned by the logistic function is treated as a probability, and the final activation of each hidden neuron is obtained by performing a stochastic binarization step. This important difference likely produces more sharp neuronal activations, driving RBMs to develop more sparse representations compared to autoencoders.

From a broader perspective, we believe that stochastic neural networks such as RBMs and their extension into hierarchical generative models will have an increasingly central role in neurocomputational modeling, because they provide a unique bridge between high-level descriptions of cognition in terms of Bayesian computation and low-level, mechanistic explanations inspired by the biophysical properties of real neuronal networks (Testolin and Zorzi, 2016). For example, generative neural networks are compatible with Bayesian approaches based on probabilistic population codes (Ma et al., 2006), which have been successfully used to simulate sensorimotor transformations with basis functions (Pouget and Sejnowski, 1997; Pouget and Snyder, 2000). RBMs extend the basis function approach by explaining how learning might shape the emergent neuronal gain fields, and they could similarly be combined with attractor dynamics to simulate optimal statistical inference over multisensory spatial representations (cf. Pouget et al., 2002) and spatial remapping in attention orienting (cf. Casarotti et al., 2012).

Moreover, the fact that generative networks can simulate both evoked (feed-forward) and intrinsic (feedback) neuronal activity makes them particularly suited to investigate spontaneous brain activity, which has been recognized as a fundamental property of the brain (Raichle, 2015) but whose computational role is still largely unknown. An intriguing hypothesis suggests that intrinsic activity could help with driving the brain close to states that are probable to be valid inferences once an external input arrives, thus potentially shortening the reaction time of the system (Fiser et al., 2010). Stochastic, generative networks are consistent with this “sampling-based” framework, and also support the idea that neuronal noise could play an important role during sampling (Kirkpatrick et al., 1983), for example by keeping the system in a metastable state that facilitates flexible settling into the most appropriate configuration (Kelso, 2012; Deco et al., 2013). Notably, we are also beginning to better understand how these powerful models could be implemented with biologically more realistic architectures, such as those incorporating temporal dynamics and spike-based communication (Buesing et al., 2011; Nessler et al., 2013).

In conclusion, we hope that the recent breakthroughs in neurally-inspired machine learning will attract the interest of the neuroscience community, as these models hold great promise for improving our understanding of how learning shapes and organizes information processing in complex neuronal networks.

AT, MD, and MZ equally contributed to the research design. AT implemented the simulations. AT and MD analyzed the data. AT and MZ wrote the paper. All the authors are accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.