Nonlinear Hebbian Learning as a Unifying Principle in Receptive Field Formation

Synapse-type-specific competitive Hebbian learning forms functional recurrent networks

Cortical networks exhibit complex stimulus-response patterns that are based on specific recurrent interactions between neurons. For example, the balance between excitatory and inhibitory currents has been identified as a central component of cortical computations. However, it remains unclear how the required synaptic connectivity can emerge in developing circuits where synapses between excitatory and inhibitory neurons are simultaneously plastic. Using theory and modeling, we propose that a wide range of cortical response properties can arise from a single plasticity paradigm that acts simultaneously at all excitatory and inhibitory connections-Hebbian learning that is stabilized by the synapse-type-specific competition for a limited supply of synaptic resources. In plastic recurrent circuits, this competition enables the formation and decorrelation of inhibition-balanced receptive fields. Networks develop an assembly structure with stronger synaptic connections between similarly tuned excitatory and inhibitory neurons and exhibit response normalization and orientation-specific center-surround suppression, reflecting the stimulus statistics during training. These results demonstrate how neurons can self-organize into functional networks and suggest an essential role for synapse-type-specific competitive learning in the development of cortical circuits.

Positive Competitive Networks for Sparse Reconstruction

We propose and analyze a continuous-time firing-rate neural network, the positive firing-rate competitive network (PFCN), to tackle sparse reconstruction problems with non-negativity constraints. These problems, which involve approximating a given input stimulus from a dictionary using a set of sparse (active) neurons, play a key role in a wide range of domains, including, for example, neuroscience, signal processing, and machine learning. First, by leveraging the theory of proximal operators, we relate the equilibria of a family of continuous-time firing-rate neural networks to the optimal solutions of sparse reconstruction problems. Then we prove that the PFCN is a positive system and give rigorous conditions for the convergence to the equilibrium. Specifically, we show that the convergence depends only on a property of the dictionary and is linear-exponential in the sense that initially, the convergence rate is at worst linear and then, after a transient, becomes exponential. We also prove a number of technical results to assess the contractivity properties of the neural dynamics of interest. Our analysis leverages contraction theory to characterize the behavior of a family of firing-rate competitive networks for sparse reconstruction with and without non-negativity constraints. Finally, we validate the effectiveness of our approach via a numerical example.

Learning what matters: Synaptic plasticity with invariance to second-order input correlations

Cortical populations of neurons develop sparse representations adapted to the statistics of the environment. To learn efficient population codes, synaptic plasticity mechanisms must differentiate relevant latent features from spurious input correlations, which are omnipresent in cortical networks. Here, we develop a theory for sparse coding and synaptic plasticity that is invariant to second-order correlations in the input. Going beyond classical Hebbian learning, our learning objective explains the functional form of observed excitatory plasticity mechanisms, showing how Hebbian long-term depression (LTD) cancels the sensitivity to second-order correlations so that receptive fields become aligned with features hidden in higher-order statistics. Invariance to second-order correlations enhances the versatility of biologically realistic learning models, supporting optimal decoding from noisy inputs and sparse population coding from spatially correlated stimuli. In a spiking model with triplet spike-timing-dependent plasticity (STDP), we show that individual neurons can learn localized oriented receptive fields, circumventing the need for input preprocessing, such as whitening, or population-level lateral inhibition. The theory advances our understanding of local unsupervised learning in cortical circuits, offers new interpretations of the Bienenstock-Cooper-Munro and triplet STDP models, and assigns a specific functional role to synaptic LTD mechanisms in pyramidal neurons.

A laminar model for the joint development of ocular dominance columns and CO blobs in the primary visual cortex

In this paper, we present a multi-layer, activity-dependent model for the joint development of ocular dominance (OD) columns and cytochrome oxidase (CO) blobs in primate V1. For simplicity, we focus on layers 4C and 2/3 with both layers receiving direct thalamic inputs and layer 4C sending vertical projections to layer 2/3. Both the thalamic and the vertical connections are taken to be modifiable by activity. Using a correlation-based Hebbian learning rule with subtractive normalization, we show how the formation of an OD map in layer 4C is inherited by layer 2/3 via the vertical projections. Competition between these feedforward projections and the direct thalamic input to layer 2/3 then results in the formation of CO blobs superimposed upon the ocular dominance map. The spacing of the OD columns is determined by the spatial profile of the intralaminar connections within layer 4, while the spacing of CO blobs depends both on the width of the OD columns inherited from layer 4 and the spatial distribution of intralaminar connections within the superficial layer. The resulting CO blob distribution is shown to be consistent with experimental data. In addition, we numerically simulate monocular deprivation and find that while the CO blob distribution is unaltered, the OD pattern undergoes modification. The OD stripes of the deprived eye narrow, whereas the OD stripes for the remaining open eye widen.

NMDA-driven dendritic modulation enables multitask representation learning in hierarchical sensory processing pathways

While sensory representations in the brain depend on context, it remains unclear how such modulations are implemented at the biophysical level, and how processing layers further in the hierarchy can extract useful features for each possible contextual state. Here, we demonstrate that dendritic N-Methyl-D-Aspartate spikes can, within physiological constraints, implement contextual modulation of feedforward processing. Such neuron-specific modulations exploit prior knowledge, encoded in stable feedforward weights, to achieve transfer learning across contexts. In a network of biophysically realistic neuron models with context-independent feedforward weights, we show that modulatory inputs to dendritic branches can solve linearly nonseparable learning problems with a Hebbian, error-modulated learning rule. We also demonstrate that local prediction of whether representations originate either from different inputs, or from different contextual modulations of the same input, results in representation learning of hierarchical feedforward weights across processing layers that accommodate a multitude of contexts.

Efficient inference of synaptic plasticity rule with Gaussian process regression

Finding the form of synaptic plasticity is critical to understanding its functions underlying learning and memory. We investigated an efficient method to infer synaptic plasticity rules in various experimental settings. We considered biologically plausible models fitting a wide range of in-vitro studies and examined the recovery of their firing-rate dependence from sparse and noisy data. Among the methods assuming low-rankness or smoothness of plasticity rules, Gaussian process regression (GPR), a nonparametric Bayesian approach, performs the best. Under the conditions measuring changes in synaptic weights directly or measuring changes in neural activities as indirect observables of synaptic plasticity, which leads to different inference problems, GPR performs well. Also, GPR could simultaneously recover multiple plasticity rules and robustly perform under various plasticity rules and noise levels. Such flexibility and efficiency, particularly at the low sampling regime, make GPR suitable for recent experimental developments and inferring a broader class of plasticity models.

Efficient neural codes naturally emerge through gradient descent learning

Human sensory systems are more sensitive to common features in the environment than uncommon features. For example, small deviations from the more frequently encountered horizontal orientations can be more easily detected than small deviations from the less frequent diagonal ones. Here we find that artificial neural networks trained to recognize objects also have patterns of sensitivity that match the statistics of features in images. To interpret these findings, we show mathematically that learning with gradient descent in neural networks preferentially creates representations that are more sensitive to common features, a hallmark of efficient coding. This effect occurs in systems with otherwise unconstrained coding resources, and additionally when learning towards both supervised and unsupervised objectives. This result demonstrates that efficient codes can naturally emerge from gradient-like learning.

Spike-Timing-Dependent Plasticity With Activation-Dependent Scaling for Receptive Fields Development

Spike-timing-dependent plasticity (STDP) is one of the most popular and deeply biologically motivated forms of unsupervised Hebbian-type learning. In this article, we propose a variant of STDP extended by an additional activation-dependent scale factor. The consequent learning rule is an efficient algorithm, which is simple to implement and applicable to spiking neural networks (SNNs). It is demonstrated that the proposed plasticity mechanism combined with competitive learning can serve as an effective mechanism for the unsupervised development of receptive fields (RFs). Furthermore, the relationship between synaptic scaling and lateral inhibition is explored in the context of the successful development of RFs. Specifically, we demonstrate that maintaining a high level of synaptic scaling followed by its rapid increase is crucial for the development of neuronal mechanisms of selectivity. The strength of the proposed solution is assessed in classification tasks performed on the Modified National Institute of Standards and Technology (MNIST) data set with an accuracy level of 94.65% (a single network) and 95.17% (a network committee)-comparable to the state-of-the-art results of single-layer SNN architectures trained in an unsupervised manner. Furthermore, the training process leads to sparse data representation and the developed RFs have the potential to serve as local feature detectors in multilayered spiking networks. We also prove theoretically that when applied to linear Poisson neurons, our rule conserves total synaptic strength, guaranteeing the convergence of the learning process.

Weight dependence in BCM leads to adjustable synaptic competition

Models of synaptic plasticity have been used to better understand neural development as well as learning and memory. One prominent classic model is the Bienenstock-Cooper-Munro (BCM) model that has been particularly successful in explaining plasticity of the visual cortex. Here, in an effort to include more biophysical detail in the BCM model, we incorporate 1) feedforward inhibition, and 2) the experimental observation that large synapses are relatively harder to potentiate than weak ones, while synaptic depression is proportional to the synaptic strength. These modifications change the outcome of unsupervised plasticity under the BCM model. The amount of feed-forward inhibition adds a parameter to BCM that turns out to determine the strength of competition. In the limit of strong inhibition the learning outcome is identical to standard BCM and the neuron becomes selective to one stimulus only (winner-take-all). For smaller values of inhibition, competition is weaker and the receptive fields are less selective. However, both BCM variants can yield realistic receptive fields.

Bridging the Functional and Wiring Properties of V1 Neurons Through Sparse Coding

The functional properties of neurons in the primary visual cortex (V1) are thought to be closely related to the structural properties of this network, but the specific relationships remain unclear. Previous theoretical studies have suggested that sparse coding, an energy-efficient coding method, might underlie the orientation selectivity of V1 neurons. We thus aimed to delineate how the neurons are wired to produce this feature. We constructed a model and endowed it with a simple Hebbian learning rule to encode images of natural scenes. The excitatory neurons fired sparsely in response to images and developed strong orientation selectivity. After learning, the connectivity between excitatory neuron pairs, inhibitory neuron pairs, and excitatory-inhibitory neuron pairs depended on firing pattern and receptive field similarity between the neurons. The receptive fields (RFs) of excitatory neurons and inhibitory neurons were well predicted by the RFs of presynaptic excitatory neurons and inhibitory neurons, respectively. The excitatory neurons formed a small-world network, in which certain local connection patterns were significantly overrepresented. Bidirectionally manipulating the firing rates of inhibitory neurons caused linear transformations of the firing rates of excitatory neurons, and vice versa. These wiring properties and modulatory effects were congruent with a wide variety of data measured in V1, suggesting that the sparse coding principle might underlie both the functional and wiring properties of V1 neurons.

Sub-Optimality of the Early Visual System Explained Through Biologically Plausible Plasticity

The early visual cortex is the site of crucial pre-processing for more complex, biologically relevant computations that drive perception and, ultimately, behaviour. This pre-processing is often studied under the assumption that neural populations are optimised for the most efficient (in terms of energy, information, spikes, etc.) representation of natural statistics. Normative models such as Independent Component Analysis (ICA) and Sparse Coding (SC) consider the phenomenon as a generative, minimisation problem which they assume the early cortical populations have evolved to solve. However, measurements in monkey and cat suggest that receptive fields (RFs) in the primary visual cortex are often noisy, blobby, and symmetrical, making them sub-optimal for operations such as edge-detection. We propose that this suboptimality occurs because the RFs do not emerge through a global minimisation of generative error, but through locally operating biological mechanisms such as spike-timing dependent plasticity (STDP). Using a network endowed with an abstract, rank-based STDP rule, we show that the shape and orientation tuning of the converged units are remarkably close to single-cell measurements in the macaque primary visual cortex. We quantify this similarity using physiological parameters (frequency-normalised spread vectors), information theoretic measures [Kullback–Leibler (KL) divergence and Gini index], as well as simulations of a typical electrophysiology experiment designed to estimate orientation tuning curves. Taken together, our results suggest that compared to purely generative schemes, process-based biophysical models may offer a better description of the suboptimality observed in the early visual cortex.

Performance of biologically grounded models of the early visual system on standard object recognition tasks

Computational neuroscience models of vision and neural network models for object recognition are often framed by different research agendas. Computational neuroscience mainly aims at replicating experimental data, while (artificial) neural networks target high performance on classification tasks. However, we propose that models of vision should be validated on object recognition tasks. At some point, mechanisms of realistic neuro-computational models of the visual cortex have to convince in object recognition as well. In order to foster this idea, we report the recognition accuracy for two different neuro-computational models of the visual cortex on several object recognition datasets. The models were trained using unsupervised Hebbian learning rules on natural scene inputs for the emergence of receptive fields comparable to their biological counterpart. We assume that the emerged receptive fields result in a general codebook of features, which should be applicable to a variety of visual scenes. We report the performances on datasets with different levels of difficulty, ranging from the simple MNIST to the more complex CIFAR-10 or ETH-80. We found that both networks show good results on simple digit recognition, comparable with previously published biologically plausible models. We also observed that our deeper layer neurons provide for naturalistic datasets a better recognition codebook. As for most datasets, recognition results of biologically grounded models are not available yet, our results provide a broad basis of performance values to compare methodologically similar models.

Sparse coding with a somato-dendritic rule

Cortical neurons are silent most of the time: sparse activity enables low-energy computation in the brain, and promises to do the same in neuromorphic hardware. Beyond power efficiency, sparse codes have favourable properties for associative learning, as they can store more information than local codes but are easier to read out than dense codes. Auto-encoders with a sparse constraint can learn sparse codes, and so can single-layer networks that combine recurrent inhibition with unsupervised Hebbian learning. But the latter usually require fast homeostatic plasticity, which could lead to catastrophic forgetting in embodied agents that learn continuously. Here we set out to explore whether plasticity at recurrent inhibitory synapses could take up that role instead, regulating both the population sparseness and the firing rates of individual neurons. We put the idea to the test in a network that employs compartmentalised inputs to solve the task: rate-based dendritic compartments integrate the feedforward input, while spiking integrate-and-fire somas compete through recurrent inhibition. A somato-dendritic learning rule allows somatic inhibition to modulate nonlinear Hebbian learning in the dendrites. Trained on MNIST digits and natural images, the network discovers independent components that form a sparse encoding of the input and support linear decoding. These findings confirm that intrinsic homeostatic plasticity is not strictly required for regulating sparseness: inhibitory synaptic plasticity can have the same effect. Our work illustrates the usefulness of compartmentalised inputs, and makes the case for moving beyond point neuron models in artificial spiking neural networks.

Towards the interpretation of complex visual hallucinations in terms of self-reorganization of neural networks

Patients suffering from dementia with Lewy body (DLB) often see complex visual hallucinations (CVH). Despite many pathological, clinical, and neuroimaging studies, the mechanism of CVH remains unknown. One possible scenario is that top-down information is being used to compensate for the lack of bottom-up information. To investigate this possibility and understand the underlying mathematical structure of the CVH mechanism, we propose a simple computational model of synaptic plasticity with particular focus on the effect of selective damage to the bottom-up network on self-reorganization. We show neurons that undergo a change in activity from a bottom-up to a top-down network framework during the reorganization process, which can be understood in terms of state transitions. Assuming that the pre-reorganization representation of this neuron remains after reorganization, it is possible to interpret neural response induced by top-down information as the sensation of bottom-up information. This situation might correspond to a hallucinatory situation in DLB patients. Our results agree with existing experimental evidence and provide new insights into data that have hitherto not been experimentally validated on patients with DLB.

Synaptic Plasticity Forms and Functions

Synaptic plasticity, the activity-dependent change in neuronal connection strength, has long been considered an important component of learning and memory. Computational and engineering work corroborate the power of learning through the directed adjustment of connection weights. Here we review the fundamental elements of four broadly categorized forms of synaptic plasticity and discuss their functional capabilities and limitations. Although standard, correlation-based, Hebbian synaptic plasticity has been the primary focus of neuroscientists for decades, it is inherently limited. Three-factor plasticity rules supplement Hebbian forms with neuromodulation and eligibility traces, while true supervised types go even further by adding objectives and instructive signals. Finally, a recently discovered hippocampal form of synaptic plasticity combines the above elements, while leaving behind the primary Hebbian requirement. We suggest that the effort to determine the neural basis of adaptive behavior could benefit from renewed experimental and theoretical investigation of more powerful directed types of synaptic plasticity.

An Adaptive Homeostatic Algorithm for the Unsupervised Learning of Visual Features

The formation of structure in the visual system, that is, of the connections between cells within neural populations, is by and large an unsupervised learning process. In the primary visual cortex of mammals, for example, one can observe during development the formation of cells selective to localized, oriented features, which results in the development of a representation in area V1 of images’ edges. This can be modeled using a sparse Hebbian learning algorithms which alternate a coding step to encode the information with a learning step to find the proper encoder. A major difficulty of such algorithms is the joint problem of finding a good representation while knowing immature encoders, and to learn good encoders with a nonoptimal representation. To solve this problem, this work introduces a new regulation process between learning and coding which is motivated by the homeostasis processes observed in biology. Such an optimal homeostasis rule is implemented by including an adaptation mechanism based on nonlinear functions that balance the antagonistic processes that occur at the coding and learning time scales. It is compatible with a neuromimetic architecture and allows for a more efficient emergence of localized filters sensitive to orientation. In addition, this homeostasis rule is simplified by implementing a simple heuristic on the probability of activation of neurons. Compared to the optimal homeostasis rule, numerical simulations show that this heuristic allows to implement a faster unsupervised learning algorithm while retaining much of its effectiveness. These results demonstrate the potential application of such a strategy in machine learning and this is illustrated by showing the effect of homeostasis in the emergence of edge-like filters for a convolutional neural network.

Biologically plausible deep learning - But how far can we go with shallow networks?

Training deep neural networks with the error backpropagation algorithm is considered implausible from a biological perspective. Numerous recent publications suggest elaborate models for biologically plausible variants of deep learning, typically defining success as reaching around 98% test accuracy on the MNIST data set. Here, we investigate how far we can go on digit (MNIST) and object (CIFAR10) classification with biologically plausible, local learning rules in a network with one hidden layer and a single readout layer. The hidden layer weights are either fixed (random or random Gabor filters) or trained with unsupervised methods (Principal/Independent Component Analysis or Sparse Coding) that can be implemented by local learning rules. The readout layer is trained with a supervised, local learning rule. We first implement these models with rate neurons. This comparison reveals, first, that unsupervised learning does not lead to better performance than fixed random projections or Gabor filters for large hidden layers. Second, networks with localized receptive fields perform significantly better than networks with all-to-all connectivity and can reach backpropagation performance on MNIST. We then implement two of the networks - fixed, localized, random & random Gabor filters in the hidden layer - with spiking leaky integrate-and-fire neurons and spike timing dependent plasticity to train the readout layer. These spiking models achieve >98.2% test accuracy on MNIST, which is close to the performance of rate networks with one hidden layer trained with backpropagation. The performance of our shallow network models is comparable to most current biologically plausible models of deep learning. Furthermore, our results with a shallow spiking network provide an important reference and suggest the use of data sets other than MNIST for testing the performance of future models of biologically plausible deep learning.

When the goal is to generate a series of activities: A self-organized simulated robot arm

Behavior is characterized by sequences of goal oriented conducts, such as food uptake, socializing and resting. Classically, one would define for each task a corresponding satisfaction level, with the agent engaging, at a given time, in the activity having the lowest satisfaction level. Alternatively, one may consider that the agent follows the overarching objective to generate sequences of distinct activities. To achieve a balanced distribution of activities would then be the primary goal, and not to master a specific task. In this setting the agent would show two types of behaviors, task-oriented and task-searching phases, with the latter interseeding the former. We study the emergence of autonomous task switching for the case of a simulated robot arm. Grasping one of several moving objects corresponds in this setting to a specific activity. Overall, the arm should follow a given object temporarily and then move away, in order to search for a new target and reengage. We show that this behavior can be generated robustly when modeling the arm as an adaptive dynamical system. The dissipation function is in this approach time dependent. The arm is in a dissipative state when searching for a nearby object, dissipating energy on approach. Once close, the dissipation function starts to increase, with the eventual sign change implying that the arm will take up energy and wander off. The resulting explorative state ends when the dissipation function becomes again negative and the arm selects a new target. We believe that our approach may be generalized to generate self-organized sequences of activities in general.

STRFs in primary auditory cortex emerge from masking-based statistics of natural sounds

We investigate how the neural processing in auditory cortex is shaped by the statistics of natural sounds. Hypothesising that auditory cortex (A1) represents the structural primitives out of which sounds are composed, we employ a statistical model to extract such components. The input to the model are cochleagrams which approximate the non-linear transformations a sound undergoes from the outer ear, through the cochlea to the auditory nerve. Cochleagram components do not superimpose linearly, but rather according to a rule which can be approximated using the max function. This is a consequence of the compression inherent in the cochleagram and the sparsity of natural sounds. Furthermore, cochleagrams do not have negative values. Cochleagrams are therefore not matched well by the assumptions of standard linear approaches such as sparse coding or ICA. We therefore consider a new encoding approach for natural sounds, which combines a model of early auditory processing with maximal causes analysis (MCA), a sparse coding model which captures both the non-linear combination rule and non-negativity of the data. An efficient truncated EM algorithm is used to fit the MCA model to cochleagram data. We characterize the generative fields (GFs) inferred by MCA with respect to in vivo neural responses in A1 by applying reverse correlation to estimate spectro-temporal receptive fields (STRFs) implied by the learned GFs. Despite the GFs being non-negative, the STRF estimates are found to contain both positive and negative subfields, where the negative subfields can be attributed to explaining away effects as captured by the applied inference method. A direct comparison with ferret A1 shows many similar forms, and the spectral and temporal modulation tuning of both ferret and model STRFs show similar ranges over the population. In summary, our model represents an alternative to linear approaches for biological auditory encoding while it captures salient data properties and links inhibitory subfields to explaining away effects.

The information carried by natural sounds enters the cortex of mammals in a specific format: the cochleagram. Instead of representing the original pressure waveforms, the inner ear represents how the energy in a sound is distributed across frequency bands and how the energy distribution evolves over time. The generation of cochleagrams is highly non-linear resulting in the dominance of one sound source per time-frequency bin under natural conditions (masking). Auditory cortex is believed to decompose cochleagrams into structural primitives, i.e., reappearing regular spectro-temporal subpatterns that make up cochleagram patterns (similar to edges in images). However, such a decomposition has so far only been modeled without considering masking and non-negativity. Here we apply a novel non-linear sparse coding model that can capture masking non-linearities and non-negativities. When trained on cochleagrams of natural sounds, the model gives rise to an encoding primarily based-on spectro-temporally localized components. If stimulated by a sound, the encoding units compete to explain its contents. The competition is a direct consequence of the statistical sound model, and it results in neural responses being best described by spectro-temporal receptive fields (STRFs) with positive and negative subfields. The emerging STRFs show a higher similarity to experimentally measured STRFs than a model without masking, which provides evidence for cortical encoding being consistent with the masking based sound statistics of cochleagrams. Furthermore, and more generally, our study suggests for the first time that negative subfields of STRFs may be direct evidence for explaining away effects resulting from performing inference in an underlying statistical model.

Theoretical Models of Neural Development

Constructing a functioning nervous system requires the precise orchestration of a vast array of mechanical, molecular, and neural-activity-dependent cues. Theoretical models can play a vital role in helping to frame quantitative issues, reveal mathematical commonalities between apparently diverse systems, identify what is and what is not possible in principle, and test the abilities of specific mechanisms to explain the data. This review focuses on the progress that has been made over the last decade in our theoretical understanding of neural development.

High-dimensional dynamics in a single-transistor oscillator containing Feynman-Sierpiński resonators: Effect of fractal depth and irregularity

Fractal structures pervade nature and are receiving increasing engineering attention towards the realization of broadband resonators and antennas. We show that fractal resonators can support the emergence of high-dimensional chaotic dynamics even in the context of an elementary, single-transistor oscillator circuit. Sierpiński gaskets of variable depth are constructed using discrete capacitors and inductors, whose values are scaled according to a simple sequence. It is found that in regular fractals of this kind, each iteration effectively adds a conjugate pole/zero pair, yielding gradually more complex and broader frequency responses, which can also be implemented as much smaller Foster equivalent networks. The resonators are instanced in the circuit as one-port devices, replacing the inductors found in the initial version of the oscillator. By means of a highly simplified numerical model, it is shown that increasing the fractal depth elevates the dimension of the chaotic dynamics, leading to high-order hyperchaos. This result is overall confirmed by SPICE simulations and experiments, which however also reveal that the non-ideal behavior of physical components hinders obtaining high-dimensional dynamics. The issue could be practically mitigated by building the Foster equivalent networks rather than the verbatim fractals. Furthermore, it is shown that considerably more complex resonances, and consequently richer dynamics, can be obtained by rendering the fractal resonators irregular through reshuffling the inductors, or even by inserting a limited number of focal imperfections. The present results draw attention to the potential usefulness of fractal resonators for generating high-dimensional chaotic dynamics, and underline the importance of irregularities and component non-idealities.

Sensory cortex is optimized for prediction of future input

Neurons in sensory cortex are tuned to diverse features in natural scenes. But what determines which features neurons become selective to? Here we explore the idea that neuronal selectivity is optimized to represent features in the recent sensory past that best predict immediate future inputs. We tested this hypothesis using simple feedforward neural networks, which were trained to predict the next few moments of video or audio in clips of natural scenes. The networks developed receptive fields that closely matched those of real cortical neurons in different mammalian species, including the oriented spatial tuning of primary visual cortex, the frequency selectivity of primary auditory cortex and, most notably, their temporal tuning properties. Furthermore, the better a network predicted future inputs the more closely its receptive fields resembled those in the brain. This suggests that sensory processing is optimized to extract those features with the most capacity to predict future input.

A large part of our brain is devoted to processing the sensory inputs that we receive from the world. This allows us to tell, for example, whether we are looking at a cat or a dog, and if we are hearing a bark or a meow. Neurons in the sensory cortex respond to these stimuli by generating spikes of activity. Within each sensory area, neurons respond best to stimuli with precise properties: those in the primary visual cortex prefer edge-like structures that move in a certain direction at a given speed, while neurons in the primary auditory cortex favour sounds that change in loudness over a particular range of frequencies.

Singer et al. sought to understand why neurons respond to the particular features of stimuli that they do. Why do visual neurons react more to moving edges than to, say, rotating hexagons? And why do auditory neurons respond more to certain changing sounds than to, say, constant tones? One leading idea is that the brain tries to use as few spikes as possible to represent real-world stimuli. Known as sparse coding, this principle can account for much of the behaviour of sensory neurons.

Another possibility is that sensory areas respond the way they do because it enables them to best predict future sensory input. To test this idea, Singer et al. used a computer to simulate a network of neurons and trained this network to predict the next few frames of video clips using the previous few frames. When the network had learned this task, Singer et al. examined the neurons’ preferred stimuli. Like neurons in primary visual cortex, the simulated neurons typically responded most to edges that moved over time.

The same network was also trained in a similar way, but this time using sound. As for neurons in primary auditory cortex, the simulated neurons preferred sounds that changed in loudness at particular frequencies. Notably, for both vision and audition, the simulated neurons favoured recent inputs over those further into the past. In this way and others, they were more similar to real neurons than simulated neurons that used sparse coding.

Both artificial networks trained to foretell sensory input and the brain therefore favour the same types of stimuli: the ones that are good at helping to grasp future information. This suggests that the brain represents the sensory world so as to be able to best predict the future.

Knowing how the brain handles information from our senses may help to understand disorders associated with sensory processing, such as dyslexia and tinnitus. It may also inspire approaches for training machines to process sensory inputs, improving artificial intelligence.

Error-Gated Hebbian Rule: A Local Learning Rule for Principal and Independent Component Analysis

We developed a biologically plausible unsupervised learning algorithm, error-gated Hebbian rule (EGHR)-β, that performs principal component analysis (PCA) and independent component analysis (ICA) in a single-layer feedforward neural network. If parameter β = 1, it can extract the subspace that major principal components span similarly to Oja’s subspace rule for PCA. If β = 0, it can separate independent sources similarly to Bell-Sejnowski’s ICA rule but without requiring the same number of input and output neurons. Unlike these engineering rules, the EGHR-β can be easily implemented in a biological or neuromorphic circuit because it only uses local information available at each synapse. We analytically and numerically demonstrate the reliability of the EGHR-β in extracting and separating major sources given high-dimensional input. By adjusting β, the EGHR-β can extract sources that are missed by the conventional engineering approach that first applies PCA and then ICA. Namely, the proposed rule can successfully extract hidden natural images even in the presence of dominant or non-Gaussian noise components. The results highlight the reliability and utility of the EGHR-β for large-scale parallel computation of PCA and ICA and its future implementation in a neuromorphic hardware.

A Unifying Framework of Synaptic and Intrinsic Plasticity in Neural Populations

A neuronal population is a computational unit that receives a multivariate, time-varying input signal and creates a related multivariate output. These neural signals are modeled as stochastic processes that transmit information in real time, subject to stochastic noise. In a stationary environment, where the input signals can be characterized by constant statistical properties, the systematic relationship between its input and output processes determines the computation carried out by a population. When these statistical characteristics unexpectedly change, the population needs to adapt to its new environment if it is to maintain stable operation. Based on the general concept of homeostatic plasticity, we propose a simple compositional model of adaptive networks that achieve invariance with regard to undesired changes in the statistical properties of their input signals and maintain outputs with well-defined joint statistics. To achieve such invariance, the network model combines two functionally distinct types of plasticity. An abstract stochastic process neuron model implements a generalized form of intrinsic plasticity that adapts marginal statistics, relying only on mechanisms locally confined within each neuron and operating continuously in time, while a simple form of Hebbian synaptic plasticity operates on synaptic connections, thus shaping the interrelation between neurons as captured by a copula function. The combined effect of both mechanisms allows a neuron population to discover invariant representations of its inputs that remain stable under a wide range of transformations (e.g., shifting, scaling and (affine linear) mixing). The probabilistic model of homeostatic adaptation on a population level as presented here allows us to isolate and study the individual and the interaction dynamics of both mechanisms of plasticity and could guide the future search for computationally beneficial types of adaptation.

Hebbian plasticity requires compensatory processes on multiple timescales

We review a body of theoretical and experimental research on Hebbian and homeostatic plasticity, starting from a puzzling observation: while homeostasis of synapses found in experiments is a slow compensatory process, most mathematical models of synaptic plasticity use rapid compensatory processes (RCPs). Even worse, with the slow homeostatic plasticity reported in experiments, simulations of existing plasticity models cannot maintain network stability unless further control mechanisms are implemented. To solve this paradox, we suggest that in addition to slow forms of homeostatic plasticity there are RCPs which stabilize synaptic plasticity on short timescales. These rapid processes may include heterosynaptic depression triggered by episodes of high postsynaptic firing rate. While slower forms of homeostatic plasticity are not sufficient to stabilize Hebbian plasticity, they are important for fine-tuning neural circuits. Taken together we suggest that learning and memory rely on an intricate interplay of diverse plasticity mechanisms on different timescales which jointly ensure stability and plasticity of neural circuits.This article is part of the themed issue 'Integrating Hebbian and homeostatic plasticity'.

Learning with three factors: modulating Hebbian plasticity with errors

Synaptic plasticity is a central theme in neuroscience. A framework of three-factor learning rules provides a powerful abstraction, helping to navigate through the abundance of models of synaptic plasticity. It is well-known that the dopamine modulation of learning is related to reward, but theoretical models predict other functional roles of the modulatory third factor; it may encode errors for supervised learning, summary statistics of the population activity for unsupervised learning or attentional feedback. Specialized structures may be needed in order to generate and propagate third factors in the neural network.

Toward an Integration of Deep Learning and Neuroscience

Neuroscience has focused on the detailed implementation of computation, studying neural codes, dynamics and circuits. In machine learning, however, artificial neural networks tend to eschew precisely designed codes, dynamics or circuits in favor of brute force optimization of a cost function, often using simple and relatively uniform initial architectures. Two recent developments have emerged within machine learning that create an opportunity to connect these seemingly divergent perspectives. First, structured architectures are used, including dedicated systems for attention, recursion and various forms of short- and long-term memory storage. Second, cost functions and training procedures have become more complex and are varied across layers and over time. Here we think about the brain in terms of these ideas. We hypothesize that (1) the brain optimizes cost functions, (2) the cost functions are diverse and differ across brain locations and over development, and (3) optimization operates within a pre-structured architecture matched to the computational problems posed by behavior. In support of these hypotheses, we argue that a range of implementations of credit assignment through multiple layers of neurons are compatible with our current knowledge of neural circuitry, and that the brain's specialized systems can be interpreted as enabling efficient optimization for specific problem classes. Such a heterogeneously optimized system, enabled by a series of interacting cost functions, serves to make learning data-efficient and precisely targeted to the needs of the organism. We suggest directions by which neuroscience could seek to refine and test these hypotheses.

A Local Learning Rule for Independent Component Analysis

Humans can separately recognize independent sources when they sense their superposition. This decomposition is mathematically formulated as independent component analysis (ICA). While a few biologically plausible learning rules, so-called local learning rules, have been proposed to achieve ICA, their performance varies depending on the parameters characterizing the mixed signals. Here, we propose a new learning rule that is both easy to implement and reliable. Both mathematical and numerical analyses confirm that the proposed rule outperforms other local learning rules over a wide range of parameters. Notably, unlike other rules, the proposed rule can separate independent sources without any preprocessing, even if the number of sources is unknown. The successful performance of the proposed rule is then demonstrated using natural images and movies. We discuss the implications of this finding for our understanding of neuronal information processing and its promising applications to neuromorphic engineering.