Our nervous system can efficiently recognize objects in spite of changes in contextual variables such as perspective or lighting conditions. Several lines of research have proposed that this ability for invariant recognition is learned by exploiting the fact that object identities typically vary more slowly in time than contextual variables or noise. Here, we study the question of how this “temporal stability” or “slowness” approach can be implemented within the limits of biologically realistic spike-based learning rules. We first show that slow feature analysis, an algorithm that is based on slowness, can be implemented in linear continuous model neurons by means of a modified Hebbian learning rule. This approach provides a link to the trace rule, which is another implementation of slowness learning. Then, we show analytically that for linear Poisson neurons, slowness learning can be implemented by spike-timing–dependent plasticity (STDP) with a specific learning window. By studying the learning dynamics of STDP, we show that for functional interpretations of STDP, it is not the learning window alone that is relevant but rather the convolution of the learning window with the postsynaptic potential. We then derive STDP learning windows that implement slow feature analysis and the “trace rule. ” The resulting learning windows are compatible with physiological data both in shape and timescale. Moreover, our analysis shows that the learning window can be split into two functionally different components that are sensitive to reversible and irreversible aspects of the input statistics, respectively. The theory indicates that irreversible input statistics are not in favor of stable weight distributions but may generate oscillatory weight dynamics. Our analysis offers a novel interpretation for the functional role of STDP in physiological neurons.
The ability to recognize objects in spite of possible changes in position, lighting conditions, or perspective is doubtlessly an advantage in everyday life. However, our brain usually performs this task with such astonishing ease that we are seldom aware of the complexity this recognition problem comprises. On the level of primary sensory signals (e. g. , light that stimulates a single retinal receptor), even small changes in the position of the object to be recognized may lead to vastly different stimuli. Our brain thus has to somehow identify rather different stimuli as representations of the same underlying cause, i. e. , it has to develop an internal representation that is invariant to irrelevant changes of the stimulus. The work presented here is motivated by the question of how such invariant representations could be established. Because of the limited amount of information in the genome as well as the apparent flexibility of the neural development in different environments, it seems unlikely that the information needed to form invariant representations is already there at the beginning of individual development. Some information must be gathered from the sensory input experienced during interaction with the environment; it has to be learned. As this learning process is likely to be at least partially unsupervised, the brain requires a heuristics as to what stimuli should be classified as being the same. One possible indicator for stimuli to represent the same object is temporal proximity. A scene that the eye views is very unlikely to change completely from one moment to the next. Rather, there is a good chance that an object that can be seen now will also be present at the next instant of time. This implies that invariant representations should remain stable over time, that is, they should vary slowly. Inverting this reasoning, a sensory system that adapts to its sensory input in order to extract slowly varying aspects may succeed in learning invariant representations. This “slowness” or “temporal stability” principle is the basis of a whole class of learning algorithms [1–7]. Most applications of this approach have focused on models of the visual system, in particular on the self-organized formation of complex cell receptive fields in the primary visual cortex [8,9]. For clarity, we will focus on one of these algorithms, slow feature analysis (SFA; [10]); a close link to the so-called “trace rule” will arise naturally. The goal of SFA is the following: given a multidimensional input signal x (t) and a finite-dimensional function space F, find the input–output function g1 (x) in F that generates the most slowly varying output signal y1 (t) = g1 (x (t) ). It is important to note that the function g1 (x) is required to be an instantaneous function of the input signal. Otherwise, slow output signals could be generated by low-pass filtering the input signal. As the goal of the slowness principle is to detect slowly varying features of the input signals, a mere low-pass filter would certainly generate slow output signals, but it would not serve the purpose. As a measure of slowness, or rather “fastness, ” SFA uses the variance of the time derivative, , which is the objective function to be minimized. Here, 〈·〉t denotes temporal averaging. For mathematical convenience and to avoid the trivial constant response, y1 (t) = const, a zero-mean, and unit variance constraint are imposed. Furthermore, it is possible to find a second function g2 (x) extracting y2 (t) = g2 (x (t) ) that again minimizes the given objective under the constraint of being uncorrelated with y1 (t), a third one uncorrelated with both y1 (t) and y2 (t), and so on, thereby generating a set of slow features of the input ordered by the degree of slowness. However, in this paper, we will consider just one single output unit. SFA has been applied to the learning of translation, rotation, and other invariances in a model of the visual system [10], and it has been shown that when applied to image sequences generated from static natural images, SFA learns functions that reproduce a wide range of features of complex cells in primary visual cortex [8]. Iteration of the same principle in a hierarchical model in combination with a sparseness objective has been used to model the self-organized formation of spatial representations resembling place cells as found in the hippocampal formation of rodents [11] (see [12] for related work). These findings suggest that on an abstract level SFA reflects certain aspects of cortical information processing. However, SFA as a technical algorithm is biologically rather implausible. There is in particular one step in its canonical formulation that seems especially odd compared with what neurons are normally thought to do. In this step the eigenvector that corresponds to the smallest eigenvalue of the covariance matrix of the time derivative of some multidimensional signal is extracted. The aim of this paper is to show how this kind of computation can be realized in a spiking model neuron. In the following, we will first consider a continuous model neuron and demonstrate that a modified Hebbian learning rule enables the neuron to learn the slowest (in the sense of SFA) linear combination of its inputs. Apart from providing the basis for the analysis of the spiking model, this section reveals a mathematical link between SFA and the trace learning rule, another implementation of the slowness principle. We then examine if these findings also hold for a spiking model neuron, and find that for a linear Poisson neuron, spike-timing–dependent plasticity (STDP) can be interpreted as an implementation of the slowness principle.
Real neurons do not transmit information via a continuous stream of analog values like the model neuron considered in the previous section, but rather emit action potentials that carry information by means of their rate and probably also by their exact timing, a fact we will not consider here. How can the model developed so far be mapped onto this scenario?
Neurons in the central nervous system display a wide range of invariances in their response behavior, examples of which are phase invariance in complex cells in the early visual system [33], head direction invariance in hippocampal place cells [34], or more complex invariances in neurons associated with face recognition [35]. If these invariances are learned, the associated learning rule must somehow reflect a heuristics as to which sensory stimuli are supposed to be categorized as being the same. Objects in our environment are unlikely to change completely from one moment to the next but rather undergo typical transformations. Intuitively, responses of neurons with invariances to these transformations should thus vary more slowly than others. The slowness principle uses this intuition and conjectures that neurons learn these invariances by favoring slowly varying output signals without exploiting low-pass filtering. SFA [10] is one implementation of the slowness principle in that it minimizes the mean square of the temporal derivative of the output signal for a given set of training data. SFA has been used to model a wide range of physiologically observed properties of complex cells in primary visual cortex [8] as well as translation, rotation, and other invariances in the visual system [10]. In combination with a sparse coding objective, SFA has also been used to describe the self-organized formation of place cells in the hippocampal formation [11]. The algorithm that underlies SFA is rather technical, and it has not yet been examined whether it is feasible to implement SFA within the limitations of neuronal circuitry. In this paper we approach this question analytically and demonstrate that such an implementation is possible in both continuous and spiking model neurons. In the first part of the paper, we show that for linear continuous model neurons, the slowest direction in the input signal can be learned by means of Hebbian learning on low-pass filtered versions of the input and the output signal. The power spectrum of the low-pass filter required for implementing SFA can be derived from the learning objective and has the shape of an upside-down parabola. The idea of using low-pass filtered signals for invariance learning is a feature that our model has in common with several others [1,4, 6]. By means of the continuous model neuron, we have discussed the relation of our model to these “trace rules” and have shown that they bear strong similarities. The second part of the paper discusses the modifications that have to be made to adjust the learning rule for a Poisson neuron. We find that in an ensemble-averaged sense it is possible to reproduce the behavior of the continuous model neuron by means of spike-timing–dependent plasticity (STDP). Our study suggests that the outcome of STDP learning is not governed by the learning window alone but rather by the convolution of the learning window with the EPSP, which is of relevance for functional interpretations of STDP. The learning window that realizes SFA can be calculated analytically. Its shape is determined by the interplay of the duration of the EPSP and the maximal input frequency νmax, above which the input signals are assumed to have negligible power. If νmax is small, i. e. , if the EPSP is sufficiently short to temporally resolve the most quickly varying components of the input data, the learning window is symmetric, whereas for large νmax or long EPSPs, it is antisymmetric. Interestingly, physiologically plausible parameters lead to a learning window whose shape and width is in agreement with experimental findings. Based on this result, we propose a new functional interpretation of the STDP learning window as an implementation of the slowness principle that compensates for neuronal low-pass filters such as the EPSP. An important question in this context is on which timescales is this interpretation valid. It is conceivable that for signals that vary on a timescale of less than a hundred milliseconds, a learning window with a width of tens of milliseconds can distinguish slower from faster signals. STDP could thus be sufficient to establish invariant representations in early sensory processing, e. g. , visual receptive fields that become invariant to microsaccades inducing small translations. Although it is unlikely that STDP alone can distinguish between signals that vary on behavioral timescales of hundreds of milliseconds or even seconds, this may not be problematic, because it is probably not sensible to order all aspects of the stimuli according to how quickly they vary. Rather, one should distinguish input components that vary so quickly that they are unlikely to be behaviorally relevant from those that vary on behavioral timescales. From this perspective, the intrinsic timescale of the learning rule should be such that its discriminative power is best on a timescale where this transition occurs. It is conceivable that this transition timescale lies on the order of several tens of milliseconds. The learning of high level invariances that correspond to behavioral timescales will probably require additional mechanisms with corresponding intrinsic timescales, e. g. , sustained firing in response to a stimulus [36]. For general learning windows and EPSPs, the convolution of the learning window with the EPSP can be split into a symmetric component and an antisymmetric component. The symmetric component picks up reversible aspects of the input statistics while the antisymmetric component detects irreversible aspects. Previous functional interpretations of STDP have mostly concentrated on the antisymmetric component, which has been interpreted, e. g. , as a mechanism for sequence learning or predictive coding [19,31] or for reducing recurrent connectivity in favor of feed-forward structures [30,32]. Other studies have neglected the phase structure of the learning window altogether and concentrated on its power spectrum, proposing that timing-dependent plasticity performs Hebbian learning on an optimal estimate of the input signals in the presence of noise [37,38]. Note that these interpretations are not necessarily contradictory to ours, because the slowness interpretation relies on the symmetric component of the learning window only and thus on the reversible aspect of the input statistics. These considerations indicate that depending on the temporal structure of the input, STDP may have different functional roles. A different approach to unsupervised learning of invariances with a biologically realistic model neuron has been taken by Körding and König [39]. In their model, bursts of backpropagating spikes gate synaptic plasticity by providing sufficient amounts of dendritic depolarization. These bursts are assumed to be triggered by lateral connections that evoke calcium spikes in the apical dendrites of cortical pyramidal cells. Of course the model presented here is not a complete implementation of SFA. We have only considered the central step of SFA, the extraction of the most slowly varying direction from a set of whitened input signals. To implement the full algorithm, additional steps are necessary: a nonlinear expansion of the input space, the whitening of the expanded input signals, and a means of normalizing the weights. When traversing the dendritic arborizations of a postsynaptic neuron, axons often make more than one synaptic contact. As different input channels may be subjected to different nonlinearities in the dendritic tree (cf. [40]), the postsynaptic neuron may have access to several nonlinearly transformed versions of the same presynaptic signals. Conceptually, this resembles a nonlinear expansion of the input signals. However, it is not obvious how these signals could be whitened within the dendrite. On the network level, however, whitening could be achieved by adaptive recurrent inhibition between the neurons [41]. This mechanism may also be suitable for extracting several slow uncorrelated signals as required in the original formulation of SFA [10] instead of just one. We assumed an explicit weight normalization in the description of our model. However, one could also use a modified learning rule that implicitly normalizes the weight vector as long as it extracts the signal with the largest variance. A possible biological mechanism is synaptic scaling [25], which is believed to multiplicatively rescale all synaptic weights according to postsynaptic activity, similar to Oja' s rule [26,42]. Thus, it appears that most of the mechanisms necessary for an implementation of the full SFA algorithm are available, but that it is not yet clear how to combine them in a biologically plausible way. Another critical point in the analytical derivation for the spiking model is the replacement of the temporal by the ensemble average, as this allows recovery of the rates that underlie the Poisson processes. The validity of the analytical results thus requires some kind of ergodicity in the training data, a condition which of course needs to be justified for the specific input data at hand. It is still open whether the results presented here can be reproduced with more realistic model neurons. The spiking model neuron used here was simplified in that it had a linear relationship between input and output firing rate. In many real neurons, highly nonlinear behavior was observed. Interestingly, Hebbian learning for nonlinear rate-based neurons has previously been associated with the detection of higher-order moments of the input statistics [43], thereby providing a mechanism for extracting statistically independent components of the input signal. Because for sparse input statistics independent component analysis is closely related to sparse coding [44], it is tempting to speculate that within a rate picture, temporally nonlocal plasticity with a nonlinear input–output relation implements a combination of sparseness and slowness. Learning paradigms that combine these two objectives are thus an interesting field for further studies [11,45]. Another nonlinearity that we have neglected is the frequency- and weight-dependence of STDP [16,46]. Additional work will be needed to examine how these interfere with the proposed functional role of STDP. Furthermore, modeling the spiking mechanism of a neuron by an inhomogeneous Poisson process is also a severe simplification that ignores basic phenomena of spike generation in biological neurons such as refractoriness and thresholding. It is not clear how these characteristics would change the learning rule that leads to an implementation of the slowness principle. It seems to be a very difficult task to answer these questions analytically. Simulations will be necessary to verify the results derived here and to analyze which changes appear and which adaptations must be made in a more realistic model of neural information processing. In summary, the analytical considerations presented here show that (i) slowness can be equivalently achieved by minimizing the variance of the time derivative signal or by maximizing the variance of the low-pass filtered signal, the latter of which can be achieved by standard Hebbian learning on the low-pass filtered input and output signals; (ii) the difference between SFA and the trace learning rule lies in the exact shape of the effective low-pass filter—for most practical purposes the results are probably equivalent; (iii) for a spiking Poisson model neuron with an STDP learning rule, it is not the learning window that governs the weight dynamics but the convolution of the learning window with the EPSP; (iv) the STDP learning window that implements the slowness objective is in good agreement with learning windows found experimentally. With these results, we have reduced the gap between slowness as an abstract learning principle and biologically plausible STDP learning rules, and we offer a completely new interpretation of the standard STDP learning window.
The methods employed in this paper rely on standard mathematical techniques as commonly used in the theory of synaptic plasticity (see, e. g. , [47]).
Neurons interact by exchanging information via small connection sites, so-called synapses. Interestingly, the efficiency of synapses in transmitting neuronal signals is not static, but changes dynamically depending on the signals that the associated neurons emit. As neurons receive thousands of synaptic input signals, they can thus “choose” the input signals they are interested in by adjusting their synapses accordingly. This adaptation mechanism, known as synaptic plasticity, has long been hypothesized to form the neuronal correlate of learning. It raises a difficult question: what aspects of the input signals are the neurons interested in, given that the adaptation of the synapses follows a certain mechanistic rule? We address this question for spike-timing–dependent plasticity, a type of synaptic plasticity that has raised a lot of interest in the last decade. We show that under certain assumptions regarding neuronal information transmission, spike-timing–dependent plasticity focuses on aspects of the input signals that vary slowly in time. This relates spike-timing–dependent plasticity to a class of abstract learning rules that were previously proposed as a means of learning to recognize objects in spite of contextual changes such as size or position. Based on this link, we propose a novel functional interpretation of spike-timing–dependent plasticity.
neuroscience/theoretical neuroscience
neuroscience/sensory systems
Slowness: An Objective for Spike-Timing–Dependent Plasticity?