In a neural network, either biological or artificial, two forms of coding can be used: localist or distributed. With localist coding, each neuron codes (i.e., maximally responds) for one and only one category of stimulus (or stimulus feature). As a result, the category of the stimulus (or the presence of a certain feature) can be inferred from the response of this sole neuron, ignoring the other neurons' responses. Conversely, with distributed coding each neuron responds to multiple stimulus categories (or features) in a similar way. Therefore, the response of each neuron is ambiguous, and the category of the stimulus, or the presence of a certain feature, can only be inferred from the responses of multiple neurons. Thus the distinction between the two schemes is the number of different stimuli to which a given neuron responds—not the number of neurons which respond to a given stimulus. Indeed, a localist network can have redundancy, and use multiple “copies” of each category specific neuron (Thorpe, 1989; Bowers, 2009).

Does the brain use localist or distributed coding? This question has been, and still is, intensively debated. In practice, discriminating between the two schemes from electrophysiological recordings is tricky (Quian Quiroga and Kreiman, 2010), since the set of tested stimuli is always limited, the responses are noisy, the thresholds are arbitrary and the boundaries between categories are fuzzy. Here we do not attempt to do a complete review of the experimental literature; but rather to summarize it. It is commonly believed that distributed coding is prevalent (Rolls et al., 1997; O'Reilly, 1998; Hung et al., 2005; Quiroga et al., 2008), but there is also evidence for localist coding, at least for familiar stimuli, reviewed in Bowers (2009, 2017), Thorpe (2009, 2011), and Roy (2017).

The question of localist vs. distributed coding is also relevant for artificial neural networks, and in particular for the recently popular deep neural networks. Most of the time, these networks are trained in a supervised manner, using the backpropagation algorithm (LeCun et al., 2015). The last layer contains exactly one neuron per category, and backpropagation forces each neuron to respond more strongly when the stimulus belongs to the neuron's category. In other words, localist coding is imposed in the last layer. Conversely, the hidden layers are free to choose their coding scheme, which is supposedly optimal for the categorization task at hand. It is thus very interesting to analyze the chosen coding scheme. It is not easy to do such analysis on the brain (as explained above), but we can do it rigorously for computational models by computing the responses to huge amounts of images, and even synthesizing images that maximize the responses. Results indicate that some hidden neurons respond to one object category only (Zhou et al., 2015; Nguyen et al., 2016; Olah et al., 2017), while others respond to multiple different objects (Nguyen et al., 2016; Olah et al., 2017). Thus it appears that both localist and distributed codes can be optimal, depending on the task, the layer number, and the network parameters (number of layers, neurons, etc.).

Let us come back to the brain, in which computation is presumably implemented by spiking neurons performing coincidence detection (Abeles, 1982; König et al., 1996; Brette, 2015). This observation raises an important question, which we tried to address in this theoretical paper: can coincidence detector neurons implement both localist and distributed codes? In this context, different stimuli correspond to different spatiotemporal input spike patterns. Here each pattern was generated randomly, leading to chance-level overlap between patterns. In addition, each pattern was jittered at each presentation, resulting in categories of similar, yet different, stimuli. Can a neuron respond to one, or several of these patterns, and not to random inputs? What is the required connectivity to do so in an optimal way? And finally, can this required connectivity emerge with spike-timing-dependent plasticity (STDP), in an unsupervised manner?

To address these questions, we extended a theory that we developed in a previous paper, but which was limited to one pattern only, i.e., localist coding (Masquelier, 2017), to the multi-pattern case. Briefly, we derived analytically the signal-to-noise ratio (SNR) of a multi-pattern detector, and investigated the conditions for its optimality. In addition, using numerical simulations, we showed that a single neuron equipped with STDP can become selective to multiple repeating spike patterns, even without supervision and that the resulting detectors can be close to the theoretical optimum. Surprisingly, a single neuron could robustly learn up to ~40 independent patterns (using parameters arguably in the biological range). This was not clear from previous simulations studies, in which neurons equipped with STDP only learned one pattern (localist coding) (Masquelier et al., 2008, 2009; Gilson et al., 2011; Humble et al., 2012; Hunzinger et al., 2012; Kasabov et al., 2013; Klampfl and Maass, 2013; Nessler et al., 2013; Krunglevicius, 2015; Sun et al., 2016; Masquelier, 2017), or two patterns (Yger et al., 2015). This shows that STDP and coincidence detection are compatible with distributed coding.

The problem we addressed is similar to the one of Masquelier (2017), but extended to the multi-pattern case. For the reader's convenience, we fully describe it below.

We addressed the problem of detecting one or several spatiotemporal spike patterns with a single LIF neuron. Intuitively, one should connect the neurons that are active during the patterns (or during subsections of them) to the LIF neuron. That way, the LIF will tend to be more activated by the patterns than by some other inputs. More formally, we note P the number of spike patterns, and assume that they all have the same duration L. We note N the number of neurons involved. For each pattern, we chose a subsection with duration Δt ≤ L, and we connect the LIF to the M neurons that emit at least one spike during at least one of these subsections (Figure 1).

We hypothesize that all afferent neurons fire according to a homogeneous Poisson process with rate f, both inside and outside the patterns. That is the patterns correspond to some realizations of the Poisson process, which can be repeated (this is sometimes referred to a “frozen noise”). At each repetition a random time lag (jitter) is added to each spike, drawn from a uniform distribution over [−T, T] (a normal distribution is more often used, but it would not allow analytical treatment, Masquelier, 2017).

We also assume that synapses are instantaneous, which facilitates the analytic calculations.

For now we ignore the LIF threshold, and we want to optimize its signal-to-noise ratio (SNR), defined as:

where V_max is the maximal potential reached during the pattern presentations, V¯noise is the mean value for the potential with Poisson input (noise period), and σ_noise is its standard deviation. Obviously, a higher SNR means a larger difference between the LIF membrane potential during the noise periods and its maximum value, which occurs during the selected Δt window of each pattern. Therefore, the higher the SNR the lower the probability of missing patterns, and of false alarms.

We consider that P, L, N, f, and T are imposed variables, and that we have the freedom to choose Δt ≤ L and the membrane time constant τ in order to maximize the SNR.

We note that this problem is related to the synfire chain theory (Abeles, 1991). A synfire chain consists of a series of pools of neurons linked together in a feed-forward chain, so that volleys of synchronous spikes can propagate from pool to pool in the chain. Each neuron can participate in several of such chains. The number of different chains that can coexist in a network of a given size has been termed capacity. This capacity can be optimized (Herrmann et al., 1995). To do so, a given neuron should respond to certain spike volleys, but not to others, which is similar to our optimization of a multi-pattern SNR. Yet it is also different: we use homogeneous Poisson activity, not spike volleys, and we ignore the threshold, while synfire chains require thresholds.

Next we investigated, through numerical simulations, if STDP could turn a LIF neuron into an optimal multi-pattern detector. More specifically, since STDP does not adjust the membrane time constant τ, we set it to the optimal value and investigated whether STDP could learn all the patterns with an optimal Δt². Here, unlike in the previous section, we had to introduce a threshold, in order to have postsynaptic spikes, which are required for STDP. As a result, the optimal V_max, computed in the previous section, was never reached. Yet a high V_max guarantees a low miss rate, and a low V¯noise guarantees a low false alarm rate. Optimizing the previously defined SNR thus makes sense.

Again, we used a clock-based approach, and the forward Euler method with a 0.1 ms time bin. The Matlab R2017a code for these simulations has been made available in ModelDB (Hines et al., 2004) at https://senselab.med.yale.edu/modeldb/.

The fact that STDP can generate selectivity to any repeating spike pattern in an unsupervised manner is a remarkable, yet well documented fact (Masquelier et al., 2008, 2009; Gilson et al., 2011; Humble et al., 2012; Hunzinger et al., 2012; Kasabov et al., 2013; Klampfl and Maass, 2013; Nessler et al., 2013; Krunglevicius, 2015; Yger et al., 2015; Sun et al., 2016; Masquelier, 2017). Here we have shown that, surprisingly, a single neuron can become optimally selective to several tens of independent patterns. Hence STDP and coincidence detection are compatible with distributed coding.

Yet one issue with having one neuron selective to multiple patterns is stability. If one of the learned pattern does not occur for a long period during which the other patterns occur many times, causing postsynaptic spikes, the unseen pattern will tend to be forgotten. This is not an issue with localist coding: if the learned pattern does not occur, the threshold is hardly ever reached so the weights are not modified, and the pattern is retained indefinitely, even if STDP is “on” all the time.

Another issue with distributed coding is how the readout is done, that is how the identity of the stimulus can be inferred from multiple neuron responses, given that each response is ambiguous? This is out of the scope of the current paper, but we suspect that STDP could again help. As shown in this study, each neuron equipped with STDP can learn to fire to multiple independent stimuli. Let's suppose that stimuli are shown one at a time. When stimulus A is shown, all the neurons that learned this stimulus (among others) will fire synchronously. Let us call S this set of neurons. A downstream neuron equipped with STDP could easily become selective to this synchronous volley of spikes from neurons in S (Brette, 2012). With an appropriate threshold, this neuron would fire if and only if all the neurons in S have fired. Does that necessarily mean that A is there? Yes, if the intersection of the sets of stimuli learned by neurons in S only contains A. In the general case, the intersection is likely to be much smaller than the typical sets of stimuli learned by the S neurons, so much of the ambiguity should be resolved.

What could determine the set of patterns to which a neuron responds? Here, we used independent, unrelated, patterns (i.e., with chance-level overlap), and yet several of these patterns could be learned by a single neuron. Of course, patterns with more overlap would be easier to group. So in the presence of multiple postsynaptic neurons, each one would tend to learn a cluster of similar patterns. Another factor is the time at which the patterns are presented: those presented at the same period are more likely to be learned by the same neuron—a neuron which was still unselective at that period. Indeed, neurons equipped with STDP have some sort of critical period, before convergence, during which they can learn new pattern easily. Conversely, after convergence, neurons tend to fire if and only if the patterns they have learned are presented (Figure 6), and thus can hardly learn any new pattern. This is interesting, because patterns presented at the same period are likely to be somewhat related. For example, a neuron could fire to the different people you have met on your first day at work. In the presence of neurogenesis, newborn neurons could handle the learning of other patterns during the subsequent periods of your life. Finally, here we did not use any reward signal. But such a signal, if available, could modulate STDP (leading to some form of supervised learning), and encourage a given neuron to fire to a particular, meaningful, set of patterns (Mozafari et al., 2018a,b), as opposed to a random set like here. For example, a single neuron could learn to fire to any animal, even if different animals cause very different sensory inputs.

Here the STDP rule we used always led to binary weights after learning. That is an afferent could be either selected or discarded. We thus could use our SNR calculations derived with binary weights, and checked that the selected set was optimal given the binary weight constraint. Further calculations in the Appendix (Supplementary Material) suggest that removing such a constraint could lead to a modest increase in SNR, of about 10%. More research is needed to see if a multiplicative STDP rule, which does not converge toward binary weights (van Rossum et al., 2000; Gütig et al., 2003), could lead to the optimal graded weights.

Our theoretical study suggests, together with others (Gütig and Sompolinsky, 2006; Brette, 2012), that coincidence detection is computationally powerful. In fact, it could be the main function of neurons (Abeles, 1982; König et al., 1996). In line with this proposal, neurons in vivo appear to be mainly fluctuation-driven, not mean-driven (Rossant et al., 2011; Brette, 2012, 2015). This is the case in particular in the balanced regime (Brette, 2015), which appears to be the prevalent regime in the brain (Denève and Machens, 2016). Several other points suggest that coincidence detection is the main function of neurons. Firstly, strong feedforward inhibitory circuits throughout the central nervous system often shorten the neurons' effective integration windows (Bruno, 2011). Secondly, the effective integration time constant in dendrites might be one order of magnitude shorter than the soma's one (König et al., 1996). Finally, recent experiments indicate that a neuron's threshold quickly adapts to recent potential values (Platkiewicz and Brette, 2011; Fontaine et al., 2014; Mensi et al., 2016), so that only a sudden potential increase can trigger a postsynaptic spike. This enhances coincidence detection. It remains unclear if other spike time aspects such as ranks (Thorpe and Gautrais, 1998) also matter.

Our results show that lower firing rates lead to better signal-to-ratio. It is worth mentioning that mean firing rates are probably largely overestimated in the electrophysiological literature, because extracellular recordings—by far the most popular technique— are totally blind to cells that do not fire at all (Thorpe, 2011). Even a cell that fire only a handful of spikes will be ignored, because spike sorting algorithms need tens of spikes from a given cell before they can create a new cluster corresponding to that cell. Furthermore, experimentalists tend to search for stimuli that elicit strong responses, and, when they can move the electrode(s), tend to look for most responsive cells, introducing strong selection biases. Mean firing rates, averaged across time and cells, are largely unknown, but they could be smaller than 1 Hz (Shoham et al., 2006). It seems like coding is sparse: neurons only fire when they need to signal an important event, and that every spike matters (Wolfe et al., 2010).

Finally, we see an analogy between our theory, and the one of neural associative memory (NAM), in which an output (data) vector is produced by multiplying an input (address) vector by a weight matrix. Unlike NAM, our framework is dynamic, yet after learning, to a first approximation, our STDP neurons count the number of input spikes arriving through reinforced synapses in a short integration window, and each one outputs a 1 (i.e., a spike) if this count exceeds a threshold, and a 0 otherwise, leading to a binary output vector, much like in a binary NAM. It is thus unsurprising that sparsity is desirable both in our theory, and in NAMs (Palm, 2013).

SK and TM did the analytical derivations. TM performed the numerical simulations. SK and TM analyzed the results. TM wrote the first version of the manuscript. SK reviewed it.