The self-organized learning of noisy environmental stimuli requires distinct phases of plasticity

Learning to distinguish between different stimuli despite high levels of noise is an important ability of living beings to ensure survival. However, the underlying neuronal and synaptic processes of this ability are largely unknown.

The brain is responsible for controlling movements of an agent’s body in response to the perceived stimulus. For instance, the agent should run away from a predator or run after the prey. To do so, the agent needs to be able to reliably classify the perceived stimulus despite its natural variability (e.g., different individuals of the same predator species) or noise (e.g., impaired vision by obstacles). In general, the sensory processing systems of the brain map the stimulus representation onto subsequent brain areas yielding successive representations which are increasingly sparse in activity and expansive in the number of neurons. If the feed-forward synaptic weights realizing this mapping are structured according to the inherent organization of the stimuli (e.g., lion versus pig), the increased sparseness and expansion lead to a significant reduction of noise and therefore to a reliable classification (Babadi & Sompolinsky, 2014). However, it remains unclear how the synapses form the required structure despite noise during learning. Furthermore, how can the system reliably adapt to varying levels of noise (e.g., being in a silent forest compared with near a loud stream)?

In the mouse olfactory system, for instance, 1,800 glomeruli receiving signals from olfactory sensory neurons project to millions of pyramidal neurons in the piriform cortex yielding an expansion of the stimulus representation (Mombaerts et al., 1996; Franks & Isaacson, 2006). Activity of the glomeruli is relatively dense with 10%–30% of glomeruli responding to a given natural odor (Vincis, Gschwend, Bhaukaurally, Beroud, & Carleton, 2012), while in the piriform cortex activity drops to 3%–15% indicating an increase in sparseness (Poo & Isaacson, 2009; Stettler & Axel, 2009). A similar picture can be observed in the Drosophila olfactory system. Here, 50 glomeruli project to about 2,500 Kenyon cells in the mushroom body (Balling, Technau, & Heisenberg, 1987; Jefferis et al., 2007). While about 59% of projection neurons respond to a given odor, only 6% of Kenyon cells do (Turner, Bazhenov, & Laurent, 2008). Similar ratios have been observed in the locust olfactory system (Perez-Orive et al., 2002). In the cat visual system, the primary visual cortex has 25 times as many outputs than it receives inputs from the LGN (Olshausen, 2003). In addition, V1-responses to natural visual stimuli are significantly sparser than in the LGN (Dan, Atick, & Reid, 1996; Vinje & Gallant, 2000). Both principles of increased expansion and sparseness of stimulus representations apply to other sensory processing systems as well (Brecht & Sakmann, 2002; Chacron, Longtin, & Maler, 2011; Deweese & Zador, 2003).

The functional roles of increased sparseness as well as expansion have already been proposed in the Marr-Albus theory of the cerebellum (Albus, 1971; Marr, 1969). Here, different representations are thought to evoke different movement responses even though the activity patterns overlap. The Marr-Albus theory demonstrates that through expansion and the sparse activity of granule cells, the overlapping patterns are mapped onto nonoverlapping patterns that can easily be classified. A recent theoretical study has focused on sparse and expansive feed-forward networks in sensory processing systems (Babadi & Sompolinsky, 2014). Here, small variations in activity patterns are caused by internal neuronal noise, input noise, or changes in insignificant properties of the stimuli. For reliable stimulus classification, these slightly varying activity patterns belonging to the same underlying stimulus should evoke the same response in a second layer (or brain area) of a sparse and expansive feed-forward network. Surprisingly, although the network is sparse and expansive, random synaptic weights increase both noise and overlap of activity patterns in the second layer. On the other hand, the same network with synaptic weights structured according to the organization of stimuli reduces the noise and overlap of activity patterns, simplifying subsequent classification. How a network is able to learn the organization of stimuli, shape its synaptic structure according to this organization, and do so even in the presence of noise is so far unknown.

The generally accepted hypothesis of learning is that it is realized by changes of synaptic weights by the process of (long-term) synaptic plasticity (Hebb, 1949; Martin, Grimwood, & Morris, 2000). Synaptic weights are strengthened or weakened depending on the activity of the pre- and postsynaptic neurons (Bi & Poo, 1998; Bliss & Lømo, 1973; Markram, Lübke, Frotscher, & Sakmann, 1997). Hebbian plasticity describes the process of increasing a synaptic weight if the activity of the two connected neurons is correlated (Hebb, 1949). Several theoretical studies indicate that Hebbian plasticity alone would lead to divergent synaptic and neuronal dynamics, thus requiring homeostatic synaptic plasticity (Triesch, Vo, & Hafner, 2018; G. G. Turrigiano, Leslie, Desai, Rutherford, & Nelson, 1998) to counterbalance and stabilize the dynamics (Miller & MacKay, 1994; Tetzlaff, Kolodziejski, Timme, & Wörgötter, 2011; Yger & Gilson, 2015; Zenke & Gerstner, 2017; Zenke, Hennequin, & Gerstner, 2013). In addition, neurons adapt their excitability by the process of intrinsic plasticity (Triesch, 2007; Zhang & Linden, 2003). Intrinsic plasticity regulates the excitability of a given neuron so as to maintain a desired average activity (Benda & Herz, 2003; Desai, Rutherford, & Turrigiano, 1999; LeMasson, Marder, & Abbott, 1993; G. Turrigiano, Abbott, & Marder, 1994), which leads to, for instance, the optimization of the input-output relation of a neuron (Triesch, 2007) or the encoding of information in firing rates (Stemmler & Koch, 1999). Several theoretical studies indicate that the interplay of intrinsic plasticity with synaptic plasticity allows neuronal systems to infer the stimulus intensity (Monk, Savin, & Lücke, 2018; Monk, Savin, & Lücke, 2016), to perform independent component analysis (Savin, Joshi, & Triesch, 2010), or to increase their computational capacity (Hartmann, Lazar, Nessler, & Triesch, 2015; Lazar, Pipa, & Triesch, 2009). However, it remains unclear whether this interplay allows sensory systems, on the one hand, to learn the organization of stimuli despite noise and, on the other hand, to adapt to variations of the noise level.

In the present study, we show that in an expansive network intrinsic plasticity regulates the neuronal activities such that the synaptic weights can learn the organization of stimuli even in the presence of low levels of noise. Interestingly, after learning, the system is able to adapt itself according to changes in the level of noise it is exposed to—even if these levels are high. To do so, intrinsic plasticity has to readapt the excitability of the neurons while the synaptic weights have to be maintained, indicating the need of a two-phase learning protocol.

In the following, first, we present the basics of our theoretical model and methods and demonstrate the ability of a feed-forward network with static random or static structured synaptic weights, respectively, to distinguish between noisy versions of 1,000 different stimuli (similar to Babadi & Sompolinsky, 2014). Then, we introduce the synaptic and intrinsic plasticity rules considered in this study. We train the plastic feed-forward network during an encoding phase without noise and test its performance afterwards by presenting stimuli of different noise levels. Intriguingly, the self-organized dynamics of synaptic and intrinsic plasticity yield a performance and network structure similar to the static network initialized with structured synaptic weights. Further analyses indicate that the performance of the plastic network to classify noisy stimuli greatly depends on the neuronal excitability, especially for high levels of noise. Hence, after learning without noise, we changed the noise level in order to test the performance but let intrinsic plasticity readapt the excitability of the neurons. This readaptation phase significantly increases the performance of the network. Note, however, if synaptic plasticity is present during this second phase, the increase in performance is impeded by a prolonged and severe performance decrease. In the next step, we show that in the encoding phase with both intrinsic and synaptic plasticity the network can also learn from noisy stimuli if the level of noise is low. Again, high levels of noise impede learning and classification performance. Interestingly, after the subsequent readaptation phase the network initially trained with low-noise stimuli performs just as well as a network trained with noise-free stimuli, demonstrating the robustness of this learning mechanism to noise.

The main question of this study concerns how sparse and expansive neural systems, such as sensory processing areas, learn the inherent organization of stimuli enabling a reduction of noise. To tackle this question, similar to a previous study (Babadi & Sompolinsky, 2014), we consider a neural network that consists of two layers of rate-based neurons, with the first layer being linked to the second layer via all-to-all feed-forward synaptic connections. The first layer, called stimulus layer, is significantly smaller (N_S = 1,000 neurons) than the second one, called cortical layer (N_C = 10,000 neurons). The activity patterns of the stimulus layer serve as stimuli or inputs to the cortical layer. These stimulus patterns are constructed of firing rates 𝒮_i ∈ {0, 1} of the stimulus neurons i ∈ {1, …, N_S} with 0 representing a silent neuron and 1 a maximally active one. Neurons belonging to the cortical layer posses a membrane potential u_j (j ∈ {1, …, N_C}) modeled by a leaky integrator receiving the inputs from the stimulus layer. The membrane potential of a cortical neuron is transformed into a firing rate 𝒞_j using a sigmoidal transfer function. Similar to the stimulus neurons, we consider the minimal and maximal firing rates 𝓕^min = 0 and 𝓕^max = 1. Note that the point of inflection of the sigmoidal transfer function ε_j, also called cortical firing threshold, is neuron-specific.

Therefore, if each pattern 𝒮^ν of a stimulus cluster ν is mapped onto a different (random) pattern 𝒞^ν in the cortical layer, ΔC = 1 and the cluster is distributed evenly over the entire cortical layer’s phase space. If each pattern of a stimulus cluster is mapped onto the same pattern of the cortical cluster (the central pattern $C-ν$⁠), ΔC = 0.

In summary, both the stimulus cluster size ΔS as well as the cortical cluster size ΔC are measures for the amount of random fluctuations of different activity patterns belonging to the same underlying stimulus. As such, a network tasked with reducing these random fluctuations should decrease the cluster size, that is, ΔC < ΔS.

Central to the performance in reducing the cluster size or noise are the feed-forward synaptic weights ω_ji between neurons. In the following, we predefine the synaptic weights and test the performance of the network for different levels of noise ΔS_test while keeping the synaptic weights fixed. For each noise level ΔS_test, we create noisy stimulus patterns for all clusters and use them to evaluate the average noise ΔC in the cortical layer. By doing so, we obtain a performance curve ΔC(ΔS_test) of the network. If the weights are initialized randomly, here drawn from a Gaussian distribution 𝒩(0, 2/$NS$⁠), the cortical cluster size ΔC is always larger than the stimulus cluster size ΔS_test, as the performance curve (red line in Figure 1C) is above the identity line (ΔC = ΔS_test; dashed line) for all values of ΔS_test. In other words, the noise of the stimuli (ΔS_test) is amplified by the network by increasing the variations between different cortical patterns of the same underlying stimulus (ΔC > ΔS_test). Note that this amplification of noise is present even though the network is expansive and sparse (Babadi & Sompolinsky, 2014).

These results show that expansive and sparse networks reduce the noise of stimuli if the synaptic weights from the stimulus to the cortical layer are structured according to the underlying organization of stimuli (here according to the central stimulus patterns $S-ν$⁠). So far, we have used Equation 5 to artificially set the synaptic weights to the correct values. The question remains how a network can learn these values from the environmental stimuli.

Training is carried out in repeated learning steps or trials. In each learning step L, we present all central stimulus patterns $S-ν$ (ν ∈ {1, …, P}) to the network once, ensuring there is no chronological information (see Methods for details). This corresponds to a stimulus cluster size ΔS_learn = 0 or noise-free learning. At different stages of learning (that is, after different numbers of learning steps), we test the performance of the network for different levels of noise ΔS_test as has been done for the static networks.

As learning progresses (Figure 2A), the performance curve develops from the random network’s (red line), which amplifies stimulus noise, into one similar to the structured network’s performance curve (blue compared with magenta line). The plasticity mechanisms (Equations 6 and 7) enable the network to encode the organization of the stimuli (existence of different clusters) in a self-organized manner, with most of the performance gained in the first L = 60,000 learning steps: During learning, the synaptic weights evolve from the initial Gaussian distribution into a bimodal distribution with peaks at about 0.033 and 0 (see Figure 2B for an example). The emergence of the bimodal weight distribution and its link to the network performance can be explained as follows: Because of the random initialization of the synaptic weights, each central stimulus pattern leads to a different membrane potential in a given cortical neuron such that all P stimuli together yield a random distribution of evoked membrane potentials (see, e.g., Figure 2D; red lines depict membrane potentials). As the target firing rate is chosen such that each neuron ideally responds to only one central stimulus pattern (as 𝓕^TP = 1), intrinsic plasticity adapts the firing threshold ε_j of a neuron such that one of the evoked membrane potentials leads to a distinctly above-average firing rate. Consequently, synapses connecting stimulus neurons being active at the corresponding stimulus pattern with the considered cortical neuron are generally strengthened the most by Hebbian synaptic plasticity. These synapses will likely form the upper peak of the final synaptic weight distribution (Figure 2B). Meanwhile, all other synaptic weights are dominated by the synaptic weight decay (second term in Equation 6) and will later form the lower peak of the distribution at zero. As the continued differentiation of the synaptic weights increases the evoked membrane potential of the most influential central stimulus pattern, these two processes of synaptic and neuronal adaptation drive each other. Interestingly, the resulting synaptic weights are correlated to the structured synapses (Figure 2C) initialized using Equation 5 (here the cortical patterns 𝓡^ν of Equation 5 were generated using the central cortical patterns 𝒞^ν of the plastic network at the corresponding learning step L; see Methods for further details). Note that the cortical firing thresholds ε_j of the plastic network become correlated to the values of the static, structured one as well (Figure 2E).

In summary, synaptic and intrinsic plasticity interact and adapt the neuronal network such that, in a noise-free environment, it learns to encode the organization of the stimuli in a way comparable to a static, prestructured network. The trained network is then able to reduce the noise of environmental stimuli even for noise levels up to about 0.6.

While being structurally similar, the performance of the trained, plastic network (Figure 2A, blue) appears significantly better than the performance of the static, structured network (magenta). This fact is not self-explanatory, since both the synaptic weights as well as the cortical firing thresholds are strongly correlated between both networks (Figure 2C, E). However, a closer look at the cortical firing thresholds and their link to the performance of the network reveals the cause of this difference:

In the trained network (L = 200,000), as mentioned before, each cortical neuron should fire in response to the central stimulus pattern $S-ν$ of exactly one cluster and stay silent otherwise. As an example, we will focus on cortical neuron j = 1, which fires in response to the central stimulus pattern $S-842$ of cluster ν = 842 and remains silent in response to all other central stimulus patterns. In general, two types of errors can occur.

False negatives (a stimulus of cluster 842 is presented and cortical neuron 1 falsely does not fire): Noisy patterns of cluster 842 elicit a distribution of membrane potentials in cortical neuron 1 (Figure 3A), which depends on the stimulus cluster size ΔS_test, that is, the level of noise. All noisy stimulus patterns 𝒮⁸⁴² that evoke a membrane potential in neuron 1 that is higher than the neuron’s firing threshold ε₁ result in a strong activation of neuron 1. The neuron therefore classifies these 𝒮⁸⁴² correctly as belonging to cluster 842. However, noisy patterns 𝒮⁸⁴² evoking a lower membrane potential than ε₁ do not elicit strong activation of cortical neuron 1. These noisy patterns are falsely classified as not belonging to cluster 842 and correspond to false negatives.

False positives (a stimulus of a cluster ν ≠ 842 is presented and cortical neuron 1 falsely fires): Similar to the analysis of false negatives, the analysis of false positives can be done with clusters whose central patterns should not elicit activity in cortical neuron 1. The distribution of membrane potentials evoked by noisy patterns of these clusters does not significantly depend on the stimulus cluster size ΔS_test (Figure 3B). Noisy stimulus patterns 𝒮^ν (ν ≠ 842) are classified correctly as not part of cluster 842 if neuron 1’s membrane potential is lower than its firing threshold ε₁. All noisy patterns evoking a higher membrane potential falsely lead to a firing of cortical neuron 1. They correspond to false positives.

The performance of the network or the total error e_tot,j thus depends on a cortical neuron’s firing threshold in a nonlinear manner. Given noise-free stimuli (ΔS_test = 0), in a large regime of different values for the firing threshold cortical neuron 1 makes almost no classification error (dashed red line in Figure 3C; gradient in shaded blue area is less than 0.001). For a higher noise level (e.g., ΔS_test = 0.7, Figure 3D), there is no such extended regime of low-error threshold values. Instead, small variations of the firing threshold can drastically change the classification performance, since the membrane potential response distributions overlap at these noise levels (Figure 3A, B).

During training without noise (ΔS_learn = 0), the neuronal firing threshold ε₁ rose to the lower bound of the low-error regime of ΔS_test = 0 (blue area; Figure 3C). In the static network, however, firing thresholds ε_1,stat were placed at the center of the highest and second highest membrane potentials in response to central stimulus patterns, leading to much higher values. Therefore, if the network performance is tested for small stimulus clusters (low noise ΔS_test; Figure 3C), the static and the plastic network have a similar total error and classification performance. For larger stimulus clusters (high noise levels ΔS_test; Figure 3D), on the other hand, the higher firing thresholds of the static network lead to considerably more misclassification and consequently to a higher cortical cluster size ΔC. Consequently, the fact that the relation between the threshold and its classification error e_j,tot depends on the noise ΔS_test provides an explanation for the large performance differences between the static structured and the plastic network (Figure 2A).

This example (Figure 3) demonstrates that the value of the neuron-specific threshold ε_j,opt optimizing a neuron’s classification performance depends on the stimulus cluster size ΔS_test or current level of noise (dotted lines in Figure 4A for neuron 1 in blue and neuron 2 in green). The firing thresholds after training (solid lines in Figure 4A), however, are independent of ΔS_test, as they are determined by the noise present during training (ΔS_learn = 0). For ΔS_test ≲ 0.5 these thresholds are within the regime of low total error (shaded areas indicate the low-error regime for each neuron marked by blue area in Figures 3C and 3D) yielding a high classification performance of the network. However, for ΔS_test ≳ 0.5 the thresholds ε_j resulting from training without noise (ΔS_learn = 0) start to deviate significantly from the optimal thresholds ε_j,opt, leading to a decreasing classification performance (Figure 2A and Figure 4C, solid lines for total error of individual neurons). Interestingly, the deviation from the optimal threshold is accompanied by a decrease of the average activity level (solid lines; Figure 4B), while the optimal thresholds would keep the cortical activity close to the target activity 𝓕^T = 0.001 (dotted lines; for ΔS_test ≳ 0.85 the total error is high and nearly independent of the threshold; see Supplementary Figure 1). We thus expect that after initial learning, intrinsic plasticity could readapt the neuronal firing thresholds according to the present level of noise such that the target activity is maintained and the thresholds ε_j,adapt approximate the optimal threshold values ε_j,opt.

We therefore considered a second learning phase, the readaptation phase, which is conducted after the initial training or encoding phase is completed. In the readaptation phase, the stimulus cluster size will be the same that the performance is tested for, that is, ΔS_test. For now, synaptic plasticity is deactivated as we will only focus on intrinsic plasticity adapting the cortical firing thresholds ε_j,adapt. To implement this readaptation phase, after the first learning phase is completed, we repeatedly presented one noisy pattern 𝒮^ν per cluster using a stimulus cluster size ΔS_test. Threshold adaptation was stopped when the mean of all cortical thresholds changed by less than 0.0001% in one step, which resulted in less than 7,000 steps for each ΔS_test. As expected, intrinsic plasticity adjusts the firing thresholds during this second phase so as to achieve the target firing rate 𝓕^T for all ΔS_test (dashed lines; Figure 4B). Furthermore, the adapted thresholds ε_j,adapt (dashed lines; Figure 4A) are similar to the optimal thresholds ε_j,opt (dotted lines). This leads to a near-optimal classification performance, which is considerably better than without a readaptation phase (Figure 4C, dashed lines lie on top of dotted line).

Importantly, if synaptic plasticity is also present during this second learning phase, ΔC increases dramatically with ongoing readaptation (solid lines in Figure 4D; different colors represent different noise levels). The initial drop of ΔC is due to intrinsic plasticity (dashed lines show final ΔC-values for intrinsic plasticity alone), while synaptic plasticity leads to a prolonged deterioration of the previously learned synaptic structure if stimuli are too noisy. We therefore conclude that the network has to maintain the synaptic weight structure during the readaptation phase, which we recreate by turning synaptic plasticity off. By doing so, the neuronal system can reliably adjust to stimuli of various noise levels using intrinsic plasticity for adapting the excitability of neurons.

Up to now, we have shown that a sparse, expansive network can learn the underlying organization of noise-free stimuli (ΔS_learn = 0) by means of synaptic and intrinsic plasticity. Afterwards, a readaptation phase with intrinsic plasticity alone enables the network to readapt to any arbitrary level of noise ΔS_test (Figures 4A–C). However, if synaptic plasticity is active during the readaptation phase, the noise of stimuli leads to a disintegration of the synaptic structure (Figure 4D). Therefore, it is unclear whether the network can also learn the organization of stimuli from noisy—instead of noise-free—stimuli by using synaptic plasticity.

To test this, we now investigate the effect of noisy stimuli during training in the encoding phase (i.e., ΔS_learn > 0). To do so, we present one noisy stimulus pattern 𝒮^ν per cluster in each learning step L using a stimulus cluster size ΔS_learn. In noisy environments with up to ΔS_learn = 0.2, cortical neurons show neuronal and synaptic dynamics (Figure 5A, B) similar to noise-free learning (Figure 2B, D). Synaptic weights and firing thresholds become correlated to the static, structured network (Figure 5E, F) to a comparable degree (Figure 2C, E). Nevertheless, because of the noise of the stimuli, some cortical neurons do not manage to separate one stimulus cluster from all others (Figure 5D, ∼24% of all neurons for ΔS_learn = 0.2). Consequently, multiple clusters trigger the Hebbian term of synaptic plasticity (Equation 6) such that all synaptic weights approach a medium value (Figure 5C). These synaptic weights diminish the correlation to the static, structured synaptic weights as the final distribution is slightly broader (Figure 5E) than the one from learning without noise (Figure 2C). Furthermore, the cortical neurons without structured incoming synaptic weights (unimodal weight distribution) on average have a lower final firing threshold (blue outliers in Figure 5F).

In general, low levels of noise (ΔS_learn ≲ 0.25) are tolerated by the network without large losses in performance (Figure 6A). The failed-learning cortical neurons (Figure 5C, D), which become more with higher noise levels (see Supplementary Figure 3), have a negative effect on the performance of the network. At ΔS_learn ≳ 0.25, the noise is so strong that the system is not able to recognize and learn the underlying organization of stimuli (that is, the existence of different clusters). However, if there is little or even no noise during learning, the network can subsequently not only classify stimuli of that same level of noise, but also classify significantly noisier stimuli (white area above orange dashed identity line). This result indicates that the network does not adapt specifically to only the noise level ΔS_learn it is learning from, but that the network generalizes across a broad variety of different noise levels ΔS_test. For instance, although the network may learn from stimulus patterns with an average noise level of ΔS_learn = 0.1, it can reliably classify stimuli of noise levels ΔS_test from 0 to about 0.6 afterwards.

Furthermore, the performance of a network that was successfully trained in a noisy environment can be drastically improved by a subsequent readaptation phase. Using this second phase in order to (re)adapt the neuronal excitabilities to the level of noise ΔS_test that will subsequently be tested for enables the network to classify stimuli up to even higher noise levels of ΔS_test ≈ 0.8 (Figure 6B). Consequently, the readaptation phase provides a significant advantage for a large regime of stimulus cluster sizes (red area in Figure 6C). Even more so, stimulus clusters with sizes ΔS_test ∈ (0.6, 0.8) can only be classified by using the readaptation phase. The decrease in performance for noise levels between ΔS_learn ∈ (0.2, 0.3) and ΔS_test ∈ (0.8, 1.0) (blue area) is not crucial given the low level of performance (Figure 6A).

In summary, sparse, expansive networks can learn the clustered organization of noisy stimuli (underlying stimuli might be triangle, circle, and cross like in Figure 7) by the interplay of synaptic and intrinsic plasticity in a self-organized manner. During the initial encoding phase, low levels of noise ΔS_learn can be tolerated by the system, while higher levels of noise obstruct the network’s ability to learn the organization of stimuli. After the encoding phase, the network can reliably classify noisy patterns of up to ΔS_test ≈ 0.6 if synaptic weights and neuronal firing thresholds are fixed (ẇ = 0; $ε̇$ = 0). On the other hand, the performance decreases significantly if both synaptic and intrinsic plasticity are allowed to modify the network’s structure during the presentation of these noisy stimuli (ẇ ≠ 0; $ε̇$ ≠ 0). Interestingly, if the synaptic structure is maintained while the excitability of the cortical neurons can adapt (ẇ = 0; $ε̇$ ≠ 0), the network can successfully classify stimuli even in the presence of very high levels of noise (see Figure 7 bottom for examples). These results suggest that learning in the presence of noise requires two distinct phases of plasticity: initial learning of the organization of environmental stimuli via synaptic and intrinsic plasticity in the encoding phase followed by the readaptation phase using only intrinsic plasticity in order to readapt to the current level of noise.

How do neuronal systems learn the underlying organization of the surrounding environment in realistic, noisy conditions? In this study, we have shown that sparse and expansive networks can reliably form the required neuronal and synaptic structures via the interplay of synaptic and intrinsic plasticity. Among others, our results indicate that after learning the classification of diverse environmental stimuli in the presence of high levels of noise works best if the synaptic structure is more rigid than the neuronal structure, namely the excitabilities of the neurons. Thereby, our model predicts that higher levels of noise lead to lower firing thresholds or (on average) increased neuronal excitabilities (Figure 4A).

Furthermore, our model predicts that classification performance is highest if the system is adapted to the perceived level of noise. We propose the following psychophysical experiment related to pattern recognition in order to test this prediction: First, subjects have to learn a set of previously unknown patterns, such as visual or auditory patterns. Second, they have to identify noisy versions of these patterns. We propose that the classification performance of a given noisy pattern depends on the history of patterns the subject perceived beforehand. Specifically, our model predicts that a given noisy pattern is classified most reliably if the previously perceived patterns had the same level of noise. By transferring this protocol to an animal model, the predicted course of the adaptation of the firing thresholds could be verified, too.

After the successful learning of the inherent organization of stimuli, in this study we changed the synaptic variability by “turning off” the dynamics of synaptic plasticity (Figure 4). This change of the timescale of synaptic plasticity between the encoding and the readaptation phase could be related to the dynamics during the development of the visual system (Daw, 2003; Daw, Fox, Sato, & Czepita, 1992; Hensch, 2004; Hooks & Chen, 2007). During the critical period, the early visual system is quite susceptible to new sensory experiences and the system is very plastic. In addition, the visual range during early developmental phases is limited, which could imply lower levels of noise. Thus, the encoding phase in our model could be linked to the critical period. By contrast, the matured visual system is quite rigid, matching the requirements of the readaptation phase, which predicts that the sensory system should be able to adapt to different levels of noise by (only) changing the neuronal excitabilities (Figure 6).

One of the major assumptions of this work, similar to a previous study (Babadi & Sompolinsky, 2014), is that environmental stimuli are organized such that they can be grouped into clusters. Each of these clusters has the same Gaussian noise level ΔS. Natural stimuli, however, have much more structured noise statistics. Nevertheless, the mechanisms considered here that enable the network to compensate for noisy stimuli (i.e., synaptic and intrinsic plasticity) do not specifically rely on the noise being Gaussian. Intrinsic plasticity will still maintain the target firing rate independent of precisely how the membrane potential distributions (Figure 3A, B) are shaped by different types of noise. Given our results (Figure 4), we expect that the neuronal thresholds resulting in the target firing rate will be close to the optimal threshold. Furthermore, the exponential synaptic decay may lead to less reliable presynaptic stimulus neurons having a smaller impact on a cortical neuron’s firing. In addition to clusters not being Gaussian shaped, in a natural environment each underlying stimulus may also have a different overall level of noise such that ΔS^ν depends on the cluster ν. However, if the synaptic structure has already been learned during the encoding phase, we expect that cluster-specific $ΔStestν$ do not have an impact on the classification performance, as each cortical neuron becomes selective to only one stimulus cluster (Figure 2D). In addition, only the noise level of this selected cluster defines the optimal firing threshold (Figure 3). Therefore, the firing threshold of each neuron can be tuned to its distinct, optimal threshold value, which is independent of the noise levels of other clusters. On the other hand, we expect that different $ΔSlearnν$ during the encoding phase will lead to over- and underrepresentations of stimulus clusters in the network. Since noise attenuates competition between clusters (Figure 5C, D), clusters with high $ΔSlearnν$ are less competitive and will subsequently be underrepresented. Nevertheless, the underrepresentation could be an advantage, as stimuli that are too noisy are less informative about the environment than others; consequently, the neuronal system attributes a smaller amount of resources (neurons and synapses) to them. However, the effect of cluster-specific noise on the neuronal and synaptic dynamics have to be investigated further.

Additionally, some stimulus clusters might be perceived more often than others. The corresponding representations would become larger than average, since their relevant synapses are strengthened more often by Hebbian synaptic plasticity, leading to a competitive advantage. Larger representations of more frequently perceived stimulus clusters might provide a behavioral advantage, as these clusters also need to be classified more often. However, the discrepancy between the frequency of such a cluster and the target firing rate of a cortical neuron responding to it might pose a problem. As intrinsic plasticity tries to maintain the target activity, the firing threshold would be placed so high that even slight noise could not be tolerated. One solution might be that neurons could have different target activities (G. G. Turrigiano, 2008) and clusters are selected such that target activity and presentation frequency match. A different mechanism could be global inhibition. A single inhibitory neuron or population of neurons connected to all relevant cortical neurons could homeostatically regulate the activity of the cortical layer by providing inhibitory feedback. Such a mechanism has been identified, for instance, in the Drosophila mushroom body (Eichler et al., 2017; Faghihi, Kolodziejski, Fiala, Wörgötter, & Tetzlaff, 2013).

In this study, only one combination of three different plasticity rules was investigated. Of course, many more plasticity mechanisms are conceivable and have been widely studied (Dayan & Abbott, 2001; Miner & Triesch, 2016; Tetzlaff, Kolodziejski, Markelic, & Wörgötter, 2012; Zenke, Agnes, & Gerstner, 2015). One mechanism could be synaptic scaling regulating the synaptic weights instead of the neuronal excitability such that the neurons reach a certain target firing rate (Desai, Cudmore, Nelson, & Turrigiano, 2002; Hengen, Lambo, Van Hooser, Katz, & Turrigiano, 2013; Keck et al., 2013; Tetzlaff et al., 2011; G. G. Turrigiano et al., 1998). However, the timescale of synaptic scaling is significantly slower than the timescale of intrinsic plasticity, which could increase the duration of the readaptation phase required by the neuronal system to adapt to new levels of noise. On the other hand, faster homeostatic mechanisms (Zenke & Gerstner, 2017) could result in a shorter duration of readaptation. However, the influence of further plasticity mechanisms on the dynamics of sparse, expansive networks has to be analyzed in future studies.

It is usually assumed that homeostatic synaptic plasticity is required for competition (Abbott & Nelson, 2000; Miller, 1996). In the present study, however, competition arises from the interactions of Hebbian synaptic plasticity and homeostatic intrinsic plasticity alone. Homeostatic intrinsic plasticity maintains a certain activity of a given cortical neuron. Stimuli compete for this activity. If one stimulus gains an activity advantage, it will see synapses activated by it strengthened. This leads to less strengthening of other synapses, because the occurrence of Hebbian synaptic plasticity is limited by homeostatic intrinsic plasticity. Synapses will only subsequently be weakened due to homeostatic synaptic plasticity (exponential decay term), which does not interfere in the interaction between Hebbian synaptic and homeostatic intrinsic plasticity generating competition (see Supplementary Figure 2). Consequently, the widely held opinion that homeostatic synaptic plasticity is required for competition might have to be revised.

Even though expansion is a common feature of sensory processing networks, it is not a prerequisite for the results presented here. Nonexpansive networks, too, can learn to distinguish different clusters, although they do not reach the performance of an expansive network (see Supplementary Figure 4). This means that nonexpansive networks as well profit from a two-phase learning protocol as suggested here.

Overall, this study suggests the following answer to how networks learn to classify stimuli in noisy environments: Learning takes place in two distinct phases. The first phase is the encoding phase. Hebbian synaptic and homeostatic intrinsic plasticity structure synaptic weights so as to represent the organization of stimuli, with each neuron becoming selectively responsive to a single stimulus cluster. Optimal synaptic structure is achieved if stimuli are noise-free. The second learning phase, called readaptation phase, ensues in an arbitrarily noisy environment. Here, synaptic weights have to be maintained in order to preserve the previously learned synaptic structure. Meanwhile, homeostatic intrinsic plasticity regulates the activity of neurons. The firing thresholds are thereby adapted to their optimal values, maximizing classification performance in the current environment (Figure 7).

In this study, a two-layered feed-forward network of rate-based neurons is investigated (Figure 1A). The first layer, called stimulus layer, consists of N_S = 1,000 neurons, while the second layer, called cortical layer, consists of N_C = 10,000 neurons. Feed-forward synaptic connections exist from all stimulus to all cortical neurons. Their synaptic strengths are given by ω_ji where j ∈ {1, …, N_C} denotes the postsynaptic cortical neuron and i ∈ {1, …, N_S} the presynaptic stimulus neuron. No recurrent connections are present.

The parameter κ = 1 ⋅ 10⁻² determines the adaptation speed of intrinsic plasticity. If the firing rate 𝒞_j of cortical neuron j is larger than the target firing rate 𝓕^T, the threshold ε_j increases such that 𝒞_j decreases (assuming the input stays constant), and vice versa.

The parameters μ = 1 ⋅ 10⁻⁵ and η = 3 ⋅ 10⁻⁸ determine the speed of the Hebbian correlation learning term and the exponential decay of synaptic weights, respectively. We assume that synaptic plasticity acts much slower than the presentation time of a single input pattern such that the fixed point of the leaky integrator given by Equation 9 does not significantly change during the presentation of a single input and the simplification thus still holds.

This normalization quantifies the average overlap two random cortical patterns with the same firing rates would have.

In the following, the classification errors e_fp,j (false positives) and e_fn,j (false negatives) of single neurons in a trained network will be set into relation with the cortical cluster size ΔC. To do so, we will first discuss the cortical cluster distance d_C and the ergodicity of the network. We will then use the results to derive the relation between the cortical cluster size and the classification errors.

That is, for each ΔS every cortical pattern ν and every cortical neuron j must have the same average firing rate. This is true given the assumptions we have already discussed: The cortical neurons are homogeneous, that is, they all have a bimodal weight distribution and so forth, and each cortical neuron is responsive to a random cluster.

When initialized randomly, the synaptic weights ω_ji are drawn from a Gaussian distribution with mean 0 and variance $2NS$⁠. The synaptic weights can also be initialized in a structured manner according to ω_ji = $100NS∑ν=1P$(⁠$S-iν$ − $12$⁠)(⁠$Rjν$ − 𝓕^T) (similar to Babadi & Sompolinsky, 2014; Tsodyks & Feigelman, 1988). The factor 100 scales the synaptic weights such that they are in the same order of magnitude as the synaptic weights in the trained dynamic network ensuring that they are comparable. 𝓡^ν are cortical patterns that are generated using one of the following methods: For Figure 1 𝓡^ν are random patterns of ones and zeros where each pattern ν and each cortical neuron j has an activity of 𝓕^T. For all other results, 𝓡^ν are computed via $Rjν$ = Θ(⁠$C-jν$ − 𝒯_j), where Θ denotes the Heaviside function and the thresholds 𝒯_j are chosen such that each cortical neuron j achieves an activity of 𝓕^T. This results in cortical patterns 𝓡^ν that are correlated to the central cortical patterns $C-ν$ of an existing network.

The cortical membrane thresholds ε_j are then initialized such that each cortical neuron j achieves an average firing rate of the target firing rate 𝓕^T at the central cortical patterns. In order to find the corresponding membrane thresholds ε_j, the secant method is used with initial values of 0 and the mean of the highest and second highest (as 𝓕^TP = 1) membrane potentials of cortical neuron j. If structured synaptic weights are used, this leads to ε_j close to the mean of the highest and second highest membrane potentials of neuron j.

Training is done by repeatedly presenting stimulus patterns for one time step Δt = 1 each. The high computational demand of simulating a network with 11,000 neurons for 200,000 training steps each containing 1,000 patterns for just a single parameter set made parallel simulation of patterns necessary and required the usage of a computer cluster. To this end, each training step consists of parallel simulation of one stimulus pattern per cluster. Stimulus patterns are generated using a stimulus cluster size ΔS_learn and the current learning step is denoted by L. The synaptic weights ω_ji and cortical thresholds ε_j are updated at the end of each training step according to Δω_ji = $∑ν=1Pω̇ji$(𝒞^ν) and Δε_j = $∑ν=1Pε̇j$(𝒞^ν), where 𝒞^ν denotes the cortical pattern of cluster ν that was simulated in this learning step. This is a reasonable approximation if a single learning step changes the network’s state only slightly, as is the case in this study.

The cortical cluster size ΔC (cf. Equation 14 and Equation 16) in response to a stimulus cluster size ΔS_test is approximated using 10 noisy patterns per cluster.

If intrinsic plasticity is active during the testing phase, repeatedly, one noisy pattern per cluster is simulated using a stimulus cluster size ΔS_test. After the mean of all cortical thresholds changed by less than 0.0001%, the cortical cluster size ΔC is calculated for the given stimulus cluster size ΔS_test. In order to speed up its computation, we used the central cortical patterns $C-ν$ and cortical cluster distance d_C from ahead of the adaptation phase and were able to verify that this does not influence the results. The thresholds are reset to their previous values afterwards. The entire procedure is performed for all ΔS_test, each requiring fewer than 7,000 learning steps for the thresholds to converge.

Supporting information for this article is available at https://doi.org/10.1162/netn_a_00118.

Steffen Krüppel: Conceptualization; Data curation; Formal analysis; Investigation; Methodology; Software; Validation; Visualization; Writing - Original Draft; Writing - Review & Editing. Christian Tetzlaff: Conceptualization; Funding acquisition; Investigation; Methodology; Project administration; Resources; Supervision; Writing - Original Draft; Writing - Review & Editing.

Christian Tetzlaff, Horizon 2020 Framework Programme (http://dx.doi.org/10.13039/100010661), Award ID: 732266. Christian Tetzlaff, Deutsche Forschungsgemeinschaft (http://dx.doi.org/10.13039/501100001659), Award ID: CRC 1286 (project C1).

Synaptic weights:

The average transmission efficacy of a synapse quantified as a single number being adapted by diverse plasticity processes.

Olfactory system:

The olfactory system includes all brain areas processing sensory information related to the sense of smell.

Piriform cortex:

One brain area in the cerebrum processing sensory information of the sense of smell.

Mushroom body:

A brain area in insects that is important for odor-related learning and memory.

Cerebellum:

A brain area responsible for the control of movement-related functions such as coordination, timing, or precision.

Synaptic plasticity:

General term for different kinds of biological mechanisms adapting the weights of synapses depending on neuronal activities.

Homeostatic synaptic plasticity:

Synaptic plasticity mechanism adapting the synaptic weights such that the neuronal dynamics remain in a desired “healthy” regime.

Intrinsic plasticity:

General term for different kinds of biological mechanisms adapting the firing threshold or excitability of a neuron.