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

Abstract

Along sensory pathways, representations of environmental stimuli become increasingly sparse and expanded. If additionally the feed-forward synaptic weights are structured according to the inherent organization of stimuli, the increase in sparseness and expansion leads to a reduction of sensory noise. However, it is unknown how the synapses in the brain form the required structure, especially given the omnipresent noise of environmental stimuli. Here, we employ a combination of synaptic plasticity and intrinsic plasticity-adapting the excitability of each neuron individually-and present stimuli with an inherent organization to a feed-forward network. We observe that intrinsic plasticity maintains the sparseness of the neural code and thereby allows synaptic plasticity to learn the organization of stimuli in low-noise environments. Nevertheless, even high levels of noise can be handled after a subsequent phase of readaptation of the neuronal excitabilities by intrinsic plasticity. Interestingly, during this phase the synaptic structure has to be maintained. These results demonstrate that learning and recalling in the presence of noise requires the coordinated interplay between plasticity mechanisms adapting different properties of the neuronal circuit.

Keywords: Intrinsic plasticity; Learning; Noise-robustness; Sensory pathways; Synaptic plasticity.

Figure 1.

Network model and mathematical approach to quantify the ability of an expansive, sparse network to reduce noise. (A) The feed-forward network consists of two layers of rate-based neurons with the stimulus layer projecting stimuli onto the cortical layer via all-to-all feed-forward synaptic connections. Stimuli are organized in P = 1,000 clusters, with each cluster ν consisting of a characteristic central pattern ${\stackrel{-}{\mathcal{S}}}^{\nu }$ (black dots) and noisy patterns 𝒮^ν (blue halos around dots). The size of the stimulus clusters ΔS corresponds to the level of noise and is indicated schematically by the size of the blue halos. Stimulus clusters are mapped by the synaptic connections to cortical clusters containing central cortical patterns ${\stackrel{-}{\mathcal{C}}}^{\nu }$ and noisy cortical patterns 𝒞^ν. (B) Illustration of different patterns and measures used in this study. The activity of each neuron (box) is indicated by its gray scale (left: stimulus layer; right: cortical layer). The central pattern ${\stackrel{-}{\mathcal{S}}}^{\nu }$ of each stimulus cluster (underlying stimulus) evokes a specific central pattern ${\stackrel{-}{\mathcal{C}}}^{\nu }$ in the cortical layer. Noisy versions of a central stimulus pattern (here 𝒮²) activate different cortical patterns with their average distance Δc from the original pattern depending on the structure of the feed-forward synaptic weights. (C) Random synaptic weights increase the cluster size for all stimulus cluster sizes, that is, ΔC > ΔS_test; the noise in the stimuli is thus amplified by the network. (D) Synapses that are structured in relation to the organization of the underlying stimuli (stimulus central patterns ${\stackrel{-}{\mathcal{S}}}^{\nu }$) decrease the size of clusters, that is, the noise, up to a medium noise level (ΔS_test ≈ 0.45). (C, D) Dashed line indicates ΔC = ΔS_test.

Figure 2.

Self-organization of the synaptic and neuronal structure via synaptic and intrinsic plasticity in a noise-free environment. (A) By repeatedly presenting one stimulus pattern 𝒮^ν per cluster per learning step L using a stimulus cluster size ΔS_learn = 0 (i.e., presenting the central stimulus patterns ${\stackrel{-}{\mathcal{S}}}^{\nu }$), the network’s performance develops from the noise-amplification of a random network (red, equal to Figure 1C) to a performance significantly decreasing the level of noise for ΔS_test up to about 0.6 (blue). (B, C) During learning, the synaptic weights develop into a bimodal distribution (B; only the weights connecting to neuron 1 are shown) that is correlated to the distribution of the static, structured network (C). (D) For each cortical neuron (here shown for neuron 1), the firing threshold (green) increases such that only one central stimulus pattern can evoke a membrane potential larger than the threshold (red lines depict membrane potentials). (E) Similar to the synaptic weights (C), the firing thresholds tend to become correlated to the ones of the static, structured network.

Figure 3.

The Classification performance of each neuron depends on its firing threshold. In a single cortical neuron (here neuron j = 1), multiple noisy stimulus patterns of the same stimulus cluster elicit a distribution of membrane potentials. Two distinct distributions can be identified: (A) The distribution of membrane potentials evoked by noisy stimulus patterns belonging to the cluster whose central pattern elicits firing in the given cortical neuron (blue; here cluster ν = 842). For any ΔS_test, all stimuli yielding a membrane potential that is below the neuron’s firing threshold (dashed line; ε₁) do not elicit a strong neuronal response representing false negatives. The distribution significantly depends on the level of noise ΔS_test. (B) The membrane potential distribution in response to noisy stimulus patterns of the clusters the neuron is not tuned to (ν ≠ 842). Here, all stimuli yielding a membrane potential above the firing threshold are false positives. (C) ΔS_test = 0: A higher firing threshold ε leads to more false negatives (orange) but fewer false positives (magenta) and vice versa for a lower threshold. The sum of errors (dashed red) is negligible in a large regime (blue area: gradient is less than 0.001). (D) ΔS_test = 0.7: With higher levels of stimulus noise, the total error and the classification performance depend critically on the firing threshold. (C, D) ε_1,opt: optimal value of the firing threshold for the given level of noise ΔS_test yielding the lowest total error; ε₁: value of the firing threshold after learning with noise-free stimuli (ΔS_test; Figure 2); ε_1,stat: firing threshold in the static network (Figure 1D).

Figure 4.

A second learning phase—the readaptation phase—enables the neuronal system to readapt to arbitrary noise levels using intrinsic plasticity. (A–C) After learning without noise, a second learning phase with the noise level ΔS_test and only intrinsic plasticity active enables the thresholds to readapt from the values after the first learning phase ε_j (solid lines) to adapted values ε_j,adapt (dashed lines), close to the optimal threshold values ε_j,opt (dotted lines) increasing performance. Blue: neuron 1; green: neuron 2. (A) ΔS_test-dependency of cortical thresholds; shaded areas indicate regimes of low error gradient (Figure 3C); (B) ΔS_test-dependency of average activities; (C) ΔS_test-dependency of total error (dashed lines lie on top of dotted lines). Solid red line shows performance of whole network (from Figure 2A), confirming Equation 8. (D) If synaptic plasticity is present during the second learning phase as well, ΔC initially drops because of intrinsic plasticity and then increases with ongoing presentation of noisy stimuli, indicating a disintegration of the synaptic structure (solid lines; different colors represent different noise levels). Dashed lines indicate ΔC-values for a second learning phase with intrinsic plasticity alone.

Figure 5.

Self-organization of the synaptic and neuronal structure in a noisy environment. The dynamics of synaptic and intrinsic plasticity enable the sparse, expansive network to learn the underlying organization of stimuli from noisy stimulus patterns (here ΔS_learn = 0.2). (A, B) The majority of cortical neurons develop a distribution of incoming synaptic weights (A, blue lines) and membrane potential responses (B, red lines) similar to the ones learning without noise (Figure 2B, D). Here shown for neuron 2. Green line in (B) denotes the threshold. (C, D) However, the noise prevents some neurons (∼24%) to form a proper synaptic structure (C), yielding a firing threshold (D) that does not separate the membrane potential evoked by one cluster from the others. Therefore, these neurons are not tuned to one specific cluster. Here shown for neuron 1. (E, F) Overall, the network trained by noisy stimuli develops synaptic weights (E) and firing thresholds (F) similarly correlated to the static, structured network than the network trained without noise (Figure 2C, E). The few neurons that failed learning lead to a minor broadening of the distributions.

Figure 6.

The network can reliably learn from noisy stimuli with and without a readaptation phase. (A) Despite the presence of noise ΔS_learn during learning, the network can learn the organization of stimuli and, after encoding, classify stimuli of even higher noise levels ΔS_test. However, higher levels of ΔS_learn decrease the performance. Color code depicts ΔC, green line marks ΔC = ΔS_test. (B) If the learning phase is followed by a readaptation phase using only intrinsic plasticity and the level of noise ΔS_test with which the system is tested, the overall classification performance increases drastically. Now, stimuli with a noise level of up to ΔS_test ≈ 0.8 can be classified. (C) The readaptation phase leads to a large performance gain for medium and high noise levels ΔS_test. Color code depicts the difference between the network without and with a readaptation phase. Red area represents a benefit by using the readaptation phase. (A–C) Orange dashed line: identity line ΔS_learn = ΔS_test.

Figure 7.

Schematic summary of results. Noisy patterns 𝒮^ν are repeatedly generated from original stimuli ${\stackrel{-}{\mathcal{S}}}^{\nu }$ (e.g., a triangle, a circle, and a cross) and imprinted on the stimulus layer (encoding phase). If the noise ΔS_learn is sufficiently small, synaptic and intrinsic plasticity lead to the formation of structure encoding the organization of stimuli (existence of different geometrical forms). After this initial learning phase, a second learning or readaptation phase enables the network to classify stimuli even in the presence of very high levels of noise ΔS_test. Here, only intrinsic plasticity should be present (ẇ = 0; $\dot{\epsilon }$ ≠ 0). This suggests that learning is carried out in two phases: In the first phase, the encoding phase, synaptic weights develop to represent the basic organization of the environmental stimuli. This structuring of synaptic weights is most efficient if the noise ΔS_learn is low. In the second phase, the readaptation phase, learning is dominated by intrinsic plasticity while synaptic weights have to be maintained. The cortical firing thresholds are then able to quickly adapt to the current level of noise ΔS_test. Thereby, intrinsic plasticity approximates the optimal thresholds for a given value of ΔS_test maximizing performance.