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Abstract
The brain's ability to tell time and produce complex spatiotemporal motor patterns is critical for anticipating the next ring of a telephone or playing a musical instrument. One class of models proposes that these abilities emerge from dynamically changing patterns of neural activity generated in recurrent neural networks. However, the relevant dynamic regimes of recurrent networks are highly sensitive to noise; that is, chaotic. We developed a firing rate model that tells time on the order of seconds and generates complex spatiotemporal patterns in the presence of high levels of noise. This is achieved through the tuning of the recurrent connections. The network operates in a dynamic regime that exhibits coexisting chaotic and locally stable trajectories. These stable patterns function as 'dynamic attractors' and provide a feature that is characteristic of biological systems: the ability to 'return' to the pattern being generated in the face of perturbations.
Figures
A: A random recurrent network (left panel) in the chaotic regime is stimulated by a brief input pulse (small black rectangle at t=0 in right panel) to produce a complex pattern of activity in the absence of noise. Color-coded raster plot of the activity of 100 out of 800 recurrent units (right panel). Color-coded activity ranges from −1 (blue) to 1 (red). B: Time series of three sample recurrent units (top panel), and the output unit (bottom panel). In the pre-training (left) the blue traces comprised the innate trajectory subsequently used for training. The divergence of the blue and red lines demonstrates that two different initial conditions (before the input) lead to diverging trajectories before training, even in the absence of ongoing noise. After training (right), however, the time series are reproducible during the trained window (2.25 s; shaded area). That is, despite different initial conditions the blue and red lines trace very similar paths, while still diverging outside of the trained window. The output unit was trained to “pulse” after 2 s. C: Five different runs of the network above, perturbed with a 10-ms pulse at t=0.5 s (dashed line) from an additional input unit randomly connected to the recurrent network. The trained network (right) robustly reproduces the trained trajectory, recovering from the perturbation resulting in the timed response of the output unit at t=2 s.
A: Blue traces represent 10 test trials in response to In1 (left panel) or In2 (right) after training; the background gray line shows the output target. These test trials were run over different initial conditions in the presence of continuous noise in all of the 800 recurrent units (0.001). Time is represented by uniformly placed colored circles (Δt≅18 ms). B: Test trials run under the same initial condition in the presence of continuous noise, but with the addition of a perturbation at 300 ms (open square). The perturbation was produced by an additional 10 ms input pulse (not diagrammed) with an amplitude of 0.2.
A: An input pulse (black trace) triggers a chaotic innate” neural trajectory, displayed as a color-coded raster plot (only 20 out of 800 units shown). The linear readout unit receives input from all the recurrent units (blue trace), showing irregular pre-training activity. After the RRN is trained to the innate trajectory (training window defined by dashed lines), the readout unit is trained to reproduce a flat target with a pulse at a given interval (green trace; 5-s duration in this example). An unsuccessful simulation from a 6 s interval training is also included as an example. B: Performance across different architectures. Ten RRNs were trained in each of the three displayed architectures, parametrically varying the delay. The performance (goodness of reproduction) is quantified by the Pearson correlation coefficient R2 between target and actual output (green and blue traces in A); mean ± SEM across networks.
A: (Top panel) Time traces of three sample units over two different trials (blue and red) (N=800, g=1.5, pc=0.25, 1.3 s training window). Gaussian noise with a standard deviation of I0=1.5 was continuously injected into all recurrent units. As in Figs. 1 and 3 the output unit was trained to generate a timed pulse (1000 ms after the onset of the 50 ms input pulse, middle panel). The lower panel shows the neural variance. The variance of each unit was calculated over 8 trials, and then averaged over all 800 units. There was a sharp decrease in variance produced by the onset of the stimulus, which persisted over many seconds before gradually ramping back up to baseline (not shown). The dashed line shows the neural variance before training: because the input “clamps” network activity stimulus onset also produced a decrease in the variance, but it rapidly increased after stimulus offset. The mean Std across units at the input of the input pulse was 0.037 and 0.024, before and after training respectively. B: Example of two simulations in which the output unit were trained to produce events at 250, 500, 750, 1000, and 1250 ms (upper panels). Variance across trials was estimated by calculating the time of the peak of each response. The relationship between variance and t2 was well fit by a linear function (lower panels). I0=1.0.
A: Activity of three sample units in the recurrent network at three different levels of noise. Blue: “template” trajectory (no noise); Red: “test” trajectory (continuous noise in each unit). The standard deviation of the noise current I0 was 0.001, 0.1, and 1.0 (top to bottom panels; noise amplitude as a fraction of total absolute incoming synaptic weights to each unit averaged across units is 0.007%, 0.7%, and 7%, respectively). B: Average data from 10 different networks. Performance was measured as the averaged Pearson correlation coefficient between template (blue) and test trajectories (red) for each condition (after Fisher transformation), mean ± SEM across networks.
A: Average logarithmic distance between original and perturbed trajectories for each of ten networks, for the trajectories triggered by Input1 (the trained input) before and after training. A straight portion with a positive slope indicates chaotic dynamics; the value of the slope is the estimate for the Largest Lyapunov Exponent (λ). After training, the original and perturbed trajectories no longer diverge (except for one network). B: The pre-training trajectories triggered by both inputs displayed positive λ, indicative of chaotic dynamics (Input1: λ=7.12 ± 0.35, mean ± SEM across the ten networks, values significantly different from zero t-test p=10−8; Input2: λ=7.29 ± 0.45, p=4×10−8; all reported λs have units of 1/s). After training, the trajectory triggered by Input1 was locally stable, as indicated by a non-positive mean λ (λ=0.05 ± 0.45, p=0.90); Input2, however, still produced diverging trajectories as evidence by λ significantly above zero (λ=3.05 ± 0.70, p=0.0016). After training the trajectories outside the trained window had a positive mean λ in response to both inputs (Input1: λ=2.75 ± 0.70, p=0.0035; Input2: λ=2.27 ± 0.60, p=0.0039), with some networks displaying chaotic activity (8/10) and some entering limit cycles (2/10). The interaction effect is significant (F2,18=20.7, p=2×10−5, a 2×3 two-way ANOVA with repeated measures, factors “Input” and “Training”). In addition to this stimulus-specific effect of training, there was a global nonspecific effect of decreased divergence of trajectories after training, represented by a lower though still positive λ for Post-train Input2 and Post-outside Input1 and Input2.
A: Distribution of the nonzero recurrent weights. Thin lines represent the distributions of the weights of ten networks before (blue) and after (red) training. Thick lines represent the averages across the 10 networks. Pre-training: networks are Gaussian by construction. Post-training: all networks are non-Gaussian (Lilliefors test, p<0.001 for each of the ten networks). Median absolute synaptic weights significantly increased after training. B: Numerical simulation of one trained network before and after shuffling the weights of its recurrent matrix WRec (two runs each, without noise), showing that the stability properties of the shuffled network are lost despite having the same weight distribution and the same connectivity. C: Distribution of local weighted cyclic clustering coefficients. Training leads to an increase in the cyclic clustering coefficients. Shuffling (green) of the weights of the Post-train recurrent matrix WRec significantly changed the cyclic clustering distribution. Insets reflect the possible circuit motifs in relation to a reference unit shown in gray. D: Distribution of local weighted non-cyclic clustering coefficients. Training also increased the median non-cyclic clustering coefficients.
Comment in
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Recurrent networks learn to tell time.Renart A. Renart A. Nat Neurosci. 2013 Jul;16(7):772-4. doi: 10.1038/nn.3441. Nat Neurosci. 2013. PMID: 23799465 No abstract available.
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