Fig 1.
Schematic of model and analysis methods.
(A) Raster plot of 3-layer network model trained to a 4 Hz + 20 Hz sum of sines stimulus. Red and blue indicate the x- and y-components for both the stimulus and readout (left). Depiction of the procedure used to process population spike trains before feeding them to the decoder to estimate the stimulus (right). T = 50 ms is the width of the sliding window used here and Δt is the bin size (B) Sketch of the information theoretic method used to validate the 5-layer network model against previous results from hawkmoth data. A window of duration T = 50 ms is used in this analysis. The variable denotes the spike count response and
denotes the spike timing response.
Fig 2.
Structural convergence to the output layer promotes timing codes across stimulus frequencies.
(A) Stimuli are sums of sines with fixed frequency component Hz and variable component
(B) Decoding accuracy based on output layer spikes binned at time resolution Δt. The blue points show the
Hz result for the network with
hidden neurons and the red points for the network with
hidden neurons. Lines are the best line fit, averaged over all network simulations. The slopes of these fits are plotted in C and E. (C) Slope of R2 v.s. Δt fits as a function of the high frequency stimulus component
. Asterisks denote where a one-sided Wilcoxon rank-sum test is significant at p<0.05. (D) Mutual information rate
based on the output layer spikes binned at time resolution Δt. (E) Slope of
v.s. Δt curves as a function of
, the high frequency stimulus component. Asterisks denote where a one-sided rank-sum test is significant at p<0.05. Error bars represent distributions of the results over 10 independent network simulations.
Fig 3.
Bottlenecks have more to gain from temporal codes than expansion layers.
(A) Example reconstructions from the hidden layer spikes binned at Δt = 5 ms (left) and Δt = 50 ms (right) resolution for (top) and
(bottom). Thin traces show reconstructions from individual network seeds. Thick colored traces show means across all network seeds. (B) Decoding accuracy from the hidden layer spikes as a function of bin size Δt for the bottleneck (left) and expansion (right) network. Gray points denote which bin sizes were used to compute accuracy gain
. Error bars denote standard errors of the mean over network seed distributions. (C) Accuracy gain of the temporal code over count code when reconstructing the stimulus based on spikes from the hidden layer, for bottleneck (red) and expansion (blue) networks. One-sided Wilcoxon rank-sum test
. Results are shown for 25 independent network simulations.
Fig 4.
Temporal codes capture high-frequency stimulus components more accurately in layers following structural convergence.
Decoding accuracy versus bin size for each layer of the bottleneck and expansion networks receiving a 4 Hz + 20 Hz sum of sines stimulus. The 4 Hz (top) and 20 Hz (bottom) components are decoded separately here. Error bars represent standard errors of the mean over 10 independent network seeds.
Fig 5.
Stimulus-dependence of spike coding as shaped by convergent/divergent structure.
(A) Each row shows the stimulus used for the corresponding plots on the right. Filtered white noise was used instead of pure white noise since much of the variation in pure white noise is low-pass filtered by the membrane voltage of the neurons. (B) Decoding accuracy v.s. the number of hidden neurons at Δt = 5 ms and Δt = 50 ms for the hidden layer (left) and output layer (right). (C) Accuracy gain (R2 at Δt = 5 ms minus R2 at Δt = 50 ms) v.s number of hidden neurons. Asterisks denote where a one-sided Wilcoxon rank sums test is significant (* for p<0.05, ** for p<0.01, and *** for p<0.001). All boxplots represent distributions of the results over 25 independent network simulations. For the filtered white noise stimulus, the mean drop in accuracy gain for the hidden layer when going from to
is 0.14; the mean drop in accuracy gain for the output layer when going from
to
is 0.04. For the binary stimulus, the accuracy gain drop is 0.04 for the hidden layer and 0.06 for the output layer.
Fig 6.
Experimental system and network model
Diagram of the central nervous system of the hawkmoth Manduca sexta and a schematic of the 5-layered spiking neural network developed here to model its visuomotor pathway. Numbers in parentheses denote the number of neurons in each population for the moth (orders of magnitude, left) and the model (exact, right).
Fig 7.
Single-neuron information during 1 Hz stimulus.
(A) Raster plot of the 5-layer network model trained to a 1 Hz sinusoidal stimulus. (B) Single neuron information rate in each layer, decomposed into spike count and spike timing contributions. Each dot represents the result of a single network seed, averaged across all neurons in the layer. Lines connect the means of the distributions. (C) Mutual information in spike count and spike timing from the hawkmoth motor program (top) and the 10 neurons in the output layer of the model (bottom). The plots on the right show mutual information pooled from the output muscles (top) and output layer of the model (bottom). Asterisks denote where a one-sided Wilcoxon rank sums test is significant at p<0.01. For the model, mutual info is taken between stimulus and response; for the moth data, mutual info
is taken between motor output m and response. The single-neuron method depicted in Fig 1B and described in Methods was used here to compute mutual information, consistent with ref. [25], which is where the moth data was originally published. For the moth muscle results, boxplots represent distributions over 7 individual moths. For the model results, boxplots represent distributions over 25 independent network simulations.
Fig 8.
Decoding analysis of a noisy 4 Hz + 20 Hz stimulus
(A) The 5-layer network receives a sum of sines corrupted by noise, but is trained to encode the noiseless version at the output. The decoding is done with respect to the noiseless stimulus. (B) Decoding accuracy from spikes binned at resolution Δt, in each layer of the 5-layer model. Each gray trace represents an individual network seed. Black traces are the means across all network seeds (top). Distribution of slopes of best line fits to the R2 v.s. Δt curves (bottom). (C) Slope distributions versus layer. Results are shown for 25 independent network simulations.
Fig 9.
Maximum entropy of population spike codes.
(A) Slope of the entropy v.s. population size curves, as a function of the number of time bins. The purple curve is simply the linear function for the spike timing code and the teal curve is the function
+ 1 ) for the spike count code. (B) Example of the entropy rate v.s. population size for both types of spike code. We set T = 15 ms and
ms here so that
.
Table 1.
Parameter values for the neurons in the alpha neuron model.
The symbol U ( A , B ) denotes the uniform distribution between A and B.
Table 2.
Parameter values for the neurons in the LIF neuron model.
The symbol U ( A , B ) denotes the uniform distribution between A and B.
Fig 10.
(A) Reduction of MSE loss through training with BPTT. Thin gray traces show individual network seeds, thick black trace shows the average across all 25 seeds. (B) Readout after training 3-layer networks with Nin = Nh = Nout = 100 to the 4 Hz + 20 Hz sum of sines stimulus. Colored traces are for the readout; the black trace denotes the true stimulus presented to the network. The top shows the x-dimension of the stimulus and the bottom shows the y-dimension.