Fig 1.
A diagram illustrating the BSN.
Neurons receive stimulus input projected onto the transpose of a set of linear weights, W⊤, and the output is reconstructed by filtering spikes through the same weights, W. Neurons are connected via two coupling weights: fast synapses, W⊤W, which instantaneously propagate individual spikes through the network, and slow synapses, , which implement network dynamics by feeding the filtered spike trains back into all neurons in the network. The network is divided into two equal populations of positive (red) and negative (blue) output weights, whose spikes have opposite effects on network output. Self-connections for these neurons are shaded in their respective colors, for visualization, but are always negative.
Fig 2.
Balanced spiking network implementing an exact integrator.
The network consists of 400 neurons, divided into two populations with output weights of + 0.1 (red) and −0.1 (blue). (A) Simulation results under the condition that only one neuron is allowed to fire per discrete time bin. (B) Simulation results when all neurons whose membrane potential is above threshold in a single time bin are allowed to fire, leading to “ping-pong” behavior. Insets show that the read-out (yellow) is alternating between large over- and under-estimates of the target (in black). Insets show, in order from top to bottom: the voltage traces of neurons in both positively and negatively weighted populations for a small time window, the resulting spikes in each time bin, and the resulting read-out (yellow) and target (black). Since the weights and inputs are identical across populations, so are the voltage traces. Ping-ponging results, as all neurons within a population cross the threshold in the same time bin, spike, and cause the read-out to oscillate between over- and under-estimates of the target.
Fig 3.
Schematic of two neurons in a BSN with conditionally Poisson neurons.
The stimulus influences each neuron’s membrane potential vi via a set of input weights W⊤. The neurons reset themselves via instantaneous, fast synapses. Fast connections to other neurons propagate the effects of spikes with a synaptic time delay d. The desired linear dynamics are implemented via slow weights (through spike trains filtered by an exponential) also with a time delay d. Within each neuron, spiking is probabilistic with an instantaneous probability of firing λi(t) = f(vi(t)), where f(⋅) is a nonlinear function of voltage. Self-connections are only shown for neuron i.
Fig 4.
(A) The conditional intensity for the exponential non-linearity (dashed lines) and the sigmoid non-linearity (solid lines). The conditional intensity of the sigmoidal non-linearity closely follows that of the exponential non-linearity for sub-threshold voltages, but levels off after threshold, keeping firing rates stable. (B) Family of nonlinearities with varying Fmax. Increasing Fmax raises the firing rate at which the nonlinearity saturates. (C) Family of nonlinearities with varying α. Increasing α increases the steepness of the nonlinearity, which approaches a hard-threshold function as α → ∞ (like the BSN). (D) Simulation of the original BSN implementing an exact integrator, showing membrane potential and spikes of a single example neuron. (E) Spikes and membrane potential of the same neuron in a local Poisson BSN implementation of the same system. High α simulations (yellow) replicate the behavior of the BSN integrator. Lowering α to 50 (blue) or 10 (red) results in a spread of spikes centered around the deterministic BSN spikes.
Fig 5.
(A) Simulations of local Poisson model showing the effects of varying the parameters of the soft-threshold nonlinearity on performance. Relevant parameters are the slope α, maximal firing rate Fmax, and baseline firing rate Fmin. Network dynamics implemented an exact 1D integrator and the stimulus was the same as Fig 2. Red and blue dots indicate spikes from neurons with positive and negative output weights, respectively. (B) Network performance as quantified by R2 across a range of parameter settings with baseline fixed at Fmin = 0 (log-scale). Red asterisk indicates the values for the rightmost column of A (α = 1000, Fmax = 100). Accuracy remains high across a broad range of parameter values, falling substantially below 1 when slope and maximum firing rate are both large or very low. (C) Percent of the neural population active as a function of α and Fmax, with Fm in = 0. The network shows ping-ponging behavior in upper right corner, where the model approaches a deterministic, hard-threshold firing rule.
Fig 6.
Simulations of the local and population frameworks implementing a 1D and 2D dynamical system.
(A) The target was a 1-dimensional integrator: . Left side shows spikes and outputs from local Poisson model, while right side shows spikes and outputs for population Poisson model. As in previous figures, red dots indicate spikes from neurons with positive output weights, blue dots indicate spikes from neurons with negative weights. (B) The target was a 2-dimensional oscillator
. For the population model, the time window for computing expected spike count was κ = 5ms (50 time bins). Weights were randomized to be positive or negative in either dimension, such that neurons are no longer divided into strictly positive- or negative-weight groups. (C) Accuracy (measured by root-mean-squared error) of the two models for 1D and 2D systems. (D) Number of spikes emitted by each model during simulations (log scale).
Fig 7.
Illustration of local and population conditionally Poisson BSN frameworks with synaptic delays.
(A) Spike trains simulated from the local Poisson framework implementing a 1D exact integrator, both without (top) and with a 1-ms synaptic delay (middle). The network output accurately tracked the target variable for both models (bottom). As before, red/blue spike trains indicate neurons with positive/output weights. (B) Analogous plots for population Poisson framework. (C) Stimulus used for simulations shown in A and B. (D) Coefficient of determination (R2) computed using 50 simulations of each framework. Black trace indicates the maximum possible R2 value that could be obtained given the exponential Euler integration rule (see Methods).
Fig 8.
Cross- and auto-correlations for the original BSN, local Poisson and population Poisson BSN models with synaptic delay.
The top row shows average auto-correlations across both populations of neurons. The bottom row shows average cross-correlations for pairs of neurons with the same sign output weight (i.e., both positive or both negative, in purple) and for pairs of neurons with opposite-sign output weights (e.g., one positive and one negative neuron, in yellow). The original BSN network exhibits negative correlations between neurons with the same sign, and positive correlations between neurons with opposite sign. The local and Population Poisson models show the opposite pattern, which more closely resembles correlations found in neural populations in (e.g.) visual cortex.
Fig 9.
Scaling of error and spike count with population size (weights held fixed).
(A) Relative root-mean-square error (RMSE) decreases approximately linearly with the network size for all three models. R2 values for the fit to 1/N were .98, .91, and .82 for the BSN, local and population models, respectively. (B) Total spike count as a function of network size. The results shown are the average of five simulations of the networks performing exact integration of a signal formed by a sum of two sinusoids.
Fig 10.
Simulation showing robustness of local and population Poisson models to silencing of a subset of neurons.
Left: We created a local Poisson BSN model with 400 neurons, with weights set to perform exact integration, and presented it with a slowly varying 1D stimulus (top). From time t = .4 to t = 1s we artificially silenced 50% of neurons in the negative-weight (blue) population, preventing them from spiking by setting p(spike) = 0 (silencing period indicated by the grey bar). From time t = 1.2 to t = 1.8s we did the same for 50% of neurons in the positive-weight (red) population. Right: Likewise for the population framework.
Table 1.
Simulation parameters.