Fig 1.
Tight-balance spiking network dynamics and readout.
(a) Nominal dynamics. When there are no spike propagation delays and zero noise, the membrane potentials (purple) follow the same trajectory in time. When the population reaches threshold, one neuron’s spike (red) instantaneously inhibits all the neurons, preventing further spikes. This produces perfectly regular spikes, like clockwork with an approximate period of . Each spike contributes
to the readout, creating a tight, zig-zag approximation
(blue) for the encoded signal, x(t) = 1 in this case. (b) The effect of delays and noise. When delays are present and noise is added to the membrane potentials (left, multicolor), two effects appear that decrease the fidelity of the readout
(right, blue): variation in spike-timing and synchronous spurious spikes. The noise on the membrane potentials (σ) creates variations in the time it takes membrane potentials to reach threshold—a deviation from the perfectly regular spikes in (a). And after the first neuron in the population crosses threshold and spikes (red), there is a delay Δ until the other neurons receive inhibition, and thus some extra neurons may spike—spurious synchronous spikes (yellow). Given a fixed delay, too little noise σsmall does not spread the membrane potentials enough to prevent a large number of spurious spikes, and too much noise σlarge destroys the fidelity of the code altogether. An optimal noise level exists, σ*. (c) Soft-threshold model. In the soft-threshold model, neurons spike probabilistically once their membrane potentials surpass threshold, with a spiking probability rate ρ. As ρ is varied (not illustrated), one finds a relationship analogous to the noise level trade-off for the LIF model shown in (b): too small ρ creates large variations in spike times, and too large ρ creates many spurious spikes during the delay. Thus, an optimal ρ* exists.
Fig 2.
(a) The soft-threshold and the delay create variations in the membrane potential dynamics Vi(t), which in turn create variations in the readout . When the membrane potentials (top, purple) surpass threshold, the neurons spike probabilistically, and the first-spike time is an exponential random variable with standard deviation
. After a first spike, the number of spurious spikes that occur during the delay is a Poisson random variable, with standard deviation
, and each spike inhibits the membrane potentials Vi(t) by 1 through recurrent connectivity (see the first paragraph of Results for recurrent connectivity). These variations in spike-timing and spurious spikes carry through to the readout
(bottom, blue). Note that since the network input is constant, the readout encoding this input should produce a constant output as closely as possible; however, these variations instead increase the deviation (light blue shaded) from the mean readout
(black horizontal line). (b) Readout error as a function of the mean number of spurious spikes λ and the delay δ. Top: for three different values of delay (blue, red, purple), λ is varied in computer simulations (N = 32, dots) and Eq 10 (solid curves), revealing both the U-shaped dependence of the readout error σreadout and an excellent match between theory and experiment. Bottom: the optimal readout error
(black) and the associated optimal λ* increase as a function of delay δ according to Eqs 12 and 11.
Fig 3.
(a) Spike raster plots from simulations (N = 32) for three different values of the membrane potential leak λV. Notably, for small λV, we observe long runs in which the same neuron repeatedly spikes. Thus in the small λV limit, the next spike time is well-approximated by considering only the possibility that the same neuron spikes again. (b) Readout (blue) and its deviation
(light blue shaded) from the mean
for a single interspike interval. The variations in interspike interval durations accumulate to produce a variation in the readout with standard deviation
, using the approximation from (a). (c) Readout error σreadout as a function of noise σ for different values of λV. Integrating the deviation illustrated in (b) across time yields the readout error in the small λV limit, Eq 19 (black). Simulations (dots on dashed lines, N = 64) with larger λV are upper-bounded by Eq 19.
Fig 4.
(a) Calculation of the number of spurious spikes. For a nonzero noise level σ1, the membrane potentials travel in a Gaussian packet with density P(V, t) (blue, top left) toward threshold (vertical red dotted line). The typical position of the packet at the time t when the first neuron spikes is determined by ensuring the tail probability (red, shaded area) above threshold equals 1/N. During the spike propagation delay Δ, the Gaussian packet continues traveling toward threshold (blue, bottom left), and the mean number of spurious spikes λ is given by the additional probability density that crosses threshold (yellow, shaded area). A larger noise level σ2 spreads out the Gaussian packet (green, right), thus reducing λ. (b) Readout (blue) and its deviation
(light blue shaded) from the mean
. Similar to Fig 3b, the accumulated spike-time variation creates fluctuations in
with standard deviation upper-bounded by
, but in addition, spurious spikes introduce a Poisson variation in the readout with standard deviation
. (c) Integrating the deviations illustrated in (b) yields an approximate upper-bound for σreadout, Eq 24 (blue). Conceptually, σreadout receives contributions from noise (Eq 24 without the λ term; yellow, dashed), and synchronous spurious spikes (Eq 24 without the
term; green, dashed). (d) Readout error σreadout for varying levels of noise. For zero delay (top), noise is not necessary to prevent spurious spikes, and thus it strictly increases σreadout. For non-zero delays (bottom), σreadout has a U-shaped dependence on σ, and an optimal noise level σ* exists. The dots/squares on dashed lines represent σreadout from simulations (N = 64), and solid lines are Eq 24, with the region below shaded, indicating upper-bound. Blue signifies λV = 0.1; yellow λV = 1.0. (e) Minimal readout error
and optimal noise level σ* as a function of delay δ. Minimizing Eq 24 (with higher-order terms, see Eq S91 in S1 Appendix) with respect to σ yields
(left, solid lines) and σ* (right, solid lines).
and σ* asymptotically approach (δ/τ)2/3 and (δ/τ)1/3, respectively (dashed black lines). We take the minimal σreadout from the simulations in (d) and the associated optimal noise level to generate the dots/squares on the blue/yellow dashed lines, observing that our theory indeed provides an upper-bound for
and a good estimate for the optimal noise level σ* in finite-sized simulations.