Signal buffering in random networks of spiking neurons: Microscopic versus macroscopic phenomena

In randomly connected networks of pulse-coupled elements a time-dependent input signal can be buffered over a short time. We studied the signal buffering properties in simulated networks as a function of the networks’ state, characterized by both the Lyapunov exponent of the microscopic dynamics and the macroscopic activity derived from mean-field theory. If all network elements receive the same signal, signal buffering over delays comparable to the intrinsic time constant of the network elements can be explained by macroscopic properties and works best at the phase transition to chaos. However, if only 20% of the network units receive a common time-dependent signal, signal buffering properties improve and can no longer be attributed to the macroscopic dynamics.

Figure 1

Dashed line: Fixed points of the population activity ${\nu }_0$ as a function of the Poisson background drive resulting from Eqs. (3, 4, 5). Note the sharp transition at ${\nu }_{\mathrm{exc}}=420\mathrm{Hz}$. The network switches from an almost quiescent state to a state of sustained activity. Dotted line: largest Lyapunov exponent (see definition in text) as a function of the external drive ${\nu }_{\mathrm{exc}}$ in a network of 800 neurons. For ${\nu }_{\mathrm{exc}}<420\mathrm{Hz}$ the population rate ${\nu }_0$ is low (“quiescent state”) and the largest Lyapunov exponent is negative. For a stronger drive, the exponent switches to a positive value, reflecting the chaotic behavior of the membrane potential trajectories. Solid line: Signal reconstruction error (arbitrary scale; for quantitative values see Fig. 2), as defined in text, for a delay of 20 ms in a network of 800 neurons. The error is minimal near the transition from the quiescent to a chaotic regime. Inset: Fixed points of the population rate [Eq. (5)] in absence of test signal (solid line), and with increasing signal variance. The system exhibit a first-order phase transition if the signal is weak.

Figure 2

(A) Signal reconstruction error $E$ as a function of the background firing rate in a network of 800 neurons for three different information buffer delays: $\Delta =10\mathrm{ms}$ (solid), $\Delta =15\mathrm{ms}$ (dashed), $\Delta =20\mathrm{ms}$ (dotted). For sufficiently long delays, optimal performance is located near the transition between the qui- escent and the chaotic state; cf. Fig. 1. Deeper in the chaotic phase the error goes back to the chance level whereas in the almost quiescent regime we can see the effects of overfitting $(E>1)$, because the num- ber of action potentials is insufficient to build an accurate model of the past events. (B) Comparison of the errors for different network sizes. Top three lines: Reconstruction error based on a microscopic readout in a network with $N=800$ neurons (solid line), $N=400$ neurons (dashed line), and $N=200$ neuron (dotted line) for a delay $\Delta =20\mathrm{ms}$. The fourth line from top (solid line with filled circles) shows the error based on a macroscopic readout of the network of $N=800$ neurons. The vertical shift of the error curves is not significant but due to overfitting because of limited amount of data. Vertical bars indicate the mean differ- ence between errors on the data used for parameter optimization (training set) and that on an independent test set for $N=800$ (left bar), $N=400$ (second bar), $N=200$ (third bar) and macroscopic readout (right bar). A representative curve of errors on the training set for $N=800$ neurons is shown by the dot-dashed line (bottom). The location of minimal error is independent of the number of neurons or readout method and coincides with the phase transition indicated in Fig. 1.

Figure 3

The input is injected to 20% of the neurons only. A macroscopic readout assuming a single population (dotted line) performs well near the phase transition. However, deeper in the chaotic phase it is outperformed by the microscopic readout (solid line). A macroscopic readout based on a two-population assumption (dashed line) explains only part of the increased performance. The signal buffering delay for this figure is $\Delta =20\mathrm{ms}$.