Relation between Single Neuron and Population Spiking Statistics and Effects on Network Activity

To simplify theoretical analyses of neural networks, individual neurons are often modeled as Poisson processes. An implicit assumption is that even if the spiking activity of each neuron is non-Poissonian, the composite activity obtained by summing many spike trains limits to a Poisson process. Here, we show analytically and through simulations that this assumption is invalid. Moreover, we show with Fokker-Planck equations that the behavior of feedforward networks is reproduced accurately only if the tendency of neurons to fire periodically is incorporated by using colored noise whose autocorrelation has a negative component.

Figure 1

Autocorrelation of single and composite spike trains. (a) left: Spike trains recorded intracellularly from a repetitively firing neuron using an in vitro slice preparation at rat cortex. Neurons were made to fire with input that mimicked synaptic barrages; see Ref. 4. Scale bars: $\mathrm{\text{vertical}}=20\mathrm{mV}$, $\mathrm{\text{horizontal}}=200\mathrm{ms}$. Middle: interspike interval distribution compiled over repeated stimulation of the neuron. Right: autocorrelation of the spike trains. (b) Superimposed traces of normalized conductance autocorrelation calculated analytically with Eq. (3) (solid line) and with simulations of LIF neurons with a single input ($G_1(t)$) (dashed line) and with $N=44$ inputs [$G(t)$ divided by $N$] (circles). Inset shows the interspike distribution of LIF neurons [see Fig. 2a] fitted with a gamma distribution ($\theta =150$ and $\lambda =37\mathrm{Hz}$). (c) Normalized autocorrelation of the conductance for a heterogeneous population of neurons. We use 44 gamma distributions with $\lambda$ varying randomly between 26 and 48 Hz to generate multiple spike trains.

Figure 2

Simulated and predicted behavior of feedforward networks. (a) Dot rasters and associated histograms for the first five layers of a simulated feedforward network. Note that full synchrony develops after the third layer and is maintained for the duration of the stimulus (200 ms). (b) Probability distributions calculated using Fokker-Planck equations with Poisson, Eq. (9), (gray line) and colored (black line) noise, Eq. (13), with $\alpha =0.8$.