Poisson-like spiking in circuits with probabilistic synapses

Abstract

Neuronal activity in cortex is variable both spontaneously and during stimulation, and it has the remarkable property that it is Poisson-like over broad ranges of firing rates covering from virtually zero to hundreds of spikes per second. The mechanisms underlying cortical-like spiking variability over such a broad continuum of rates are currently unknown. We show that neuronal networks endowed with probabilistic synaptic transmission, a well-documented source of variability in cortex, robustly generate Poisson-like variability over several orders of magnitude in their firing rate without fine-tuning of the network parameters. Other sources of variability, such as random synaptic delays or spike generation jittering, do not lead to Poisson-like variability at high rates because they cannot be sufficiently amplified by recurrent neuronal networks. We also show that probabilistic synapses predict Fano factor constancy of synaptic conductances. Our results suggest that synaptic noise is a robust and sufficient mechanism for the type of variability found in cortex.

Figure 1. Poisson-like output firing requires Poisson-like inputs in a single neuron.

(a) Firing rate as a function of mean input drive with constant noise (dashed lines) and Poisson-like input noise (solid lines). (b,c) Fano factor as a function of mean input drive (b) and firing rate (c). Vertical red line indicates threshold current , defined as the minimal current to elicit firing in absence of input noise. (d) Mean membrane potential as a function of mean input drive. (e) Membrane potential traces corresponding to color dots in the previous panels: low (light green) and high (dark green) firing rate with constant input noise, and low (light blue) and high (dark blue) firing rate with Poisson-like input noise. Low and high firing rates conditions were chosen such that firing rates were comparable for the two input noise types.

Figure 2. Approximate Fano factor constancy with probabilistic synapses.

(a) Scheme of a balanced recurrent network with excitatory and inhibitory neurons driven by non-Poisson-like inputs. Bottom: the network is embedded with probabilistic synaptic transmission. The scheme shows how a presynaptic spike train generates stochastic currents on several postsynaptic neurons. (b) Mean firing rate for excitatory (red) and inhibitory (green) populations for a network with probabilistic synapses and noiseless inputs (solid lines) and for a network without probabilistic noise and constant input noise (dashed lines) as a function of the mean input drive. (c–d) Spike count variance and Fano factor as a function of firing rate. Open circles correspond to mean values, and black dots correspond to individual neurons. Line and color codes are as in panel b. (e) Raster plots of 20 randomly selected excitatory and inhibitory neurons for the high firing rate network corresponding to the point marked in blue in panels b–d. Center: sample traces of excitatory and inhibitory current leading to the net input current (black), magnified on the right. Yellow line corresponds to zero net current, and blue trace shows the membrane potential of a randomly selected excitatory neuron. (f) Coefficient of variation of the ISIs, _,as a function of the mean ISI. (g) Distribution of ISIs for the selected neuron. (h) Auto-correlogram (ACG) of the spike train for that neuron.

Figure 3. Sparse connectivity, high reset voltage or deterministic STD does not necessarily produce Poisson-like firing.

(a) Sparse and randomly connected networks display low spiking variability at high rates. (b) Raising the reset membrane potential of the neurons increases the Fano factor at low rates but does not generate Poisson-like firing for a broad range of firing rates. (c) Networks with deterministic STD fail to generate Poisson-like variability at high firing rates. (d) Networks with random spike jittering display low firing variability at high rates. (e) Exact analytical predictions for networks with probabilistic synapses without STD (blue lines) for the firing rate (left) and Fano factor of the spike counts (right). Red and green points correspond to simulations results for excitatory and inhibitory neurons, respectively. Blue solid lines correspond to theoretical predictions.

Figure 4. Poisson-like variability from probabilistic synapses does not require fine-tuning of the parameters.

The plots display the iso-Fano factor lines on the synaptic scaling factor g vs. input drive plane for a network with (top) and without (bottom) probabilistic synapses. The region for which the Fano factor is high and sustained (shaded area) is broad for a network with probabilistic synapses, but this region vanishes at moderately high rates for a network without probabilistic synapses. Network parameters are as in Fig. 2. The shaded areas are defined as the areas of the planes with Fano factors lying between 0.8 and 1.2.

Figure 5. The mechanism for Poisson-like variability in a network with probabilistic synapses.

(a) Scheme of the transformation between input variance in the spike counts of the presynaptic spike trains and output variance of the post-synaptic currents in an open loop network with probabilistic synapses. (b) Precise balancing of two competing forces in a closed-loop network: the integration step tends to lower spiking variance, while the probabilistic synaptic step increases spiking variability. (c) Output variance (solid red line) and the variance of the spike train, (dashed) increase linearly as a function of input variance for fixed input firing rates. Solid line is vertically shifted respect to the dashed line due to the increase of variance by probabilistic synapses, which is uniform for all input variances. The equilibrium point of the network (red point) corresponds to the state where the input and output variances match. (d) The equilibrium point moves linearly with firing rate because the vertical shift induced by probabilistic synapses increases linearly with rate. (e) Spike count variance increases linearly with rate, leading to Fano factor constancy.

Figure 6. Theoretical predictions: Fano factor constancy of synaptic conductances.

(a) The standard deviation of the membrane potential is approximately constant as a function of firing rate for networks with (full line) and without (dashed) probabilistic synapses. (b) The mean excitatory (red) and inhibitory (green) conductances increase linearly with firing rate. (c) The Fano factor of the synaptic conductances (FF, variance to mean ratio) for a network with probabilistic synapses is constant as a function of the firing rate (full lines), indicating that the variance of the conductance is proportional to the mean conductance. The FF of the synaptic conductances for a network without probabilistic synapses is lower than in the previous case and strongly decreases with firing rate (dashed lines). For all panels, open circles correspond to mean values and black dots correspond to sampled neurons. Error bars represent s.e.m.