Firing statistics and correlations in spiking neurons: A level-crossing approach

We present a time-dependent level-crossing theory for linear dynamical systems perturbed by colored Gaussian noise. We apply these results to approximate the firing statistics of conductance-based integrate-and-fire neurons receiving excitatory and inhibitory Poissonian inputs. Analytical expressions are obtained for three key quantities characterizing the neuronal response to time-varying inputs: the mean firing rate, the linear response to sinusoidally modulated inputs, and the pairwise spike correlation for neurons receiving correlated inputs. The theory yields tractable results that are shown to accurately match numerical simulations and provides useful tools for the analysis of interconnected neuronal populations.

Figure 1

(a) Steady-state firing rate of the IF model. Solid line, Eq. (6). Symbols, numerical simulation. Parameters are $J_e=6.3\text{nS},J_i=4.5\text{nS},E_0=-55\text{mV},v_{\theta }=-48\text{mV},v_r=-60\text{mV},{\tau }_v=5\text{ms},{\tau }_e=3\text{ms},{\tau }_i=10\text{ms}$, and where not otherwise mentioned, $R_e=2.66\text{kHz}$, $R_i=2.22\text{kHz}$, and $c_{ei}=0$. (b) Firing-rate response to time-varying inputs. Top: Solid line, Eq. (5); symbols, numerical simulation. Middle: Input spike rates, with mean $R_{e0}=2.66\text{kHz}$ for excitation and $R_{i0}=2.22\text{kHz}$ for inhibition. Bottom: Instantaneous correlation coefficient between excitatory and inhibitory inputs. Other parameters identical to those in (a).

Figure 2

Frequency response of the IF model to sinusoidal modulations in presynaptic excitatory and inhibitory firing rates. The response is illustrated in three typical cases: at low frequency, $r_1^e>r_1^i$ (a), $r_1^e=r_1^i$ (b), or $r_1^e<r_1^i$ (c). Top two graphs: The components of the firing-rate response due to excitatory ($r_1^e$) and inhibitory ($r_1^i$) modulations are shown as functions of the frequency of the modulation. The summed response is determined from the amplitude (top graph) and the phase difference (bottom graph) of the two components. Bottom two graphs: Amplitude $|r_1|$ and phase lag $\text{phase}\left(r_1\right)$ of the summed firing-rate response (solid line, theory; symbols, numerical simulation). For (c) a phase difference of 29${}^{\circ }$ is introduced between excitatory and inhibitory modulations to maximize the trough.

Figure 3

Response of correlated neuronal populations. (a) Schematic representation of the network. In addition to background inputs that are uncorrelated across neurons, each population receives shared inputs that are identical for all neurons. Shared inputs to population 1 and 2 are also cross correlated (indicated by blue arrows). (b) Firing rate of population 1 (solid line with open circles) and 2 (dashed line with squares). Lines, Eq. (5); symbols, numerical simulations. Parameters for population 1 are ${\tau }_v=5\text{ms},J_e=7.3\text{nS},J_i=4.5\text{nS},v_{\theta }=-53\text{mV},v_r=-60\text{mV}$, and for population 2 ${\tau }_v=4.65\text{ms},J_e=7.5\text{nS},J_i=5.4\text{nS},v_{\theta }=-50\text{mV},v_r=-60\text{mV}$. (c) Cross-correlation coefficient of the outputs of the two populations [solid line, expression (9); symbols, numerical simulation]. (d) Excitatory (solid lines) and inhibitory (dashed lines) input firing rates (population 2 data marked with arrow). (e) Cross-correlation coefficients between excitatory (solid) and inhibitory (dashed) inputs.

Figure 4

Synchronized spiking from pairwise correlations. (a) Schematic representation of the network. In addition to background inputs that are uncorrelated across neurons, the population is subjected to common inputs that are shared by all neurons. (b) Mean firing rate and Fano factor of the pooled output of the population. Black dashed lines, theory [top, Eq. (5); bottom, Eq. (10)]. Red solid lines, numerical simulation. Shared inputs with constant spike rate are activated between $t=250$ and $750\text{ms}$; background activity is adjusted to maintain a constant overall firing rate. (c)–(e) Population firing rate (green solid line in top graphs) for three single runs of the numerical simulation. Raster plots showing the spikes of 200 randomly chosen neurons are also shown, in which synchronous events are clearly seen.

Figure 5

Range of validity of the level-crossing approximation. (a) Two-dimensional plot of the accuracy of the approximation as a function of the dimensionless parameters $\Lambda$ and ${\sigma }_v/v_{\vartheta }$, for the case of the two-dimensional system (11). The accuracy is plotted for the two limits ${\tau }_e\to 0$ (white noise limit) and $+\infty$ (static noise limit). The plot is color coded according to the scale bar on the right. Level curves are also shown as black solid lines. (b) The accuracy is represented here as a function of $\Lambda$ and the dimensionless firing rate $r/\gamma$. (c) The accuracy of the approximation in the case of the three-dimensional system (2, 3, 4) is shown for four different values of the parameter $\Lambda$. Solid lines shows the theoretical prediction, and shaded regions correspond to the accuracy expected from the analysis in (b). Symbols show the result of 100 numerical simulations with randomly chosen parameters.